)) to be small. The conflict is resolved by relaxing each of these specifications such that they apply only at the appropriate dynamical bandwidths. In the design of feedback controllers for physical systems it is common to find that the robustness requirements are important only at the high dynamical frequencies uo where the small model gain and large model uncer-tainty combine such that the relative uncertainty is greater than 100%. One then proceeds by designing K(z) conservatively at high dynamical frequencies uo with o (K(ew)) small. Disturbance attenuation is then achieved by designing K(z) aggressively at low frequen-cies UJ with o (K(e1\")) large. 1 These concepts may be found in a variety of sources and the interested reader is referred to [16, 44, 56, 66] and references therein. Two Dimensional Loop Shaping 76 For the relevant bandwidths, one can obtain open-loop approximations for the closed-loop specifications listed above, 1. Disturbance attenuation requires o(GK(elU1)) to be large where \u00a3 ( G ( e m ) ) is large, typically at low frequencies to. 2. Limited control action requires o (Kfe1\")) to be small where o (G(el<\")) is small, typically at high frequencies to. 3. Robust stability for additive plant uncertainty GP = G + SGA requires o (K(elu)) to be small where a{G{eiu)) \u00ab a(8GA(eiu)) and\/or a(G(eiu)) < a(8GA(ei(V)), typically at high frequencies to. 4. Robust stability for multiplicative plant uncertainty GP = (I + 8GM)G requires a(GK(eiw)) to be small where a ( J G M ^ ) ) ~ 1 and\/or a {8GM{e^)) > 1, typically at high frequencies u>. The open-loop specifications 1-4 may be summarized as the requirement that the loop gain a_(GK) be large at low frequencies and that a(GK) be small at high frequencies. These requirements are illustrated graphically in Figure 4.1, where the feedback K(z) must be designed such that the singular values a(GK) and a(GK) avoid the regions indicated. The design procedure is complicated by the fact that the singular value loop shaping must be performed with an internally stabilizing K(z). The requirement of closed-loop internal stability requires consideration of the phase of the system via the eigenvalues of GK and places limits on the achievable loop shape [16]. For example, time delay does not affect the singular value spectrum of a plant, since aj(z~2DG) = Oj(z~DG) for all singular values j. However it is well-known that the achievable performance bandwidth of a closed-loop system with the plant (^(z) = z~2DG is roughly half of the bandwidth that can be achieved for a closed-loop system with the plant G\\(z) = z~DG. The specifications on the loop shape must consider the properties of the plant G(z) and the capabilities of the controller K(z). Two Dimensional Loop Shaping 77 Figure 4.1: Traditional multivariable open-loop singular value shaping. The performance requirement places a lower bound on o(GK) for low frequencies u) < UJ\\. The safety requirements (robust stability, limited control action, etc.) place an upper bound on o(GK) for high frequencies OJ > ojh-As the requirement of internal stability is difficult to handle when shaping the open-loop singular values a(GK), o(GK), o(K), o(K) of the system, several loop shaping methodologies have emerged to handle this problem. These include the L Q G \/ L T R tech-niques in which either (\/ - GK)~\\ GK{I - GK)~\\ or (\/ - KG)'1, KG(I - KG)-1, are shaped through the design of performance weights in an L Q G optimization problem in which nominal stability is guaranteed [16]. The more sophisticated % 2 and Hoc mixed sensitivity techniques allow all relevant closed-loop transfer functions to be included directly in an optimization problem. As with the L T R methods, the loop shape is modified through the use of weighting functions in the optimization. With the added flexibility of these methods comes the risk of over-specification. One must still respect the trade off (Figure 4.1) required when designing Two Dimensional Loop Shaping 78 the optimization weights [8, 56, 66]. Finally the Hoo loop shaping [44, 66] technique performs controller design in two steps. The first step is to shape the open-loop singular values according to rules such as outlined above and in Figure 4.1. The second step uses Tioo synthesis to robustly stabilize the closed-loop. 4.2 Two Dimensional Frequency Domain Specifications The rules described above for the shaping of MIMO loops are quite practical and apply to a wide variety of control design problems. However the general multivariable design problem as described above is not achievable for many spatially distributed problems. It can be seen from Figure 4.1 that the control loop's performance specifications must be satisfied for all singular values Oj(GK) for j G {l , . . . ,n}. However, there exist many industrial examples of spatially-distributed control systems in which the process model G(z) is very ill-conditioned. The majority of cross-directional paper machine control systems have Oj (G(eiu>)) \u00ab 0 for several singular values, even at the steady-state with UJ = 0. See Figure 2.4 for such an example. The model uncertainty compounds this problem and it is not uncommon to find industrial applications in which the relative uncertainty is larger than 100%, with the additive uncertainty Oj (G(eiu)) < o (\u00a3GU(e\"'')) at UJ = 0, for over half of the n singular values. In some multivariable control applications, the ill-conditioning of a plant transfer ma-trix G(z) is an indication that the process has been poorly designed, and that efforts may be better spent on process redesign rather than feedback compensation [55, 56]. However, for cross-directional control applications, it has been shown that if a process is well-designed in the sense that there are enough actuators to control the low spatial fre-quency components of the error, then it is likely the process model G(z) is ill-conditioned with vanishingly small gain for input directions corresponding to high spatial frequencies [19, 34]. The modification of the traditional MIMO loop shaping requirements of the previous section for ill-conditioned spatially-distributed processes may be interpreted as the relax-Two Dimensional Loop Shaping 79 ation of the performance requirement for some of the singular values (corresponding to high spatial frequency modes Uj) at low dynamical frequency OJ [62]. It was shown in Chapter 3, that a symmetric circulant structure for feedback con-trollers K(z) in (2.9) is sufficient for a wide variety of control applications for spatially-distributed systems modelled by symmetric circulant transfer matrices G(z) in (2.1). Restricting the feedback controller K(z) in (2.9) to be symmetric circulant was shown in Chapter 3 to reduce the large design problem to that of designing a family of n SISO controllers one for each spatial frequency Vj \u20ac {y\\,..., vn}. In addition to the design, the analysis of the closed-loop system is also simplified to consideration of the family of SISO problems. Repeated here are the expressions for the open-loop plant (analogous to (2.1)) y(uj, z) = jftuj, Z)U(UJ,Z) (4.1) and the feedback controller (analogous to (2.9)) u(vj1z) = k(vj,z)v(vj,z), (4.2) where the feedback signal is the deviation from the reference signal v(fj,z) \u2014 y(uj,z) \u2014 f(uj,z). The notation of Chapter 3 has been used and the circulant symmetric transfer matrix G(z) in (2.1) is decoupled by the real Fourier matrix F, such that FG(z)FT = diag{y(^i, z),..., g(un, z)}. The corresponding definition is used for k(uj, z) in (4.2). Since, as discussed in Section 3.7, the singular values of a circulant symmetric transfer matrix are given by magnitude of SISO transfer functions such as (4.1), (4.2), then the multivariable loop shaping design specifications listed in Section 4.1 can be restated in terms of the individual feedback loops. The design specifications for these systems are then rewritten in terms of their spatial and dynamical frequencies, 1. Disturbance attenuation requires 1\/[1 \u2014 gk(vj,exw)] to be small. 2. Limited control action requires fc(i\/j,ew)\/[l \u2014 gk(uj,eiw)] to be small. Two Dimensional Loop Shaping 80 3. Robust stability for additive plant uncertainty GP = G + 8GA requires to be small. 4. Robust stability for multiplicative plant uncertainty GP = (I + 8GM)G requires gkfae^\/ll-gkfae*\")] to be small. As with the MIMO case in section 4.1, the closed-loop specifications for the decoupled system are in conflict and a trade off is required between performance and robustness. The design problem is to maintain the best performance possible without compromising the robustness margins of the closed-loop. Qualitatively speaking, the performance specification may be satisfied, for a closed-loop stable system, by designing \\k(uj,eiu)\\ to be large at those frequencies {VJ,LO} where \\g(vj, e\"\")| is large and the relative uncertainty is small. The robustness of the closed-loop is a matter of designing \\k(uj, eiu)\\ to be small at those spatial and dynamical frequencies {VJ,LO} where \\g(uj,eiM)\\ is small and\/or the relative uncertainty is large. In this way, it will be shown that one can accomodate the large number of ill-conditioned spatially distributed applications exhibiting gain roll-off for high spatial frequencies vi as well as high dynamical frequencies LO. With these trade-offs in mind, the closed-loop performance specifications above may be approximated (as with the MIMO case) in terms of open-loop design objectives. 