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Discrete-time closed-loop control of a hinged wavemaker Hodge, Steven Eric 1986

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DISCRETE-TIME CLOSED-LOOP CONTROL OF A HINGED WAVEMAKER by STEVEN ERIC HODGE B.A.Sc, The University of British Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1986 © STEVEN ERIC HODGE, 1986 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 ^ g C H A I M tcVH-- ^ M C l N g g R t M ^ The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) ABSTRACT The waves produced by a flap-type wavemaker, hinged in the middle, are modelled using first-order linear wavemaker theory. A simplified closed-loop, discrete-time system is proposed. This includes a propor-tional plus integral plus derivative (PID) controller, and the wavemaker in order to compare the actual wave spectral density with the desired wave spectral density at a single frequency. Conventional discrete-time control theory is used with the major difference being the use of a relatively long timestep duration between changes in waveboard motion. The system response is calculated for many controller gain combina-tions by the computer simulation program CBGANES. System stability is analyzed for the gain combinations by using two different methods. One method is an extension of the Routh criterion to discrete-time and the other is a state-space eigenvalue approach. The computer simulation and the stability analysis provide a means for selecting possible controller gains for use at a specific frequency in an actual wave tank experiment. The computer simulation performance response and the two stability analyses predict the same results for varying controller gains. It is evident that integral control is essential in order to achieve a desired response for this long duration timestep application. The variation in discrete timestep duration and in desired spectral density (an indirect indication of frequency variation) provide variation in the constraints on controller gain selection. The controller gain combinations yielding the fastest stable response at a single frequency are for large propor-tional gain and small integral and derivative gains. - i i -TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES vi LIST OF SYMBOLS ix ACKNOWLEDGEMENT x i i 1. INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Review of Literature 3 1.3 Purpose and Scope 5 2. CLOSED-LOOP CONTROL SYSTEM FORMULATION 8 2.1 Physical System Description 8 2.2 Error Formulation 10 2.2.1 Waves and Spectral Density 10 2.2.2 Discrete-Time Error 11 2.3 Control Strategy 14 2.3.1 PID Control Description 14 2.3.2 NRC Control Description 17 3. SIMULATION 18 3.1 Wave Generation System Model 18 3.1.1 Wave tank Model and Assumptions 18 3.1.2 Wavemaker Theory and Assumptions 19 3.2 Computer Simulation and Controller Details 25 3.2.1 Controller Gain Orders of Magnitude 25 3.2.2 Computer Simulation 29 3.3 Results and Discussion 35 3.3.1 Proportional Control 35 3.3.2 Proportional-Derivative Control 37 3.3.3 Proportional-Integral Control 37 3.3.4 Proportional-Integral-Derivative Control 38 - i i i -TABLE OF CONTENTS (Continued) Page 4. STABILITY ANALYSIS 40 4.1 State Space Formulation and Eigenvalue Stability Analysis 40 4.2 Routh Stability Analysis 44 4.3 Results and Discussion 49 4.3.1 Routh Stability Analysis Results 49 4.3.2 State Space and Eigenvalue Stability Analysis Results 51 5. PERFORMANCE ANALYSIS 53 5.1 The Measure of Performance 53 5.2 Controller Gain Selection 56 5.3 Results and Discussion 58 6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 60 6.1 Conclusions 60 6.2 Suggested Subsequent Investigation 61 6.2.1 Proposed Practical Simulation Based Upon the Present Work 61 REFERENCES 92 APPENDICES 94 A - COMPUTER SIMULATION PROGRAM LISTING '.. 95 B - EIGENVALUE AND ROUTH PROGRAM LISTINGS 100 C - COMPUTER SIMULATION OUTPUT AND EIGENVALUE OUTPUT 108 D - GRAPHICS EXAMPLE AND PROGRAM LISTING 135 - iv -LIST OF TABLES Page Table 1. Controller Gain Values Tested 36 - v -LIST OF FIGURES Page Figure 1 Towing Tank Configuration 2 2. Simple Closed-Loop System Block Diagram 7 3. Possible Hardware Configuration for Closed-Loop System 9 4. Specific Closed-Loop Block Diagram for Computer Simulation 13 5. Qualitative Sequence of Waveboard Motions vs. Discrete Time 15 6. Geometry of Waveboard and Wave tank 26 7. Computer Simulation Flowchart 32 8. Error vs. Discrete-Time, KP(a>) = -0.158, KI(UJ) = -0.0 KD(co) = -1.8 64 9. Error vs. Discrete-Time, KP(co) = -0.158, KI (co) = -0.0 KD(a>) = -5.4, Initial Actual Spectral Density = 0.001 m2.sec 65 10. Error vs. Discrete-Time, KP(OJ) = -0.079, KI(w) = -0.002, KD(u) = -1.8 66 11. Error vs. Discrete-Time, KP(OJ) = -0.079, KI(<D) = -0.002, KD(u) = -5.4 67 12. Error vs. Discrete-Time, KP(to) = -0.079, KI(UJ) = -0.002, KD(u>) = -9.0 68 13. Error vs. Discrete-Time, KP(to) = -0.079, KI(OJ) = -0.004, KD(oj) = -1.8 69 14. Error vs. Discrete-Time, KP(co) = -0.079, KI(OJ) = -0.004, KD(<JO) = -3.6 70 15. Error vs. Discrete-Time, KP(to) = -0.158, KI(u>) = -0.002, KD(u)) = -0.0 71 16. Error vs. Discrete-Time, KP(u>) = -0.158, KI(co) = -0.002, KD(u>) = -1.8 72 17. Error vs. Discrete-Time, KP(OJ) = -0.158, KI(w) = -0.002, KD(u)) = -5.4 73 - vi -LIST OF FIGURES (Continued) Page Figure 18. Error vs. Discrete-Time, KP(co) = -0.158, KI(co) = -0.002, KD(OJ) = -9.0 74 19. Error vs. Discrete-Time, KP(ui) = -0.158, KI(OJ) = -0.004, KD((JJ) = -1.8 75 20. Error vs. Discrete-Time, KP(ai) = -0.237, KI(w) = -0.002, KD(uj) = -0.0 76 21. Error vs. Discrete-Time, KP(to) = -0.237, KI(ai) = -0.002, KD(to) = -1.8 77 22. Error vs. Discrete-Time, KP(UJ) = -0.237, KI(w) = -0.002, KD(aO = -9.0 78 23. Error vs. Discrete-Time, KP(to) = -0.237, KI(co) = -0.004, KD(aj) = -1.8 79 24. Error vs. Discrete-Time, KP(u)) = -0.237, KI(w) = -0.008, KD((JO) = -1.8 80 25. Error vs. Discrete-Time, KP(w) = -0.316, KI(co) = -0.002, KD(a>) = -0.0 81 26. Error vs. Discrete-Time, KP(ai) = -0.316, KI(cu) = -0.002, KD(aj) = -3.6 82 27. Error vs. Discrete-Time, KP(w) = -0.316, KI(<D) = -0.004, KD(io) = -3.6 83 28. Error vs. Discrete-Time, KP(ai) = -0.316, KI(co) = -0.006, KD(u) = -3.6 84 29. Error vs. Discrete-Time, KP(a>) = -0.316, KI(to) = -0.008, KD(w) = -3.6 85 30. Routh Stability Constraints, |KI(io)| vs |KD(o))| For Various |KP(co)| 86 31. Routh Stability Constraints, |KI(OJ)| VS |KD(to)| For Various T 87 32. Routh Stability Constraints, |KI(to) | vs |KD(a>)| For Various DSPD(u)) 88 33. Largest Eigenvalue vs |KP(ai)| For Various |KI(a))| and |KD(co)| 89 - v i i -LIST OF FIGURES (Continued) Page Figure 34. Largest Eigenvalue vs |KI(UJ)| For Various |KP(ID) | and |KD(u>)| 90 35. Largest Eigenvalue vs |KD(OJ) | For Various |KP(co)| and | K I ( U ) | 91 - v i i i -LIST OF SYMBOLS a variable in quadratic equation for Routh analysis a coefficient for output value of conventional difference equation n a element of system matrix F mn — A complex eigenvalue magnitude A coefficient from partial fraction expansion n ASPD actual wave spectral density b variable in quadratic equation for Routh analysis b^ coefficient for input value of conventional difference equation B^  coefficient of system characteristic equation in 3-domain c variable in quadratic equation for Routh analysis C output in conventional difference equation C coefficient in wave equation n d depth of s t i l l water in wave tank DSPD desired wave spectral density E error operated on by controller E' error from summing/difference junction E^ energy due to each frequency in a seaway E measurement error m E g energy of a particular wave per unit area of sea surface E^ , total energy of seaway per unit area of sea surface f waveboard flap draft measured from s t i l l water level to waveboard hinge F_ system matrix g gravitational constant h waveboard fixed portion length measured from bottom of tank to hinge H transfer function - ix -H wave height amplitude Si i subscript indicating a particular frequency j variable in summation K integer discrete timestep KD derivative gain Kl integral gain KP proportional gain m wavenumber of generated wave n n subscript, integer 1,2 r input in conventional difference equation R input for discrete-time control system S^  spectral density of wave energy at a particular frequency ST instantaneous waveboard stroke AST change in waveboard stroke at end of discrete timestep K ST waveboard stroke amplitude a STMAX maximum stroke spectral density STSPD waveboard stroke spectral density ASTSPD change in stroke spectral density at end of discrete timestep K t time t sampling period s T discrete timestep duration between controller action WTF wave transfer function x horizontal coordinate axis with origin at s t i l l water level at waveboard x state variable n 3C_ state vector y vertical coordinate axis with origin at s t i l l water level at waveboard z variable - x Z z-transformation a arbitrary constant g transformation plane variable for Routh analysis 6 iteration factor n instantaneous water surface elevation measured positive upwards from s t i l l water level X eigenvalue of characteristic equation eigenvalue of characteristic equation p fluid density <j> temporal and spatial velocity potential cj) spatial velocity potential P 6 angle in imaginary plane, for complex vector oi frequency Ao),A<i)^  spectral frequency bandwidth co maximum frequency of interest max n J w , frequency at which peak in wave spectrum occurs - xi -ACKNOWLEDGEMENT The author wishes to thank Professor Dale B. Cherchas for his guidance and patience throughout this work. Some funding was provided for this work through the NSERC Grant No. A4682. - x i i -1. CHAPTER I  INTRODUCTION 1.1 Preliminary Remarks Discrete-time control systems have been studied for many years, but most analyses deal with discrete timesteps which are short, i.e. the timestep duration is on the order of seconds as opposed to minutes. A particular application has created interest in examining discrete-time control in which the discrete timestep duration between controller action is relatively long. The application is that of automatically adjusting the driving signal of a waveboard in a ship model towing tank. From an i n i t i a l state for waveboard position and water surface height the waveboard is driven to produce a spectrum of waves travelling down the length of the tank (see Figure 1). A certain time must elapse for the slowest wave in the spectrum to reach a sensing probe before correc-tive action, in the event that the actual wave characteristics do not emulate the desired wave characteristics, can take place. Hence, the system inherently contains a long duration discrete timestep. The system responds dynamically for the timestep duration until the result-ant or actual wave characteristics are compared to the desired wave characteristics. The system is not one of instantaneous control action requiring dynamic and ongoing changes. The control of this class of discrete-time system is achieved by the use of a proportional plus integral plus derivative (PID) controller with P, PI, PD and PID conditions examined. Ultimately the system is operated by a user wishing to achieve desired wave characteristics in the towing tank in a reasonably short time and within specified FIGURE 1 TOWIMG TANK CONFIGURATION HYDRAULIC CYLINDER WAVE HEIGHT SENSING PROBE WAVE ABSORBING BEACH S S S V S S S S SSS WAVE SYNTHESIZER WAVEBOARD DRIVING SIGNALS WAVE HEIGHT HISTORY OR SPECTRUM INFORMATION 3. tolerances. This work develops a suitable control strategy and examines the stability of the system as i t relates to the controller gains and desired wave characteristics. As well, the performance of a simple system model based upon first order linear wavemaker theory is examined. The desire to experimentally achieve specific wave characteristics is not new and has been studied by many researchers. This work examines a small area of this field, that of automatically adjusting the motion of a waveboard to correct for irregularities between desired and actual wave characteristics. This work has been motivated by the desire to decrease the turnaround time between towing tank model tests and the desire to eliminate the need for operator feedback in the control loop. 1.2 Review of Literature Other researchers have examined the control of wavemakers but none have closed the control loop automatically by using conventional discrete-time techniques. The author is not aware of any research in the area of controls that has examined dynamic systems with long duration discrete timesteps. Much research has been published regarding small step discrete-time systems including adaptive control, optimal control and deadbeat control. The study of discrete-time systems has been ongoing primarily since 1950. Techniques used in the study of continuous-time control systems have been applied successfully to discrete-time control systems. Kuo (1970) mentions the areas in which the commonalities exist, whereas Dorato (1983) discusses results in discrete-time systems that do not extend from continuous-time theory. Practical situations studied for discrete-time include that of radar tracking, for example, Benedict and 4. Bordner (1962). This example is one which is inherently discrete and does not extend from continuous analysis. The reference input varies for successive discrete timesteps and the timestep duration between controller action is short. In comparison, the case in this work is one with a constant reference input over a l l timesteps at each frequency of interest in the spectrum and a long timestep duration between controller action. The study of wavemakers began with Havelock (1929) as he discussed forced surface waves on water. In work more specific to wave tanks, Biesel et al. (1951) discussed the theory for a particular type of wave-maker. Ursell, Dean and Yu (1959) compared theory with experiment in a study on forced small amplitude water waves. Considering wavemakers similar to the one studied in this work, the flap—type, Galvin (1965; 1966) discussed experimental and theoretical aspects of wave generation. Hyun (1976) studied a flap-type wavemaker hinged in the middle of a wall for first order linear theory, the same situation as in this work. Flick and Guza (1980) studied first and second order Stokes theory for a flap-type wavemaker and Hudspeth and Chen (1981) extended the work of Hyun (1976) for design forces and moments to generate design curves for several wavemaker and wave tank geometries. The control aspect of wave generation is not as widely documented, however, a notable contribution is given by Anderson and Johnson (1977). A wave generation system is presented in which the feedback from an observed wave characteristic or response is provided by an operator who makes the decision of whether to perform more iterations in the attempt to achieve better agreement between actual and desired wave characteris-tics. The transfer function between the waveboard driving spectrum and 5. the wave height response is modified, i f desired, on each iteration until satisfactory wave characteristics are achieved. This system is claimed to be effective but i t does not close the control loop auto-matically. The corrective action is achieved by changing the transfer function relating the waveboard motion to the wave height response and not by u t i l i z i n g the error between desired and actual wave characteristics directly in the control scheme. Funke and Mansard (1984) describe the wave generation package at the Canadian National Research Council (NRC) Hydraulics laboratory in Ottawa, Canada. They outline an elaborate system for controlling spec-tral energy distribution, energy distribution in the time domain, wave repetition period and phase, statistical wave profile characteristics and wave group characteristics as well as the synthetic generation of certain waves and the propagation of wave trains. Also outlined is the feature that the waveboard control signals are generated using linear wavemaker theory and that the waveboard driving spectrum is modified to compensate for energy shift due to wave propagation. This compensation is achieved by using an iteration based upon the differences between actual and desired wave spectrums including the use of a transfer func-tion, dissimilar to that used by Anderson and Johnson (1977) for correcting the waveboard control signal. These last two features of the NRC wave generation package are of particular interest in this work. 1.3 Purpose and Scope This work examines a small area of the subject of wave generation, that of producing waveboard driving signals based upon the error between the desired spectral density of the wave and the actual achieved 6. spectral density of the wave at each frequency of interest. The intent of the work is to select a control scheme and to examine the result of varying the parameters or gains for this scheme at each frequency. The work therefore considers the problem as one of discrete-time control and examines the stability of the system using two methods, a state space eigenvalue formulation and the Routh stability criterion extended from continuous-time control theory. Bounds for the controller parameters are determined from the stability analysis. The work examines the performance of the system for variations in controller parameters by simulating a closed-loop system including the controller, the waveboard, the desired spectral information and the actual achieved spectral infor-mation (see Figure 2). Also the eigenvalue analysis is extended to the selection of controller parameters which tend toward a quicker response. This work does not attempt to develop new wavemaker theory nor does i t attempt to examine every facet of a wave generation system. Discus-sions on the control of many wave characteristics can be found in the report written by Funke and Mansard (1984). The emphasis in this work is to extend discrete-time control theory to the specific application in which the discrete timestep duration is long. The response time is therefore in terms of minutes as opposed to seconds. The simulation is based upon the towing tank geometry of the system in place at the B.C. Research Ocean Engineering Center, in Vancouver, Canada. The simulation focusses on the system performance in terms of a spectral density error at each spectrum frequency for various controller parameters. FIGURE 2 SIMPLE CLOSED-LOOP SYSTEM BLOCK DIAGRAM ERROR WAVEBOARD STROKE INFORMATION CONTROLLER / WAVEBOARD ->~ ASPD ACTUAL ACHIEVED WAVE SPECTRAL INFORMATION 8. CHAPTER 2 CLOSED LOOP CONTROL SYSTEM FORMULATION 2.1 Physical System Description The wave generation system considered in this work is currently open-loop with the human operator providing the closure of the loop. A flap-type waveboard which can be pivoted at its base or at its mid-section is situated at one end of a 67 meter long towing tank (see Figure 1). This waveboard is driven by an hydraulic cylinder with the stroke controlled by a spool valve. The spool valve is controlled by wave synthesizer voltage signals. Waves generated by the waveboard travel down the tank to a twin wire probe at which the wave height is sensed. A wave absorbing beach is situated at the end of the tank opposite the waveboard. The time domain wave height information obtained from the sensing probe can be analyzed using Fourier Transform techniques to obtain spectral information. With the actual wave spectrum information obtained, the wave synthesizer control signals can be manually altered in order to provide better agreement with the desired wave spectral information. A possible hardware configuration for a closed-loop control system, eliminating the manual alteration of wave synthesizer control signals, is shown in Figure 3. The waveboard generates waves which are sensed and sampled every t seconds (a period chosen to eliminate any aliasing), converted to digital signals and transmitted to the computer in which the Fourier Transformation and control algorithm calculations are done. The updated wavemaker signals are then passed through a latch with period T, the long timestep of interest in this work, converted to 9. FIGURE 3 POSSIBLE HARDWARE CONFIGURATION for CLOSED-LOOP SYSTEM TRANSDUCER D/A HOLD V POSITION FEEDBACK COMPUTER (VAX, IBM) WAVE HEIGHT SENSOR SAMPLER ,\ts A/D WATER SURFACE DATA ACQUISITION UNIT I TRANSMITTER SOFTWARE FOURIER TRANSFORMATION, CONTROL, etc. 10. analog data and fed to the waveboard spool valve for the next iteration of wave generation. Of primary interest in this work is the controller algorithm used in the computer. 