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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  degree  at  this  the  thesis  in  partial  University of  fulfilment  of  of  department publication  this or of  thesis for by  his  requirements  British Columbia, I agree  freely available for reference and study. I further copying  the  that the  her  representatives.  It  this thesis for financial gain shall not  is  ^ g C H A I M  tcVH--  The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6(3/81)  Library shall make  by the  understood  that  be allowed without  permission.  Department of  an advanced it  agree that permission for extensive  scholarly purposes may be granted  or  for  ^ M C l N g g R t M ^  head of copying  my or  my written  ABSTRACT  The waves produced by a flap-type wavemaker, hinged i n the middle, are modelled using first-order linear wavemaker theory. closed-loop, discrete-time system i s proposed.  A simplified  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. control theory  Conventional discrete-time  i s used with the major difference being the use of a  relatively long timestep duration between changes in waveboard motion. The system response i s calculated for many controller gain combinations by the computer simulation program CBGANES.  System stability i s  analyzed for the gain combinations by using two different methods. One method i s an extension of the Routh criterion to discrete-time and the other i s 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  i n an actual  wave  tank  experiment. The computer simulation performance response and the two stability analyses predict the same results for varying controller gains.  It i s  evident that integral control is essential in order to achieve a desired response for this long duration timestep application.  The variation i n  discrete timestep duration and i n desired spectral density (an indirect indication of frequency variation) provide variation i n the constraints on controller gain selection.  The controller gain combinations yielding  the fastest stable response at a single frequency are for large proportional gain and small integral and derivative gains. - ii -  TABLE OF CONTENTS Page ABSTRACT  i i  LIST OF TABLES  v  LIST OF FIGURES  vi  LIST OF SYMBOLS  ix  ACKNOWLEDGEMENT  xii  1.  2.  INTRODUCTION  1  1.1  Preliminary Remarks  1  1.2  Review of Literature  3  1.3  Purpose and Scope  5  CLOSED-LOOP CONTROL SYSTEM FORMULATION  8  2.1 Physical System Description  8  2.2 Error Formulation 2.2.1 2.3  3.  10  Waves and Spectral Density  10  2.2.2 Discrete-Time Error  11  Control Strategy  14  2.3.1  PID Control Description  14  2.3.2 NRC Control Description  17  SIMULATION  18  3.1 Wave Generation System Model  18  3.2  3.1.1 Wave tank Model and Assumptions  18  3.1.2  19  Wavemaker Theory and Assumptions  Computer Simulation and Controller Details  25  3.2.1 Controller Gain Orders of Magnitude  25  3.2.2  29  Computer Simulation  3.3 Results and Discussion  35  3.3.1 Proportional Control  35  3.3.2  37  Proportional-Derivative Control  3.3.3 Proportional-Integral Control  37  3.3.4  38  Proportional-Integral-Derivative Control - iii -  TABLE OF CONTENTS (Continued) Page 4.  STABILITY ANALYSIS 4.1  5.  6.  40  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  PERFORMANCE ANALYSIS  53  5.1  The Measure of Performance  53  5.2  Controller Gain Selection  56  5.3  Results and Discussion  58  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  - v -  36  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  Qualitative Sequence of Waveboard Motions vs. Discrete Time  15  5. 6.  Geometry of Waveboard and Wave tank  7.  Computer Simulation Flowchart  8.  Error vs. Discrete-Time, KD(co) = -1.8  9.  KP(a>)  26 32  = -0.158, KI(UJ) = -0.0 64  Error vs. Discrete-Time, KP(co) = -0.158, K I ( c o ) = -0.0 KD(a>) = -5.4, I n i t i a l Actual Spectral Density = 0.001 m .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  2  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  Error vs. Discrete-Time, KP(OJ) = -0.158, KI(w) = -0.002, KD(u)) = -5.4  73  17.  - vi -  LIST OF FIGURES (Continued)  Page  Figure 18.  Error vs. Discrete-Time, KP(co) = -0.158, KI(co) = -0.002, KD(OJ) = -9.0  19.  74  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  24.  Error vs. Discrete-Time, KP(u)) = -0.237, KI(w) = -0.008,  79  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, 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, Various |KP(co)|  |KI(io)| vs |KD(o))| For 86  Routh Stability Constraints, Various T  |KI(OJ)| VS |KD(to)| For  32. Routh Stability Constraints, Various DSPD(u))  |KI(to) | vs |KD(a>)| For  31.  33.  KI(<D)  = -0.004,  87  Largest Eigenvalue vs |KP(ai)| For Various |KI(a))| and |KD(co)| - vii-  88 89  LIST OF FIGURES (Continued) Page Figure 34.  Largest Eigenvalue vs and  35.  For Various  |KP(ID) |  |KD(u>)|  Largest Eigenvalue vs and  |KI(UJ)|  90 |KD(OJ)  | For Various  |KI(U)|  |KP(co)| 91  - viii -  LIST OF SYMBOLS  a  variable in quadratic equation for Routh analysis  a n  coefficient for output value of conventional difference equation  a mn  element of system matrix F —  A  complex eigenvalue magnitude  A  n  coefficient from partial fraction expansion  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 i n 3-domain  c  variable i n quadratic equation for Routh analysis  C  output in conventional difference equation  C n  coefficient i n wave equation  d  depth of s t i l l water i n wave tank  DSPD  desired wave spectral density  E  error operated on by controller  E'  error from summing/difference  E^  energy due to each frequency i n a seaway  E m E  measurement error  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  g  junction  energy of a particular wave per unit area of sea surface  - ix -  H  wave height amplitude  i  subscript indicating a particular frequency  j  variable in summation  K  integer discrete timestep  KD  derivative gain  Kl  integral gain  KP  proportional gain  m n  wavenumber of generated wave  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 a  waveboard stroke amplitude  STMAX  maximum stroke spectral density  Si  STSPD waveboard stroke spectral density ASTSPD change i n stroke spectral density at end of discrete timestep K t t  time s  sampling period  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 n  state variable  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) P  spatial velocity potential  6  angle in imaginary plane, for complex vector  oi  frequency  Ao),A<i)^ spectral frequency bandwidth co max w  maximum frequency of interest n  ,  J  frequency at which peak in wave spectrum occurs  - xi -  ACKNOWLEDGEMENT The author wishes  to thank Professor Dale B. Cherchas  guidance and patience throughout this work.  Some funding was provided  for this work through the NSERC Grant No. A4682.  - xii -  for his  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 i s 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  action i s relatively long.  duration  between controller  The application i s 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 i s 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 i n the spectrum to reach a sensing probe before corrective action, in the event that the actual wave characteristics do not emulate the desired wave characteristics, can take place. system inherently  contains  a long  duration  Hence, the  discrete timestep.  The  system responds dynamically for the timestep duration until the resultant or actual wave characteristics are compared to the desired wave characteristics.  The system i s not one of instantaneous  control action  requiring dynamic and ongoing changes. The control of this class of discrete-time system i s achieved by the use of a proportional plus integral plus derivative (PID) controller with P, PI, PD and PID conditions examined. operated the  by a user wishing  towing  tank  Ultimately the system i s  to achieve desired wave characteristics i n  in a reasonably  short  time  and within  specified  FIGURE 1  TOWIMG TANK CONFIGURATION  WAVE HEIGHT SENSING PROBE  HYDRAULIC CYLINDER  SSS V  SS SS  WAVE ABSORBING BEACH  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 f i r s t order linear wavemaker theory i s 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 f i e l d , 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  discrete-time techniques. the  area  of  controls  that  step discrete-time  by  using  conventional  The author is not aware of any research in  duration discrete timesteps. small  automatically  has  examined dynamic systems with  long  Much research has been published regarding  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 i s one which i s inherently discrete and  does not extend from continuous analysis. for  successive discrete  timesteps and  controller action i s short.  The reference input varies  the timestep duration between  In comparison, the case i n this work i s 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 a l . (1951) discussed the theory for a particular type of wavemaker.  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 f i r s t 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 i s not as widely documented, however, a notable contribution i s given by Anderson and Johnson (1977). A wave generation system i s presented in which the feedback  from an  observed wave characteristic or response i s provided by an operator who makes the decision of whether to perform more iterations i n the attempt to achieve better agreement between actual and desired wave characteristics.  The transfer function between the waveboard driving spectrum and  5.  the wave height response i s modified, i f desired,  on each iteration  until satisfactory wave characteristics are achieved.  This system i s  claimed to be effective but i t does not close the control loop automatically.  The corrective action i s achieved by changing the transfer  function relating the waveboard motion to the wave height response and not  by  utilizing  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 Ottawa, Canada.  (NRC) Hydraulics  laboratory i n  They outline an elaborate system for controlling spec-  t r a l energy distribution, energy distribution i n 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 i s modified to compensate for energy shift due to wave propagation.  This compensation  i s achieved by using an iteration based upon the differences between actual and desired wave spectrums including the use of a transfer function,  dissimilar to that  used  by Anderson and Johnson  correcting the waveboard control signal.  (1977) for  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 i s 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 information (see Figure 2).  Also the eigenvalue analysis i s 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  i s to extend discrete-time control theory to the specific application in which the discrete timestep duration i s long.  The response time i s  therefore i n terms of minutes as opposed to seconds.  The simulation i s  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 i n terms of a spectral density error at each spectrum frequency for various controller parameters.  FIGURE 2 SIMPLE CLOSED-LOOP SYSTEM BLOCK DIAGRAM  ERROR  CONTROLLER  WAVEBOARD STROKE INFORMATION  /  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 i n this work i s currently  open-loop with the human operator providing the closure of the loop. A flap-type waveboard which can be pivoted at i t s base or at i t s midsection i s situated at one end of a 67 meter long towing tank (see Figure 1).  This waveboard i s driven by an hydraulic cylinder with the  stroke controlled by a spool valve. wave synthesizer voltage  signals.  The spool valve i s controlled by Waves generated by the waveboard  travel down the tank to a twin wire probe at which the wave height i s sensed.  A wave absorbing  beach i s 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 i n 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 i n Figure 3.  The waveboard generates waves which are sensed  and  seconds (a period chosen to eliminate any  sampled every  t  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 i n this work, converted to  9.  FIGURE 3 POSSIBLE HARDWARE CONFIGURATION for CLOSED-LOOP SYSTEM  WATER SURFACE POSITION FEEDBACK TRANSDUCER  WAVE HEIGHT SENSOR SAMPLER ,\ts  D/A HOLD  A/D  DATA ACQUISITION UNIT  V  I  TRANSMITTER COMPUTER (VAX, IBM)  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 i n this work i s the controller  algorithm used i n 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 i n 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), i s transformed via Fourier Transform techniques i n t o an amplitude spectrum.  The amplitude, H , of a a  particular sinusoidal wave at a specific frequency i s used to express the energy of the particular wave as, E  =1 pgH  per unit area of sea surface  2  (2.1)  The total energy per unit area of wave surface for an irregular seaway, one composed of superimposed sinusoids, i s given as, l  E  T 1  n  = y P g Z (H 2) i=l i Z  (2.2)  a  The total energy of a given seaway gives an indication of the severity of the seaway (Bhattacharyya, 1978) . in the seaway i s given by  The energy due to each frequency  11.  E  =  ±  \  p  (2.3)  gH  where Aio^ i s the bandwidth about a particular frequency. An extension of the energy spectrum i s the wave spectrum.  This i s  obtained by dividing out pg from the energy spectrum leaving an expression for the spectral density of wave energy at a particular frequency in the seaway,  S  i  =  (2.4)  Y a' "i H  / A  Standard spectral density, i.e. wave spectrum, information i s used i n towing tank experiments when prototype sea state information i s not available.  Therefore, i t i s reasonable to utilize the spectral density  formulation for the comparison of desired and actual wave characteri s t i c s i n this work.  The advantage of using the wave energy spectrum  method for describing a seaway i s that the time dependence of the wave i s dropped and both the wave amplitude and frequency can be taken into account.  The phases of each sinusoid i n a sea state w i l l differ from  one test to another but the amplitude/frequency information w i l l remain the same.  2.2.2 Discrete-Time Error For  this work an irregular seaway, i.e. a seaway composed of many  sinusoids  each  characteristic  at a  different  of interest  density of each sinusoid.  frequency  i s assumed.  