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Sampled-data generalized predictive control (SDGPC) Lu, Guoqiang 1996-02-19

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Sampled-Data Generalized Predictive Control (SDGPC) by Guoqiang Lu B.Sc. Beijing Institute of Technology, 1984 M.Sc. Beijing Institute of Technology, 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1996 © Guoqiang Lu, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of EUyty^L %M*W/<j» The University of British Columbia Vancouver, Canada Date Tel. % «* ; (9? & Abstract This thesis develops a novel predictive control strategy called Sampled-Data Generalized Pre dictive Control (SDGPC). SDGPC is based on a continuous-time model yet assumes the projected control profile to be piecewise constant, i.e. to be compatible with zero order hold circuit. It thus enjoys both the advantage of continuous-time modeling and the flexibility of digital implementation. SDGPC is shown to be equivalent to an infinite horizon LQ control law under certain conditions. For well-damped open-loop stable systems, the piecewise constant projected control scenario adopted in SDGPC is shown to have benefits such as reduced computational burden, increased numerical robustness etc. When extending SDGPC to tracking design, it is shown that future knowledge of the setpoint significandy improves tracking performance. A two-degree-of-freedom SDGPC based on optimization of two performance indices is proposed. Actuator constraints are considered in an anti-windup framework. It is shown that the nonlinear control problem is equivalent to a linear time-varying problem. The proposed anti-windup algorithm is also shown to have attractive stability properties. Time-delay systems are treated later. It is shown that the Laguerre-filter-based adaptive SDGPC has excellent performance controlling systems with varying time-delay. An algorithm for continuous-time system parameter estimation based on sampled input output data is presented. The effectiveness and the advantages of continuous-time model estimation and the SDGPC algorithm over the pure discrete-time approach are highlighted by an inverted pendulum experiment. Iable of Contents Abstract . ii List of Tables v List of Figures vAcknowledgment ix 1 Introduction 1 1.1 Background and Motivation. 1 1.2 Literature Review 2 1.3 Contribution of the Thesis 4 1.4 Outline of the Thesis . 5 2 Sampled-Data Generalized Predictive Control (SDGPC) 7 2.1 Formulation of SDGPC 7 2.2 Stability Properties of SDGPC 14 2.2.1 Stability of SDGPC with control execution time Texe = Tm 14 2.2.2 Property of SDGPC with control execution time Texe <Tm 22 2.3 Interpretation and Stability Property of the Integral Control Law 24 2.4 Simulations and Tuning Guidelines of SDGPC 29 2.5 Conclusion 40 3 SDGPC Design for "Tracking Systems 41 3.1 The Servo SDGPC Problem3.2 The Model Following SDGPC Problem 51 3.3 The Tracking SDGPC Problem 5 3.4 The Feedforward Design of SDGPC . 60 3.5 Conclusion 66 iii 4 Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC 67 4.1 The Direct Approach 64.2 The Laguerre Filter Modelling Approach 71 4.3 Conclusion 77 5 Anti-windup Design of SDGPC by Optimizing Two Performance Indices 78 5.1 SDGPC Based on Two Performance Indices 79 5.1.1 Optimizing Servo Performance 80 5.1.2 Optimizing Disturbance Rejection Performance 83 5.2 Anti-windup Scheme 91 5.3 Conclusion 104 6 Continuous-time System Identification Based on Sampled-Data 105 6.1 The Regression Model for Continuous-time Systems 108 6.2 TheEFRA 110 6.3 Dealing with Fast Time-varying Parameters 112 6.4 Identification and Control of an Inverted Pendulum 115 6.4.1 System Model 116 6.4.2 Parameter Estimation 118 6.4.3 Controller Design 122 6.5 Conclusion 125 7 Conclusions 7 A Stability Results of Receding Horizon Control 130 A.l The Finite and Infinite Horizon RegulatorA.2 The Receding Horizon Regulator 132 References 138 iv List of Tables 2.1 Comparison of SDGPC and discrete-time receding horizon LQ control List of Figures 2.1 The projected control derivative 8 2.2 The implementation scheme 12 2.3 The integral control law2.4 Zero placing strategy 3 2.5 Comparison of SDGPC and GPC strategy 22 2.6 Zero placement in SDGPC 29 2.7 The projected control derivative 30 2.8 Step response of example 1 1 2.9 Simulation 1 of example 1 32 2.10 Simulation 2 of example 1 3 2.11 Simulation 3 of example 1 34 2.12 Simulation 4 of Examplel: infinite horizon LQR 35 2.13 Simulation of plant (2.68) 36 2.14 Simulation of plant (2.69) 7 2.15 Simulation of plant (2.69) 38 2.16 Simulation of plant (2.73) 9 3.17 The projected control derivative 43 3.18 The servo SDGPC controller 5 3.19 The dynamic feedforward controller implementation 43.20 Servo SDGPC of plant (3.110)-double integrator 50 vi 3.21 Servo SDGPC of plant (3.110)-single integrator 51 3.22 Desired trajectory for model-following problem3.23 Model following control of unstable third-order system 53 3.24 Model following control of stable third-order system 5 3.25 Servo SDGPC of plant (3.128) 58 3.26 Tracking SDGPC of plant (3.128)3.27 Servo SDGPC of plant (3.128) 59 3.28 Tracking SDGPC of plant (3.128) 60 3.29 Disturbance feedforward design 4 3.30 The effect of disturbance model . 66 4.31 Graphical illustration of SDGPC for systems with delay 68 4.32 Laguerre Filter Network 72 4.33 Simulation of plant (4.169) 6 4.34 Simulation of plant (4.169) with measurement noise 75.35 Graphical illustration of (5.189) 83 5.36 Control subject to actuator constraints 91 5.37 Example 5.1: Control law (5.247) without anti-windup compensation 97 5.38 Example 5.1: Control law (5.247) with anti-windup compensation 98 5.39 Example 5.2: Control law (5.251) with anti-windup compensation 100 5.40 Example 5.2: Control law (5.251) without anti-windup compensation 101 5.41 Conventional anti-windup 105.42 Conventional anti-windup scheme vs proposed 102 6.43 Graphical illustration of numerical integration 109 vii 6.44 Estimation of time-varying parameters 115 6.45 The inverted pendulum experimental setup 116 6.46 Downward pendulum 117 6.47 Upward pendulum 8 6.48 Parameter estimation of model (6.283) 119 6.49 Parameter estimation of model (6.283) 120 6.50 Step response of the estimated discrete-time model (6.292) 121 6.51 Changing dynamics 124 6.52 SDGPC of pendulum subject to disturbance 125 viii Acknowledgment I would like to express my deep gratitude to Professor Guy A. Dumont for giving me the opportunity to pursue a PhD degree in process control under his supervision. I wish to thank him for his invaluable guidance and instruction throughout my study in UBC. I thank him for helping me selecting a project for his self-tuning control course which led to the research reported in this thesis. I would also like to thank Professor Michael S. Davies, Dr. Ye Fu and other members of the Control Group at the Pulp and Paper Centre for their helping hands. I also appreciate the constructive suggestions and criticisms of the members of my examination committee for making the draft more readable. Financial support from Professor Guy A. Dumont and the Woodpulp Network of Centres of Excellence is greatly acknowledged. I would like to thank my parents, brother and sister for their love and care. They were always there especially during the hard times. I am grateful to my wife Weiling, who always believed in me and offered unlimited support. I would like to thank her for her love and patience during those years. Chapter 1: Introduction Chapter 1 Introduction 1.1 Background and Motivation Model Based Predictive Control (MBPC) has achieved a significant level of success in industrial applications during the last ten years. This has inspired the academic community to investigate the theoretical foundations of MBPC. As a result, a wealth of exciting stability results have been obtained for the last couple of years. It is safe to say that a solid theoretical foundation for model predictive control has now been established. One of the many explanations of the success of MBPC is that predictive control is an open methodology. That is, within the framework of predictive control, the predictive controller can be closely tailored to meet different requirements of a particular problem. As a result, quite a few predictive controllers have been proposed. Some of the well-known predictive controllers are GPC ( Generalized Predictive Control [13]), DMC ( Dynamic Matrix Control [15]), Model Predictive Heuristic Control [61], etc. All of these controllers are developed in a discrete-time context. That is, all the controller designs start with a discrete-time model which can be obtained either by direct identification from the discrete input output data or by discretizing a continuous-time model. Although most of the industrial processes are continuous in nature, the discrete-time approach of MBPC is a natural choice since most of the MBPC algorithms need computer implementation. However, the selection of the sampling interval in digital control is not a trivial task. Moreover, it has been pointed out that in applications where fast sampling is needed, the discrete-time model in ^-domain is not a good description of the underlying continuous-time process since the poles and zeros of the continuous-time system are mapped to the unit circle as the sampling interval A goes to zero. It is thus not a surprise to see a resurgence of interest in continuous-time model based methods [33] [32]. Efforts have also recently been made to unify discrete-time and continuous-time methods under the name of 6-operator [50]. 1 •r Chapter 1: Introduction One of the few continuous-time results on MBPC is the work done by H.Demircioglu and P.J.Gawthrop [18] in which Continuous-time Generalized Predictive Control (CGPC) was developed based on Laplace transfer function model. Multivariable CGPC [19] and modified CGPC with guaranteed stability are also available [16]. However, results on continuous-time MBPC are still very limited compared with its discrete-time counterpart. There is still no reported real life application of continuous-time MBPC to the best of the author's knowledge. This is perhaps partly due to the fact that it assumes the projected future control inputs to be of a polynomial type which is not compatible with the widely used zero-order hold device in digital control equipment. As a result, the digital implementation of CGPC unavoidably introduces approximations which often demand a small sampling interval. This demand will result in computation difficulties in some applications. Nonetheless, continuous-time modelling is still appealing even for the purpose of digital control since physical relevance of the model parameters is retained and it is easier to identify partially-known systems in a continuous-time setting. This motivates us to develop a MBPC algorithm based on continuous-time modelling while assuming the projected future control scenario to be piecewise constant, i.e. to be compatible with the zero-order hold device. The model form is chosen to be a continuous-time state-space equation instead of a continuous-time transfer function for two reasons. First, it is easier to deal with time-delay in time domain. Second, Laguerre network naturally has a state space form in time domain. Actuator constraints are not considered in the problem formulation initially, rather they are incorporated into the scheme later in the framework of anti-windup design. 1.2 Literature Review Historical background as well as current trends in Model Based Predictive Control ( MBPC ) are reviewed in this section. The concept of predictive control originated in the late seventies with the seminal papers on DMC [15] by Cutler and Ramaker and on Model Predictive Heuristic Control [61], by Richalet et al. The common features of predictive control are: 1. At each "present moment" t, a forecast of the process output over a long-range time horizon is made. This forecast is based on a mathematical model of the process dynamics, and on the future control scenario one proposes to apply from now on. 2 ' Chapter 1: Introduction 2. The control strategy is selected such that it brings the predicted process output back to the setpoint in the "best" way according to a specific control objective. Most often this is done by minimizing a quadratic performance index. 3. The resulting control is then applied to the process input but only at the present time. At the next sampling instant the whole procedure is repeated leading to an updated control action with corrections based on the latest measurements. This is called a receding horizon strategy. Another school of thought in predictive control, whose objective is to design the underlying controllers in an adaptive control context, emerged almost independently at about the same time. Peterka's predictive controller [58], Ydstie's extended-horizon control [84], Mosca etaWs MUSMAR [53] and the GPC [13] of Clarke et al. are all in this category. The continuous-time counterpart of GPC called CGPC is reported in [18]. However, the completely continuous-time design seems to limit its applicability. The structures of all the MBPC algorithms are the same but differ in details. For example, the DMC [15] uses a finite step response model and Model Predictive Heuristic Control [61] uses impulse response model while GPC [13] on the other hand uses an ARTMAX model. Many application of MBPC are reported in the literature and several companies offer MBPC software. The survey paper by Garcia [31] et al. examines the relationship between several MBPC algorithms and industrial applications are also reported. A more recent paper by Richalet [62] presented two classical applications of MBPC. By the late eighties, MBPC had secured a widespread acceptance in process industry despite the lack of firm theoretical foundation, which is quite remarkable. It is acknowledged [51] that there is no useful general stability results for the original formulation of MBPC. In fact it was shown in [4] that GPC has difficulty controlling systems with nearly cancelled unstable poles and zeros. Although such kind of systems are difficult to control for any control methods, it nonetheless showed that GPC has some serious shortcomings. Bitmead et al. [4] suggested using the traditional infinite horizon LQG instead. The infinite horizon approach, albeit with guaranteed stability property, is less appealing in applications where some input and/or state constraints exist. A finite horizon with terminal state constraints is proposed independently by a group of researchers [14, 60, 52, 54]. The survey paper [11] by Clarke covers the most recent advances in MBPC. A bibliography of MBPC and related topics from 1965 to 1993 is also appended 3 Chapter 1: Introduction in that paper. A book entitled " Advances in Model-Based Predictive Control " [11], edited by Clarke is based on the presentations made at a conference wholly devoted to recent advances in MBPC. It is a complete collection of the latest results on MBPC. As pointed out by Clarke [11], MBPC can handle real-time state and actuator constraints in a natural way. This is an active research topic which has important practical implications. It is predicted [51] that MBPC will emerge as a versatile tool with many desirable properties and with a solid theoretical foundation. It is worth pointing out at this point that most of the MBPC algorithms are not robust synthesis methods in the sense that there is no explicit incorporation of realistic plant uncertainty description in the problem formulation. Recent developments in the theory and application (to control) of convex optimization involving Linear Matrix Inequalities (LMI) [7] have opened a new avenue for research in MBPC. Much of the existing robust control theory can be recast in the framework of LMIs and the resulting convex optimization problem can be solved very efficiently using the recent interior-point methods. It is thus not surprising to see that results on MBPC using convex optimization ( as opposed to conventional linear or quadratic programs ) have begun to appear in the literature [40, 75]. This is certainly a promising research filed for MBPC. Literature reviews on related topics such as receding horizon LQ control, Laguerre filter based modelling and control, anti-windup scheme, control of time-delay systems and continuous-time system identification based on sampled input output data will be given when these topics are introduced. 1.3 Contribution of the Thesis The contributions of this thesis can be summarized as follows. 1. A new predictive control strategy is developed in a sampled-data framework. The resulting algorithm, SDGPC, has guaranteed stability property. Its relationship with infinite horizon LQ regulator is established clearly. SDGPC enjoys the advantage of continuous-time modeling and the flexibility of digital implementation. 2. A two-degree-of-freedom SDGPC based on optimization of two performance indices is proposed. Its servo performance and disturbance rejection performance can be tuned separately. Based on 4 Chapter 1: Introduction this design, an and-windup scheme is developed with guaranteed stability properties. The novel approach used here is to transform the nonlinear problem into a time-varying linear problem. This scheme has important practical implications as well as theoretical interests. 3. The one-degree-of-freedom SDGPC is extended to tracking system design. 4. Control of time-delay systems is treated in detail. A practically appealing Laguerre filter based adaptive SDGPC algorithm is developed. 5. An algorithm to estimate the parameters of continuous-time system based on sampled input output data is presented. Fast time-varying parameters can also be estimated under this framework. The effectiveness and the advantage of continuous-time model estimation and the SDGPC algorithm over the pure discrete-time approach are highlighted by an inverted pendulum experiment. 1.4 Outline of the Thesis Chapter 2 presents the Sampled-Data Generalized Predictive Control algorithm SDGPC. Its relationship with infinite horizon LQ regulator and stability property are analyzed in detail. Simulation and tuning guidelines are also given by examples. Chapter 3 extends the One-Degree-of-Freedom (ODF) SDGPC to tracking problems resulting in a Two-Degree-of-Freedom (TDF) design formulation. The TDF-SDGPC can track non-constant reference trajectories and/or disturbances with zero steady state error. When the future setpoint information is available, the TDF-SDGPC has a concise form and the tracking performance can be improved dramatically. Chapter 4 considers control of time-delay systems. The direct approach, in which time-delay appears explicitly in,the model, and Laguerre filter modeling approach are proposed. The Laguerre filter based adaptive SDGPC is particularly appealing in that its computation burden is independent on the prediction horizon. Chapter 5 deals with another important issue in process control: actuator constraints. A SDGPC algorithm based on two performance indices is proposed. The control problem is interpreted as a nominal servo performance design plus an integrator compensation for disturbances and modeling 5 Chapter 1: Introduction error. This algorithm under the framework of anti-windup design effectively transforms the con strained control problem into an unconstrained time-varying control problem whose stability can be guaranteed—a pleasant result. Examples are presented to show the effectiveness of the algorithm. Chapter 6 proposes a method to estimate the parameters of a continuous-time model based on sampled input output data. It is argued that even if the controller design is based on discrete-time model, it is always desirable to estimate the continuous-time model before discretization. An inverted pendulum is successfully controlled by SDGPC based on a continuous-time model estimated using the algorithm developed in this chapter. Chapter 7 summarizes the thesis and gives suggestions for future research. 6 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Chapter 2 Sampled-Data Generalized Predictive Control (SDGPC) The poor numerical property of discrete-time models based on shift operator for fast sampling applications was shown by Middleton and Goodwin [50, pp. 44]. This is no surprise since the discrete-time model coefficients could be badly conditioned under fast sampling [50, pp. 46]. One solution is to use the 8 operator. The 8 operator offers superior numerical property and has great resemblance in model coefficients with its continuous-time counterpart [50, pp. 46]. Gawthrop [32] on the other hand argued that a continuous-time process is best represented by a continuous-time model and took the complete continuous-time approach, for example, in the formulation of Continuous-time Generalized Predictive Control (CGPC) [18] in which the user selected future control scenario is of a polynomial form. This approach requires approximation in digital implementation and may cause unacceptable errors for large sampling interval. The SDGPC approach given in this chapter will be based on continuous-time modeling while assuming a piecewise constant projected control scenario thus enjoying the advantages of both sides. This chapter is organized as follows. SDGPC is formulated in section 2.1. Section 2.2 studies the stability properties of SDGPC. Section 2.3 gives interpretations for the SDGPC law in its integral form. Simulations are presented in section 2.4 to give tuning guidelines of SDGPC. Section 2.5 concludes the chapter. The work in this chapter was summarized in [46]. 2.1 Formulation of SDGPC In order to highlight the basic ideas behind SDGPC, we only consider SISO systems here. However, the extension to MIMO systems is straightforward. The system being considered is described by a state-space equation x(t) = Ax(t)+Bu(t) y(t) = cTx(t) (2.1) dim(x) — n 1 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) In order to introduce integral action in the control law, an integrator is inserted before the plant to give the augmented system if = AfXf + Bfud IJf = CTfXf (2.2) Where xf = dim(xf) = nf = n + 1 Xd(t) = i(t), ud(t) = ii(t), e(t) = y(t)-w 'xd' 'A 0' 'B~ _ e _cT 0_ .0. 0 1] (2.3) And w is the constant setpoint. We further assume that the projected future control derivative ud(t) is piecewise constant over the period of Tm = jf- with values ud.