2.2 Error Formulation To control the motion of the wavemaker automatically, a corrective action must be computed based upon an error. This error should be the difference between a desired wave characteristic and an actual wave characteristic. 2.2.1 Waves and Spectral Density Sea states are often described in terms of an energy spectrum or spectral density. A time history of wave heights can be converted to a frequency domain amplitude spectrum, then into an energy spectrum and subsequently into a wave spectrum. The wave height at a fixed location as a function of time, n(x,t), is transformed via Fourier Transform techniques into an amplitude spectrum. The amplitude, H , of a a particular sinusoidal wave at a specific frequency is used to express the energy of the particular wave as, E = 1 p g H 2 per unit area of sea surface (2.1) The total energy per unit area of wave surface for an irregular seaway, one composed of superimposed sinusoids, is given as, l n E T = y P g Z (H 2) (2.2) 1 Z i=l a i The total energy of a given seaway gives an indication of the severity of the seaway (Bhattacharyya, 1978) . The energy due to each frequency in the seaway is given by 11. E ± = \ p g H (2.3) where Aio^  is the bandwidth about a particular frequency. An extension of the energy spectrum is the wave spectrum. This is obtained by dividing out pg from the energy spectrum leaving an expres-sion for the spectral density of wave energy at a particular frequency in the seaway, Standard spectral density, i.e. wave spectrum, information is used in towing tank experiments when prototype sea state information is not available. Therefore, i t is reasonable to utilize the spectral density formulation for the comparison of desired and actual wave character-istics in this work. The advantage of using the wave energy spectrum method for describing a seaway is that the time dependence of the wave is dropped and both the wave amplitude and frequency can be taken into account. The phases of each sinusoid in a sea state will differ from one test to another but the amplitude/frequency information will remain the same. 2.2.2 Discrete-Time Error For this work an irregular seaway, i.e. a seaway composed of many sinusoids each at a different frequency is assumed. The wave characteristic of interest for comparison purposes is the spectral density of each sinusoid. The error required for the generation of new waveboard driving signals is expressed as the difference between the actual spectral density and the desired spectral density at each frequency of interest in the wave spectrum. For generality, the error is expressed as, S i = Y H a ' / A " i (2.4) 12. E(KT,iu) = (ASPD(KT.u) - DSPD(KT,UJ) )/DSPD(KT,u)) + EM(KT,u>) (2.5) where ASPD(KT,u)) = actual measured spectral density at the end of timestep K DSPD(KT,OJ) = desired spectral density at end of timestep K E (KT,u>) = measurement error m K = integer number of timestep. T = discrete timestep duration (o = frequency In an actual wave tank experiment the wave spectral density can only be represented by an estimate of the true or actual spectral density at each frequency. This estimate arises from the process of sampling the wave height time series. The spectral density estimate often has some bias error and variance error dependent upon how many sample records are taken and the actual sampling rate. Therefore, strictly speaking, the error in the control equation would include this measurement error, E (KT,u)), denoting the difference between the true or m actual wave spectral density in the tank and the estimated wave spectral density. For the remainder of this work the measurement error is assumed to be zero. This implies that the following work cannot be applied directly for the control of actual modelled sea states. Figure 2 is now altered to include the description of E(KT,co) given in Eq. (2.5) with Em(KT,u))=0. Figure 4 reflects this modification. The linearity of the system is not affected since in this work DSPD(KT,u)) is chosen to be constant with respect to time, K. A typical sequence of events for the wavemaking system begins with the waveboard moving in a fashion prescribed by the Initial state of the FIGURE 4 SPECIFIC CLOSED-LOOP BLOCK DIAGRAM for COMPUTER SIMULATION E'(KT,OJ) E(KT,OJ) STSPD(KT,c )^ DSPD(KT,6o) 1/DSPD(KT,(Uj) CONTROLLER ASPD(KT,a>) WAVEBOARD 14. system for a timestep of duration, T. The actual wave characteristics are sensed and compared to the desired wave characteristics and the error is formed between the spectral densities at each frequency of interest in the spectrum. The waveboard motion is then adjusted based upon this discrete-time error and generates waves for another timestep of duration T in the newly specified manner. The cycle repeats itself in a discrete fashion until the discrete-time error is a satisfactory value. Figure 5 shows the sequence of waveboard motions qualitatively versus discrete-time. As mentioned, the motion of the waveboard is changed after each timestep based upon the discrete-time error. It is the task of the controller to calculate the new waveboard motion by operating on this error. 2.3 Control Strategy 2.3.1 PID Control Description A simple closed-loop configuration for the system is shown in Figure 4. In order to generate a specific wave amplitude or spectral density value at each frequency or sinusoid the waveboard must move with a specific amplitude at each frequency. The waveboard motion can then be expressed as a spectrum of amplitudes. These waveboard amplitudes can also be expressed in terms of spectral density. As a result, at each frequency of interest a waveboard stroke amplitude corresponds to a water wave amplitude. The water wave amplitude is then converted to a spectral density. By considering the control of the amplitudes a conventional and simple controller scheme can be employed. For this work a PID Control strategy was adopted to operate on the discrete-time error and calculate changes in the waveboard motion. The output of the controller is given as the change in the controlled variable, in this case the waveboard motion, as follows, FIGURE 5 QUALITATIVE SEQUENCE OF WAVEBOARD MOTIONS VERSUS DISCRETE-TIME 16. AST(KT,co) = KP(to).E(KT,co) K + KI(u).( S (E(jT,aj) + E((j-l)T,a)))).T/2 j - l + KD(d))»(E(KT,u)) - E((K-l)T,co) )/T (2.6) where KP(co) = proportional gain at frequency to KI(co) = integral gain at frequency co KD(co) = derivative gain at frequency co The proportional part of the output provides a change in the waveboard motion proportional to the error described in Eq. (2.5). The integral part provides a change dependent on a l l previous errors over time, specifically, the rate of change of the controlled variable with respect to discrete-time is proportional to the error. The derivative part of the controller output provides a change in the controlled variable proportional to the rate of change of the error with respect to discrete-time, an anticipative component. Therefore the new waveboard motion or stroke at the beginning of a discrete timestep is, ST(KT,co) = ST((K-l)T,co) + AST( (K-l)T,co) (2.7) where AST((K-l)T,co) = output of the PID controller given in Eq. (2.6). The new waveboard motion equals the previous motion plus the newly calculated change in waveboard motion. 17. 2.3.2 NRC Control Description The NRC Hydraulics laboratory wave generation system uses a similar control strategy in an attempt to correct for the nonlinear effects present in an experimental towing tank setup. In this system, the new waveboard driving spectrum is equal to the old driving spectrum plus a change in the driving spectrum. The change is given by an iteration factor multiplied by the error between the desired and actual sea state spectrums. The iteration factor used is given by, 6 = (j)-(o)/a» p e a k) a ; c o p e a k < u < V x (2.8) 1 j ; otherwise where Upeak = f r e c l u e n c y a t which the peak in the spectrum occurs to = maximum frequency of interest max n J a = arbitrary constant usually equal to 1/2 This control scheme is essentially proportional control with the propor-tional gain expressed as a function of frequency. 18. CHAPTER 3  SIMULATION The PID gains used in this work are frequency dependent but do not have a specific functional form. As an aid in the determination of which gain combinations to use at each frequency, whether under P, PI, PD or PID control, a simple simulation is performed which yields the performance of the system based upon the discrete-time error. 3.1 Wave Generation System Model The relationship between waveboard motion and wave height has been studied by many researchers in an attempt to achieve realistic sea states in laboratory wave tanks. Galvin (1965, 1966) concluded that wave heights predicted by the complete hydrodynamic theory agree well with experimental results. Madsen (1970) concluded that differences between predicted and observed wave heights could be attributed to leak-age past the waveboard sides as opposed to an inadequacy of the linear theory. Dean and Dalrymple (1984) state that linear wave theory predicts wave motion induced by wavemakers reasonably well. Therefore, in this work, linear wave theory is adopted for the simulation. The theory adopted is based upon the solution of Laplace's equation in terms of a velocity potential, <(), i.e. the linear theory of Airy (Le Mehaut€, 1976). Of course other wave theories exist and Dean (1970) presents a comparison of these in terms of the two free surface boundary conditions. 3.1.1 Wave tank Model and Assumptions A description of the actual wave tank and the model based upon this wave tank is necessary. The wave tank housed in the B.C. Research Ocean 19. Engineering Center is 67 meters long, 3.6 meters wide and 2.4 meters deep for its entire length. A maneouvering tank is adjacent to the wave tank at the paddle end and shares the same body of water. A partial wall separates the two tanks but during the waveboard motion there is water transfer between the tanks. For the purposes of this work i t is assumed that the wave tank is completely separate from the maneouvering tank. Figure 6 shows the geometry of the actual waveboard including the variables used in the wavemaker theory. The actual waveboard dimensions are incorporated into the simulation model. The dimensions are as follows: d = 2.44 meters, f = 1.40 meters, h = 1.04 meters maximum hydraulic cylinder stroke = 0.46 meters maximum stroke of waveboard at free surface of water = 0.35 meters In this work the tank is assumed to be infinitely long and there-fore no reflection waves are present in the simulation. In reality, the wave tank does have a wave absorbing beach at one end which will produce some reflective waves. There is leakage around the sides and bottom of the actual waveboard in both directions since water exists on both sides of the waveboard. The simulation model assumes no leakage past the waveboard. 3.1.2 Wavemaker Theory and Assumptions The relationship between the waveboard motion and the resulting wave height profile in the tank is obtained from the theory developed by 2 0 . Havelock (1929), Biesel (1951) and Hyun (1976), among others. The theory is based upon two-dimensional waves in a perfect incompressible, irrotational, inviscid fluid to a first order approximation. For these conditions, Laplace's equation holds for the velocity potential in the fluid. Therefore, The linearized boundary conditions for the specific waveboard and wave tank shown in Figure 6 are 32<|>/3x2 + 92<t>/3y2 = 0 (3.1) 3*/3y|y = _ d = 0 (3.2) : no flow through tank bottom 3())/9x| . = 3x/3t • v = n (3.3) : water particles at the waveboard move with the same velocity as the waveboard : the waveboard motion is given by x(y,t) = ST(y) sin(cot) 32<|>/3t2 + g«3<t>/3y = 0 @ y = 0 (3.4) : the dynamic free surface boundary condition stating that there exists a constant pressure on the water surface. 3c()/3y = 3n(x,t)/3t @ y = n (3.5) : the kinematic free surface boundary condition stating that the water particles at the surface do not leave the surface. 21. An additional condition and assumption is that a 'radiation* condition exists, i.e. waves travel in one direction only, that of increasing x, particularly at large x. Another assumption is that the potential § is finite everywhere within the region, x > 0, -d < y < 0. In reality, there will be reflection off the wave absorbing beach at one end of the tank which will send waves back towards the waveboard. Ursell, Dean and Yu (1959) and Patel and Ioannou (1980) discuss how the potential § may be altered to account for reflection from a wave absorbing beach. However, for this work i t is assumed there is no reflection. Using the boundary conditions specified and assuming a potential of the form, <t>(x,y,t) = <|>p(x,y) cos(cot) (3.6) then Eq. (3.2) can be re-written as, a V 8 y l y = -d = ° ( 3' 7 ) and Eq. (3.4) can be re-written as, <|> - (g/o)2)«3<t)p/3y = 0 @ y = 0 (3.8) Assuming that d> i s f i n i t e then there are an inf i n i t e number of P solutions to these conditions, specifically, <b = (uj/m )»cosh(m (y+d))«sin(m x) , n = 0 o o o o (3.9) 22. and -m x 4>n = -(a)/mn)«cos(mn(y+d) )«e , n = 1,2,... (3.10) so d> = Z C d> = C •(co/m )»cosh(m (y+d))«sin(m x) p r t n n o o o o n=0 °° -m x + I - C .(ui/m )«cos(m (y+d))«e n (3.11) . n n n n=l where m is the positive solution to u 2 = m g»tanh(m d), o o o and m is the positive solution to co2 = -m g«tan(m d) , n = 1,2, n n n The C and C coefficients in Eq. (3.11) are determined by using o n the lateral boundary condition at the waveboard, Eq. (3.3). The waveboard motion is given as x(y,t) = ST(y) sin(o)t) for each frequency where, ST(y) = ST «((y/h) +1) ; -h < y < 0 (3.12) 3L =0 ; -d < y < -h Therefore from Eq. (3.3) and the fact that cosh(m (y+d)) and o cos(mn(y+d)), n = 1, ...,<», form a complete set of orthogonal functions, the coefficients can be written as follows, 23. 0 ((i)/m )•/ ST(y) cosh(m (y+d)) dy o ' , o C = ~ d o 0 / cosh2(m (y+d)) dy -d 2u>ST •f((sinh(m d))/m ) - ((cosh(m d))/hm 2) + ((cosh(m (d-h)))/h m2)l a v o o 2 2 2 o m d + sinh(m d)«cosh(m d) o o o (3.13) and 0 -(to/m )•/ ST(y) •cos(m (y+d)) dy n ' n C = " d n 0 / cos2(m (y+d)) dy -d -2OJST .f((cos(m d))/hm2) + ((sin(m d))/m ) - ((cos(m (d-h)))/hm2)) a v n n n n n n ' m d + sin(m d)»cos(m d) n n n (3.14) Now the potential <))(x,y,t) can be written, <j>(x,y,t) = Co«cosh(mo(y+d))«sin(ut - ™ Q X ) » -m x + E C »e n .cos(m (y+d)).cos(wt) (3.15) , n n n=l This potential is composed of two parts, a progressive wave, the C o term, and a series of standing waves which decay with increasing distance, x, away from the waveboard. It has been shown, Dean (1984) that after a distance, x, of three tank depths away from the waveboard, the f i r s t term in the series is practically negligible. The wave height, n(x,t), can be written by using Eqs. (3.4) and (3.5) to yield, n(x,t) = -(l/g ) -8Kx,y,t)/3t| y = 0 (3.16) Now using Eq. (3.15) in Eq. (3.16) the wave height can be written as, ri(x,t) = -(u)/g)«C «cosh(m (y+d) )•cos(o)t - m x) o o o °° -m x + E wC »e n •cos(mn(y+d)).sin(cot) (3.17) n=l or, for x > 3d, n(x,t) = -((o/g) »CQ«cosh(mo(y+d) )«cos(u)t - mox) (3.17a) a progressive wave. The required relationship between the waveboard motion and the resulting wave form has now been determined. For use in the simulation i t is necessary only to look at the relation between the wave height amplitude, H , and the amplitude (stroke), ST , of the waveboard a a motion. Therefore, 2 5 . H /ST = a a 2u>2. cosh(mod) • ( ( ( s i n n e d ) )/mo)-(cosh(mod) )/hm2) )+( (cosh(mo(d-h) ) )/hm|) ) g»(m d + sinh(m d)»cosh(m d)) o o o (3.18) for each frequency of interest. 3.2 Computer Simulation and Controller Details 3.2.1 Controller Gain Orders of Magnitude The computer simulation was written primarily to test many controller gain combinations and to examine the associated effect on system performance. To determine a set of PID gains to use for each frequency of the wave spectrum an approximate range is calculated based upon the physical dimensions of the system. It is assumed that initially there are no waves present in the tank and the wavemaker is at rest in the vertical position. The simulation is done for many gain combinations within the calculated range. To determine the order of magnitude of a range for the KP(co) values consider KI(OJ) and KD(to) equal to zero. This condition gives the output of the controller as, AST(KT,u>) = E(KT,o))«KP((o), and therefore the stroke for the next iteration is the in i t i a l stroke, assumed equal to zero, plus the change in stroke. As mentioned in Section 2.3.1 the waveboard amplitudes or strokes can be expressed in terms of spectral density also. For convenience, this is done in the computer simulation. The controller gains are therefore calculated with this in mind. The output from the controller is then, FIGURE 6 GEOMETRY of WAVEBOARD and WAVE TANK I STROKE,^ ST@y=0 HYD. CYL. STROKE • WAVE AMPLITUDE, Ha s s s S y—/////// ON ASTSPD(KT,co) = E(KT,w)'KP(u)) (3.19) and the new waveboard motion in terms of a stroke spectral density is given as, STSPD(KT,oo) = STSPD((K-l)T,a>) + ASTSPD( (K-l)T,io) (3.20) where STSPD indicates that the stroke is expressed in spectral density form. The maximum physical stroke for the B.C. Research Ocean Engineering Center waveboard at the free surface of the water is 0.356 meters. In spectral density form, the maximum stroke spectral density is, STSPD(KT,u>) = ST(KT,u))2/(2Aa>) = (0.356)2/(2)(0.2) = 0.316 m2sec where Aco = 0.2 sec"1 is assumed as typical. The error given by Eq. (2.5) is initially equal to -1 since ASPD(0,u>) is assumed equal to zero. Therefore from Eq . (3.20) we can say that, IKP(co) I = 0.316/E(0,w) = max 0.316. This would be a maximum gain value starting from rest since the waveboard cannot exceed its maximum physical stroke. If the actual spectral density of the waves, ASPD(KT,u>) is greater than desired then the error in Eq. (2.5) is positive. Therefore, in order to decrease the stroke and accordingly ASPD(KT,oo), the change given by the controller output must be negative. Similarly, i f ASPD(KT,a)) is less than desired then the error would be negative. To increase the stroke, and ASPD(KT,u>), the change must be positive which again means that KP(w) 2 8 . must be negative. Therefore, an approximate range and order of magnitude for the proportional gain, KP(u>) is, -0.316 < KP(to) < 0 (3.21) To determine an approximate range and order of magnitude for the integral gain, KI (UJ ) , i t is convenient to eliminate KP(u)) and KD(ui) from the controller output. The change in waveboard motion is thus given by, ASTSPD(KT,u)) = KI(UD).