The wave  for comparison purposes i s the spectral The error required for the generation of new  waveboard driving signals i s expressed as the difference between the actual  spectral  density  and the desired  frequency of interest i n the wave spectrum. is expressed as,  spectral  density  at each  For generality, the error  12. E(KT,iu) = (ASPD(KT.u) - DSPD(KT,UJ) )/DSPD(KT,u)) + E (KT,u>) M  (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  K  = integer number of timestep.  T  = discrete timestep duration  (o  = frequency  m  In an actual wave tank experiment the wave spectral density can only  be represented  by an estimate of the true  density at each frequency.  or actual spectral  This estimate arises from the process of  sampling the wave height time series.  The spectral density estimate  often has  error dependent upon how many  some bias error and variance  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 i n the tank and the estimated wave spectral density.  For the remainder of this work the measurement error i s  assumed to be zero.  This implies that the following work cannot be  applied directly for the control of actual modelled sea states. Figure 2 i s now altered to include the description of E(KT,co) given in Eq. The  (2.5)  linearity  with E (KT,u))=0. m  of the system  Figure  4 reflects this  i s not affected  since  modification. i n 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 I n i t i a l state of the  FIGURE 4 SPECIFIC CLOSED-LOOP BLOCK DIAGRAM for COMPUTER SIMULATION  E'(KT,OJ)  STSPD(KT,c^)  E(KT,OJ)  DSPD(KT,6o)  ASPD(KT,a>) 1/DSPD(KT,(Uj)  CONTROLLER  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 i s formed between the spectral densities at each frequency of interest i n the spectrum.  The waveboard motion i s 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 i t s e l f  in a discrete fashion until the discrete-time error i s a satisfactory value.  Figure 5 shows the sequence of waveboard motions qualitatively  versus discrete-time.  As mentioned, the motion of the waveboard i s  changed after each timestep based upon the discrete-time error.  It i s  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  Figure 4.  is shown in  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 i n terms of spectral density.  As a result, at  each frequency of interest a waveboard stroke amplitude corresponds to a water wave amplitude. spectral density.  By  The water wave amplitude i s then converted to a 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 i n the waveboard motion. output  of the controller  The  i s 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 i n Eq. (2.5). part  provides  a change dependent on a l l previous  The integral  errors over time,  specifically, the rate of change of the controlled variable with respect to discrete-time i s proportional to the error. the  controller output  proportional  provides  to the rate  The derivative part of  a change in the controlled variable  of change of the error with  discrete-time, an anticipative component.  respect  to  Therefore the new waveboard  motion or stroke at the beginning of a discrete timestep i s ,  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 calculated change in waveboard motion.  motion plus  the newly  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 i s equal to the old driving spectrum plus a change in the driving spectrum.  The change i s 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»  peak  )  a  ;  co  p e a k  <  u  <  V  x  (2.8) 1 j  ;  otherwise  where U  peak  =  f  r e c  l  u e n c  y  which the peak i n the spectrum occurs  a t  to max  = maximum frequency of interest  a  = arbitrary constant usually equal to 1/2  n  J  This control scheme i s essentially proportional control with the proportional gain expressed as a function of frequency.  18.  CHAPTER 3 SIMULATION  The PID gains used i n this work are frequency dependent but do not have a specific functional form.  As an aid i n the determination of  which gain combinations to use at each frequency, whether under P, PI, PD or PID control, a simple simulation i s 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 i n an attempt to achieve realistic sea states i n 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 leakage 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 i s adopted for the simulation.  The  theory adopted i s based upon the solution of Laplace's equation i n terms of a velocity potential, <(), i.e. 1976).  the linear theory of Airy (Le Mehaut€,  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 i s necessary. The wave tank housed i n the B.C. Research Ocean  19.  Engineering Center i s 67 meters long, 3.6 meters wide and 2.4 meters deep for i t s entire length.  A maneouvering  tank i s 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 i s water transfer between the tanks.  For the purposes of this  work i t i s assumed that the wave tank i s completely separate from the maneouvering tank. including  Figure 6 shows the geometry of the actual waveboard  the variables used i n 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 i s assumed to be infinitely long and therefore no reflection waves are present i n the simulation.  In reality, the  wave tank does have a wave absorbing beach at one end which w i l l produce some reflective waves.  There i s leakage around the sides and bottom of  the actual waveboard i n 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 i n the tank i s obtained from the theory developed by  20.  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 f i r s t order approximation.  For these  conditions, Laplace's equation holds for the velocity potential in the fluid.  Therefore, 3<|>/3x + 9<t>/3y = 0 2  The  linearized  boundary  2  2  conditions for  (3.1)  2  the  specific  waveboard  and  wave tank shown in Figure 6 are  3*/3y| = _ y  d  =  (3.2)  0  : no flow through tank bottom  3())/9x|• v = n.  = 3x/3t  (3.3)  : water particles at the waveboard move with the same velocity as the waveboard : the waveboard motion i s given by x(y,t) = ST(y) sin(cot)  3<|>/3t + g«3<t>/3y = 0 2  2  @  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 i s that a 'radiation* condition exists, i.e. waves travel in one direction only, that of increasing x, particularly at large x.  Another assumption i s that the potential § i s  finite everywhere within the region, x > 0, -d < y < 0.  In reality,  there w i l l be reflection off the wave absorbing beach at one end of the tank which w i l l 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) = <|>(x,y) cos(cot)  (3.6)  p  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))«3<t)/3y = 0 2  p  @ y = 0  (3.8)  Assuming that d> i s f i n i t e then there are an i n f 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  4> = -(a)/m )«cos(m (y+d) )«e n  n  -m x  n  , n = 1,2,...  (3.10)  so d> = p  Z C d> = C •(co/m )»cosh(m (y+d))«sin(m x) n n o o o o n=0  r  t  +  °° -m x I - C .(ui/m )«cos(m (y+d))«e . n n n n=l  (3.11)  n  where m i s the positive solution to u o  = m g»tanh(m d), o o  2  m i s the positive solution to co = -m g«tan(m d) , n = 1,2, n n n  and  2  The C and C c o e f f i c i e n t s i n Eq. (3.11) are determined by using o n the  lateral  boundary  condition  at the waveboard,  Eq. (3.3).  The  waveboard motion i s 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  ;  T h e r e f o r e from Eq. (3.3) and the f a c t  -d < y < -h  that cosh(m (y+d)) and o cos(m (y+d)), n = 1, ...,<», form a complete set of orthogonal functions, n  the coefficients can be written as follows,  23.  0  ((i)/m )•/  o  Co =  ', 0~  ST(y) cosh(m (y+d)) dy o  d  / cosh (m (y+d)) dy -d 2  2u>ST •f((sinh(m d))/m ) - ((cosh(m d))/hm ) + ((cosh(m (d-h)))/h m )l a o o 2 2 2 o m d + sinh(m d)«cosh(m d) o o o (3.13) 2  2  v  and  0  -(to/m )•/  n  Cn =  ST(y) •cos(m (y+d)) dy ' n " 0 / cos (m (y+d)) dy -d d  2  -2OJST .f((cos(m d))/hm ) + ((sin(m d))/m ) - ((cos(m (d-h)))/hm )) a n n n n n n ' m d + sin(m d)»cos(m d) n n n 2  2  v  (3.14)  Now the potential <))(x,y,t) can be written,  <j>(x,y,t) = C «cosh(m (y+d))«sin(ut - ™ X ) o  +  o  Q  » -m x E C »e .cos(m (y+d)).cos(wt) , n n n=l n  (3.15)  This potential i s composed of two parts, a progressive wave, the C term, and a series  of standing  waves which decay with  o 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 i n the  series i s 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 ) - 8 K x , y , t ) / 3 t |  (3.16)  y=0  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 °° +  E wC  -m x »e  n  •cos(m (y+d)).sin(cot)  (3.17)  n  n=l or, for x > 3d,  n(x,t) = -((o/g) »CQ«cosh(m (y+d) )«cos(u)t - m x) o  o  (3.17a)  a progressive wave. The required  relationship between the waveboard motion and the  resulting wave form has now been determined. it  For use i n the simulation  i s necessary only to look at the relation between the wave height  amplitude, H , and the amplitude (stroke), ST , a a motion.  Therefore,  of the waveboard  25.  H /ST = a a 2u> . cosh(mod) • 2  (  ((  s i n n e d )  )/m )-(cosh(m d) )/hm2) )+( (cosh(m (d-h) ) )/hm|) ) o  o  o  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 i s calculated based upon  the physical  dimensions  of the system.  It i s assumed  that  i n i t i a l l y there are no waves present i n the tank and the wavemaker i s at rest i n the vertical position.  The simulation i s 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. of  the controller  This condition gives the output  as, AST(KT,u>) = E(KT,o))«KP((o), and therefore the  stroke for the next iteration i s the i n 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 i n terms of spectral density also. this  i s done i n the computer simulation.  therefore calculated with this i n mind. is then,  For convenience,  The controller gains are  The output from the controller  FIGURE 6  GEOMETRY of WAVEBOARD and WAVE TANK I  HYD. CYL. STROKE  STROKE,^  ST@y=0  •  WAVE AMPLITUDE, Ha s s s S  y—///////  ON  ASTSPD(KT,co) = E(KT,w)'KP(u))  (3.19)  and the new waveboard motion i n terms of a stroke spectral density i s given as,  STSPD(KT,oo) = STSPD((K-l)T,a>) + ASTSPD( (K-l)T,io)  (3.20)  where STSPD indicates that the stroke i s expressed i n spectral density form. The maximum physical stroke for the B.C. Research Ocean Engineering Center waveboard at the free surface of the water i s 0.356 meters. In spectral density form, the maximum stroke spectral density i s ,  STSPD(KT,u>) = ST(KT,u))/(2Aa>) = (0.356) /(2)(0.2) = 0.316 m sec 2  2  where Aco = 0.2 sec" i s assumed as typical. 1  2  The error given by Eq.  (2.5) i s i n i t i a l l y equal to -1 since ASPD(0,u>) i s assumed equal to zero. Therefore 0.316.  from Eq . (3.20) we can say that, IKP(co) I = 0.316/E(0,w) = max  This would be a maximum gain value starting from rest since the  waveboard cannot exceed i t s maximum physical stroke.  If the actual  spectral density of the waves, ASPD(KT,u>) i s greater than desired then the error i n Eq. (2.5) i s positive.  Therefore, i n order to decrease the  stroke and accordingly ASPD(KT,oo), the change given by the controller output must be negative. then  the error  would  Similarly, i f ASPD(KT,a)) i s less than desired be negative.  To increase  the stroke, and  ASPD(KT,u>), the change must be positive which again means that KP(w)  28. must  be negative.  Therefore,  an approximate  range  and order of  magnitude for the proportional gain, KP(u>) i s ,  -0.316 < KP(to) < 0  (3.21)  To determine an approximate range and order of magnitude for the integral gain, K I ( U J ) , i t i s convenient to eliminate KP(u)) and KD(ui) from the controller output.  The change i n waveboard motion i s thus given  by,  ASTSPD(KT,u)) = KI(UD).(E(KT,U)) + E( (K-l)T,u>) ) «T/2  where a discrete timestep duration, T, must be assumed. in  reality  i s dependent  (3.22)  This duration  upon the time for the slowest wave i n the  spectrum to travel the distance from the waveboard to the sensing probe. Several values for T can be used i n the simulation but a value of 60 seconds  i s 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))|  max  = 0.316/f(-1+0)(60/2)1 = 0.011  Again, as for KP(u), KI(u>) w i l l be negative which means that an approximate range and order of magnitude for the integral gain, K I ( w ) i s ,  -0.011 < K I < 0  for T = 60 seconds.  (3.23)  Similarly, an approximate range and order derivative gain, KD(io) i s found. but  29.  of magnitude for the  Derivative control i s never used alone  assuming so does aid in determining possible KD(io) values.  The  change in waveboard motion i s 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>) i s ,  -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 i s necessary to discuss the details of the closed-loop system shown in Figure 4 and how each component i s included i n the simulation. As  mentioned  i n Section  controller i s chosen as  2.2.2  the error  operated  on by the  30. = (ASPD(KT.a))  E(KT,UJ)  - DSPD(KT, co) )/DSPD(KT, cu)  (2.5)  Using the conventional summing/difference junction shown i n Figure 4 , then DSPD(KT,<o) - ASPD(KT,co) = E'(KT,co). in Eq. ( 2 . 5 ) i t i s necessary  In order to obtain the error by (-1/DSPD(KT, co)).  to multiply E'(KT.co)  Normally, this would introduce a nonlinearity with respect to time but since DSPD(KT,co) i s 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 i s given i n Eq. ( 3 . 2 0 ) .  This output  i s a new waveboard stroke spectral density for each frequency.  The  relationship between the waveboard stroke and the wave amplitude i s given i n Eq. ( 3 . 1 8 ) .  The computer simulation i s written such that the  relationship between the waveboard motion and the resulting wave height i s in terms of spectral density.  This relationship between STSPD(KT,oo)  and ASPD(KT,co) i s obtained by converting the waveboard stroke and wave amplitude relationship, H /ST 3.  signifies,  = WTF.  This i s Eq.  ( 3 . 1 8 ) where WTF  3.  'wave transfer function' .  given by, H /2Aco at each frequency. 2  The wave spectral density, i s  Therefore,  H /2Aco = (ST •WTF) /(2Aco) = (ST /2Aco) • (WTF ) a a a 2  2  2  2  = (stroke spectral density)•(WTF ) 2  and i t can be written,  ( S T S P D ( K T , O J ) ) . ( W T F ) = ASPD(KT,co) 2  (3.26)  31.  Equation  (3.26)  i s used  i n the computer  simulation.  Now,  this  ASPD(KT,(o) i s 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 i n Appendix A and the flowchart i n Figure 7, the simulation begins with the user selecting the desired frequency  of interest and the frequency bandwidth for spectral  density calculations, both i n radians/sec.  The maximum stroke spectral  density, STMAX, i s then calculated from this information using the maximum  stroke for the waveboard, STMAX = 0.063/AOJ.  The 'wave transfer  function', WTF, relating waveboard stroke to wave amplitude, larly  or simi-  waveboard stroke spectral density to the actual wave spectral  density i s then calculated using the f i r s t order linearized wave theory discussed i n Section 3.1.2.  