(l),u<i(2) • • •ua(Aru) as in Fig.2.1. The benefit of assuming piecewise constant control derivative is that it will result in a continuous control signal. Setpoint Predicted output SDGPC projected controls Ud Texe Tm p Time Figure 2.1: The projected control derivative We call Tp the prediction horizon or prediction time and Nu the control order which is the allowable control maneuvers over the prediction horizon. In Fig.2.1, Tm is called the design sampling interval since the resulting SDGPC law, as will be shown in section 2.2, is equivalent to a discrete-time receding horizon control law based on (2.2) with sampling interval Tm provided that the first control u,i(l) is injected into the plant for a duration of Tm. However this is not necessarily so, the first control Ud(l) can actually be injected into the plant for a shorter time interval Texe which we Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) will call it the execution sampling interval. Texe is the implementation sampling interval (in contrast to design sampling interval ) and can take any value on [0, Tm]. Similar to all other model based predictive control approaches, SDGPC is based on minimizing a performance index: Note that the above optimization problem is a standard finite time linear quadratic regulator problem in terms of the augmented plant model (2.2). One of the key concepts in the formulation of model based predictive control is the receding horizon strategy. However, special to SDGPC is that there are two ways to implement the receding horizon strategy. That is, after the projected control vector [ud(l),Ud(2) • •-Ud(iVu)] is obtained, either of the following strategies can be used: 1. The first control ud(l) is applied to the plant for a time duration of Tm. 2. The first control Ud(l) is applied to the plant for a time duration of Texe which is a fraction of the design sampling interval Tm. The first case is equivalent to a digital control law with sampling interval Tm as will be shown in the next section. In the second case, Texe can be smaller than Tm and when the execution time TeXe 0, it will become a continuous time control law. This approach thus has the potential to solve the numerical problem for the pure discrete-time approach, as we mentioned at the beginning of this chapter, in fast sampling applications. With the above preparations, we are in a position to derive the SDGPC law. The projected future control derivative in Fig.2.1 can be described mathematically as: (2.4) o Subject to : xf(t + Tp) = 0 (2.5) ud{t) = H(t)ud 9 (2.6) Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) where H(t) = [H1(t) H2(t) •••Hi(t)---HN.(t)} Ud = [Ud(l) Ud(2) • • • ud(i) • • • Ud(iV„)] (2.7) Hi(t) fl (i-l)Tm<t<iT„ { 0 otherwise i = 1, 2,---Nu T — IjL m " Nu (2.8) Based on the system model (2.2) and the projected control scenario (2.6), we have the following T-ahead state prediction: Where i xd(t + T) = eATxd(t) + J eA{?-^Bud{r)dT o T = eATxd(t) + (J eA^BH{r)dr)xxA o T T = eATxd(t) + [ f eA^BH1 (r)dr ••• J eA(T-^BEK(r)dT]nd eAI xd(t) + T(T)ud r(T) nxN„ l l J eW-^BH^dT-.- j eA^T-^BHNtt(T)di nxiV„ (2.9) (2.10) r(T)B xiV„ jeA(T-r)dTB. 0. Q...0 0 TfeA(T-r)dTB. j eA{T-r)(WB. Q . . . 0 0 T,„ /" e^-^drB; ••• j e^^drB .0 (7V„-l)Xm 0 < T < Tm 7?JI ^ 7" < 2T'7n (iVu - l)rm <T <Tf (2.11) 10 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) With xd(t + T), e(t + T) can be obtained: Where i e(t + T) = e(t) + cT j xd(t + r)dr o T = e(t) + J eATdTxd(t) + cTro(T)ud o T T0(T) = J T(T)dr o Recalling the cost (2.4), we define the Hamiltonian: H(t,rj) = J(t) + riTxf(t + Tp) = J [e(t) + cTA-l(eAT - l)xd{i) + cTro(T)ud]2 dT + j XujHT(t)H(t)uddT + T]'1 eA1»xd{t) + T(Tp)ud e(t) + (?A-1(eAT> - I)xd{t) + cTT0(Tp)nd_ ^et iu7 = ®i iftf = 0> we nave tne optimal solution for ud: ud = Kdxd(t) + Kee(t) (2.12) (2.13) (2.14) (2.15) where Kd = -K1^K3Td + TgK2 eATr cTA-l(eATp _ty -0-< KzTe + TgK2 • 0 .1. 4 Td = Jrl(T)ccT{eAT - IJA-'dT, Te = JTT0{T)cdT T9 = r(T) -i Ki J Tl(T)ccTT0(T)dT + \ J HTHdT K2 = [TjKiTg) \ 7C3 = I - TgK2TgK\ (2.16) 11 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) ob>—^ x=Ax+Bu X T c' 1 Observer Figure 2.2: The implementation scheme Fig.2.2 shows the SDGPC law (2.15) in a block-diagram form: As we mentioned earlier, the control law (2.15) does not necessarily need to be implemented with the design sampling interval Tm. When the execution sampling interval Texe goes to zero, we can take integration on both sides of (2.15) to obtain an integral control law (2.17) in terms of the state and the control signal of the original systems (2.1). u(t) = Kdx(t) + Ke f e{r)dr + The block diagram of control law (2.17) is shown in Fig. 2.3 (2.17) <x>EHI] & u x=Ax+Bu X J -TK51-Figure 2.3: The integral control law The constant term 770 in (2.17) is unspecified and has no bearing oh the problem in the sense that it neither affects the closed loop eigenvalue nor the asymptotic property of e(t) —• 0 as t —• 0 provided that the integral control law (2.17) is stabilizing. However, we can make use of the above fact and let rjo be proportional to the constant setpoint w. The effect is that a system zero can be placed in a desired location to improve the transient response of the closed system under control law (2.17). The scheme is depicted in Fig.2.4. Details on how to select the feedforward gain Kw in Fig.2.4 will be discussed in section 2.3. 12 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) w =Ax+Bu Figure 2.4: Zero placing strategy SDGPC was developed above by minimizing the cost (2.4) subject to end point state constraints (2.5). Another approach is to include the end point state in the cost functional: J{t) = j' [e2(t + T) + Xu2(t + T)]dT + jxT(t + Tp)xf(t + Tp) (2.18) o Substitute equations (2.6) (2.9) (2.12) into performance index (2.18), we have J{t) = J e{t) + J eATdrxd(t) + cTT0(T)ud o L o dT+ J \u/HT(t)H(t)uddT+ 0 eAT>xd(t) + r(T>H left) + J eA*dTxd{t) + cTro{Tp)xid 0 0 Let gjj^- = 0, we have the solution for u(1: ud = -K(Tdxd(t) + Tee(t)) eAT"xd(t) + r(Tp)ud e(t)+ feATdrxd(t) + cTT0(Tp)ud -1 K=\J TTccTY0dT + \J HTHdT + Trr(Tp)r(Tp) + 7r^(rp)ccTro(Tp) ,0 (2.19) (2.20) Td = JTT0ccTA-1 (eAT -I)dT + yrT(Tp)eAT> + (Tp)ccT A~'(eAT> - I) o Te = J TT(T)dTc + 1rT(Tp)c o Where T, Fa, H are defined by equation (2.10), (2.13), (2.7) respectively. Obviously, when 7 —• oo, control laws (2.20) and (2.15) become equivalent. 13 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) The main point of this section is that by selecting the projected control derivative scenario to be piecewise constant, a predictive control law SDGPC can be designed based on continuous-time modelling without causing any difficulty for digital implementation. This is in sharp contrast to CGPC. 2.2 Stability Properties of SDGPC Stability results for GPC with terminal states constraints (or weighting) are available both for discrete-time [11, 12, 52] and continuous-time [16]. A natural question is whether SDGPC possesses such stability properties. This question will be answered in this section. The basic idea is to show that SDGPC is equivalent to a stabilizing discrete-time receding horizon LQ control law. The important work of Bitmead et al. [4] is included in Appendix A for completeness. Those results are used to establish the stability property of SDGPC in section 2.2.1. 2.2.1 Stability of SDGPC with control execution time Texe = Tm The SDGPC stability problem is attacked by first applying a transformation to convert the SDGPC problem to a discrete-time receding horizon problem, then making use of the stability results summarized in Theorem A. 10 and Corollary A.2. The transformation is based on the work of Levis et al. [43] in which the infinite horizon problem was treated. Recall the state augmented system described by equation (2.2) dim(xf) = nf = n + 1 Assuming that the execution sampling interval Texe under SDGPC control is the same as the design sampling interval Tm, the discrete-time equivalent of the augmented system (2.21) is then if = AfXf + BfUd (2.21) Xf(i+l) = <f>Xf(i) + rud(i) 2//(0 = Cfxf(}) (2.22) With T, (2.23) o 14 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Recall the cost functional (2.4) with Q = CfCT-J(t) = J [xT(t + T)Qxf{t + T) + uT(t + T)Rud{t + T))dT (2.24) o Subject to : xf(t + Tp) = 0 With the projected control scenario described by (2.6) as in Fig.2.1, the cost (2.24) can be expressed as the sum of Nu integrals: J(t) = f [xT(t + T)Qxf(t + T) + uT(t + T)Rud(t + T)]dT = / [xTf{t + T)Qxf(t + T) + u^(t + T)Rud{t + T)]dT • A «/ (2.25) i=0 iTm Define xf(i) = xf(t + iTm), ud(i) = ud(t + iTm), i = 0,l,---Nu-l (2.26) The integrals in (2.25) can be expressed as (••+i)rm J [xj(t + T)Qxf{t + T) + uTd{t + T)Rud(t + T)] dT = J [xT(i + r)Qxf(i + T) + uj(i + r)Rud(i + r)] dr (2.27) The inter-sampling behavior Xf(i + r) of system (2.22) is a function of x(i) and ud(i) as follows T xf(i + T) = eAfTx(i) + J eA>{T-^B}ud(i)ds (2.28) o Substitute equation (2.28) into equation (2.27), we have 15 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) (••+1)T„ J [xT(t + T)Qxf(t + T) + uT(t + T)Rud(t + T)] dT iTm = J [xT(i + T)Qxf(i + T) + uJ(i + T)Rud(i + T)]dT 0 = xT(i)Qxf(i) + 2xT(i)Mud(i) + uJ(i)Rud(i) (2.29) where Q = J eAfTQeA'Tdr, M = J eAJTQ J eA'f o o Lo Tm r T -I r T R = TmR + Bj J J eAVdt Q J eA'fdt Bfdr 0 LO J LO Finally the continuous-time cost (2.24) has the form (2.30) drBt J(t) = J [xT(t + T)Qxf(t + T) + uT(t + T)Rud{t + T)]dT o = YI UT(i)Qxf(i)+2xT(i)Mnd(i) + nJ(i)Rud(i) (2.31) i=0 Remarks: 1. These weighting matrices are time-invariant as long as Tm is constant. The symmetric and positive semi-definite or positive definite properties of Q, R are preserved in Q, R. 2. Even if the control weighting R = 0 in the original cost functional, there always is a non-zero weighting term R in the equivalent discrete-time cost. Note that there is a cross-product term in the discrete-time cost (2.25) involving Xf(i) and ud(i). However, by some transformation [43], the cross-product term can be removed to form a standard discrete-time cost. Define Q = Q- MRTlMT # = $ - virlMT v(i) = R-1MTxf{i) + ud(i) (2.32) 16 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) By substituting equation (2.32) into system equation (2.22) and the associated cost (2.25), we obtain xf(i + 1) = $xf(i) + Tv(i) yf(i) = cjxf(i) (2.33) dim(xf) = rif = n + 1 and cost functional, iV„-l i=0 J(t)= £ \^(i)Qxf(i) + vT(i)Rv(i) xf(Nu) = 0 (2.34) For clarity, the above derivation is summarized in Table 2.1 17 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Problem Formulation SDGPC Discrete-time receding horizon LQ control System equation xf — AfXf + Brud dim(xf) = rif = n + 1 x/(i + l) = $x/(i) + rv(i) »/(0 = cJx/(0 dim(x}) = rif = n + 1 Performance index m = y + T)Qxf(t + T) + uTd{t + T)R.ud(t + T)]dT o J(t) = £ [xf(i)Qxf(i)+vT(i)Rv(ij\ Final state constraint xf(T„) = 0 xf(Nu) = 0 $ = e AfTm^ T= J eAjrBfdT in o Bfdr Relationships T Q T fe^dt drBf J .0 J .0 Q = Q- MR-lMT $ = * - rj^M3, v(i) = R-1MTxf(i) + ud(t) Table 2.1 Comparison of SDGPC and discrete-time receding horizon LQ control We summarize the above results as follows: lemma 2.1 When the execution time interval Texe is equal to the design sampling interval Tm, the SDGPC problem can be transformed to a standard discrete-time receding horizon LQ control problem as summarized in Table 2.1. Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) From lemma 2.1, it is clear that the stability problem of SDGPC boils down to finding the conditions in terms of system (2.1) under which Theorem A.10 holds. We have following results to serve this purpose. Lemma 2.2 investigates the controllability and observability of the integrator augmented system (2.2). The proof of the controllability part can be found in [59]. The proof of the observability part is straightforward as given below. lemma 2.2 ( Power et al. [59] ) If the original system (2.1) with triple (A,B,cT) is a. both controllable and observable b. there is no system zeros at the origin then the augmented system (2.2) with triple (^Af, Bf, cT^j is also controllable and observable. Proof: The proof for controllability of (Af,Bf) can be found in Power and Porter [59]. The observability matrix of (A, cT) is 0AcT = [ cT; AcT; A2cT • • • An~1cT]nxn, with rank(OAcT) = n-/ rp\ Olxn 1 / \ The observability matrix of (Af,cT) is 0AfCr — . Obviously, rank\0AjCTJ = / [OACT 0„xij n + 1, and the pair ^Af,cT^j is observable. • Remark: Condition b is intuitively obvious. If violated, there is no way that the system output of (2.1) can be driven to a nonzero setpoint. Or in terms of the augmented system (2.2), the state e(t) with nonzero initial value can not be driven to the origin. The following theorem is due to Kalman et al [38]. Theorem 2.1 ( Kalman et al [38] ) Let the continuous-time system (2.2) be controllable. Then the discrete-time system (2.22) is completely controllable if: Im(\i{A] - \j{A}) * np-Jm (2.40) n = ±1,±2,.... whenever Re(Xi{A} - Xj{A}) = 0. 19 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) lemma 2.3 ( Anderson et al. [3, pp. 354] ) Assume ($,T) given by equation (2.23) is controllable, then ($,r) given by (2.32) is also controllable. Proof: The proof is obvious. Recall $ = $ - TR~1MT, the controllability of a controllable pair ($,r) can not be changed by state feedback. • lemma 2.4 ( Levis et al. [43] ) Q > o. Proof: Since Q > 0, R > 0, so every integrand in (2.25) (i+l)Tm Ii= j [xJ(t + T)Qxf(t + T) + uT(t + T)Rud(t + T)]dT = xj(i)Qxf(i) + 2xTf(i)MvLA(i) + uJ(i)Rud(i) is nonnegative for any ud(i). Let u<i(i) = —R~1MTXf(i), We have from equation (2.41) = + 2xJ(i)Mud(z') + uJ(i)Rud(i) = xTf(i) (Q - irlMT)xf(i) (2.42) = xJ(i)Qxf(i) > 0 for any Xf(i). So Q > 0. • Lemma 2.5 establishes the observability of the pair ($, <5) and the observability of the augmented system (2.2). This is a special case of the results for periodic time-varying systems given by Al-Rahmani and Franklin [2] in which multi-rate control strategy is used. A simpler proof based on the if-controllability and observability concept [6] [37] is given in the following. lemma 2.5 Assume the controllability conditions of Theorem 2.1 hold, then ($, Q) is observable if and only if the pair (^Af,cJ^j of equation (2.2) is observable . Sufficiency: Assume ($,<Q) is observable but (Af,Q) is not, then there exists an eigenvalue A of $ associated with a nonzero eigenvector z such that <3>z = Xz and QeAftz = 0 for any r > 0 20 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) [6]. It then follows from equation (2.30) that Qz = 0, MTz = 0. From equation (2.32), we have ®z = Xz, Qz = 0. So A is unobservable in ($,Q) l37!- This contradicts the assumption. Necessity: Assume (A/, Q) is observable but ($, Q) is not. Let A be an unobservable eigenvalue of $ and z ^ 0 be an associated eigenvector. We have $z = Xz, Qz = 0. Recall equation (2.41), let Xf(i) = z, \id(i) = —R~lMTz, we have (t+i)r,„ Ii= J [xT(t + T)Qxf{t + T) + uT(t + T)Rud(t-rT)]dT «T» (2.43) = xT(i)Qxf(i) + 2xT(i)Mud(i) + uJ(i)Rud(i) = zTQz = 0 Since Q > 0, R > 0, equation (2.43) implies / xT(T)Qxf(r)dT = J uT(T)Rud(T)dT = 0. o o Tm Further, J uT(T)Rud(r)dT = zTMR~1 (TmR)R~1MT2 = 0. Since £-1(Tmi?).R-1 > 0, we 0 have MT2 = 0. From equation (2.32), zTQz - zTMRr1MTz - zTQz = 0 and $z = - TR-^M^z = $z = A*. From equation (2.30), zrQz = 0 implies QeAt*z = 0. But the existence of z ^ 0 such that $2 = A2, QeAftz = 0 contradicts the observability assumption of(A/,Q). • Now, we are in a position to state the main stability property of SDGPC. Theorem 2.2 For systems described by equation (2.1), if a. The triple (A,B,cT) is both controllable and observable. b. There is no system zero at the origin. c. The control execution time Tm is selected such that the condition in Theorem 2.1 is fulfilled, then the resulting closed loop system under SDGPC is asymptotically stable for Nu > n + 1. Proof: According to lemma 2.1, SDGPC of system (2.1) is equivalent to receding horizon control of discrete-time system (2.33). Thus we need only to prove the stability of the receding horizon control problem for system (2.33) with performance index (2.34). Conditions a. and b. guarantee the controllability and observability of the integrator augmented system (2.2) according to 21 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) lemma 2.2. From condition c. and Theorem 2.1, it is apparent that the discrete-time counterpart of (2.2) given by (2.22) is also controllable and observable. Applying lemma 2.3-2.5, it is obvious that Q > 0, R > 0 in (2.34), is controllable and is observable. Apply Theorem A.10 proves the theorem. • 2.2.2 Property of SDGPC with control execution time Texe < Tm In section 2.1, we mentioned that the execution sampling time interval Texe, i.e. the time' interval with which the plant is actually being sampled, can take any value on [0, Tm]. The case of Texe < Tm will be analyzed in this section. This strategy is very similar in spirit to the well known GPC design practice of selecting a smaller control horizon than the prediction horizon in which case the computation burden can be greatly reduced. Fig.2.5 illustrates these two closely related strategies. Setpoint Figure 2.5: Comparison of SDGPC and GPC strategy GPC [13] design is based on minimization of the following performance index J(N) = (y(t + w(t + i)f + A [Au{t + i)}2 (2.44) i=N. i=0 Ni can always be selected as zero. The prediction horizon N2 corresponds to the prediction time Tp in SDGPC. The control weighting A has the same meaning in SDGPC but the control horizon Nu has a quite different interpretation as is clearly illustrated in Fig.2.5. In SDGPC, Nu is the number of controls that will cover the whole prediction horizon Tp, while in GPC it is the number of 22 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) controls that only cover a portion of the prediction horizon after which the control is kept constant or the increment of controls is kept zero. And in SDGPC the control execution time Texe is not necessarily equal to the design sampling time Tm. It is also possible to assume the projected controls in SDGPC design have the same form as that of GPC, or any other form, say a polynomial up to certain degree. However, the advantage of choosing piecewise constant equally spaced controls over the entire prediction horizon is that in doing so the SDGPC problem can be transformed into a discrete-time receding horizon LQ problem for which powerful stability analysis methods in optimal control theory can be utilized and improved numerical property can be expected because a larger design sampling interval Tm is used. Refer to Fig.2.5, both SDGPC and GPC use Nu = 4. Both SDGPC and GPC update their control every Texe seconds. Both SDGPC and GPC use the same prediction horizon: N2*Texe = Tp. The difference is that the design sampling interval in SDGPC is Tm = 4 * Texe, i.e. four times as large as the execution time. Both of them have the effects of reducing computational burden and damping the control action. However, SDGPC will have superior numerical property when Texe is small because SDGPC is computed based on a larger design sampling interval Tm while GPC is still based on Texe. Another advantage of SDGPC is that although neither of them has stability claim when Nu < N2 or Texe < Tm, we know that the same SDGPC law does have guaranteed stability when Texe = Tm. While it is very natural to choose Texe < Tm in SDGPC design under the framework of receding horizon strategy, it is almost unthinkable for any other controller synthesis method to design a stabilizing control law for one sampling interval but to apply it to the process with another sampling interval. It is well known that discrete-time design methods based on z-transform will encounter numerical problems when the sampling interval is small [50]. In SDGPC, a larger design sampling interval Tm can be used to improve numerical property while implementing it with a shorter sampling interval Texe. Although there are no general stability results for Texe < Tm, extensive simulation examples will be presented in next section to offer guidelines of selecting Tm and Texe. As a by-product, those simulations will also shed some light on the selection of sampling interval in digital control in general. 