(E(KT,U)) + E( (K-l)T,u>) ) «T/2 (3.22) where a discrete timestep duration, T, must be assumed. This duration in reality is dependent upon the time for the slowest wave in the spectrum to travel the distance from the waveboard to the sensing probe. Several values for T can be used in the simulation but a value of 60 seconds is used as typical for the calculations. Therefore, using STSPD(KT,to) = 0.316 and Eqs. (3.20) and (3.22) we have, max |KI(o))| = 0.316/f(-1+0)(60/2)1 = 0.011 max Again, as for KP(u), KI(u>) will be negative which means that an approxi-mate range and order of magnitude for the integral gain, KI (w) is, -0.011 < KI < 0 (3.23) for T = 60 seconds. 29. Similarly, an approximate range and order of magnitude for the derivative gain, KD(io) is found. Derivative control is never used alone but assuming so does aid in determining possible KD(io) values. The change in waveboard motion is assumed as, ASTSPD(KT,to) = KD(co).(E(KT,aj) - E((K-l)T,o>) )/T (3.24) This equation and Eq. (3.20) give |KD(u)| = 0.316/((-l-0)(l/60)) = 18.97 max v ' Therefore an approximate range and order of magnitude for the derivative gain, KD(u>) is, -18.97 < KD(io) < 0 (3.25) for T = 60 seconds. It should be noted that the given ranges for KP(<JO), KI(CO) and KD(<o) are applicable for a l l frequencies of interest. Also, an obvious point is that the maximum values for Kl(w) and KD(o>) are dependent upon the timestep duration chosen. 3.2.1 Computer Simulation It is necessary to discuss the details of the closed-loop system shown in Figure 4 and how each component is included in the simulation. As mentioned in Section 2.2.2 the error operated on by the controller is chosen as 30. E(KT,UJ) = (ASPD(KT .a)) - DSPD(KT, co) )/DSPD(KT, cu) ( 2 . 5 ) Using the conventional summing/difference junction shown in Figure 4 , then DSPD(KT,<o) - ASPD(KT,co) = E'(KT,co). In order to obtain the error in Eq. ( 2 . 5 ) i t is necessary to multiply E'(KT.co) by (-1/DSPD(KT, co)) . Normally, this would introduce a nonlinearity with respect to time but since DSPD(KT,co) is actually assumed to be constant for a l l K at each frequency then multiplying by the block (-1/DSPD(KT,w), does not change the system linearity. The output from the controller is given in Eq. ( 3 . 2 0 ) . This output is a new waveboard stroke spectral density for each frequency. The relationship between the waveboard stroke and the wave amplitude is given in Eq. ( 3 . 1 8 ) . The computer simulation is written such that the relationship between the waveboard motion and the resulting wave height is in terms of spectral density. This relationship between STSPD(KT,oo) and ASPD(KT,co) is obtained by converting the waveboard stroke and wave amplitude relationship, H /ST = WTF. This is Eq. ( 3 . 1 8 ) where WTF 3. 3. signifies, 'wave transfer function' . The wave spectral density, is given by, H2/2Aco at each frequency. Therefore, H2/2Aco = (ST •WTF)2/(2Aco) = (ST 2/2Aco) • (WTF2) a a a = (stroke spectral density)•(WTF2) and i t can be written, (STSPD(KT,OJ)).(WTF 2) = ASPD(KT,co) (3.26) 31. Equation (3.26) is used in the computer simulation. Now, this ASPD(KT,(o) is compared with DSPD(KT,<JJ), (actually DSPD(o))) to form the error and continue through the closed-loop system until a satisfactory error is achieved. Referring to the printout of program CBGANES in Appendix A and the flowchart in Figure 7, the simulation begins with the user selecting the desired frequency of interest and the frequency bandwidth for spectral density calculations, both in radians/sec. The maximum stroke spectral density, STMAX, is then calculated from this information using the maxi-mum stroke for the waveboard, STMAX = 0.063/AOJ. The 'wave transfer function', WTF, relating waveboard stroke to wave amplitude, or simi-larly waveboard stroke spectral density to the actual wave spectral density is then calculated using the first order linearized wave theory discussed in Section 3.1.2. The maximum ASPD(KT,u>) achievable at this frequency is calculated and displayed on the terminal screen as a guide for the selection range of DSPD(KT,io). In reality, a standard spectrum will likely be predetermined, for instance, from a Pierson-Moskowitz (Hudspeth, Jones and Nath, 1978) or an International Towing Tank Conference (ITTC)(Bhattacharyya, 1978) wave spectrum but for this work the DSPD(KT,w) value is chosen by the user. Initial and maximum controller gain values are chosen as well as the number of increments with which to step through the controller gain range. Finally, the discrete timestep duration, i.e. the time between successive changes to the waveboard motion, is chosen. The i n i t i a l conditions for the water wave height and the waveboard stroke are given, then the control loop is cycled through for each set of controller gains. The actual wave spectral density is calculated using the difference equation for the system. This equation is obtained by combining the discrete-time relationships for each part of the system and is written as, START t enter cu enter Act i calculate: STMAX, WTF. ADSPOMAX display: WTF. ASPOMAX enter DSFO enter initial KPM. NM. KDM enter tt d each gain to test enter maximum KFV), NM, KDM enter T initialize wave and Stroke terms calculate E<KW> calculate ASPD((K+1)W) calculate E((K+1)T,a>) calculate STSP0((K+1)Xo.) Set indicator set indicator® STSPD = STMAX 0 increment timestep, K F i g u r e 7. C o m p u t e r S i m u l a t i o n F l o w c h a r t 33. Store KP, N, KD combination Store indicators2 & 1 Store STSPO, ASPD, E (or all K Store UJAU>, WTF, DSPO.T select next KD select nextKJ select next KP Format Stored Data • C - ) Figure 7. Computer Simulation Flowchart (Continued) 34. ASPD((K+l)T,a)) = ((WTF2/DSPD(KT,u)).(KP(a))+KI(o))«T/2 + KD(u))/T)+2).ASPD(KT,a)) + ((WTF2/DSPD((K-l)T, co) ) • (-KP( oi)+KI( a)) • T/2 )-1) ) • ASPD( (K-l)T, co) + ((WTF2/DSPD((K-2)T,«))).(-KD(a))/T))-ASPD((K-2)T,a)) + (-KI(UJ).T.WTF2) (3.27) The error and the stroke spectral density are calculated at each time-step. For each gain combination, the number of timesteps, K, taken to achieve a consistent error value with magnitude less than 3% (0.03) is an indication of how well the particular combination performs. Thirty timesteps are used for each gain combination in order to examine the error versus discrete-time response. If the maximum achievable waveboard stroke spectral density is exceeded during the control loop an indicator (specifically, the last K at which STSPD > STMAX) is set and printed with the error versus time numerical results. For actual experiments, i f such a condition occurs, the new stroke spectral density value could be set to the maximum allowable value based upon the maximum allowable waveboard stroke, as shown in Figure 7. For the purposes of the computer simulation, i f the waveboard stroke spectral density value exceeds the maximum allowable, the simulation is permitted to continue cycling with no restriction on the stroke spectral density. If the stroke spectral density value becomes negative during the control looping i t is considered an unstable state since spectral density values are always positive. The control looping would be terminated for such a condition in an actual experiment but in this simulation i t is allowed to continue cycling with an indicator set at the last K value that a negative stroke spectral density occurred. An unstable state is assumed when the error value Increases for successive discrete timesteps. 35. The computer simulation cycles through as many gain combinations as desired and stores the error, E(KT,u>), the actual s p e c t r a l density, ASPD(KT,a>) and the stroke s p e c t r a l density, STSPD(KT ,to) , i n column form with respect to the number of di s c r e t e timesteps, K. The other pertinent parameters such as frequency, t o , desired s p e c t r a l density, DSPD (co) , wave transf e r function, WTF, timestep duration, T, frequency bandwidth, ACD, the c o n t r o l l e r gain values and the i n d i c a t o r s f o r the stroke s p e c t r a l density condition are also stored i n the same data f i l e . Appendix C contains the numerical output from the computer simulation f o r some of the many gain combinations tested. Figures 8 through 29 show the system response i n terms of E(KT ,u>) vs. discrete-time, K f o r the corresponding numerical output combinations shown i n Appendix C. 3.3 Results and Discussion The r e s u l t s from running the computer simulation at a s i n g l e frequency to = 1 rad/sec, a desired s p e c t r a l density DSPD(cio) = 0.004 m 2.sec and a d i s c r e t e timestep duration, T = 60 seconds w i l l now be discussed. Combinations of four proportional, s i x i n t e g r a l and s i x deriv a t i v e gains are used y i e l d i n g E(KT,w), ASPD(KT,oo) and STSPD(KT ,to) . The gain values tested are shown i n Table 1. The 144 combinations tested give the following c o n t r o l modes: P, PD, PI and PID. 3.3.1 Proportional Control The simulation predicts no change i n response f o r increased propor-t i o n a l gain i f the i n t e g r a l and d e r i v a t i v e gains are zero. In f a c t , f o r the i n i t i a l conditions s t a t i n g that no waves e x i s t i n the tank, the response i n i t i a l l y depends upon the Integral gain alone. This i s 36. KPM -0.079 -0.158 -0.237 -0.316 TABLE 1 CONTROLLER GAIN VALUES TESTED KIM 0 -0.002 -0.004 -0.006 -0.008 -0.010 144 COMBINATIONS IN TOTAL 37. evident from Eq. (3.27). Figure 8 shows a case for which KI(OJ) = -0.0, KD(OJ) = -1.8 and the i n i t i a l actual spectral density is zero. This response, or lack of i t , is typical for proportional control alone also. 3.3.2 Proportional-Derivative Control As indicated in the previous Section, the i n i t i a l conditions used in the simulation dictate that the i n i t i a l response depends upon the integral gain alone. Therefore, for proportional-derivative control, KI(to) = -0.0, and there is effectively no response, the same situation as for strictly proportional control and shown in Figure 8. If the in i t i a l actual spectral density is non-zero and the integral gain is zero, the response changes initially but quickly converges to a non-zero error value. Figure 9 shows a typical response for this condition. The response shows more oscillation in the first few timesteps as the derivative gain is increased and also as the proportional gain is increased. In general i f no integral gain is used in the controller, a steady-state error exists and the desired response is never achieved. 3.3.3 Proportional-Integral Control The results from this mode of control show a response which is mainly oscillatory. The period of oscillation, in terms of K, decreases as |KI (o ) ) | increases for a fixed |KP(u)) j value, i.e. the response oscillates more quickly. At low |KP(co) | values the response becomes unstable sooner with lower |KI(<JI))| values than at high |KP(u))| values. In general, the error response takes longer to reach 3% as |KI(u>)| increases and the larger |KP(w)| values tend to cause a more stable response at the higher |KI(w)| values. Upon comparison of Figures 15, 3 8 . 20 and 25 these trends are evident. As well, comparison of Figures 10 and 13, 16 and 19, 21 and 23, and 26 through 29 for various KI(co) at constant KP(w) and KD(co) show these trends. In order to determine whether the error has reached 3% i t must exhibit such a condition for successive timesteps. One timestep at which the error is less than 3% does not necessarily indicate that the response has stabilized. The system response Figures indicate the timestep after which the error magnitude remains below 3%. 3.3.4 Proportional-Integral-Derivative Control The response for this mode varies depending upon the combination of KP(OJ), KIO) and KD(oo) values tested. For low |KP(u>) I values the response is very slow and often unstable. It is best at low |KI(u>) | values and, for a constant KI( < o ) , as |KD(u))| increases, the response initially becomes quicker but then slows again at high |KD(co)| values. As the | K P ( U ) | value increases the response becomes more stable and is quicker. Again, at larger |KI(oo) | values the response becomes unstable for low |KD(OJ)| values. As |KD(to) | increases, the response is slow initially, then becomes quicker but slows again at the higher |KD ( w ) | values. At even larger |KP(oo)| values the response is more stable in general. Notably, for a constant KI(OJ), as |KD(OJ)| varies from small to large, the response becomes progressively slower. Therefore, depending upon the magnitude of the KP(to) value the response is not always the same for increasing |KI(OJ) | or increasing |KD(OJ)| values. In most cases, however, the response is oscillatory. The combinations in which 39. |KI(W)| are zero does not show any response, as mentioned in previous Sections. Six other combinations show a monotonically decreasing response with time, K. These combinations are for KP((o) = -0.237, KI(ui) = -0.002 and KD(io) = -1.8, -3.6, -5.4 and for KP(OJ) = -0.316, K K » = -0.002 and KD(OJ) = -1.8, -3.6, -5.4. Figures 21 and 25 show this type of response while Figure 22 shows how the response changes from Figure 21 as |KD(O))| is increased. In general, the quickest responses are for the largest |KP(<D)| values, small |KI(u>)| values and small |KD(co)| values. The best response times to achieve a consistent 3% error are on the order of ten timesteps. At the timestep duration T = 60 seconds used, a desired response is achieved at this frequency in ten minutes. In particular, the fastest responses were for the combination KP(co) = -0.316, KI(OJ) = -0.002, KD(to) = -0.0 (5 timesteps to achieve 3% error) and the combina-tion, KP(co) = -0.316, KI(u>) = -0.004, KD(u) = -3.6 (5 timesteps to achieve 3% error) in Figures 25 and 26, respectively. 40. CHAPTER 4 STABILITY ANALYSIS A proper selection of controller gains requires a stability analysis of the closed-loop system response. Of particular interest in this work is the stability of the system using a relatively long discrete timestep duration. The goal in this analysis is to provide some bounds on the controller gains, KP(u>), KI(co) and KD(OJ). 4.1 State Space Formulation and Eigenvalue Stability Analysis Referring to Figure 4 . for a closed-loop block diagram, the difference equations describing each part of the system are obtained. Consider the relationship between error, E(KT,OJ) and the stroke spectral density, STSPD(KT,u>) given as, Using this form and re-writing as a difference equation obtained through addition, the relationship can be written as, STSPD(KT,u>) = STSPD((K-l)T,u>) + KP(to)-E( (K-l)T,oo) K-l + KI(o)).( I (E((j)T,a)) T) - E((K-l)T,0)).T/2) j - o + KD(u>).(E((K-l)T,u>) - E((K-2)T,U)))/T (4.1) STSPD((K+l)T,a>) - 2 STSPD(KT,oo) + STSPD((K-l)T,u>) = (KP(w) + KI(OJ)-T/2 + KD(O))/T).E(KT,OJ) + (-KP(a)) + KI(OJ).T/2).E((K-1)T,OJ) + (-KD(oj)/T)^E((K-2)T,a>) (4.2) This equation can be compared to the di f f e r e n c e equation, C(KT) + a..C((K-l)T) + a •C((K-2)T) + ... a .C((K-n)T) >• *• n = b n.r(KT) + b . r ( ( K - l ) T ) + ... b «r((K-n)T) (4.3) i n order to write a state equation v i a the method of d i r e c t programming (Cadzow, 1970). By d i r e c t programming the state equation f o r Eq. (4.3) i s given as, x ((K+1)T) x J((K+l ) T ) x 3((K+l)T) " 31 _ a 2 *3 1 0 0 0 1 0 x (KT) x,(KT) x 3(KT) r(KT) (4.4) and C(KT) = ( b l f b 2 , b 3) x x (KT) x (KT) x, (KT) + b Q.r(KT) (4.5) where A . b l A b 2 A b 0 b l " b 0 a l b 2 - b Q a 2 b 3 " b 0 a 3 Therefore, f o r the c o n t r o l l e r given by Eq. (4.2), the state equation i s written as, 42. x ((K+l)T,co) x2((K+l)T,a)) x3((K+l)T,to) 2 - 1 0 1 0 0 0 1 0 x (KT.co) x (KT,co) x (KT.co) + E(KT,co) (4.6) and STSPD(KT,co) = [(KP(to) + KI(u)»T/2 + KD(io)/T), (-KP(co) + KI(o))«T/2), (-KD(co)/T)] x (KT , co ) x (KT , co) x , (KT , co) (4.7) The same procedure is done for the relationship between STSPD(KT,co) and ASPD(KT,co). The relationship is, ASPD(KT.co) = WTF2 • STSPD(KT, to), which by direct programming gives, and x. ((K+l)T,co) = [0] x (KT,co) + [1] STSPD(KT,co) ASPD(KT.co) = [0] x (KT.ui) + WTF2« STSPD(KT, co) (4.8) (4.9) Now introducing, E(KT,co) = (ASPD(KT,co) - DSPD(KT,co))/DSPD(KT,co) and combining Eqs. (4.6), (4.8) and (4.9) a state equation can be written as, x ((K+l)T,co) x (<K+l)T,io) x3((K+l)T,a)) x ((K+l)T,co) - 1 0 0 0 0 0 1 0 0 0 0 0 x (KT.co) x2(KT,co) x (KT.co) x (KT.co) WTF2/DSPD(KT,co) 0 0 1 STSPD(KT,co) 1 0 0 0 DSPD(KT,co) /DSPD(KT, co) (4.10) 43. Now introducing Eq. (4.7) and re-writing Eq. (4.10) for the system, Xl((K+l)T,co) x2((K+l)T,a>) x3((K+l)T,u)) Xl(((K+l)T,a)) l21 l31 l12 l22 l32 \2 l13 *2 3 l33 \ 3 x (KT , c o ) 1 x 2(KT,ai) 0 x 3(KT,w) 0 X 4 ( K T , O J ) 0 (4.11) or x((K+l)T,co) = F(T,0))«x(KT,tD) + G(T,co) • r(KT, to) in general, where, a u = 2 + (WTF2/DSPD(KT,co)). (KP(co) + KI(co).T/2 + KD(co)/T) a 1 2 = -1 - (WTF2/DSPD(KT,o))).(KP(co) - KI((o)«T/2) a u = -(WTF2/DSPD(KT,co)).(KD(o))/T) a l k = a 2 2 = a 2 3 = a2k = a 3 1 a 3 3 = 3 34 = = 0 a 2 1 = a 3 2 = 1 a 1 + 1 = KP(co) + KI(co).T/2 + KD(co)/T a^2 = -KP(co) + KI(co)«T/2 &hZ = -KD(co)/T The stability of the system can now be determined by examining the homogeneous case, x( (K+l)T,co) = F_(T,u))»x(KT,a)), and finding the eigen-values of the matrix F_(T,co). For stability, the magnitude of the eigen-values of F_(T,co) must be less than unity, that is, a l l eigenvalues must li e within the unit circle. This is analogous to a continuous-time system for which a l l eigenvalues must lie in the left half of the complex plane. 44. A short progam was written to calculate the eigenvalues of the F_(T,io) matrix, in Eq. (4.11), for the same combinations of controller gains used in the computer simulation. The problem reduces from a fourth order equation in X, the eigenvalue, to a cubic equation in A. The routine SGEEV on the UBC Mechanical Engineering VAX 11/750 computer system is used for determining the eigenvalues. The program EIGVALS incorporates SGEEV to determine the stability of the gain combinations. This program listing is contained in Appendix B. 4.2 Routh Stability Analysis An often used technique for determining system stability is the Routh criterion. The method eliminates the time consuming task of solv-ing for system characteristic equation poles or eigenvalues. Instead the values of the elements in the Routh array are examined for changes in sign. Traditionally, this method is used for continuous-time systems but a transformation of the unit circle in the z-plane to the imaginary axis in the 3-plane allows the Routh criterion to be applied to a discrete-time system. Thus, examining the coefficients of the system characteristic equation and the values of the elements in the first column of the Routh array will provide constraints for which to deter-mine stable controller gains. The characteristic equation of the entire system is obtained by transforming the discrete-time relationships for the system to z-domain expressions. Taking the z-transform of Eq. (4.2) and simplifying yields, 45. •p ( ~\ ( z 3 2z 2 + z).STSPD(z)  K } (KP(o))+KI(o)).T/2+KD((i))/T).z;i+(-KP(a))+KI(cu).T/2).z+(-KD(a))/T) (4.12) Also, taking the z-transform of the error equation, E(KT,co) = (ASPD(KT,(D) - DSPD(KT,OJ) )/DSPD(KT,ui) gives DSPD((o)«E(z) = ASPD(z) - DSPD(z) (4.13) considering DSPD(KT,co) = DSPD(w) as constant. Similarly, the relation, STSPD(KT,oo).WTF2 = ASPD(KT,OJ) (3.26) becomes, STSPD(z).WTF2 = ASPD(z) (4.14) Combining Eqs. (4.12), (4.13) and (4.14) gives the overall system transfer function as, (WTF2/DSPD). [a z 2 + a][z + aQ] ASPD(z)/DSPD(z) = (4.15) 3 9 b3Z + b 2z , i + bxz + b 0 where, 46. a £ = KP(co) + KI(o))«T/2 + KD(io)/T ax = -KP(OJ) + KI(co).T/2 a Q = -KD(a))/T b 3 = -1 b 2 = (WTF 2/DSPD).