1. Disturbance attenuation requires \\gk(vj, elu)\\ to be large where |(7(i\/j,elw)| is large, typically at low frequencies v and LO. 2. Limited control action requires |fc(i\/,-, e\"\")| to be small where \\g(uj,elu,)\\ is small, typically at high frequencies v and LO. 3. Robust stability for additive plant uncertainty GP = G + 8GA requires \\k(vj, e\u2122)\\ to be small where \\g(vj, eiuJ)\\ \u00ab a(8GA) and\/or \\g(uj,ei,J)\\ < o(8GA), typically at high frequencies v and LO. 4. Robust stability for multiplicative plant uncertainty GP \u2014 (I + 8GM)G requires \\gk(yj,eiu)\\ to be small where a(8GM) ~ 1 and\/or a(8GM) > 1, typically at high frequencies LO. Two Dimensional Loop Shaping 81 70 >,.\u2022\u2022\u2022\u2022\"': 6(K 50 -0 0 Figure 4.2: The analogous open-loop uw-surface shaping. Note that contrary to tradi-tional loop shaping, the performance constraint is not selected to cover all singular values j G {1 , . . . , n}. The roll-off of the gain of the plant g(vj, z) for high spatial frequencies re-places a limit on the spatial bandwidth of a closed-loop system. The open-loop specifications 1-4 above are analogous to the multivariable design re-quirements discussed in Section 4.1. The two dimensional design specifications are illus-trated in Figure 4.2. The lower bound on the loop gain for performance at low frequencies is illustrated as applying to spatial frequencies v < 0.6 and dynamical frequencies to < 0.6. The upper bound on the loop gain that applies at high spatial and dynamical frequencies {UJ, to} (analogous to Figure 4.1) is omitted for legibility. In order to better illustrate the two dimensional frequency domain, contour plots in {vj,to} are introduced. In an analogy to Figure 4.1, the relevant contours of the two dimensional loop gain is to be plotted on the same diagram as the design constraints indicated in requirements 1-4 above. The two dimensional performance specification requires that the loop gain lie above a Two Dimensional Loop Shaping 82 performance boundary \\g(vj,v>i (4.3) for low spatial and dynamical frequencies {VJ,OJ} G fi\/. In order to satisfy the requirement of robust stability, the loop gain is designed to lie below a robustness bound, \\g(vj,ei\u00bb)~k(vj,ei\u00bb)\\)k(vj,e\"\")| = u>i and \\g(i\/j,eiu)k(vj,eiM)\\ = Wh avoid the shaded areas fi\/ and fi^. The performance condition (4.3) is satisfied if the wi contour does not intersect the set fi\/. The robustness condition (4.4) is satisfied if the wh contour does not intersect the set fi^. Figure 4.3 illustrates an aggressive design satisfying the performance condition (4.3) but not the robustness condition (4.4). Figure 4.4 illustrates an conservative design satisfying the robustness condition (4.4) but not the performance condition (4.3). Figure 4.5 illustrates a design which has successfully traded off the conflicting requirements. As with traditional loop shaping, these design specifications apply only to closed-loop systems that are nominally stable - a property only accessible through the eigenvalues of the system. The two dimensional loop shaping approach for these symmetric circulant systems has the advantage that the eigenvalues are very closely related to the singu-lar values of the closed-loop. Section 3.7 showed that the eigenvalues of GK(el\") are g(vj, eiu,)k(uj, eiu) for Vj G . . . , fn}, while the singular values of GK(exu) are given by \\g(vj,eiw)k(i>j,ei\")\\. This feature allows one to consider the loop shaping design of the large multivariable system in terms of the SISO problems defined by g(uj, z) and k(uj, z) for Vj G {ui,..., vn}. The multivariable closed-loop internal stability of G(z) and K(z) is ensured by the internal stability of all n pairs of transfer functions g(uj,z) and k(uj,z), while the closed-loop performance and robustness are analyzed by applying the above Two Dimensional Loop Shaping 83 Figure 4.3: The LOV contour plot shows that this design was too aggressive. The robustness condition (4.4) is not satisfied for all {VJ, LO} G \u00a32\/,, as the \\g(vj, ew)k(vj, e\"\")| = u>h contour intersects \u00a32\/,. specifications to the open-loop singular values |p(^j,e lu,)fc(z^-,eIu;)| and \\k(vj,ellJ)\\. Chapter 5 contains a design example which relies on this close relationship between closed-loop eigenvalues and singular values to ensure stability while shaping the closed-loop transfer matrices in spatial and dynamical frequencies Vj and LO. 4.3 Controller Spatial Order Reduction with Stability Requirement In Chapter 3 it was shown that the two-dimensional frequency domain is an appropriate domain for the analysis and design of symmetric circulant feedback controllers for sym-metric circulant processes. A wide range of practical design specifications in terms of performance and robustness may be specified in the LOV domain. In Chapter 2 it was stated that the goal was to design localized feedback controllers Two Dimensional Loop Shaping 84 V 0 0 CO 71 Figure 4.4: The OJU contour plot shows that this design was too conservative. The perfor-mance condition (4.3) is not satisfied for all {VJ,OJ} \u00a3 fi\/, as the \\g(uj,elu>)k(i\/j,ezu)\\ = wi contour intersects f2\/. K(z) = [I + \u00a3(.2:)]-1C7(z) in which the factors C(z) and S(z) are band-diagonal Toeplitz symmetric transfer matrices. As discussed in Section 3.1, the first step in the design of the band-diagonal Toeplitz system is the design of banded symmetric circulant matrices C{z) and S(z). As shown in Chapter 3, the goal of the spatial frequency domain design techniques is the synthesis of a family of n single variable controllers k(Pj,z) \u2014 c(i\/j,z)\/[l + s(fj,z)] are designed, one for each Uj \u00a3 {v\\,... ,vn}. However, this frequency-by-frequency de-sign generally results in a multivariable feedback controller K(z) defined by full (i.e. not banded) transfer matrix factors C(z) = FTdiag{c(ui,z),... ,c(un,z)}F and S(z) = FTdiag{s(i>i, z),..., s(vn, z)}F, in spite of the localization of the process [4]. The de-velopment of optimization-based problem statements which directly synthesize an imple-mentable feedback controller of low spatial order is currently the subject of much active Two Dimensional Loop Shaping 85 TC V 0 Figure 4.5: The uv contour plot illustrates a design which has successfully traded off the conflicting requirements. research, but is still essentially an open problem [4, 11, 46]. In this work, a practical approach to low spatial order controller design is followed and is analogous to that used in the design of low dynamical order controllers [66]. One first synthesizes a high-order feedback controller to satisfy performance and robustness requirements and then approximates it with a lower order feedback controller. In this section, it is assumed that we are given an internally stabilizing symmetric circulant feedback controller K(z) which satisfies all design requirements (possibly result-ing from a two-dimensional loop shaping design), but whose factors C(z) and S(z) are full symmetric circulant matrices. A spatial order reduction is then achieved by approx-imating the factors C(z) and S(z) with banded symmetric circulant matrices Ci(z) and Si(z). A simple approximation technique is proposed, and a condition is derived for which closed-loop stability of the system with Kt(z) = [J + Si(z)]~lCi(z) is guaranteed. First, write the factors of the high-order, high-performance controller K(z) = [7 + Two Dimensional Loop Shaping 86 S{z))lC(z) as, toeplitz{ci, c 2 , . . . , c n \/ 2 , c ( n + 2 ) \/ 2 , c n \/ 2 , . . . , c2} n even G(zj = < (4.5) toeplitz{c!, c 2, : . . , C ( n + 1 ) \/ 2 , c ( 7 l ; +i ) \/2 , \u2022 \u2022 \u2022, c2} n odd where each element Ci(z) is generally non-zero (the argument z has been supressed in (4.5) for legibility considerations). A similar definition holds for the denominator transfer matrix S(z). Next, if the full-matrix C(z) and S(z) in (4.5) are approximated with banded sym-metric circulant transfer matrices by simply truncating the elements, Ci(z) = toeplitz{ci, c 2 , . . . , Cnc, 0, . . . , 0, c \u201e c , . . . , c2} Si(z) = toeplitz{si, s2,.. . , sn >, 0, . . . , 0, s n \u201e , . . . , s2} (4.6) where again the argument z has been supressed in the elements in (4.6) to preserve legibility. The controller perturbation introduced by the spatial order reduction is then given by, 6Ct(z) = C(z)-Q(z) _ toeplitz{0,..., 0, c n c + i , . . . , c n \/ 2 , c ( n + 2 ) \/ 2 , cn\/2,..., c n c + 1 , 0 , . . . , 0} n even toeplitz{0,..., 0, c n c + 1 , . . . , c ( n + i ) \/ 2 , C ( n + 1 ) \/ 2 , . . . , c \u201e c + i , 0, . . . , 0} n odd (4.7) a similar definition is used for 8Si(z) = S(z) \u2014 Si(z). The following theorem presents a sufficient condition to be satisfied by the eliminated elements ck{z) for k > nc + 1 and Si(z) for ns + 1 in order to guarantee the preservation of closed-loop stability for the localized controller K~i(z) \u2014 [I + Si(z)]~1Ci(z). Two Dimensional Loop Shaping 87 Theorem 3 (Stability-Preserving Controller Localization) \/ \/ the symmetric cir-culant controller K(z) = [I 4- 5 , (z)] _ 1 C(z) in (4-5) is an internally stabilizing controller for a symmetric circulant plant G(z), normalized such that WG^z)^ \u2014 1, then the closed-loop system with the symmetric circulant localized controller Ki{z) = [I + 5\/(z)]_1C;(2;) in (4-6) is stable if the truncated elements in (4.