23 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) 2.3 Interpretation and Stability Property of the Integral Control Law The integral control law (2.17) was obtained by integrating both sides of (2.15) under the assumption that Texe —• 0. However, (2.17) itself can be interpreted as a solution of a well formulated t predictive control problem for system (2.1). Define integral Ie = J (y(r) — w)dr, where the lower limit of the integral was left blank to indicate that Ie can take any initial value, as the new state of system (2.1), the augmented system becomes: xj = AfXj + BfU + Bvw y=[cT 0]xx dim(xi) = n + 1 (2.45) Where X 'A 0' 'B' ' 0 ' XJ = , Bf = , BV = Je. cT 0. .0. -1. (2.46) Where w is the constant setpoint. Notice that the augmented system matrices Af,Bf are exacdy the same as of that in (2.2). The objective of the control is to let the output y(t) of system (2.1) track the constant setpoint w without steady state error. Thus at equilibrium, the following relations hold: lim y(t) = i/oo = w t—too and lim u(i) = tia t—too lim Ie(t) = I0 t—*oo lim x(t) = XQ, t—too J/oo — W — C XQQ 0 = AXQO + BUQO (2.47) (2.48) where Uoo-Zooj^oo are constants whose value can not be determined a priori based on the nominal plant parameter matrices (A, B,cT) and the setpoint because of the unavoidable modelling errors. A sensible approach is thus to define the shifted input, the shifted state respectively as 24 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) u'(t) — u(t) — u, •oo x'(t) = x(t) — X, 00 (2.49) I'e(t) = Ie{t) - f 00 y'(t) = y(t) - w Solving (2.49) for u,x,Ie,y, substituting the results into (2.45), and using (2.48) it is not difficult to find that the shifted variables satisfy the equations The shifted equilibrium of (2.50) is at zero as that in (2.2) and a predictive control problem can be well formulated by minimizing a quadratic performance index And at the end of the prediction horizon Tp, the state of (2.50) should be constraint to be zero, that is x'j(t + Tp) = 0. Although the above problem is well defined, it is still very inconvenient, to say the least, to obtain the control law due to the unknown equilibrium point UQO, x^, 1^. A more effective formulation should thus have a model which accommodates the fact that at the equilibrium, the input, output and the state are all constant but at the same time should not explicitly have those unknown constants in the model. Taking derivative of both sides of the first equation of (2.45) will just do that. The resulting equivalent system model has the form x' (2.50) (2.51) o if = AfXf + Bfiij + Bvw (2.52) where Xd(t)-x(t), ud(t) = u(t) eW = 2/(0 ~w, xf = (2.53) e 25 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) For constant setpoint as we assumed, w — 0 and (2.52) is exacdy the same as (2.2). The only modification needs to be made is that the observation matrix should be cj in (2.3). The SDGPC problem for (2.2) and the associated performance index (2.4) can thus be interpreted as a sensible way to circumvent the unknown equilibrium difficulty encountered in the control problem defined by (2.50) and (2.51). According to Theorem 2.2, the control law (2.15) stabilize system (2.1). Similar results can be said about control law (2.17): Theorem 2.3 For systems described by equation (2.1) and the integral control law (2.17), if a. The triple (A,B,cT) is both controllable and observable. b. There is no system zero at the origin. c. The control execution time Texe is equal to the design sampling time Tm and is selected such that the condition in Theorem 2.1 is fulfilled. d. Zero-th order hold is used when applying (2.17) to system (2.1). then the resulting closed loop system under the integral control law (2.17) is asymptotically stable for Nu > n + 1. Proof: When the integral control law (2.17) is applied to (2.1) with zero order hold, the resulting closed loop system matrix will be the same as that of by applying (2.15) to (2.1). This can be seen readily by comparing equations (2.2) and (2.45) considering that fact that the disturbance term Bvw in (2.45) will not affect the stability of the closed loop system. Since system (2.1) is stable under the control of (2.15) according to Theorem 2.2, it will be stable as well under the control of (2.17). • We mentioned in section 2.1 that the unspecified term in 770 in (2.17) can be used to place a zero to improve the transient response of the closed loop system. In the following we show that there is a sound mathematical basis for doing so. Consider system (2.52) and the cost (2.18), the T-ahead state predictor is described by (2.54) with u<* given by (2.7). 26 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) xf(t + T) = eA'Txf(t) + Du(T)w(t) + T(AfBf,T)ud T DU{T) = j eA*T-^BvdT (2.54) where T(Af,Bf,T) is given by (2.11) with A, B replaced by Af,Bf. Without detailed derivation, the optimal control to system (2.52) can be obtained as K = Kxf Le(<) + Kpw{t) where K = Kxt = KHXJ Kp = KHp J TT(T)cfcjT{T)dT + \J HT(T)H(T)dT + 7rT(Tp)r(Tp) \° 0 J N*xN, Hxt — — HB = -J TT(T)cfcJeAfTdT + 7rT(Tp)eA'T» J TT(T)cfc^Du(T)dT + jTT(Tp)Du(Tp) (2.55) (2.56) The counterpart of the optimal control sequences (2.55) with respect to system (2.45) is given by u* = K Xf x(t) + l<0w(t) lJe(T)dT_ The first control which is the only one being applied to the plant is (2.57) u*(l) = Kxx(t) + Ke j e{r)dT + K0(l)w(t) (2.58) where Kx denotes the first n entries of the first row of the Nu x (n + 1) matrix KXf, Ke is the last element of the first row of Kr 27 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) The effect of Kp{l) in (2.58) is to add a zero at Ke/Kp(l) from reference w(s) to output y(s) [28 , p.559]. Since Kp(l) does not affect the eigenvalues of the closed loop system matrix, meaning that it can take any value in addition to the one being computed by equation (2.56). This provides one extra degree of freedom in the design. Example 2.3.1: In this simulation, the plant with transfer function G{s) = 3_ (2.59) is being controlled using control law (2.58) with the following design parameters The resulting feedback gains are: Nu = l Tp — 6s Texe = 0.1s (2.60) A = 10"4 7 = 100 Kx = -[0.2839 0.8628 0.8776] (2.61) Ke = -0.2993 The eigenvalues of the augmented closed-loop matrix Af + Bf[Kx Ke] are at: •1.0523 ±0.0653t -0.8697 -0.3096 (2.62) 28 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Fig. 2.6 shows the control results for two different values of the feedforward term Kp, i.e. Kp = 0 and Kp = _0*c096 = 0.9668. The latter Kp places a zero which cancels the last pole -0.3096 of the closed-loop system matrix resulting a faster response which can be seen from Fig. 2.6. 1 1 1 Setpoint Output: Kp = 0.9668 w > / > / I / ;y X \ / »/\ Output: Kp = 0 * \ / \ s / 1 1 1 1 1 - -10 20 30 40 50 60 (S) Control: Kp = 0.9668 Control: Kp = 0 _5I 1 1 1 1 1 1 0 10 20 30 40 50 60 (s) Figure 2.6: Zero placement in SDGPC 2.4 Simulations and Tuning Guidelines of SDGPC Refer to Fig.2.7, the design parameters of SDGPC are: Prediction time Tp, design sampling interval Tm, execution sampling interval Texe, and control weighting A. The control order Nu is related to prediction time and design sampling interval by -/V„ = If the final states weighting is used other than final states constraint as in performance index (2.18), there is an additional design parameter 7. This is the approach used in [17] where 7 served as the tuning parameter to damp the control action. However in SDGPC, we would rather fix 7 to a very large value which corresponds to the states constraint case since this is crucial to guarantee stability. The task of reducing excessive control action can be accomplished by selecting Texe < Tm, which is equivalent to putting infinite 29 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) weighting on controls with sampling interval Texe and only allowing control to vary every Tm time units. This will be shown later by example. w Setpoint SDGPC projected controls Ud(Nu) Texe Tm Tp TilTIG Figure 2.7: The projected control derivative Example 1: The aim of the first example is to show the effects of the SDGPC design parameters on the control performance, and compare SDGPC with infinite horizon LQ control. The process being controlled is G(s) = (2.63) It is assumed that this process has to be controlled with a relatively fast sampling interval Texe = 0.2s in order to have fast disturbance rejection property. It is also assumed the states of the process is available for measurements and the derivatives of the states are computed by the state space equivalent of system model (2.63) -3 -3 -r "1" xd(t) = 1 0 0 a:(*) + 0 . 0 1 0 . .0. u(t) (2.64) Fig.2.8 shows the step response of the plant. 30 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Time (sees) Figure 2.8: Step response of example 1 Simulation 1: SDGPC of plant (2.63) with the following design parameters Nu = 6 (2.65) Tp = 1.2s Tm — Texe — 0.2s A = 10~5,0.01,0.1,0.5,1010 According to Theorem 3 in section 2.2, Nu should not be smaller than 4 to ensure stability. Also from Fig.2.8, the final prediction horizon Tp = 1.2s is very short for this plant. Fig.2.9 shows the control results. 31 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) 2 1 0 -1 1000 500 0 -500 -1000. It is obvious from Fig.2.9 that this control law is unacceptable in practice because of the large magnitude of the control action. Also notice that increasing the control weighting is not effective in damping the control since the prediction horizon Tp is too short. It can be seen from Fig.2.9 that between 40 (s) and 50 (s) even a control weighting of 1010 can not penalize the control action. This is because when Tp is small, the end point states constraints dominate the control law calculation whereas the performance index (2.4) has little effect on the controls. Since we can not reduce the control order Nu because of the stability requirements, the only option is to increase the prediction horizon Tp. Simulation 2 shows the results. Simulation 2: SDGPC of plant (2.63) with Nu = 21 Tp = 4.2s (2.66) Tm — Texe = 0.2s A = 10-5,0.01,0.1,0.5,2 32 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) The prediction horizon is selected to cover the significant part of the step response. See Fig.2.8. The design sampling interval and the execution sampling interval are the same as in simulation 1. Fig.2.10 shows the results. ^=0.00001 1 X=0.01 1 1 X=0A X=0.5 X=2 / \ / \ -1 t 1 1 10 20 30 40 50 (S) 50 (s) Figure 2.10: Simulation 2 of example 1 The results shown in Fig.2.10 are good except that the control law involves calculation of a 21 x 21 matrix inversion, a significant increase in computation burden compared with simulation 1. Simulation 3: Simulation 3 shows the SDGPC of plant (2.63) with N„ Tp = 4.2s Tm = 0.7s ^exe = 0.2$ A = 10-5,0.01,0.1,0.5,2. (2.67) 33 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) The prediction horizon is the same as in simulation 2 but the control order equals the one in simulation 1. That means the design sampling interval T,„ = = 0.7s and the execution sampling interval remains to be 0.2s as in simulation 1 and 2. X=0.00001 \ X=0.01 h=0.S X=2 / \ / \ • 1 1 1 1 0 10 20 30 40 50 (s) (a) Figure 2.11: Simulation 3 of example 1 The good results in Fig.2.11 suggest that selecting Tm > Texe is a useful strategy to reduce computation burden and at the same time damping the control action. More simulations will be presented to support this claim in example 2. Simulation 4: It is interesting to compare SDGPC with infinite horizon LQR with the performance index OO r -I J = ~~ w) + m which the only tuning parameter is control weighting A since infinite horizon is used. Plant (2.63) is discretized with sampling interval 0.2s as previous simulations. Control weighting varies as indicated in Fig.2.12 34 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) ^=0.00001 X=0.01 X=0.1 X=0.5 X=2 / \ / \ • I 1 1 1 10 20 30 40 50 (s) 50(s) Figure 2.12: Simulation 4 of Examplel: infinite horizon LQR Compare Fig.2.12 with Fig.2.10 and Fig.2.11, it can be seen that infinite horizon LQR has visible overshoot for small control weighting A whereas increase A slows down the response significantly. However, this is not suggesting that SDGPC has inherent advantage over infinite horizon LQR, after all they are the same as analyzed in section 2.2.1. However, it might be easier to tune SDGPC than LQR since there are fewer design parameters in SDGPC ( prediction horizon, control order etc. ) than that in LQR ( all entries of the weighting matrices ). Example 2: Two plants are simulated in this example. The first one is a non-minimum phase well damped open loop stable system. Gi(s) = (2.68) (s + 1) The second one is an open loop unstable system with imaginary poles. G'"> = (»-+0.4, + 9) <"» Simulation 1: Plant (2.68) is controlled by SDGPC with the following design parameters: 35 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Nu = 5 Tp = 5s Tm = Is A = 0.1 (2.70) In the first 15 seconds, the execution sampling interval Texe is equal to Tm, and after that Texe is reducing every 15 seconds as illustrated in Fig.2.13 1 Texe =1s \ 1 1 Texe =0-8s 1 1 ATexe=0.5s r Texe=0.2s 1 1 ATexe=0-01s / i U i i i i i i 10 20 30 40 50 60 70 (s) (a) Figure 2.13: Simulation of plant (2.68) This simulation shows that for well damped stable system (low pass plant) when fast sampling is needed, SDGPC can offer both low computation load and high implementation sampling rate by selecting Texe < Tm. Simulation 2: Plant (2.69) is studied. First the following group of design parameters is used. 36 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) Nu = 5 Tp = 5s Tm = Is A = 0.1 The execution time Texe varies as illustrated in Fig.2.14 2i 1 , 1 Texe =1s Texe =0.5s Texe =0.2s (2.71) 100(s) Figure 2.14: Simulation of plant (2.69) 100(s) Fig.2.14 shows that when the execution sampling interval Texe is equal to the design sampling interval Tm, the performance is good. As Texe decreases the performance deteriorates and the system becomes unstable when Texe = 0.2s. Considering the plant (2.69) has an unstable pole with time constant of 1 second and has a lightly damped mode with resonance frequency of 0.4759Hz, the design sampling interval Tm = Is is relatively large. Two things can be told by the results in Fig.2.14 for unstable and/or lightly damped systems. First, when the sampling interval Tm is relatively large, selecting Texe < Tm can cause performance deterioration or even instability. Second, even Texe = Tm is not a good choice for such a system. Since changing sampling interval can be viewed as a perturbation to the sampled-data system, the performance deterioration in Fig.2.14 means that the closed4oop system under SDGPC with sampling interval Tm — Is is sensitive to plant uncertainties. 37 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) The next simulation suggests that for systems with unstable and/or lightly damped poles the design sampling interval Tm should be at most one third of the unstable pole time constant or the sampling rate be 6 times that of the resonant frequency. Simulation 3: Plant (2.69) is controlled with the following design parameters. Nu = l Tp - 2.1s Tm = 0.3s (2.72) A = 0.1 The sampling interval Tm is reduced to one third of the unstable pole time constant. Fig.2.15 shows the results. Texe =0.3s r i 1 1 1 Texe=0.1s i Texe=0.01s A / till! 10 20 30 40 50 60 (s) Figure 2.15: Simulation of plant (2.69) 60 (s) It can be seem from Fig.2.15 that when Tm is reduced to 0.3s for this plant, good results are obtained regardless of the changing of the execution time. The conclusion drawn from these two examples is that for stable well damped systems, the design parameters can be selected primarily based on performance and computation load considerations and 38 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) the execution time can be selected flexibly. For unstable and/or lightly damped systems, in addition to performance and computation load considerations, there is an upper bound on the design sampling interval Tm restricted by the unstable pole time constant or the resonant frequency. There is no explicit formula available for the bound yet. But a rule of thumb is to select Tm less than one third of the unstable time constant or make ^ larger than 6 times the highest resonance frequency. Example 3: This example shows the ability of SDGPC to control systems with nearly cancelled unstable zeros and poles. GPC will encounter difficulty controlling this kind of systems [4, pp. 102]. The plant being controlled is s - 0.9999 Design parameters: G2(s) (s - l)(2s + 1) Nu = 6 Tp = 2.4s (2.73) T — T OAs (2.74) A = 0.5 30 (s) 25 30 (s) Figure 2.16: Simulation of plant (2.73). 39 Chapter 2: Sampled-Data Generalized Predictive Control (SDGPC) SDGPC can control this plant without any difficulty. 2.5 Conclusion At the beginning of this chapter, we mentioned that there are some problems with pure discrete-dme approach when fast sampling is needed. A new predictive control algorithm SDGPC was thus formulated based on continuous-time modeling while assuming a piecewise constant control scenario. Better numerical property can be expected since when fast sampling is needed, the control law can actually be designed based on a larger sampling interval. SDGPC relates continuous-time control and discrete-time control in a natural way thus enjoys the advantage of continuous-time modeling and the flexibility of digital implementation at the same time. Under mild condition, SDGPC is shown to be equivalent to an infinite horizon LQ control law thus it has the inherent robustness property of an infinite horizon LQ regulator. However, the finite horizon formulation of SDGPC makes it convenient to handle various input and states constraints. Moreover, like other predictive control methods, the tuning of SDGPC is easy and intuition based. The design of SDGPC in this chapter is a one-degree-of-freedom design. Various extension will be made in the following chapters. 40 Chapter 3: SDGPC Design for Tracking Systems Chapter 3 SDGPC Design for Tracking Systems In process control, the main objective is to regulate the process output at a constant level subject to various types of disturbances which is known as regulator problem. There is another type of control problem known as tracking or servo problem where it is required that the output of a system follow a desired trajectory in some optimal sense. This servo problem does occur in process industry , albeit not as often, such as the change of paper grade from a basis weight of 80g/m2 to lOOg/m2 in paper production. Another important class of problem in process control which also fits into the framework of tracking control is the feed forward design problem when the disturbance information is available. The optimal tracking problem in the linear quadratic optimal control context was well formulated [41] [3]. But they were either in continuous-time or discrete-time framework. It is thus worthwhile to formulate the tracking problem in the context of sampled-data generalized predictive control. The SDGPC algorithm we developed in chapter 2 is a special case of tracking system design in which the desired trajectory is a constant setpoint. A wider class of trajectories will be considered here. Trajectory following problems were classified in three categories in Anderson and Moore [3]. We follow the same treatment in this chapter. If the plant outputs are to follow a class of desired trajectories, for example, all polynomials up to a certain order, the problem is referred to as a servo problem; if the desired trajectory is a particular prescribed function of time, the problem is called a tracking problem. When the outputs of the plant are to follow the response of another plant (or model), it is referred to as the model-following problem. However, the differences between them are rather subde in principle. 3.1 The Servo SDGPC Problem Given the n-dimensional SISO linear system having state equations x(t) = Ax(t)+Bu(t) (3.75) y(t) = cTx(t) 41 Chapter 3: SDGPC Design for Tracking Systems The augmented system is described by Where Xf = if — AfXf + BfUd Vf = °Tfxf dim(xf) = rif = n + 1 xd(t) = i(t), ud(t) = u(t) ' Cf (3.76) 'xd~ 'A 0" 'B' , *f = . Bf = .y. _cT 0. .0. [0 0 1] (3.77) Note that the system output y(t) rather than the tracking error e(i) = y(i) — r(t) is augmented as the system state since the setpoint r(t) is no longer constrained to be constant. Suppose the reference signal is the output of a p-dimensional linear reference model w = Fw (3.78) r(t) = RTw with the pair [F, RT] completely observable. Assume that the future projected control derivative is piecewise constant in the time interval [t,t + Tp] as illustrated in Fig. 3.17, the SDGPC servo problem is to find the optimal control vector ud = [ud(l) ud(2) • • • ud(Nu)]T such that the following performance index is minimized. J = l(y{t + Tp) - r(t + Tp))2 + 1XTd{t + Tp)xd(t + Tp) + J [(y(t + T)-r(t + T))2 + \u2d(t-rT) dT (3.79) To solve this optimization problem, we need the T-ahead state prediction for both the plant (3.76) and the reference (3.78). Recall that the projected control scenario in Fig. 3.17 can be written as ud(t) = H(t)ud (3.80) where H(t) = [E1(t) H2(t) •••Hi(t).--HN.