The maximum ASPD(KT,u>) achievable at this  frequency i s calculated and displayed on the terminal screen as a guide for the selection range of DSPD(KT,io).  In reality, a standard spectrum  w i l l likely be predetermined, for instance, from a Pierson-Moskowitz (Hudspeth,  Jones  and Nath,  1978) or an International Towing Tank  Conference (ITTC)(Bhattacharyya, the  DSPD(KT,w) value  1978) wave spectrum but for this work  i s 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, i s 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 i s cycled through for each set of controller gains.  The actual wave spectral density i s calculated  using the difference equation for the system.  This equation i s obtained  by combining the discrete-time relationships for each part of the system and i s written as,  START  t  enter cu enter A c t i calculate: STMAX, WTF. ADSPOMAX display: WTF. ASPOMAX enter DSFO enter initial KPM. NM. KDM enterttd 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 F i g u r e  increment timestep, K  7.  Computer  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 indicators & Store STSPO, ASPD, E (or all K Store UJAU>, WTF, DSPO.T 2  1  select next KD  select nextKJ  select next KP  Format Stored Data •  C -  Figure 7.  )  Computer Simulation Flowchart (Continued)  34. ASPD((K+l)T,a)) = ((WTF /DSPD(KT,u)).(KP(a))+KI(o))«T/2 2  + KD(u))/T)+2).ASPD(KT,a)) + ((WTF /DSPD((K-l)T, co) ) 2  • (-KP( oi)+KI( a)) • T/2 )-1) ) • ASPD( (K-l)T, co) + +  ((WTF /DSPD((K-2)T,«))).(-KD(a))/T))-ASPD((K-2)T,a)) 2  (-KI(UJ).T.WTF ) 2  (3.27)  The error and the stroke spectral density are calculated at each timestep.  For each gain combination,  the number of timesteps, K, taken to  achieve a consistent error value with magnitude less than 3% (0.03) i s an indication of how well the particular combination performs. timesteps are used for each gain combination error  versus  discrete-time  response.  Thirty  i n order to examine the  If the maximum  achievable  waveboard stroke spectral density i s exceeded during the control loop an indicator (specifically, the last K at which STSPD > STMAX) i s 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 i n Figure 7.  For the purposes of  the computer simulation, i f the waveboard stroke spectral density value exceeds the maximum allowable, the simulation i s 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 i s considered  density  values  terminated  an unstable  are always positive.  state since spectral  The control  looping would be  for such a condition i n an actual experiment but i n this  simulation i t i s allowed to continue cycling with an indicator set at the last K value that a negative stroke spectral density occurred. An unstable state i s assumed when the error value Increases for successive discrete timesteps.  35. The desired  computer s i m u l a t i o n c y c l e s through as many g a i n combinations as and  stores  the e r r o r ,  ASPD(KT,a>) and the s t r o k e with  respect  pertinent DSPD(co),  to  the  parameters wave  bandwidth,  E(KT,u>),  the a c t u a l  spectral  density,  s p e c t r a l d e n s i t y , STSPD(KT,to) , i n column form  number such  of  discrete  to, desired  as frequency,  transfer function,  WTF,  ACD, t h e c o n t r o l l e r g a i n  timesteps,  timestep  values  K.  The  spectral  duration,  T,  other  density, frequency  and t h e i n d i c a t o r s f o r the  s t r o k e s p e c t r a l d e n s i t y c o n d i t i o n a r e a l s o s t o r e d i n t h e same d a t a Appendix  C contains  for  o f the many g a i n  some  show t h e system  response  the c o r r e s p o n d i n g  3.3  Results The  m .sec  from  gain  t h e computer  tested.  Figures  of E(KT,u>)  simulation  8 through  vs. discrete-time,  29  K for  output combinations shown i n Appendix C.  running  1 rad/sec,  Combinations  derivative  from  and D i s c u s s i o n  to =  discussed.  The  i n terms  a  gains values  of  the  desired  and a d i s c r e t e t i m e s t e p  2  output  combinations  numerical  results  frequency  the n u m e r i c a l  file.  computer  spectral density  duration,  four  simulation  a  DSPD(cio)  T = 60 seconds  proportional,  at  will  six integral  single =  0.004  now  be  and s i x  a r e used y i e l d i n g E(KT,w), ASPD(KT,oo) and STSPD(KT,to) . tested  a r e shown  i n Table  1.  The  144  combinations  t e s t e d g i v e t h e f o l l o w i n g c o n t r o l modes: P, PD, PI and PID.  3.3.1  Proportional Control The  s i m u l a t i o n p r e d i c t s no change i n response f o r i n c r e a s e d  t i o n a l g a i n i f t h e i n t e g r a l and d e r i v a t i v e g a i n s a r e z e r o . the  initial  response  conditions  initially  stating  depends  upon  that  no waves  the I n t e g r a l  exist gain  propor-  In f a c t , f o r  i n the tank, the alone.  This  is  36.  TABLE 1  CONTROLLER GAIN VALUES TESTED  KPM  KIM  -0.079 -0.158 -0.237 -0.316  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 i s zero.  This  response, or lack of i t , i s typical for proportional control alone also.  3.3.2  Proportional-Derivative Control  As indicated i n 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 i s effectively no response, the same situation as for strictly proportional control and shown i n Figure 8.  If the  i n i t i a l actual spectral density i s non-zero and the integral gain i s zero, the response changes i n i t i a l l y but quickly converges to a non-zero error value.  Figure 9 shows a typical response for this condition. The  response shows more oscillation i s increased  in the f i r s t  few timesteps as the  derivative  gain  and also as the proportional gain i s  increased.  In general i f no integral gain i s used i n the controller, a  steady-state error exists and the desired response i s never achieved.  3.3.3  Proportional-Integral Control The  results from this mode of control show a response which i s  mainly oscillatory. as  |KI(o))|  The period of oscillation, i n terms of K, decreases  increases  for a fixed  oscillates more quickly.  |KP(u)) j value,  At low |KP(co) | values  i.e. the response  the response becomes  unstable sooner with lower |KI(<JI))| values than at high |KP(u))| values. In general,  the error response takes longer  increases and the larger |KP(w)| values response at the higher  |KI(w)| values.  to reach 3% as |KI(u>)|  tend to cause a more stable Upon comparison of Figures 15,  38.  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 constant  KP(w) and  KD(co)  show these trends.  KI(co)  at  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 i s less than 3%  does not necessarily indicate that the response has stabilized. system response Figures indicate the timestep  The  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  slow  and often  unstable.  It i s best at low  as  increases, the response i n i t i a l l y becomes quicker but then  |KD(u))|  slows again at high  |KD(co)|  |KI(u>) |  i s very  values and, for a constant  KI(<o),  values.  As the | K P ( U ) | value increases the response becomes more stable and is  quicker.  Again,  at larger  the response becomes  |KD(OJ)|  slow i n i t i a l l y ,  then becomes quicker but slows again at the higher  At general.  As  | values  unstable for low  |KD(w)|  values.  |KI(oo)  | increases, the response i s  |KD(to)  values. even larger  |KP(oo)|  values  Notably, for a constant  the response i s more stable i n  KI(OJ),  as  |KD(OJ)|  large, the response becomes progressively slower.  varies from small to Therefore, depending  upon the magnitude of the KP(to) value the response i s not always the same for increasing  |KI(OJ)  | or increasing  cases, however, the response i s oscillatory.  |KD(OJ)|  values.  In most  The combinations i n which  39.  |KI(W)| are zero does not show any response,  Sections.  Six other  response with time, K.  combinations  show  as mentioned i n previous  a monotonically  decreasing  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 values,  small  |KI(u>)|  values  and small  are for the largest |KP(<D)| |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 i s achieved at this frequency i n 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 combination, 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  analysis of the closed-loop system response. this  work  i s the stability  discrete timestep duration.  requires  a  stability  Of particular interest i n  of the system using  a relatively  long  The goal i n this analysis i s 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,  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)  Using this form and re-writing as a difference equation obtained through addition, the relationship can be written as,  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)) + K I ( O J ) . T / 2 ) . E ( ( K - 1 ) T , O J )  + (-KD(oj)/T)^E((K-2)T,a>)  (4.2)  T h i s e q u a t i o n can be compared t o the d i f f e r e n c e e q u a t i o n ,  C(KT) + a . . C ( ( K - l ) T ) + a •C((K-2)T) + ... a .C((K-n)T) >• *• n  = b . r ( K T ) + b . r ( ( K - l ) T ) + ... b «r((K-n)T)  (4.3)  n  i n o r d e r t o w r i t e a s t a t e e q u a t i o n v i a the method  of d i r e c t  programming  (Cadzow, 1970). By  direct  programming  the s t a t e  equation  f o r Eq.  (4.3) i s g i v e n  as,  x ((K+1)T) xJ((K+l)T) x ((K+l)T)  "1 3  1 0  3  _ a  2  x (KT) x,(KT) x (KT)  *3  0 1  0 0  r(KT)  (4.4)  3  and  x (KT) x (KT) x, (KT) x  C(KT) = ( b  l f  b , 2  b ) 3  +  b .r(KT) Q  (4.5)  where A. b  l  A b  2  b  l  b  " 0 b  2  - b  a  Q  l a  2  A  b  0  b  3  Therefore, w r i t t e n as,  "  b  0  a  3  f o r t h e c o n t r o l l e r g i v e n by Eq. ( 4 . 2 ) , t h e s t a t e e q u a t i o n i s  42. x ((K+l)T,co) x ((K+l)T,a)) x ((K+l)T,to) 2  3  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 ( K T , co ) x ( K T , co) x , ( K T , co)  (4.7)  The same procedure i s done for the relationship between STSPD(KT,co) and ASPD(KT,co).  The relationship i s , ASPD(KT.co) = WTF • STSPD(KT, to), 2  which by direct programming gives,  x. ((K+l)T,co) = [0] x (KT,co) + [1] STSPD(KT,co)  (4.8)  ASPD(KT.co) = [0] x (KT.ui) + WTF « STSPD(KT, co)  (4.9)  and 2  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) x ((K+l)T,a)) x ((K+l)T,co) 3  STSPD(KT,co)  -10 0 0 1 0 0 0 1 0 0 0  0 0 0 0  x (KT.co) x (KT,co) x (KT.co) x (KT.co) 2  WTF/DSPD(KT,co) 0 0 1 2  DSPD(KT,co) /DSPD(KT, co)  (4.10)  43. Now introducing Eq. (4.7) and re-writing Eq. (4.10) for the system,  ((K+l)T,co) x ((K+l)T,a>) x ((K+l)T,u)) ((K+l)T,a))  12 22 32 \2  2  l  3  l  21 31  l l  Xl(  (KT,co)  l  x  l  x (KT,ai) x (KT,w)  13 *2 3 33 \  l  Xl  2  3  X (KT,OJ) 4  3  1 0 0 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 + (WTF /DSPD(KT,co)). (KP(co) + KI(co).T/2 + KD(co)/T)  a  1 2  = -1 - (WTF /DSPD(KT,o))).(KP(co) - KI((o)«T/2)  a  u  = -(WTF /DSPD(KT,co)).(KD(o))/T)  a  a  2  2  2  lk 21  a  1+1  =  a  22  =  =  a  32  =  a  23  =  2k  a  =  a  31  =  3  34  =  =  0  = KP(co) + KI(co).T/2 + KD(co)/T = -KP(co) + KI(co)«T/2  &  = -KD(co)/T  hZ  33  1  a^  2  a  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 eigenvalues of the matrix F_(T,co).  For stability, the magnitude of the eigen-  values of F_(T,co) must be less than unity, that i s , a l l eigenvalues must l i e within the unit circle.  This i s analogous to a continuous-time  system for which a l l eigenvalues must complex plane.  l i e in the left  half of the  44.  A short progam was  written to calculate the eigenvalues of the  F_(T,io) matrix, i n Eq. (4.11), for the same combinations of controller gains used  i n the computer simulation.  The problem reduces from a  fourth order equation i n X, the eigenvalue, to a cubic equation i n A. The routine SGEEV on the UBC Mechanical Engineering VAX 11/750 computer system i s used for determining the eigenvalues.  The program EIGVALS  incorporates SGEEV to determine the stability of the gain combinations. This program l i s t i n g i s contained i n Appendix B.  4.2  Routh Stability Analysis An often used  Routh criterion.  technique for determining system stability i s the  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 i s 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 discrete-time system.  Thus, examining  to be applied to a  the coefficients of the system  characteristic equation and the values of the elements in the f i r s t column of the Routh array w i l l provide constraints for which to determine stable controller gains. The  characteristic equation of the entire system i s obtained by  transforming the discrete-time relationships for the system to z-domain expressions. yields,  Taking  the  z-transform  of Eq.  (4.2)  and  simplifying  45.  •p ( ~\ K  }  3 2 z + z).STSPD(z) (KP(o))+KI(o)).T/2+KD(( ))/T).z +(-KP(a))+KI(cu).T/2).z+(-KD(a))/T) 2  ( z  ;i  i  (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).WTF  =  2  (3.26)  ASPD(KT,OJ)  becomes,  STSPD(z).WTF = ASPD(z)  (4.14)  2  Combining  Eqs. (4.12),  (4.13) and  (4.14) gives  the overall  system  transfer function as,  (WTF /DSPD). [a z + a z + a ] 2  2  ][  ASPD(z)/DSPD(z) = b3Z  where,  3  Q  (4.15) + bz 2  9  ,i  + bz + b x  0  46. a  = KP(co) + KI(o))«T/2 + KD(io)/T  £  a  = -KP(OJ) + KI(co).T/2  x  a  Q  = -KD(a))/T  b  3  = -1  b  2  = (WTF /DSPD).(KP((o) + KI(u>)«T/2 + KD(u>)/T) + 2  b  x  b  n  2  = ( W T F / D S P D ) . (-KP(co) + KI(a)).T/2) - 1 2  = (WTF /DSPD)«(-KD(o))/T) 2  T h e r e f o r e , the c h a r a c t e r i s t i c  •  b z  _  »  + b z  3  3  « where,  b  g  = b /b , 3  use  b  3  2  equation  '  »  L  Q  + b z + b  2  2  t  characteristic 3-plane  e q u a t i o n f o r the system i s ,  =0  (4.16)  i  t  = b /b , 2  must  3  be  b  ]L  = bj/bg  transformed  and b  from  3  This  t h e z-plane  t o the  Q  = b /b .  using the b i l i n e a r transformation, z = ( g + l ) / ( $ - l ) ,  t h e Routh c r i t e r i o n  forstability.  The c h a r a c t e r i s t i c  Q  i n order t o equation i n  terms o f 3 i s then,  (B e  3  3  + B 3 2  2  + BjB + B ) / ( B - 1 )  = 0  3  0  (4.17)  where,  B  3  = b +b +b|+bQ = -(WTF /DSPD).KI(o)).T  B  2  = 3b +b -b^-3bQ = (-2 WTF /DSPD).KP(u))-(4 WTF /DSPD) »KD(u))/T  B  1  = 3b -b -b^+3bQ = 4 + (WTF /DSPD)«KI(u)) «T+(4 WTF /DSPD)  2  2  3  2  2  2  3  0  =  b  3~ 2 l"" 0 b  (4.17b)  2  2  • KD(aj)/T  B  (4.17a)  2  3  + b  b  (4.17c)  =  4  +  (  2  WTF /DSPD).KP(a>) 2  (4.17d)  According  t o t h e Routh  c r i t e r i o n the c o e f f i c i e n t s ,  must be p o s i t i v e  and t h e elements o f t h e f i r s t  must be p o s i t i v e  i n order  forstability  B  Q  through  B  3  column i n t h e Routh a r r a y  to p r e v a i l .  