-7) are small enough such that p p 1 Yl \\ck(z)\\+ Yl \\8i(z)\\<2\\1 + 5(vi>z)~9(ui>z)\u00a3(uJ>z) fc=nc+l i=n,+l (4.8) for z = e\"\" and all OJ G [\u20147r, U] and Vj G {v\\,..., vn}. The term p = (n + l)\/2 ifn is odd, and p = (n + 2)\/2 if n is even. Proof. The small gain theorem is used to determine a sufficient condition for closed-loop stability in terms of the controller perturbations 8Ci(z) and 5Si(z) defined by (4.7) [66], -G [6Ci 6St}-I (I \u2014 KG)~X(I + 5 ) _ 1 < 1 (4.9) Since each of these transfer matrices in (4.9) is symmetric circulant, then the real unitary Fourier matrix F in (A.4), allows the condition (4.9) to be written in terms of the diagonal system by pre- and post-multiplication (i.e. F(-)FT ). Then, Sctiuj, e\u2122) \u2022 g(uj,e*\") + \u00ab,(!\/,\u2022,e i w)| < |1 + 3(1\/,-,c**) - c(uj, e^vj, e*\")| (4.10) for all Vj G {vx,...,vn} and OJ G [\u2014IT,TT]. Then using the fact that \\g(vj,e\"\")| < 1, we can write the left-hand-side of (4.10) as | - SSiiuj^) \u2022 S(i\/ i,e t o) + Wifa-.e*\")! < {Sc^e^l + | < ^ ; , O I (4.11) Then using the result in [13], the singular values of a symmetric circulant matrix 8Ci(z) = Two Dimensional Loop Shaping 88 C(z) \u2014 Ci(z) defined by (4.7), are bounded by maX\\Scl(uj,z)\\=a(8Cl(z)) = o(C(z)-Cl(z))<2 \u00a3 \\ck(z)\\ (4.12) 3 fe=nc-)-l where p = (n+l)\/2 if n is odd, andp = (n+2)\/2 if n is even. Writing a similar expression for SSi(z) allows to write, \\8cl{vj^)\\ + \\8sl{uj^)\\<2 \u00a3 \\ck(z)\\+2 \u00a3 \\st(z)\\ (4.13) fc=nc+l i=n3+l Combining results (4.10), (4.11), and (4.13) completes the proof. \u2022 Remarks: 1. The RHS of condition (4.8) is defined in the LOV domain and is calculated using the properties of the plant and the full matrix non-localized controller k(vj, z). The LHS is given directly in terms of the truncated matrix elements in the spatial domain. 2. Since the spatial order reduction only guarantees nominal stability of the closed-loop (4.8), the designer is still obliged to re-calculate the LOV components of the system with the localized control Ki(z), and verify that the low-order controller satisfies the design requirements. 3. The industrial rule of thumb is that the elements of the controller factors Ci(z) and Si(z) in (4.5) are less important for larger i. This has recently been justified theoretically in [4] in which it is shown that a quadratic optimal problem statement for a spatially localized process will synthesize a feedback controller K(z) for which the gain of its elements decreased exponentially as a function of distance from the main diagonal. 4. Theorem 3 is a special case of the more general problem of analyzing controller perturbations which may be due to uncertainty in its implementation or due to deliberate reduction by the designer. This result can easily be expanded to include dynamical order reduction of C(z) and S(z) and techniques for stable order reduction may be found in [27, 66]. Two Dimensional Loop Shaping 89 4.4 Two Dimensional Loop Shaping Design Procedure In this section, a procedure is presented for the design of a feedback controller K(z) in (2.9) to satisfy performance and robustness requirements, in which the feedback is localized such that each actuator's input is restricted to depend only upon information from nearby sensors and actuators. However, it is not straightforward to directly design a localized controller K(z), al-though currently there is much interest in the topic [4, 10, 46]. The proposed design procedure is more closely related to the traditional loop shaping procedure. The con-trollers produced by traditional loop shaping techniques (especially through Ti2 and Hoo synthesis) result in controllers that have orders comparable to the generalized plant. But for practical reasons, a lower order is favored over a high order. The design often proceeds in two steps. First one synthesizes a high-performance, high order controller through a loop shaping procedure, and subsequently reduce the design to obtain a low order con-troller [56, 66]. The two dimensional loop shaping design proceeds in an analogous fashion. First, a high-performance controller is designed to satisfy the performance and robustness spec-ifications described in Section 4.2 above. This controller will generally not be localized and the second step of the design is to reduce the spatial distribution of the feedback controller by approximating it with a localized controller that maintains closed-loop sta-bility (Section 4.3). The localized controller is required to not significantly degrade the performance of the high-order controller obtained from the loop shaping. Theorems 1 and 2 show that in terms of performance and robustness it is sufficient to consider symmetric circulant feedback controllers K(z) for many practical problem statements concerning symmetric circulant plants G(z). Two dimensional loop shaping is used to trade off the performance and robustness specifications as described in section 4.2 and results in a family of feedback controllers k(vj, z) one for each Uj G {v\\,..., un}. Each member of the family of controllers is factored as k(uj, z) \u2014 C(UJ, z)\/[l + S(VJ, z)] for each Vj G {ux,..., vn}. The multivariable feedback controller K(z) = [I + S(z)]~1C(z) is obtained by computing the factors C(z) = FTdiag{c(ui,z),...,c(un,z)}F and S(z) = Two Dimensional Loop Shaping 90 F rdiag{s( tv 1, z),..., S(i\/\u201e, z)}F. Localized control was defined by restricting each actuator's input to depend only upon information from nearby sensors and actuators. As was discussed in Section 2.2 localized control is preferrable over non-localized control for many practical reasons related to the implementation of the control law. However, the loop shaping design typically results in a controller K(z) that is not localized. In this localized controller K(z) = [\/ + S()) and Oj(GK(elw)) while maintaining internal stability of the closed-loop. 2. Diagonalize the problem. Using Chapter 3 write G(z) and K(z) as the decoupled family g(fj, z) and k(vj, z) for Vj G {y\\,..., un}. 3. Controller synthesis. Shape the closed-loop transfer matrices by manipulat-ing the internally stabilizing k(i>j,z) such that the open-loop approximations \\g(v},eiu)k(vj,eiw)\\ and \\k(i\/j,elu')\\ satisfy the specifications of Section 4.2. 4. Factor the family of SISO controllers such that k(vj, z) = c(i\/j, z)\/[l + s(fj, z)] and construct transfer matrices C(z) := FTdiag{c(ui, z),..., c(un, z)}F and S(z) := FTdiag{s(u1, z),..., S(vn, z)}F. 5. Approximate the full matrix factors C(z) and S(z) with banded Ci(z) and Si(z) such that Theorem 3 is satisfied. Verify that the symmetric circulant controller Ki(z) satisfies the loop shaping design requirements. 6. Check the associated Toeplitz system satisfies condition (3.6) for internal stability, and implement the band-diagonal Toeplitz factors Kt(z) = [I + St{z)]~lCt{z). The second option, described below, is a closed-loop shaping procedure in which the performance weighting functions in an optimization problem are used to shape the closed-loop singular values. The 'controller synthesis' step can potentially be done in a number of ways depending on the specific problem requirements. Examples which consider some or all of these issues include generalized minimum variance predictive control [59, 61], mixed-sensitivity H2 synthesis [4, 60], mixed-sensitivity Hoo synthesis [4], Hoo loop shaping [47], or \/\/-synthesis [37]. The advantage over open-loop shaping is that these optimization techniques have been developed to guarantee an internally stable closed-loop system. Two Dimensional Loop Shaping 92 There is no need to select an appropriate controller structure. The controller design is achieved via careful shaping of the optimization weights. The remainder of this work is dedicated to the industrial control problem and uses the open-loop shaping design approach described above. Therefore the closed-loop shaping approach is not expanded on further. An example of its application to a mixed-sensitivity 7^ 2 loop shaping design procedure is described in [60]. Closed-Loop Shaping 1. Set up the problem. Select appropriate inputs and outputs and write the general-ized plant P{z) in terms of G{z) and weighting functions. 2. Diagonalize the problem. Using Chapter 3 write the generalized transfer matrix ^i(P(z), K{zj) as the decoupled family Fi(p{vj, z), k(fj, z)) for Vj e {\/vx,..., un}. 3. Controller synthesis. Shape the closed-loop transfer matrices to satisfy the specifications of Section 4.2, by adjusting the relevant performance weights in Ti{$>(yj,z), k(fj, z)) -> min, for each Uj 6 {ux,..., un}. 4. Factor the family of SISO controllers such that k(fj, z) = c(i\/j, z)\/[l + S(VJ, z)] and construct transfer matrices C(z) :\u2014 FTdiag{c(vi,z),... ,c(vn,z)}F and S(z) :\u2014 F T diag{s(\/Vi, z),..., 5(i\/\u201e, z)}F. 5. Approximate the full matrix factors C(z) and S(z) with banded Ci(z) and Si(z) such that Theorem 3 is satisfied. Verify that the symmetric circulant controller Ki(z) satisfies the loop shaping design requirements. 