(t)) Ud=Ki(i) ud(2) • • • ud(i) • • • ud(iV„)] (3.81) 42 Chapter 3. SDGPC Design for Tracking Systems Setpoint SDGPC projected controls Ud (t) Tp Time We have where Figure 3.17: The projected control derivative 1 (t - l)Tm < t < iTm Hi(t) = L 0 otherwise i = l,2,---Nu T m ~ Na Xf(t + T) = eA'Txf(t) + J eA'(T-^Bfud{r)di = eA'Txf{t) + T(AfBf,T)vLd T(Af,BfT) = T feAf(T-T)dTBf. Q.Q...Q 0 TJ eAf(T-r)dTBf. j eAj(T-T)dTBf. 0 . . . o 0 Tm TfeAf(T-r)dTBf.... J eAAT-r)drBf o (ivB-i)rm And the reference state prediction is simply 0 < T < Tm Tm<T< 2Tm (Nu - l)Tm < T < Tf (3.82) (3.83) (3.84) w(t + T) = eiflw{t) 43 (3.85) Chapter 3: SDGPC Design for Tracking Systems Consequently the output predictions are as follows. y(t + T) = cTxf(t + T) Xd(t + T) = CTXf(t + T) = [Inxn0)nxnxf(t + T) r(t + T) = RTw(t + T) Substitute equations (3.86) and (3.87) into the performance index (3.79) (3.86) (3.87) J = J [cTfeA'Txf{t) + cTT{Af,Bf,T)ud - RT eFT w{t)]2 dT o T +A J [uTHT(T)H(T)ud]dT o +7[cTeA'T>xf(t) + cTT(Af,Bf,Tp)ud - RTeFT"w(t)]2 +1[cTeA'T>>xf(t) + cTr(Af,Bf,Tp)ud]T[cTeA'T*xf(t) + cTT(Af,Bf,Tp)ud] Take the derivative of J with respect to u^1 dJ _ dud j [TT{T)cfcTT{T)ud + TT{T)cf{cTeA'Txf{t) - RTeFTw(t))]dT +A j HT(T)H(T)dT +j[TT(Tp)cfcTr(Tp)ud + rT(Tp)cf{cTeA'T>xf(t) - RTeFT*w(t))} +7[rT(rp)cdCJr(2>d + rT(Tp)CdcJeA^xf(t)] The optimal SDGPC tracking control solution u*d is given by ~xd(ty ud = Kww(t) + KXs ly(t) J (3.88) (3.89) (3.90) Af,Bf are dropped (tomT(Af,Bf,T) for clarity 44 Chapter 3: SDGPC Design for Tracking Systems where Kw — -ft Hw KXf = KHX/ y1 = I J VT(T)cfRTeFTdT + jFT(Tp)cfR TeFTp JV„xp K=\J rT(T)cfcjT(T)dT+X f HT(T)H(T)dT + jTT(Tp)F(Tp) \ (3.91) I HXJ = -\J TT(T)cfcJeA'TdT + >yTT(Tp)eA'T> NuXTlf As shown in Fig. 3.18, the servo SDGPC law (3.90) has one feedforward term in addition to a usual feedback term as in the regulator case. This is what is known as a two-degree-of-freedom design method. Also note that equation (3.91) clearly shows that the feedback gain KXf is independent of the trajectory reference model (3.78). zoh w= Fw w(t) r(t)= RTw H^^'^V^y'iHIl^ x=Ax+Bu KY, Observer xt Figure 3.18: The servo SDGPC controller So far we have assumed that the state w of the reference signal is available for measurement. In practice, however, often only an incoming signal is at hand. For this case, a state estimator may be constructed with the pair [F, RT] completely observable. Then the state estimator and the static feed forward gain Kw can be combined to give a dynamic feedforward controller as illustrated in Fig. 3.19 r(t) Feedforward controller zoh x=Ax+Bu X T c' Ky, Observer Figure 3.19: The dynamic feedforward controller implementation 45 Chapter 3: SDGPC Design for Tracking Systems When the reference model (3 0 • 1 • 78) is given by • 0 0 " • 0 0 1 ^ -* pxp ,RT = {0 0 l]j (3.92) the reference trajectory will be the class of signals consisting of all polynomials of degree (p— 1). The state w(t) will be consisting of the incoming signal r(t) and its derivatives up to the order (p — 1) = •- ^ <t)f- (3.93) Clearly, the SDGPC algorithm developed in chapter 2 is a special case where r(t) — 0, that is, F = 0,RT = 1. As mentioned earlier, for general F and RT, a state estimator may be needed to construct the state of the incoming signal. However, when F and RT are given by (3.92) with p = 2, a simple structure can be obtained. According to the receding horizon strategy, only u^(l), the first element of the optimal control ud, is applied to the plant. For 0 0 1 0 RT = [0 1] (3.94) , from equation (3.90), we have uj(l) = [7^(1,1) 1^(1,2)] r(t) r('). .y(') (3.95) where KXf(l,:) denotes the first row of matrix KXJ in equation (3.91). A closer look at Hw and HXf in equation (3.91) reveals that the last columns of eFT and eA>T are [0 T]T and [0 0 • • • T]£ xl because the last columns of F and Af are all zeros. Thus, the last columns of CfRTeFT in Hw is equal to the last column of matrix CfCJ"eAfT in HXf. As an immediate consequence, Kw(l,2) and A'X/(l,n/) in equation 3.95 are of the same amplitude but with opposite signs. Let KTd = Kw(l, l),KTty = Kw(l,2),KXd = KXf(l, 1 : n), we have uj(l) = Krdf(t) + Kr,y[r(t) - y(t)) + KXdxd(t) (3.96) 46 Chapter 3: SDGPC Design for Tracking Systems Here KXf(l, 1 : n) denotes the first n elements of the first row of matrix KXf. For small execution sampling time, the optimal servo SDGPC law (3.96) can be written in another form as Compare with the optimal SDGPC solution (2.17) in chapter 2, control law (3.97) has an additional feedforward term KTdr(t). It should be noted that although the servo SDGPC solution (3.97) is derived for ramp signal f(t) = 0, it does not necessarily yield zero tracking error for a ramp even asymptotically. The reason for this is that there is only one integrator in the controller (3.97) which can only track constant reference with zero steady state error [73]. According to the internal model principle, there must be a model of the exogenous signal included in the control law for robust zero error tracking and disturbance rejection. As most of the discrete-time predictive control algorithms, SDGPC has the ability to track a general class of reference trajectory with zero steady state error. The spirit of this approach is state augmentation. That is, by including the equations satisfied by the external signal into the system model, a new system model in the error space [28] with new coordinates can be obtained and the SDGPC design procedure can then be applied. In the following, the servo SDGPC problem which incorporates double integrators in the control law is presented to show the procedure. Consider the plant described by (3.97) x{t) = Ax(t)+Bu(t) y{t) = cTx(t) (3.98) dim(x) = n The augmented system is described by xz = Azxz + Bzuz (3.99) dim(xz) = nz = n + 2 47 Chapter 3: SDGPC Design for Tracking Systems Where uz(t) = u(t) "x~ • A Onxl Onxl ' •B-y , AZ = T C1 0 0 , Bz = 0 .y. -Olxn 1 0 . .0. ,£ = [0 0 1] (3.100) Assume that the reference trajectory is given by w = Fw r(t) = RTw (3.101) where 0 0 0' 1 0 0 .0 1 0. ,RT = [0 0 l],w(t) = r(t) r(t) r(t). (3.102) Similarly, assume that the future projected uz(t) over [t,t + Tp] is piecewise constant as illus trated in Fig. 3.17, the objecdve is to find the optimal control vector uz = [u*(l) uz(2) • • •uz(Nu)]T such that the following performance index is minimized. \ 2 i t ' (4. i T> \ * / ± i T1 \ *\ 2 J = 7(y(t + Tp) - r(t + TP)Y + 7(y(t + Tp) - f(t + TP)Y +JxT(t + Tp)x{t + Tp) + J {(y(t + T)-r(t + T))2+ \u2z(t + T) dT (3.103) Again, we need the T-ahead state prediction for both the plant (3.99) and the reference (3.101) which are given by l cx(t + T) — eA'Txz(t) + J eA'(T-^Bzuz(r)dr o = eA>Txz(t) + r{Az,Bz,T)uz and (3.104) (3.105) w(t + T) = eFTw(t) T(AZ,BZ,T) is given by equation (3.84) with Af,Bf replaced by AZ,BZ. The optimal solution to the performance index (3.103) is given below without detailed derivation: 48 Chapter 3: SDGPC Design for Tracking Systems •f(ty •x(ty u* = Kw r(t) + Kx. m -y(t). (3.106) where Kw = KHU Kx, = KHX K=\J TT(T)czcTT(T)dT + X J HT {T)H(T)dT + ^ {Tp)T(Tp) -l NuxNu Hw = I jrT(T)c2RTeFTdT + jTT(Tp) 0 -^2x2 (3.107) eFTr n,x3 JV„x3 H*. = -[f FT (T)czcT eA*T dT + jTT (Tp)eA'Tp Nuxnf We are concerned about the first row of Kw and KXz since only the first element of the optimal control vector u* is applied to the plant. Consider the equalities Kw(l,2) = -KXa(l,nz-l),Kw(l,3) = -Kx,(l,nz) and let KT = Kw{l,l),Kty = Kw(l,2),Kr,v = KW(1,3),KX = KXl(l, 1 : n), the first control in (3.106) has a simplified form: uj(l) = Krr(t) + KrA+M ~ »(*)] + KrM*) ~ y(01 +Kxx(t) (3.108) Or in terms of the control input to the original plant (3.98) when the execution time goes to zero, we have by integrating both sides of equation (3.108) twice, t t V u*(l) = Krr(t) + Kr,y J V(T) - y(r)}dr + Kr,y J J [r(r) - y(r))dTdv (3 1QQ) +Kxx(t) Following is an example of servo SDGPC design to track a ramp reference signal. 49 Chapter 3: SDGPC Design for Tracking Systems Example 3.1.1 The plant being controlled has the transfer function Control law (3.109) is used with the design parameter Text* = O.lS Tp = 3s (3.110) (3.111) Nu = e A = 0.001 7 = 1000 Fig. 3.20 shows the reference and the output, tracking error and the control input. Clearly zero steady state error is obtained. Reference and output -10 0 5 10 15 20 25 30 35 40 45 50 Figure 3.20: Servo SDGPC of plant (3.110)-double integrator For comparison, the servo SDGPC law with single integrator (3.97) is designed with the same group of design parameters given in (3.111). The control results are illustrated in Fig. 3.21 in which steady state error can be clearly observed. 50 Chapter 3: SDGPC Design for Tracking Systems Reference and output 5 10 15 20 25 30 35 40 45 50 Figure 3.21: Servo SDGPC of plant (3.110)-single integrator 3.2 The Model Following SDGPC Problem There is another kind of tracking system called the model following problem. It is a mild generalization of the servo problem of section 3.1. In the framework of SDGPC, the problem is to find the control vector for the system (3.76) which minimizes the performance index (3.79), where r(t) is the response of a linear system model *!(*) = Ai*i(t) +Sit (*) r(t) = Cfz1(t) to command input i(t), which, in turn, is the zero input response of the system i2(*) = A2z2(t) *'(<) = CTz2(t) as indicated in Fig. 3.22 Command Signal Desired Trajectory (3.112) (3.113) z2 = A2 z2 /'= CJ2z2 z, = A,z, + B,/ r= CTz, Figure 3.22: Desired trajectory for model-foil owing problem 51 Chapter 3: SDGPC Design for Tracking Systems The two systems described by ( 3.112 ) and ( 3.113 ) can be combined into a single linear system with state space equation where Z = [ZT z2 ]> A z(t) = Az(t) r(t) = CTz(t) Ax B&f ,CT=[Cf 0] (3.114) (3.115) 0 A2 With equation (3.114 ), the model following problem is identical to the servo problem in section 3.1. The following example shows the design procedure of the model following problem and the control results. Example 3.2.1 The plant being controlled is an unstable third order process with transfer function G(s) 1 s3 - 1 The reference model has the following transfer function (3.116) G(s) = s2 + 4.5s + 2 The step response r(t) of the reference model to input co is given by w(t) = "-4.5 -2' '2" w(t) + 1 o . 0. co r(t) = [0 l]w(t) w(t) (3.117) (3.118) Or in the form of equation (3.114), the above state space equation can be rewritten as •o 0 0 ' x(t) = 2 -4.5 -2 x(t) .0 1 0 . (3.119) r(t) = [0 0 l]x(t) 52 Chapter 3: SDGPC Design for Tracking Systems (3.120) with x(t) = [co r r]T. Now the control law (3.75) can be applied. The design parameters are: Tp = 4s Nu = 20 A = 0.001 7 = 1000 7m = Texe = 0.2s where Tp is the prediction horizon, iV„ is the control order and A, 7 are the control weighting and final state weighting respectively. The execution sampling interval Texe is set to equal to the design sampling interval Tm since the plant being controlled is unstable. Fig. 3.23 shows the control results. Setpoint and outputs setpoint model response — plant output 80 (S) Figure 3.23: Model following control of unstable third-order system 53 Chapter 3: SDGPC Design for Tracking Systems Example 3.2.1 The second example is a third order stable plant with transfer function 1 (*+l)a The reference model is a second order under damped plant G(s) = (3.122) K J 4s2 + 2.4s + 1 The design parameters are again Tp = 4s Nu = 20 A = 0.001 (3 123) 7 = 1000 Tm = Texe = 0.2s Fig. 3.24 shows the setpoint, reference model response, plant output and the control signal. 54 Chapter 3: SDGPC Design for Tracking Systems Setpoint, reference model and plant outputs 0 10 20 30 40 50 60 70 80(s) Control signal 51 1 1 1 1 1 1 1 _51 1 1 1 1 1 1 1 1 0 10 20 30 40 50 60 70 80 (s) Figure 3.24: Model following control of stable third-order system 3.3 The Tracking SDGPC Problem It is well known that when the future setpoint is available the tracking performance can be improved radically. Similarly, future values of disturbance can be utilized for better disturbance rejection. Practical examples for which future setpoints are available can be found in areas such as robot manipulator applications, high speed machining of complex shaped work pieces and vehicle lateral guidance control problems [77, 76, 57]. Predictive control is a natural candidate in these applications since it explicitly accommodates the future values of setpoint in its formulation. However, the setpoint preview capacity of predictive control has not been fully exploited before since in process control applications, where predictive control has blossomed, disturbance rejection is the major concern and the future disturbances are often unknown. The SDGPC tracking problem is formulated as follows. 55 Chapter 3: SDGPC Design for Tracking Systems For the system (3.75) and its augmented plant (3.76), with the desired trajectory r(t) available in the range [t,t + Tp], the SDGPC tracking problem is to find the optimal control by minimizing the performance index (3.79). Assuming that the projected ud(t) in [t, t + Tp] is given by (3.80) as illustrated in Fig. 3.17, the performance index (3.79) can be written as J = J [cTfeA'Txf{t) + cJT(Af,Bf,T)nd - r(t + T)]2dT o +\ J [uTHT(T)H(T)ud]dT o +1[cT}eA^xf{t) + cTT(Af,Bf,Tp)ud - r(t + Tp)]2 +j[<$eA'T'xf(t) + cTdT{Af,BhTp)nd}T[cTdeA^xf{t) + c^T(Af,Bf,Tp)nd] (3.124) Take the derivative of J with respect to u^, we have dJ dud J [YT{T)cfcTT{T)ud + YT(T)cf{cTeA'Tx}{t) - r(t + T))]dT +A J HT (T)H(T)dT +1[rT(Tp)cfcTT(Tp)ud + rT(Tp)cf(cTeA'T>xf(t) - r(t + Tp))] +y[rT(Tp)cdcJr(Tp)ud + TT(Tp)cdcJeA^xf(t)] The optimal SDGPC tracking control solution urf is given by nd = fr(t) + KXf xd(t) Ly(0 (3.125) (3.126) 56 Chapter 3: SDGPC Design for Tracking Systems where fr(t) = KHt Kxf = KHXf K= \J YT{T)c}cTfT{T)dT + \ J HT(T)H(T)dT + -yrT (TP)T(TP) /Tr \ (3-127) Ht = \J rT(T)cfr(t + T)dT + 7TT(Tp)cfr(t + Tp) j TT (T)cfCT eAfT dT + jTT (Tp)eAfTp With receding horizon strategy, the feedforward term fr(t) needs to be computed at every time instant. Simple numerical integration algorithm such as Euler approximation can be used without compromising the performance of the controller. As we mentioned at the beginning of this section, use of the future setpoint information can improve the tracking performance, sometimes significandy. The following example compares the tracking performance of two controllers one of which utilizes the future setpoint information and the other one does not. Example 3.3.1 The plant in Example 3.1.1 is used again with the following transfer function 1_ (* + l)a First, control law (3.97) is used with the design parameter G(s) = \ 3 (3.128) T^e.xe. — 0.25 Tv = 6s Nu = 10 (3.129) A = 0.001 7 = 10000 Fig. 3.25 shows the setpoint and the output, the tracking error and the control input under control law (3.97). 57 Chapter 3: SDGPC Design for Tracking Systems Figure 3.25: Servo SDGPC of plant (3.128) Now the tracking control (3.126) which utilizes the future setpoint information is designed with same design parameters given in (3.129). Fig. 3.26 shows the results. Reference and output —i 1 1 r l l l l I 1 i_ 0 10 20 30 40 50 60 70 Tracking error 0 -1 Figure 3.26: Tracking SDGPC of plant (3.128) 58 Chapter 3: SDGPC Design for Tracking Systems Compare Fig. 3.25 and Fig. 3.26, the improvements in tracking error are obvious. Also notice that the control effort in Fig. 3.26 is smoother due to the preview ability. The improvements can be explained as follows. At current time t, knowing the future setpoint information r(t + T) is equivalent to knowing the current setpoint and all its derivatives up to an arbitrarily large order. Indeed any future setpoint value r(t + T) can be calculated using Maclaurin oo series expansion r(t + T) = r(t) + ^ r (tfjr- 1° control law (3.97), it was assumed that the future setpoint is a ramp. In another words, only the first derivative of the setpoint is assumed to be available. It is thus natural to expect performance improvements for complex setpoint when tracking control law (3.126) is used. However, these two control laws will not differ from each other for ramp signal. This can be confirmed by comparing Fig. 3.27 and Fig. 3.28 which show the results of plant (3.128) being controlled by (3.97) and (3.126). It can be seen that the tracking errors are the same for these two control laws at steady state while the tracking errors under control (3.126) at the transition region around time 10s are smaller since the setpoint here is no longer pure ramp signal. 59 Chapter 3: SDGPC Design for Tracking Systems Reference and output 40 20 0 0 5 10 15 20 25 30 35 40 45 50 Tracking error 0.11 i 1 1 1 1 1 1 1 1 1 Control input 40 20 0 0 5 10 15 20 25 30 35 40 45 50 Figure 3.28: Tracking SDGPC of plant (3.128) It should be pointed out that the tracking performance of the servo control law (3.97) can not be improved significantly by simply increasing the order of the reference model (3.94) without knowing the future setpoint information. For model order p > 2, the setpoint derivatives will be needed in the computation of control law (3.97). In such case a state observer of the reference model (3.94) can be constructed with desired dynamics. However, no matter how fast the dynamics of the state observer is, there is still no anticipation ability in this approach and thus the transient tracking error can not be reduced efficiently. On the other hand, the knowledge of the future setpoint can also be used in the design of control law (3.97) in which case the setpoint derivatives can be estimated using CO the Maclaurin series expansion r(t + T) = r(t) + r^(*)7T i« a least squares sense. k=i 3.4 The Feedforward Design of SDGPC When the disturbances can be measured, the control performance can be improved radically by utilizing this information compared with the use of feedback only. The reason is that there are inherent delays in all dynamic systems. It is always better to cancel the disturbance before it is observed at the output. Feedforward disturbance rejection also alleviates the burden of feedback disturbance rejection so that the design of the feedback loop can concentrate on robustness issues. 60 j I | | | | L i r J I I 1 I I I 1 L Chapter 3: SDGPC Design for Tracking Systems Here is the formulation of the feedforward design of SDGPC. Given the n-dimensional SISO linear system having state equations x(t) = Ax(t)+Bu(t) + Bvv(t) y(t) = cTx(t) where v(t) is a measurable disturbance satisfying state space equation /?(«) = W(3(t) = DT(3{t) with dimension np. The integrator augmented system is described by if = AfXf + BfUd + Bfuvd XJf = cjxf dim(xf) = rif = n + 1 xd(t) = i(t), ud(t) = ii(t), ud(t) = i>(t) (3.130) (3.131) (3.132) Where xf = ~xd' 'A 0" B' 'Bu .y. cT 0. .0. . 0 . (3.133) cf = [0 • • • 0 1 ] Following the arguments in section 3.1, the T-ahead state predictor based on equation (3.132) can be written as follows: T T J V — V f 0 0 l 1 Xf(t + T) = eA'Txf(t) + J eA^T-^BfUd(r)dr + J eA^T-^Bhvd{r)di = e */(*) + J cA^T-T^BfVd(t + r)dr + T{AfBf,T)ud (3.134) Where F(Af,Bf,T) is given by equation (3.84), and ud is piecewise constant as illustrated in Fig. 3.17. vd(t + T) can be obtained from state equation (3.131) as r3(t + r) = eWr$(t) vd(t + r) = DTeWrp(t) (3.135) 61 Chapter 3: SDGPC Design for Tracking Systems The T-ahead state predictor can be obtained by substituting equation (3.135) into (3.134): xf(t + T) = eA'Txf(t) + Dv{T)P{t) + T(AfBf,T)ud T DV{T) = j eA^T-^Bfi/DTeWTdr (3.136) With the above state predictor, the feedforward SDGPC problem is to minimize performance index (3.79) subject to the measurable disturbance v{t). Based on (3.136) and (3.87), the performance index (3.79) can be written as J = T, J [cTfeA'Txf{t) + cTDv{T)P(t) + cJF(T)nd - RTeFTw(t)f'dT +A J [udHT(T)H(T)ud] dT o +y[cTeA>T*xf{t) + cTfDu{Tp)P{t) + cTT{Tp)ud - RTeFT"w(t) +1\cTeA'T>xf{t) + cTdDv{Tp)P{t) + cTdY{Tp)nd^[cTdeA'T-xf{t) + <$Du(Tp)p(t) + cTdT{Tp)ud (3.137) Take the derivative of J with respect to u,j and let it be zero 8J dnd *p J [rr(T)c/CJr(T)ud + rT(T)cf(cTeA'Txf(t) + cTDu(T)!3(t) - RTeFTw(t))]dT +A j HT(T)H(T)dT (3.138) +j[TT(Tp)cfcTT(Tp)ud + rT(Tp)cf(cTeA'T>xf{t) + cTDv{Tp)p(t) - RTeFT>w(t)}} +7 [TT(Tp)cdcjT(Tp)ud + TT{Tp)cdcTdeA'T>xf(t) + rT(Tp)cdcjDu(TPP(t)\ The optimal SDGPC tracking control solution urf is given by ~xd(ty u*d = Kww(t) + KXf y{t) J (3.139) 62 Chapter 3: SDGPC Design for Tracking Systems where Kw — KHW KXf = KHXJ Kp = KHp T, K=\J TT{T)cfcTfT(T)dT + \ J HT(T)H(T)dT + yTT (TP)T(TP) Hw=\J VT(T)cfRTeFTdT + yTT (Tp)cfRTeFT" -l N„xNa Nuxp HX} = -\j TT (T)cfCT eAfT dT + \Tp)eA^ I N„XTlf \ ) ) (3.140) H0 = -IJ TT{T)c}cTfDv{T)dT + 7TT(Tp)Du(TpNuxriff Notice again that, like the setpoint feedforward term Kw, the inclusion of the disturbance feedforward termKp does not affect the state feedback gain KXf. The control action given by (3.139) is the derivative of the control to the original plant. For small sampling interval, the direct control acdon can be obtained by integrating both sides of (3.139) resulting in, z u*(l) = KXf(l, 1 : n)x(t) + J [KWW(T) + KXf(l,n + l)y(r)]dr + Kp(3(t) (3.141) Here KXJ(1,1 : n) denotes the first n elements of the first row of matrix KXf. Normally, the states (3(t) of the disturbance model (3.131) are not available. A state observer with gain L can be constructed to give the states estimates p(t) as follows J3(t) = W(}(t) + L(v(t) - DTp{t)) (3.142) The observer gain L is selected such that the matrix W — LDT is stable and has desired dynamics. Following are some examples which show the effects of feedforward disturbance rejection. 63 Chapter 3: SDGPC Design for Tracking Systems Example 3.4.1 The plant being controlled is G(s) = (3.143) The disturbance is generated by a white noise passed through an integrator. However, in the design of the control law, the disturbance is assumed to be constant. That is v(t) — 0. The design parameters are Texe — 0.2s Tp = 6s (3.144) A = 0.001 7 = 1000 The control law takes the form of (3.141). Fig. 3.29 shows the control results. The effect of the disturbance feedforward can be seen clearly by comparing the first 50 seconds of the figure where feedforward gain Kp was set to zero and the rest of the figure where Kp is set to the value as computed. Output and setpoint 0.5 o -0.5 -1 • without feedforward • with feedforward-10 20 30 40 50 60 70 80 90 100 (s) Control signal 10 20 30 40 50 60 70 80 90 100 (s) Figure 3.29: Disturbance feedforward design 64 Chapter 3: SDGPC Design for Tracking Systems This example shows how disturbance feedforward can improve the control performance dramat ically even with a simple and inaccurate disturbance model. The next example shows that further performance improvements can be made with more accurate disturbance model. Example 3.4.2 The plant being controlled is G(s) = -L- (3.145) The disturbance is sinusoidal with known frequency. The state space model of the disturbance is: fct) 0 -4 1 0 !/(*) = [0 1 ]/?(*) (3.146) The observer gain LT = [1 4.5] is selected such that the eigenvalues of the observer closed loop matrix W — LDT are set to -2,-2.5 respectively. The control law design parameters are 0.2s Tp = 3.5s Nu = 10 A = 0.001 7 = 1000 (3.147) First, the correct disturbance model (3.146) was used to design the control law (3.141) and then a constant disturbance model i>{i) — 0 was used. The corresponding control results can be seen from the Fig. 3.30. As expected, the performance deteriorates in the time span between 50 seconds and 100 seconds as a wrong disturbance model is used. 65 Chapter 3: SDGPC Design for Tracking Systems 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25 Sinusoidal model-I H I r. | 1 i! T7T 111 ji - Constant model I V I I 'l I i i t * i, " :i i! ,i,i ilLd •I« M ,i 'i ,i M ,i II J II .i H i i. i ii i, • i, i' n 11 'llll i . * 11 i i .i i I i i. •" • i1. i. i " II i.,11.,11.i.i *i •I •I •I •I II i, • I, I. '"I l| I, I, ' | ' I I I I I J ' ' l| II '' I' l| I I ' 'I I I' ' ' l| '| •j • •! j,1 i| j! •! n i • i '»J •' ,i ii i ! ;i,ii i,' i 11! n 0 10 20 30 40 50 60 70 80 90 100 (s) Figure 3.30: The effect of disturbance model 3.5 Conclusion In this chapter, various tracking problems are formulated in the framework of SDGPC. Tracking control problems generally have two-degree-of-freedom design structure. However, the feedback part of the tracking problem is equivalent to the regulator problem which has one-degree-of-freedom design structure provided that the design parameters are the same. When information about the future setpoint are available, tracking performance can be radically improved. This is because knowing the future setpoint is equivalent to knowing the exact states information of the state equation describing the setpoint. When the disturbance is available for measurement, the disturbance rejection performance can be improved dramatically by using feedforward design. 66 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC Chapter 4 Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC Time-delay, or dead time, occurs frequently in industrial processes and in some cases, is rime-varying. Time-delay poses one of the major challenges for the design of robust process control systems. In discrete-time, time-delay systems have the form: G{q) = vM q-*i*r>+ +- + (4148) where k — integer(Tj/A) is the delay in samples and Td, A are the delay time and sampling rime respectively. For unknown time-delay, k can be either estimated directly [24] or via the extended B polynomial approach in which the leading coefficients of the B polynomial in (4.148) up to order k would be all zero. In continuous-time, time-delay can be approximated by a low order rational approximation such as Pad6 approximation [66]. Laguerre filter was introduced into systems theory first by Wiener in the fifties [82] and has been popular recently [87, 80, 48, 49]. In particular it can approximate time-delay systems efficiently. With the time-delay known or being modeled properly, model based predictive control strategies provide an effective way of controlling such systems. In this chapter, we give two approaches to the control of time-delay systems. The direct approach in section 4.1 is based on general state space model and assumes the time-delay is known. Emphasis will be placed on the Laguerre filter based adaptive control given in section 4.2. 