(KP((o) + KI(u>)«T/2 + KD(u>)/T) + 2 b x = (WTF 2/DSPD). (-KP(co) + KI(a)).T/2) - 1 b n = (WTF 2/DSPD)«(-KD(o))/T) Therefore, the c h a r a c t e r i s t i c equation for the system i s , • _ » ' » b 3 z 3 + b 2 z 2 + b Lz + b Q =0 (4.16) « t t i where, b g = b 3 / b 3 , b 2 = b 2 / b 3 , b ] L = bj/bg and b Q = b Q / b 3 . T h i s c h a r a c t e r i s t i c equation must be transformed from the z-plane to the 3-plane using the b i l i n e a r transformation, z = ( g + l ) / ( $ - l ) , i n order to use the Routh c r i t e r i o n f o r s t a b i l i t y . The c h a r a c t e r i s t i c equation i n terms of 3 i s then, ( B 3 e 3 + B 2 3 2 + BjB + B 0 ) / ( B - 1 ) 3 = 0 (4.17) where, B 3 = b3+b2+b|+bQ = -(WTF2/DSPD).KI(o)).T (4.17a) B 2 = 3b3+b2-b^-3bQ = (-2 WTF2/DSPD).KP(u))-(4 WTF2/DSPD) »KD(u))/T (4.17b) B 1 = 3b3-b2-b^+3bQ = 4 + (WTF2/DSPD)«KI(u)) «T+(4 WTF2/DSPD) • KD(aj)/T (4.17c) B0 = b 3 ~ b 2 + b l " " b 0 = 4 + ( 2 WTF2/DSPD).KP(a>) (4.17d) According to the Routh c r i t e r i o n the c o e f f i c i e n t s , BQ through B3 must be p o s i t i v e and the elements of the f i r s t column i n the Routh array must be p o s i t i v e i n order for s t a b i l i t y to p r e v a i l . The Routh array i s written as, < B 2 B r B 3 V / B 2 0 B„ 0 0 0 R e c a l l i n g that the c o n t r o l l e r gains KP(OJ) ,KI(OJ) and KD(to) are a l l negative, the c o e f f i c i e n t s BQ through B3 can be re-written i n terms of the magnitudes of the gains. Therefore, B0 = Bo = B, = B„ = (WTF2/DSPD). |KI(o») | «T (2 WTF 2/DSPD)« |KP((o) | + (4 WTF2/DSPD) • |KD(co) | /T 4 - (WTF2/DSPD). |KI(a)) | «T - (4 WTF2/DSPD) • |KD(u)) | /T 4 - (2 WTF 2/DSPD)«|KP(w)| (4.18a) (4.18b) (4.18c) (4.18d) The Routh element, (B2B1-B3BQ)/B2, then becomes, [8(WTF 2/DSPD)« |KP(u))| - 4(WTF 2/DSPD) • | Kl(u>) | • T - 8(WTFVDSPD 2). |KP(OJ) I • |KD(OJ) |/T + 16(WTF 2/DSPD). |KD(a>) 1/T - 4(WTF' T/DSPD 2). |KI(a))| • |KD(a))| - 16(WTF 1*/DSPD 2) • |KD(OJ) | 2 / T 2 ] / B 2 (4.18e) 48. With the conditions that BQ,B1,B2 and Bg and B^-B^g be greater than zero the constraints on the controller gains are determined for each frequency, desired wave spectral density and timestep duration. The constraints are shown on a gain chart of |KI(uo) | versus |KD(OJ)| for various |KP(io) | values. The coefficients, B3 and B 2 are always greater than zero and therefore provide no constraint on the controller gains. The coefficient, Bn, provides the constraint that, |KP (o») | < 2 DSPD/WTF 2 (4.19) and the coefficient, B provides the constraint that, (DSPD/WTF 2)• T - ( T 2 / 4 ) . |KI(o))| > JKD(CC) | (4.20) The expression (B„B -B Bn)/B„, provides the constraint that, (4.21) where a = 16(WTF 2/DSPD) 2/T 2 b = -(16(WTF 2/DSPD)/T - 8(WTF 2/DSPD) 2» |KP(to) | /T -4(WTF 2/DSPD) 2.|KI(u)|) c = -(8(WTF 2/DSPD). |KP(OJ) | - 4(WTF 2/DSPD)• |KI(OJ) |-T) These constraints are plotted in Figures 30, 31 and 32 and are discussed in the following Section. 49. 4 .3 Results and Discussion 4 . 3 . 1 Routh Stability Analysis Results The program, ROUTH, contained in Appendix B, is used to calculate the Routh stability constraints. The Routh stability constraints for the controller gains are plotted on Figure 30 for a specific case of frequency, to = 1 rad/sec, desired spectral density, DSPD = 0 . 0 0 4 m2.sec and discrete timestep duration, T = 60 seconds. The constraint, ( B 2 B 1 ~ B 3 B Q ) / B 2 > 0 is given by the four concave-down traces, each for different |KP(to)| values. The stable region lies below the curves at each |KP( t o ) | . Also plotted on the Figure is the constraint > 0 given by the straight line relationship between |KI(<o) | and |KD(to)| for a l l |KP(to) I values; points below this curve are again in the stable region. This Bj; constraint proves less restrictive than the (B 2B 1-B 3B Q)/B 2 constraint. Not plotted are the B 3, B 2 and B Q relationships since B 3 and B 2 are always greater than zero and B Q states that |KP(<o)| < 2 DSPD/WTF2 = 0 .586 independent of |KI(to)| and |KD(to)|. If KI(to) = 0 , the |KD(to)| axis in the plot, the constraint, B g is zero which indicates instability. The plot shows that i f |KP(to)| decreases, the range of values for the |KI(to)| and |KD(to)| gains becomes more limited for a stable response. As stable points approach the curves, the response tends more towards instability and conversely, points farther away from the constant |KP(io)| curves are more stable. The eigenvalue results presented in Section 4 . 3 . 2 agree with this entirely which is not surprising since the two analyses are based upon the same system characteristic equation. As well, the computer simula-tion responses agree with these stability results. 50. Figure 31 shows the result of varying the discrete timestep duration while holding |KP(to)|, to and DSPD fixed. For shorter timestep duration, T, the plot indicates more limitation on the range of |KD(co)| values and more freedom on the range of |KI(to)| values for a stable response. The opposite holds for longer timestep duration, T. This result can also be inferred from the PID controller relationship of Eq. (2.6). For large T the contribution to the change in waveboard motion is more dominant for the integral term than the derivative term and therefore, in the extreme, more prone to causing ins t a b i l i t y . Similarly, for short T the derivative term is more dominant than the integral term and more freedom in the range of |KD(to)| is afforded. The variation in the system stability resulting from a change in a particular frequency can be examined using the relationship between the waveboard stroke or stroke spectral density and the wave height response or wave spectral density. A change in frequency directly changes the 'wave transfer function', WTF, via the hyperbolic functions in the relationship. As to increases, WTF increases. Also, accordingly, the maximum wave spectral density achievable changes with frequency via ASPD(KT,to) = WTF2«STMAX. Recalling the controller algorithm and the max Routh constraints, the ratio, WTF2/DSPD is prominent. Hence, a change in frequency will change WTF and therefore the response. Similarly, i f the desired spectral density is changed the response will change also. However, i f the ratio, WTF2/DSPD remains unchanged after a change in frequency and desired spectral density at that frequency, then the response will remain unchanged from the previous frequency. The variation in the Routh stability curves for a change in desired spectral density, DSPD, is shown in Figure 32. As DSPD decreases, the range of stable gain values becomes more limited. For a decrease in DSPD the ratio, WTF2/DSPD, increases. Therefore, an increase in frequency, w, will have the same effect on the range of stable gains that a decrease in DSPD will. Also note that instability will prevail at a particular combination of frequency, |KP(oi) | and T i f the condition, DSPD < |KP(OJ) | •WTF2/2 is satisfied (recall the Routh constraint, B Q). Therefore, at a particular frequency, there is a maximum achievable wave spectral density dictated by the waveboard maximum stroke and there is a minimum wave spectral density dictated by the Routh stability constraint for a particular |KP((O) I value. However, i t is of more interest and also more practical to choose the gains knowing the desired spectral density and not vice-versa. 4.3.2 State Space and Eigenvalue Stability Results The eigenvalues for various controller gains are given in Appendix C for a frequency u> = 1 rad/sec, desired spectral density DSPD = 0.004 m2.sec and timestep duration T = 60 seconds. Again, these values are used to provide a comparison with the computer simulation performance and the Routh stability analysis. It is evident that for proportional control alone, one eigenvalue decreases with increasing |KP(u)) | but the other remains fixed and equal to one. For proportional-derivative control, KI(u)) = 0 and again one eigenvalue remains fixed and equal to one for a l l gain combinations. These results indicate instability just as the Routh analysis does. The variation of eigenvalues with changes in controller gains coincides 52. directly with the stability constraints dictated by the Routh Analysis. For example, for particular |KP(d>)| and |KI((JJ) | values while traversing with increasing |KD ( w )| on Figure 30, the eigenvalue becomes smaller as the (|KI(O))|, |KD(OL))|) point moves farther away from the constant |KP(u>) I curve. Therefore, the system should become more stable. Conversely, as a traverse along a constant |KI(u>) | line continues and the (|KI(u))|, |KD(CO)|) point moves closer to the constant |KP(o))| curve, the eigenvalue becomes larger with magnitude closer to unity. The fact that the two stability analyses agree is not surprising, as mentioned earlier. Due to the nature of the state equations three eigenvalues are determined by the program, EIGVALS. Two of the three eigenvalues are complex conjugates for every combination of gain values except for strictly proportional control and for the combination, KP(<A>) = -0.316, KI(o)) = -0.002, KD( c d ) = -0.0 for which the eigenvalues are real. The variation of eigenvalues with discrete timestep duration shows the same results as those mentioned for the Routh Analysis. At low |KI(to)| values the eigenvalues decrease with increasing timestep duration. But at larger |KI(<D) | values, the eigenvalues increase with increasing timestep duration. The derivative portion of the controller is dominant for short timestep duration whereas the integral portion of the controller is dominant for long timestep duration. Restated, as the timestep duration increases, the eigenvalues begin to decrease, but to a lesser extent at larger |KI(u)| values. As the timestep duration increases further s t i l l the eigenvalues then increase, especially at larger |KI(a))| values. The degree to which a system is stable is best determined by examining the performance of the system and how i t relates to the eigen-values . 53. CHAPTER 5 PERFORMANCE ANALYSIS The analysis of a control system would be incomplete i f stability alone were analyzed. In order to choose appropriate controller gains i t is necessary to examine how quickly a desired response is achieved. The fact that a combination of controller gains provides a stable response outwardly provides no information as to whether the response is oscilla-tory, highly damped or of any specific form. However, closer examina-tion of the equations governing the system reveal such information. 5.1 The Measure of Performance In a conventional discrete-time control system the relationship between the input and the output may be written as, C(z) = H(z)R(z), where H(z) is a ratio of polynomials in z determined from the system difference equation. The output, C(z), may also be written in terms of a partial fraction expansion, C(z) = Al/(z-\l) + A2/(z-X2) + + A /(z-X ) + expansion poles of R(z) n n (5.1) Now by inverse transformation, the time response is given by, C(KT) = A 1 \ l K-l K-l + + A X K-l + Z _ 1 (expansion of poles of R(z)) (5.2) n n 54. where X, ,X0...X are the poles or eigenvalues of the system character-J- ^ n ist i c equation. Hence, i f the magnitude of the eigenvalues are less than unity then as time, K, increases the magnitude of the response decreases. The nature of the eigenvalues indicates the type of response that can be expected. If the eigenvalue is real, positive and with magnitude less than one, the response is a monotonically decreasing sequence. If the eigenvalue is real, negative and with magnitude less than one the response is an oscillatory, decreasing sequence. If the eigenvalue is real with a magnitude of one then the response remains at a fixed oscillatory or constant value, i.e. a case of limited stability prevails. Another possibility for the nature of the eigenvalue exists, that of complex values and in particular complex conjugate pairs. For this case the response is written as follows: A i K—1 , , , K—1 . . 10 >K— 1 . . ,. — i0«K— 1 + k2 2 = A1»(Ae ) + A2« (Ae ) The real portion is comprised of the magnitude, A of the eigenvalue, the coefficients from the partial fraction expansion, A^,A2,...t and an oscillatory term. Thus, the response is a decaying oscillatory one for A < 1. How fast the response oscillates depends upon the value of 8 determined by the inverse tangent of the imaginary part of the eigen-value divided by the real part. As 6 increases, the response oscillates more quickly in terms of discrete-time, K. 55. For t h i s work, the response can be written knowing the o v e r a l l system trans f e r function i n the z-domain, ASPD(z)/DSPD(z), as given i n Section 4.2. Now, ASPD(z) = Al/(z-Xl) + A 2 / ( z - A 2 ) + A 3 / ( z - A 3 ) + expansion poles of D(z) (5.3) by p a r t i a l f r a c t i o n expansion, where \^ , A 2 and A G are the system eigenvalues and A^.Ag, and A 3 are the c o e f f i c i e n t s from the expansion. The transient response i n the time domain i s therefore, ASPD(KT,OJ) = A j X ^ " 1 + A 2 X 2 K - 1 + A ^ ^ " 1 (5.4) This expression i s a solution of the difference equation, ASPD(KT,w)b 3 + ASPD((K-l)T,w)b 2 + ASPD( (K-2)T,w)b]L + ASPD((K-3)T , o))b 0 = DSPD( (K-l)T, a>) • (WTF 2/DSPD)a 2 + DSPD((K-2)T,w)-(WTF 2/DSPD)a 1 + DSPD( (K-3)T,u>) • (WTF 2/DSPD)a 0 (5.5) where the c o e f f i c i e n t s , a Q . . . a 2 and b Q...b 3 are those from Eq. (4.15). The Eq. (3.27) i s a s i m p l i f i c a t i o n of Eq. (5.5). The c o e f f i c i e n t s , A ^ A 2 and A G i n Eq. (5.4) could be solved f o r by employing the i n i t i a l conditions f o r the system. For instance, f or no waves i n the tank i n i t i a l l y , ASPD(-2T ,OJ) , ASPD(-T,a)) and ASPD(0,a)) are assumed equal to zero. Therefore, using Eq. (5.4) the c o e f f i c i e n t s could be determined. These c o e f f i c i e n t s w i l l be i n terms of the eigenvalues of the system. 56. It is sufficient for this work to examine the form of the solution for different eigenvalues. The response ASPD(K.T,co) forms part of the error expression E(KT , o o ) so the performance or response can be measured in terms of E(KT,OJ). This is what the computer simulation, discussed in Section 3.2.1, determined. 5.2 Controller Gain Selection As an aid in selecting the controller gains which yield a desired, stable response in an operator-determined reasonable time, the magnitude of the largest eigenvalue can be plotted against one controller gain for given values of the remaining two gains. This gives a performance indi-cation since as eigenvalue magnitudes decrease the desired response will be achieved more quickly. Figures 33, 34, and 35 show the variation of the largest eigenvalue versus controller gain for several gain combinations, a frequency u> = 1 rad/sec, a desired spectral density DSPD = 0.004 m2.sec, and a timestep duration T = 60 seconds. The general trends will now be discussed. The magnitude of the largest eigenvalue remains fixed at unity for proportional control alone and for PD control, as |KP(oo)| is increased. For PID control the eigenvalues generally decrease with increasing |KP(co)|. However as seen in Figure 33, some combinations yield a concave-up trace. In such cases, the i n i t i a l decrease is due to complex conjugate pair magnitude decreasing until the third real eigenvalue magnitude becomes larger after a certain KP(OJ) gain value. Examples showing this are |KI(u)| = 0.002, |KD(OJ)| = 1.8 and |KI(u>)| = 0.002, |KD(OJ)| = 7.2. This could be interpreted that the response tends toward 57. instability in a manner described by the real eigenvalue for larger proportional gains. In general, the eigenvalues increase with increasing |KI(oo) | values. The eigenvalue at |KI(<o)| = 0 is equal to one which causes a concave-up trace but as well, some cases show that the real eigenvalue is larger at low |KI(to) | , and decreases until the complex conjugate pair becomes larger at higher |KI(w)| (see Figure 34). In other words, for these cases the response tends towards instability in a manner dictated by the complex conjugate eigenvalues for larger integral gains. The combinations, |KP(u)| = 0.316, |KD(io)| = 1.8 and |KP(u>) | = 0.316, |KD(u))| = 3.6 exhibit this behaviour. The eigenvalues both increase and decrease with increasing |KD(u>) | depending upon the gain combination. In each case, however, the complex conjugate eigenvalues are larger than the real eigenvalues except for the two cases, |KP(a>)| = 0.237, |KI(ui)| = 0.002 and |KP (OJ ) | = 0.316, |KI((o) | = 0.002. For these two cases the eigenvalues increase with increasing |KD(w)|. For the cases in which the complex conjugate eigen-values are larger, the traces are concave-up (see Figure 35). This behaviour agrees with the Routh analysis results, discussed in Section 4.3.1, depicting a traverse along a constant | KI (co) | line under the constraint curve. The computer simulation results also agree with this behaviour. As the eigenvalues increase, the error response takes more time to reduce below 3%. In general, the complex conjugates dictate the stability and performance for |KD(OJ)| variation. An additional method of gain selection which is not employed in this work is that of controller gain optimization. The gains are optimized based upon a performance index such as the sum of the squared 5 8 . error or the sum of the error since i t is generally desirable to mini-mize the error. This approach has been adopted in continuous-time systems. The same approach could be attempted for the case for a long discrete timestep duration but is beyond the scope of this work. An operator could use the eigenvalue versus gain trends to select gains that will yield a satisfactory response. In practise i t is essen-t i a l that some integral gain is included to eliminate any steady-state error. 5.3 Results and Discussion Upon comparison of the computer simulation results for error, E(KT,to) (or equivalently ASPD(KT,u))) a positive agreement is evident. When the Routh and eigenvalue analyses predict a stable or unstable response, the computer simulation exhibits respectively a stable or unstable response. As a stable eigenvalue becomes larger, a (|KI(cu)|, |KD(U))|) point on the Routh stability chart moves closer to the constraint curve and the computer simulation response takes more time to achieve a consistent error below 3%. The reverse also holds true. This agreement is not surprising since the two stability analyses and the computer simulation are based upon the same system characteristic equation and difference equations. The type of response predicted by the nature of the eigenvalues agrees with the actual response obtained by the computer simulation. The response is oscillatory when two of the eigenvalues are complex conjugates. The response is monotonically decreasing when the eigen-values are purely real (see Figure 25). For oscillatory responses, as the angle given by the inverse tangent of the eigenvalue imaginary over 5 9 . real parts increases, the response oscillates more quickly. These results are predicted by the theory in Section 5.1. The gain combinations for which KI(OJ) = 0 show no wave response and a consistent error of -1.0. The eigenvalues for this condition predict that part of the response due to the real eigenvalue of magnitude equal to one, should be a sequence of constant values. The eigenvalues also predict that the response should show some oscillation due to the complex conjugate pair eigenvalues for these particular gain combina-tions. However, the i n i t i a l conditions for wave height dictate the subsequent response and for the particular case where no waves exist initi a l l y , a KI(u) value of zero produces no response. As mentioned in Section 3.3.2, i f the in i t i a l conditions are such that waves do exist, the error response does show some oscillation for the first few timesteps and then converges to a sequence of constant values. Six gain combinations produce monotonically decreasing error responses when the associated complex conjugate eigenvalues predict that some oscillation should exist. It is likely that the coefficients, A n discussed in Section 5.1, are very small or zero for the complex eigen-values, leaving the real eigenvalue to dominate the response. The six gain combinations are mentioned in Section 3.3.4. In general, the computer simulation performance agrees well with the stability analysis. 60. CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK 6.1 Conclusions A control strategy for the discrete-time control of a particular wave generation system has been modelled using conventional discrete-time control theory. The main difference from conventional discrete-time control theory in this work is the use of a long duration discrete timestep between successive controller actions. This work has shown that stability analysis is possible and useful for determining control parameters. From a performance standpoint, the type of response predic-ted by the conventional discrete-time theory used agrees with the response obtained from the computer simulation model for a long duration timestep, T. In general the response based upon the simulation is the same as conventional theory in that an oscillatory response indeed usually prevails including varying speeds of oscillation depending upon the controller parameters. Based upon the discrete-time required to obtain an error response less than 3% and upon the stability of the response, the best controller gain combinations can be selected. For a system with a long duration discrete timestep starting from rest the system controller must include some integral gain in order for a desired response to be realized. This work assumes that for each frequency in the desired wave spectrum, there be a unique controller gain combination as opposed to using one gain combination for a l l frequencies. 61. 6.2 Suggested Subsequent Investigation Several areas could be examined to further this work: 1) For performance analysis the method of gain optimization based upon performance indices with respect to error could be developed for this discrete-time case. 2) An examination of the eigenvalue coefficients in the ASPD(KT,a>) response formulation to reinforce the response results obtained by the computer simulation. 3) Coordination of an automatic control package for the actual wave generation system at B.C. Research including a l l software and hard-ware necessary to operate the system. The adoption of methods similar to those employed at the NRC Hydraulics Laboratory would be useful combined with the controller theory contained in this work. 4) As a preliminary to an actual implementation of a control package at B.C. Research, a more detailed computer simulation could be developed. This simulation could include the effects of other physical components in the system such as A/D and D/A converters and associated measurement errors, transmission cables with the associated time lag, the waveboard servo system, the wave absorbing beach including reflections and a user interface. A more elaborate version of the computer simulation performance in this work will be discussed in the next Section. 6.2.1 Proposed Practical Simulation Based Upon the Present Work The computer simulation in this work tests one frequency at a time including one desired spectral density at that frequency. This is done 62. to simplify the performance and stability analysis for variations in the system parameters. To this end the simulation is very useful and infor-mative. In reality a spectrum of frequencies comprise a sea state. It is therefore desirable to discuss a simulation which can incorporate many frequencies along with the associated desired spectral densities. Initially, an operator enters the number of frequency components in the spectrum to be modelled along with the actual frequencies. The 'wave transfer function' is then calculated for each frequncy. The operator enters the desired value for the wave spectral density at each frequency based upon a previously successful waveform or upon a standard model spectrum. The discrete timestep duration is then calculated knowing the slow-est wave component in the spectrum. This timestep duration is used for a l l frequencies. As a reference the maximum controller gain values are displayed to the operator to aid in the gain selection. Also, past experience based upon stability at each frequency would be a guide to gain selection. Once a specific set of controller gains is selected at each frequency, the control iteration commences automatically. The iteration requires the calculation of the required waveboard amplitudes based upon the stroke spectral density at each frequency. For each discrete timestep, K, the superposition of new waveboard amplitudes over the frequency range is calculated based upon the new spectral densities over the frequency range. The iteration proceeds automatically until satisfactory actual characteristics are achieved at a l l frequencies or until the operator interrupts. As the iteration proceeds the spectral density curves could be displayed to the operator. The decision for whether the actual wave spectrum is satisfactory can be 63. made automatically based upon the condition that a l l immediately previous positive and/or negative error values at every frequency be less than a desired tolerance perhaps 3% (a typical tolerance suggested by towing tank personnel). Once a satisfactory wave spectral density spectrum is achieved the associated waveboard motion in terms of waveboard amplitudes at each frequency is stored for future use along with the actual wave spectrum. The software will automatically manipulate the data and task files to form an operator oriented system. An option in the software for simulations and for in-tank experi-ments is the graphical, real-time representation of the moving waveboard and the waveform travelling down the tank. This animation would serve as a visual aid to an operator at a remote computer terminal. In this work a crude example of the graphical depiction of the waveboard and waveform is attempted. The animation is not in real-time and depends upon the speed of the graphics package used. The program and a typical screen layout are presented in Appendix D. This graphical representa-tion serves as an example of what can be done to assist an operator. FIGURE 8 ERROR vs DISCRETE-TIME KPM = -0.158 KIM= -0.0 KDM- -1.8. n i i i i r 5 10 15 20 25 30 DISCRETE TIME, K FIGURE 9 ERROR vs DISCRETE-TIME 1.0-1 K P M = -0.158 K I M = -0.0 KD(ct>) = -5.4 * - INITIAL ACTUAL SPECTRAL DENSITY = 0.001 m2 sec. 0.5-I g DISCRETE g o , , r- 1 — r r T , M E ' K <jg 5 10 15 20 25 30 -0.5-ON ON FIGURE 12 ERROR vs DISCRETE-TIME K P M = -0.079 K I M = -0.002 KD(co)= -9.0 ERROR <0.03 DISCRETE >TIME, K 30 FIGURE 14 ERROR vs DISCRETE-TIME K P M = -0.079 K I M = -0.004 K D M = -3.6 o FIGURE 21 ERROR vs DISCRETE-TIME KP(co)= -0.237 K I M = -0.002 KD(co)= -1.8 ERROR <0.03 i 10 15 20 25 DISCRETE -jTIME, K 30 -^ 1 FIGURE 26 ERROR vs DISCRETE-TIME K P M = -0.316 K I M = -0.002 K D M = -3.6 ERROR <0.03 15 20 25 DISCRETE - fTIME,K 30 - 1 . 0 H > FIGURE 27 ERROR vs DISCRETE-TIME K P M = -0.316 KI(co) = -0.004 K D M = -3.6 15 20 25 DISCRETE ^TIME.K 30 0 0 FIGURE 29 ERROR vs DISCRETE-TIME 1.0 - I K P M - -0.316 K I M - -0.008 KD(co) - -3.6 FIGURE 30. ROUTH STABILITY CONSTRAINTS. W(a>)[ vs K D M I FOR VARIOUS K P M I K D M I IKIMI 0.013-0.012-0.011-0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 FIGURE 32. ROUTH STABILITY CONSTRAINTS, IKIMI vs IKDMI FOR VARIOUS DSPDM K D M I 90. 3mVAN390 JLS39HV1 RGUHE35. LARGEST EIGENVALUE vs ¥D(co)\ FOR VARIOUS KPMI AND KIMI KDMI 92. REFERENCES 1. Anderson, CH. and Johnson, B., "A Computer Controlled Wave Generation System for the U.S. Naval Academy", Proceedings of the 18th American Towing Tank Conference, 1977. 2. Bendat, J.S., "Statistical Errors in Measurement of Coherence Functions and Input/Output Quantities", Journal of Sound and Vibration, Vol. 59, No. 3, 1978, pp. 405-421. 3. Benedict, T.R. and Bordner, G.W., "Synthesis of an Optimal Set of Radar Track While Scan Smoothing Equations", IRE Transactions on Automatic Control, Vol. AC-7, No. 4, July 1962, pp. 27-32. 4. Biesel, F., "Etude theorique d'un certain type d'appareil a houle", La Houille Blanche, No. 2, 1951. English Translation in Project Report 39, March 1954, St. Falls Hydraulic Laboratory, University of Minnesota, Minneapolis, Minn. 5. Bhattacharyya, R., Dynamics of Marine Vehicles, John Wiley & Sons, New York, 1978. 6. Cadzow, J.A. and Martens, H.R., Discrete-Time and Computer Control  Systems, Prentice-Hall, Englewood Cliffs, N.J., 1970. 7. Cherchas, D.B., "Advanced Mechanical Systems Control", UBC Course  Notes, 1982. 8. Dean, R.G., "Relative Validities of Water Wave Theories", Proceedings of the ASCE, Journal of the Waterways and Harbours Division, Feb. 1970, pp. 105-119. 9. Dean, R.G. and Dalrymple, R.A., Water Wave Mechanics for Engineers  and Scientists, Prentice-Hall, Englewood Cliffs, N.J., 1984. 10. Dorato, P., "Theoretical Developments in Discrete-Time Control", Automatica, Vol. 19, No. 4, July 1983, pp. 395-400. 11. Flick, R.E. and Guza, R.T., "Paddle Generated Waves in Laboratory Channels", Journal of the Waterways, Port, Coastal and Ocean Division, ASCE, Vol. 106, Feb. 1980. 12. Funke, E.R. and Mansard, E.P.D., "The NRC 'Random' Wave Generation Package", Technical Report 1984/04, Hydraulics Laboratory, National Research Council, Ottawa, Canada, 1984. 13. Galvin, C.J. Jr., "Heights of Waves Generated by a Flap-Type Wave Generator", Report of Research Progress, U.S. Army Coastal Engineering Research Center, Washington, D.C., 1965/66, pp. 54-59. 14. Harrison, H.L. and Bollinger, J.G., Introduction to Automatic  Controls, Harper and Row, N.Y. 2nd ed., 1969. 93. 15. Havelock, T.H., "Forced Surface-Waves on Water", Philosophical Magazine, Series F, Vol. 8, 1929, pp. 569-576. 16. Hudspeth, R.T., Jones, D.F. and Nath, J.H., "Analysis of Hinged Wavemakers for Random Waves", Proceedings of the 16th Coastal Engineering Conference, ASCE, Hamburg, West Germany, 1978, pp. 372-387. 17. Hudspeth, R.T. and Chen, M.C., "Design Curves for Hinged Wavemakers: I-Theory", Journal of the Hydraulics Division, ASCE, Vol. 107, No. HY5, Proc. Paper 16236, May 1981, pp. 533-552. 18. Hyun, J.M., "Theory for Hinged Wavemakers of Finite Draft in Water of Constant Depth", Journal of Hydronautics, Vol. 10, No. 1, Jan. 1976, pp. 2-7. 19. Kuo, B.C., Analysis and Synthesis of Sampled Data Control Systems, Prentice-Hall, Englewood Cliffs, N.J., 1963. 20. Kuo, B.C., Discrete-Data Control Systems, Prentice-Hall, Englewood Cliffs, N.J., 1970. 21. Le Mehaute, B., An Introduction to Hydrodynamics and Water Waves, Springer-Verlag, New York, 1976. 22. Madsen, O.S., "Waves Generated by a Piston-Type Wavemaker", Proceeding of the Twelfth Coastal Engineering Conference, Vol. 1, Sept. 1920, pp. 589-607. 23. Patel, M.H. and Ioannou, P.A., "Comparative Performance Study of Paddle and Wedge-Type Wave Generators", Journal of Hydronautics, Vol. 14, No. 1, Jan. 1980, pp. 5-9. 24. Raven, F.H., Automatic Control Engineering, McGraw-Hill Book Co., N.Y., 3rd ed., 1978. 25. Ursell, F., Dean, R.G. and Yu, Y.S., "Forced Small-Amplitude Water Waves: A Comparison of Theory and Experiment", Journal of Fluid Mechanics, Vol. 7, Part 1, 1959, pp. 33-52. 94. APPENDICES APPENDIX A COMPUTER SIMULATION PROGRAM LISTING 96. 17-Sep-1986 22:37 17-Sep-1986 22:37 0001 C. PROGRAM CBGANES 0002 C 0003 C. STEVE HODGE 31712771 0004 C 0005 C . ..PROGRAM TO TEST MANY CONTROLLER GAIN COMBINATIONS FOR A SPECIFIC 0006 C. ..FREQUENCY, W, A DESIRED SPECTRAL DENSITY, DSPD, AND A TIMESTEP 0007 C. ..DURATION, I , IN ORDER TO EXAMINE THE PERFORMANCE OF THE MODEL 0008 c. ..AND CONTROLLER. THE ERROR IS CALCULATED AS A FUNCTION OF DISCRETE 0009 c. ..TIME. 0010 c 0011 c. ..DECLARE THE VARIABLES USED IN THIS PROGRAM 0012 c 0013 REALAS Z<500,3),ASPD<50,500),Q,DKP,DKI,DKD,DU, 0014 A ERR2,ERR0R(5O,500),KP(2O),KK20) ,KD(20),MO,MM(21), 0015 A PRODO,STMAX,STSPD(50,500),T,W,WTF,KPMAX,KIMAX, 0016 k KDMAX,ASPDMX,SPUTF,B1,B2 , B3 0017 c 0018 INTEGER K,IEND,JEND,LEND,M/1/,MMAX,KAY(50 ) ,KEND(500) , 0019 k S(500,3)/1500A0/ 0020 c 0021 c. ..READ IN THE FREQUENCY AND THE SPD FREQUENCY BANDWIDTH. 0022 c. ..CALCULATE THE MAXIMUM STROKE SPECTRAL DENSITY. 0023 c 0024 WRITE(6,A), 'ENTER THE FREQUENCY, W=' 0025 READ<6,5), W 0026 WRITE<6,A), 'ENTER FREQUENCY BANDWIDTH FOR SPD, DW=' 0027 READ<6,5), DW 0028 STMAX=.063226/DU !! < STROKE SPD FROM MAX. WAVEMAKER STROM 0029 c 0030 c. ..CALCULATE THE TRANSFER FUNCTION, WTF, RELATING WAVEMAKER STROKE 0031 c. ..TO WATER WAVE AMPLITUDE. DISPLAY WTF AND THE MAXIMUM WAVE SPECTRAL 0032 c. ..DENSITY FOR THIS FREQUENCY. 0033 c 0034 c. ..NEWTON'S METHOD IS REQUIRED 0035 c 0036 MM<1)=.5 0037 Q=<WAA2)A.249 0038 DO 30 N=l,20 0039 MM(N+1)=MM<N) + <<Q-MM<N)ADTANH<MM<N)) ) /<DTANH(MM(N)) + O040 A (MM(N)/(DCOSH(MM<N))AA2) ) ) ) 0041 IF(MM<N+1).LE.O. ) THEN 0042 MM<N + 1)=MM(N) + . 1 0043 GOTO 30 0044 END IF 0045 ERR2=ABS(MM(N+1>-MM(N)) 0046 IF(ERR2.LE..001) GOTO 35 0047 30 CONTINUE 0048 35 M0=MM(N+1)/2.44 0049 PR0D0=M0A2.44 0050 c 0051 WTF = 2.AMOADSINH<PRODO)A(<DS INH<PRODO)/MO)-(DCOSH<PRODO)/ 0052 A ( 1 . 04ACMOAA2))) + (DCOSH<MOA 1 .4)/(1.04A<M0AA2) ) ) ) / 0053 A (PRODO + DSINH<PRODO)ADCOSH(PRODO) ) 0054 SPWTF=WTFAA2 0055 c 0056 WRIIE<6,15), WTF 0057 ASPDMX=STMAXA(SPUTF) 97. CBGANES$MAIN 17-Sep-l986 22:37 17-Sep-1986 22:37 0053 URITE<6,27), ASPDMX 0059 C 0060 c. ..READ IN THE DSPD, INITIAL ASPD, INITIAL GAINS, NUMBER OF GAIN INCREMENT 0061 c. ..MAXIMUM GAIN VALUES AND THE TIMESTEP DURATION. 0062 c 0063 URITE<6 , A > , 'ENTER THE DESIRED SPECTRAL DENSITY, DSPD=' 0064 READ(6,10), DSPD 0065 WRITE<6 ,A>, 'ENTER THE INITIAL ACTUAL SPECTRAL DENSITY, ASPD=' 0066 R E A D ( 6 , 1 0 ) , ASPD(2,1) 0067 U R I I E ( 6 , A > , 'ENTER THE INITIAL CONTROLLER GAINS, KP, K I , KD' 0068 READ(6,A ), K P ( 1 ) , K I ( 1 ) , K D ( 1 ) 0069 U R I T E ( 6 , A ) , 'ENTER * OF GAIN INCREMENTS, IEND, JEND, LEND ' 0070 READ(6,25>, IEND,3 END,LEND 0071 U R I T E ( 6 , A ) , 'ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX, KDMAX' 0072 READ<6 , A ) , KPMAX,KIMAX,KDMAX 0073 URITE<6 , A > , 'ENTER DISCRETE TIMESTEP DURATION, T=' 0074 READ(6,20), T 0075 5 FORMAT(F4.2 ) 0076 10 FORMAT(F5.4) 0077 15 FORMAT(' WAVE TRANSFER FUNCTION=',T25,E7.5) 0078 20 FORMAT(F5.2) 0079 25 FORMAT < 312) 0080 ' 27 FORMAT(' MAXIMUM WAVE SPECTRAL DENS IIY= ' ,T32,F6.4) 0081 c 0082 MMAX=IENDkJENDkLEND 0033 DKP = KPMAX/<FLOAT( IEND) ) 0084 DK I=KIMAX/<FLOAT(JEND-1>> 0035 DKD=KDMAX/<FLOAT(LEND-l)) 0086 c 0087 c. ..COMMENCE THE LOOPING THROUGH THE CONTROL SYSTEM 0088 c 0089 c 0090 DO 300 1=1,IEND 0091 DO 200 J=1,JEND 0092 DO 100 L = 1,LEND 0093 ASPD<1,M)=0. 0094 ASPD(2,M)=0. 0095 ASPD(3,M)=0. 0096 STSPD<2,M)=0. 0097 DO 80 K=3,33 0093 KAY(K)=K-3 0099 c 0100 c. ..CALCULATE COEFFICIENTS FOR THE SYSTEM DIFFERENCE EQUATION 0101 c 0102 B1=KP(I)+KI<J)AT/2.+KD(L)/T 0103 B 2=-KP<I)+KI<J)AT/2. 0104 B3=-KD<L)/T 0105 c 0106 c. . .SET UP THE DIFFERENCE EQUATION IN TERMS OF ASPD 0107 c 0108 ERROR<K,M> = <ASPD<K,M)/DSPD)-1 . 0109 c 0110 ASPDCK+l,M)=<BlAWTFAA2/DSPn+2.)AASPD<K,M) 0111 A +CB2AUTFAA2/DSPD-1.)AASPD(K -1,M) 0112 A +(B3AWTFAA2/DSPD)AASPD<K-2,M) 01 13 k -<B1+B2+B3)AWTFAA2 0114 c 98. CBGANES*MAIN 17-Sep-198& 22:37 17-Sep-1986 22:37 0115 C. ..CALCULATE THE ERROR ASSOCIATED WITH THE ASPD 0116 C 0117 ERROR<K+l,M)=<ASPD<K+l,M)/nSPD)-l. 0118 c 0119 c. .-CALCULATE THE STROKE SPECTRAL DENSITY 0120 c 0121 STSPD(K+1,M>=ASPD(K+1,M)/<WTFA*2> 0122 c 0123 c. ..CHECK IF WAVEMAKER STROKE SPD HAS BEEN FORCED -VE...IF SO THEN 0124 c. ..SET INDICATOR AND TRY ANOTHER GAIN. 0125 c 0126 IF(STSPD(K+1,M).LT.-0.000001) THEN 0127 S(M,2)=K-1 !! <0 INDICATOR 0128 KEND(M)=K+1 0129 c GOTO 81 0130 END IF 0131 c 0132 c. ..CHECK IF WAVEMAKER STROKE SPD HAS EXCEEDED PHYSICAL LIMITATIONS 0133 c. ..OF THE WAVEMAKER MOTION..' IF SO SET INDICATOR % SET STSPD TO STMAX 0134 c \-NOT IN THIS PROGRAM' 0135 IF(STSPD<K+1,M>.GT.STMAX) THEN 0136 S(M,3)=K-1 !! >STMAX INDICATOR 0137 c STSPDOC + l , M ) =STMAX 0138 END IF 0139 c 0140 80 CONTINUE 0141 c 0142 K E N D < M > = K -1 0143 c 0144 c 0145 c. ..ASSIGN THE GAIN VALUES TO MEMORY 0146 c 0147 81 Z(M,1)=KP(I) 0148 Z(M,2)=KI(J) 0149 Z(M,3)=KD(L) 0150 M = M+1 0151 c 0152 c. ..COMBINATION M HAS NOW BEEN TESTED 0153 c 0154 c. ..TRY ANOTHER KD VALUE 0155 c 0156 KB(L+l>=KD(L)+DKD 0157 100 CONTINUE 0158 c 0159 c. ..TRY ANOTHER KI VALUE 0160 c 0161 KI(J+1)=KI(J)+DKI 0162 200 CONTINUE 0163 c 0164 c. ..TRY ANOTHER KP VALUE 0165 c 0166 KP( 1 + 1 )=KP( D+DKP 0167 300 CONTINUE 0168 c 0169 c. ..PRINT THE RESULTS OF THE GAIN COMBINATIONS 0170 c 0171 0PEN(UNIT=6,FILE='CB0UT.DAT;l',STATUS='OLD') 99 . CBGANES$MAIN 17-Sep-1986 22:37 17-Sep-1986 22:37 0172 WRITE (6,500),W,DSPD,WTF,T,DW 0173 HO 700 M=1,MMAX 0174 WRITE(6,550),(M,Z(M,1),Z<H,2),Z(M,3>,S(M,2>,S(M,3)) 0175 C 0176 500 F0RMAI(/,T2,'FREQUENCY=',T12,F5.2,T17,' DESIRED SPECTRAL DENSITY 5? 0177 k ',T45,F6.4,T51, ' WAVE TRANSFER FUNCTION=',T76,F7.4,/, 0178 k T2,'TIMESTEP DURATION=',T20,F5.2,T27,'SPD BANDWIDTH*', 0179 k T41,F5.2) 0180 550 F0RMAT(/,T2,'COMBINATION',T13, ' KP ',T20,' KI ',128, 0181 k ' KD ',T39,' <0', T43,' >STMAX',/,T6,14,T15,F5.3 , 0182 A T22,F6.4,T29,F6.3,T41,I2,T46,12) 0183 C 0184 WRITE(6,600) 0185 WRITE(6,650),(KAY<K>,STSPD(K,M),ASPD<K,M>,ERROR<K,M),K=3.33> 0136 C 0187 600 F0RMAT(/,T9,' K',T13,'STROKE SPECTRAL DENSITY',T39,'ACTUAL', 0183 k T46,'SPECTRAL DENS ITY',T63,' ERROR') 0189 C 0190 650 F0RMAKT8, 12 , T 1 6 , F12 . 6 , T42 , F12 . 6 , T62 , F12 . 6 ) 0191 C 0192 700 CONTINUE 0193 0194 ' STOP 0195 END APPENDIX B EIGENVALUE AND ROUTH PROGRAM LISTINGS 101. 6-Sep-1986 10:57:1 6-Sep-1986 10:48:3 0001 C. PROGRAM EIGVALS 0002 C 0003 C. STEVE HODGE 31712771 0004 C 0005 C. ..PROGRAM TO TEST MANY CONTROLLER GAIN COMBINATIONS FOR A SPECIFIC 0006 C. . .FREQUENCY, U, A DESIRED SPECTRAL DENSITY, DSPD, AND A TIMESTEP 0007 C. ..DURATION, T, IN ORDER TO EXAMINE THE STABILITY OF THE SYSTEM 0008 C. ..BY FINDING THE EIGENVALUES OF THE CHARACTERISTIC MATRIX IN 0009 C. ..DISCRETE STATE SPACE 0010 C 0011 C. ..DECLARE THE VARIABLES USED IN THIS PROGRAM 0012 C 0013 REAL A(4,4),Z(500,3),Q,DKP,DK I,DKD,DU . 0014 A ERR2,KP(20),KI(20),KD(20>,M0,MM(21), 0015 k PRODO,STMAX,T,M,UTF,KFMAX,KIMAX, 0016 k KDMAX,ASPDMX,SPUIF,W0RK(8),EMAG(500,4) 0017 C 0018 INTEGER K,IEND,JEND,LEND,M/1/,MMAX,KAY(50),KEND(500> , 0019 * S(500,3)/l500*0/,JOB,INFO,LDV,N,LDA,CONT 0020 C 0021 COMPLEX E(4),V(4,4),CMPLX,CONJG,EI (500,4> 0022 c 0023 c. ..READ IN THE FREQUENCY AND THE SPD FREQUENCY BANDWIDTH. 0024 c. ..CALCULATE THE MAXIMUM STROKE SPECTRAL DENSITY. 