The Routh a r r a y i s w r i t t e n a s ,  0 0 <  2  B  B  r  B 3  V  /  B  2  0  B„  Recalling  that  negative,  0  the c o n t r o l l e r  the c o e f f i c i e n t s  B  Q  the magnitudes of the g a i n s .  gains  KP(OJ) ,KI(OJ)  through B  3  and  KD(to)  are  a l l  can be r e - w r i t t e n i n terms o f  Therefore,  B = (WTF /DSPD). |KI(o») | «T  (4.18a)  2  0  Bo = (2 WTF /DSPD)« |KP((o) | + (4 WTF /DSPD) • |KD(co) | /T  (4.18b)  B, = 4 - (WTF /DSPD). |KI(a)) | «T - (4 WTF /DSPD) • |KD(u)) | /T  (4.18c)  B„ = 4 - (2 W T F / D S P D ) « | K P ( w ) |  (4.18d)  2  2  2  2  2  The Routh element,  (B B -B B )/B , then becomes, 2  1  3  Q  2  [ 8 ( W T F / D S P D ) « |KP(u))| - 4(WTF /DSPD) • | Kl(u>) | • T 2  2  - 8 ( W T F V D S P D ) . |KP(OJ) I • |KD(OJ) |/T + 1 6 ( W T F / D S P D ) . |KD(a>) 1/T 2  2  - 4(WTF' /DSPD ). |KI(a))| • |KD(a))| - 16(WTF */DSPD ) • |KD(OJ) | / T ] / B T  2  1  2  2  2  2  (4.18e)  48.  With the conditions that B ,B ,B Q  1  and Bg and B ^ - B ^ g be greater than  2  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 for various  |KP(io)  | values.  |KI(uo)  The coefficients, B  3  | versus  and B  2  |KD(OJ)|  are always  greater than zero and therefore provide no constraint on the controller gains.  The coefficient, B , provides the constraint that, n  |KP(o»)|  and the coefficient, B  < 2 DSPD/WTF  (4.19)  2  provides the constraint that,  (DSPD/WTF )• T 2  ( T / 4 ) . |KI(o))| 2  >  JKD(CC) |  (4.20)  The expression (B„B -B B )/B„, provides the constraint that, n  (4.21)  where a =  16(WTF /DSPD) /T 2  2  2  b = - ( 1 6 ( W T F / D S P D ) / T - 8 ( W T F / D S P D ) » |KP(to) | /T 2  2  2  4(WTF /DSPD) .|KI(u)|) 2  2  c = - ( 8 ( W T F / D S P D ) . |KP(OJ) | - 4 ( W T F / D S P D ) • |KI(OJ) |-T) 2  2  These constraints are plotted in Figures 30, 3 1 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 i n Appendix B, i s 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, and  to  discrete  (B B ~B BQ)/B 2  1  3  different each  rad/sec, desired spectral density, DSPD =  = 1  T = 60 seconds.  timestep duration,  values.  |KP(to)|  The stable region lies below the curves at  by the straight line relationship between |KP(to) I  This  constraint  Bj;  B  2  proves less restrictive  Not plotted are the B , B 3  are always greater  DSPD/WTF = 2  If  0.586  KI(to)  = 0,  the  | and  |KD(to)|  |KD(to)|  than the  and B B  0  given  for a l l  Q  2  1  3  instability.  3  < 2  |KP(<o)|  shows that  and  if  B  g  is  |KP(to)|  |KD(to)|  gains becomes  As stable points  approach the  |KI(to)|  the response tends more towards instability  results presented  2  |KD(to)|.  The plot  points farther away from the constant  Q  relationships since B  axis i n the plot, the constraint,  more limited for a stable response.  eigenvalue  (B B -B B )/B  states that  Q  and  |KI(to)|  decreases, the range of values for the  The  2  than zero and  independent of  zero which indicates  curves,  |KI(<o)  >  values; points below this curve are again i n the stable region.  constraint. and  The constraint,  Also plotted on the Figure is the constraint  |KP(to)|.  2  i s given by the four concave-down traces, each for  > 0  2  m .sec  0.004  and conversely,  |KP(io)|  curves are more stable.  i n Section  4 . 3 . 2 agree with this  entirely which i s 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  duration while holding |KP(to)|, to and DSPD fixed.  timestep  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 i n waveboard motion  i s more dominant for the integral term than the derivative term and therefore,  i n the extreme, more prone  to causing  instability.  Similarly, for short T the derivative term i s more dominant than the integral term and more freedom in the range of |KD(to)| i s afforded. The variation i n the system stability resulting from a change i n 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 i n frequency directly changes the  'wave transfer function', WTF, relationship.  via the hyperbolic functions in the  As to increases, WTF increases.  maximum wave spectral density achievable ASPD(KT,to)  = WTF «STMAX. 2  max  Also, accordingly, the  changes with  frequency via  Recalling the controller algorithm and the  Routh constraints, the ratio, WTF/DSPD i s prominent. 2  Hence, a change  in frequency w i l l change WTF and therefore the response.  Similarly, i f  the desired spectral density i s changed the response w i l l change also. However, i f the ratio, WTF/DSPD remains unchanged after a change i n 2  frequency  and desired  spectral density at that frequency, then the  response will remain unchanged from the previous frequency.  The variation i n the Routh stability curves for a change i n desired spectral density, DSPD, i s shown i n Figure 32. As DSPD decreases, the range of stable gain values becomes more limited. DSPD the ratio,  WTF/DSPD, 2  increases.  For a decrease i n  Therefore,  an increase in  frequency, w, w i l l have the same effect on the range of stable gains that a decrease i n DSPD w i l l . Also note that instability w i l l prevail at a particular combination of frequency, |KP(oi)| and T i f the condition, DSPD < |KP(OJ) | •WTF/2 i s 2  satisfied (recall the Routh constraint, B ) . Q  Therefore, at a particular  frequency, there i s a maximum achievable wave spectral density dictated by the waveboard maximum stroke and there i s a minimum wave spectral density dictated by the Routh stability constraint for a particular |KP((O) I value.  However, i t i s of more interest and also more practical  to choose the gains knowing the desired spectral density and not viceversa.  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 m .sec and timestep duration T = 60 seconds. 2  Again, these values are  used to provide a comparison with the computer simulation performance and the Routh stability analysis. It i s 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 with increasing the  (|KI(O))|,  |KP(u>) I  and  |KP(d>)|  |KI((JJ)  on Figure 30, the eigenvalue becomes smaller as  |KD(w)|  |KD(OL))|) point moves farther  curve.  | values while traversing  Therefore,  the  system  away from the constant  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 i s not surprising, as mentioned earlier. Due  to the nature of the state equations three eigenvalues are  determined by the program, EIGVALS. complex conjugates  for every  Two  of the three eigenvalues are  combination  of gain values except for  strictly proportional control and for the combination, KI(o)) = -0.002,  KD(cd)  KP(<A>)  = -0.316,  = -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. |KI(to)|  values  duration.  the  eigenvalues  But at larger  |KI(<D)  increasing timestep duration.  decrease  with  increasing  At low timestep  | values, the eigenvalues increase with  The derivative portion of the controller  i s dominant for short timestep duration whereas the integral portion of the controller i s dominant for long timestep duration. Restated, as the timestep duration increases, the eigenvalues begin to decrease, but to a lesser  extent at larger  increases further s t i l l  |KI(u)| values.  As  the timestep duration  the eigenvalues then increase, especially at  larger |KI(a))| values. The  degree to which a system i s stable i s best determined by  examining the performance of the system and how i t relates to the eigenvalues .  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  i s necessary to examine how quickly a desired response i s achieved. The fact that a combination of controller gains provides a stable response outwardly provides no information as to whether the response i s oscillatory, 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) i s a ratio of polynomials i n z determined from the system difference equation.  The output, C(z), may also be written i n terms of  a partial fraction expansion,  C(z) = A /(z-\ ) l  l  + A /(z-X ) + 2  2  + 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 \ K-l 1  K-l  l  K-l + Z + A X n n  _ 1  +  (expansion of poles of R(z))  (5.2)  54. where X, ,X ...X are the poles or eigenvalues of the system characterJ- ^ n 0  i s t 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 i s real, positive and with magnitude  less than one, the response i s a monotonically decreasing sequence. If the eigenvalue i s real, negative and with magnitude less than one the response i s an oscillatory, decreasing sequence. real with a magnitude of one oscillatory  or  constant  If the eigenvalue i s  then the response  value,  i.e. a  case  of  remains at a fixed 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 i s written as follows: Ai K1 —  +  , , , K—1 . . 10 >K— 1 . . ,. — i0«K— 1 2 2 = A »(Ae ) + A « (Ae ) k  1  2  The real portion i s comprised of the magnitude, A of the eigenvalue, the coefficients  from  oscillatory term. A < 1.  How  determined  the partial fraction  expansion,  A^,A ,... 2  t  and  an  Thus, the response i s a decaying oscillatory one for  fast the response oscillates depends upon the value of 8  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 system  this  work,  transfer  S e c t i o n 4.2.  the  response  function  can  be  written  knowing  the  overall  i n the z-domain, A S P D ( z ) / D S P D ( z ) , as g i v e n i n  Now,  ASPD(z) = A /(z-X ) l  + A /(z-A ) + A /(z-A )  l  2  2  3  3  + expansion poles of D(z)  by  partial  fraction  expansion,  e i g e n v a l u e s and A^.Ag, The  transient  and  A  where  \^ ,  A  (5.3)  and  2  are the c o e f f i c i e n t s  3  A  are  G  from  the  system  the e x p a n s i o n .  response i n the time domain i s t h e r e f o r e ,  ASPD(KT,OJ) = A j X ^ "  This expression i s a s o l u t i o n  ASPD(KT,w)b  3  + A X  1  2  K - 1 2  + A ^ ^ "  (5.4)  1  of the d i f f e r e n c e e q u a t i o n ,  + ASPD((K-l)T,w)b  2  + ASPD( (K-2)T,w)b  ]L  + A S P D ( ( K - 3 ) T , o ) ) b = DSPD( ( K - l ) T , a>) • ( W T F / D S P D ) a 2  0  + DSPD((K-2)T,w)-(WTF /DSPD)a  + DSPD( (K-3)T,u>) •  2  (WTF /DSPD)a  2  1  (5.5)  2  where t h e c o e f f i c i e n t s , The Eq. (3.27) A  2  and  A  G  0  a ...a Q  2  and  i n Eq.  (5.4)  could  system.  be  f o r the  initially,  ASPD(-2T ,OJ) , ASPD(-T,a))  These  Q  3  a r e those from Eq.  i s a s i m p l i f i c a t i o n o f Eq. ( 5 . 5 ) .  conditions  zero.  b ...b  T h e r e f o r e , u s i n g Eq. coefficients  For  solved  f o r by  instance, and  The c o e f f i c i e n t s , employing  f o r no  ASPD(0,a))  (5.4) t h e c o e f f i c i e n t s  (4.15).  waves  the in  a r e assumed  A^  initial the  tank  equal to  c o u l d be determined.  w i l l be i n terms of the e i g e n v a l u e s of the system.  56.  It i s 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 ( K T , o o ) so the performance or response can be measured in terms of  E(KT,OJ).  This i s what the computer simulation, discussed i n  Section 3.2.1, determined.  5.2  Controller Gain Selection As an aid i n selecting the controller gains which yield a desired,  stable response i n 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 w i l l 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 m .sec, and a timestep 2  duration T = 60 seconds. The general trends w i l l 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 However as seen i n Figure 33, some combinations yield a  |KP(co)|.  concave-up trace. conjugate  In such cases, the i n i t i a l decrease i s due to complex  pair magnitude decreasing  until  magnitude becomes larger after a certain showing this are |KI(u)| = 0.002, |KD(OJ)|  = 7.2.  |KD(OJ)|  the third  KP(OJ)  real  gain value.  eigenvalue Examples  = 1.8 and |KI(u>)| = 0.002,  This could be interpreted that the response tends toward  57.  instability  in a manner described by the real eigenvalue for larger  proportional gains. In values.  general,  the  eigenvalues  increase with  increasing  |KI(oo)|  The eigenvalue at |KI(<o)| = 0 i s 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. combinations,  |KP(u)|  = 0.316,  |KD(io)|  = 1.8  and  The  |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  |KI((o) | = 0.002.  For these two  | K P ( O J ) | = 0.316,  cases the eigenvalues increase with  increasing |KD(w)|. For the cases i n which the complex conjugate eigenvalues 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 constraint curve. behaviour.  | KI (co) | line under the  The computer simulation results also agree with this  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 i s not employed in this  work i s that of controller gain optimization.  The  gains are  optimized based upon a performance index such as the sum of the squared  58.  error or the sum of the error since i t i s generally desirable to minimize the error. systems.  This approach has been adopted  i n continuous-time  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 w i l l yield a satisfactory response.  In practise i t i s essen-  t i a l that some integral gain i s 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 i s evident. When the Routh and eigenvalue analyses predict a stable or unstable response,  the computer simulation exhibits respectively  unstable response. |KD(U))|)  a stable or  As a stable eigenvalue becomes larger, a (|KI(cu)|,  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 i s 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  conjugates.  i s oscillatory when two of the eigenvalues are complex The response i s 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  59.  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. predict  that the response  should  The eigenvalues also  show some oscillation  due to the  complex conjugate pair eigenvalues for these particular gain combinations.  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 i n i t i a l l y , a KI(u) value of zero produces no response.  As mentioned i n  Section 3.3.2, i f the i n i t i a l conditions are such that waves do exist, the  error  response  does  show  some  oscillation  for the f i r s t  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 o s c i l l a t i o n should exist.  It i s likely that the coefficients, A n  discussed i n Section 5.1, are very small or zero for the complex eigenvalues, leaving the real eigenvalue to dominate the response. gain combinations  are mentioned i n Section 3.3.4.  The six  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 discretetime control theory.  The main difference from conventional discrete-  time control theory i n this work is the use of a long duration discrete timestep between successive controller actions.  