4.1 The Direct Approach The SDGPC approach to deal with time-delay systems is formulated as follows. The system model considered is: x(t) = Ax(t)+Bu(t - rd) y(t) = cTx(t) (4.149) dim(x) = n And the augmented system is given by 67 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC Where Xf(t) = AfXf(t) + BfUd(t - Td) yf(t) = cTxf(t) dim(xf) = rif — n + 1 xd(t) = x(t), ud(t - rd) - u(t - Td), e(t) = y(t)-w xf = ~xd~ 'A 0' 'B' » Bf = _ e 0. .0. ,cT = {0 0 1] w is the constant setpoint and rd is the time delay in the system. Consider the performance index (4.150) (4.151) J(t) = J [e2(t + T) + Xud(t-rT)]dT-rjxT(t + Tp)xf(t + Tp) (4.152) w Past controls \ Setpoint Predicted output Projected control derivatives Time -Td Xd Tp-Td Tp Figure 4.31: Graphical illustration of SDGPC for systems with delay Assume the projected control signal to be piecewise constant as illustrated in Fig.4.31. For simplicity, we assume the dme delay rd has an integral number Nd of the design sampling interval Tm. That is Nd = At present time t = 0, for a prediction horizon Tp, define •* m ud(T) = H(T)ud 0<T<Tp-rd ud{T) = HTd{T)uTi — Td <T < 0 (4.153) 68 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC where H(T) = [H1(T) H2(T) •••Hi{T)---HNST)} HTi(T) = [H(_Nd){T) H(_Nj+1)(T) ...2T(0(r) •.•*•<_!)( T)] Ud=hi(i) ud(2) • • • ud(i) • • • ud(7Vu)]T Urd=[uTd (-Nd) urd {-Nd+l) • • • uTd (i) • • • uTd (-1)]T 1 (t - l)Tm <T<iTm Hi(T) = H(i){T) (0 otherwise i = 1,2,---NU T -TP~T f 1 iTm<T < (i+l)Tn (4.154) (4.155) . 0 otherwise i=-Nd, -Nd + 1, 2,-1 With the system equation (4.150) and the control scenario (4.153), the T-ahead states prediction can be obtained: xd(t + T) = eATxd(t) + rTd(T)uTd + ru(T)ud where (4.156) rr,(r) = u u J eA^BH(_Nd){r)dr... J eA^) BH^^dt (4.157) T-Td T—Td -I nxNu r„(T)= J eA(T-T'-T)BH1(T)dT--- J eA(T-T<-T)BHNa{r)di 0 0 T> rd r„(T) = [0 0 0---0]BXjVii 0<T<rd The predicted error between system output and the setpoint is: e(t + T) = e(t) + cT eArdr j xd(t) + cTT{Td)o{T)nT<i + cTT(u)o(T)nd (4.158) (4.159) 69 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC Where o T r(u)o(T) = J Tu{r)di \ (4.160) Substitute equations (4.156) (4.159) into cost function (4.152), the optimal control vector can be obtained as: ud = -K(Tdxd(t) + Tee(t) + TTduTd) -1 K=\J TUoccTr(u)odT + \TmiNM + 7r^Tp)ru(Tp) + 7rfu)o(rp)ccTr(u)o(Tp) Td = J rfu)ocJA-' (e^ - I)dT + 7^(Tp)e^ + lTju)o{Tp)ccT(e^ - /) Td Te = /rfu)o(r)drc + 7rfu)o(rp)c Td T ?Td = J Tfu)occTr{Td)odT + 1Tl(Tp)T^ (4.161) Remark: Systems with delay can also be treated in a LQR setting. For example, the continuous-time system model (4.150) can be first transformed into a discrete one without delay by augmenting the past controls up to time rd as the new system states, resulting in a system of order Nd + rif as equation (4.162). WhereA^ = rd is the time-delay and Tm is the design sampling interval 70 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC as illustrated in Fig.4.31. 'xf(k + l) .UTd(*+l) / e^Bfdri 0 0 0 0 0 0 1 0 0 1 0 0 yf(k) = [cj 0] '0" 0 "*/(*)" 0 + .UTd(fc). .1. ud(k) (4.162) xf(k) uTd (k) u£ (k) = [ud(k - Nd) ud(k -Nd + 1) ud(k-l)} Based on system equation (4.162), either finite horizon or infinite horizon discrete-time LQR solution can be found. The problem with the infinite horizon case is the singularity of the transition matrix of system (4.162) [55]. For the finite horizon case, the computation time could increase significantly due to the increase of the system order. Also notice in Fig.4.31 that although the projected controls in the time interval [Tp — Td,Tp] have no effect on the performance index (4.152), but due to the reverse-time iteration of the associated Riccati Difference Equation, the recursion has to go through the whole horizon whereas in the SDGPC approach , only the controls in the time interval [0, Tp — rd] are computed. 4.2 The Laguerre Filter Modelling Approach The use of orthogonal series expansion, particularly Laguerre expansion has become increasingly popular in system identification and control particularly for control of systems with long and time-varying time delay. Briefly speaking, given a open loop stable system with transfer function G(s), its Laguerre filter expansion is G(,) = f>^(±Z*)^ (4.163) ^ s +p s+p 71 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC The truncated expression of equation (4.163) can be expressed in a network form as depicted in Fig.4.32. u(t) /2p x, (t) s-p X , (t) ^ s-p (t) ^ s+p • s+p s+p y(t) Summing Circuit Figure 4.32: Laguerre Filter Network Where p is the time scale selected by the user. The Laguerre network consists of a first-order low-pass filter followed by a bank of identical all-pass filters. Its input u(t) is the process input. The Laguerre states defined in Fig.4.32 as xi(t),X2(t), • • • xj^(t) are weighted to produce the output which matches the output of the process being modeled. The set of Laguerre functions is particularly appealing because it is simple to represent, is similar to transient signals, and closely resembles Pad6 approximants. The continuous Laguerre functions, a complete orthonormal set in L2[0, oo), i.e. will allow us to represent with arbitrary precision any stable system [21]. Any stable process can be expanded exactly in an infinite Laguerre series regardless of the value of the time scale p. However, when a truncated series with expansion number N is used, an immediate problem is the choice of the time scale used to ensure a fast convergence rate. Parks [56] gave a procedure to determine the optimal value of the time scale based on two measurements of the impulse response of the process being approximated. For open loop stable non-oscillatory systems with possibly long and time-varying time delays a real number p is sufficient to provide good convergence results. Fu and Dumont [29] gave an optimal time scale for discrete-time Laguerre network and proposed an optimization algorithm to search for the optimal complex time scale p when the process being modeled is highly under damped. Since the Laguerre network has a state space representation, any state space design method can be applied to the controller design. Dumont and Zervos [22] proposed a simple one step ahead predictive controller based on discrete-time Laguerre filters. This algorithm has been commercialized and is 72 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC routinely used in pulp and paper mills [21]. Dumont, Fu and Lu [23] proposed a GPC algorithm based on nonlinear Laguerre model in which the linear dynamic part is represented by a series of Laguerre filters and the nonlinear part is a memoryless nonlinear mapping. Simulation shows it has superior performance over the linear approach for systems with severe nonlinearity such as the chip refiner motor load control problem and the pH control problem. Recently, Finn, Wahlberg and Ydstie [27] reformulated Dynamic Matrix Control (DMC) based on a discrete-time Laguerre expansion. In this section, we propose the SDGPC algorithm based on continuous-time Laguerre filter modelling. It is shown that the resulting control law is particularly suitable for adaptive control applications in that its computation burden is independent on the prediction horizon used in the SDGPC design. Define xT(t) = [xi(t) 13(f) ... xN(t)]. From Fig.4.32, we have the following state space expression of the Laguerre network: where x(t) = Ax(t)+Bu(t) y(i) = cTx(t) A = -p 0 -2p -p . -2p -2p . 0 0 ~P-* NxN B = V2p-(4.164) (4.165) (4.166) JVxl cT = [ci c2 • • -CJV]1XJV With the time scale p being properly selected, the Laguerre spectrum cT = [ci C2 • • • CJV ] can be estimated based on the input output of the process. In fact, notice that the equation y(t) = cTx(t) in (4.164) is already in a regression model form, various Recursive Least Squares algorithm can be applied. We settled for the recent EFRA ( Exponential Forgetting and Resetting Algorithm ) [68] which will be described in chapter 6, section 6.1. 73 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC Now the SDGPC design procedure can be readily applied to the Laguerre state space equation (4.164). The intended application of Laguerre filter based SDGPC is adaptive control where the Laguerre coefficients cT — \c\ ci •• • c^]lxN are estimated on-line using EFRA [68]. We show in the following that the on-line computation burden is actually independent on the prediction horizon Tp. The main computation involved in the calculation of (2.16) is integration. For example the integration / TT(T)ccTT0(T)dT in K\. By some simple manipulations, we have: o Tp J TT(T)ccTT0(T)dT = o T, cT'JiafdTc 0 (4.167) where 7^ jj are the ith and jth column of ro(T) = [71 72 • • • 7ivJnXjyu. The integral / njJdT 0 can be computed off-line and stored. The other integrals in the calculation of (2.16) can be treated similarly so that the on-line computation burden of the control law (2.15) is independent on the prediction horizon Tp. Although Laguerre filter modeling is known for its capability of dealing with dead time, however for long time-delay, it requires a large number of Laguerre filters causing problems such as slow convergence rate in the Laguerre coefficients estimation and increasing the control law computation burden. An effective way of dealing with this problem is to use the delayed control u(t — rd) as the input to the Laguerre network in Fig.4.32 resulting in a system model: x(t) = Ax(t)+Bu(t - rd) (4.168) 3/(*) = cTx(t) Where rd can be either estimated or just a rough guess based on prior knowledge about the process. Only the uncertainty of the time-delay needs to be taken care of by Laguerre network modeling. Based on system model (4.168), the SDGPC for time-delay systems in Section can be applied. Example 4: Adaptive Laguerre based SDGPC is shown in this example. The plant is given by 74 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC G4(s) = e -sT 0.5s (4.169) Where the dead time T varies from 5.5s to 4.5s as illustrated in Fig.4.33. Four Laguerre filters with pole p = 1.2 are used. The delay is assumed to be 5s in the design. Thus the system model can be described as x(t) = •-1.2 0 0 o - 1.5492--2.4 -1.2 0 0 1.5492 x(t) + -2.4 -2.4 -1.2 0 1.5492 .-2.4 -2.4 -2.4 -1.2. .1.5492. u(t - 5) (4.170) y(t) = c1 x(t) where cT is the Laguerre filter coefficients vector which will be estimated using RLS. The algorithm developed in section 4.1 will be used. The design parameters are: Td - 5s Texe — 0.25s Tp = lis (4.171) Nu = 6 A = 1 7 = 1000 where 7 is the end point states weighting. Fig.4.33 shows the plant output and the reference, the control input to the plant and the control derivative , the estimated Laguerre coefficients. Between 0 to 80 seconds, the dead time is 5.5 seconds, between 80 to 160 seconds T is changing to 5 seconds and after 160 seconds it is reduced to 4.5 seconds. As can be seen, the performances are very good in all three cases. By using the prior delay knowledge, less Laguerre filters can be used resulting in quick convergence in the coefficients estimation and thus less transients in the response. 75 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC -5. T=5.5s' T=5s -T=4.5s Jl 0 50 100 ^ 150 200 (s) Figure 4.33: Simulation of plant (4.169) The simulation was performed with the same conditions as the previous run except measurement noise was added at the output. Fig.4.34 shows the system output, setpoint and the estimated Laguerre parameters. The algorithm still performs well in the presence of noise. Output and setpoint 50 100 150 200 Estimated Laguerre coeffi. with forgetting:0.98 50 100 150 200 Figure 4.34: Simulation of plant (4.169) with measurement noise 76 Chapter 4: Control of Time-delay Systems and Laguerre Filter Based Adaptive SDGPC 4.3 Conclusion Model based predictive control can deal with time-delay systems effectively and SDGPC is no exception. Laguerre network requires little a priori information about the system and is able to model varying dead times. The adaptive Laguerre filter based SDGPC developed in section 4.2 is a suitable candidate for most process control applications. 77 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Chapter 5 Anti-windup Design of SDGPC by Optimizing Two Performance Indices Actuation limits exist in almost all practical control systems. For example, a motor has limited speed, a valve can only operate between fully open and fully close etc. Other than these physical actuator limitations, there are constraints which are imposed by production requirements. On the other hand, most of the available controller design methods ignore the existence of the saturation nonlinearity. When large disturbances occur or, the operating conditions are changed over a wide range, it may happen that the theoretical controller output goes beyond the actuator limits. As a result, the control system is effectively operating in open-loop as the input to the plant remains at its limit regardless of the controller output. When this happens, the controller output is wrongly updated. For example, if the controller has integral action, the error will continue to be integrated resulting in a very large integral term. This effect is called controller windup. The windup difficulty was first experienced in the integrator part of PID controllers. It was recognized later that integrator windup is only a special case of a more general problem. In fact, any controller with relatively slow or unstable modes will experience windup problems if there are actuator constraints [20]. The consequence of windup is either significant performance deterioration, large overshoots or sometimes even instability [8]. Various compensation schemes have been proposed. The anti-reset windup method was proposed by Fertik and Ross [26]. Astrom and Wittenmark [63 pp. 184-185] suggested resetting the integrator at saturation to prevent integrator windup for PID controllers. A general approach where an observer is introduced into the controller structure to prevent windup was proposed by Astrom and Wittenmark [63 pp. 369-373]. The " conditioning technique" was proposed by Hanus [36] and it was found that the conditioned controller can be derived as a special case of the observer-based approach [8]. However, as pointed out in [39], many of these schemes are by and large intuition based. Rigorous stability analyses are rare and there is no general analysis and synthesis theory. Several attempts have been made to unify anti-windup schemes notably by Walgama and Stemby [81] and Kothare et al. [39]. 78 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Since the SDGPC control law (2.17) has integral action it will also encounter windup difficulties in the case of actuator saturation unless measures are taken. A systematic approach that takes into account the input constraints right from the start of the problem formulation is the constrained model based predictive control [69]. This is in fact one of the main advantages of using model based predictive control strategy. However, one difficulty which comes with this approach is the increased complexity. The more common approach which most of the afore mentioned schemes adopt is the two steps paradigm. That is, the linear controller is designed ignoring control input nonlinearities first and then anti-windup algorithm is added to compensate for the performance degradation due to constraints. This will also be the method of attack used here. This chapter is organized as follows, in section 5.1, a SDGPC algorithm based on two perfor mance indices is given in which the "nominal" response of the closed-loop system and the integral compensation performance can be designed independently. This was motivated by the work reported in [1, 34, 35, 30] , in which a control algorithm with the structure of state feedback plus integral action was developed where the state feedback and the integral feedback gain can be tuned sepa rately. However, that work was in the framework of continuous-time infinite horizon LQ control. The SDGPC approach, however, has a quite different formulation procedure and an interpretation which naturally leads to a novel anti-windup compensation scheme presented in section 5.2. The importance of this anti-windup scheme is that under some reasonable assumptions, the overall "two-degree-of-freedom" SDGPC and the anti-windup scheme guarantee closed-loop stability. Section 5.3 concludes the chapter. 5.1 SDGPC Based on Two Performance Indices The primary goal of introducing integral action in the design of SDGPC is to ensure zero static error for systems tracking a non-zero constant setpoint subject to constant disturbances and modeling error to some degree. If there are neither modeling error nor disturbances, there is no need to introduce an additional integral state. However, models are inevitably wrong and there are always disturbances acting on the plant thus integral action is always needed. Nevertheless, the argument is that the controller can be designed for good servo step response performance assuming perfect modeling and 79 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices no disturbance. Integral action, on the other hand can be added on to compensate for step and impulse disturbances and for modeling error. The key idea is to have the servo performance and disturbance rejection performance tuned independently. In other words, changing the servo performance shall not affect the disturbance rejection performance or vice versa. 5.1.1 Optimizing Servo Performance Consider system (2.1) x(t) = Ax(t)+Bu(t) y{t) = cTx(t) (5.172) dim(x) = n The system is required to track a constant setpoint rH. If there is no system zero at the origin, a constant uo can be found for any ro to hold the system state at xo such that [41]: 0 = AXQ + Bua yQ - cTx0 = r0 (5.173) Define the shifted input, the shifted state and the shifted output respectively as u(t) = u(t) — UQ x(t) = x(t) - x0 (5.174) !/(0 = 2/(0 - ro Substitute (5.174) into (5.172) and using (5.173), the shifted variables satisfy the equations x(t) = Ax(t)+Bu(t) y(t) = cTx(t) (5.175) dim(x) - n A sensible control objective for the system (5.175) is to minimize the following performance index 1v J(t) = j3xT(t + Tp)x(t + Tp) + J {[y(t + T)]2 + X3[u(t + T)]2}dT (5.176) 80 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices where U(T),T € [£,/ + Tp] is confined to be piecewise constant as u<i(t) in (3.76) which is depicted in Fig. 3.17. Follow the same arguments as that of servo SDGPC in chapter 3, section 3.1, the optimal control vector u* which minimizes (5.176) can be written as u* = Ksx(t) (5.177) with (5.178) where T(T) is a n x Nu matrix given by (2.11) and H(T) is given by (2.7). Nu is the control order as defined before. Since only the first element of u* is applied to the plant according to receding horizon strategy, define F as the first row of the Nu x n feedback matrix Kx F = Kx(l,l:n) (5.179) The time-invariant control law for the shifted system is thus u(t) = Fx(t) (5.180) Control law (5.180) has guaranteed stability when applied to (5.175) with sampling interval Tm according to Theorem 2.3. When the design sampling interval Tm is small, the continuous-time control law also stabilizes the system as shown by simulation in chapter 2, section 2.4. This can be thought of as a procedure of designing continuous-time control law based on sampled-data control in a reverse way of the conventional approach of designing discrete-time control law based on continuous-time design method. For the sake of clarity, (5.180) will be applied to (5.175) with Texe —* 0. This 81 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices way, the spirit of the anti-windup scheme which will be given in the next section can be shown more clearly and comparisons can be made with that of the familiar three term PID control law. In terms of the original system variables, control law (5.180) can be written from (5.174): u(t) = Fx(t) + UQ- FxQ (5.181) It is easy to see from (5.181) that there is a constant term u'0 = uo — FXQ in the control law. It can be shown [41pp.270-276] that u'0 = -i i-i cT{-A)-LB r0 (5.182) where A is the closed-loop system matrix A=A+BF (5.183) The term cT(—A)~lB in (5.182) is the static gain of the close-loop system with transfer function Hc(s) = cT(sI-A) lB (5.184) from the constant term u0 to the output. That is Hc(0) = cT(-A)~lB (5.185) The nonexistence of system zero at the origin ensures that Hc(0) is nonsingular and thus guarantees the existence of such a constant term u'0 given by (5.182) which makes (5.173) hold at steady state. The transfer function from the constant setpoint ro to the output is y(s) — i7~1(0)cT(s/ - A) lB = H-l($)Hc{s) (5.186) ro(s) where the steady state gain is one. Thus the optimal control law without integral action for (5.172) is 82 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices u{t) = Fx(t) + #c_1(0)r0 (5.187) The control law (5.187) is not "robust" in the sense that when there are either disturbances or modeling error, the output of system (5.172) at steady state will differ from the setpoint ro. This is where integral control given in the next subsection kicks in. 5.1.2 Optimizing Disturbance Rejection Performance Define the integral state z(t): The integral state augmented system of (5.172) is (5.188) •i(ty ' A 0" 'B~ u(t) + '0' = + At). -cT 0. A*). .0. _1_ ro A*) At)} (5.189) y(t) = {cT o] Compare with (2.3) where the integrator was inserted before the plant, the integrator in (5.189) is added after the plant. See. Fig.5.35. x=Ax+Bu <sHIr y.A Figure 5.35: Graphical illustration of (5.189) The last term [0 l]Tro in (5.189) can be ignored since the control law for (5.189) will have integral term which will eliminate any constant disturbance like [0 1 ]Tro. Thus we will work on the disturbance rejection control law based on the following equations At)' ' A 0" At)' + 'B' At). -cT 0. At). .0. At) y(t) = [cT 0] (5.190) 83 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices What we are looking for is a control law that consists of two terms un(t),v(t). The first term un(t) is the nominal control given by (5.187) which ensures nominal servo performance. The second term v(t) is responsible for disturbance rejection and modeling error compensation. That is u(t) = un(t) + v(t) = Fx(t)+H-1(0)rQ + v(t) Substitute (5.191) into (5.190), we have (5.191) ' A 0" At)' At). T —c1 0. At). + [F 0] At) At)} + tf-i(0)r0 + v(t) Again ignoring the constant disturbance term [B 0]THC a(0)ro, we have from (5.192) (5.192) At)' 'A + BF 0" Aty B' = + At). -cT 0. At). .0. v(t) The disturbance rejection state feedback control law for (5.193) will have the form of I At)' (5.193) v(t) = [L1 L2] At) Without loss of generality, assume L2 = £,Li = £_L. i.e. \x{t) v(t) = f[L 1] The closed-loop system of (5.193) under control (5.195) is (5.194) (5.195) At)' At). A + BF + fBL fB •T 0 —c The criteria for the disturbance rejection control (5.195) are that (5.196) 1. It should not alter the nominal servo performance given by (5.187). i.e. the eigenvalue of the closed loop system (5.196) should contain the eigenvalues of A given by (5.183) 2. It should at the same time give good disturbance rejection properties 84 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Defining the new state CW as CW = Lx(t) + z(t) and substituting (5.197) into (5.193), the new state equation is (5.197) .CW. A + BF 0 •X(ty ' B ' + .CW. LB. v(t) (5.198) Substitute control law (5.195), where £, L are yet to be decided, into (5.198). The closed loop system is A + BF (5 1 \x(t)' L(A + BF) - cT $LB J LC(0. To fulfill criterion 1, it is obvious that L(A + BF) — cT has to be a zero vector, i.e. L should be given by .CW. (5.199) L = cT(A + BF)'1 (5.200) Since (A + BF) is the nominal closed loop system matrix which is stable, it is always possible to find L from (5.200). The closed loop system equation is thus .CW. det .CW. = det[sln - (A + BF)](s - £LB) = 0 (5.201) 'A + BF £B 0 t\LB_ The eigenvalues of (5.201) are the solutions of Sln - (A + BF) -£B 0 s-£LB_ That is, the eigenvalues of (5.201) are those of (A + BF) and one at £LB. Given the desired pole location pz, the feedback gain £ can be easily decided by (5.202) i=Pt{LB)-1 The gain £ can also be given by applying SDGPC method to the following equation 85 (5.203) Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices C(t) = LBv(t) (5.204) The performance index is J = JrC2(t + Tp) + J [C2(t + T) + Xrv2(t + T)]dT (5.205) o Applying the formulae (5.178) with design parameters Nu = l Xr = 0 7r = 0 (5.206) to (5.205), we have ^-^(LB)-1 (5.207) The closed loop pole is at pz = -^A. The overall control law (5.191) which takes into account both servo performance and disturbance rejection performance is thus u{t) = (F + £L)x(*) + t;z(t) + Krr0 (5.208) Where F is given by (5.179) and L = cT(A + BF)-1 1 5 i = -^riLB)~l (5.209) J-P KR = H~l(0) = - ]cT(A + BF)-1B\ _1 = -(LB)~L We summarize the above results in the following theorem. 