6. Check the associated Toeplitz system (3.5) satisfies condition (3.6) for inter-nal stability, and implement the band-diagonal Toeplitz factors Kt{z) \u2014 [I + St(z)]-lCt(z). Chapter 5 Industrial Paper Machine Control This chapter presents many of the issues involved when applying the loop shaping design approach developed in Chapter 4 to the control of industrial paper machine processes as described in Chapter 1. Section 5.1 presents an overview of the functioning of a prototype software tool devel-oped for the tuning of cross-directional paper making processes. The process is modelled by existing software [32] as the linear transfer matrix G(z) described in (2.1)-(2.8). The performance and robustness specifications in (2.24) and (2.27) are diagonalized as in Chapter 3 and then restated in terms of the loop shaping criteria in Chapter 4. The function of the tuning tool is to generate parameters of the industrial cross-directional controller K(z) in Figure 2.3 and (2.9), (2.15) according to the principles of the open-loop shaping procedure described in Section 4.4. Section 5.2 describes the inaugural field trial of the prototype tuning tool for tuning the feedback controller for the basis weight of a newsprint machine in a Canadian paper mill. The main steps in the execution of the testing procedure of the controller tuning tool are presented. The success of this field test provides an industrial validation of the controller analysis and design concepts presented in Chapters 3 and 4. 5.1 Prototype Tuning Tool A prototype tuning tool has been developed in Matlab for the industrial cross-directional control problem. Currently there exists a software tool (described in [32]) that identifies the parameters of the open-loop process model G(z) in (2.7)-(2.8). However, there existed no tools that allow the design of the free parameters of the industrial feedback controller K(z) in (2.15)-(2.19). The industrial problem requirements are such that the tuning tool 93 Industrial Paper Machine Control 94 must be capable of using the knowledge provided by the identified process model G(z) in order to generate parameters of the controller K(z) such that acceptable closed-loop performance is achieved for any of the CD processes in Section 1.1. Above all, an acceptable controller design requires guarantees of stability for the closed-loop. In an industrial setting, this means, not only the nominal stability of the closed-loop is satisfied, but that some margin for model uncertainty has been allowed. Secondary to the requirement of closed-loop stability is the performance specification in which the feedback controller should counteract the effect of the disturbances on the pa-per sheet. This section provides an overview of the functioning of the prototype tuning tool. Details of its operation may be found in Section 5.2.1, which presents the data from the tuning tool's inaugural field trial. The tuning tool relies on the two dimensional loop shaping concepts developed in Chapters 3 and 4. Following a successful model identification session, the parameters of the process model G(z) are available and controller design may begin. 1. The variables of the tuning tool are first initialized with the identified parameters of the process model G(z) defined in (2.7)-(2.8). 2. The circulant extension of the process model is constructed (see Section 3.1), and the process model is diagonalized such that subsequent design may proceed with the family of SISO process models g(uj, z), one for each v3- G {i^,..., un}, where n is the number of actuators in the process. 3. The tuning tool calculates default values for the parameters of the diagonalized feedback controller k(uj,z). These default values are based on the diagonalized process model g(fj,z), and are quite conservative. 4. The designer is presented with several tuning 'knobs' through which the performance and robustness of the closed-loop design may be accessed. Section 5.2.3 explains the functioning of the tuning knobs in some detail. 5. (Automated) The tuning tool then calculates parameters for the controller k(uj,z) based on the process model g(i\/j,z), and the positions of the tuning knobs. This Industrial Paper Machine Control 95 calculation is performed such that the (user-specified) robust stability margin is achieved (as defined by the additive unstructured perturbation in Section 2.4). 6. (Automated) The tuning tool then automatically forms the multivariable controller K(z), from an inverse Fourier transform of the family of SISO controllers k(vj,z), and truncates the high-order elements in order to obtain a (user-specified) spatial order, as described in Section 4.3. 7. The user evaluates the design based on the trade-off between the performance, robustness, and spatial order of the controller. If the design is unacceptable, then return to Step 4. 8. Following the completion of a successful design in terms of the spatial frequency components of the symmetric circulant extension, the software then performs a final stability check on the 'true' truncated Toeplitz system (see Section 3.1). 9. The tuning tool then saves the tuning parameters for the spatial filter, the Dahlin controller, and the actuator profile smoothing into a file tune.mat. Following the successful generation of the file tune.mat in Step 9 above, the designer is then free to implement the generated controller tuning parameters. As the prototype tuning tool exists only as Matlab m-files, the procedure for this implementation is cur-rently 'manual' (i.e. the designer must walk over to the operator station and type the numbers in by hand). The following section presents a demonstration of the procedure for the tuning of a CD controller for an industrial paper machine process in a working paper mill. More detail for the inner workings of each of the software-implemented operations of the controller design will be presented. 5.2 Field Test: Consistency Profiling for Newsprint This section describes the first field trial of the prototype tuning tool described in Section 5.1. The purpose of such a trial is to begin the validation procedure for the controller analysis and design techniques introduced in Chapters 3 and 4. Industrial Paper Machine Control 96 The inaugural test site for the prototype tuning tool was selected to be a Canadian paper mill producing newsprint. As described in Chapter 1, newsprint is a lightweight paper product and mill in question was producing 45gsm (grams per square metre) paper for the duration of our site visit. The testing of the tuning tool was limited to the design of the control for the basis weight profile only. This paper machine uses the consistency profiling actuators (see Section 1.1.1) to flatten the basis weight of the produced paper sheet. This particular machine has n \u2014 226 consistency profiling actuators spaced on xa = 35mm centres and distributed across the 7.91m wide paper sheet. The actuators change the weight profile by injecting low consistency Whitewater into the pulp slurry as it exits the headbox. An increase in the flow of water injected by an actuator reduces the local concentration of pulp fibres and thus locally reduces the basis weight. A commonly-occurring form of closed-loop instability in industrial CD control occurs as a slowly developing steady-state actuator profile of a very high spatial frequency [36, 65]. This phenomenon is well known to papermakers and is referred to as actuator picketing, due to the fact that the actuators profile slowly develops a steady-state 'picket-fence' appearance. In the language of Chapters 3 and 4 this signal has a high spatial frequency v and a low dynamical frequency OJ. It is caused by the application of a large control signal in a low gain direction of the process that is swamped by the model uncertainty, typically at high spatial frequency v. Cross-directional controllers with large gain in the direction of high spatial frequency modes result from the application of a controller design rule-of-thumb without due con-sideration of the process. Usually feedback control is expected to remove the steady-state error from a closed-loop process. It is well-known that this may be accomplished by de-signing a feedback controller K(z) with integral action [56]. A multivariable controller with integral action has infinite gain in all directions at steady-state OJ = 0. However, the closed-loop stability of such a configuration cannot be guaranteed for the majority of industrial CD control processes. It was discussed in Section 2.4 that the sign of the gain of the C D process is uncertain at certain input directions. In Chapter 4 it was stated that robust stability requires the loop gain to be small at those spatial and dynamical frequen-cies {VJ,OJ} for which the gain of the process \\g{y^eiw)\\ is small. Figure 5.2 illustrates Industrial Paper Machine Control 97 the gain roll-off of g(fj, z) for the newsprint model considered here. Indeed, initially this mill had tuned their CD basis weight controller in Figure 2.3 with integral control action. The subsequent appearance of a picketing actuator profile resulted in the field engineers' implemention of the 'Actuator Profile Smoothing' feature of the industrial controller (see Figure 1.7). The functioning of this feature may be observed from Figure 2.3 in which it may be seen that an integrating controller corresponds to setting the n x n matrix S \u2014 I. The smoothing function is included in the industrial controller specifically to combat actuator picketing. Setting S ^ I according to the spatial filter parameters described in (2.18)-(2.19) in Section 2.2 removes integral control action at high spatial frequencies. The smoothing function, defined by S is very effective at reducing controller gain at high spatial frequencies. Care must be taken not to introduce an overly-conservative controller while attempting to reduce actuator picketing. Such an example is presented in Figure 5.7 where the closed-loop performance was unnecessarily degraded. The configuration of the