This work has shown  that stability analysis i s possible and useful for determining control parameters. ted  From a performance standpoint, the type of response predic-  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 i s the same as conventional prevails  theory  in that  an oscillatory  response indeed usually  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. frequency  This work assumes that for each  i n the desired wave spectrum, there be a unique controller  gain combination frequencies.  as opposed  to using one gain combination  for  all  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 hardware 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  developed.  This simulation could  include the effects of  physical components in the system such as A/D and  and D/A  associated measurement errors, transmission  be  other  converters  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 i s done  62.  to simplify the performance and stability analysis for variations i n the system parameters. mative. is  To this end the simulation i s very useful and infor-  In reality a spectrum of frequencies comprise a sea state. It  therefore desirable to discuss a simulation which can incorporate  many frequencies along with the associated desired spectral densities. I n i t i a l l y , an operator enters the number of frequency components i n the spectrum to be modelled along with the actual frequencies.  The  'wave transfer function' i s 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 i s then calculated knowing the slowest wave component i n the spectrum. a l l frequencies.  As a reference the maximum controller gain values are  displayed to the operator experience  This timestep duration is used for  to aid i n the gain selection.  Also, past  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 i s calculated based upon the new spectral densities over the frequency range.  The iteration proceeds  automatically until satisfactory actual characteristics are achieved at all  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 i s satisfactory can be  63.  made automatically previous  based  upon  the condition  positive and/or negative  that  a l l immediately  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 i s achieved the associated waveboard motion i n terms of waveboard amplitudes at each frequency i s stored for future use along with the actual wave spectrum. The software w i l l automatically manipulate the data and task f i l e s to form an operator oriented system. An option i n the software for simulations and for in-tank experiments i s 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 i s attempted.  The animation  i s not i n real-time and depends  upon the speed of the graphics package used. screen layout are presented i n Appendix D.  The program and a typical 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  n  5  i 10  KIM= -0.0  i 15  KDM- -1.8.  i 20  i 25  DISCRETE TIME, K r 30  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 g o <jg -0.5-  , 5  , 10  r15  1 — 20  r 25  DISCRETE r ' 30 T , M E  K  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 DISCRETE - TIME,K 30 f  15  20  25  FIGURE 27 ERROR vs DISCRETE-TIME  K P M = -0.316  KI(co) = -0.004  15  K D M = -3.6  20  25  DISCRETE ^TIME.K 30  -1.0H>  00  FIGURE 29 ERROR vs DISCRETE-TIME 1.0  -I  K P M - -0.316  K I M - -0.008  KD(co) - -3.6  F I G U R E 3 0 . ROUTH STABILITY CONSTRAINTS. W(a>)[ vs K D M I  FOR VARIOUS K P M I  KDMI  IKIMI 0.0130.0120.011-  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  FIGURE 32. ROUTH STABILITY CONSTRAINTS, IKIMI vs IKDMI FOR VARIOUS DSPDM  16  17  18 KDMI  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., New York, 1978.  6.  Cadzow, J.A. and Martens, H.R., Discrete-Time and Computer Control Systems, Prentice-Hall, Englewood C l i f f s , N.J., 1970.  7.  Cherchas, D.B., Notes, 1982.  8.  Dean, R.G., "Relative V a l i d i t i e s 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 C l i f f s , 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 Controls, Harper and Row, N.Y. 2nd ed., 1969.  Dynamics of Marine Vehicles, John Wiley & Sons,  "Advanced Mechanical Systems Control", UBC Course  Automatic  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 C l i f f s , N.J., 1963.  20.  Kuo, B.C., Discrete-Data Control Systems, Prentice-Hall, Englewood C l i f f s , 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 17-Sep-1986 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 O040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 0055 0056 0057  22:37 22:37  PROGRAM CBGANES C. C C. S T E V E HODGE 3 1 7 1 2 7 7 1 C C . ..PROGRAM TO T E S T MANY CONTROLLER GAIN COMBINATIONS FOR A S P E C I F I C C. ..FREQUENCY, W, A D E S I R E D S P E C T R A L D E N S I T Y , DSPD, AND A T I M E S T E P C. ..DURATION, I , IN ORDER TO EXAMINE THE PERFORMANCE OF THE MODEL c. ..AND CONTROLLER. THE ERROR IS C A L C U L A T E D AS A FUNCTION OF D I S C R E T E c. ..TIME.  c c. c  ..DECLARE  A A k  c  THE  c c. c. c  c c. c. c. c c. c  IN T H I S  W R I T E ( 6 , A ) , 'ENTER READ<6,5), W WRITE<6,A), 'ENTER READ<6,5), DW STMAX=.063226/DU  AND THE SPD FREQUENCY BANDWIDTH. STROKE S P E C T R A L D E N S I T Y . THE  FREQUENCY,  FREQUENCY !! <  . . C A L C U L A T E THE TRANSFER FUNCTION, ..TO WATER WAVE A M P L I T U D E . D I S P L A Y ..DENSITY FOR T H I S FREQUENCY. ..NEWTON'S METHOD  30 35  c A A  PROGRAM  K,IEND,JEND,LEND,M/1/,MMAX,KAY(50 ) ,KEND(500) , S(500,3)/1500A0/  ..READ IN THE FREQUENCY . . C A L C U L A T E THE MAXIMUM  A  c  USED  REALAS Z < 5 0 0 , 3 ) , A S P D < 5 0 , 5 0 0 ) , Q , D K P , D K I , D K D , D U , E R R 2 , E R R 0 R ( 5 O , 5 0 0 ) , K P ( 2 O ) , K K 2 0 ) ,KD(20),MO,MM(21), PRODO,STMAX,STSPD(50,500),T,W,WTF,KPMAX,KIMAX, KDMAX,ASPDMX,SPUTF,B1,B2 , B3 INTEGER  k  VARIABLES  W='  BANDWIDTH  FOR  STROKE SPD  SPD,  FROM  DW='  MAX.  WAVEMAKER  WTF, R E L A T I N G WAVEMAKER STROKE WTF AND THE MAXIMUM WAVE S P E C T R A L  IS REQUIRED  MM<1)=.5 Q=<WAA2)A.249 DO 30 N = l , 2 0 MM(N+1)=MM<N) + <<Q-MM<N)ADTANH<MM<N)) ) /<DTANH(MM(N)) + (MM(N)/(DCOSH(MM<N))AA2) ) ) ) IF(MM<N+1).LE.O. ) THEN MM<N + 1)=MM(N) + . 1 GOTO 30 END IF ERR2=ABS(MM(N+1>-MM(N)) I F ( E R R 2 . L E . . 0 0 1 ) GOTO 35 CONTINUE M0=MM(N+1)/2.44 PR0D0=M0A2.44 WTF = 2.AMOADSINH<PRODO)A(<DS INH<PRODO)/MO)-(DCOSH<PRODO)/ ( 1 . 0 4 A C M O A A 2 ) ) ) + (DCOSH<MOA 1 .4)/(1.04A<M0AA2) ) ) ) / (PRODO + DSINH<PRODO)ADCOSH(PRODO) ) SPWTF=WTFAA2 WRIIE<6,15), WTF ASPDMX=STMAXA(SPUTF)  STROM  97.  CBGANES$MAIN  0053 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 ' 0081 0082 0033 0084 0035 0086 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0093 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 01 13 0114  17-Sep-l986 17-Sep-1986  U R I T E < 6 , 2 7 ) , ASPDMX C c . ..READ IN THE DSPD, I N I T I A L ASPD, I N I T I A L G A I N S , NUMBER OF GAIN c. ..MAXIMUM GAIN VALUES AND THE TIMESTEP DURATION.  c  5 10 15 20 25 27 c  22:37 22:37  INCREMENT  U R I T E < 6 , A > , 'ENTER THE DESIRED SPECTRAL DENSITY, DSPD=' R E A D ( 6 , 1 0 ) , DSPD WRITE<6,A>, 'ENTER THE I N I T I A L ACTUAL SPECTRAL DENSITY, ASPD=' R E A D ( 6 , 1 0 ) , ASPD(2,1) U R I I E ( 6 , A > , 'ENTER THE I N I T I A L CONTROLLER G A I N S , KP, K I , KD' READ(6,A), K P ( 1 ) , K I ( 1 ) , K D ( 1 ) U R I T E ( 6 , A ) , 'ENTER * OF GAIN INCREMENTS, IEND, JEND, LEND ' R E A D ( 6 , 2 5 > , IEND,3 END,LEND U R I T E ( 6 , A ) , 'ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX, KDMAX' R E A D < 6 , A ) , KPMAX,KIMAX,KDMAX U R I T E < 6 , A > , 'ENTER D I S C R E T E TIMESTEP DURATION, T=' READ(6,20), T FORMAT(F4.2 ) FORMAT(F5.4) FORMAT(' WAVE TRANSFER FUNCTION=',T25,E7.5) FORMAT(F5.2) FORMAT < 3 1 2 ) FORMAT(' MAXIMUM WAVE SPECTRAL DENS I I Y = ' , T 3 2 , F 6 . 4 ) MMAX=IENDkJENDkLEND DKP = KPMAX/<FLOAT( IEND) ) DK I=KIMAX/<FLOAT(JEND-1>> DKD=KDMAX/<FLOAT(LEND-l))  c c . ..COMMENCE THE LOOPING THROUGH THE CONTROL SYSTEM c c DO 300 1=1,IEND DO 200 J = 1 , J E N D DO 100 L = 1,LEND ASPD<1,M)=0. ASPD(2,M)=0. ASPD(3,M)=0. STSPD<2,M)=0. DO 80 K=3,33 KAY(K)=K-3 c c . ..CALCULATE C O E F F I C I E N T S FOR THE SYSTEM D I F F E R E N C E EQUATION c B1=KP(I)+KI<J)AT/2.+KD(L)/T B2=-KP<I)+KI<J)AT/2. B3=-KD<L)/T c c . . .SET UP THE D I F F E R E N C E EQUATION IN TERMS OF ASPD c ERROR<K,M> = <ASPD<K,M)/DSPD)-1 . c ASPDCK+l,M)=<BlAWTFAA2/DSPn+2.)AASPD<K,M) A +CB2AUTFAA2/DSPD-1.)AASPD(K-1,M) A +(B3AWTFAA2/DSPD)AASPD<K-2,M) -<B1+B2+B3)AWTFAA2 k c  98.  CBGANES*MAIN  0115 0116 0117 0118 0119 0120 0121 0122 0123 0124 0125 0126 0127 0128 0129 0130 0131 0132 0133 0134 0135 0136 0137 0138 0139 0140 0141 0142 0143 0144 0145 0146 0147 0148 0149 0150 0151 0152 0153 0154 0155 0156 0157 0158 0159 0160 0161 0162 0163 0164 0165 0166 0167 0168 0169 0170 0171  17-Sep-198& 17-Sep-1986  22:37 22:37  C. ..CALCULATE THE ERROR ASSOCIATED WITH THE ASPD C ERROR<K+l,M)=<ASPD<K+l,M)/nSPD)-l.  c c. .-CALCULATE THE STROKE SPECTRAL DENSITY c  STSPD(K+1,M>=ASPD(K+1,M)/<WTFA*2>  c c. ..CHECK IF WAVEMAKER STROKE SPD HAS BEEN FORCED -VE...IF SO THEN c. ..SET INDICATOR AND TRY ANOTHER GAIN. c IF(STSPD(K+1,M).LT.-0.000001) THEN S(M,2)=K-1 !! <0 INDICATOR KEND(M)=K+1 GOTO 81 END IF  c  c c. ..CHECK IF WAVEMAKER STROKE SPD HAS EXCEEDED PHYSICAL LIMITATIONS c. ..OF THE WAVEMAKER MOTION..' IF SO SET INDICATOR % SET STSPD TO STMAX c \-NOT IN THIS PROGRAM' IF(STSPD<K+1,M>.GT.STMAX) S(M,3)=K-1 STSPDOC + l , M ) =STMAX END IF  c c  c  80  THEN !! >STMAX  INDICATOR  CONTINUE K E N D < M > = K -1  c c  c. ..ASSIGN THE GAIN VALUES TO MEMORY c 81  Z(M,1)=KP(I) Z(M,2)=KI(J) Z(M,3)=KD(L) M = M+1  c c. ..COMBINATION M HAS NOW c c. ..TRY ANOTHER KD VALUE c 100  BEEN TESTED  KB(L+l>=KD(L)+DKD CONTINUE  c c. ..TRY ANOTHER KI VALUE c  200  KI(J+1)=KI(J)+DKI CONTINUE  300  KP( 1 + 1 )=KP( D+DKP CONTINUE  c c. ..TRY ANOTHER KP VALUE c  c c. ..PRINT THE RESULTS OF THE GAIN COMBINATIONS c 0PEN(UNIT=6,FILE='CB0UT.DAT;l',STATUS='OLD')  99 .  CBGANES$MAIN  0172 0173 0174 0175 0176 0177 0178 0179 0180 0181 0182 0183 0184 0185 0136 0187 0183 0189 0190 0191 0192 0193 0194 ' 0195  1 7 - S e p - 1 9 8 6 22:37 1 7 - S e p - 1 9 8 6 22:37 WRITE (6,500),W,DSPD,WTF,T,DW HO 700 M=1,MMAX WRITE(6,550),(M,Z(M,1),Z<H,2),Z(M,3>,S(M,2>,S(M,3))  C 500  F0RMAI(/,T2,'FREQUENCY=',T12,F5.2,T17,' DESIRED SPECTRAL DENSITY ? ',T45,F6.4,T51, ' WAVE TRANSFER FUNCTION=',T76,F7.4,/, T 2 , ' T I M E S T E P DURATION=',T20,F5.2,T27,'SPD BANDWIDTH*', T41,F5.2) 550 F 0 R M A T ( / , T 2 , ' C O M B I N A T I O N ' , T 1 3 , ' KP ',T20,' KI ',128, k ' KD ',T39,' <0', T 4 3 , ' >STMAX',/,T6,14,T15,F5.3 , A T22,F6.4,T29,F6.3,T41,I2,T46,12) 5  k k k  C WRITE(6,600) WRITE(6,650),(KAY<K>,STSPD(K,M),ASPD<K,M>,ERROR<K,M),K=3.33> C 600 k  F0RMAT(/,T9,' K',T13,'STROKE SPECTRAL T 4 6 , ' S P E C T R A L DENS I T Y ' , T 6 3 , '  DENSITY',T39,'ACTUAL', ERROR')  C 650  F 0 R M A K T 8 , 12 , T 1 6 , F12 . 6 , T42 , F 1 2 . 6 , T62 , F 1 2 . 6 )  C 700  CONTINUE STOP END  APPENDIX B EIGENVALUE AND ROUTH PROGRAM LISTINGS  101.  6-Sep-1986 6-Sep-1986 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 O035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 0055 0056 0057  10:57:1 10:48:3  C. PROGRAM E I G V A L S C C. S T E V E HODGE 31712771 C C. ..PROGRAM TO TEST MANY CONTROLLER GAIN COMBINATIONS FOR A S P E C I F I C C. . .FREQUENCY, U, A DESIRED SPECTRAL DENSITY, DSPD, AND A TIMESTEP C. ..DURATION, T, IN ORDER TO EXAMINE THE S T A B I L I T Y OF THE SYSTEM C. ..BY FINDING THE EIGENVALUES OF THE C H A R A C T E R I S T I C MATRIX IN C. ..DISCRETE S T A T E S P A C E C C. ..DECLARE THE VARIABLES USED IN T H I S PROGRAM C REAL A ( 4 , 4 ) , Z ( 5 0 0 , 3 ) , Q , D K P , D K I , D K D , D U . A ERR2,KP(20),KI(20),KD(20>,M0,MM(21), k PRODO,STMAX,T,M,UTF,KFMAX,KIMAX, k KDMAX,ASPDMX,SPUIF,W0RK(8),EMAG(500,4) C INTEGER K,IEND,JEND,LEND,M/1/,MMAX,KAY(50),KEND(500> , * S(500,3)/l500*0/,JOB,INFO,LDV,N,LDA,CONT C COMPLEX E ( 4 ) , V ( 4 , 4 ) , C M P L X , C O N J G , E I (500,4>  c c. c. c  ..READ IN THE FREQUENCY AND THE SPD FREQUENCY BANDWIDTH. ..CALCULATE THE MAXIMUM STROKE SPECTRAL D E N S I T Y . URITE<6,A>, 'ENTER READ(6,5), U WRITE(6,A>, 'ENTER R E A D ( 6 , 5 ) , DU STMAX=.063226/DW  THE  FREQUENCY,  FREQUENCY BANDWIDTH FOR !! < — - S T R O K E  c c. ..CALCULATE THE TRANSFER FUNCTION, c. ..TO WATER WAVE AMPLITUDE. DISPLAY c. ..DENSITY FOR T H I S FREQUENCY. c c. ..NEWTON'S METHOD IS REQUIRED. c  A  30 35  c A A  c  U='  SPD  SPD,  DW='  FROM MAX.  WAVEMAKER  STROKE  U T F , RELATING WAVEMAKER STROKE UTF AND THE MAXIMUM WAVE SPECTRAL  MM(1)=.5 Q=(UAA2>A.249 DO 30 N = l , 2 0 MM(N+1)=MM(N)+((Q-MM(N)ATANH(MM(N)))/(TANH(MM(N))+ (MM(N)/(COSH(MM(N))AA2)))> I F ( M M ( N + l ) . L E . O . ) THEN MM(N + 1) = MM(N) + . 1 GOTO 30 END IF ERR2=ABS(MM(N + 1)-MM(N> > I F ( E R R 2 . L E . . 0 0 1 ) GOIO 35 CONTINUE M0=MM(N+1)/2.44 PR0D0=M0A2.44 WTF=2.AM0ASINH<PRODO)A((SINH(PRODO)/MO)-(COSH(PRODO)/ ( 1 . 0 4 A ( M 0 A A 2 ) ) ) + (COSH(MOA1 . 4 > / ( 1 . 0 4 A ( M 0 A A 2 ) ) ) ) / (PRODO+SINH(PRODO)ACOSH(PRODO)) SPUTF=WTFAA2  102.  EIGVALS*MAIN  0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0060 0081 0082 0083 0084 0085 0066 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  6-Sep-1986 6-Sep-1986  10:57:1 10:48:3  W R I I E ( 6 , 1 5 > , WTF ASPDMX=STMAXA(SPUTF) W R I T E ( 6 , 2 7 ) , ASPDMX C C. ..READ IN I H E DSPD, I N I T I A L ASPD, I N I I I A L GAINS, NUMBER OF GAIN INCREMENT C. ..MAXIMUM GAIN VALUES AND THE T I M E S T E P DURATION. C U R I I E ( 6 , A > , 'ENTER THE DESIRED SPECTRAL DENSITY, DSPD=' R E A D ( 6 , 1 0 ) , DSPD U R I T E ( 6 , A > , 'ENTER THE I N I T I A L CONTROLLER GAINS, KP, K I , KD' READ(6, A >, K P ( 1 ) , K I ( 1 ) , K D ( 1 ) WRIIE(6,A>, 'ENTER * OF GAIN INCREMENTS, IEND, JEND, LEND ' READ(6,25), IEND,JEND,LEND U R I T E ( 6 , A ) , 'ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX, KDMAX' READ(6,A >, KPMAX,KIMAX,KDMAX U R I T E ( 6 , A ) , 'ENTER D I S C R E T E T I M E S T E P DURATION, T=' READ(6,20), T 5 FORMAT(F4.2) 10 F0RMAT(F5.4) 15 F O R M A T C WAVE TRANSFER FUNCTION=',T25,F7.5) 20 FORMAT(FS.2) 25 FORMAT(312) 27 FORMAT(' MAXIMUM WAVE SPECTRAL DENS I T Y = ' , T 3 2 , F 6 . 4 ) C MMAX=IENDAJENDALEND DKP=KPMAX/(FLOAI(IEND >) DKI=KIMAX/(FLOAT(JEND-1> > DKD=KDMAX/(FL0AT<LEND-1)) C C . ..COMMENCE THE LOOPING C DO 300 1=1,IEND DO 200 J=1,JEND DO 100 L=1,LEND C C. ..SET UP THE MATRIX VALUES C A(1 ,1>=2. + ( S P U T F / D S P D > A ( ( K P < I ) ) + ( K I ( J > A I / 2 . ) + ( K D < L ) / T ) ) A(1,2>=-l.-((SPWTF/DSPD)A(KP(I)-(KI(J)AT/2.))) A<1,3)=-(SPUTFAKD(L))/<DSPDAT) A<1,4)=0. A(2,1>=1. A(2,2>=0. A(2,3>=0. A<2,4)=0. A(3,l)=0. A<3,2)=1. A<3,3)=0. A(3,4>=0. A(4,1)=KP(I)+KI<J)AT/2.+KD(L)/T A(4,2)=-KP( D + K K J ) A T / 2 . A(4,3)=-KD(L)/T A<4,4)=0. C c. ..SET UP THE SUBROUTINE PARAMETERS REQ'D  c  LDA=4  103.  