86 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Theorem 5.1 For the system given by (5.172) and the augmented system (5.190) 1. The eigenvalues of the closed loop system under control law (5.208) are those of A + BF and pz = ((LB). Where F and ( are obtained by minimizing two performance indices (5.176) and (5.205) respectively using SDGPC method. L is given by (5.209). 2. The system transfer function from reference ro to system output y(i) under control law (5.208) is the same as that of the nominal case (5.186). i.e. KrcT(sIn — (A + BF))-1B. Proof: 1. The closed loop system equations of (5.190) under control law (5.208) are: ~A + BF + (BL B(' -cT 0 y(t) = [cT 0] x(0" At)' 'B' + At). .0. x(t) [z(t) (5.210) Apply the similarity transformation defined by nonsingular matrix T = loop system matrix ~A + BF + (BL B(' In 0 -L 1 Adose — C 0 Aclose — T Ac\oseT A + BF + £BL - ZBL L(A + BF) - cT + £LBL - £LBL (LB _ A + BF £B 0 £LB The eigenvalues of Aciose are given by det sln+i -A + BF £B 0 (LB •sIn-(A + BF) -(B • 0 s-(LB. det[sln - (A + BF)]det(s - (LB) = 0 det to the closed (5.211) (5.212) (5.213) 87 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices So the eigenvalues of Adose are eigenvalues of A+BF and (LB. Since similarity transformation does not change the eigenvalues of a matrix, it is clear that the eigenvalues of Aaoae = A + BF + t]BL Bf are those of A^ogg. This completes the proof of statement 1. -cT 0 J 2. Substitute control law (5.208) into system equations (5.189), we have •x(ty At). A + BF + tBL B£ 0 'BKr' + At). 1 y(t) = [cT 0] •(0 (5.214) Apply transformation -*(<)• 'In 0' At)' —L 1 At). L*(*)J to (5.214), we have •x(ty At). A + BF B( 0 LBt] y(t) = [cT o] •x(ty ' BKr ' + X(t). _LBKr + l_ ro At) At) (5.215) Using (5.209), we have LBKT + 1 = 0, the closed loop transfer function from ro to y(t) is thus rS/n - (A + BF) -B£ [cT 0] -] -1 'BKr' 0 0 s- LBt] • [sln -(A + BF))-1 [sln - (A + BF^B^s - LB£)~ (s - LBZY BKT 0 (5.216) = l °31 = cT[sIn -(A + BF)]-1BKT This completes the proof of statement 2. • Remark: Statement 2 implies that the integral term in control law (5.208) adds a systems pole as well as a system zero at (LB. However, the integral term adds a blocking zero at the origin from the reference (or disturbance) to the tracking error e(t) = ro — y(t) thus ensures zero steady state error for constant reference signal and/or constant disturbance. Theorem 5.1 says for a changing £, £t, the eigenvalues of the closed loop system are those of A + BF, which are constant, and a time-varying pole atp*, = (tLB. For stable A + BF and a stable pz given by £, as long as the sign of £t stays the same as £, then the eigenvalues of the closed loop Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices system matrix (5.211) are stable at any instant of time. However, this does not guarantee the stability \A + BF + £tBL B^t' of the rime-varying A£,OS(. = eigenvalues of A(t) have rea [9, pp.411]. For the particular Alclo,e = That is, for time-varying x(t) = A(t)x(t), if the negative parts for each t, then x(t) = A(t)x(t) is not necessarily stable 'A + BF + StBL Bit' , we have the following theorem. o Theorem 5.2 For the system (5.172) and the augmented system (5.190) under control law (5.208) with constant setpoint and with constant F, which stabilizes A + BF, and time-varying £t, if 1. & is such that the time-varying pole pZt = (tLB is always negative, i.e. pZt = (tLB < 0, and f LBZt(T)dT f LB£t(r)dT 2. £t(<)e° is bounded and lim ft(i)e° = constant exponentially, t—too then the closed loop system (5.214) is exponentially stable. Proof: For the closed loop system we have •±(ty .C(0. A + BF B(t{t) 0 LB(t(t) ^ = LB£t(t)dt •*(*)• .CO. C(t) = C(0)e° / LBUr)dr (5.217) (5.218) Condition 1. ensures lim J LB(t(r)dT is a negative constant ( not necessarily -oo ) which means t—too lim ((t) = constant t—too (5.219) from (5.218). For the states x(t), we have x(t) = (A + BF)x(t) + B$t(t)C{t) t x(t) = e(A+BF»x(0) + J e(A+BF^-^B(t(T)C(T)dT (5.220) 89 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Because A + BF is stable and lim = constant (or 0) according to condition 2., which t—»oo means we have a stable system subject to an input which is either exponentially vanishing or goes to a nonzero constant exponentially. Thus i.e. exponentially stable at its equilibrium point. • Theorem 5.2 will be used to construct an anti-windup scheme in next subsection. In the process industries, most processes are well damped open-loop stable systems. For such systems, the " mean level " control [13], i.e. the control law which does not alter the process poles, may be desirable since more aggressive control requires large input amplitude. The following theorem gives the SDGPC version of " mean level " control. Theorem 5.3 For system (5.172) with stable A and the augmented system (5.190), the mean level control law (5.223), i.e. the one with F = 0, is obtained with design parameters lim x(t) = a;^ [constant) (5.221) u 1 OO (5.222) A 0 7 0 The mean level control law is: u(t) = {(L)x(t) + (z(t) + Krr0 (5.223) where £ is given by (5.207) and Kr = - [cTA~lB] -l Proof: For stable A and iV„ = 1, F(T) in (5.178) has a concise form: r(,4,T) = (eAT - ^A^B 90 (5.224) Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Substitute (5.224) into (5.178), we have -l K = J [(trV^A^B)2 - 2cTeATA-1BcTA-1B + {<FA~lB)A dT (5.225) = (h -I2 + hy1 Since A is stable, lim eAT* = 0, thus both the first and the second integrals in (5.225) Tr->oo approach constant while 1$ approaches infinity as Tp —• 00. Thus lim K = 0. Similarly, it can be Tp—»oo shown that Hx approaches constant as Tv —+ 00. So lim F = lim K * Hx = 0. • Remark: Although the mean level control law (5.223) does not change the system dynamics of the setpoint response from the open loop one, its disturbance rejection response can be tuned by selecting different £. Larger £ results in faster disturbance rejection rate. 5.2 Anti-windup Scheme Fig. 5.36 shows the control law (5.208) applied to a system subject to actuator constraints where ' u(t), Umin < u(t) < Umax u(t) = sat[u(t)] = < lmax: "•max (5.226) — u x=Ax+Bu T — ' c Figure 5.36: Control subject to actuator constraints As we mentioned earlier, u(t) consists of the nominal control term un(t) and the disturbance rejection control term v(t). i.e. u(t) = un(t) + v(t) un(t) = Fx(t) + Krro (5.227) v(t) = £(Lx(t) + z(t)) 91 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices We only consider open loop stable plants here as it is meaningless to design anti-wind up scheme for strictly unstable systems. It is impossible to stabilize a strictly unstable system regardless of whatever control strategy is applied when the system disturbance causes the input to saturate [47]. We first consider the case when u(t) is over the upper limit umax. Case 1: Do not reset integrator when both u(t) and un(t) exceed control limit. That is: ««»el(0 = Z(*) {Zrese<(<) = u(t)=um u(t) > umax and un(t) > um Jf< fJilteo (5-228) u(t) < umin and un(t) < umin or un(t) > umal, theni I u(t)=umin where u(r) is defined as the new controller output after integrator reset. Case 1 happens either when an unreachable setpoint is asked or the system suffers too large a disturbance. The system is effectively operated in open loop. If the input is saturated long enough, the output of the system will be —cTA~1Bunm. Case 2: Reset integrator state when u(t) exceeds control limit but un(t) does not. Stop reset integrator when saturation is over. That is: 1. When saturation occurs: i *re..i(«) = *(*) + "mV(t) u{t) > umax and umin < u„(t) < umax, theni u(t) < umin and umin < uH(t) < umax, thenl {u(t) = un(t) + ttLx(t) + zre3Ct(t)) 2. After saturation is over, the control should be updated according to: z(t) = z(t) (5.230) u(t) = un(t) + £(Lx(t) + z(t)) We have the following theorem regarding scheme (5.229). Theorem 5.4 Scheme (5.229) guarantees u(t) = u(t) where u(t) = sat[u(t)] is defined in (5.226). Proof: 92 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Consider the case when u(t) > umax and umin < un(t) < umax, by direct calculation we have umax u(t) u(t) = un(t) + £ (^Lx(t) + z(t) + i J (5.231) = Umax + Un(t) + Lx(t) + z(t) - z(t) = Umax Thus u(t) = sat[u(t)] = u(t). •. Remark: Theorem 5.4 essentially claims that the nonlinear control problem with saturated u(t) can be transformed into a linear control problem by scheme (5.229). Case 3: Reset the feedback gain f but keep the integrator state unchanged when u(t) exceeds control limit but un(t) does not. After saturation is over, changing £ according to equation (5.233). That is: 1. When saturation occurs: -»(*) I t £ _i_ Umax— "UJ u(t) > umax and umin < u„(t) < umax, then< u(t) < umin and u„in < un(t) < umax, then «(*) = «„(*) + (rc.ct{Lx(t) + *(*)) (5 2n) e C _1_ »m.n-»(t) preset — I -u(t) = u„(*) + U..t(Lx(t) + z(t)) 2. After saturation is over, the disturbance feedback gain £(t) should fulfill the following equation: f LBl(r)dr l{t)e<° = £re„t(<o)<^B('-'o) (5-233) where to denotes the time just when saturation is over, £ is the originally designed feedback gain without considering saturation. £reset{to) is the reset £ at to obtained from (5.232). lemma 5.1 The integrator reset algorithm given by (5.229) and (5.230) is equivalent to the gain reset algorithm given by (5.232) and (5.233). We only prove the case when the control exceeds the upper limit umax. The case when u < umtn can be proved in the same manner. Proof: 93 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices (5.235) 1. When control is saturated, i.e. u(t) > umax. From Theorem 5.4, it was shown that (5.229) gives u(t) = umax. While in (5.232), we have = un(t) + ((Lx(t) + z(t)) + umax - u(t) (5-234) — Umax So, both (5.229) and (5.232) give the same control action when u(t) > umax. 2. After saturation is over, denote that moment as to, we have Ul(tn) = Un(to) + ((Lx(t0) + Zreset(to)) = Un(tn) + (C,reset{to) U2(to) = Un(to) + (reset(to){Lx(t0) + z(to)) = Un(t0) + (reset (*o)C(*o) where ui(*o),"2(<o) are the control obtained from (5.229) and (5.232) respectively. Since "i(*o) = "2(^0) as we just proved, we have £Crese<(*f)) = (reset(to)C(t(l) (5.236) The control given by (5.229) after to is "1 (0 = «n(<) + Z(reset(t) (5.237) where Creset(t) is given by (5.201) as: ^reset(t) = LB((]reset(t) Creset{t) = (reset^13^ (5.238) Ol(t) = «»(*) + (Uset{t^LB{t-U) Similarly, the control given by (5.232) after to is *2 (*) = «»(*)+ £(*)<(*) (5.239) where at time to, ((to) = (reset{to)- After to, ((t) is described by (5.233). Finally, we have J LBi{r)dT at) = LB|(OC(*). m = «to)e>° (5240) / LBl{T)dr u2(t) = un(t) + ({t)ato)^ For ((t) given by (5.233), it is easy to show that ui(t) = u2(t) using (5.236). 94 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices We conclude that the integral state reset algorithm described by (5.229) and (5.230) and the gain reset algorithm described by (5.232) and (5.233) give the same unsaturated control action during saturation period and thereafter and are thus equivalent. • Remark: Lemma 5.1 claims that the nonlinear control problem with saturated u(t) is equivalent to a linear time-varying control problem with the time-varying feedback gain given by (5.232) and (5.233). lemma 5.2 For the gain reset scheme (5.232) and (5.233), the sign of€reset(t) during saturation and the sign of£(t) after saturation are the same as the sign of the originally designed £. Proof: 1. During saturation, from (5.232), we have C (4\ /• , umax ~ u(t) Umax — Un(t) /CO/11\ Uset{t) ~ * + Lx(t) + z(t) = Lx(t) + z(t) (5-241) Since umax — un(t) > 0 by assumption, we have freset(t)(Lx(t) + z(t))> 0 (5.242) We also have u(t) = un(t) + f(La:(f) + z(i)) > umax, thus i(Lx(t) + z(t))> (t) > 0 (5.243) Equations (5.242) and (5.243) hold at the same time only if £ and fTe3et(t) have the same sign. 2. After the saturation, from (5.233), it is obvious that eSLB(t-U) iresetih) J LBftT)dr e'<> > 0 (5.244) j LBS(r)dT since both e^-B^-M) e<0 are greater than zero. Thus <f (t) has the same sign as £reset(to)• 95 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices We conclude that the signs of (, (Teset{t) and ((t) are all the same. • Theorem 5.5 The gain reset anti-windup scheme described by (5.232) and (5.233) and the equivalent integrator reset anti-windup scheme described by (5.229) and (5.230) are exponentially stable and gives zero steady state error for constant setpoint subject to constant disturbance. Proof: According to lemma 5.2, the time-varying (t has the same sign as £. Since (LB < 0 by design, we havep2, = (tLB < 0 during all the stages of the algorithm. Further (t "is bounded during saturation J LBi(T)dr and lim ((t)e*<> = lim (resetito)^13^'^ = 0 after saturation according to (5.233). Thus t—too t—too both condition 1. and condition 2. of Theorem 5.2 are met. The resulting time-varying linear control law is exponentially stable for the overall system, i.e. the linear system (5.172) with saturation nonlinearity as depicted in Fig. 5.36. The equivalent integrator state reset algorithm (5.229) and (5.230) is also exponentially stable. Further, since lim = lim (Lx(t) + z(i)) = constant t—too t—»oo t and lim x(t) = constant, we have lim z(t) = lim J (ro — y(r))dT = constant, which means t—too t—too t—too lim e(t) = lim (r0 - y(t)) = 0. • t—too t—too Some examples are presented here to show the effectiveness of the TDF-SDGPC and the anti-windup algorithm. Example 5.1 The first example is a simple integrator process. G(s) = - (5.245) s The actuator limits are ±0.1. Design the SDGPC algorithm with two performance indices using the same following design parameters Nu = l Tp = 1.5sec (5.246) A = 7 = 0 96 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices gives the control law u(t) = un(t) + v(t) un(t) = -y(t) + r0 (5.247) v(t) = -y(t) + z(t) Fig.5.37 shows the control results of control law (5.247) without anti-windup compensation. The overshoot in Fig.5.37(a) can be clearly observed. 60 (S) 60 (S) integral of tracking error 1 1 1 1 1 0 10 20 30 40 (c) nominal control un(t) SO 60 - I 1 i 1 -- I 1 1 1 -I 1 1 , 1 1 1 0 10 20 30 40 SO 60 (S) (d) Figure 5.37: Example 5.1: Control law (5.247) without anti-windup compensation Fig. 5.38 shows the control results of control law (5.247) with anti-windup scheme. From Fig. 5.38(c), the integral state is reset just after the 20 seconds mark at which time the nominal control is within the control limits. The effectiveness of the algorithm is obvious. 97 setpoint and output 30 (a) \ control I after saturation \ ter^sa -2 - before saturation 30 (b) Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices setpoint and output control integral of tracking error -integrator reset (c) 60 (S) 1 0.5 0 -0.5 nominal control 30 (d) 60 (S) Figure 5.38: Example 5.1: Control law (5.247) with anti-windup compensation Example 5.2 The process for the second example is described by -_3 _3 _r T x(t) = 1 0 0 x(t) + 0 .0.1 0 . .0. y(t) = [o o l]x{t) The actuator limits are: ±3.5 For the nominal control law un(t), the design parameters are 98 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Nu = 10 Tp = 2sec. (5.249) A = 0.0001 7 = 1000 The resulting nominal control law is given by un(<) = Fx(t) + Krro (5.250) F = [-4.7599 -25.9369 -51.4516 ],Kr = 52.4516 For the control v(t), the prediction horizon Tp is selected such that the pole is placed at —2 and f = 104.9031. the overall control law is un(t) = (F + £L)x(t) + Krr0 + £z(t) (5.251) = [-6.7599 -41.4567 -109.3254 }x(t) + 52.4516r0 + 104.90312(f) The control results for the control law (5.251) with anti-windup algorithm are shown in Fig. 5.39. The effectiveness of the anti-windup algorithm can be again observed by comparing Fig. 5.39 with Fig. 5.40 in which no anti-windup algorithm is used. Notice the integral state reset in Fig. 5.39(c) after each setpoint change. 99 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices setpoint and output 60 (s) control 10 20 30 (b) 60 (S) integral of tracking error 60 (s) Nominal control 100 l\ -0 100 7 V -0 10 20 30 40 60 & (d) 60 (S) Figure 5.39: Example 5.2: Control law (5.251) with anti-windup compensation 100 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices setpoint and output - 1 — r, -1 / / "\ \ ,s V ^ \J --10 20 30 (a) control 40 50 60 (S) integral of tracking error 50 60 (s) nominal control Figure 5.40: Example 5.2: Control law (5.251) without anti-windup compensation The integrator reset algorithm has a strong similarity with the conventional anti-windup PID controller whose structure is depicted in Fig.5.41. Figure 5.41: Conventional anti-windup The performances of the conventional anti-windup algorithm and the proposed scheme are compared in next example. 101 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices Example 5.3 The plant being controlled is a simple integrator with random disturbance injected at the input. The saturation limits are ±0.1 and the control law is given in equation (5.252) G(s) = I -0.1 < u(t) < 0.1 t u(t) = -1.9998y(i) + 0.9998r„ + 0.9998 J (r0 - y(r))dr u„(t) = -0.9998j/(t) + 0.9998r0 (5.252) t v{t) = -y{t) + 0.9998 J (r0 - y(n ))dr The reset gain K as depicted in Fig.5.41 is selected to ensure good performance for the conventional algorithm. Fig.5.42 shows the results. The first three plots are for the conventional methods with different reset gains. The observation is that although good performance can be obtained for properly selected gain K, it is nonetheless a nontrivial trial and error procedure. Moreover, the effect of changing K on the performance is "non-monotonic". This makes the tuning more difficult. On the other hand, the proposed scheme gives excellent results in a straightforward manner as can be seen from the fourth plot of Fig.5.42. K=0.02 -^Xy2(t) = 0.7618 0.2 0.4 0.6 0.8 ^ 1 1.2 1.4 1.6 1.8 K=0.5 l£y2(t) = 0-2130 x1° 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 1 K=10 -^xy2(t) = °-3575 x 10 Figure 5.42: Conventional anti-windup scheme vs proposed 102 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices We make some remarks to conclude this section. 1. The stability result stated in Theorem 5.5 also applies to unstable systems provided that umin < un(t) < umax holds. However, for unstable plants, it is always possible to find a system state XQ such that un(t) = FXQ + Krro > umax or un{i) < umin- This will drive the system in open loop mode thus destabilize the system. 2. For open loop stable systems, it is also desirable to have un(t) = Fx(t) + KTrQ (5.253) Umin *N f n(i) <C Umax hold. This way, the anti-windup scheme leads to a graceful performance degradation compared with the unconstrained linear design. However, to fulfill (5.253), a smaller feedback gain F is required. The extreme case is the "mean-level" control given by Theorem 5.3 where F — 0, i.e. no control effort is made to make the servo response faster. This means that when input limits exist, a trade-off must be made to balance servo performance and disturbance rejection performance even for a two-degree-of-freedom design. 