control system at the mill was standard. The operator's computer (an N T station) is connected to the control processor via a L A N network con-nection. The operator's computer contains an 'operator station' and the industrial model identification software. The operator station is used to implement the day-to-day mainte-nance of the control system. It is mainly used to monitor the scanned paper profiles and to take the controller off-line in the event of problems (sheet breaks etc.). The tuning pa-rameters of the feedback controller are also accessed via the operator station. The model identification software (described in [32]) is an off-line identification tool. Its function is to send excitation signals to the actuators profile and to record the measured response of the paper profile. Prior to departing for the field trial, the prototype tuning tool was installed on a laptop N T machine and tested at Honeywell-Measurex's Devron division in Vancouver. All of the tuning calculations were to be performed on this laptop station. The procedure followed for the field trial of the prototype tuning tool is outlined as follows: 1. A local TCP-IP address was secured within the mill in order to connect the laptop computer to the network. Industrial Paper Machine Control 98 2. The industrial model identification software tool was used to log closed-loop data for the system running with the controller tuned by mill personnel. 3. The real-time control system was taken off-line by freezing the consistency profiling actuators at a constant profile. 4. The model identification software tool [32] was then used to send an excitation signal to the actuator profile (see Figure 5.1) and the profile response was logged. 5. The control system (still with the original controller tuning) was placed back on-line to maintain the paper quality. 6. The industrial model identification software, was used to identify the parameters of the spatial and dynamical response of the process model G(z) in (2.1) with parameters (2.7), (2.8). 7. The output of the model identification experiment was saved as model.mat. 8. The file model.mat was transferred across the network into the laptop computer containing the prototype tuning tool m-files. 9. A new set of feedback controller parameters were generated and saved as tune.mat, using the prototype tuning tool described in Section 5.1. 10. The real-time control system was taken off-line by freezing the consistency profiling actuators at a constant profile. 11. The new controller tuning parameters were keyed into the industrial controller database using the operator station. 12. The control system was placed back on-line, this time with the feedback controller defined by the new tuning numbers. 13. Closed-loop data were logged using the model identification tool over the course of several hours. The data collected from this field trial are reported in the following subsections. Industrial Paper Machine Control 99 5.2.1 Process M o d e l The nominal model for the weight process has been identified from industrial paper ma-chine data and has the output disturbance form (2.20) shown in Figure 2.2, y(z) = Gt(z)u(z) + Dt(z)d(z) (5.1) where y(z),u(z) G C 2 2 6 are the ^-transforms of the measurement vector (error profile) and the control vector (actuator profile) respectively, Dt(z) G C 2 2 6 x 2 2 6 represents the transfer matrix shaping filter through which the vector of white noise disturbances d(z) G Cn enters the process, Gt(z) G \u00a3 2 2 6 x 2 2 6 is the process transfer matrix containing both the dynamic and the spatial responses of the system to the actuator array. The two transfer matrices in (5.1) are given by the factors, Gt(z) = (J - Az-1)-1 (Bt \u2022 z~d) Dt(z) = (I- Htz'1)-1 (I - Etz~x) (5.2) where the dead time d = 3 (the sample time was T = 25s). The matrices At,Bt,Ht,Et G 72-226x226 are all symmetric band-diagonal constant matrices, (5.3) Bt \u2014 toepl i tz^ , . . . , b5,0..., 0} Et = toeplitz{ai, 0, . . . , 0} At = toeplitz{ax, 0, . . . , 0} Ht = toeplitz-f\/ix, 0, . . . , 0} where ax = a x = 0.8221, hx = 0.9990, and h = -0.0814 b2 - -0.0455 h = -0.0047 bi = 0.0017 h = 0.0003 ( 5 4 ) The circulant extension to the process model (5.2), is obtained by writing the circulant Industrial Paper Machine Control 100 symmetric matrices according to the procedure in Section 3.1), B = toeplitz{6i,..., 6 6 ,0, . . . , 0,6 5 , . . . , b2} E = Et A = At H = Ht (5.5) The design will then proceed based on the system defined with the circulant symmetric model, G(z) = [I- Az-\\xBz~z D{z) = Dt(z) (5.6) where the circulant symmetric process model in (5.6) has been normalized such that IIGOsOHoo = 1. The true process Gp(z) is assumed to belong to the set Tlg that is defined w or. P < r -o < 201 1 1 1 r -20 n 1 1 r 20 40 60 80 100 120 140 160 180 200 220 3 4 5 CD POSITION [m] Figure 5.1: Model identification: The upper plot illustrates the actuator profile shape used to excite the process during the model identification. The second plot indicates the 'true' measured basis weight response profile. The last plot indicates the modelled response. The lower plot contains the residual signal due to process disturbances and model uncertainty. Industrial Paper Machine Control 101 as in (2.26) by an additive unstructured perturbation on the nominal model, nff = {G(z) + 8GA(z) : o (8GA{J\u00bb)) < 1(UJ)} (5.7) where G(z) is the nominal circulant model given in (5.6). In this example the level of model uncertainty is estimated as \/(w) = 0.1-||C?(z)||oe = 0.1 (5.8) The relationship between the nominal process model and the uncertainty is illustrated in Figure 5.2 where the contour plot of the two-dimensional frequency response is shown. For the process under consideration, a process model-mismatch of 1(UJ) = 0.1 \u2022 HG^z)!^ defines the 100% relative uncertainty bound on the model by the 0.1 contour. This bound is especially relevant to the design of control systems, as no benefit can be guaranteed from the feedback for those spatial and dynamical frequencies Uj and UJ outside the 0.1 contour, and provides an upper bound on the spatial and dynamical closed-loop bandwidths that may be achieved with feedback control. 5.2.2 Design Specifications The industrial requirements for the feedback in the cross-directional control of a paper-making process may be summarized as \"tune the existing controller to make the paper sheet as uniform as possible\". 1. Controller Structure. The controller structure is given by (2.11) in this case, the transfer matrix factors are 226 x 226. The industrial implementation of the feedback controller is illustrated in Figure 2.3 and given by (2.15) Kt(z) = [! + St(z)} lCt{z) (5.9) Ct(z) = SfKw-c{z) = Ct-c{z) St(z) = -St-z - l (5.10) Industrial Paper Machine Control 102 14F 0.02 co [Hz] Figure 5.2: Contour plot of the open-loop basis weight frequency response \\g{vj,elw)\\ for the model in (5.1)-(5.8). The area outside the 0.1 contour indicates the region of the LOv-xAane, for which there is more than 100% relative model uncertainty. Even at steady-state, over a quarter of the spatial frequencies (60 out of 226 singular vectors) are uncontrollable. where S,KW G U n > < ' 1 are symmetric band-diagonal Toeplitz matrices, defined in (2.15)-(2.19). The design procedure requires the use of the circulant extension to the feedback controller Kt(z) in (5.9). This is obtained by writing the circulant symmetric ma-trices, C = toep l i t z {c i , . . . , c n c , 0 . . . , 0 , c\u201e ( : , . . . ,C2} S = t o e p l i t z { a i , . . . , a n j > 0 . . . , 0 , a \u201e . , . . . > a 2 } (5-H) The loop shaping design will proceed based on the controller defined with the cir-Industrial Paper Machine Control 103 culant symmetric matrices in (5.11) K(z) = [I - Sz'^C \u2022 c(z) (5.12) with C, S, and c(z) as denned in (5.11) and (2.15)-(2.19). 2. Performance. As described in Chapter 1, paper is sold based on the quadratic variance of the error profile v(t) = y(t) \u2014 r(t). This metric for paper quality lead to the measure for feedback controller performance in (2.24) in terms of the T^-norm of the closed-loop transfer matrix This requirement is a restatement of the loop shaping specifications of Section 4.1 that the sensitivity function bances are important. In the case of the disturbance transfer matrix with parameters as in (5.2), the gain of D(z) is largest at low dynamical frequencies LO. Thus, for the exogenous white noise signal d(t) in (5.1), the performance condition (5.14) is most important at low dynamical frequencies LO. 3. Robust Stability. As stated in Section 4.1, an advantage of the loop shaping control design techniques is their ability to quantify the trade off between performance and robustness. It is not possible to satisfy the performance condition (5.14) for all singular values j G {1,..., 226} and all dynamical frequencies LO G [\u20147r, 7r]. Distur-bance attenuation will always be sacrificed for closed-loop stability. An internally stable closed-loop defined by G(z) and K(z) is robustly stable for all Gp(z) G II9 in [\/ - G(z)K(z)]-1D(z) -> small (5.13) (5.14) for singular values j G {1,. ,226} and LO G [\u20147r,7r], where the exogenous distur-(5.7) if a (K(eiu)[I - G{eiu)K(eiu3)]-1) 1 1 (5.15) < 1{LO) 0.1 = 10.0 Industrial Paper Machine Control 104 for > 0 in (5.7) and all UJ \u20ac [\u20147r,7r]. 