EIGVALS*MAIN 0115 0116 0117 0118 0119 0120 0121 0122 0123 0124 0125 0126 0127 0128 0129 0130 0131 0132 0133 0134 0135 0136 0137 0138 0139 0140 0141 0142 0143 0144 0145 0146 0147 0148 0149 0150 0151 0152 0153 0154 0155 0156 0157 0158 0159 0160 0161 0162 0163 0164 0165 0166 0167 0168 0169 0170 0171  6-Sep-1986 6-Sep-1986 LDV = 4 JOB=0 N=4  C C. ..DETERMINE THE EIGENVALUES USING THE SUBROUTINE, SGEEV C CALL SGEEV(A,LDA,N,E,V,LDV,WORK,JOB,INFO) C C . ..CALCULATE THE MAGNITUDE OF THE EIGENVALUES,E C EI(M,1)=E(1) EI(M,2)=E(2> EI(M,3)=E(3) EI< M,4)=E < 4) C EMAG(M,1)=CABS(EI(M,1)> EMAG(M,2)=CABS(EI(M,2) ) EMAG(M,3)=CABS(EI(M,3) ) EMAG(M,4)=CABS(EI(M,4)) C C . ..LOOK AT THE INFO VALUE RETURNED BY THE SUBROUTINE C URITE(6,A) ,INFO c READ<6,1000),CONT c clOOO FORMAT(11 ) C C C. ..ASSIGN THE GAIN VALUES TO MEMORY  c  Z(M,1)=KP <I) Z (M , 2)=KI(J) Z<M,3)=KD(L> M=M+1  90  c c. ..COMBINATION c c. . .TRY ANOTHER c  .TRY ANOTHER KI VALUE K K J + 1)=KI(J) + DKI CONTINUE  200  c c. . c  KD VALUE  KD(L+1)=KD(L)+DKD CONTINUE  100  c c. . c  M HAS NOW BEEN TESTED  .TRY ANOTHER KP VALUE  300  KP<I+1)=KP(I)+DKP CONTINUE  c c. ..PRINT c  THE RESULTS OF THE GAIN  COMBINATIONS  0PEN(UNIT=6,FILE='ZERR0UT.DAT;l',STATUS='OLD' ) write(6,500),U,DSPD,UTF,T,DU DO 700 M=1,MMAX WRITE(6,550),<M,Z<M,1),Z<M,2>,Z<M,3)>  10:57:1 10:48:3  104.  E IGVALS*MA IN 0172 0173 0174 0175 0176 0177 0178 0179 0160 0181 0182 0183 0184 0185 0186 0187 0188 0189 0190 0191 0192  6-Sep-1986 6-Sep-1986  10:57 10:48  C 500 F 0 R M A T ( / , T 2 , ' F R E Q U E N C Y = ' , T 1 2 , F 5 . 2 , T 1 7 ' DESIRED SPECTRAL DENSITY = A ',T45,F6.4,T51 , ' WAVE TRANSFER F U N C T I 0 N = ' , T 7 6 , F 7 . 4 , / , A T2,'TIMESTEP DURATI0N=',T20,F5.2,T27,'SPD BANDUIDTH=', A 141,F5.2) f  C 550 F0RMAT</,T2,'COMBINATION',T13, ' KP ' , T 2 0 , ' KI ' , T 2 8 , A ' KD ' , / , T 6 , 1 4 , T l 5 , F 5 . 3 , T 2 2 , F 6 . 4 , T 2 9 , F 6 . 2 ) C WRITE(6,600),<EI<M,l>,EI<M 2>,EI<M,3),EI<M,4)) WRITE(6,650),(EMAGCM,1),EMAG <M,2),EMAG< M , 3 ),EMAG < M , 4 > ) f  C 600  A  FORMAT<T3,'<',T4,2F7.2,T19,')',T21 '<',T22,2F7.2,T37, ' ) ' ,139,'(',I40,2F7.2,T55,')',T57,'(',T58,2F7.2,T73,') ') f  C 650  F0RMAI(T7,F7.2,T25,F7.2,T43,F7.2,T61,F7.2)  C 700  CONTINUE STOP END  105.  6-Sep-1986 11:27:2 6-Sep-1986 11:26:5 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  PROGRAM ROUTH C.. C C.. STEVE HODGE 31712771 C C . . .PROGRAM TO CALCULATE GAIN VALUES FROM THE STABILITY CONSTRAINTS C . . .DERIVED FROM THE ROUTH STABILITY CRITERION FOR DISCRETE TIME WITH c . . .THE INTENT OF GRAPHING THE RESULTANT GAIN CHARTS FOR STABILITY. c . . .PARTICULAR FREQUENCY, W, DESIRED SPECTRAL DENSITY, DSPD, AND c . . .TIMESTEP DURATION, T , VALUES ARE CHOSEN FOR EACH CHART.  c c . . .DECLARE THE VARIABLES USED IN THIS PROGRAM c REAL  A A A  c  AA,BB,CC,DIS,XX,DISCR,Q,DKP,DKI,DU, ERR2,KP(20),KD1(20,50),KD2(20,50),M0,MM(21>, K1(20,50),KDNU(20,100),KINU<20,100),PRODO,STMAX, KPMAX,KIMAX,ASPDMX,SPWTF,T,W,WTF,RAT,A,B,C,X  INTEGER K, IEND,JEND,JAX  c c . . .READ IN THE FREQUENCY AND THE SPD FREQUENCY BANDWIDTH. c . . .CALCULATE THE MAXIMUM STROKE SPECTRAL DENSITY. c  WRITE(6,A>, 'ENTER THE FREQUENCY, W=' READ(6,5>, W WRITE(6,A>, 'ENTER FREQUENCY BANDWIDTH FOR SPD, DW=' READ(6,5), DW STMAX=.063226/DW !! < STROKE SPD FROM MAX. WAVEMAKER STROKE  c c . . .CALCULATE THE TRANSFER FUNCTION, UTF, RELATING WAVEMAKER STROKE c . . .TO WATER WAVE AMPLITUDE. DISPLAY WTF AND THE MAXIMUM WAVE SPECTRAL c . . .DENSITY FOR THIS FREQUENCY. c  A  30 35 C A A  MM <1) = .5 Q=(WAA2)A.249 DO 30 N=l,20 MM(N+1>=MM(N)+((Q-MM(N)ATANH(MM(N)))/(TANH(MM(N))+ (MM(N)/(COSH(MM(N)>AA2> ) ) ) IF(MM(N+1).LE.O.) THEN MM(N+l)=MM(N)+.l GOTO 30 END IF ERR2=ABS(MM(N + 1)-MM(N> > IF(ERR2.LE..001> GOTO 35 CONTINUE H0=MM(N+1)/2.44 PR0D0=M0A2.44  WTF=2.AM0ASINH(PRODO)A((SINH(PRODO)/MO)-(COSH(PRODO)/ (1.04A(M0AA2)> > + (COSH(MOA1.4>/(1.04A(MOAA2)))>/ (PRODO+SINH(PROD0)AC0SH(PR0DO>> SPWTF=WTFAA2  C WRITE(6,15>, WTF ASPDMX=STMAXA(SPWTF) WRITE(6,27), ASPDMX C C . . .READ IN THE DSPD,  INITIAL ASPD,  INITIAL GAINS, NUMBER OF GAIN  INCREMENT  106.  ROUTHtMAIN 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  C• C  6-Sep-1986 11:27:2 6-Sep-1986 11:26:5 • •MAXIMUM  GAIN VALUES AND THE TIMESTEP  DURATION.  W R I T E ( 6 , A ) , 'ENTER THE DESIRED SPECTRAL DENSITY, DSPD=' R E A D ( 6 , 1 0 ) , DSPD U R I T E ( 6 , A > , 'ENTER THE INITIAL CONTROLLER GAINS, KP, K I ' READ(6 A), KP(1),KI(1,1> WRITE<6,A), 'ENTER * OF GAIN INCREMENTS, IEND, JEND' READ<6,25), IEND,JEND W R I T E ( 6 , A ) , ' ENTER MAXIMUM GAIN VALUES, KPMAX, KIMAX' R E A D ( 6 , A ) , KPMAX,KIMAX W R I T E ( 6 , A ) , 'ENTER DISCREIE TIMESTEP DURATION, T = ' READ<6,20), T FORMAT(F4.2 > FORMAT(F6.5 > FORMAT(' WAVE TRANSFER FUNCTION=',T25,F7.5) FORMAT(F5.2) FORMAT < 312) FORMAT(' MAXIMUM WAVE SPECTRAL D E N S I T Y = ' , T 3 2 , F 6 . 4 ) r  5 10 15 20 25 27 C  DKP=KPMAX/<FLOAT<IEND)) DK I=KIMAX/(FLOAT(JEND-1)> C C• C  • •COMMENCE  THE LOOPING  0PEN<UNIT=6,FILE='R0UD.DAT;l',STATUS='OLD') RAT=(SPWTF/DSPD) DO 300 1=1,IEND DO 200 J=1,JEND C C• C  > •CALCULATE  KD VALUES FROM THE CONSTRAINT:  BIAB2-B0AB3  A=16. A ( ( R A T / D A A 2 ) B = -(16.ARAT/T-8.A(RATAA2/T)AKP(I)-4.ARATAA2AKI<I,J)) C=-(8.ARATAKP<I)-4.ARATATAKI(I,J>) C X=-B/(2.AA) DISCR=BAB-4.AAAC IF(DISCR.GI.O.> THEN KDM I , J)=X+(SQRT(DISCR>)/<2.AA> KD2(I,J)=X-(SQRT(DISCR))/(2.AA> C • • •TRY ANOTHER KI VALUE KI< I , J + l)=KI<I,J)+DKI GOTO 200 ENDIF GOTO 205 200 CONTINUE C C • • •CALCULATE THE PEAK OF THE KI VS KD CURVE C AA=16.A < RATAA4) 205 BB=64.A<RATAA4)AKP(I)/T-384.A(RATAA3>/T CC=256.A<RAT/T)AA2+256.A(RATAA3)AKP<I)/(TAA2>+ A 64.A(RAIAA4)A<(KP(I)/T)AA2) XX=-BB/(2.AAA) DIS=BBABB-4.AAAACC C KI( I , J ) = X X - ( S Q R T ( D I S ) ) / ( 2 . A A A )  107.  ROUTHtMAIN 0115 0116 0117 0118 0119 0120 0121 0122 0123 0124 0125 0126 0127 0128 0129 0130 0131 0132 0133 0134 0135 0136 0137 0138 0139 0140 0141 0142 0143 0144 0145 0146 0147 0148 0149 0150 0151 0152 0153 0154 0155  6-Sep-1986 6-Sep-1986  C  *  KD1<I, J) = <16.ARAT/T-8.A<RATAA2/T>AKP(I)4.ARATAA2AKK1,3))/<32.A(RAT/T)AA2) KD2<I,J)=KD1<1,3)  C C . .REARRANGE THE KD AND KI VALUES FOR EASE OF GRAPHING C N = 0. JAX = J DO 33 K=1,JAX N=N+ 1 KDNU( I,N)=KD2< I,K). KINU(I,N)=KI<I,K> 33 CONTINUE DO 36 K=1,JAX KDNU<I,N)=KD1<I,JAX-K+1) KINU( I,N)=KI<I,JAX-K+l) N=N+1 36 CONTINUE C C . .STORE THE COMPUTED GAIN VALUES C WRITE(6,A),KP<I) WRITE(6,A),(KINUCI,J),J=1,N) WRITE(6,A),<KDNU(I,J>,J=1,N) C C . . . TRY ANOTHER KP VALUE C KP< I + l )=KP( D+DKP 300 CONTINUE C write*. 6,500),W,DSPD,WTF,T,DU C 5 00 FORMAT*./,T2, 'FREQUENCY=' , T 1 2 , F 5 . 2 , 1 1 7 . ' DESIRED ! A ',T45,F6.4,T51,' WAVE TRANSFER FUNCIION=' A T2,'TIMESTEP DURATION=',T20,F5.2,T27,'SPD ] A T41,F5.2) C STOP END  11:27 11:26  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 n  BANDWIDTH FOR SPD,  DW=  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 8 K 0 1 2 3 4 5 6 7 3 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  COMBINATION 10 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 24 25 26 27 23 29 30  KP -.079 SIROKE  KP -.079  KI -.0020  SPECTRAL 0.000000 0.120000 0.290780 0.421609 0.460384 0.410710 0.317151 0.234425 0.199097 0.216601 0.266078 0.316505 0.343795 0.340543 0.315553 0.285836 0.266568 0.264383 0.276243 0.293205 0.306017 0.309609 0.304477 0.295125 0.286983 0.283615 0.285480 0.290448 0.295430 0.2980&6 0.2976G6  KI -.0020  STROKE SPECTRAL 0 000000 0 120000 0 266204 0 332117 0 312564 0 272015 0 261144 0 279813 0 300267 0 304391 0 296022 0 288253 0 287880 0 292120 0 295249 0 294980 0 .293030 0 291867 0 .292190 0 293073 0 .293489 0 293269 0 292834 0 292743 0 292875 0 293038 0 .293077 0 293010 0 .292943 0 .292935 0 .292968  KD -1.800 DENSITY  KD -5.400 DENSITY  <0 0  >STMAX 14  ACTUAL SPECTRAL 0.000000 0.001638 0.003970 0.005756 0.006286 0.005607 0.004330 0.003201 0.002718 0.002957 0.003633 0.004321 0.004694 0.004649 0.004308 0.003903 0.003639 0.003610 0.003772 0.004003 0.004173 0.004227 0.004157 0.004029 0.003918 0.003872 0.003393 0.003965 0.004033 0.004069 0.004064  DENSITY  ERROR -1.000000 -0.590412 -0.007501 0.439050 0.571399 0.401849 0.082511 -0.199352 -0.320434 -0.260690 -0.091315 0.030304 0.173453 0.162354 0.077075 -0.024374 -0.090140 -0.097598 -0.057100 0.000776 0.044508 0.056769 0.039250 0.007329 -0.020461 -0.031954 -0.025591 -0.008634 0.008372 0.017369 0.016005  ACTUAL SPECTRAL DENSITY 0. 000000 0. 001638 0. 003634 0. 004534 0. 004267 0. 003714 0 .003565 0. 003820 0. 004100 0. 004156 0. 004042 0. 003926 0. 003930 0. 003988 0 .004031 0. 004027 0. 004001 0. 003985 0. 003939 0. 004001 0. 004007 0. 004004 0. 003999 0. 003997 0. 003999 0. 004001 0 .004001 0. 004000 0 004000 0 003999 0 004000  ERROR -1 .000000 -0. 590412 -0 091382 0 133594 0. 066854 -0 071549 -0 103653 -0 044933 0 024882 0 038958 0 010393 -0 016125 -0 017398 -0 002925 0 007752 0 006836 0 000181 -0 003791 -0 002688 0 000326 0 001746 0 000997 -0 000318 -0 000783 -0 000351 0 000207 0 000340 0 000113 -0 000117 -0 000144 -0 000031  <0 0  >STMAX 4  Ill  COMBINATION 12 K 0 1 2 3 4 C *J  6  7  8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 23 29 30  COMBINATION 14 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 23 29 30  KP -.079  KI KD -.0020 -9.000  STROKE SPECTRAL DENSITY 0.000000 0.120000 0.241629 0.252691 0.211601 0.221880 0.276388 0.300986 0.280207 0.264051 0.279701 0.300033 0.297734 0.284404 0.283451 0.293775 0.298305 0.292754 0.288177 0.291049 0.295397 0.294776 0.291550 0.291088 0.293317 0.294358 0.293092 0.291976 0.292562 0.293552 0.293435  KP -.079  KI KD -.0040 -1.800  STROKE SPECIRAL DENSITY 0.000000 0.240000 0.532409 0.615084 0.407060 0.089306 -0.052235 0.116237 0.450309 0.650791 0.531479 0.191785 -0.065552 -0.004068 0.329407 0.639265 0.642939 0.328342 -0.027413 -0.101855 0.180773 0.573550 0.722296 0.484719 0.065921 -0.158275 0.021650 0.453552 0.751707 0.641051 0.210576  <0 0  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.000000 0.001638 0.003299 0.003450 0.002889 0.003029 0.003774 0.004109 0.003826 0.003605 0.003819 0.004096 0.004065 0.003883 0.003870 0.004011 0.004073 0.003997 0.003934 0.003974 0.004033 0.004025 0.003981 0.003974 0.004005 0.004019 0.004002 0.003936 0.003994 0.004008 0.004006  <0 32  ERROR -1.000000 -0.590412 -0.175264 -0.137505 -0.277756 -0.242672 -0.056623 0.027335 -0.043587 -0.098731 -0.045316 0.024081 0.016237 -0.029264 -0.032515 0.002722 0.018183 -0.000763 -0.016384 -0.006583 0.008258 0.006138 -0.004871 -0.006450 0.001161 0.004714 0.000390 -0.003418 -0.001416 0.001961 0.001562  >STMAX 30  ACTUAL SPECTRAL DENSITY 0.000000 0.003277 0.007269 0.008398 0.005558 0.001219 -0.000713 0.001587 0.006148 0.008885 0.007256 0.002618 -0.000395 -0.000056 0.004497 0.008728 0.008778 0.004483 -0.000374 -0.001391 0.002468 0.007831 0.009861 0.006618 0.000900 -0.002161 0.000296 0.006192 0.010263 0.008752 0.002875  ERROR -1.000000 -0.180824 0.817235 1 .099426 0.389391 -0.695179 -1.178290 -0.603256 0.537010 1 .221300 0.314063 -0.345395 -1 .223745 -1.013887 0.124344 1.181961 1.194502 0.120709 -1.093566 -1.347655 -0.382930 0.957661 1.465364 0.654460 -0.774995 -1.540228 -0.926103 0.548079 1.565752 1. 188056 -0.281255  COMBINATION 15 K 0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  COMBINATION 38 K 0 1 **i  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 28 29 30  KP -.079  KI KD -.0040 -3.600  STROKE SPECTRAL DENSITY 0 . 000000 0.240000 0.507833 0.533142 0.305221 0.076869 0.087033 0.312008 0.504722 0.465409 0.247927 0.089504 0.152575 0.359014 0.435053 0.403297 0.210660 0.114691 0.210445 0.337251 0.455755 0.350263 0.190657 0.146824 0.258245 0.399869 0.421927 0.307918 0.134546 0.181348 0.295067  KP -.158  KI KD 0.0000-1.800  STROKE SPECTRAL DENSITY 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000  <0 0  >STMAX 32  ACTUAL SPECTRAL DENSITY 0.000000 0.003277 0.006933 0.007279 0.004167 0.001049 0.001188 0.004260 0.006891 0.006354 0.003385 0.001222 0.002083 0.004902 0.006622 0.005506 0.002376 0.001566 0.002873 0.005287 0.006222 0.004782 0.002603 0.002005 0.003526 0.005459 0.005761 0.004204 0.002520 0.002476 0.004029  ERROR -1.000000 -0.180824 0.733354 0.819738 0.041792 -0.737629 -0.702937 0.064956 0.722734 0.588548 -0.153769 -0.694502 -0.479226 0.225398 0.655598 0.376546 -0.2S0969 -0.608532 -0.281702 0.321779 0.555597 0.195531 -0.349244 -0.498857 -0. 118549 0.364847 0.440136 0.050997 -0.370101 -0.331018 0.007131  <0 >STMAX 0 0 ACTUAL SPECTRAL DENSITY 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000  ERROR -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.oooooo -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.000000 -1.oooooo -1.000000 -1.000000 -1.000000 -1.oooooo -1.000000 -1.oooooo -1.000000 -1.000000 -1.000000 -1.oooooo -1.000000 -1.000000 -1.000000 -1.000000  113.  COMBINATION 43 K 0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 19 20 21 2 *^ 23 24 25 26 27 28 29 30  COMBINATION 44 K 0 1 o 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 28 29 30  KP -.158  KI -.0020  KD 0.000  STROKE SPECTRAL DENSITY 0.000000 0.120000 0.270710 0.380128 0.417250 0.391055 0.333450 0.278537 0.247907 0.245982 0.263951 0.287797 0.305789 0.312515 0.308989 0.300084 0.291247 0.286075 0.285460 0.288130 0.291892 0.294840 0.296040 0.295583 0.294212 0.292794 0.291925 0.291778 0.292171 0.292763 0.293245  KP -.158  KI KD -.0020-1.800  STROKE SPECTRAL DENSITY 0.000000 0.120000 0.258422 0.338234 0.350466 0.325635 0.297023 0.281798 0.280733 0.286707 0.292749 0.295633 0.295550 0.294155 0.292895 0.292362 0.292443 0.292762 0.293021 0.293117 0.293087 0.293015 0.292963 0.292946 0.292955 0.292971 0.292982 0.292984 0.292982 0.292978 0.292976  <0 0  5STMAX 7  ACTUAL SPECTRAL DENSITY 0.000000 0.001638 0.003696 0.005190 0.005697 0.005339 0.004553 0.003803 0.003385 0.003358 0.003604 0.003929 0.004175 0.004267 0.004219 0.004097 0.003976 0.003906 0.003897 0.003934 0.003985 0.004025 0.004042 0.004036 0.004017 0.003997 0.003986 0.003984 0.003989 0.003997 0.004004  <0 0  ERROR -1 OOOOOO -0. 590412 -0 076004 0 297466 0 424173 0. 334761 0 138143 -0. 049289 -0. 153837 -0. 160404 -0 099075 -0 017680 0 043730 0 066687 0 054651 0 024257 -0 005906 -0 023561 -0 025660 -0 016547 -0 003705 0 006359 0 010452 0 008895 0 004215 -0 000626 -0 003591 -0 004094 -0 002751 -0 000731 0 000913  >STMAX 6  ACTUAL SPECTRAL DENSITY 0.000000 0.001638 0.003528 0.004618 0.004785 0.004446 0.004055 0.003847 0.003333 0.003914 0.003997 0.004036 0.004035 0.004016 0.003999 0.003992 0.003993 0.003997 0.004001 0.004002 0.004001 0.004001 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000  ERROR -1.OOOOOO -0.590412 -0.117945 0.154473 0.196223 0.111468 0.013810 -0.038159 -0.041794 -0.021401 -0.000780 0.009063 0.008782 0.004019 -0.000231 -0.002099 -0.001823 -0.000735 0.000151 0.000476 0.000374 0.000130 -0.000050 -0.000106 -0.000076 -0.000022 0.000014 0.000023 0.000015 0.000003 -0.000004  114.  