3. Although the integral state reset scheme (5.229), (5.230) is equivalent to the gain reset scheme (5.232), (5.233), it is clear that the integrator reset scheme is much easier to implement. The gain reset scheme is also very important since it clearly shows that the original time-invariant nonlinear problem is equivalent to a time-varying linear problem which possesses nice stability property under reasonable assumption. 4. Integral reset can also be used for bumpless transfer in process control. For manual control signal uman(t), the integral z(t) in control law (5.208) can be set as z(t) = U™an{t) - {F + ZL)x{t) - Krr(i ^ ^ This way, the control law (5.208) tracks the operator action in manual mode and takes over the control task from the operator based on the last operator entered value whenever in auto mode thus realizing bumpless transfer. 103 Chapter 5: Anti-windup Design of SDGPC by Optimizing Two Performance Indices 5.3 Conclusion In this chapter, we developed a two-degree-of-freedom SDGPC algorithm based on two per formance indices. Based on this TDF-SDGPC algorithm, an anti-windup scheme was proposed in section 5.2. It is shown that the linear control law plus saturation nonlinearity is effectively equivalent to an unconstrained linear time-varying control law which leads to graceful performance degradation compared with the original linear unconstrained design while guarantees stability. It also clearly poses a trade-off design problem between the servo performance and the disturbance rejection per formance. The attractive features of this anti-windup scheme are its elegant simplicity, effectiveness and most importandy, guaranteed stability property. 104 Chapter 6: Continuous-time System Identification Based on Sampled-Data Chapter 6 Continuous-time System Identification Based on Sampled-Data System model is the centerpiece in all Model Based Predictive Control algorithms. In the case of various SDGPC algorithms we developed in previous chapters, the system models being used are continuous-time state space ( or high order differential equation ) models. The continuous-time Laguerre filter identification and its associated Laguerre filter based adaptive SDGPC problem were treated in chapter 4. The parameter estimation problem for general (stable or unstable) continuous-time differential equation is discussed in this chapter. There is no doubt that discrete-time models have received more attention than their continuous-time counterparts in the development of both identification theory and techniques. The theoretical results and algorithms available for discrete-time model parameter estimation are overwhelming [65, 25, 45, 44, 72]. The continuous-time model identification problem using digital computers, on the other hand, has yet to reach the same level although the relevance and importance of continuous-time system identification have been increasingly recognized in the recent years. An earlier survey solely devoted to this subject can be found in P. C. Young [85]. A comprehensive review of recent developments in the identification of continuous-time systems was given by Unbehauen and Rao [79]. A book written by Unbehauen and Rao [78] attempts to present a simple and unifying account of a broad class of identification techniques for continuous-time models. Least squares (LS) is a basic technique for parameter estimation. The least squares principle, formulated by Gauss at the end of eighteenth century, says that the unknown parameters of a mathematical model should be chosen in such a way that the sum of the squares of the differences between the actually observed and the computed values, multiplied by numbers that measure the degree of precision, is a minimum [64J. The method is particularly simple if the model has the property of being linear in the parameters. That is if a model has the following form y{t) = y>i(i)0i + ip2(t)92 + ••• + M*)0n = ¥>(OT0 105 (6.255) Chapter 6: Continuous-time System Identification Based on Sampled-Data where y is the observed variable, #1, #2, • • •, #n are unknown parameters, and , <pi, <p%, • • •, <pn are known functions that may depend on other known variables. The model is indexed by t, which often denotes time. The variables <pi are called the regression variables or the regressors and the model described by (6.255) is called a regression model. The vectors <pT(t),9T are defined as <pT(t) = [^(t) <p2(t) ••• <pn(t)} (6.256) eT = [el{t) e2(t) ••• en(t)] With pairs of observations and regressors {(y(i),<pi(i)), i = 1,2, • • • t}, the parameter estimation problem is to determine the parameters in such a way that the outputs computed from the model described by (6.255) agree as closely as possible with the measured variables y(i). The solution has a analytical form (6.257) where Y(t) = [y(l) y(2) ••• y(t)]T Vr(i)' (6.258) and the symbol "A" denotes estimates throughout die thesis. In view of real time application, the computation based on (6.257) can also be done recursively resulting in the so called Recursive Least Squares (RLS) algorithm. The regression model (6.255) is an algebraic (non-dynamic) equation which has two important properties: it is realizable in the parameters 0T and contains only realizable functions <pT(t) of the data. Usually there are two phases in applying the above least squares method to parameter estimation of dynamic models. The primary phase involves converting the dynamic equation into an algebraic equation. And the secondary phase involves solving these algebraic equations for the unknown parameters which is given by (6.257) or its recursive version. For systems described by a difference equation y(') u(t) acq71 + aiqn~l H h an (6.259) 106 Chapter 6: Continuous-time System Identification Based on Sampled-Data or equivalently y(t) _ frp + feig *H + bnq : n (6.260) u(t) aa + aiq-1 H b anq n where q is the difference operator, i.e. qy(t) — y(t + 1), the solution to the primary phase is obvious since the dynamic equation (6.260) already fulfills the requirement of the primary phase. For a continuous-time model given by where s is the differential operator, i.e. sy(t) = dy(t)/dt (also loosely interpreted here as the Laplace operator), the solution to the primary phase is not as simple since derivative operation sny(t) = dny(i)/dnt is not feasible. One way to circumvent this difficulty is to make continuous-to-discrete conversion of the continuous-time model (6.261) first, and then estimate the parameters of the resulting discrete-time model. The parameters of the original continuous-time model can then be ob tained by a discrete-to-continuous time transformation. However, obtaining a continuous-time model from its identified discrete-time form is not without difficulties [70, 71] as the choice of sampling interval is not trivial. On the other hand several methods are available to make the continuous-time model (6.261) compatible with the requirement of the primary phase without changing the parameters (an • • • a„ ,bo---bn) [79, 79, 85]. Perhaps the most direct of these is the low-pass filter approach. The key idea is to choose a low-pass filter H(s) with a transfer function of sufficient relative degree to make snH(s) proper so that all the signals sny(t), sn~1y(t), • • •, y(t), snu(t), sn-1u(t), • • •, u(t) involved in (6.261) will be feasible by passing through H(s). Thus a realizable, linear-in-the-parameters form of equation (6.261) can be obtained. One particularly simple choice of the filter is the multiple integration filter l/sn. The initial condition problem associated with integration opera tion can be overcome by integrating the input/output signal over a moving time interval [to, to + T] [67]. In section 6.1, the continuous-time model (6.261) is transformed into a regression model by passing sny(t), sn~1y(t), • • •, y(t), snu(t), sn-1u(f), • • •, u(t) through a multiple integration filter 1/s". And the numerical integration formulae will also be given. Section 6.2 introduces the recursive least squares algorithm EFRA [68] which stands for exponential forgetting and resetting algorithm. In y(t) _ b0sn + b1sn-1 + ••• + &, 'n (6.261) u(t) a0sn + a!*""1 -| \-a, •n 107 Chapter 6: Continuous-time System Identification Based on Sampled-Data section 6.3, we develop a new algorithm to estimate fast time-varying parameters. A real life inverted pendulum experiment in section 6.4 shows the effectiveness of the algorithm presented in this chapter. 6.1 The Regression Model for Continuous-time Systems Considering the system model (6.261), assuming the leading coefficient OQ is equal to 1. Let y(n\t) denotes the ra-th derivative of y(t), the system model being considered has the form y(n)(0 + «iy(n_1)(0 + • • • + any(t) = &ou(n)(0 + M(n_1)(*) + ••• + bnu(t) (6.262) Define the multiple integral of signal y(t) over [to,to + T) as U+TU+T tj-i+T Iny(ta) = f J "' f y{tj)dtjdti-X---dt1 (6 263) to tl tj — 1 j = l,2,---n Apply In defined in (6.263) to both sides of (6.262), the resulting regression model is Iny(n)(to) = <pT(t0)9 (6.264) where <pT(to) = [-Iny(n-1)(to) ••• -Iny{to) J„U<»)(*o) ••• /„U(*0)] (6.265) 0 = [ai ••• an bo ••• bn] As can be seen from (6.265), all the regressor entries are numerically feasible since derivation is no longer necessary. Since we are interested in computer implementation of the algorithm, the input/output data are only available at discrete sampling instants. However, this will be enough to compute the regressor numerically. The trapezoidal rule will be used for its simplicity in form and its satisfactory accuracy. Rather than giving the general formulae to compute the integrals in (6.265), we only consider the case when n = 2 for the purpose of not hindering the basic principle. Assume / +1 samples of signal y(t), y(0), y(l), • • • y(l), are available on the interval [to, to + T] with sampling interval A = j as illustrated in Fig.6.43, then the integral hy(to) = / y(h)dti can be given numerically using trapezoidal rule as 108 0 Chapter 6: Continuous-time System Identification Based on Sampled-Data •t0+T hy(to) = J y{h)dh « A £y(0-0.5(y(0) + y(7)) Li=0 (6.266) to to+A U+T Figure 6.43: Graphical illustration of numerical integration Similarly, the double integration of y(t) over [to,to + T] can be given by t0+T U+T U+T hy(h) = J dtx J y(t2)dt2 = J [hy{h))dt to ti to • / i A hy(to + t'A) - 0.5Jiy(«0) - 0.5/iy(t0 + T) Li=0 where each hy(to + iA), i — 0,1, • • • / in (6.267) can be computed using (6.266). Double integration of y(t) over [to, to + T] is given by U+T U+T U+T hy{to)= J dh J y{t2)dt2= J (yCti + T)-^!))*! to ti to = y{t0 + 2T) - 2y(tQ + T) + y(to) Double integration of y(t) over [to, to + T] is given by U+T U+T U+T hy(to)= J dtx J y(t2)dt2= j (y(*i+r)-y(ti))d<i to t] to = hy(to + T)-Iiy{to) (6.267) (6.268) (6.269) With these formulae (6.266)-(6.269), the regressors in (6.265) can be computed for n < 2. For n > 2, the corresponding equations can be obtained in a similar fashion. It is obvious that both the sampling interval A and the integration interval T affect the estimates. Sampling interval A directly affects the accuracy of the numerical integration (6.266) thus A should 109 Chapter 6: Continuous-time System Identification Based on Sampled-Data keep small. However, too small A may also lead to inaccurate estimates due to round off error. As a rule of thumb, the sampling interval should be chosen according to |rJ<A<! (6'270) where T3 = ^- is the Shannon maximum sampling period [86]. In obtaining the regression model (6.264), the multiple integral operation In defined in (6.263), which functions as a pre-filter, is applied to both sides of system model (6.262). Intuitively, the bandwidth of the this pre-filter should match that of the system (6.262) so that the noise in the measurement data can be depressed and at the same time, without rounding off the "richness" of the input/output signal which contains the information about the dynamics of the system (6.262). The Laplace transform of the multiple integrator (6.263) over a time length of T is [67] (1 _ e-sT)n 4«01 - J-^Uoi (6.27I) = =(»)£[(•)] It is clear from (6.271) that the integration span T of the multiple integrator /„ should be selected such that the bandwidth of the multiple integrator transfer function E(s) matches the bandwidth of the system being identified. 6.2 The EFRA For the regression model (6.264), the standard RLS algorithm is given by the following for mulae^] 8(t0) = §(to - 1) + /<•(*(>) (WB)(*o) - /(to)^fl - 1)) P(to)={l-K(t0)<pT(ta))P(t(i-l) where P(0) is a large enough positive definite matrix. The statistical interpretation of the least-squares method is that the initial covariance of the parameters is proportional to -P(O). 110 Chapter 6: Continuous-time System Identification Based on Sampled-Data The RLS algorithm (6.272) can not track time-varying parameters effectively. A simple extension to (6.272) is to use a forgetting factor 0 < A < 1 to give more recent data more weight. The modified algorithm is given by 0(tQ) = e{tQ -1) + tf(*0)(WB)(*o) - vT('oW*o -1)) K(to) = P(t0M*o) = ir+^Tp^l <6-273> A7 + y>r(to)P(*o-lM<o) P(t0) = (I - K(t0)<pT{to))P(to - 1)/A A disadvantage of the exponential forgetting (6.273) is that the covariance matrix P may eventually blow up when the excitation is poor. The Exponential Forgetting and Resetting Algorithm (EFRA) [68] of Salgado et al has been shown to have superior performance. It guarantees bounded covariance matrix P even when the excitation is poor. The EFRA is given by 0(<o) = 0(<o - 1) + Km){l2y{n\to) - <PT(to)Hto ~ 1)) K(to)-I + ^o)P(t0-lMto) (6-2?4) P(*o) = \p(tQ - 1) - K(to)<pT(t0)P(t0 -i) + pi- 6P2(t0 - 1) There are four parameters in this algorithm to be chosen by the user. However, it is straightfor ward to select them in practice. The general guidelines are: 1. a adjusts the gain of the estimators, typically a e [0.1,0.5] 2. (5 is small, directly related to the smallest eigenvalue of P, typically (3 G [0,0.01] 3. A is the usual forgetting factor, A e [0.9,0.99] 4. 6 is small, inversely proportional to the maximum eigenvalue of P, typically S £ [0,0.01] The desirable features of EFRA are: 1. Exponential forgetting and resetting 2. An upper bound for P, i.e. a nonzero lower bound for P-1 3. An lower bound for P-1, i.e. a nonzero lower bound for P 111 Chapter 6: Continuous-time System Identification Based on Sampled-Data 6.3 Dealing with Fast Time-varying Parameters RLS with forgetting factor can deal with slow time-varying parameters effectively but will encounter difficulties for fast time-varying parameters. In such cases it is advantageous to assume the parameters to be time-varying right from the start of the problem formulation. Xie and Evans [83] proposed an algorithm in a discrete-time setting assuming that the parameters are of the form of offset linear ramp. The moving horizon multiple integrator approach developed in section 6.1 will be used to deal with the time-varying parameter case. Again for simplicity, a second order differential equation with time varying parameters is considered. Consider the equation y + a!(t)y + a2(t)y = bi(t)u (6.275) Assume that the time-varying coefficients ai(t), 6,-(t) have the form di(t) = aw + ant bi(t) = bi0 + but (6.276) te[0,Tres] over interval [to, to + Tres\. Obviously, equation (6.276) would be a very good approximation of ai(t), bi(t) if Tres is reasonably small. Note that Tre3 is not necessarily the same as the integration span T in (6.263). Usually Tre3 > T. Substitute equation (6.276) into equation (6.275), we have V + amy + ^202/ + a\\ty + a2\ty = bwu + bntu (6.277) Apply I2 defined in (6.263) on both sides of (6.277) over [t0, to + T], the following regression model can be obtained hy(to) = <pT(ta)e 112 (6.278) Chapter 6: Continuous-time System Identification Based on Sampled-Data where <pT(ta) = •hy{h) - hy(to) hu(ta) : - hty(to) - hty{h): htu(t0) aio a 20 ^10 : an c-21 hi (6.279) The integrals l2y(to),l2y(ta),I2y(t(\),l2u(tQ) can be computed using formulae (6.266)-(6.269), while l2ty(t<)),hty(h),htu{to) are given as.follows t0+T U+T u+T t,+r hty{h) = j dh J ry{T)dr = J dta J rdy(r) to tj to U+TT u+T = J (Tvir))$+T- J vi*)* to L ti U+T = J [(*! +T)y(ta + T)-<!!/(*!)-/u/^ijjdt! U U+T U+T hty(to) = J dh J ry(T)dr u u U+T U+T I2tu(to) = J dti J Tu(r)dT (6.280) With the regression model (6.278), either the standard RLS (6.272) or the EFRA (6.274) can be used. Recall equation (6.276), apparently the offset linear ramp approximation of time-varying co efficients is valid only when Tre3 is small. Define Tres as the resetting period . At each kTres, k = 0, 1, 2 • • •, it is necessary to reset parameter vector and covariance matrix as follows 113 Chapter 6: Continuous-time System Identification Based on Sampled-Data aw(k + 1) = a1Q(k) + Tre3an(k) an(k + 1) = an(k) G2o(k + 1) = a2(\(k) + Tresa2\{k) a2i(k + 1) = a2i(k) (6.281) b1Q(k + l) = b10(k) + TTesbu(k) hi(k + I) = bu(k) P(k + 1) = KiP(k), I<! > 1 It is important to select the resetting period Tres properly, the principle is that Tres must be chosen large enough to allow reasonable convergence of the parameters but the variation of the real parameters over the period of Tres should stay small so that the offset linear ramp is still a good approximation. The following example shows the effectiveness of this algorithm. Example 6.3.1 Consider system (6.275) with parameters described by b(t) = 2 + 0.It, 0<i<30sec The simulation is performed in open-loop with PRBS signal as input, and the following settings: sampling interval A = 0.01 sec, integration interval T = 0.05sec and resetting period Tres = 0.08sec. The standard RLS algorithm (6.272) is used. The results are depicted in Fig.6.44. ai(f) = 2 + 1.5 * sin(0.27r*), 0 < t < ZQsec (6.282) a2 = 1, 0 < t < 30sec 114 Chapter 6: Continuous-time System Identification Based on Sampled-Data 0U 1 1 1 1 1 « 1 0 5 10 15 20 25 30 (S) Figure 6.44: Estimation of time-varying parameters As can be seen from Fig.6.44, even the sinusoidally time-varying parameter can be tracked satisfactorily. This verifies the validity of our assumption in equation (6.276). 6.4 Identification and Control of an Inverted Pendulum The control of an inverted pendulum is a classic topic in control engineering. There are many solutions available to this problem, for example PID, LQG, fuzzy logic etc. The SDGPC solution will be given in this section. The advantage of continuous-time parameter estimation over the discrete-time method is highlighted by the comparison between the two different approaches. Fig 6.45 is a picture of the experimental setup2 which is built on a used printer. The pendulum rod is mounted on the printer head and able to freewheel 360 degrees around its axis. The printer head is driven by a DC motor along the x axis. Both the printer head position x and the pendulum angel 6 are available for measurement through a LVDT and an encoder attached to the printer head. The control input to the system is the voltage applied to the DC motor. The purpose of the control is to keep the pendulum rod upward and at the same time keep the printer head at the center position. 2 The author thanks Dr. A. Elnaggar who was then a research engineer at the Pulp & Paper Centre for making this experimental setup available. 115 Chapter 6: Continuous-time System Identification Based on Sampled-Data Figure 6.45: The inverted pendulum experimental setup 6.4.1 System Model The printer head position is proportional to the angular displacement of the DC motor. Thus the transfer function from the input voltage u(t) to the printer head position x(t) has the form Gm(s) = bo %\8j km u(s) s(rms + l) s2 + ais (6.283) Only two parameters bo,a-[ need to be estimated. As for the relation between the printer head position x(t) and the pendulum angle 0(t), let us consider the downward pendulum first. Fig. 6.46 is an idealized sketch of the downward pendulum. Notice that at the equilibrium point Oo = 0, only the acceleration of the printer head M will break the equilibrium. Suppose M is moving with acceleration x(t), observing on the moving M, the effect of x(t) on m is that it seems 116 Chapter 6: Continuous-time System Identification Based on Sampled-Data as though there is a force mx(t) being applied on the other direction. Applying Newton's law in the 9 direction as indicated in Fig. 6.46, the torque balance is —mg * L sin0 — e9 + mx(t) * L cos 9 = ml?9 (6.284) where L, m are the length and the mass of the idealized pendulum respectively, g is the gravity constant and e is the friction coefficient. Assuming small 9, and linearizing it around 9$ = 0, equation (6.284) becomes Gdoum(^) — 9 + ai9 + a29 = b'x(t) 9{s) bs2 x(s) s2 + a\ s + a2 £ 9 , 1 (6.285) t mg Figure 6.46: Downward pendulum Similarly, the torque balance for the upward pendulum case is mg * L sin 9 — e9 + mx(t) * L cos 9 = mL29 (6.286) And the linearized model around 9o — 0 can be written as 117 Chapter 6: Continuous-time System Identification Based on Sampled-Data 6 + aiO - a26 = bx(t) (6.287) Figure 6.47: Upward pendulum Comparing Gdown(s) in (6.285) and Gup(s) in (6.287), it is easy to see that the parameters in . GdoWn(s) and Gup(s) are the same except for a sign difference in a2. This important a priori information is only preserved in the continuous-time model! Since the upward pendulum is open loop unstable, it is very difficult, if not impossible, to estimate Gup(s) direcdy without stabilizing it first. However, with continuous-time modeling, it is possible to estimate ai, a2 and b for the downward pendulum which is open-loop stable. The estimation results will be presented in the next subsection. 6.4.2 Parameter Estimation The experiment was conducted on the downward pendulum. The input voltage applied to the DC motor is PRBS ( Pseudo Random Binary Sequence) signal with an amplitude of ±1 volt and a length of 28 - 1 = 255 samples. The sampling interval A = O.lsec. Both the printer head position x(t) and the angle 6(t) are measured. For the model structure given by (6.283), the continuous-time regression model (6.264) and the standard RLS (6.272) can be readily applied. The integration 118 Chapter 6: Continuous-time System Identification Based on Sampled-Data interval T = 0.3sec. The input/output data and the estimated parameters bo,ai are depicted in Fig. 6.48 where So = 10250, fii = 12.59. Input voltage to DC motor Parameter estimates 201 1 1 : r— Figure 6.48: Parameter estimation of model (6.283) The estimated model from input voltage to printer head position is thus - 10250 814.09 L'm~ + 12.59s " s(0.0794s + 1) Since the time constant rm = 0:0794sec is very small, (6.288) can be reasonably approximated by 814.09 Gm = (6.289) s For the identification of the system model from printer head position to pendulum angle, a 255 sample PRBS signal with an amplitude of Iv is applied to the motor. The sampling interval is again A = O.lsec. The model parameters a\, a2 and b of G<*oum(s) in (6.285) are estimated using algorithms given in previous sections in this chapter. The printer head position, the angle output data and the 119 Chapter 6: Continuous-time System Identification Based on Sampled-Data estimated parameters a\, £2, b are depicted in Fig. 6.49 with di = 0.01418, a2 = 43.9627, b — 0.1125. 