0025 c 0026 URITE<6,A>, 'ENTER THE FREQUENCY, U=' 0027 READ(6,5), U 0028 WRITE(6,A>, 'ENTER FREQUENCY BANDWIDTH FOR SPD, DW=' 0029 READ(6,5), DU 0030 STMAX=.063226/DW !! <—-STROKE SPD FROM MAX. WAVEMAKER STROKE 0031 c 0032 c. ..CALCULATE THE TRANSFER FUNCTION, UTF, RELATING WAVEMAKER STROKE 0033 c. ..TO WATER WAVE AMPLITUDE. DISPLAY UTF AND THE MAXIMUM WAVE SPECTRAL 0034 c. ..DENSITY FOR THIS FREQUENCY. O035 c 0036 c. ..NEWTON'S METHOD IS REQUIRED. 0037 c 0038 MM(1)=.5 0039 Q=(UAA2>A.249 0040 DO 30 N=l,20 0041 MM(N+1)=MM(N)+((Q-MM(N)ATANH(MM(N)))/(TANH(MM(N))+ 0042 A (MM(N)/(COSH(MM(N))AA2)))> 0043 IF(MM(N+l).LE.O.) THEN 0044 MM(N + 1) = MM(N) + . 1 0045 GOTO 30 0046 END IF 0047 ERR2=ABS(MM(N + 1)-MM(N> > 0048 IF(ERR2.LE..001 ) GOIO 35 0049 30 CONTINUE 0050 35 M0=MM(N+1)/2.44 0051 PR0D0=M0A2.44 0052 c 0053 WTF=2.AM0ASINH<PRODO)A((SINH(PRODO)/MO)-(COSH(PRODO)/ 0054 A (1.04A(M0AA2))) + (COSH(MOA1 . 4>/(1.04A(M0AA2))))/ 0055 A (PRODO+SINH(PRODO)ACOSH(PRODO)) 0056 SPUTF=WTFAA2 0057 c 102. EIGVALS*MAIN 6-Sep-1986 10:57:1 6-Sep-1986 10:48:3 0058 WRIIE(6,15>, WTF 0059 ASPDMX=STMAXA(SPUTF) 0060 WRITE(6,27), ASPDMX 0061 C 0062 C. ..READ IN IHE DSPD, INITIAL ASPD, INIIIAL GAINS, NUMBER OF GAIN INCREMENT 0063 C. ..MAXIMUM GAIN VALUES AND THE TIMESTEP DURATION. 0064 C 0065 URIIE(6,A>, 'ENTER THE DESIRED SPECTRAL DENSITY, DSPD=' 0066 READ(6,10), DSPD 0067 URITE(6,A>, 'ENTER THE INITIAL CONTROLLER GAINS, KP, KI, KD' 0068 READ(6, A >, KP(1),KI(1),KD(1) 0069 WRIIE(6,A>, 'ENTER * OF GAIN INCREMENTS, IEND, JEND, LEND ' 0070 READ(6,25), IEND,JEND,LEND 0071 URITE(6,A), 'ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX, KDMAX' 0072 READ(6,A >, KPMAX,KIMAX,KDMAX 0073 URITE(6,A), 'ENTER DISCRETE TIMESTEP DURATION, T=' 0074 READ(6,20), T 0075 5 FORMAT(F4.2) 0076 10 F0RMAT(F5.4) 0077 15 FORMATC WAVE TRANSFER FUNCTION=',T25,F7.5) 0078 20 FORMAT(FS.2) 0079 25 FORMAT(312) 0060 27 FORMAT(' MAXIMUM WAVE SPECTRAL DENS ITY=',T32,F6.4) 0081 C 0082 MMAX=IENDAJENDALEND 0083 DKP=KPMAX/(FLOAI(IEND >) 0084 DKI=KIMAX/(FLOAT(JEND-1> > 0085 DKD=KDMAX/(FL0AT<LEND-1)) 0066 C 0087 C. ..COMMENCE THE LOOPING 0088 C 0089 DO 300 1=1,IEND 0090 DO 200 J=1,JEND 0091 DO 100 L=1,LEND 0092 C 0093 C. ..SET UP THE MATRIX VALUES 0094 C 0095 A(1 ,1>=2. + (SPUTF/DSPD>A((KP<I)) + (KI(J>AI/2.) + (KD<L)/T)) 0096 A(1,2>=-l.-((SPWTF/DSPD)A(KP(I)-(KI(J)AT/2.))) 0097 A<1,3)=-(SPUTFAKD(L))/<DSPDAT) 0098 A<1,4)=0. 0099 A(2,1>=1. 0100 A(2,2>=0. 0101 A(2,3>=0. 0102 A<2,4)=0. 0103 A ( 3 , l ) = 0 . 0104 A<3,2)=1. 0105 A<3,3)=0. 0106 A(3,4>=0. 0107 A(4,1)=KP(I)+KI<J)AT/2.+KD(L)/T 0108 A(4,2)=-KP( D + K K J ) A T / 2 . 0109 A(4,3)=-KD(L)/T 0110 A<4,4)=0. 0111 C 0112 c. ..SET UP THE SUBROUTINE PARAMETERS REQ'D 0113 c 0114 LDA=4 103. EIGVALS*MAIN 6-Sep-1986 10:57:1 6-Sep-1986 10:48:3 0115 LDV = 4 0116 JOB=0 0117 N = 4 0118 C 0119 C. ..DETERMINE THE EIGENVALUES USING THE SUBROUTINE, SGEEV 0120 C 0121 CALL SGEEV(A,LDA,N,E,V,LDV,WORK,JOB,INFO) 0122 C 0123 C. ..CALCULATE THE MAGNITUDE OF THE EIGENVALUES,E 0124 C 0125 EI(M,1)=E(1) 0126 EI(M,2)=E(2> 0127 EI(M,3)=E(3) 0128 EI< M,4)=E < 4) 0129 C 0130 EMAG(M,1)=CABS(EI(M,1)> 0131 EMAG(M,2)=CABS(EI(M,2) ) 0132 EMAG(M,3)=CABS(EI(M,3) ) 0133 EMAG(M,4)=CABS(EI(M,4)) 0134 C 0135 C. ..LOOK AT THE INFO VALUE RETURNED BY THE SUBROUTINE 0136 C 0137 c URITE(6,A) ,INFO 0138 c READ<6,1000),CONT 0139 clOOO FORMAT(11 ) 0140 C 0141 C 0142 C. . .ASSIGN THE GAIN VALUES TO MEMORY 0143 c 0144 90 Z(M,1)=KP <I) 0145 Z (M , 2)=KI(J) 0146 Z<M,3)=KD(L> 0147 M=M+1 0148 c 0149 c. ..COMBINATION M HAS NOW BEEN TESTED 0150 c 0151 c. . .TRY ANOTHER KD VALUE 0152 c 0153 KD(L+1)=KD(L)+DKD 0154 100 CONTINUE 0155 c 0156 c. . .TRY ANOTHER KI VALUE 0157 c 0158 K K J + 1)=KI(J) + DKI 0159 200 CONTINUE 0160 c 0161 c. . .TRY ANOTHER KP VALUE 0162 c 0163 KP<I+1)=KP(I)+DKP 0164 300 CONTINUE 0165 c 0166 c. . .PRINT THE RESULTS OF THE GAIN COMBINATIONS 0167 c 0168 0PEN(UNIT=6,FILE='ZERR0UT.DAT;l',STATUS='OLD' ) 0169 write(6,500) ,U,DSPD,UTF,T,DU 0170 DO 700 M=1,MMAX 0171 WRITE(6,550),<M,Z<M,1),Z<M,2>,Z<M,3)> 104. E IGVALS*MA IN 6-Sep-1986 10:57 6-Sep-1986 10:48 0172 C 0173 500 F0RMAT(/ ,T2 , 'FREQUENCY=' ,T12 ,F5 .2 ,T17 f ' DESIRED SPECTRAL DENSITY = 0174 A ' , T 4 5 , F 6 . 4 , T 5 1 , ' WAVE TRANSFER FUNCTI0N=' ,T76,F7.4 , / , 0175 A T2,'TIMESTEP DURATI0N=',T20,F5.2,T27,'SPD BANDUIDTH=', 0176 A 141,F5.2) 0177 C 0178 550 F0RMAT</,T2,'COMBINATION',T13, ' KP ' , T 2 0 , ' KI ' , T 2 8 , 0179 A ' KD ' , / , T 6 , 1 4 , T l 5 , F 5 . 3 , T 2 2 , F 6 . 4 , T 2 9 , F 6 . 2 ) 0160 C 0181 WRITE(6,600),<EI<M,l>,EI<M f2>,EI<M,3),EI<M,4)) 0182 WRITE(6,650),(EMAGCM,1),EMAG <M,2),EMAG< M , 3 ),EMAG < M , 4 > ) 0183 C 0184 600 F O R M A T < T 3 , ' < ' , T 4 , 2 F 7 . 2 , T 1 9 , ' ) ' , T 2 1 f ' < ' , T 2 2 , 2 F 7 . 2 , T 3 7 , ' ) ' 0185 A , 1 3 9 , ' ( ' , I 4 0 , 2 F 7 . 2 , T 5 5 , ' ) ' , T 5 7 , ' ( ' , T 5 8 , 2 F 7 . 2 , T 7 3 , ' ) ' ) 0186 C 0187 650 F0RMAI(T7 ,F7 .2 ,T25 ,F7 .2 ,T43 ,F7 .2 ,T61 ,F7 .2 ) 0188 C 0189 700 CONTINUE 0190 0191 STOP 0192 END 105. 6-Sep-1986 11:27:2 6-Sep-1986 11:26:5 0001 C . . PROGRAM ROUTH 0002 C 0003 C . . STEVE HODGE 31712771 0004 C 0005 C . . .PROGRAM TO CALCULATE GAIN VALUES FROM THE STABILITY CONSTRAINTS 0006 C. . .DERIVED FROM THE ROUTH STABILITY CRITERION FOR DISCRETE TIME WITH 0007 c . . .THE INTENT OF GRAPHING THE RESULTANT GAIN CHARTS FOR STABILITY. 0008 c . . .PARTICULAR FREQUENCY, W, DESIRED SPECTRAL DENSITY, DSPD, AND 0009 c . . .TIMESTEP DURATION, T, VALUES ARE CHOSEN FOR EACH CHART. 0010 c 0011 c . . .DECLARE THE VARIABLES USED IN THIS PROGRAM 0012 c 0013 REAL AA,BB,CC,DIS,XX,DISCR,Q,DKP,DKI,DU, 0014 A ERR2,KP(20),KD1(20,50),KD2(20,50),M0,MM(21>, 0015 A K1(20,50),KDNU(20,100),KINU<20,100),PRODO,STMAX, 0016 A KPMAX,KIMAX,ASPDMX,SPWTF,T,W,WTF,RAT,A,B,C,X 0017 c 0018 INTEGER K, IEND,JEND,JAX 0019 c 0020 c . . .READ IN THE FREQUENCY AND THE SPD FREQUENCY BANDWIDTH. 0021 c . . .CALCULATE THE MAXIMUM STROKE SPECTRAL DENSITY. 0022 c 0023 WRITE(6,A>, 'ENTER THE FREQUENCY, W=' 0024 READ(6,5>, W 0025 WRITE(6,A>, 'ENTER FREQUENCY BANDWIDTH FOR SPD, DW=' 0026 READ(6,5), DW 0027 STMAX=.063226/DW ! ! < STROKE SPD FROM MAX. WAVEMAKER STROKE 0028 c 0029 c . . .CALCULATE THE TRANSFER FUNCTION, UTF, RELATING WAVEMAKER STROKE 0030 c . . .TO WATER WAVE AMPLITUDE. DISPLAY WTF AND THE MAXIMUM WAVE SPECTRAL 0031 c . . .DENSITY FOR THIS FREQUENCY. 0032 c 0033 MM <1) = .5 0034 Q=(WAA2)A.249 0035 DO 30 N=l,20 0036 MM(N+1>=MM(N)+((Q-MM(N)ATANH(MM(N)))/(TANH(MM(N))+ 0037 A (MM(N)/(COSH(MM(N)>AA2> ) ) ) 0038 IF(MM(N+1).LE.O.) THEN 0039 MM(N+l)=MM(N)+.l 0040 GOTO 30 0041 END IF 0042 ERR2=ABS(MM(N + 1)-MM(N> > 0043 IF(ERR2.LE..001> GOTO 35 0044 30 CONTINUE 0045 35 H0=MM(N+1)/2.44 0046 PR0D0=M0A2.44 0047 C 0048 WTF=2.AM0ASINH(PRODO)A((SINH(PRODO)/MO)-(COSH(PRODO)/ 0049 A (1.04A(M0AA2)> > + (COSH(MOA1.4>/(1.04A(MOAA2)))>/ 0050 A (PRODO+SINH(PROD0)AC0SH(PR0DO>> 0051 SPWTF=WTFAA2 0052 C 0053 WRITE(6,15>, WTF 0054 ASPDMX=STMAXA(SPWTF) 0055 WRITE(6,27), ASPDMX 0056 C 0057 C . . .READ IN THE DSPD, INITIAL ASPD, INITIAL GAINS, NUMBER OF GAIN INCREMENT 106. ROUTHtMAIN 6-Sep-1986 11:27:2 6-Sep-1986 11:26:5 0058 C • • • MAXIMUM GAIN VALUES AND THE TIMESTEP DURATION. 0059 C 0060 WRITE(6 ,A) , 'ENTER THE DESIRED SPECTRAL DENSITY, DSPD=' 0061 READ(6,10), DSPD 0062 URITE(6 ,A>, 'ENTER THE INITIAL CONTROLLER GAINS, KP, K I ' 0063 R E A D ( 6 r A) , KP(1),KI(1,1> 0064 WRITE<6,A), 'ENTER * OF GAIN INCREMENTS, IEND, JEND' 0065 READ<6,25), IEND,JEND 0066 WRITE(6 ,A) , ' ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX' 0067 R E A D ( 6 ,A) , KPMAX,KIMAX 0068 WRITE(6 ,A) , 'ENTER DISCREIE TIMESTEP DURATION, T=' 0069 READ<6,20), T 0070 5 FORMAT(F4.2 > 0071 10 FORMAT(F6.5 > 0072 15 FORMAT(' WAVE TRANSFER FUNCTION=',T25,F7.5) 0073 20 FORMAT(F5.2) 0074 25 FORMAT < 312) 0075 27 FORMAT(' MAXIMUM WAVE SPECTRAL DENSITY=',T32,F6.4 ) 0076 C 0077 DKP=KPMAX/<FLOAT<IEND)) 0078 DK I=KIMAX/(FLOAT(JEND-1)> 0079 C 0080 C • • • COMMENCE THE LOOPING 0081 C 0082 0PEN<UNIT=6,FILE='R0UD.DAT;l',STATUS='OLD') 0083 RAT=(SPWTF/DSPD) 0084 DO 300 1=1,IEND 0085 DO 200 J=1,JEND 0086 C 0087 C • > • CALCULATE KD VALUES FROM THE CONSTRAINT: BIAB2-B0AB3 0088 C 0089 A=16. A ( (RAT/DAA2) 0090 B = - (16.ARAT/T-8 .A (RATAA2/T)AKP(I)-4 .ARATAA2AKI<I,J)) 0091 C=-(8.ARATAKP<I)-4.ARATATAKI(I,J>) 0092 C 0093 X=-B/(2.AA) 0094 DISCR=BAB-4.AAAC 0095 IF(DISCR.GI.O.> THEN 0096 KDM I, J)=X+(SQRT(DISCR>)/<2.AA> 0097 KD2(I,J)=X-(SQRT(DISCR))/(2.AA> 0098 C • • • TRY ANOTHER KI VALUE 0099 KI< I , J + l)=KI<I,J)+DKI 0100 GOTO 200 0101 ENDIF 0102 GOTO 205 0103 200 CONTINUE 0104 C 0105 C • • • CALCULATE THE PEAK OF THE KI VS KD CURVE 0106 C 0107 205 AA=16.A < RATAA4) 0108 BB=64.A<RATAA4)AKP(I)/T-384.A(RATAA3>/T 0109 CC=256.A<RAT/T)AA2+256.A(RATAA3)AKP<I)/(TAA2>+ 0110 A 64 .A (RAIAA4 )A< (KP(I) /T)AA2) 0111 XX=-BB/(2.AAA) 0112 DIS=BBABB-4.AAAACC 0113 C 0114 KI( I ,J)=XX-(SQRT(DIS))/(2.AAA) 107. ROUTHtMAIN 6-Sep-1986 11:27 6-Sep-1986 11:26 0115 C 0116 KD1<I, J) = <16.ARAT/T-8.A<RATAA2/T>AKP(I)-0117 * 4.ARATAA2AKK1,3)) /<32.A(RAT/T)AA2) 0118 KD2<I,J)=KD1<1,3) 0119 C 0120 C . .REARRANGE THE KD AND KI VALUES FOR EASE OF GRAPHING 0121 C 0122 N = 0. 0123 JAX = J 0124 DO 33 K=1,JAX 0125 N = N + 1 0126 KDNU( I,N)=KD2< I,K). 0127 KINU(I,N)=KI<I,K> 0128 33 CONTINUE 0129 DO 36 K=1,JAX 0130 KDNU<I,N)=KD1<I,JAX-K+1) 0131 KINU( I,N)=KI<I,JAX-K+l) 0132 N=N+1 0133 36 CONTINUE 0134 C 0135 C . .STORE THE COMPUTED GAIN VALUES 0136 C 0137 WRITE(6,A),KP<I) 0138 WRITE(6,A),(KINUCI,J),J=1,N) 0139 WRITE(6,A),<KDNU(I,J>,J=1,N) 0140 C 0141 C . . . TRY ANOTHER KP VALUE 0142 C 0143 KP< I + l )=KP( D+DKP 0144 300 CONTINUE 0145 C 0146 write*. 6,500),W ,DSPD,WTF,T,DU 0147 C 0148 5 00 FORMAT*./,T2, 'FREQUENCY=' ,T12 ,F5 .2 ,117 . ' DESIRED ! 0149 A ' , T 4 5 , F 6 . 4 , T 5 1 , ' WAVE TRANSFER FUNCIION=' 0150 A T2,'TIMESTEP DURATION=',T20,F5.2,T27,'SPD ] 0151 A T41,F5.2) 0152 C 0153 0154 STOP 0155 END APPENDIX C COMPUTER SIMULATION OUTPUT AND EIGENVALUE OUTPUT 109. A typical input session for running the computer simulation program, CBGANES. RUN CBGANES ENTER THE EREQUENCY, W= 1 . ENTER FREQUENCY BANDWIDTH FOR SPD, DW= n WAVE TRANSFER FUNCTI0N=0.11685 MAXIMUM WAVE SPECTRAL DENSITY=0.0043 ENTER THE DESIRED SPECTRAL DENSITY, DSPD= .004 ENTER THE INITIAL ACTUAL SPECTRAL DENSITY. ASPD= 0. ENTER THE INITIAL CONTROLLER GAINS, KP, KI, KD -.079.0.,0. ENTER # OF GAIN INCREMENTS. IEND, JEND, LEND 4,6,6 ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX, KDMAX -.316.-.01,-9. ENTER DISCRETE TIMESTEP DURATION, T= 60. FORTRAN STOP $ The following pages show the computer simulation output coinciding with Figures 8 through 29. 110. COMBINATION KP KI KD <0 >STMAX 8 -.079 -.0020 -1.800 0 14 K SIROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.120000 0.001638 -0.590412 2 0.290780 0.003970 -0.007501 3 0.421609 0.005756 0.439050 4 0.460384 0.006286 0.571399 5 0.410710 0.005607 0.401849 6 0.317151 0.004330 0.082511 7 0.234425 0.003201 -0.199352 3 0.199097 0.002718 -0.320434 9 0.216601 0.002957 -0.260690 10 0.266078 0.003633 -0.091315 11 0.316505 0.004321 0.030304 12 0.343795 0.004694 0.173453 13 0.340543 0.004649 0.162354 14 0.315553 0.004308 0.077075 15 0.285836 0.003903 -0.024374 16 0.266568 0.003639 -0.090140 17 0.264383 0.003610 -0.097598 18 0.276243 0.003772 -0.057100 19 0.293205 0.004003 0.000776 20 0.306017 0.004173 0.044508 21 0.309609 0.004227 0.056769 2 2 0.304477 0.004157 0.039250 23 0.295125 0.004029 0.007329 24 0.286983 0.003918 -0.020461 25 0.283615 0.003872 -0.031954 2 6 0.285480 0.003393 -0.025591 27 0.290448 0.003965 -0.008634 28 0.295430 0.004033 0.008372 29 0.2980&6 0.004069 0.017369 30 0.2976G6 0.004064 0.016005 COMBINATION KP KI KD <0 >STMAX 10 -.079 -.0020 -5.400 0 4 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0 000000 0. 000000 -1 . 000000 1 0 120000 0. 001638 -0. 590412 2 0 266204 0. 003634 -0 091382 3 0 332117 0. 004534 0 133594 4 0 312564 0. 004267 0. 066854 5 0 272015 0. 003714 -0 071549 6 0 261144 0 . 003565 -0 103653 7 0 279813 0. 003820 -0 044933 8 0 300267 0. 004100 0 024882 9 0 304391 0. 004156 0 038958 10 0 296022 0. 004042 0 010393 11 0 288253 0. 003926 -0 016125 12 0 287880 0. 003930 -0 017398 13 0 292120 0. 003988 -0 002925 14 0 295249 0 . 004031 0 007752 15 0 294980 0. 004027 0 006836 16 0 .293030 0. 004001 0 000181 17 0 291867 0. 003985 -0 003791 18 0 .292190 0. 003939 -0 002688 19 0 293073 0. 004001 0 000326 20 0 .293489 0. 004007 0 001746 21 0 293269 0. 004004 0 000997 0 292834 0. 003999 -0 000318 23 0 292743 0. 003997 -0 000783 24 0 292875 0. 003999 -0 000351 25 0 293038 0. 004001 0 000207 26 0 .293077 0 . 004001 0 000340 27 0 293010 0. 004000 0 000113 23 0 .292943 0 004000 -0 000117 29 0 .292935 0 003999 -0 000144 30 0 .292968 0 004000 -0 000031 I l l COMBINATION KP KI KD <0 >STMAX 12 -.079 -.0020 -9.000 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.120000 0.001638 -0.590412 2 0.241629 0.003299 -0.175264 3 0.252691 0.003450 -0.137505 4 0.211601 0.002889 -0.277756 C *J 0.221880 0.003029 -0.242672 6 0.276388 0.003774 -0.056623 7 0.300986 0.004109 0.027335 8 0.280207 0.003826 -0.043587 9 0.264051 0.003605 -0.098731 10 0.279701 0.003819 -0.045316 11 0.300033 0.004096 0.024081 12 0.297734 0.004065 0.016237 13 0.284404 0.003883 -0.029264 14 0.283451 0.003870 -0.032515 15 0.293775 0.004011 0.002722 16 0.298305 0.004073 0.018183 17 0.292754 0.003997 -0.000763 13 0.288177 0.003934 -0.016384 19 0.291049 0.003974 -0.006583 20 0.295397 0.004033 0.008258 21 0.294776 0.004025 0.006138 2 2 0.291550 0.003981 -0.004871 23 0.291088 0.003974 -0.006450 24 0.293317 0.004005 0.001161 25 0.294358 0.004019 0.004714 26 0.293092 0.004002 0.000390 27 0.291976 0.003936 -0.003418 23 0.292562 0.003994 -0.001416 29 0.293552 0.004008 0.001961 30 0.293435 0.004006 0.001562 COMBINATION KP KI KD <0 >STMAX 14 -.079 -.0040 -1.800 32 30 K STROKE SPECIRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.240000 0.003277 -0.180824 2 0.532409 0.007269 0.817235 3 0.615084 0.008398 1 .099426 4 0.407060 0.005558 0.389391 5 0.089306 0.001219 -0.695179 6 -0.052235 -0.000713 -1.178290 7 0.116237 0.001587 -0.603256 8 0.450309 0.006148 0.537010 9 0.650791 0.008885 1 .221300 10 0.531479 0.007256 0.314063 11 0.191785 0.002618 -0.345395 12 -0.065552 -0.000395 -1 .223745 13 -0.004068 -0.000056 -1.013887 14 0.329407 0.004497 0.124344 15 0.639265 0.008728 1.181961 16 0.642939 0.008778 1.194502 17 0.328342 0.004483 0.120709 13 -0.027413 -0.000374 -1.093566 19 -0.101855 -0.001391 -1.347655 20 0.180773 0.002468 -0.382930 21 0.573550 0.007831 0.957661 2 2 0.722296 0.009861 1.465364 23 0.484719 0.006618 0.654460 24 0.065921 0.000900 -0.774995 25 -0.158275 -0.002161 -1.540228 26 0.021650 0.000296 -0.926103 27 0.453552 0.006192 0.548079 23 0.751707 0.010263 1.565752 29 0.641051 0.008752 1. 188056 30 0.210576 0.002875 -0.281255 COMBINATION KP KI KD <0 >STMAX 15 -.079 -.0040 -3.600 0 32 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0 . 000000 0.000000 -1.000000 1 0.240000 0.003277 -0.180824 0.507833 0.006933 0.733354 3 0.533142 0.007279 0.819738 4 0.305221 0.004167 0.041792 5 0.076869 0.001049 -0.737629 6 0.087033 0.001188 -0.702937 7 0.312008 0.004260 0.064956 8 0.504722 0.006891 0.722734 9 0.465409 0.006354 0.588548 10 0.247927 0.003385 -0.153769 11 0.089504 0.001222 -0.694502 12 0.152575 0.002083 -0.479226 13 0.359014 0.004902 0.225398 14 0.435053 0.006622 0.655598 15 0.403297 0.005506 0.376546 16 0.210660 0.002376 -0.2S0969 17 0.114691 0.001566 -0.608532 18 0.210445 0.002873 -0.281702 19 0.337251 0.005287 0.321779 20 0.455755 0.006222 0.555597 21 0.350263 0.004782 0.195531 2 2 0.190657 0.002603 -0.349244 23 0.146824 0.002005 -0.498857 24 0.258245 0.003526 -0. 118549 25 0.399869 0.005459 0.364847 26 0.421927 0.005761 0.440136 27 0.307918 0.004204 0.050997 28 0.134546 0.002520 -0.370101 29 0.181348 0.002476 -0.331018 30 0.295067 0.004029 0.007131 COMBINATION KP KI KD <0 >STMAX 38 -.158 0.0000-1.800 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.000000 0.000000 -1.000000 **i 0.000000 0.000000 -1.000000 3 0.000000 0.000000 -1.000000 4 0.000000 0.000000 -1.000000 5 0.000000 0.000000 -1.000000 6 0.000000 0.000000 -1.000000 7 0.000000 0.000000 -1.000000 8 0.000000 0.000000 -1.oooooo 9 0.000000 0.000000 -1.000000 10 0.000000 0.000000 -1.000000 11 0.000000 0.000000 -1.000000 12 0.000000 0.000000 -1.000000 13 0.000000 0.000000 -1.000000 14 0.000000 0.000000 -1.000000 15 0.000000 0.000000 -1.000000 1 6 0.000000 0.000000 -1.oooooo 17 0.000000 0.000000 -1.000000 13 0.000000 0.000000 -1.000000 19 0.000000 0.000000 -1.000000 20 0.000000 0.000000 -1.oooooo 21 0.000000 0.000000 -1.000000 2 2 0.000000 0.000000 -1.oooooo 23 0.000000 0.000000 -1.000000 24 0.000000 0.000000 -1.000000 25 0.000000 0.000000 -1.000000 26 0.000000 0.000000 -1.oooooo 27 0.000000 0.000000 -1.000000 28 0.000000 0.000000 -1.000000 29 0.000000 0.000000 -1.000000 30 0.000000 0.000000 -1.000000 113. COMBINATION KP KI KD <0 5STMAX 43 -.158 -.0020 0.000 0 7 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1 OOOOOO 1 0.120000 0.001638 -0. 590412 0.270710 0.003696 -0 076004 3 0.380128 0.005190 0 297466 4 0.417250 0.005697 0 424173 5 0.391055 0.005339 0. 334761 6 0.333450 0.004553 0 138143 7 0.278537 0.003803 -0. 049289 8 0.247907 0.003385 -0. 153837 9 0.245982 0.003358 -0. 160404 10 0.263951 0.003604 -0 099075 11 0.287797 0.003929 -0 017680 12 0.305789 0.004175 0 043730 13 0.312515 0.004267 0 066687 14 0.308989 0.004219 0 054651 15 0.300084 0.004097 0 024257 16 0.291247 0.003976 -0 005906 17 0.286075 0.003906 -0 023561 19 0.285460 0.003897 -0 025660 19 0.288130 0.003934 -0 016547 20 0.291892 0.003985 -0 003705 21 0.294840 0.004025 0 006359 2 *^  0.296040 0.004042 0 010452 23 0.295583 0.004036 0 008895 24 0.294212 0.004017 0 004215 25 0.292794 0.003997 -0 000626 2 6 0.291925 0.003986 -0 003591 27 0.291778 0.003984 -0 004094 28 0.292171 0.003989 -0 002751 29 0.292763 0.003997 -0 000731 30 0.293245 0.004004 0 000913 COMBINATION KP KI KD <0 >STMAX 44 -.158 -.0020-1.800 0 6 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.OOOOOO 1 0.120000 0.001638 -0.590412 o 0.258422 0.003528 -0.117945 3 0.338234 0.004618 0.154473 4 0.350466 0.004785 0.196223 5 0.325635 0.004446 0.111468 6 0.297023 0.004055 0.013810 7 0.281798 0.003847 -0.038159 8 0.280733 0.003333 -0.041794 9 0.286707 0.003914 -0.021401 10 0.292749 0.003997 -0.000780 11 0.295633 0.004036 0.009063 12 0.295550 0.004035 0.008782 13 0.294155 0.004016 0.004019 14 0.292895 0.003999 -0.000231 15 0.292362 0.003992 -0.002099 16 0.292443 0.003993 -0.001823 17 0.292762 0.003997 -0.000735 13 0.293021 0.004001 0.000151 19 0.293117 0.004002 0.000476 20 0.293087 0.004001 0.000374 21 0.293015 0.004001 0.000130 2 2 0.292963 0.004000 -0.000050 23 0.292946 0.004000 -0.000106 24 0.292955 0.004000 -0.000076 25 0.292971 0.004000 -0.000022 2 6 0.292982 0.004000 0.000014 27 0.292984 0.004000 0.000023 28 0.292982 0.004000 0.000015 29 0.292978 0.004000 0.000003 30 0.292976 0.004000 -0.000004 114. COMBINATION KP KI KD <0 >STMAX 46 -.158 -.0020 -5.400 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1 . 000000 1 0.120000 0.001638 -0. 590412 0.233847 0.003193 -0. 201826 3 0.261996 0.003577 -0. 105747 4 0.249799 0.003410 -0 . 147378 5 0.254467 0.003474 -0. 131445 6 0.275660 0.003764 -0. 059109 7 0.288912 0.003945 -0. 013874 3 0.288816 0.003943 -0. 014204 9 0.236414 0.003910 -0. 022400 10 0.288272 0.003936 -0. 016061 11 0.291602 0.003981 -0. 004693 12 0.292788 0.003997 -0. 000645 13 0.292263 0.003990 -0. 002423 14 0.292007 0.003987 -0. ,003311 15 0.292471 0.003993 -0. 001727 16 0.292925 0.003999 -0. 000179 17 0.292966 0.004000 -0. 000038 18 0.292846 0.003998 -0. ,000447 19 0.292844 0.003998 -0. 000454 20 0.292935 0.003999 -0. 000145 21 0.292985 0.004000 0. 000027 2 2 0.292972 0.004000 -0. 000017 23 0.292954 0.004000 -0. 000079 24 0.292961 0.004000 -0. 000055 25 0.292976 0.004000 -0. 000005 2 6 0.292980 0.004000 0. 