COMBINATION 46 K 0 1 3 4 5 6 7 3 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  COMBINATION 48 K 0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 28 29 30  KP -.158  KI KD -.0020 -5.400  STROKE SPECTRAL DENSITY 0.000000 0.120000 0.233847 0.261996 0.249799 0.254467 0.275660 0.288912 0.288816 0.236414 0.288272 0.291602 0.292788 0.292263 0.292007 0.292471 0.292925 0.292966 0.292846 0.292844 0.292935 0.292985 0.292972 0.292954 0.292961 0.292976 0.292980 0.292975 0.292974 0.292976 0.292978  KP -.153  KI KD -.0020-9.000  STROKE SPECTRAL DENSITY 0.000000 0.120000" 0 . 209272 0.195823 0.187846 0.236567 0.271236 0.260519 0.254418 0.274762 0.288469 0.282004 0.278488 0.287193 0.292699 0.239201 0.287392 0.291193 0.293433 0.291645 0.290769 0.292455 0.293376 0.292491 0.292083 0.292839 0.293221 0.292792 0.292607 0.292950 0.293109  <0 0  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.000000 0.001638 0.003193 0.003577 0.003410 0.003474 0.003764 0.003945 0.003943 0.003910 0.003936 0.003981 0.003997 0.003990 0.003987 0.003993 0.003999 0.004000 0.003998 0.003998 0.003999 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000  <0 0  ERROR -1 .000000 -0. 590412 -0. 201826 -0. 105747 -0 .147378 -0. 131445 -0. 059109 -0. 013874 -0. 014204 -0. 022400 -0. 016061 -0. 004693 -0. 000645 -0. 002423 -0.,003311 -0. 001727 -0. 000179 -0. 000038 -0.,000447 -0. 000454 -0. 000145 0. 000027 -0. 000017 -0. 000079 -0. 000055 -0. 000005 0. 000008 -0. 000006 -0.,000013 -0. 000005 0.,000001  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.000000 0.001638 0.002857 0.002674 0.002565 0.003230 0.003703 0.003557 0.003474 0.003751 0.003933 0.003850 0.003802 0.003921 0.003996 0.003948 0.003924 0.003976 0.004006 0.003982 0.003970 0.003993 0.004005 0.003993 0.003983 0.003998 0.004003 0.003997 0.003995 0.004000 0.004002  ERROR -1.000000 -0.590412 -0.285707 -0.331610 -0.358338 -0.192540 -0.074208 -0.110789 -0..131611 -0.062173 -0.015387 -0.037454 -0.049454 -0.019743 -0.000951 -0.012888 -0.019063 -0.006091 0.001556 -0.004548 -0.007538 -0.001784 0.001361 -0.001660 -0.003054 -0.000470 0.000832 -0.000631 -0.001264 -0.000094 0.000448  115.  COMBINATION 50 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 8 19 20 21 22 23 24 25 26 27 28 29 30  COMBINATION 79 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  KP -.158  KI KD -.0040 -1.800  STROKE SPECTRAL DENSITY 0.000000 0.240000 0.467694 0.474841 0.308036 0.166879 0.178861 0.295998 0.382246 0.363354 0.282365 0.230801 0.250431 0.305638 0.335650 0.318087 0.280958 0.264091 0.278604 0.303250 0.312274 0.300373 0.234726 0.230253 0.288396 0.299334 0.301257 0.294883 0.238230 0.287663 0.292262  KP -.237  KI -.0020  KD 0.000  STROKE SPECTRAL DENSITY 0.000000 0 .1200000.238352 0.307577 0.329000 0.322726 0.308058 0.296074 0.290062 0.288876 0.290086 0.291750 0.292911 0.293398 0.293418 0.293246 0.293068 0.292960 0.292925 0.292932 0.292954 0.292972 0.292931 0.292983 0.292982 0.292979 0.292977 0.292977 0.292977 0.292977 0.292977  <0 0  >STMAX 16  ACTUAL SPECTRAL DENSITY 0.000000 0.003277 0.006385 0.006483 0.004206 0.002278 0.002442 0.004041 0.005219 0.004961 0.003855 0.003151 0.003419 0.004173 0.004583 0.004343 0.003836 0.003606 0.003804 0.004140 0.004263 0.004108 0.003387 0.003826 0.003944 0.004087 0.004113 0.004026 0.003935 0.003927 0.003990  <0 0  ERROR -1.OOOOOO -0.180824 0.596348 0.620743 0.051398 -0.430404 -0.389504 0.010310 0.304694 0.240213 -0.036222 -0.212222 -0.145219 0.043215 0.145652 0.085704 -0.041026 -0.098596 -0.049059 0.035063 0.065365 0.026949 -0.023164 -0.043430 -0.013932 0.021695 0.023261 0.006504 -0.016204 -0.018139 -0.002441  >STMAX 6  ACTUAL SPECTRAL DENSITY 0.OOOOOO 0.001638 0.003254 0.004199 0.004492 0.004406 0.004206 0.004042 0.003960 0.003944 0.003961 0.003983 0.003999 0.004006 0.004006 0.004004 0.004001 0.004000 0.003999 0.003999 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000  ERROR -1.000000 -0.590412 -0.186448 0.049831 0.122954 0.101540 0.051473 0.010571 -0.009950 -0.013998 -0.009867 -0.004190 -0.000227 0.001435 0.001505 0.000916 0.000308 -0.000059 -0.000130 -0.000154 -0.000081 -0.000019 0.000014 0.000021 0.000015 0.000007 0.000001 -0.000002 -0.000002 -0.000001 -0.000001  116.  COMBINATION 80 K 0 1  KP -.237 STROKE  *?  3 4 5  & 7 !3  9 10 11 12 13 14 15 16 17 18 19 20 21 2 2 23 24 25 26 27 28 29 30  COMBINATION 84 K 0 1 3 4 5  & 7 8 9 10 11 12 13 1 4 15 16 17 18 19 20 21 9 2 23 24 25 2 6 27 29 29 30  KP -.237 STROKE  KI -.0020 SPECTRAL 0.000000 0.120000 0.226065 0.272309 0.283485 0.285917 0.288378 0.290735 0.292093 0.292612 0.292775 0.292853 0.292910 0.292946 0.292964 0.292971 0.292974 0.292975 0.292976 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977  KB -1.800 DENSITY  ACTUAL  >STMAX 0 SPECTRAL 0.000000 0.001638 0.003086 0.003718 0.003870 0.003904 0.003937 0.003969 0.003988 0.003995 0.003997 0.003998  DENSITY  ERROR -1.000000 -0.590412 -0.228388 -0.070544 -0.032399 -0.024098 -0.015697 -0.007653 -0.003018 -0.001245 -0.000690 -0.000426 -0.000231 -0.000106 -0.000046 -0.000022 -0.000012 -0.000007 -0.000003 -0.000002 -0.000001 0.000000 0.000000 0.oooooo 0.000000 0.000000 0.oooooo 0.000000 0.oooooo 0.000000 0.oooooo  DENSITY  ERROR -1.OOOOOO -0.590412 -0.396151 -0.466154 -0.366553 -0.192143 -0.184692 -0.199205 -0.119742 -0.072495 -0.088972 -0.074807 -0.037376 -0.033666 -0.039472 -0.024530 -0.013245 -0.016781 -0.015274 -0.007383 -0.006047 -0.007765 -0.005069 -0.002426 -0.003120 -0.003115 -0.001484 -0.001068 -0.001514 -0.001055 -0.000448  .  0.003999 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000  KI KD -.0020-9.000 SPECTRAL 0.000000 0.120000 0.176914 0.156405 0.185586 0.236684 0.238867 0.234615 0.257896 0.271738 0.266911 0.271061 0.282027 0.283114 0.281413 0.285791 0.239097 0.288061 0.238502 0.290814 0.291206 0.290702 0.291492 0.292267 0.292063 0.292065 0.292542 0.292664 0.292534 0.292668 0.292346  <0 0  DENSITY  <0 0 ACTUAL  >STMAX 0 SPECTRAL 0.000000 0.001633 0.002415 0.002135 0.002534 0.003231 0.003261 0.003203 0.003521 0.003710 0.003644 0.003701 0.003850 0.003865 0.003842 0.003902 0.003947 0.003933 0.003939 0.003970 0.003976 0.003969 0.003930 0.003990 0.003988 0.003988 0.003994 0.003996 0.003994 0.003996 0.003998  117.  COMBINATION 86 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 29 30  COMBINATION 93 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  KP -.237  KI KD -.0040 -1.900  STROKE SPECTRAL DENSITY 0.000000 0.240000 0.402979 0.369498 0.273443 0.245014 0.279975 0.310957 0.308085 0.291106 0.284473 0.289873 0.295786 0.295878 0.292942 0.291499 0.292291 0.293396 0.293522 0.293026 0.292726 0.292833 0.293036 0.293078 0.292996 0.292936 0.292943 0.292984 0.292995 0.292982 0.292971  KP -.237  KI KD -.0080 -1.800  STROKE SPECTRAL DENSITY 0.000000 0.480000 0.609355 0.159301 -0.043515 0.369738 0.639913 0.274471 -0.054638 0.253519 0.631624 0.388507 -0.027433 0.144799 0.586517 0.488971 0.034077 0.055264 0.510536 0.565245 0.122223 -0.005817 0.412337 0.609643 0.226634 -0.032541 0.304702 0.618196 0.335444 -0.022926 0.193244  <0 0  >STMAX 4  ACTUAL SPECTRAL DENSITY 0.000000 0.003277 0.005502 0.005045 0.003733 0.003345 0.003822 , 0.004245 0.004206 0.003974 0.003334 0.003958 0.004033 0.004040 0.004000 0.003980 0.003991 0.004006 0.004007 0.004001 0.003997 0.003998 0.004001 0.004001 0.004000 0.003999 0.004000 0.004000 0.004000 0.004000 0.004000  <0 30  ERROR -1.000000 -0.180824 0.375461 0.261183 -0.066673 -0.163710 -0.044380 0.061368 0.051567 -0.006387 -0.029027 -0.010595 0.009586 0.009902 -0.000119 -0.005047 -0.002342 0.001429 0.001860 0.000165 -0.000859 -0.000492 0.000199 0.000342 0.000063 -0.000143 -0.000100 0.000025 0.000062 0.000017 -0.000023  >STMAX 32  ACTUAL SPECTRAL DENSITY 0.000000 0.006553 0.008319 0.002175 -0.000594 0.005048 0.008737 0.003747 -0.000746 0.003461 0.008624 0.005304 -0.000375 0.001977 0.008008 0.006676 0.000465 0.000755 0.006970 0.007717 0.001669 -0.000079 0.005636 0.008323 0.003094 -0.000444 0.004160 0.008440 0.004530 -0.000313 0.002707  ERROR -1.000000 0.638352 1.079873 -0.456270 -1.148525 0.262001 1.184172 -0.063166 -1.186492 -0.134680 1.155879 0.326064 -1.093634 -0.505768 1.001921 0.668971 -0.883688 -0.811373 0.742580 0.929312 -0.582825 -1.019856 0.409110 1.030871 -0.226445 -1.111072 0.040018 1.110046 0.144948 -1.078253 -0.323346  118.  COMBINATION 115 K 0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  COMBINATION 117 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 28 29 30  KP -.316  KI -.0020  KB 0.000  STROKE SPECTRAL BENS IT Y 0.000000 0.120000 0.205995 0.252475 0.274931 0.285157 0.289651 0.291580 0.292396 0.292737 0.292879 0.292937 0.292961 0.292971 0.292975 0.292976 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977 0.292977  KP -.316  KI KB -.0020-3.600  STROKE SPECTRAL DENSITY 0.000000 0.120000 0.181420 0.197711 . 0.222372 0.246273 0.258411 0.266823 0.274383 0.279685 0.233164 0.285825 0.287833 0.289237 0.290248 0.290999 0.291543 0.291934 0.292219 0.292427 0.292578 0.292687 0.292766 0.292824 0.292S66 0.292897 0.292919 0.292935 0.292946 0.292955 0.292961  <0 0  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.000000 0.001633 0.002312 0.003447 0.003754 0.003893 0.003955 0.003981 0.003992 0.003997 0.003999 0.003999 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000  <0 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 0 0 0 0  ERROR 000000 590412 296891 138242 061596 026694 011355 004768 001934 000820 000337 000133 000056 000023 000009 000004 000002 000001 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.000000 0.001638 0.002477 0.002699 0.003043 0.003362 0.003523 0.003643 0.003746 0.003819 0.003866 0.003902 0.003930 0.003949 0.003963 0.003973 0.003980 0.003986 0.003990 0.003992 0.003995 0.003996 0.003997 0.003998 0.003998 0.003999 0.003999 0.003999 0.004000 0.004000 0.004000  ERROR -1.000000 -0.590412 -0.380772 -0.325168 -0.239235 -0.159413 -0.117984 -0.089272 -0.063448 -0.045370 -0.033496 -0.024412 -0.017559 -0.012766 -0.009317 -0.006754 -0.004895 -0.003561 -0.002588 -0.001878 -0.001364 -0.000991 -0.000720 -0.000523 -0.000380 -0.000276 -0.000200 -0.000145 -0.000106 -0.000077 -0.000056  119  COMBINATION 123 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 23 29 30  COMBINATION 129 K 0 1 •-t  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  KP -.316  KI KD -.0040 -3.600  STROKE SPECTRAL DENSITY 0.000000 0.240000 0.313689 0.256872 0.264187 0.300330 0.297371 0.285996 0.290885 0.295546 0.293029 0.291715 0.293099 0.293443 0.292822 0.292800 0.293070 0.293032 0.292927 0.292963 0.293001 0.292979 0.292967 . 0.292978 0.292981 0.292976 0.292976 0.292978 0.292978 0.292977 0.292977  KP -.316  KI KD -.0060-3.600  STROKE SPECTRAL DENSITY 0.000000 0.360000 0.396807 0.207682 0.242350 0.354766 0.308953 0.251136 0.292794 0.318649 0.237132 0.278537 0.299896 0.300193 0.287062 0.289921 0.297313 0.293347 0.290117 0.293107 0.294701 0.292498 0.292031 0.293490 0.293439 0.292556 0.292792 0.293279 0.293021 0.292782 0.292996  <0 0  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.000000 0.003277 0.004283 0.003507 0.003607 0.004100 0.004060 0.003905 0.003971 0.004035 0.004001 0.003983 0.004002 0.004006 0.003998 0.003998 0.004001 0.004001 0.003999 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000 0.004000  <0 0  ERROR -1.000000 -0.180824 0.070693 -0^123235 -0.098269 0.025097 0.014996 -0.023828 -0.007140 0.008766 0.000175 -0.004310 0.000414 0.001590 -0.000532 -0.000605 0.000316 0.000188 -0.000171 -0.000050 0.000080 0.000006 -0.000035 0.000004 0.000014 -0.000004 -0.000005 0.000003 0.000002 -0.000001 0.OOOOOO  >STMAX 10  ACTUAL SPECTRAL DENSITY 0.000000 0.004915 0.005418 0.002835 0.003309 0.004844 0.004218 0.003429 0.003997 0.004350 0.003920 0.003804 0.004094 0.004099 0.003919 0.003953 0.004059 0.004012 0.003961 0.004002 0.004024 0.003993 0.003997 0.004007 0.004006 0.003994 0.003997 0.004004 0.004001 0.003997 0.004000  ERROR -1 .OOOOOO 0.228764 0.354396 -0.291132 -0.172802 0.210899 0.054528 -0.142813 -0.000627 0.087622 -0.019953 -0.049116 0.023613 0.024645 -0.020191 -0.010433 0.014799 0.002968 -0.009762 0.000442 0.005884 -0.001635 -0.003229 0.001751 0.001574 -0.001433 -0.000632 0.001028 0.000149 -0.000665 0.000063  120.  COMBINATION 135 K 0 1 2 3 4 5 6 7 3 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  COMBINATION 40 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 23 29 30  KP -.316  KI KD -.0080 -3.600  STROKE SPECTRAL DENSITY 0.000000 0.480000 0.430775 0.080338 0.251033 0.482978 0.261009 0.146952 0.370539 0.386622 0.196027 0.249451 0.388420 0.295570 0.213114 0.318794 0.350007 0.251654 0.260266 0.338615 0.304060 0.251342 0.298386 0.325527 0.277107 0.271606 0.313588 0.303441 0.272263 0.291573 0.310605  KP -.158  KI KD 0.0000 -5.400  STROKE SPECTRAL DENSITY 0.000000 0.084489 0.063715 0.057072 0.062433 0.065297 0.064090 0.063025 0.063232 0.063591 0.063582 0.063471 0.063456 0.063488 0.063498 0.063489 0.063435 0.063487 0.063489 0.063488 0.063488 0.063488 0.063488 0.063438 0.063483 0.063488 0.063483 0.063488 0.063488 0.063483 0.063483  <0 0  >STMAX 24  ACTUAL SPECTRAL DENSITY 0.000000 0.006553 0.005881 0.001097 0.003427 0.006594 0.003564 0.002006 0.005059 0.005279 0.002676 0.003406 0.005303 0.004035 0.002910 0.004352 0.004779 0.003436 0.003553 0.004623 0.004151 0.003432 0.004074 0.004444 0.003783 0.003703 0.004231 0.004143 0.003717 0.003981 0.004241  <0 0  ERROR -1 .000000 0.638352 0.470336 -0.725788 -0.143167 0.648518 -0. 109115 -0.498419 0.264737 0.319630 -0.330914 -0.148564 0.325767 0.008843 -0.272594 0.088117 0.194657 -0.141047 -0.111651 0.155773 0.037828 -0.142110 0.018460 0.111099 -0.054168 -0.072944 0.070349 0.035714 -0.070701 -0.004792 0.060167  >STMAX 0  ACTUAL SPECTRAL DENSITY 0.001000 0.001154 0.000370 0.000779 0.000852 0.000891 0.000875 0.000860 0.000863 0.000868 0.000863 0.000367 0.000866 0.000867 0.000867 0.000367 0.000367 0.000867 0.000867 0.000867 0.000867 0.000867 0.000867 0.000367 0.000867 0.000867 0.000367 0.000867 0.000867 0.000867 0.000367  ERROR -0. 750000 -0 711620 -0 782526 -0 305201 -0 786901 -0 777126 -0 781247 -0 784882 -0 734175 -0 782949 -0 782978 -0 783359 -0 783409 -0 783299 -0 733267 -0 783296 -0 783310 -0 783304 -0 733293 -0 733300 -0 793301 -0 783301 -0 783301 -0 733301 -0 783301 -0 783301 -0 783301 -0 733301 -0 733301 -0 783301 -0 783301  121.  The following pages show the output from the eigenvalue program, EIGVALS.  122.  FREQUENCY= 1.00 DESIRED SPECTRAL DENS ITY= 0.0040 TIMESTEP DURATION=60.0O SPD BANDWIDTH= 0.20  WAVE TRANSFER FUNCTION' 0.1168  COMBINATION KP KI KD 1 -.079 0.0000 0.00 < 0.00 0.00 ) < 0.00 0.00 ) < 0.000 0.000  1.00 0.00 ) ( 1.000  COMBINATION KP KI KD 2 -.079 0.0000 -1.80 ( 0.00 0.00 ) ( 1.00 0.00 ) < 0.000 1.000  0.31 0.06 ) < 0.320  0.73 0.00 ) 0.730  0.31 -0.06 ) 0.320  COMBINATION KP KI KD 3 -.079 0.0000 -3.60 < 0.00 0.00 ) < 0.26 0.37 > < 0.000 0.453  0.26 -0.37 > ( 0.453  1.00 0.00 ) 1.000  COMBINATION KP KI KD 4 -.079 0.0000 -5.40 < 0.00 0.00 ) ( 0.21 0.51 > < 0.000 0.554  0.21 -0.51 ) ( 0.554  1.00 0.00 ) 1.000  COMBINATION KP KI KD . 