1000 -1000 100 Printer head position 10 15 Pendulum angle 10 15 Parameter estimates 25 (s) 25 (s) Figure 6.49: Parameter estimation of model (6.283) The identified model Gdown(s) is thus Gdown{s) — 0.1125s2 s2 + 0.01418s + 43.9627 with poles at -0.0071 ± 6.6304z. The open loop unstable upward pendulum model (6.287) can be readily written as r (*\ °-1125s2 Gup{s) = (6.290) (6.291) s2 + 0.01418s - 43.9627 with poles at -6.6375,6.6233 which correspond to a time constant of approximately 0.15sec. The sampling interval of A = O.lsec we used in the experiment is relatively large for this system, either for the downward or the upward case. Unfortunately, that was the smallest sampling interval we can get due to the computer system limitations. 120 Chapter 6: Continuous-time System Identification Based on Sampled-Data It is interesting to see how discrete-time estimation will perform with the same experimental data. The discrete-time counterpart of system model (6.285) has the form: 9(k) + adl9(k - 1) + adi9{k - 2) = bdix(k) + bd2x(k - 1) + bd3x(k - 2) 9{k) bdi + bd,q-1 + bd3q~2 (6.292) x(k) 1 + adlq~l + ad3q~2 The MATLAB function ARX in the system identification toolbox was used on the same data set to identify the parameters in (6.292), i.e.: arx([9 x],[na nb nk]) na = 2,nb = 3,nk = 0 The parameter estimates are [bd, bit bd3] = [0.093 -0.1863 0.0923] [adl ad2] = [-1.5685 0.9663] Fig. 6.50 shows the step response of model (6.292) with estimated parameters. (6.293) (6.294) Figure 6.50: Step response of the estimated discrete-time model (6.292) Fig. 6.50 tells us that the system has a natural undamped frequency of 1.05. This agrees quite satisfactorily with what we have from the continuous-time identification approach. See (6.290) where 121 Chapter 6: Continuous-time System Identification Based on Sampled-Data f = ^43.9627 _ 10553. However, the damping factor is quite different from what we obtained in (6.290). From Fig. 6.50, the pendulum should settle in about 30 seconds when there is a step position input. Experiment shows that the pendulum oscillation will last about 400 seconds after a hit on the printer head which agrees quite well with the continuous-time estimation results. Also notice from Fig. 6.50 that there is a nonzero steady state gain in the estimated discrete-time model. This is obviously wrong. It has been shown that the system has two zeros at the origin, see (6.285). This important a priori information is also lost in the discrete-time modelling. As a matter of fact, the continuous-time counterpart of the estimated discrete-time model with sampling interval A = 0.1 sec and zero-order-hold is: 0.093s2 + 0.2004s - 0.1036 r (o.zio) s2 + 0.3428s + 41.91 Compare (6.295) with (6.290), the superior performance of continuous-time identification for this example is obvious. 6.4.3 Controller Design Define the states of the pendulum system and the input voltage to the DC motor as [0(t) 6(t) x(t) x(t)] and u(t) respectively, the state space description of the system can be written based on (6.289) and (6.291) •o-0 X .X. -0.01418 43.9627 0 0 1 0 0 0 0 0 0 0 0 0 10 "1 •o--91.5851" 0 0 X 814.09 .X. . 0 . ud(t) (6.296) where ud(t) = ii(t). 122 Chapter 6: Continuous-time System Identification Based on Sampled-Data In the SDGPC framework, system (6.296) can be regarded as the integrator augmented system of • e • t = -0.01418 43.9627 0 1 0 0 0 0 0. • 9 • -91.5851" t /' + 0 .814.09 . «(*) • e t Jo x x(t) = [0 0 1 ] Apply the final state weighting SDGPC (2.20) to (6.296) with design parameters: Nu = 6 Tp = 1.2sec A = 1 7 = 1 The resulting control law is ud(t) = -[0.1905 1.2643 -0.0069 -0.0120] The control law (6.299) is implemented in the following form: •01 9 x .x. u(t) = -0.1905 * 9(t) - 1.2643 * J 9(T)<1T t +0.0069 * (x(t) - 1100) + 0.0120 * J (X(T) - 1100)<*7 (6.297) (6.298) (6.299) (6.300) The number 1100 in (6.300) is the LVDT reading corresponding to the center position. The picture in Fig.6.52 shows the pendulum being successfully controlled by (6.300). Notice that the control law is fairly robust against a disturbance ( a plastic bag was placed on the top of the pendulum after (6.300) was applied ). The same control law can also stabilize the pendulum when the rod is stretched to twice the original length. See Fig. 6.51. 123 Chapter 6: Continuous-time System Identification Based on Sampled-Data Chapter 6: Continuous-time System Identification Based on Sampled-Data Figure 6.52: SDGPC of pendulum subject to disturbance It is found in the experiment that it is not difficult to stabilize the pendulum, i.e. keeping it upward. But it is not easy to control the printer head exacdy in the center position while keeping the pendulum upward. Also notice that the system model we developed in subsection 6.4.1 is an idealized one with the assumption that the pendulum rod is a rigid body. As the length of the rod increases, so does its flexibility. An adaptive version of the SDGPC would have been more interesting. Unfortunately the experiment setup was only available for a limited period of time. Nevertheless, the experiment results show that the continuous-time model parameter estimation algorithm and the SDGPC algorithm are quite effective solving practical control problems. 6.5 Conclusion Continuous-time system identification based on sampled-data is considered in this chapter. The moving horizon integration approach given in section 6.1 is a simple yet powerful method for 125 Chapter 6: Continuous-time System Identification Based on Sampled-Data parameter estimation of a continuous-time model. Based on the regression model (6.264), various available recursive estimation algorithms such as the EFRA developed in discrete-time context can be readily applied. The algorithm we proposed in 6.2 can deal with fast time-varying parameters as was shown by simulation. A real life inverted pendulum experiment in section 6.3 showed the benefits of continuous-time identification, namely, effect use of a priori information etc. The identification method in this chapter together with the SDGPC algorithm offer an effective way for solving complicated control problems. In this way, the insight about the underlying inherent continuous-time process is never lost during the whole design process. It is the author's belief that even if the control law is designed in discrete-time domain, it is always beneficial to identify the underlying continuous-time process first and then discretize it. The experiment presented in section 6.3 can be regarded as a supportive example. 126 Chapter 7: Conclusions Chapter 7 Conclusions A new predictive control approach was taken in this thesis. The important issue such as actuator saturation in practical applications was taken into account. The resulting algorithms have very important practical interests as well as nice theoretical properties. The work can be summarized as follows. 1. A new predictive control algorithm, SDGPC, has been developed. It possesses the inherent robustness ( gain and phase margin ) and stability property of infinite horizon LQ regulator and at the same time, has the constraint handling flexibility of the finite horizon formulation, a feature unique to MBPC. SDGPC distinguishes itself from the rest of the MBPC family in that it is based on continuous-dme modelling yet assumes digital implementation. This formulation stresses the connection rather than the differences between continuous-time and discrete-time control. It has been shown by simulation that for a stable well damped process, the execution time Texe can vary from 0, which corresponds to continuous-time control, to the design sampling interval Tm, which can be quite large, without affecting the servo performance significantly. This means that for a given prediction horizon Tp and desired sampling interval Texe, a larger Tm can be selected to reduce computation burden in adaptive applications. For unstable and/or lightly damped processes, however, Texe should be equal to Tm. 2. SDGPC for tracking systems has a two-degree-of-freedom design structure. This is achieved by assuming a different model for the reference signal and the disturbance. However, only one performance index is used to obtain the control law. Tracking performance can be improved radically when the future setpoint information is available. This is because knowing the future setpoint is equivalent to knowing the complete state information of the reference signal at present time. 3. Another two-degree-of-freedom design extension to SDGPC was made. Contrary to the approach taken in tracking system design in which different models for reference and disturbance were assumed, two performance indices are used but assuming the reference and the disturbance 127 Chapter 7: Conclusions have the same model ( in this thesis, it is a simple constant). The servo performance and the disturbance rejection performance can be tuned separately by using different design parameters ( prediction horizon, control order, control weighting etc. ) for the two performance indices. The nonlinearity due to actuator constraints is considered in the framework of anti-windup design. The resulting scheme effectively transforms the nonlinear problem into a time-varying linear problem and was shown to have guaranteed stability property. Simulation results confirmed the effectiveness of the scheme. 4. Control of time-delay systems was considered. Laguerre filter based adaptive SDGPC was shown to be particularly effective in dealing with time-delay systems. A priori information about the time-delay can be utilized to improve the control performance significandy. 5. A continuous-time model parameter estimation algorithm based on sampled data was developed. Numerical integration on a moving interval was used to eliminate the initial condition problem. It was argued that even if the controller design is purely in discrete-time, it is always beneficial to identify the underlying continuous-time model first before discretizing. A priori information about the physical system is best utilized in continuous-time modelling. The continuous-time model identification method and the SDGPC algorithm were applied to an inverted pendulum experiment. The results confirmed the benefits of continuous-time modelling and identification. Some future research suggestions are: 1. Extend the work to multi-input multi-output systems. Although the author sees no major obstacles in doing so for most of the topics covered in the thesis, some efforts are needed to formulate the anti-windup scheme for MDMO case. 2. Dealing with the trade-off between good tracking and disturbance rejection performance and, good noise suppression performance. This is a basic trade-off in any control systems design [3, pp. 112]. SDGPC was formulated in a deterministic framework. Deterministic disturbances such as impulse, step, ramp, sinusoidal etc. can all be handled in a straightforward manner in the framework of SDGPC. The trade-off between good noise suppress performance and good disturbance rejection performance can be obtained by proper tuning of SDGPC to give the desired 128 Chapter 7: Conclusions closed-loop system bandwidth. For stochastic noise, the well known Kalman filter theory can be applied to estimate the system states. The deterministic treatment of SDGPC does not prevent it from using these results because of the Separation Theorem or Certainty Equivalence Principle [3, pp. 218]. For systems with colored noise, which is more often than not, the optimal Kalman filter can be designed based on the noise model provided that the noise model is known [4, pp. 54]. Unfortunately, the noise model is often unknown and difficult to estimate. Estimation of the "true" system states subject to unknown colored noise poses one of the biggest challenges in process control applications. Thus how to utilize the available results and develop new one, in the framework of SDGPC, is certainly a. topic worth pursuing. 3. Adaptive SDGPC. We only considered adaptive Laguerre filter based SDGPC in the thesis. Since the parameter estimation algorithm has been developed, it would be nice to see an adaptive version of SDGPC based on general transfer function description of systems. 4. Apply the SDGPC algorithm to practical problems. Although initial experiment on an inverted pendulum showed the effectiveness of SDGPC and the associated continuous-time identification algorithm, only large scale industrial applications can be the final judge. 129 Appendix A Stability Results of Receding Horizon Control Appendix A.l and appendix A.2 are based on [4]. A.l The Finite and Infinite Horizon Regulator Given a state-space model of a linear plant where F,G,H have proper dimensions. x(k + 1) = Fx(k) + Gu(k) (A.301a) y(k) = Hx(k) (A.301bThe finite horizon LQ regulator problem can be posed as follows. The performance index: J(N, x(k)) = xT(k + N)P0x(k + N)+ N-l {xT{k + j)QxT{k + j) + uT(k + j)Ru(k + j)} (A.302) The solution to the above optimal LQ problem may be given by iterating the Riccati Difference Equation ( RDE ), Pj+1 = FTPjF - FTPjG(GTPjG + R)~1GTPjF + Q (A.303) j = 0,l,-.-JV-2 from the initial condition PQ and implements the feedback control sequences given by u{k + N-j) = -(GTPj-1G + R)~lGTPj-iFx(k + N - j) (A.304) = Kj-.lX(k + N-j), j = 1,2, - •• N, where Pj is the matrix solution of RDE (A.303). Notice from the control sequence (A.304) that it iterates reversely in time compared with the direction of evolution of the plant (A.301). That is, in order to obtain the current control u(k), PN-I has to be solved first by iterating (A.303). The accumulated cost of (A.302) is given by PN which itself does not appear in the control law, 130 J{N,x(k))F = xT(k)PNx(k) (A.305) Similarly, the infinite horizon LQ regulator problem may be posed as the limiting case of the finite horizon LQ problem (A.302), J(x(k))= lim J(N,x(k)) (A.306) N—»co And the optimal solution can be obtained by iterating (A.303) indefinitely. Under mild assump tions, Pj converges to its limit P^ which is the maximal solution of the Algebraic Riccati Equation ( ARE ), P00 = FTP00F-FTP00G(GTP00G + R) 1GTPCX)F + Q (A.307) And a stationary control law is obtained as u(k) = - (GTPO0G + R)~1GTP00Fx(k) = Kx(k) (A.308) The following theorem regarding the stability property of the infinite horizon LQ control law (A.308) is due to De Souza et al [74]. 131 Theorem A.6 (De Souza et al [74]) Consider an infinite horizon LQ regulator problem with plant (A.301) and performance index (A.306), for the associated ARE, P = FTPF - FTPG(GTPG + R)~1GTPF + Q (A.309) if • [F,G] is stabilizable, • [F, Q1/2] has no unobservable modes on the unit circle, • Q > 0 and R > 0 , then • there exists a unique, maximal, nonnegative definite symmetric solution P. • P is a unique stabilizing solution, i.e. F -G(GTPG + R)~1GTPF (A.310) has all its eigenvalues strictly within the unit circle. The solution P above is called the stabilizing solution of the ARE (A.309). Also note that the matrix (A.310) is the state transition matrix of the closed loop system when the stationary control law (A.308 ) is applied to plant (A.301). Theorem A.6 is the fundamental closed loop stability result for infinite horizon LQ control which will be utilized to prove the stability result of receding horizon control in the following and the stability property of SDGPC thereafter. A.2 The Receding Horizon Regulator From the discussions in appendix A.l, a number of facts about the finite horizon and infinite horizon discrete-time LQ regulator problem are clear. For the finite horizon case, the optimization task with cost function (A.302) is merely to find N control values which, in principle, may be found by finite-dimensional optimization which is referred as the "one shot" algorithm in most model based 132 predictive controllers. The control sequences may also be obtained by iterating the RDE (A.303) explicitiy from Pa to PN-2 using simple linear algebra. The resulting control law in feedback form (A.304) is time-varying even if the plant being controlled is time invariant. By contrast, the infinite horizon problem involves an infinite-dimensional optimization or the solution of an ARE (A.307) which is computationally burdensome especially in adaptive applications. However, the control law of the infinite horizon problem is stationary and have guaranteed stability properties under mild assumptions. Receding horizon control is one method proposed to inherit the simplicity of the finite horizon LQ method while addressing an infinite horizon implementation and preserving the time-invariance of the infinite horizon feedback. In this formulation only the first element u(k) in the control sequences u(k),u(k + 1), • • • u(k + N — 1) is applied to the plant at time k and at time k + 1 the first control u(k + 1) in the control sequences u(k + l),u(k + 2), • • • u(k + N) is applied and so on. In terms of the finite horizon feedback control law (A.304), one has for the receding horizon strategy u(k) = K~N-.ix(k) -i ^ (A311> = - {GTPN-1G + R) GTPN-XFx{k) which is a stationary control law. Note that there is still no word having been said about the stability of control law (A.304). In fact, receding horizon strategy does not guarantee stability itself. Motivated by the facts that the infinite horizon LQ control law has guaranteed stabilizing property and there are strong similarities between receding horizon control law (A.304) and infinite horizon LQ control law (A.308), i.e. both are stationary and have the same form, one has enough reason to wonder if the stability result of infinite horizon LQ control summarized as Theorem A.6 could be of any help to the stability problem of receding horizon control. For this we go to the important work of Bitmead et al [4]. Consider the RDE (A.303) Pj+i = FTPjF - FTPjG(GTPjG + R)~1GTPjF + Q (A.312) j = 0,l,---JV-2 define QJ = Q-{PJ+I-PJ) (A.313) 133 , the RDE (A.312) will have a form of an ARE, Pj = FTPjF - FTPjG(GTPjG + R)~1GTPjF + Qj (A.314) which is called Fake Algebraic Riccati Equation ( FARE ) [4]. From Theorem A.6, the stability property of the solution of the above FARE can be immediately established as follows. Theorem A.7 (Bitmead et al. [4, pp. 87] ) Consider the FARE (A.314) or (A.313) defining the matrix Qj. If Qj > 0, R > 0, [F,G] is stabilizable, -1/2 F,Qj is detectable, thenPj is stabilizing, i.e. Fj = F-G (GTPj G + R)~1GTPjF (A.315) has all its eigenvalues strictly within the unit circle. Clearly, if the conditions in Theorem A.7 are met for j = N - 1, then the receding horizon control law (A.304) will be stabilizing. However, further work needs to be done to relate the design parameters, i.e. the matrices Po,Q, R in the performance index (A.302), to the conditions in Theorem A.7. The following results from [4] can serve this purpose. lemma A.3 (Bitmead et al. [4, pp. 88] ) r 1/2 Given two nonnegative definite symmetric matrices Q\ and Qi satisfying Q\ < Q2 then F, Q^ detectable implies F,Qf detectable. The following corollary [5] which is an immediate result of lemma A.3 tells that if the solution of the RDE is decreasing at time j then the closed loop state transition matrix of (A.315) is stable. Corollary A.1 ( Bitmead et al. [5] ) r 1 /21 If the RDE with [F, G] stabilizable, F, Qy' detectable and if Pj in (A.312) is non-increasing at j, i.e. Pj+i < Pj, then Fj defined by (A.315) is stable. 134 Also from [5], we have the following theorem regarding the monotonicity properties of the solution of the RDE (A.312). Theorem A.8 ( Bitmead et al. [5]) If the nonnegative definite solution Pj of the RDE (A.312) is monotonically non-increasing at one time, i.e. Pj+\ < Pj for some j, then Pj is monotonically non-increasing for all subsequent times, Pj+k+i < Pj+k, far all k > 0. The following result is immediate by combining Corollary A.1 and Theorem A.8. Theorem A.9 (Bitmead et al. [4, pp. 90] ) Consider the RDE (A.312), if • [F,G] is stabilizable [F, Qll2] is detectable • Pj+i < Pj for some j then Fk given by (A.315) with Pk is stable for all k > j. As an immediate consequence of Theorem A.9, we see that if Po in the design of receding horizon controller is selected in such a way that one iteration of the RDE will result in Pi < Po, then we have QQ = Q - (P\ - PQ) > Q and Qj > Q for any subsequent j > 0, this implies that Fj given by (A.315) is stable for any j > 0. A clever choice of Po which will guarantee the monotonically non-increasing solution of RDE is to let Po = oo as first proposed by Kwon and Pearson [42] albeit in a very different framework. The result can be summarized as follows: 135 Theorem A.10 ( Kwon et al. [42] and Bitmead et al. [4, pp. 97] ) Consider system x(k + 1) = Fx(k) + Gu(k) (A.316a) y(k) = Hx(k) (A.316b) and the associated receding horizon control problem, i.e. minimize performance index J(N,x(k)) = JV-l (A.317) Y, {xT(k + j)QxT(k + j) + uT(k + j)Ru(k + j)} subject to final state constraint x(k + N) = 0 (A.318) assume Q > 0, R > 0, F is nonsingular and [F, G] is controllable, [F, Q] is observable, then the optimal solution exists and stabilizes the system (A.316) whenever N > n, where n is the dimension of system (A.316). The nonsingularity condition of F was removed in a recent paper by Chisci and Mosca [10]. The following corollary is a natural consequence of Theorem A. 10 using the argument given by Demircioglu et al. [16]. Corollary A.2 For system described by equation (A.316) with performance index there exists a positive number 7 such that for Po > 71, the closed loop system under the control law obtained by minimizing (A.319) is also stable. 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