000008 27 0.292975 0.004000 -0. 000006 28 0.292974 0.004000 -0. ,000013 29 0.292976 0.004000 -0. 000005 30 0.292978 0.004000 0. ,000001 COMBINATION KP KI KD <0 >STMAX 48 -.153 -.0020-9.000 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.120000" 0.001638 -0.590412 0 . 209272 0.002857 -0.285707 3 0.195823 0.002674 -0.331610 4 0.187846 0.002565 -0.358338 5 0.236567 0.003230 -0.192540 6 0.271236 0.003703 -0.074208 7 0.260519 0.003557 -0.110789 8 0.254418 0.003474 -0..131611 9 0.274762 0.003751 -0.062173 10 0.288469 0.003933 -0.015387 11 0.282004 0.003850 -0.037454 12 0.278488 0.003802 -0.049454 13 0.287193 0.003921 -0.019743 14 0.292699 0.003996 -0.000951 15 0.239201 0.003948 -0.012888 16 0.287392 0.003924 -0.019063 17 0.291193 0.003976 -0.006091 13 0.293433 0.004006 0.001556 19 0.291645 0.003982 -0.004548 20 0.290769 0.003970 -0.007538 21 0.292455 0.003993 -0.001784 22 0.293376 0.004005 0.001361 23 0.292491 0.003993 -0.001660 24 0.292083 0.003983 -0.003054 25 0.292839 0.003998 -0.000470 26 0.293221 0.004003 0.000832 27 0.292792 0.003997 -0.000631 28 0.292607 0.003995 -0.001264 29 0.292950 0.004000 -0.000094 30 0.293109 0.004002 0.000448 115. COMBINATION KP KI KD <0 >STMAX 50 -.158 -.0040 -1.800 0 16 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.OOOOOO 1 0.240000 0.003277 -0.180824 2 0.467694 0.006385 0.596348 3 0.474841 0.006483 0.620743 4 0.308036 0.004206 0.051398 5 0.166879 0.002278 -0.430404 6 0.178861 0.002442 -0.389504 7 0.295998 0.004041 0.010310 8 0.382246 0.005219 0.304694 9 0.363354 0.004961 0.240213 10 0.282365 0.003855 -0.036222 11 0.230801 0.003151 -0.212222 12 0.250431 0.003419 -0.145219 13 0.305638 0.004173 0.043215 14 0.335650 0.004583 0.145652 15 0.318087 0.004343 0.085704 16 0.280958 0.003836 -0.041026 17 0.264091 0.003606 -0.098596 1 8 0.278604 0.003804 -0.049059 19 0.303250 0.004140 0.035063 20 0.312274 0.004263 0.065365 21 0.300373 0.004108 0.026949 2 2 0.234726 0.003387 -0.023164 23 0.230253 0.003826 -0.043430 24 0.288396 0.003944 -0.013932 25 0.299334 0.004087 0.021695 2 6 0.301257 0.004113 0.023261 27 0.294883 0.004026 0.006504 28 0.238230 0.003935 -0.016204 29 0.287663 0.003927 -0.018139 30 0.292262 0.003990 -0.002441 COMBINATION KP KI KD <0 >STMAX 79 -.237 -.0020 0.000 0 6 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.OOOOOO -1.000000 1 0 .120000- 0.001638 -0.590412 2 0.238352 0.003254 -0.186448 3 0.307577 0.004199 0.049831 4 0.329000 0.004492 0.122954 5 0.322726 0.004406 0.101540 6 0.308058 0.004206 0.051473 7 0.296074 0.004042 0.010571 8 0.290062 0.003960 -0.009950 9 0.288876 0.003944 -0.013998 1 0 0.290086 0.003961 -0.009867 11 0.291750 0.003983 -0.004190 12 0.292911 0.003999 -0.000227 13 0.293398 0.004006 0.001435 1 4 0.293418 0.004006 0.001505 15 0.293246 0.004004 0.000916 16 0.293068 0.004001 0.000308 17 0.292960 0.004000 -0.000059 18 0.292925 0.003999 -0.000130 19 0.292932 0.003999 -0.000154 20 0.292954 0.004000 -0.000081 21 0.292972 0.004000 -0.000019 2 2 0.292931 0.004000 0.000014 23 0.292983 0.004000 0.000021 24 0.292982 0.004000 0.000015 25 0.292979 0.004000 0.000007 26 0.292977 0.004000 0.000001 27 0.292977 0.004000 -0.000002 28 0.292977 0.004000 -0.000002 29 0.292977 0.004000 -0.000001 30 0.292977 0.004000 -0.000001 1 1 6 . C O M B I N A T I O N KP K I KB <0 >STMAX 80 - . 2 3 7 - . 0 0 2 0 - 1 . 8 0 0 0 0 K S T R O K E S P E C T R A L D E N S I T Y A C T U A L S P E C T R A L D E N S I T Y ERROR 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 - 1 . 0 0 0 0 0 0 1 0 . 1 2 0 0 0 0 0 . 0 0 1 6 3 8 - 0 . 5 9 0 4 1 2 *? 0 . 2 2 6 0 6 5 0 . 0 0 3 0 8 6 - 0 . 2 2 8 3 8 8 3 0 . 2 7 2 3 0 9 0 . 0 0 3 7 1 8 - 0 . 0 7 0 5 4 4 4 0 . 2 8 3 4 8 5 0 . 0 0 3 8 7 0 - 0 . 0 3 2 3 9 9 5 0 . 2 8 5 9 1 7 0 . 0 0 3 9 0 4 - 0 . 0 2 4 0 9 8 & 0 . 2 8 8 3 7 8 0 . 0 0 3 9 3 7 - 0 . 0 1 5 6 9 7 7 0 . 2 9 0 7 3 5 0 . 0 0 3 9 6 9 - 0 . 0 0 7 6 5 3 !3 0 . 2 9 2 0 9 3 0 . 0 0 3 9 8 8 - 0 . 0 0 3 0 1 8 9 0 . 2 9 2 6 1 2 0 . 0 0 3 9 9 5 . - 0 . 0 0 1 2 4 5 10 0 . 2 9 2 7 7 5 0 . 0 0 3 9 9 7 - 0 . 0 0 0 6 9 0 11 0 . 2 9 2 8 5 3 0 . 0 0 3 9 9 8 - 0 . 0 0 0 4 2 6 12 0 . 2 9 2 9 1 0 0 . 0 0 3 9 9 9 - 0 . 0 0 0 2 3 1 13 0 . 2 9 2 9 4 6 0 . 0 0 4 0 0 0 - 0 . 0 0 0 1 0 6 14 0 . 2 9 2 9 6 4 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 4 6 15 0 . 2 9 2 9 7 1 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 2 2 16 0 . 2 9 2 9 7 4 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 1 2 17 0 . 2 9 2 9 7 5 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 0 7 18 0 . 2 9 2 9 7 6 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 0 3 19 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 0 2 20 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 - 0 . 0 0 0 0 0 1 21 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . 0 0 0 0 0 0 2 2 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . 0 0 0 0 0 0 23 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . o o o o o o 24 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . 0 0 0 0 0 0 25 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . 0 0 0 0 0 0 26 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . o o o o o o 27 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . 0 0 0 0 0 0 28 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . o o o o o o 29 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . 0 0 0 0 0 0 30 0 . 2 9 2 9 7 7 0 . 0 0 4 0 0 0 0 . o o o o o o C O M B I N A T I O N KP K I KD <0 >STMAX 84 - . 2 3 7 - . 0 0 2 0 - 9 . 0 0 0 0 0 K S T R O K E S P E C T R A L D E N S I T Y A C T U A L S P E C T R A L D E N S I T Y ERROR 0 0 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 - 1 . O O O O O O 1 0 . 1 2 0 0 0 0 0 . 0 0 1 6 3 3 - 0 . 5 9 0 4 1 2 0 . 1 7 6 9 1 4 0 . 0 0 2 4 1 5 - 0 . 3 9 6 1 5 1 3 0 . 1 5 6 4 0 5 0 . 0 0 2 1 3 5 - 0 . 4 6 6 1 5 4 4 0 . 1 8 5 5 8 6 0 . 0 0 2 5 3 4 - 0 . 3 6 6 5 5 3 5 0 . 2 3 6 6 8 4 0 . 0 0 3 2 3 1 - 0 . 1 9 2 1 4 3 & 0 . 2 3 8 8 6 7 0 . 0 0 3 2 6 1 - 0 . 1 8 4 6 9 2 7 0 . 2 3 4 6 1 5 0 . 0 0 3 2 0 3 - 0 . 1 9 9 2 0 5 8 0 . 2 5 7 8 9 6 0 . 0 0 3 5 2 1 - 0 . 1 1 9 7 4 2 9 0 . 2 7 1 7 3 8 0 . 0 0 3 7 1 0 - 0 . 0 7 2 4 9 5 10 0 . 2 6 6 9 1 1 0 . 0 0 3 6 4 4 - 0 . 0 8 8 9 7 2 11 0 . 2 7 1 0 6 1 0 . 0 0 3 7 0 1 - 0 . 0 7 4 8 0 7 12 0 . 2 8 2 0 2 7 0 . 0 0 3 8 5 0 - 0 . 0 3 7 3 7 6 13 0 . 2 8 3 1 1 4 0 . 0 0 3 8 6 5 - 0 . 0 3 3 6 6 6 1 4 0 . 2 8 1 4 1 3 0 . 0 0 3 8 4 2 - 0 . 0 3 9 4 7 2 15 0 . 2 8 5 7 9 1 0 . 0 0 3 9 0 2 - 0 . 0 2 4 5 3 0 16 0 . 2 3 9 0 9 7 0 . 0 0 3 9 4 7 - 0 . 0 1 3 2 4 5 17 0 . 2 8 8 0 6 1 0 . 0 0 3 9 3 3 - 0 . 0 1 6 7 8 1 18 0 . 2 3 8 5 0 2 0 . 0 0 3 9 3 9 - 0 . 0 1 5 2 7 4 19 0 . 2 9 0 8 1 4 0 . 0 0 3 9 7 0 - 0 . 0 0 7 3 8 3 20 0 . 2 9 1 2 0 6 0 . 0 0 3 9 7 6 - 0 . 0 0 6 0 4 7 21 0 . 2 9 0 7 0 2 0 . 0 0 3 9 6 9 - 0 . 0 0 7 7 6 5 9 2 0 . 2 9 1 4 9 2 0 . 0 0 3 9 3 0 - 0 . 0 0 5 0 6 9 23 0 . 2 9 2 2 6 7 0 . 0 0 3 9 9 0 - 0 . 0 0 2 4 2 6 24 0 . 2 9 2 0 6 3 0 . 0 0 3 9 8 8 - 0 . 0 0 3 1 2 0 25 0 . 2 9 2 0 6 5 0 . 0 0 3 9 8 8 - 0 . 0 0 3 1 1 5 2 6 0 . 2 9 2 5 4 2 0 . 0 0 3 9 9 4 - 0 . 0 0 1 4 8 4 27 0 . 2 9 2 6 6 4 0 . 0 0 3 9 9 6 - 0 . 0 0 1 0 6 8 2 9 0 . 2 9 2 5 3 4 0 . 0 0 3 9 9 4 - 0 . 0 0 1 5 1 4 29 0 . 2 9 2 6 6 8 0 . 0 0 3 9 9 6 - 0 . 0 0 1 0 5 5 3 0 0 . 2 9 2 3 4 6 0 . 0 0 3 9 9 8 - 0 . 0 0 0 4 4 8 117. COMBINATION KP KI KD <0 >STMAX 86 -.237 -.0040 -1.900 0 4 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.240000 0.003277 -0.180824 2 0.402979 0.005502 0.375461 3 0.369498 0.005045 0.261183 4 0.273443 0.003733 -0.066673 5 0.245014 0.003345 -0.163710 6 0.279975 0.003822 , -0.044380 7 0.310957 0.004245 0.061368 8 0.308085 0.004206 0.051567 9 0.291106 0.003974 -0.006387 10 0.284473 0.003334 -0.029027 11 0.289873 0.003958 -0.010595 12 0.295786 0.004033 0.009586 13 0.295878 0.004040 0.009902 14 0.292942 0.004000 -0.000119 15 0.291499 0.003980 -0.005047 16 0.292291 0.003991 -0.002342 17 0.293396 0.004006 0.001429 18 0.293522 0.004007 0.001860 19 0.293026 0.004001 0.000165 20 0.292726 0.003997 -0.000859 21 0.292833 0.003998 -0.000492 2 2 0.293036 0.004001 0.000199 23 0.293078 0.004001 0.000342 24 0.292996 0.004000 0.000063 25 0.292936 0.003999 -0.000143 26 0.292943 0.004000 -0.000100 27 0.292984 0.004000 0.000025 23 0.292995 0.004000 0.000062 29 0.292982 0.004000 0.000017 30 0.292971 0.004000 -0.000023 COMBINATION KP KI KD <0 >STMAX 93 -.237 -.0080 -1.800 30 32 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.480000 0.006553 0.638352 2 0.609355 0.008319 1.079873 3 0.159301 0.002175 -0.456270 4 -0.043515 -0.000594 -1.148525 5 0.369738 0.005048 0.262001 6 0.639913 0.008737 1.184172 7 0.274471 0.003747 -0.063166 8 -0.054638 -0.000746 -1.186492 9 0.253519 0.003461 -0.134680 10 0.631624 0.008624 1.155879 11 0.388507 0.005304 0.326064 12 -0.027433 -0.000375 -1.093634 13 0.144799 0.001977 -0.505768 14 0.586517 0.008008 1.001921 15 0.488971 0.006676 0.668971 16 0.034077 0.000465 -0.883688 17 0.055264 0.000755 -0.811373 18 0.510536 0.006970 0.742580 19 0.565245 0.007717 0.929312 20 0.122223 0.001669 -0.582825 21 -0.005817 -0.000079 -1.019856 2 2 0.412337 0.005636 0.409110 23 0.609643 0.008323 1.030871 24 0.226634 0.003094 -0.226445 25 -0.032541 -0.000444 -1.111072 26 0.304702 0.004160 0.040018 27 0.618196 0.008440 1.110046 28 0.335444 0.004530 0.144948 29 -0.022926 -0.000313 -1.078253 30 0.193244 0.002707 -0.323346 118. COMBINATION KP KI KB <0 >STMAX 115 -.316 -.0020 0.000 0 0 K STROKE SPECTRAL BENS IT Y ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1 000000 1 0.120000 0.001633 -0 590412 0.205995 0.002312 -0 296891 3 0.252475 0.003447 -0 138242 4 0.274931 0.003754 -0 061596 5 0.285157 0.003893 -0 026694 6 0.289651 0.003955 -0 011355 7 0.291580 0.003981 -0 004768 8 0.292396 0.003992 -0 001934 9 0.292737 0.003997 -0 000820 10 0.292879 0.003999 -0 000337 11 0.292937 0.003999 -0 000133 12 0.292961 0.004000 -0 000056 13 0.292971 0.004000 -0 000023 14 0.292975 0.004000 -0 000009 15 0.292976 0.004000 -0 000004 16 0.292977 0.004000 -0 000002 17 0.292977 0.004000 -0 000001 18 0.292977 0.004000 0 000000 19 0.292977 0.004000 0 000000 20 0.292977 0.004000 0 000000 21 0.292977 0.004000 0 000000 2 2 0.292977 0.004000 0 000000 23 0.292977 0.004000 0 000000 24 0.292977 0.004000 0 000000 25 0.292977 0.004000 0 000000 26 0.292977 0.004000 0 000000 27 0.292977 0.004000 0 000000 28 0.292977 0.004000 0 000000 29 0.292977 0.004000 0 000000 30 0.292977 0.004000 0 000000 COMBINATION KP KI KB <0 >STMAX 117 -.316 -.0020-3.600 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.120000 0.001638 -0.590412 2 0.181420 0.002477 -0.380772 3 0.197711 . 0.002699 -0.325168 4 0.222372 0.003043 -0.239235 5 0.246273 0.003362 -0.159413 6 0.258411 0.003523 -0.117984 7 0.266823 0.003643 -0.089272 8 0.274383 0.003746 -0.063448 9 0.279685 0.003819 -0.045370 1 0 0.233164 0.003866 -0.033496 11 0.285825 0.003902 -0.024412 12 0.287833 0.003930 -0.017559 13 0.289237 0.003949 -0.012766 14 0.290248 0.003963 -0.009317 15 0.290999 0.003973 -0.006754 16 0.291543 0.003980 -0.004895 17 0.291934 0.003986 -0.003561 13 0.292219 0.003990 -0.002588 19 0.292427 0.003992 -0.001878 20 0.292578 0.003995 -0.001364 21 0.292687 0.003996 -0.000991 2 2 0.292766 0.003997 -0.000720 23 0.292824 0.003998 -0.000523 24 0.292S66 0.003998 -0.000380 25 0.292897 0.003999 -0.000276 26 0.292919 0.003999 -0.000200 27 0.292935 0.003999 -0.000145 28 0.292946 0.004000 -0.000106 29 0.292955 0.004000 -0.000077 30 0.292961 0.004000 -0.000056 119 COMBINATION KP KI KD <0 >STMAX 123 -.316 -.0040 -3.600 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1.000000 1 0.240000 0.003277 -0.180824 2 0.313689 0.004283 0.070693 3 0.256872 0.003507 -0^123235 4 0.264187 0.003607 -0.098269 5 0.300330 0.004100 0.025097 6 0.297371 0.004060 0.014996 7 0.285996 0.003905 -0.023828 8 0.290885 0.003971 -0.007140 9 0.295546 0.004035 0.008766 10 0.293029 0.004001 0.000175 11 0.291715 0.003983 -0.004310 12 0.293099 0.004002 0.000414 13 0.293443 0.004006 0.001590 14 0.292822 0.003998 -0.000532 15 0.292800 0.003998 -0.000605 16 0.293070 0.004001 0.000316 17 0.293032 0.004001 0.000188 13 0.292927 0.003999 -0.000171 19 0.292963 0.004000 -0.000050 20 0.293001 0.004000 0.000080 21 0.292979 0.004000 0.000006 2 2 0.292967 0.004000 -0.000035 23 . 0.292978 0.004000 0.000004 24 0.292981 0.004000 0.000014 25 0.292976 0.004000 -0.000004 26 0.292976 0.004000 -0.000005 27 0.292978 0.004000 0.000003 23 0.292978 0.004000 0.000002 29 0.292977 0.004000 -0.000001 30 0.292977 0.004000 0.OOOOOO COMBINATION KP KI KD <0 >STMAX 129 -.316 -.0060-3.600 0 10 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1 .OOOOOO 1 0.360000 0.004915 0.228764 •-t 0.396807 0.005418 0.354396 3 0.207682 0.002835 -0.291132 4 0.242350 0.003309 -0.172802 5 0.354766 0.004844 0.210899 6 0.308953 0.004218 0.054528 7 0.251136 0.003429 -0.142813 8 0.292794 0.003997 -0.000627 9 0.318649 0.004350 0.087622 10 0.237132 0.003920 -0.019953 11 0.278537 0.003804 -0.049116 12 0.299896 0.004094 0.023613 13 0.300193 0.004099 0.024645 14 0.287062 0.003919 -0.020191 15 0.289921 0.003953 -0.010433 16 0.297313 0.004059 0.014799 17 0.293347 0.004012 0.002968 18 0.290117 0.003961 -0.009762 19 0.293107 0.004002 0.000442 20 0.294701 0.004024 0.005884 21 0.292498 0.003993 -0.001635 2 2 0.292031 0.003997 -0.003229 23 0.293490 0.004007 0.001751 24 0.293439 0.004006 0.001574 25 0.292556 0.003994 -0.001433 26 0.292792 0.003997 -0.000632 27 0.293279 0.004004 0.001028 28 0.293021 0.004001 0.000149 29 0.292782 0.003997 -0.000665 30 0.292996 0.004000 0.000063 120. COMBINATION KP KI KD <0 >STMAX 135 -.316 -.0080 -3.600 0 24 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.000000 -1 .000000 1 0.480000 0.006553 0.638352 2 0.430775 0.005881 0.470336 3 0.080338 0.001097 -0.725788 4 0.251033 0.003427 -0.143167 5 0.482978 0.006594 0.648518 6 0.261009 0.003564 -0. 109115 7 0.146952 0.002006 -0.498419 3 0.370539 0.005059 0.264737 9 0.386622 0.005279 0.319630 10 0.196027 0.002676 -0.330914 11 0.249451 0.003406 -0.148564 12 0.388420 0.005303 0.325767 13 0.295570 0.004035 0.008843 14 0.213114 0.002910 -0.272594 15 0.318794 0.004352 0.088117 16 0.350007 0.004779 0.194657 17 0.251654 0.003436 -0.141047 18 0.260266 0.003553 -0.111651 19 0.338615 0.004623 0.155773 20 0.304060 0.004151 0.037828 21 0.251342 0.003432 -0.142110 22 0.298386 0.004074 0.018460 23 0.325527 0.004444 0.111099 24 0.277107 0.003783 -0.054168 25 0.271606 0.003703 -0.072944 26 0.313588 0.004231 0.070349 27 0.303441 0.004143 0.035714 28 0.272263 0.003717 -0.070701 29 0.291573 0.003981 -0.004792 30 0.310605 0.004241 0.060167 COMBINATION KP KI KD <0 >STMAX 40 -.158 0.0000 -5.400 0 0 K STROKE SPECTRAL DENSITY ACTUAL SPECTRAL DENSITY ERROR 0 0.000000 0.001000 -0. 750000 1 0.084489 0.001154 -0 711620 2 0.063715 0.000370 -0 782526 3 0.057072 0.000779 -0 305201 4 0.062433 0.000852 -0 786901 5 0.065297 0.000891 -0 777126 6 0.064090 0.000875 -0 781247 7 0.063025 0.000860 -0 784882 8 0.063232 0.000863 -0 734175 9 0.063591 0.000868 -0 782949 10 0.063582 0.000863 -0 782978 11 0.063471 0.000367 -0 783359 12 0.063456 0.000866 -0 783409 13 0.063488 0.000867 -0 783299 14 0.063498 0.000867 -0 733267 15 0.063489 0.000367 -0 783296 16 0.063435 0.000367 -0 783310 17 0.063487 0.000867 -0 783304 18 0.063489 0.000867 -0 733293 19 0.063488 0.000867 -0 733300 20 0.063488 0.000867 -0 793301 21 0.063488 0.000867 -0 783301 2 2 0.063488 0.000867 -0 783301 23 0.063438 0.000367 -0 733301 24 0.063483 0.000867 -0 783301 25 0.063488 0.000867 -0 783301 2 6 0.063483 0.000367 -0 783301 27 0.063488 0.000867 -0 733301 2 3 0.063488 0.000867 -0 733301 29 0.063483 0.000867 -0 783301 30 0.063483 0.000367 -0 783301 121. The following pages show the output from the eigenvalue program, EIGVALS. 1 2 2 . FREQUENCY= 1.00 DESIRED SPECTRAL DENS ITY= 0.0040 WAVE TRANSFER FUNCTION' 0.1168 TIMESTEP DURATION=60.0O SPD BANDWIDTH= 0.20 COMBINATION KP KI KD 1 -.079 0.0000 0.00 < 0.00 0.00 ) < 0.00 0.00 ) < 1.00 0.00 ) ( 0.73 0.00 ) 0.000 0.000 1.000 0.730 COMBINATION KP KI KD 2 -.079 0.0000 -1.80 ( 0.00 0.00 ) ( 1.00 0.00 ) < 0.31 0.06 ) < 0.31 -0.06 ) 0.000 1.000 0.320 0.320 COMBINATION KP KI KD 3 -.079 0.0000 -3.60 < 0.00 0.00 ) < 0.26 0.37 > < 0.26 -0.37 > ( 1.00 0.00 ) 0.000 0.453 0.453 1.000 COMBINATION KP KI KD 4 -.079 0.0000 -5.40 < 0.00 0.00 ) ( 0.21 0.51 > < 0.21 -0.51 ) ( 1.00 0.00 ) 0.000 0.554 0.554 1.000 COMBINATION KP KI KD . 5 -.079 0.0000 -7.20 ( 0.00 0.00 > < 0.16 0.62 ) < 0.16 -0.62 ) < 1.00 0.00 > 0.000 0.640 0.640 1.000 COMBINATION KP KI KD 6 -.079 0.0000 -9.00 ( 0.00 0.00 ) < .0.11 0.71 ) ( 0.11 -0.71 ) < 1.00 0.00 ) 0.000 0.716 0.716 1.000 COMBINATION KP KI KD 7 -.079 -.0020 0.00 < 0.00 0.00 ) ( 0.00 0.00 ) < 0.76 ' 0.59 > ( 0.76 -0.59 ) 0.000 0.000 0.967 0.967 COMBINATION KP KI KD 8 -.079 -.0020 -1.80 < 0.00 0.00 ) < 0.64 0.59 ) < 0.64 -0.59 ) < 0.13 0.00 ) 0.000 0.873 0.373 0.134 COMBINATION KP KI KD 9 -.079 -.0020 -3.60 ( 0.00 0.00 ) ( 0.49 0.60 > ( 0.49 -0.60 ) ( 0.34 0.00 ) 0.000 0.775 0.775 0.341 COMBINATION KP KI KD 10 -.079 -.0020 -5.40 ( 0.00 0.00 ) ( 0.34 0.67 ) ( 0.34 -0.67 ) ( 0.54 0.00 ) 0.000 0.754 0.754 0.540 COMBINATION KP KI KD 11 -.079 -.0020 -7.20 < 0.00 0.00 ) ( 0.23 0.76 > ( 0.23 -0.76 ) ( 0.65 0.00 > 0.000 0.795 0.795 0.648 1 2 3 . COMBINATION KP KI KB 12 -.079 -.0020 -9.00 ( 0.00 0.00 ) ( 0.15 0.83 ) ( 0.000 0.849 0.15 -0.83 > ( 0.71 0.00 ) 0.849 0.711 COMBINATION KP KI KD 13 -.079 -.0040 0.00 < 0.00 0.00 ) ( 0.00 0.00 ) ( 0.000 0.000 0.66 0.84 ) < 0.66 -0.84 ) 1.068 1.068 COMBINATION KP KI KD 14 -.079 -.0040 -1.80 < 0.00 0.00 ) ( 0.56 0.85 ) ( 0.000 1.014 0.56 -0.85 ) < 0.10 0.00 ) 1.014 0.100 COMBINATION KP KI KD 15 -.079 -.0040 -3.60 < 0.00 0.00 ) ( 0.45 0.86 ) ( 0.000 0.972 0.45 -0.86 ) < 0.22 0.00 ) 0.972 0.217 COMBINATION KP KI KD 16 -.079 -.0040 -5.40 ( 0.00 0.00 ) < 0.34 0.89 ) ( 0.000 0.955 0.34 -0.89 ) ( 0.34 0.00 ) 0.955 0.337 COMBINATION KP KI KD 17 -.079 -.0040 -7.20 ( 0.00 0.00 > ( 0.24 0.94 > < 0.000 0.966 0.24 -0.94 ) ( 0.44 0.00 ) 0.966 0.439 COMBINATION KP KI KD 18 -.079 -.0040 -9.00 < 0.00 0.00 > ( 0.15 0.98 ) ( 0.000 0.995 0.15 -0.98 ) < 0.52 0.00 > 0.995 0.518 COMBINATION KP KI KD 19 -.079 -.0060 0.00 ( 0.00 0.00 ) ( 0.00 0.00 > < 0.000 0.000 0.56 1.02 ) < 0.56 -1.02 ) 1.160 1.160 COMBINATION KP KI KD 20 -.079 -.0060 -1.80 < 0.00 0.00 ) ( 0.47 1.03 ) < 0.000 1.127 0.47 -1.03 > ( 0.08 0.00 ) 1.127 0.081 COMBINATION KP KI KD 21 -.079 -.0060 -3.60 < 0.00 0.00 ) ( 0.37 1.04 ) ( 0.000 1.105 0.37 -1.04 ) < 0.17 0.00 ) 1.105 0.168 COMBINATION KP KI KD 22 -.079 -.0060 -5.40 < 0.00 0.00 ) < 0.28 1.06 ) ( 0.000 1.097 0.28 -1.06 > < 0.26 0.00 ) 1.097 0.255 COMBINATION KP KI KD 23 -.079 -.0060 -7.20 < 0.00 0.00 ) ( 0.19 1.09 > ( 0.000 1.105 0.19 -1.09 ) ( 0.34 0.00 ) 1.105 0.336 124 . COMBINATION KP K I KB 24 - . 0 7 9 - . 0 0 6 0 - 9 . 0 0 ( 0 . 0 0 0 . 0 0 > ( 0 . 1 0 1.12 > ( 0 . 0 0 0 1.124 0 . 1 0 - 1 . 1 2 ) ( 0 .41 0 . 0 0 ) 1.124 0 . 4 0 5 COMBINATION KP K I KD 25 - . 0 7 9 - . 0 0 8 0 0 . 0 0 ( 0 . 0 0 0 . 0 0 ) < 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 0 . 0 0 0 0 . 4 6 1.16 ) ( 0 . 4 6 - 1 . 1 6 ) 1 .245 1.245 COMBINATION KP K I KD 26 - . 0 7 9 - . 0 0 8 0 - 1 . 8 0 ( 0 . 0 0 0 . 0 0 > < 0 . 0 0 0 0 . 3 7 1.17 ) < 1.224 0 . 3 7 - 1 . 1 7 ) ( 0 . 0 7 0 . 0 0 ) 1.224 0 . 0 6 8 COMBINATION KP 27 - . 0 7 9 • ( 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 K I KD . 0 0 8 0 - 3 . 6 0 0 . 2 8 1.18 ) ( 1.213 0 . 2 8 - 1 . 1 8 ) ( 0 .14 0 . 0 0 ) 1 .213 0 . 1 3 9 COMBINATION KP 28 - . 0 7 9 • ( 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 K I KD . 0 0 8 0 - 5 . 4 0 0 . 2 0 1 .19 ) ( 1.211 0 . 2 0 - 1 . 1 9 > ( 0 .21 0 . 0 0 ) 1.211 0 . 2 0 9 COMBINATION KP K I KD . 29 - . 0 7 9 - . 0 0 8 0 - 7 . 2 0 < 0 . 0 0 0 . 0 0 ) < 0 . 0 0 0 0 .11 1.21 ) ( 1 . 220 0 . 1 1 - 1 . 2 1 ) < 0 . 