5.2.3 Two Dimensional Open-Loop Shaping Set up the problem The structure of the industrial multivariable paper machine controller K(z) was described in (2.15)-(2.19) above. It remains to be shown that the existing controller structure is capable of shaping the singular values of Oj{K{e%w)) and Oj(GK(el<\")) while maintaining internal stability of the closed-loop. In order to demonstrate these features, the plant G(z) in (5.5), (5.6) and the controller K(z) in (5.11), (5.12) will be diagonalized with the real Fourier matrix F in (A.4). As noted in Section 3.7.1, this diagonalization allows the simultaneous examination of both the eigenvalues and singular values of a circulant symmetric system. First, the plant G(z) in (5.5), (5.6) is diagonalized = T ^ < 5 1 6 > where FG(z)FT = diag{g(ux, z ) , g { u 2 2 6 , z)} is formed by computing FBFT = diag{6(i\/i),..., 6(-v226)} and FAFT = diag{o(^),..., o(i \/ 2 2 6 )} . Next, the industrial controller K(z) in Section 2.2 is diagonalized as = idS^-*) (5-17) where FK(z)FT = diag{fc(i\/i, z),..., k(u22e, z)} is formed by diagonalizing FCFT \u2014 diag{c(i\/i),..., c(^22e)} and FSFT = diag{s(i\/i),..., 5 ( i \/ \u201e ) } . Next, it is important to determine bounds on the spectra C(UJ) and S(VJ), as well as tuning parameters {dc,ac,ac} for the dynamical part c(z) in (5.17) such that closed-loop stability is guaranteed with the plant g(vj,z) in (5.16). The dynamical part of the controller c(z) is restricted to setting the controller pa-rameters {dc,ac,ac} in (2.16) in terms of the model parameters {dc,ac,ac} \u2014 {d, a i , a i } Industrial Paper Machine Control 105 given in (5.2) as {d ,Oi ,a i } = {3,0.8221,0.8221}. This setting of the parameters for c(z) is motivated by stability considerations and the fact that minimum variance control is achieved for certain values of C(UJ) and S(UJ) in (5.17) [18]. In fact, the proposed loop shaping design procedure may be interpreted as the detuning of a minimum variance controller for robustness considerations [59]. The loop shaping will proceed by assigning the remaining degrees of freedom; the two spectra C{VJ) and S(VJ) in (5.17) for Uj \u20ac { u i f . . . , f 2 26}- ft m a y be shown that closed-loop stability is guaranteed for k(vj,z) and g(vj,z) in (5.16) and (5.17) if, 0 < S(UJ) < 1, 0 < f (j\/,) (5.18) with **) = y 1 ( 5 1 9 ) for all spatial frequencies Vj \u20ac {vi,..., ^226} The design can then safely proceed by shaping the open-loop transfer functions \\g(uj,z)k(vj,z)\\ and |fc(i\/j,z)| via the spectra {P(VJ),S(I\/J)} according to the closed-loop stability condition (5.18). Two Dimensional Frequency Domain Specifications This step presents the performance and robust stability design specifications which are required to be met during the loop shaping step of the design. The frequency domain design specifications on the feedback are derived as bounds on the shape of controller gain |A,(i^ ,2:)| rather than loop gain \\g(fj,z)k(vj,z)\\ as was developed in Figures 4.3-4.5. The design procedure is identical, but the robustness re-quirement for an ill-conditioned plant with additive uncertainty is more easily stated in terms of requirements on |fc(^-,2;)| [44, 56]. 1. Performance. The design procedure requires the performance specification in (5.14) to be rewritten in terms of the open-loop transfer function k(uj, z). Industrial Paper Machine Control 106 First, the requirement is quantified by requiring 90% attenuation of disturbances, l-g(u3,eiu)k(u3,eiu) < p(u3,u>) \u2014 0.1 (5.20) for the low spatial and dynamical frequencies in {VJ,LO} G \u00a32\/ where, in engineering units, Qi = {{U3,LO} : \\VJ\\ < 5.5m - 1 , |u;| < 10 _ 4- 6Hz} (5.21) The closed-loop specification (5.20) is satisfied by the open-loop requirement | ^ . , 0 | > vn = (Jj + l ) \u2022 \u00b1 = 22.0 (5.22) for the low spatial and dynamical frequencies in {VJ,LO} G \u00a32> in (5.21). The set of low frequencies \u00a32\/ in (5.21) for which condition (5.22) must be satisfied is illustrated in Figure 5.3. The design satisfies the open-loop performance condition (5.22) if the contour(s) of |fc(i\/j,e*w)| = 22.0 lie in the white space of Figure 5.3. 2. Robust stability. Next, the requirement of closed-loop robust stability (5.15) is rewritten as the two dimensional closed-loop condition < = 10.0 (5.23) \\\\ - g(v3^)k(v3^)\\ for spatial frequencies v and dynamical frequencies LO. The closed-loop specification (5.23) is satisfied for the high spatial and dynamical frequencies in {VJ,LO} G \u00a32\/I where, \u00a32, = {{^,a;}: | fe ,e i w ) |<0. l} (5.24) by the open-loop requirement \\k{vj,eiu)\\ 0 with Vj \u00a3 {vx,..., 1*226} in (5.18), (5.19). Following the selection of the order of the Blackman window na in (2.18), the design of the control system is specified completely by the value of the tuning knob A in (2.18). The only remaining degree of freedom is the spectrum r(v3) in (5.19). It is then automatically calculated such that the conflicting specifications imposed by (5.18), (5.22), (5.25) are simultaneously satisfied. 1. Performance. In order to satisfy the performance condition in (5.22), it is required that r{vj) ->\u2022 0+ s{u3) -> 1\" (5.26) for low spatial frequencies v3- < 5.5m _ 1 as defined by the set \u00a32\/ in (5.21) and Figure 5.3. A value of f(v3) = 0 corresponds to a model-inverse gain for the controller given by c(v3) = b(vj)~x in (5.19) and should only be used at low spatial frequencies v3 where \\b(uj)\\ is large. Setting s(u3) = 1 leads to integrating dynamics in k(v3,z) in (5.17). 2. Robust Stability. In order to satisfy the robust stability condition in (5.25), it is required that f{u3) > 0 s{v3) < 1 (5.27) for high spatial frequencies u3 > 5.5m _ 1 as defined by the set Cln in (5.24) and Figure 5.3. A value of r(v3) > 0 corresponds to a more conservative controller gain as \\c(u3)\\ < |6(^) _ 1 | in (5.19) and is especially important at high spatial frequencies for which Industrial Paper Machine Control 109 the open-loop process gain rolls off and \\b(uj)\\ \u2014v 0. The quantity S{UJ) < 1 cor-responds to the removal of integral control action in k(i>j,z) in (5.17), as the con-troller pole S(VJ) is moved away from the unit circle. Integrating control results in \\k(vj,elw)\\ \u2014> oo as u \u2014> 0, this is undesirable at high spatial frequencies Vj for which an integrating k(vj,z) violates the robust stability condition (5.25). For practical controller design, it has been found sufficient to restrict the robust stability analysis to evaluating condition (5.25) along the i\/-axis of Figure 5.3. The gain of the Dahlin controller c(z) in (2.16) rolls off quickly for high dynamical frequencies UJ. The contours of \\k(i\/j,eiu)\\ = Wh are still examined for all UJU, but the robust design of the spectrum f(i\/j) only requires to examine the i\/-axis. It can be shown that |fc(^-,ei0)| will satisfy (5.25) for {^ -,0} \u20ac fi^ in (5.24) if, for z = eiw and UJ = 0, r(vj) > 2 \u2022 1(UJ) c(z) \u2022 b(uj) 1 - 8{VJ)Z-I \u00ab ^ - f e ) | - ^ ; ) 2 (5-28) where l(w) is the level of model uncertainty in (5.8), c(z) in (2.16) tuned as in (5.17), and S(UJ) PH 1 \u2014 A for high spatial frequencies Vj > 10m _ 1 (see Figure 5.4). 3. Controller Localization. The matrix S = F Tdiag{s(^i),... ,s(u226)}F is automat-ically a banded circulant matrix denned by the Blackman window of order ns in (2.18). However, the localization of the matrix C = FTdiag{c(\u00abvi),...,c(\u00abv226)}-F with C(UJ) defined by spectra b(vj) and r(i>j) in (5.19) remains an issue. Theorem 3 shows that it is desirable to keep the magnitude of the elements of 8C = C \u2014 Ci small, in other words the full matrix C be as close as possible to a banded matrix Ci. An empirical rule for this is that the localization of C is related to the smooth-ness of C(UJ). Therefore, in anticipation of the truncation of the elements of C, it is desirable to design r(vj) such that \\fiuj) -f(vj+i)\\ - 4 small (5.29) Industrial Paper Machine Control 110 for j G {1,..., 225}. Note that the satisfaction of conditions (5.26)-(5.29) requires some trade off. The performance specification of (5.26) requires s(v3) \u2014> 1, which means that the tuning knob A \u2014>\u2022 0. However, notice that A appears in the denominator of the robust stability condition (5.28), meaning that a small value of A results in a large value of f(v3) needed at high frequencies v3 to satisfy the robust stability condition. However, the performance requirement (5.26) already has f(u3) = 0 at low frequencies v3. Such a large difference in f{vj) at the low spatial frequencies and the high spatial frequencies leads to a violation of the localization constraint (5.29). The controller tuning proceeded as follows: \u2022 the order of the Blackman window in (2.18) (and hence S) was selected as ns = 4 for spatial bandwidth considerations. \u2022 for each iteration of A, the spectrum r(v3) was automatically calculated such that the performance condition (5.28) was satisfied at low v3 < 5.5m - 1 and the robust stability condition (5.28) was satisfied at high Vj > 10m - 1 . \u2022 the tuning knob A in (2.18) was manually iterated until an acceptable trade off between performance (5.26), and the localization of C in (5.29) was achieved. The resulting tuning parameters for the spectrum s{v3) were ns = 4 and A = 0.01. The spectrum s(u3) is illustrated in Figure 5.4. The tuning spectrum f(v3) and the corresponding controller gain spectrum c(u3) in (5.19) are both illustrated in Figure 5.4. The central row of the full circulant matrix C \u2014 FTdiag{c(ui),..., c(u226)}F is plotted in Figure 5.5. The size of the elements of C decay