2 8 0 . 0 0 ) 1 .220 0 .275 COMBINATION KP K I 30 - . 0 7 9 - . 0 0 8 0 ( 0 . 0 0 0 . 0 0 ) ( 0 . 000 KD 9 . 0 0 0 . 0 3 1.24 ) ( 1 . 236 0 . 0 3 - 1 . 2 4 ) < 0 .34 0 . 0 0 ) 1.236 0 .335 COMBINATION KP K I KD 31 - . 0 7 9 - . 0 1 0 0 0 . 0 0 ( 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 0 . 0 0 0 0 . 3 5 1.23 ) ( 0 .35 - 1 . 2 8 ) 1 .325 1.325 COMBINATION KP 32 - . 0 7 9 K I KD - . 0 1 0 0 - 1 . 8 0 ( 0 . 0 0 0 . 0 0 ) < 0 . 2 7 1.28 ) ( 0 . 000 1 .31: 0 . 2 7 - 1 . 2 8 ) < 0 .06 0 . 0 0 ) 1.312 ' 0 . 059 COMBINATION KP K I 33 - . 0 7 9 - . 0 1 0 0 < 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 KD 3 . 6 0 0 . 1 9 1.29 ) < 1 . 3 0 7 0 . 1 9 - 1 . 2 9 ) ( 0 . 1 2 0 . 0 0 ) 1 .307 0 . 1 2 0 COMBINATION KP 34 - . 0 7 9 < 0 . 0 0 0 . 0 0 ) < 0 . 0 0 0 K I KD . 0 1 0 0 - 5 . 4 0 0 . 1 1 1.30 ) < 1.310 0 . 1 1 - 1 . 3 0 ) ( 0 . 18 0 . 0 0 ) 1 .310 0 .179 COMBINATION KP 35 - . 0 7 9 • < 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 K I KD . 0 1 0 0 - 7 . 2 0 0 . 0 3 1 .32 > < 1 . 3 1 9 0 . 0 3 - 1 . 3 2 ) ( 0 .24 0 . 0 0 ) 1 .319 0 .235 125. COMBINATION KP KI KD 36 -.079 -.0100 -9.00 < 0.00 0.00 > ( -0.05"1.33" ) ( 0.000 1.335 -0.05 -1.33 ) ( 0.29 0.00 ) 1.335 0.287 COMBINATION KP KI KD 37 -.158 0.0000 0.00 < 0.00 0.00 ) ( 0.00 0.00 ) ( 0.000 0.000 1.00 0.00 ) < 0.46 0.00 ) 1.000 0.461 COMBINATION KP KI KD 33 -.158 0.0000 -1.80 ( 0.00 0.00 ) ( 0.18 0.27 ) ( 0.000 0.320 0.18 -0.27 ) < 1.00 0.00 ) 0.320 1.000 COMBINATION KP KI KD 39 -.158 0.0000 -3.60 ( 0.00 0.00 > ( 0.13 0.43 > ( 0.000 0.453 0.13 -0.43 > ( 1.00 0.00 ) 0.453 1.000 COMBINATION KP KI KD 40 -.158 0.0000 -5.40 < 0.00 0.00 ) < 0.08 0.55 > ( 0.000 0.554 0.08 -0.55 > ( 1.00 0.00 ) 0.554 1.000 COMBINATION KP KI KD • 41 -.153 0.0000 -7.20 < 0.00 0.00 ) ( 0.03 0.64 ) ( 0.000 0.640 0.03 -0.64 > < 1.00 0.00 ) 0.640 1.000 COMBINATION KP KI KD 42 -.153 0.0000 -9.00 < 0.00 0.00 > < -0.03 0.72 > ( 0.000 0.716 -0.03 -0.72 > ( '1.00 0.00 ) 0.716 1.000 COMBINATION 43 KP KI -.158 -.0020 KD 0.00 0.00 0.00 ) < 0 .000 0.00 0.00 ) < 0.000 0.63 0.52 ) ( 0.63 -0.52 ) 0.316 0.316 COMBINATION KP 44 -.153 • ( 0.00 0.00 ) ( O.000 KI KD .0020 -1.80 0.46 0.49 ) ( 0.676 0.46 -0.49 ) ( 0.22 0.00 ) 0.676 0.224 COMBINATION KP 45 -.158 • < 0.00 0.00 ) ( 0.000 COMBINATION KP 46 -.159 • < 0.00 0.00 ) < 0 .000 KI KD -.0020 -3.60 0.26 0.57 > < 0.624 K I K D .0020 -5.40 0.15 0.67 > ( 0.637 0.26 -0.57 ) ( 0.53 0.00 ) 0.624 0.526 0.15 -0.67 > ( 0.65 0.00 ) 0.687 0.651 COMBINATION KP 47 -.158 • < 0.00 0.00 ) ( 0 .000 KI KD .0020 -7.20 0.07 0.75 ) < 0.756 0.07 -0.75 ) ( 0.72 0.00 ) 0.756 0.716 126. COMBINATION KP KI KD 48 -.158 -.0020 -9.00 ( 0.00 0.00 > ( -0.01 0.82 ) ( 0.000 0.822 -0.01 -0.82 > ( 0.76 0.00 ) 0.322 0.758 COMBINATION KP KI KD 49 -.153 -.0040 0.00 ( 0.00 0.00 ) ( 0.00 0.00 ) ( 0.000 0.000 COMBINATION KP KI KD 50 -.158 -.0040 -1.80 ( 0.00 0.00 ) ( 0.41 0.77 > < 0.000 0.372 COMBINATION KP KI . KD 51 -.158 -.0040 -3.60 ( 0.00 0.00 ) ( 0.28 0.79 > < 0.000 0.842 COMBINATION KP KI KD 52 -.158 -.0040 -5.40 ( 0.00 0.00 ) < 0.16 0.84 ) < 0.000 0.857 0.53 0.77 ) ( 0.53 -0.77 ) 0.933 0.933 0.41 -0.77 ) ( 0.13 0.00 ) 0.872 0.135 0.28 -0.79 > ( 0.29 0.00 ) 0.842 0.289 0.16 -0.84 ) < 0.42 0.00 > 0.857 0.413 COMBINATION KP KI KD • 53 -.153 -.0040 -7.20 < 0.00 0.00 > ( 0.07 0.89 ) ( 0.000 0.896 COMBINATION KP KI KD 54 -.153 -.0040 -9.00 < 0.00 0.00 ) < -0.02 0.94 ) < 0.000 0.944 COMBINATION KP KI KD 55 -.158 -.0060 0.00 ( 0.00 0.00 ) < 0.00 0.00 ) < 0.000 0.000 0.07 -0.89 ) < 0.51 0.00 ) 0.896 0.510 -0.02 -0.94 ) < 0.57 0.00 ) 0.944 0.575 0.42 0.95 ) ( 0.42 -0.95 ) 1.037 1.037 COMBINATION KP KI KD 56 -.158 -.0060 -1.80 ( 0.00 0.00 ) < 0.32 0.95 > ( 0.000 1.005 COMBINATION KP KI KD 57 -.158 -.0060 -3.60 ( 0.00 0.00 ) ( 0.22 0.97 ) ( 0.000 0.992 COMBINATION KP KI KD 58 -.158 -.0060 -5.40 < 0.00 0.00 ) ( 0.12 1.00 ) < 0.000 1.002 COMBINATION ' KP KI KD 59 -.158 -.0060 -7.20 < 0.00 0.00 ) < 0.02 1.03 ) < 0.000 1.028 0.32 -0.95 ) ( 0.10 O.OO ) 1.005 0.101 0.22 -0.97 ) < 0.21 0.00 ) 0.992 0.208 0.12 -1.00 ) < 0.31 0.00 ) 1.002 0.306 0.02 -1.03 ) ( 0.39 0.00 > 1.028 0.388 127. COMBINATION KP 60 -.158 ( 0.00 0.00 ) ( 0.000 KI KD -.0060 -9.00 -0.06 1.06 ) ( 1 .063 -0.06 -1.06 ) ( 0.45 0.00 ) 1.063 0.453 COMBINATION KP 61 -.159 ( 0.00 0.00 ) ( 0.000 KI KD -.0030 0.00 0.00 0.00 ) < 0.000 0.32 1.08 > < 0.32 -1.08 ) 1.131 1.131 COMBINATION KP 62 -.158 • ( 0.00 0.00 ) < 0 . 000 KI KD -.0080 -1.80 0.23 1.09 ) < 1.115 0.23 -1.09 ) ( 0.08 0.00 ) 1.115 0.082 COMBINATION KP 63 -.153 • ( 0.00 0.00 > < 0 .000 KI KD -.0080 -3.60 0.14 1.10 > < 1.111 0.14 -1.10 > ( 0.17 0.00 ) 1.111 0.166 COMBINATION KP 64 -.153 • ( 0.00 0.00 ) ( 0 .000 KI KD .0080 -5.40 0.05 1.12 ) ( 1 . 122 0.05 -1.12 ) < 0.24 0.00 ) 1.122 0.244 COMBINATION KP . 6 5 -.158 • ( 0.00 0.00 ) ( 0.000 KI KD .0080 -7.20 -0.04 1.14 ) ( 1.143 -0.04 -1.14 ) ( 0.31 0.00 ) 1.143 0.314 COMBINATION KP 66 -.153 • ( 0.00 0.00 > < 0 . 000 KI KD .0080 -9.00 -0.12 1.16 ) < 1.171 -0.12 -1.16 ) ( 0.37 0.00 ) 1.171 0.373 COMBINATION KP 67 -.158 • < 0.00 0.00 ) ( 0.000 KI KD .0100 0.00 0.00 0.00 ) < 0.000 0.22 1.20 ) < 0.22 -1.20 ) 1.218 1.218 COMBINATION KP 63 -.153 • < 0.00 0.00 ) ( 0.000 KI KD -.0100 -1.80 0.13 1.20 ) < 1.211 0.13 -1.20 ) ( 0.07 0.00 ) 1.211 0.070 COMBINATION KP 69 -.158 • ( 0.00 0.00 ) < 0.000 K I K D .0100 -3.60 0.05 1.21 . ( 1.213 0.05 -1.21 ) < 0.14 0.00 ) 1.213 0.139 COMBINATION KP 70 -.153 • ( 0.00 0.00 ) < 0 .000 KI KD .0100 -5.40 -0.04 1.22 > ( 1 . 225 -0.04 -1.22 ) ( 0.20 0.00 ) 1.225 0.205 COMBINATION KP 71 -.158 • < 0.00 0.00 ) ( 0 . 000 KI KD .0100 -7.20 -0.12 1.24 ) ( 1.244 -0.12 -1.24 ) ( 0.26 0.00 ) 1.244 0.265 128. COMBINATION KP 72 -.158 • ( 0.00 0.00 ) ( 0.000 KI KD .0100 -9.00 -0.20 1.25 ) ( 1 .269 -0.20 -1.25 ) ( 0.32 0.00 ) 1.269 0.318 COMBINATION 73 ( 0.00 0.00 ) ( 0. 000 KP KI KD .237 0.0000 0.00 0.00 0.00 ) < 0.000 1.00 0.00 ) ( 0.19 0.00 ) 1.000 0.191 COMBINATION KP KI 74 -.237 0.0000 ( 0.00 0.00 ) ( 0.000 KD 1 .80 0.04 0.32 ) ( 0.320 0.04 -0.32 ) < 1.00 0.00 ) 0.320 1.000 COMBINATION 75 < 0.00 0.00 ) < 0.000 KP KI KD .237 0.0000 -3.60 0.01 0.45 ) ( 0.453 -0.01 -0.45 ) ( 1.00 0.00 ) 0.453 1.000 COMBINATION 76 KP KI -.237 0.0000 KD -5.40 ( 0.00 0.00 ) ( -0.06 0 .000 0.554 -0.06 -0.55 ) ( 1.00 0.00 ) 0.554 1.000 COMBINATION 77 KP KI KD -.237 0.0000 -7.20 < 0.00 0.00 ) ( -0.11 0.63 ) ( 0 .000 0.640 -0.11 -0.63 > ( 1.00 0.00 ) 0.640 1.000 COMBINATION 73 KP KI -.237 0.0000 ( 0.00 0.00 ) < -0.16 KD -9.00 0.70 ) ( 0.000 0.716 -0.16 -0.70 ) ( 1.00 0.00 ) 0.716 1.000 COMBINATION 7 9 ( 0.00 0.00 ) ( 0 . 000 KP KI KD -.237 -.0020 0.00 0.00 0.00 > ( 0.000 0.49 0.39 ) ( 0.49 -0.39 > 0.629 0.629 COMBINATION KP KI KD 80 -.237 -.0020 -1.30 ( 0.00 0.00 ) ( 0.19 0.41 ) ( 0.000 0.452 0.19 -0.41 ) < 0.50 0.00 ) 0.452 0.502 COMBINATION KP 31 -.237 • ( 0.00 0.00 ) ( 0.000 KI KD .0020 -3.60 0.06 0.56 ) ( 0.559 0.06 -0.56 > ( 0.65 0.00 ) 0.559 0.655 COMBINATION KP 82 -.237 < 0.00 0.00 ) ( 0 . 000 KI KD .0020 -5.40 -0.02 0.65 ) ( 0.653 -0.02 -0.65 ) ( 0.72 0.00 ) 0.653 0.721 COMBINATION KP 83 -.237 • ( 0.00 0.00 ) ( 0. 000 KI KD .0020 -7.20 -0.09 0.73 > ( 0. 733 -0.09 -0.73 > ( 0.76 0.00 ) 0.733 0.762 129. COMBINATION KP KI KD . 84 -.237 -.0020 -9.00 ( 0.00 0.00 ) < -0.16 0.79 ) ( -0.16 -0.79 ) ( 0.79 0.00 ) 0.000 0.804 0.804 0.791 COMBINATION KP KI KD 85 -.237 -.0040 0.00 ( 0.00 0.00 ) ( 0.00 0.00 ) ( 0.39 0.67 > < 0.39 -0.67 ) 0.000 0.000 0.775 0.775 COMBINATION KP KI KD 86 -.237 -.0040 -1.80 ( 0.00 0.00 ) < 0.24 0.67 ) ( 0.24 -0.67 ) < 0.20 0.00 ) 0.000 0.710 0.710 0.203 COMBINATION KP KI KD 87 -.237 -.0040 -3.60 ( 0.00 0.00 ) ( 0.09 0.72 ) ( 0.09 -0.72 ) ( 0.39 0.00 ) 0.000 0.726 0.726 0.388 COMBINATION KP KI KD 88 -.237 -.0040 -5.40 ( 0.00 0.00 > ( -0.01 0.78 ) ( -0.01 -0.78 ) < 0.50 0.00 > 0.000 0.783 0.783 0.500 COMBINATION KP KI KD . 89 -.237 -.0040 -7.20 ( 0.00 0.00 > ( -0.10 0.84 > < -0.10 -0.84 ) ( 0.57 0.00 ) 0.000 0.846 0.846 0.572 COMBINATION KP KI KD 90 -.237 -.0040 -9.00 < 0.00 0.00 > ( -0.18 0.89 ) ( -0.18 -0.89 ) < 0.62 0.00 > 0.000 0.906 0.906 0.623 COMBINATION KP KI KD 91 -.237 -.0060 0.00 ( 0.00 0.00 ) ( 0.00 0.00 ) ( 0.29 0.85 ) ( 0.29 -0.85 ) 0.000 0.000 0.897 0.897 COMBINATION KP KI KD 92 -.237 -.0060 -1.80 ( 0.00 0.00 > < 0.17 0;85 > < 0.17 -0.85 ) ( 0.13 0.00 ) 0.000 0.872 0.872 0.135 COMBINATION KP KI KD 93 -.237 -.0060 -3.60 ( 0.00 0.00 ) ( 0.05 0.88 ) ( 0.05 -0.88 > < 0.26 0.00 ) 0.000 0.881 0.881 0.264 COMBINATION KP KI KD 94 -.237 -.0060 -5.40 < 0.00 0.00 ) ( -0.05 0.92 ) ( -0.05 -0.92 ) ( 0.37 0.00 ) 0.000 0.917 0.917 0.365 COMBINATION KP KI KD 95 -.237 -.0060 -7.20 ( 0.00 0.00 ) ( -0.14 0.95 ) ( -0.14 -0.95 ) < 0.44 0.00 ) 0.000 0.963 0.963 0.442 130. COMBINATION KP KI KD 96 -.237 -.0060 -9.00 < 0.00 0.00 ) ( -0.22 0.99 ) ( -0.22 -0.99 > < 0.50 0.00 ) 0.000 1.012 1.012 0.500 COMBINATION KP KI KD 97 -.237 -.0080 0.00 < 0.00 0.00 ) ( 0.00 0.00 ) < 0.19 0.99 ) ( 0.19 -0.99 ) 0.000 0.000 1.005 1.005 COMBINATION KP KI KD 98 -.237 -.0080 -1.80 ( 0.00 0.00 > ( 0.08 0.99 > ( 0.08 -0.99 ) ( 0.10 0.00 ) 0.000 0.997 0.997 0.103 COMBINATION KP KI KD 99 -.237 -.0080 -3.60 ( 0.00 0.00 ) ( -0.02 1.01 ) ( -0.02 -1.01 ) ( 0.20 0.00 ) 0.000 1.009 1.009 0.201 COMBINATION KP KI KD 100 -.237 -.0080 -5.40 ( 0.00 0.00 ) < -0.11 1.03 ) < -0.11 -1.03 > ( 0.29 0.00 > 0.000 1.036 1.036 0.286 COMBINATION KP KI KD .101 -.237 -.0080 -7.20 < 0.00 0.00 ) < -0.20 1.05 ) < -0.20 -1.05 ) < 0.36 0.00 ) 0.000 1.073 1.073 0.356 COMBINATION KP KI KD 102 -.237 -.0080 -9.00 ( 0.00 0.00 ) ( -0.28 1.03 ) ( -0.28 -1.08 ) < 0.41 0.00 ) 0.000 1.113 1.113 0.413 COMBINATION KP KI KD 103 -.237 -.0100 0.00 < 0.00 0.00 ) ( 0.00 0.00 ) ( 0.08 1.10 ) < 0.08 -1.10 ) 0.000 0.000 1.102 1.102 COMBINATION KP KI KD 104 -.237 -.0100 -1.80 ( 0.00 0.00 ) ( -0.01 1.10 ) ( -0.01 -1.10 ) < 0.08 0.00 ) 0.000 1.103 1.103 0.034 COMBINATION KP KI KD 105 -.237 -.0100 -3.60 ( 0.00 0.00 ) < -0.10 1.11 ) ( -0.10 -1.11 ) < 0.16 0.00 ) 0.000 1.117 1.117 0.164 COMBINATION KP KI KD 106 -.237 -.0100 -5.40 < 0.00 0.00 > ( -0.19 1.13 ) < -0.19 -1.13 ) ( 0.24 0.00 ) 0.000 1.142 1.142 0.236 COMBINATION KP KI KD 107 -.237 -.0100 -7.20 ( 0.00 0.00 ) ( -0.27 1.14 ) ( -0.27 -1.14 ) ( 0.30 0.00 ) 0.000 1.173 1.173 0.293 1 3 1 . COMBINATION KP KI KD 108 - . 2 3 7 - . 0 1 0 0 - 9 . 0 0 ( 0 . 0 0 0 .00 ) ( - 0 . 3 5 1 .16 ) ( - 0 . 3 5 - 1 . 1 6 > ( 0 .35 0 .00 ) 0 . 0 0 0 1 .208 1.208 0.351 COMBINATION KP KI KD 109 - . 3 1 6 0 .0000 0 . 0 0 < 0 . 0 0 0 . 0 0 > ( 0 . 0 0 0 . 0 0 ) ( 1.00 0 .00 ) ( - 0 . 0 8 0 .00 ) 0 . 000 0 .000 1.000 0 .079 COMBINATION KP KI KB 110 - . 3 1 6 0 .0000 - 1 . 8 0 < 0 . 0 0 0 . 0 0 > ( - 0 . 0 9 0 .31 . < - 0 . 0 9 - 0 . 3 1 > ( 1.00 0 . 0 0 ) 0 . 0 0 0 0 . 3 2 0 0 .320 1.000 COMBINATION KP KI KD 111 - . 3 1 6 0 .0000 - 3 . 6 0 ( 0 . 0 0 0 . 0 0 ) < - 0 . 1 4 0 . 4 3 ) < - 0 . 1 4 - 0 . 4 3 ) ( 1.00 0 .00 ) 0 . 0 0 0 0 . 4 5 3 0 .453 1.000 COMBINATION KP KI KD 112 - . 3 1 6 0 .0000 - 5 . 4 0 < 0 . 0 0 0 . 0 0 ) ( - 0 . 1 9 0 .52 ) ( - 0 . 1 9 - 0 . 5 2 ) ( 1 .00 0 . 0 0 ) 0 . 0 0 0 0 .554 0 .554 1.000 COMBINATION KP KI KD . 1 1 3 - . 3 1 6 0 .0000 - 7 . 2 0 ( 0 . 0 0 0 . 0 0 ) < - 0 . 2 4 0 . 5 9 ) < - 0 . 2 4 - 0 . 5 9 ) < 1.00 0 . 0 0 ) 0 . 000 0 .640 0 .640 1.000 COMBINATION KP KI KD 114 - . 3 1 6 0 .0000 - 9 . 0 0 < 0 . 0 0 0 . 0 0 ) < - 0 . 3 0 0 . 6 5 ) ( - 0 . 3 0 - 0 . 6 5 > ( 1.00 0 . 0 0 ) 0 . 0 0 0 0 .716 0 .716 1.000 COMBINATION KP KI KD 115 - . 3 1 6 - . 0 0 2 0 0 . 0 0 < 0 . 0 0 0 . 0 0 ) ( 0 . 0 0 0 . 0 0 ) ( 0 . 4 0 0 . 0 0 > ( 0.31 0 .00 > 0 . 0 0 0 0 .000 0 .405 0 .312 COMBINATION KP KI KD 116 - . 3 1 6 - . 0 0 2 0 - 1 . 3 0 < 0 . 0 0 0 . 0 0 ) ( - 0 . 0 2 0 . 3 9 ) ( - 0 . 0 2 - 0 . 3 9 ) ( 0 .66 0 .00 ) 0 . 0 0 0 0 .394 0 .394 0 .659 COMBINATION KP KI KD 117 - . 3 1 6 - . 0 0 2 0 - 3 . 6 0 ( 0 . 0 0 0 . 0 0 ) ( - 0 . 1 1 0 . 5 2 ) ( - 0 . 1 1 - 0 . 5 2 ) ( 0 .73 0 .00 ) 0 . 0 0 0 0 .531 0.531 0 .726 COMBINATION KP KI KD 118 - . 3 1 6 - . 0 0 2 0 - 5 . 4 0 ( 0 . 0 0 0 . 0 0 ) ( - 0 . 1 8 0 .61 ) ( - 0 . 1 8 - 0 . 6 1 ) ( 0 .77 0 .00 ) 0 . 0 0 0 0 . 6 3 3 0 .633 0 .767 COMBINATION KP KI KD 119 - . 3 1 6 - . 0 0 2 0 - 7 . 2 0 ( 0 . 0 0 0 . 0 0 ) ( - 0 . 2 4 0 . 6 7 ) ( - 0 . 2 4 - 0 . 6 7 ) ( 0 .80 0 .00 > 0 . 0 0 0 0 .718 0 .718 0 .796 132. COMBINATION KP 120 -.316 • ( 0.00 0.00 ) ( 0.000 KI KD .0020 -9.00 -0.31 0.73 ) ( 0.792 -0.31 -0.73 ) ( 0.82 0.00 ) 0.792 0.817 COMBINATION KP 121 -.316 • ( 0.00 0.00 > ( 0.000 KI KD .0040 0.00 0.00 0.00 ) ( 0.000 0.26 0.52 ) < 0.26 -0.52 ) 0.575 0.575 COMBINATION KP 122 -.316 • ( 0.00 0.00 ) ( 0 .000 KI KD .0040 -1.80 0.04 0.55 ) < 0.04 -0.55 ) < 0.33 0.00 ) 0.553 0.335 COMBINATION KP 123 -.316 • < 0.00 0.00 > ( 0.000 KI KD .0040 -3.60 -0.09 0.64 ) < 0.648 -0.09 -0.64 ) ( 0.49 0.00 ) 0.648 0.488 COMBINATION 124 KP -.316 KI KD -.0040 -5.40 < 0.00 0.00 ) ( -0.18 0.71 ) ( 0 .000 0.734 -0.18 -0.71 ) ( 0.57 0.00 ) 0.734 0.570 COMBINATION KP 125 -.316 • ( 0.00 0.00 ) < 0.000 KI KD -.0040 -7.20 -0.26 0.77 ) < 0.810 -0.26 -0.77 ) ( 0.62 0.00 ) 0.810 0.624 COMBINATION 126 KP -.316 KI -.0040 ( 0.00 0.00 > ( -0.33 KD -9.00 0.31 ) ( 0 . 000 0.878 -0.33 -0.81 ) < 0.66 0.00 ) 0.873 0.664 COMBINATION KP 12 7 -.316 • < 0.00 0.00 ) < 0.000 KI KD .0060 0.00 0.00 0.00 ) ( 0.000 0.15 0.72 ) ( 0.15 -0.72 ) 0.732 0.732 COMBINATION KP 128 -.316 • ( 0.00 0.00 ) < 0.000 KI KD .0060 -1.80 0.01 0.73 ) ( 0.730 0.01 -0.73 ) ( 0.19 0.00 ) 0.730 0.192 COMBINATION KP 129 -.316 • < 0.00 0.00 ) < 0.000 KI KD .0060 -3.60 -0.12 0.77 ) ( 0.783 -0.12 -0.77 ) ( 0.33 0.00 ) 0.783 0.334 COMBINATION KP 130 -.316 KI -.0060 ( 0.00 0.00 ) < -0.21 KD -5.40 0.82 ) ( 0.000 0.848 -0.21 -0.82 ) < 0.43 0.00 ) 0.848 0.427 COMBINATION KP 131 -.316 • < 0.00 0.00 ) < 0.000 KI KD .0060 -7.20 -0.30 0.86 ; < 0.911 -0.30 -0.86 ) ( 0.49 0.00 ) 0.911 0.494 133. COMBINATION KP KI KD 132 -.316 -.0060 -9.00 ( 0.00 0.00 ) < -0.37 0.90 > ( 0.000 0.971 -0.37 -0.90 ) < 0.54 0.00 > 0.971 0.543 COMBINATION KP KI KD 133 -.316 -.0080 0.00 ( 0.00 0.00 ) < 0.00 0.00 ) < 0.000 0.000 0.05 0.86 ) ( 0.05 -0.86 ) 0.861 0.861 COMBINATION KP KI KD 134 -.316 -.0080 -1.80 < 0.00 0.00 ) < -0.07 0.87 > ( 0.000 0.871 -0.07 -0.87 ) < 0.13 0.00 ) 0.871 0.135 COMBINATION KP KI KD 135 -.316 -.0080 -3.60 < 0.00 0.00 ) < -0.18 0.89 ) ( 0.000 0.910 -0.18 -0.89 ) < 0.25 0.00 ) 0.910 0.248 COMBINATION KP KI KD 136 -.316 -.0080 -5.40 ( 0.00 0.00 ) ( -0.27 0.92 ) < 0.000 0.959 -0.27 -0.92 ) ( 0.33 0.00 ) 0.959 0.334 COMBINATION KP KI KD 137 -.316 -.0080 -7.20 < 0.00 0.00 ) ( -0.35 0.95 > ( 0.000 1.012 -0.35 -0.95 > < 0.40 0.00 ) 1.012 0.400 COMBINATION KP KI KD 133 -.316 -.0080 -9.00 ( 0.00 0.00 ) < -0.43 0.97 ) ( 0.000 1.063 -0.43 -0.97 > ( 0.45 0.00 ) 1.063 0.453 COMBINATION KP KI KD 139 -.316 -.0100 0.00 ( 0.00 0.00 ) ( 0.00 0.00 > ( 0.000 0.000 -0.05 0.97 ) ( -0.05 -0.97 ) 0.972 0.972 COMBINATION KP KI KD 140 -.316 -.0100 -1.30 ( 0.00 0.00 ) ( -0.15 0.98 ) ( 0.000 0.989 -0.15 -0.98 > ( 0.10 0.00 ) 0.989 0.105 COMBINATION KP KI KD 141 -.316 -.0100 -3.60 < 0.00 0.00 ) ( -0.25 0.99 ) ( 0.000 1.022 -0.25 -0.99 ) ( 0.20 0.00 ) 1.022 0. 196 COMBINATION KP KI KD 142 -.316 -.0100 -5.40 ( 0.00 0.00 ) ( -0.34 1.01 > ( 0.000 1.063 -0.34 -1.01 ) ( 0.27 0.00 ) 1.063 0.272 COMBINATION KP KI KD 143 -.316 -.0100 -7.20 < 0.00 0.00 ) < -0.42 1.02 ) < 0.000 1.108 -0.42 -1.02 ) ( 0.33 0.00 ) 1.103 0.334 1 3 4 . COMBINATION KP KI KB 144 - .31G -.0100 -9 .00 < 0.00 0.00 ) < -0 .50 1.04 ) < -0.50 -1.04 ) < 0.38 0.00 ) 0.000 1.153 1.153 0.385 APPENDIX D GRAPHICS EXAMPLE AND PROGRAM LISTING 136. APPENDIX D POSSIBLE GRAPHICS SCREEN LAYOUT (i SENSING PROBE LOCATION ST INPUT OUTPUT t ©Wave tank including moving waveboard and the resulting moving water wave profile. A spatial plot changing in real time. (2) Input-Output graphs, waveboard stroke and wave height profile changing with time. Alternatively, an SPD vs co plot, changing with time, could be placed here. It could show the desired spectrum with the actual spectrum changing to match. 137. 7-Sep-1986 14:06 7-Sep-1986 14:05 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 0055 0056 0057 C . C C . C C . C. C. C PROGRAM NUUAVE SIEVE HODGE 31712771 C C. C ..PROGRAM TO GRAPHICALLY DEP1CI WAVEBOARD MOTION AND . . T H E RESULTING WAVEHEIGHI PROFILE. THE WAVEBOARD STROKE ..AND THE WAVEHEIGHI ARE ALSO PLOTTED VERSUS TIME. INTEGER XEND.TEND REAL W,MM(21>,EM(21),DT,DX, C,HO,KO,STROKE,HORZ,VERT, A EX,UY,B0ARDX(30),B0ARDY(30).F(50,30),ERR0R1,ERR0R2, A KA<30),F1<50>,M(30>,FO(30,50),NU(30,50),XO(30),RAT 10, A PRODOO) 0PEN<UNIT=5,FILE='WAVDAT.DAT;2',STATUS='OLD' ) READ(5,A) U,EM1,MM(1),DT,TEND,DX,XEND,STROKE,TIME 100 FORMAT(II ,F5 .3 .T9 ,F5 .3 ,T17 ,F5 .3 ,T25 ,F5 .3 .T33 ,12 ,137 ,F5 .3 ,145 ,12 , A I49,F5.3> DATA X / 0 . / . A L 0 T / 2 0 / EM<1)=EM1 C=(WAA2)A0.249 ..CALCULATE WAVE COEFFICIENTS BY NEWTON'S METHOD DO 60 K=1,AL0T DO 70 N=l,20 EM<N+l)=EH<N>-< <C+EM<N)AIAN<EM<N>>>/<TAN<EM<N> > + A (EM<N)/C0S<EM(N)>AA2>> ) IF(EM(N+1).LE.O.O) GOTO 400 ERR0R1=ABS(EM(N+1)-EM(N)) IF(ERR0R1.LE..001) GOTO 300 GOIO 70 400 EM(l)=EM(l)+.5 EH(N+1)=EM<1> 70 CONTINUE 300 M(K)=EM<N+l)/2.44 PR0D<K)=M(K)A2.44 KA(K)=2.AM(K)ASTR0KEA((C0S(PR0D(K))/((M < K)AA2)Al.04))+ A (SIN(PR0D(K)>/H(K> >-(COS(M(K)Al.4>/((M(K)AA2)Al.04> ) ) / A (PR0D(K)+SIN(PR0D(K))AC0S(PR0D(K)) ) EM(1)=(K+1)AM<1)A2.44 60 CONTINUE DO 75 N=l,20 MM(N + 1)=MM(N) + (<C-MM<N)ATANH <MM(N)))/(TANH<MM(N)> + A <MM<N)/(C0SH<MM<N>>AA2>>)> IF(MM(N+1).LE.0.) GOTO 450 ERR0R2=ABS(MM<N+1>-MM<N)) IF(ERR0R2.LE..001) GOIO 350 GOTO 75 450 MM(N+l)=MM(N)+.l 75 CONTINUE 350 M0=MM(N+l)/2.44 PR0D0=M0A2.44 K0=2.AM0ASTR0KEA< ( SINH<PRODO>/MO)-(COSH(PRODO>/(1.04A<M0AA2 ) ) ) A •MC0SH<H0A1.4>/(1 . 04A ( MOAA2) > > ) / A (PR0D0 + SINH<PR0D0)AC0SH<PR0D0) ) 138. NUWAVE*MAIN 7-Sep-1986 14:06 7-Sep-1986 14:05: 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 0114 ! !WAVEBOARD INPUT C. . .CALCULATE ALL UAVEHEIGHTS S BOARD POSITIONS FOR X t TIME C DO 40 1=1,TEND T IME = T IME+DT " X0<I)=STR0KEAC0S<WAIIME) RATIO=XO< D / 1 . 0 4 BOARDXC I) = <15. + U.4ASIN<ATAN<RATI0>) )A9.84) B0ARDY<I) = <79. + <..37-<1.4A<l-C0S<ATAN<RATI0> >>>>A9.84> X = 0. DO 50 J=1,XEND X=X+DX F1(J)=0.0 DO 65 K=1,AL0I E(J,K)=KA(K)A<U/N<K>)AC0S<PR0D<K> >AEXP<-M<K)AX> F1<J)=F1<J)+F<J,K) 65 CONTINUE F0<I,J)=<W/M0)AC0SH<PR0D0)ASIN(UAIIME-M0AX> NU( I ,J ) = <K0AF0<I,J)+F1<J)ACOS(WATIME)>AOU/9.81> !!WAVEHEIGHT 50 40 CONTINUE CONTINUE ! ! OUTPUT C C. C. C ,DRAW WAVEBOARD POSITIONS t WAVEHEIGHT PROFILES •ALSO PLOT THE STROKE AND WAVEHEIGHT VERSUS TIME CALL GRSTRT(4027,5) !CHECK THIS CALL M0VE<5.,85.) CALL DRAU(5.,55.) CALL DRAW(145 . ,55.) CALL M0VEC15.,55.) CALL DRAW(15.,68.5) CALL M0VE<5.,78.7) CALL DRAW(148.,78.7) CALL M0VE(5.,45.) CALL DRAW(5.,5.) CALL DRAW(70.,5.) CALL MOVE(5.,25.> CALL DRAW(70.,251> CALL M0VEC80. ,45.) CALL DRAUC80.,5.) CALL DRAW(145.,5.) CALL M0VE(80.,25.) CALL DRAU(145.,25.) CALL MARKER(1.,40.,88) CALL MARKERO. ,40. ,48) CALL MARKER<76.,40.,78) CALL MARKER(78.,40.,85) CALL MARKER<69.,3.,84) CALL MARKER(144.,3.,84) TIME=0. DO 1000 1=1,TEND T IME=TIME+DI CALL MARKER(5.+TIMEA33.163,25.+(X0(I)A53.37)/2.,42) CALL MARKER<80.+TIMEA33.163,25.+<NU<I,25)A53.37)/2.,42) 1000 CONTINUE TIME=0. DO 30 1=1,TEND 139. NUWAVE$MAIN 7-Sep-1986 14:06 7-Sep-1986 14:05 0115 TIME=TIME*DT 0116 CALL M0VE<15.,68.5) 0117 CALL APPEAR 0118 CALL DRAU(B0ARDX(I) fB0ARDY<I)) 0119 DO 900 L=l ,2 0120 X=0. 0121 DO 90 J=1,XEND 0122 X=X+DX 0123 H0RZ=15.+XA9.84 0124 VERT=78.7+NU<I,J)A9.84 0125 IF(XO<I).GT.X) GOIO 90 0126 I F ( L . E Q . l ) GOIO 89 0127 CALL REMOVE 0128 89 CALL MARKER(HORZ,VERTr42> 0129 90 CONTINUE 0130 IF((L.EQ.2) .OR.<I .EQ.TEND) ) GOTO 30 0131 CALL M0VE(15.,68.5) 0132 CALL REMOVE 0133 CALL DRAU(BOARDX<I),BOARDY(I)) 0134 900 CONTINUE 0135 30 CONTINUE 0136 CALL GRSTOP 0137 STOP 0138 END 

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