quickly as a function of their distance from the central element. This almost-localized structure facilitates the elimination of the smaller elements of C to satisfy Theorem 3. In summary, at the end of this step we are left with the settings for the dynamical controller for c(z) in (2.16) and the spectra C(VJ), s(u3) in (5.17), such that the controller k(vj, z) satisfies the performance requirement (5.22) and the robust stability requirement (5.25) are satisfied as illustrated in Figure 5.3. Industrial Paper Machine Control 111 1.005 0.99 h 0.985' 1 1 1 1 ' 1 u 0 2 4 6 8 10 12 14 T 1 1 i 1 1 r v [cycles\/metre] Figure 5.4: The spectra S(UJ) and C(UJ) resulting from the controller synthesis for the controller k(uj,z) in (5.17) and (5.19). The dashed line represents the spectrum of ci{v3) obtained by spatial order reduction. Controller Spatial Order Reduction The previous step produced spectra C(VJ) and S(UJ) such that the controller k(i\/j,z) in (5.17) is stable and satisfied the design requirements for performance (5.22) and robust stability (5.25). The multivariable controller is constructed from these spectra by writing, C = FTdiag{c(u1),...,c(u226)}F, S = F T diag{ S > 1 ), . . . , 5 > 2 2 6 )}F (5.30) where C, S G 7 \u00a3 2 2 6 x 2 2 6 are both symmetric circulant matrices. It was stated in Section 5.2.2 the final goal of the design is band-diagonal Toeplitz matrices Ct and St. Section 4.3 describes the first step towards that goal as the spatial order reduction of the circulant matrices C and S by truncating elements to obtain banded Industrial Paper Machine Control. 112 O \"1 Co 4 ^ 2 0 -2 1 1 1 1 1 1 1 1 I I I . . L11J \u2022 I i - - - -i i i i i i 20 40 60 80 100 120 140 160 180 200 220 X 1 0 \" 3 I I I I i i I I i i i If i i i i i i i > i i i 20 40 60 80 100 120 140 160 180 200 220 Figure 5.5: The central row of the full circulant matrices C(113,:) and I \u2014 5(113,:) result-ing from the spectrum C(PJ) and S(UJ) in Figure 5.4 and C = FTdiag{c(ui),..., c(i\/226)}F and S = FTdiag{s(u1),s(u226)}F. circulant matrices Ci and Si. Due to the parameterization of the industrial controller, the matrix S in (2.18) is already banded with ns = 4 and is given for A = 0.01 as the 226 x 226 constant matrix, Si = S = toeplitz{si,..., S 4 , 0 , . . . , 0, s 4 , . . . , s2} (5.31) where, {si, 5 2 , ss, s4} = {0.9930, 0.0023, 0.0010, 0.0002} (5.32) However, the 226 x 226 symmetric circulant matrix C, illustrated in Figure 5.5, has no non-zero elements, C = toeplitz{ci,..., Cn3, c i 1 4 , c n 3 , . . . c2} (5.33) Industrial Paper Machine Control 113 but the magnitude of \\ck\\ in (5.33) decreases rapidly away from the centre. It is desired to truncate these smaller elements of C in (5.33) to obtain a banded 226 x 226 symmetric circulant matrix with nc = 7 in (5.11). Ci = toeplitz{c x,..., c 7 ,0 , . . . , 0, c 7 , . . . c2} (5.34) such that closed-loop stability is maintained. The perturbation on the controller K(z) introduced by the spatial order reduction is then, 6Ci = toeplitz{0, . . . ,0 ,c 8 , . . . , c i i3 , cn4 , c i i s , . . . , c 8 , 0 , ...,0} 8Si = 02 6x226 (5.35) As discussed in Section 4.3, closed-loop stability must be verified following the spatial order reduction of a controller. In this case, only the gain matrix C was required to be reduced to Cj in (5.34). This fact allows for the evaluation of the closed-loop stability based on the gain margin for each of the 226 SISO loops. Calculation of the gain margin is less conservative than the small gain theorem based condition in Theorem 3, and is used to verify the stability of the closed-loop system with Ci in (5.34) with {ci, . . . ,c7} = {-4.1226, -1.9487, 0.9150, 0.9408, 0.0027, 0.0011, 0.0002} (5.36) The multivariable controller is then written as Kl(z) = [I-Siz-1]-1Ci-c{z) (5.37) with banded circulant factors Ci and Si. The central rows of the banded circulant matrices Ci and Si are illustrated in Figure 5.6. Since only stability has been guaranteed for the closed-loop with circulant symmetric controller Ki(z) with the circulant plant defined by the transfer matrix G(z) in (5.6). The satisfaction of the design requirements of disturbance attenuation (5.22) and robust sta-bility (5.25), is verified for the reduced order controller Ki(z) by plotting the appropriate Industrial Paper Machine Control 114 1 o -2 -3 -4 n 1 r t -i i i 1 r _J I I I I L_ 20 40 60 80 100 120 140 160 180 200 220 Figure 5.6: The central row of the banded circulant matrix (7\/(113,:) resulting from the spatial order reduction (5.34). The central row of the banded circulant matrix 7\u20145\/(113,:) is also shown, due to the structure of the industrial controller Si = S and no spatial order reduction was required. LOV contours of \\ki(i\/j, elu,)| in Figure 5.3. The design requirements (5.22) and (5.25) have been satisfied with a controller Ki(z) in (5.37) which may be implemented such that each actuator in the array requires infor-mation only from 2nc \u2014 1 = 13 measurement locations and 2ns \u2014 1 = 7 actuators. In other words, to realize the control law Ki(z) each actuator requires only about 6% of the n = 226 available measurements and information from about 3% of the n = 226 actuators in the array! Implementation The previous step produced banded circulant matrices Ci in (5.34) and Si in (5.31) which, when used in the feedback controller Ki(z) in (5.37), satisfy the design requirements (5.22) and (5.25) for the circulant process model G(z) in (5.6) defined in terms of the banded Industrial Paper Machine Control 115 circulant matrices B,Ae ft226*226 in (5.2), (5.5). The final step in the controller design is to verify that the controller formed by ex-tracting the band-diagonal Toeplitz matrices from Q and Si, i.e. Ct \u2014 t oep l i t z{c i , . . . , CT, 0 , . . . , 0} St = toep l i t z{s i , . . . , s 4 , 0 , . . . , 0} (5.38) such that Kt(z) = [I- Stz'^Ctciz) (5.39) stabilizes the more accurate process model Gt(z) in (5.3) with band-diagonal Toeplitz factors given by Bt and At in (5.2). The Toeplitz system matrices {Bt, At, Ct, St} are obtained by trimming the 'ears' from the circulant matrices {B, A, Ci, Si}, as shown in Figure 3.1. In Section 3.1 it was shown that closed-loop stability of the band-diagonal Toeplitz system is given by the invertibility of the 2n x 2n transfer matrix Lt(z) defined in 3.5. These factors are realized here by the 452 x 452 transfer matrix, Hz) := I - Stz'1 Ctc(z) Btz~d I - Atz~ I - Siz'1 Cic(z) Bz~d I - Az~l -SStz'1 8Ctc(z) 8Btz~d 0 2 -\u00bb 26x226 (5.40) where B and A in (5.5), St in (5.31), Q in (5.34), and c(z) in (2.16). The matrix 8Bt = B \u2014 Bt, and similar definition apply to 8Ct and 8St. The matrix 8At = 0 226x226 due to the fact the circulant A and the band-diagonal Toeplitz At both being given by A = At = 0.8221 \u2022 \/ 2 26x226-The invertibility of the first term in (5.40) is guaranteed by the internal stability of the circulant symmetric controller Ki(z) in (5.37) and the circulant symmetric plant model G(z) in (5.6). The second term in (5.40) contains the 'ears' of each of the relevant transfer Industrial Paper Machine Control 116 matrices. Following the verification of invertibility of Lt(z) in (5.40), the two matrices Ct and St are implemented as factors in the controller Kt(z) in (5.39). 5.2.4 Paper M i l l Results This section presents data obtained from a paper machine describing and comparing the closed-loop performance obtained by tuning the industrial controller in (5.9) with two different feedback controller designs. The first of these is denoted by Kb(z) and has been designed using the empirical tuning rules for paper machine control. The second set of results displays the closed-loop performance of Kt(z) in (5.39), resulting from the two dimensional loop shaping procedure in Section 5.2.3. Both of these controllers were implemented on a paper machine whose model is described in Section 5.2.1. The controller designed with traditional empirical tuning rules is given by (5.9) with Kb(z) = [I- SiZ-^C^z) (5.41) this design has na \u2014 2 and A = 0.1 in (2.18), such that Sb = toeplitz{0.9595,0.0202,0,..., 0} (5.42) The matrix Cb \u2014 \u20146.5494 \u2022 Sb and the parameters of the dynamical controller cb(z) given by c(z) in (2.16) with parameters {dc,ac,ac} = {3,0.7316,0.8365}. The two dimensional frequency components of the controller are calculated from the circulant extension of Kb(z) as in (5.5) and diagonalizing with the Fourier matrix F in (A.4) ^=i4S^-c'(z) (5-43) The relevant ct>^-contours defining performance |fc&(i\/j,e\"\")| = 22.0 in (5.22) and robust stability \\kb(i>j,eiu})\\ = 5.0 in (5.25) are plotted in Figure 5.7 along with the sets fi\/ in (5.21) and Cth in (5.24). The performance criterion is violated since the contour \\kb(vj,e%u)\\ = 22.0 intersects the set fi\/. In this case, the empirical approach, in an effort to maintain Industrial Paper Machine Control 117 closed-loop robust stability, has resulted in an overly conservative controller. 14 CU o \u00b1 3 8 12 10 cu \u00a3 CD U er o l*\u00bb|=5.0 = 22.0 \\ 1 10\" ioJ 10\" 10\": \u00a9 [ H z ] Figure 5.7: An LOV contour plot showing the controller \\kb(vj,z)\\ in (5.43) relative to the sets fi\/ in (5.21) and Vth in (5.24). The performance criterion is violated since the contour \\kb(vj,eiw)\\ = 22.0 intersects the set fi\/. Figure 5.8 displays the closed-loop steady-state (a; = 0) error profiles for the con-trollers Kb(z) in (5.41) and Kt(z) in (5.39), respectively. The profiles shown are the high-resolution measured profiles obtained from the scanning sensor as described in Chapter 1.1. The paper sheet is measured at 693 locations evenly-spaced across the 7.91m wide paper sheet. The high resolution error profile Vh(t) G T2.693 is related to the low resolution error profile v(t) = y(t) \u2014 r(t) G 7\u00a3 2 2 6 by a linear spatial downsampling transformation, v(t) = C\u00a3tfc(i) (5.44) where Cm G 7e693x226. While it is difficult to quantitatively compare the closed-loop profiles in Figure 5.8, Industrial Paper Machine Control 118 1 2 3 4 5 CD position [metres] Figure 5.8: The steady-state of the measured paper profiles under closed-loop control, (a) with the controller Kb(z) in (5.41) which was designed according to the more traditional industrial tuning rules, and (b) with the controller Kt(z) in (5.39) which was designed according to two dimensional loop shaping procedure in Chapter 4. there is much information available in the spatial frequency content of these signals. Figure 5.9 contains the steady state spatial frequency components of the error profile v(pj,z) = y{vj, z) \u2014 f(uj,z) with z = el 8 m _ 1 . In Figure 5.2 it can be seen that the open-loop gain of the process g{yj,z) in (5.16) rolls off as a function of spatial frequency. The process is much harder to control at high spatial frequencies and both controllers have this bandwidth limitation. At very low spatial frequencies v < l m _ 1 the closed-loop performance is not signifi-cantly different for Kb(z) and Kt(z). The contour plots in Figures 5.3 and 5.7 predicted that each of these controllers would achieve over 90% attenuation of disturbances for these Industrial Paper Machine Control 119 Figure 5.9: Spatial frequency power spectra of the steady-state error profiles of the closed-loop system with the controller Kb(z) in (5.41) shown as the dashed line, and the controller Kt(z) in (5.39) shown as the solid line. low spatial frequencies. It is the mid-range spatial frequencies 2 m _ 1 < v < 7 m _ 1 where the greatest difference in closed-loop performance is realized. The contour plots in Figure 5.3 and Figure 5.7 indicated that the controller Kt(z) would achieve approximately twice the closed-loop spatial bandwidth of the more conservative controller Ki{z). It can be seen easily in Figure 5.9 the performance of the controller Kt{z) is significantly better than that of the controller Kb(z) for these mid-range spatial frequencies. Chapter 6 Conclusions This work has concentrated on the analysis and design of feedback control for dynami-cal systems that are distributed in one spatial dimension and controlled by an array of identical actuators. The cross-directional control of paper machines is an industrially important example of such a process and it has played a central role in shaping this work. This Chapter summarizes the goals, approach, and results contained in this thesis. Chapter 1 illustrated the broad range of cross-directional processes occurring in indus-trial paper machine applications. The spatial response of the process to a single actuator can be as narrow as two actuator-widths, or as wide as about a third of the paper sheet. The dynamics of these processes are also diverse. Relative to the sample time, the ac-tuators can respond almost instantaneously or be slow enough to be described as an integrating process. Process deadtime due to the transport delay of the paper sheet from the actuator array to the scanning sensor must also be considered. Time delays as long as five sample times have been observed in working mills. There exists a large body of theoretical work in advanced control which may possibly be applied to the cross-directional control problem, as discussed in Section 1.3. In addition to the theoretical work directed specifically at cross-directional control of flat sheet pro-cesses, there have also been many advances in multivariable control, robust control, and spatially distributed systems, that can be collected and applied to the cross-directional control problem. However, in spite of the existing control theory, the state of the art for industrial paper machine control is to design the cross-directional controller using empirical rules-of-thumb. This is a complex task and, as a result, many paper machines are running with poorly-tuned feedback controllers. The two most common problems associated with profile control represent the two extremes associated with feedback controller design. 120 Conclusions 121 First, the closed-loop may be conservatively designed, resulting in an underactive control which does not do enough to remove variation in the process. This was the situation faced by the mill as described in Chapter 5. The second problem faced by papermakers is that of closed-loop instability. The cross-directional controller is often designed without calculation of a stability margin, and the incautious application of empirical tuning rules has lead to a large number of paper machines with a marginally-stable closed-loop. This is a potentially dangerous situation as the process model uncertainty, inevitably occurring when modelling real-world problems, easily destabilizes such a closed-loop system. This instability always results in operator intervention and often results in financial loss for the mill due to culled paper. In order to reconcile the above issues, in Chapter 3 a general framework is presented for the analysis of dynamical systems that are distributed in one spatial dimension and controlled by an array of identical actuators. Within this framework, Chapter 4 devel-oped a constructive technique for the design of practical feedback controllers for such processes. Referred to as two dimensional loop shaping, the analogy to the traditional 'one dimensional' loop shaping is preserved in order to allow the transfer of knowledge and experience from a design technique that is familiar to most control engineers. This technique is graphical in nature and allows the designer to view the trade-off between the conflicting criteria of closed-loop stability and aggressive closed-loop performance of the control system. As spatially distributed systems tend to involve a large number of input and output variables, the feedback control can potentially involve a large amount of computation -both offline for controller design and also online for real-time implementation. Here the offline computations, involved in the design of the controller, are kept low by exploiting the natural structure of the process. The real-time computation is negligible due to the low complexity of the resulting feedback control law. As stated above, the analysis and design techniques developed and presented in this work were motivated by the need to tune cross-directional controllers for industrial paper machines. The two dimensional loop shaping technique is applied as a tool for the design of feedback controllers for paper machine cross-directional processes. This framework Conclusions 122 for analysis and design was shown to be well-suited to address the wide variety of such processes for which a cross-directional controller must perform well. The theoretical component of this work has been balanced by the requirements of tuning an industrial paper machine in a working mill. This practical control problem was maintained as an example throughout this work. Chapter 5 contains the details of the inaugural field trial on an industrial paper machine in a Canadian mill produc-ing newsprint-grade paper. The analysis of the control system indicated that the mill had tuned their controller to be overly conservative. 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Appendix A Fourier Matrices The complex Fourier matrix is defined as follows [38], 1 TH = - i [ m i m2 \u2022\u2022\u2022 mn] (A.l) where the vectors mk G C n x l are given by, mk = [1 vk vl \u2022 \u2022 \u2022 vrT, = J*\"-1*'\" (A.2) In other words, the kth row of T contains the kth spatial harmonic and has frequency vk. The complex Fourier matrix T e C n x n in (A.l) may then be used to diagonalize any nxn circulant matrix, A = THXAF, E A = d iag{a(z \/ 1 ) , . . . ,aK)} (A.3) The subset of circulant symmetric matrices may be diagonalized with a pure real Fourier matrix. The real Fourier matrix F is constructed from the complex Fourier transform matrix T in (A.l) by the following unitary operations, F(l , : ) = i[ l , 1,...,1] F(k,:) = 4= :)] - Sm[f(n + 2 - k,:)]) v2 F(n + 2-k,:) = (^e[T(k,:)} + ^ e[T{n + 2 - k,:)}) (AA) v2 for k \u2014 2,... ,p, where p = (n + l)\/2 if n is odd and p = n\/2 if n is even. The jth row of F contains the jth spatial harmonic and has frequency Uj \u2014 2ir(j \u2014 l)\/n. The real Fourier 1The unitary, complex Fourier matrix T may be created, for example, using the MATLAB command F = fft(eye(n))\/sqrt(n). 130 Fourier Matrices 131 matrix F is unitary, satisfying the property FTF \u2014 I. More intuitively, the rows of the real Fourier matrix F in (A.4) may be re-written in terms of the familiar trigonometric functions, F(j, k) yj -s inp- l )^] j = 2,...,p { V?' c o s ^ k _ j = p + 1,..., n (A.S) ","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2000-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0065298","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Electrical and Computer Engineering","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"For non-commercial purposes only, such as research, private study and education. 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