5 -.079 0.0000 -7.20 ( 0.00 0.00 > < 0.16 0.62 ) < 0.000 0.640  0.16 -0.62 ) < 0.640  1.00 0.00 > 1.000  COMBINATION KP KI KD 6 -.079 0.0000 -9.00 ( 0.00 0.00 ) < .0.11 0.71 ) ( 0.000 0.716  0.11 -0.71 ) < 0.716  1.00 0.00 ) 1.000  COMBINATION KP KI KD 7 -.079 -.0020 0.00 < 0.00 0.00 ) ( 0.00 0.00 ) < 0.000 0.000  0.76 ' 0.59 > ( 0.967  0.76 -0.59 ) 0.967  COMBINATION KP KI KD 8 -.079 -.0020 -1.80 < 0.00 0.00 ) < 0.64 0.59 ) < 0.000 0.873  0.64 -0.59 ) < 0.373  0.13 0.00 ) 0.134  COMBINATION KP KI KD 9 -.079 -.0020 -3.60 ( 0.00 0.00 ) ( 0.49 0.60 > ( 0.000 0.775  0.49 -0.60 ) ( 0.775  0.34 0.00 ) 0.341  COMBINATION KP KI KD 10 -.079 -.0020 -5.40 ( 0.00 0.00 ) ( 0.34 0.67 ) ( 0.000 0.754  0.34 -0.67 ) ( 0.754  0.54 0.00 ) 0.540  COMBINATION KP KI KD 11 -.079 -.0020 -7.20 < 0.00 0.00 ) ( 0.23 0.76 > ( 0.000 0.795  0.23 -0.76 ) ( 0.795  0.65 0.00 0.648  >  123.  COMBINATION  (  COMBINATION  <  KI  KP  KI  KP  KI  KP  KI  KP  KI  KP  KI  KP  KI  KP  KI  0.66 -0.84 ) 1.068  0.56 -0.85 ) < 1.014  0.10 0.00 ) 0.100  0.45 -0.86 ) < 0.972  0.22 0.00 ) 0.217  0.34 -0.89 ) ( 0.955  0.34 0.00 ) 0.337  0.24 -0.94 ) ( 0.966  0.44 0.00 ) 0.439  0.15 -0.98 ) < 0.995  0.52 0.00 > 0.518  0.56 1.02 ) < 1.160  0.56 - 1 . 0 2 ) 1.160  0.47 -1.03 > ( 1.127  0.08 0.00 ) 0.081  0.37 -1.04 ) < 1.105  0.17 0.00 ) 0.168  0.28 -1.06 > < 1.097  0.26 0.00 ) 0.255  0.19 -1.09 ) ( 1.105  0.34 0.00 ) 0.336  KD  KD  KD  KD  KD  KD  KD  22 -.079 -.0060 -5.40 0.00 0.00 ) < 0.28 1.06 ) ( 0.000 1.097  COMBINATION  <  KP  0.66 0.84 ) < 1.068  KD  21 -.079 -.0060 -3.60 0.00 0.00 ) ( 0.37 1.04 ) ( 0.000 1.105  COMBINATION  <  KI  20 -.079 -.0060 -1.80 0.00 0.00 ) ( 0.47 1.03 ) < 0.000 1.127  COMBINATION  <  KP  0.71 0.00 ) 0.711  KD  19 -.079 -.0060 0.00 0.00 0.00 ) ( 0.00 0.00 > < 0.000 0.000  COMBINATION  <  KI  18 -.079 -.0040 -9.00 0.00 0.00 > ( 0.15 0.98 ) ( 0.000 0.995  COMBINATION  (  KP  0.15 -0.83 > ( 0.849  KD  17 -.079 -.0040 -7.20 0.00 0.00 > ( 0.24 0.94 > < 0.000 0.966  COMBINATION  <  KI  16 -.079 -.0040 -5.40 0.00 0.00 ) < 0.34 0.89 ) ( 0.000 0.955  COMBINATION  (  KP  15 -.079 -.0040 -3.60 0.00 0.00 ) ( 0.45 0.86 ) ( 0.000 0.972  COMBINATION  (  KB  14 -.079 -.0040 -1.80 0.00 0.00 ) ( 0.56 0.85 ) ( 0.000 1.014  COMBINATION  <  KI  13 -.079 -.0040 0.00 0.00 0.00 ) ( 0.00 0.00 ) ( 0.000 0.000  COMBINATION  <  KP  12 -.079 -.0020 -9.00 0.00 0.00 ) ( 0.15 0.83 ) ( 0.000 0.849  KD  23 -.079 -.0060 -7.20 0.00 0.00 ) ( 0.19 1.09 > ( 0.000 1.105  124 .  COMBINATION KP KI KB 24 -.079 -.0060 -9.00 ( 0.00 0.00 > ( 0.10 1.12 0.000 1.124 COMBINATION KP KI KD 25 -.079 -.0080 0.00 ( 0.00 0.00 ) < 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 26 -.079 -.0080 -1.80 ( 0.00 0.00 > < 0.37 1.17 0.000 1.224 COMBINATION KP KI KD 27 - . 0 7 9 • .0080 -3.60 ( 0.00 0.00 ) ( 0.28 1.18 0.000 1.213 COMBINATION KP KI KD 28 -.079 • .0080 -5.40 ( 0.00 0.00 ) ( 0.20 1.19 0.000 1.211 COMBINATION KP KI KD . 29 -.079 -.0080 -7.20 < 0.00 0.00 ) < 0.11 1.21 0.000 1 .220 KD COMBINATION KP KI 9.00 30 -.079 -.0080 0.03 1.24 ( 0.00 0.00 ) ( 1 .236 0 . 000 COMBINATION KP KI KD 31 -.079 -.0100 0.00 ( 0.00 0.00 ) ( 0.00 0.00 0.000 0.000 KI KD COMBINATION KP -.0100 -1.80 32 -.079 ( 0.00 0.00 ) < 0.27 1.28 1 .31: 0 . 000 COMBINATION KP KI KD 33 -.079 -.0100 3.60 < 0.00 0.00 ) ( 0.19 1.29 0.000 1 .307 COMBINATION KP KI KD 34 -.079 .0100 -5.40 < 0.00 0.00 ) < 0.11 1.30 0 .000 1.310 COMBINATION KP KI KD 35 - . 0 7 9 • .0100 -7.20 < 0.00 0.00 ) ( 0.03 1.32 0.000 1 .319  > (  0.10 -1.12 1.124  ) (  0.41 0.00 0.405  )  ) (  0.46 1.16 1.245  ) (  0.46 -1.16 1.245  )  ) <  0.37 -1.17 1.224  )  (  0.07 0.00 0.068  )  ) (  0.28 -1.18 1.213  )  (  0.14 0.00 0.139  )  ) (  0.20 -1.19 1.211  > (  0.21 0.00 0.209  )  ) (  0.11 -1.21 1.220  )  <  0.28 0.00 0.275  )  ) (  0.03 -1.24 1.236  ) <  0.34 0.00 0.335  )  ) (  0.35 1.23 1.325  ) (  0.35 -1.28 1.325  )  ) (  0.27 -1.28 1.312  )  <  0.06 0.00 ' 0.059  )  ) <  0.19 -1.29 1.307  )  (  0.12 0.00 0.120  )  ) <  0.11 -1.30 1.310  )  (  0.18 0.00 0.179  )  > <  0.03 -1.32 1.319  )  (  0.24 0.00 0.235  )  125.  COMBINATION KP KI KD 36 -.079 -.0100 -9.00 < 0.00 0.00 > ( - 0 . 0 5 " 1 . 3 3 " ) ( 0.000 1.335  -0.05 -1.33 ) ( 1.335  0.29 0.00 ) 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 ) < 1.000  0.46 0.00 ) 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 ) < 0.320  1.00 0.00 ) 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 > ( 0.453  1.00 0.00 ) 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 > ( 0.554  1.00 0.00 ) 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 > < 0.640  1.00 0.00 ) 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 > ( 0.716  '1.00 0.00 ) 1.000  KD COMBINATION KP KI 0.00 43 -.158 -.0020 0.00 0.00 ) < 0.00 0.00 ) < 0 .000 0.000  0.63 0.52 ) ( 0.316  0.63 -0.52 ) 0.316  COMBINATION KP KI KD 44 -.153 • .0020 -1.80 ( 0.00 0.00 ) ( 0.46 0.49 ) ( O.000 0.676  0.46 -0.49 ) ( 0.676  0.22 0.00 ) 0.224  COMBINATION KP KI KD 45 -.158 • -.0020 -3.60 < 0.00 0.00 ) ( 0.26 0.57 > < 0.000 0.624  0.26 -0.57 ) ( 0.624  0.53 0.00 ) 0.526  COMBINATION KP KI KD 46 -.159 • .0020 -5.40 < 0.00 0.00 ) < 0.15 0.67 > ( 0 .000 0.637  0.15 -0.67 > ( 0.687  0.65 0.00 ) 0.651  COMBINATION KP KI KD 47 -.158 • .0020 -7.20 < 0.00 0.00 ) ( 0.07 0.75 ) < 0 .000 0.756  0.07 -0.75 ) ( 0.756  0.72 0.00 ) 0.716  126.  COMBINATION KP KI KD 48 -.158 -.0020 -9.00 ( 0.00 0.00 > ( - 0 . 0 1 0.82 ) ( 0.000 0.822  -0.01 -0.82 > ( 0.322  0.76 0.00 ) 0.758  COMBINATION KP KI KD 49 -.153 -.0040 0.00 ( 0.00 0.00 ) ( 0.00 0.00 ) ( 0.000 0.000  0.53 0.77 0.933  ) (  0.53 -0.77 ) 0.933  COMBINATION KP KI KD 50 -.158 -.0040 -1.80 ( 0.00 0.00 ) ( 0.41 0.77 0.000 0.372  > <  0.41 -0.77 ) ( 0.872  0.13 0.00 ) 0.135  > <  0.28 -0.79 > ( 0.842  0.29 0.00 ) 0.289  ) <  0.16 -0.84 ) < 0.857  0.42 0.00 > 0.413  ) (  0.07 -0.89 ) < 0.896  0.51 0.00 ) 0.510  ) <  -0.02 -0.94 ) < 0.944  0.57 0.00 ) 0.575  ) <  0.42 0.95 ) ( 1.037  0.42 -0.95 ) 1.037  COMBINATION KP KI KD 56 -.158 -.0060 -1.80 ( 0.00 0.00 ) < 0.32 0.95 > ( 0.000 1.005  0.32 -0.95 ) ( 1.005  0.10 O.OO 0.101  COMBINATION KP KI KD 57 -.158 -.0060 -3.60 ( 0.00 0.00 ) ( 0.22 0.97 0.000 0.992  ) (  0.22 -0.97 ) < 0.992  0.21 0.00 ) 0.208  ) <  0.12 -1.00 ) < 1.002  0.31 0.00 ) 0.306  0.02 -1.03 ) ( 1.028  0.39 0.00 > 0.388  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 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 . 0 2 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  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  )  127.  KI KD COMBINATION KP -.0060 -9.00 60 -.158 1.06 ) ( ( 0.00 0.00 ) ( -0.06 1 .063 0.000  -0.06 -1.06 ) ( 1.063  0.45 0.00 ) 0.453  KI KD COMBINATION KP -.0030 0.00 61 -.159 0.00 0.00 ) < ( 0.00 0.00 ) ( 0.000 0.000  0.32 1.08 > < 1.131  0.32 -1.08 ) 1.131  COMBINATION KP KI KD 62 -.158 • -.0080 -1.80 ( 0.00 0.00 ) < 0.23 1.09 ) < 0 . 000 1.115  0.23 -1.09 ) ( 1.115  0.08 0.00 ) 0.082  COMBINATION KP KI KD 63 -.153 • -.0080 -3.60 ( 0.00 0.00 > < 0.14 1.10 > < 0 .000 1.111  0.14 -1.10 > ( 1.111  0.17 0.00 ) 0.166  COMBINATION KP KI KD 64 -.153 • .0080 -5.40 0.05 1.12 ) ( ( 0.00 0.00 ) ( 1 . 122 0 .000  0.05 -1.12 ) < 1.122  0.24 0.00 ) 0.244  COMBINATION KP KI KD .65 -.158 • .0080 -7.20 ( 0.00 0.00 ) ( -0.04 1.14 ) ( 0.000 1.143  -0.04 -1.14 ) ( 1.143  0.31 0.00 ) 0.314  COMBINATION KP KI KD 66 -.153 • .0080 -9.00 ( 0.00 0.00 > < -0.12 1.16 ) < 0 . 000 1.171  -0.12 -1.16 ) ( 1.171  0.37 0.00 ) 0.373  COMBINATION KP KI KD 0.00 67 -.158 • .0100 0.00 0.00 ) < < 0.00 0.00 ) ( 0.000 0.000  0.22 1.20 ) < 1.218  0.22 -1.20 ) 1.218  COMBINATION KP KI KD 63 -.153 • -.0100 -1.80 < 0.00 0.00 ) ( 0.13 1.20 ) < 0.000 1.211  0.13 -1.20 ) ( 1.211  0.07 0.00 ) 0.070  COMBINATION KP KI KD 69 -.158 • .0100 -3.60 ( 0.00 0.00 ) < 0.05 1.21 . ( 0.000 1.213  0.05 -1.21 ) < 1.213  0.14 0.00 ) 0.139  KI KD COMBINATION KP 70 -.153 • .0100 -5.40 1.22 > ( ( 0.00 0.00 ) < -0.04 1 . 225 0 .000  -0.04 -1.22 ) ( 1.225  0.20 0.00 ) 0.205  KI KD COMBINATION KP 71 -.158 • .0100 -7.20 -0.12 1.24 ) ( < 0.00 0.00 ) ( 1.244 0 . 000  -0.12 -1.24 ) ( 1.244  0.26 0.00 ) 0.265  128.  KI KD COMBINATION KP -9.00 72 -.158 • .0100 -0.20 1.25 ( 0.00 0.00 ) ( 1 .269 0.000 KP KI KD COMBINATION .237 0.0000 0.00 73 0.00 0.00 ( 0.00 0.00 ) ( 0.000 0. 000 KD COMBINATION KP KI 1 .80 74 -.237 0.0000 0.04 0.32 ( 0.00 0.00 ) ( 0.320 0.000 COMBINATION KP KI KD 75 .237 0.0000 -3.60 < 0.00 0.00 ) < 0.01 0.45 0.000 0.453  ) (  -0.20 -1.25 1.269  ) (  0.32 0.00 0.318  )  ) <  1.00 0.00 1.000  ) (  0.19 0.00 0.191  )  ) (  0.04 -0.32 0.320  ) <  1.00 0.00 1.000  )  -0.01 -0.45 ) ( 0.453  1.00 0.00 1.000  )  -0.06 -0.55 0.554  ) (  1.00 0.00 1.000  )  ) (  -0.11 -0.63 0.640  > (  1.00 0.00 1.000  )  ) (  -0.16 -0.70 0.716  ) (  1.00 0.00 1.000  )  > (  0.49 0.39 0.629  ) (  0.49 -0.39 > 0.629  ) (  0.19 -0.41 0.452  ) <  0.50 0.00 0.502  )  ) (  0.06 -0.56 0.559  > (  0.65 0.00 0.655  )  ) (  -0.02 -0.65 0.653  ) (  0.72 0.00 0.721  )  > (  -0.09 -0.73 0.733  > (  0.76 0.00 0.762  )  ) (  KD COMBINATION KP KI -5.40 76 -.237 0.0000 ( 0.00 0.00 ) ( -0.06 0 .000 0.554 COMBINATION KP KI KD 77 -.237 0.0000 -7.20 < 0.00 0.00 ) ( -0.11 0.63 0 .000 0.640 KD COMBINATION KP KI 73 -.237 0.0000 -9.00 ( 0.00 0.00 ) < -0.16 0.70 0.716 0.000 COMBINATION KP KI KD 79 -.237 -.0020 0.00 ( 0.00 0.00 ) ( 0.00 0.00 0 . 000 0.000 COMBINATION KP KI KD 80 -.237 -.0020 -1.30 ( 0.00 0.00 ) ( 0.19 0.41 0.000 0.452 KI KD COMBINATION KP -3.60 31 -.237 • .0020 ( 0.00 0.00 ) ( 0.06 0.56 0.000 0.559 COMBINATION KP 82 -.237 < 0.00 0.00 ) ( 0 . 000  KI KD .0020 -5.40 -0.02 0.65 0.653  KI KD COMBINATION KP -7.20 83 - . 2 3 7 • .0020 -0.09 0.73 ( 0.00 0.00 ) ( 0. 7 3 3 0. 000  129.  COMBINATION KP KI KD . 84 -.237 -.0020 -9.00 ( 0.00 0.00 ) < -0.16 0.79 0.000 0.804 COMBINATION KP KI KD 85 -.237 -.0040 0.00 ( 0.00 0.00 ) ( 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 86 -.237 -.0040 -1.80 ( 0.00 0.00 ) < 0.24 0.67 0.000 0.710 COMBINATION KP KI KD 87 -.237 -.0040 -3.60 ( 0.00 0.00 ) ( 0.09 0.72 0.000 0.726 COMBINATION KP KI KD 88 -.237 -.0040 -5.40 ( 0.00 0.00 > ( -0.01 0.78 0.000 0.783 COMBINATION KP KI KD . 89 -.237 -.0040 -7.20 ( 0.00 0.00 > ( -0.10 0.84 0.000 0.846 COMBINATION KP KI KD 90 -.237 -.0040 -9.00 < 0.00 0.00 > ( -0.18 0.89 0.000 0.906 COMBINATION KP KI KD 91 -.237 -.0060 0.00 ( 0.00 0.00 ) ( 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 92 -.237 -.0060 -1.80 ( 0.00 0.00 > < 0.17 0;85 0.000 0.872 COMBINATION KP KI KD 93 -.237 -.0060 -3.60 ( 0.00 0.00 ) ( 0.05 0.88 0.000 0.881 COMBINATION KP KI KD 94 -.237 -.0060 -5.40 < 0.00 0.00 ) ( -0.05 0.92 0.000 0.917 COMBINATION KP KI KD 95 -.237 -.0060 -7.20 ( 0.00 0.00 ) ( -0.14 0.95 0.000 0.963  ) (  -0.16 -0.79 0.804  ) (  0.79 0.00 0.791  ) (  0.39 0.67 0.775  > <  0.39 -0.67 ) 0.775  ) (  0.24 -0.67 0.710  ) <  0.20 0.00 0.203  )  ) (  0.09 -0.72 0.726  ) (  0.39 0.00 0.388  )  ) (  -0.01 -0.78 0.783  ) <  0.50 0.00 > 0.500  > <  -0.10 -0.84 0.846  ) (  0.57 0.00 0.572  ) (  -0.18 -0.89 0.906  ) <  )  )  0.62 0.00 > 0.623  ) (  0.29 0.85 0.897  ) (  0.29 -0.85 ) 0.897  > <  0.17 -0.85 0.872  ) (  0.13 0.00 0.135  )  ) (  0.05 -0.88 0.881  > <  0.26 0.00 0.264  )  -0.05 -0.92 0.917  ) (  ) (  ) (  -0.14 -0.95 0.963  ) <  0.37 0.00 0.365  0.44 0.00 0.442  )  )  130.  COMBINATION KP KI KD 96 -.237 -.0060 -9.00 < 0.00 0.00 ) ( -0.22 0.99 0.000 1.012 COMBINATION KP KI KD 97 -.237 -.0080 0.00 < 0.00 0.00 ) ( 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 98 -.237 -.0080 -1.80 ( 0.00 0.00 > ( 0.08 0.99 0.000 0.997 COMBINATION KP KI KD 99 -.237 -.0080 -3.60 ( 0.00 0.00 ) ( -0.02 1.01 0.000 1.009 COMBINATION KP KI KD 100 -.237 -.0080 -5.40 ( 0.00 0.00 ) < -0.11 1.03 0.000 1.036 COMBINATION KP KI KD .101 -.237 -.0080 -7.20 < 0.00 0.00 ) < -0.20 1.05 0.000 1.073 COMBINATION KP KI KD 102 -.237 -.0080 -9.00 ( 0.00 0.00 ) ( -0.28 1.03 0.000 1.113 COMBINATION KP KI KD 103 -.237 -.0100 0.00 < 0.00 0.00 ) ( 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 104 -.237 -.0100 -1.80 ( 0.00 0.00 ) ( -0.01 1.10 0.000 1.103 COMBINATION KP KI KD 105 -.237 -.0100 -3.60 ( 0.00 0.00 ) < -0.10 1.11 0.000 1.117 COMBINATION KP KI KD 106 -.237 -.0100 -5.40 < 0.00 0.00 > ( -0.19 1.13 0.000 1.142 COMBINATION KP KI KD 107 -.237 -.0100 -7.20 ( 0.00 0.00 ) ( -0.27 1.14 0.000 1.173  ) (  -0.22 -0.99 1.012  > <  0.50 0.00 0.500  )  ) <  0.19 0.99 1.005  ) (  0.19 -0.99 ) 1.005  > (  0.08 -0.99 0.997  ) (  0.10 0.00 0.103  )  ) (  -0.02 -1.01 1.009  ) (  0.20 0.00 0.201  )  ) <  -0.11 -1.03 1.036  > (  0.29 0.00 > 0.286  ) <  ) (  ) (  -0.20 -1.05 1.073  -0.28 -1.08 1.113  0.08 1.10 1.102  ) <  ) <  ) <  0.36 0.00 0.356  0.41 0.00 0.413  )  )  0.08 -1.10 1.102  )  ) (  -0.01 -1.10 1.103  ) <  0.08 0.00 0.034  )  ) (  -0.10 -1.11 1.117  ) <  0.16 0.00 0.164  )  ) <  ) (  -0.19 -1.13 1.142  -0.27 -1.14 1.173  ) (  ) (  0.24 0.00 0.236  0.30 0.00 0.293  )  )  131.  COMBINATION (  KP  108 -.237 0.00 0.00 ) 0.000  KI  KD  -.0100 -9.00 ( -0.35 1.16 1.208  COMBINATION KP KI KD 109 -.316 0.0000 0.00 < 0.00 0.00 > ( 0.00 0.00 0.000 0.000 COMBINATION KP KI KB 110 -.316 0.0000 -1.80 < 0.00 0.00 > ( -0.09 0.31 0.000 0.320 COMBINATION KP 111 -.316 ( 0.00 0.00 ) 0.000  KI KD 0.0000 -3.60 < -0.14 0.43 0.453  COMBINATION KP 112 -.316 < 0.00 0.00 ) 0.000  KI KD 0.0000 -5.40 ( -0.19 0.52 0.554  COMBINATION KP .113 -.316 ( 0.00 0.00 ) 0.000  KI KD 0.0000 -7.20 < -0.24 0.59 0.640  COMBINATION KP 114 -.316 < 0.00 0.00 ) 0.000  KI KD 0.0000 -9.00 < -0.30 0.65 0.716  COMBINATION KP 115 -.316 < 0.00 0.00 ) 0.000  KI KD -.0020 0.00 ( 0.00 0.00 0.000  COMBINATION KP 116 -.316 < 0.00 0.00 ) 0.000  KI KD -.0020 -1.30 ( -0.02 0.39 0.394  COMBINATION KP 117 -.316 ( 0.00 0.00 ) 0.000  KI KD -.0020 -3.60 ( -0.11 0.52 0.531  COMBINATION KP 118 -.316 ( 0.00 0.00 ) 0.000  KI KD -.0020 -5.40 ( -0.18 0.61 0.633  COMBINATION KP 119 -.316 ( 0.00 0.00 ) 0.000  KI KD -.0020 -7.20 ( -0.24 0.67 0.718  )  (  -0.35 -1.16 1.208  )  (  1.00 0.00 1.000  .  <  )  > (  0.35 0.00 0.351  )  (  -0.08 0.00 0.079  )  -0.09 -0.31 0.320  > (  1.00 0.00 1.000  )  <  -0.14 -0.43 0.453  )  (  1.00 0.00 1.000  )  )  (  -0.19 -0.52 0.554  )  (  1.00 0.00 1.000  )  )  <  -0.24 -0.59 0.640  )  <  1.00 0.00 1.000  )  )  (  -0.30 -0.65 0.716  > (  1.00 0.00 1.000  )  )  (  0.40 0.00 0.405  > (  0.31 0.00 > 0.312  )  (  -0.02 -0.39 0.394  )  (  0.66 0.00 0.659  )  )  (  -0.11 -0.52 0.531  )  (  0.73 0.00 0.726  )  )  (  -0.18 -0.61 0.633  )  (  0.77 0.00 0.767  )  )  (  -0.24 -0.67 0.718  )  (  0.80 0.00 > 0.796  )  132.  COMBINATION KP KI KD 120 -.316 • .0020 -9.00 ( 0.00 0.00 ) ( -0.31 0.73 0.000 0.792 COMBINATION KP KI KD 121 -.316 • .0040 0.00 ( 0.00 0.00 > ( 0.00 0.00 0.000 0.000 COMBINATION KP KI 122 -.316 • .0040 ( 0.00 0.00 ) ( 0.04 0 .000  KD -1.80 0.55  COMBINATION KP KI KD 123 -.316 • .0040 -3.60 -0.09 0.64 < 0.00 0.00 > ( 0.648 0.000 COMBINATION KP KI KD 124 -.316 -.0040 -5.40 < 0.00 0.00 ) ( -0.18 0.71 0 .000 0.734 COMBINATION KP KI KD 125 -.316 • -.0040 -7.20 -0.26 0.77 ( 0.00 0.00 ) < 0.810 0.000 COMBINATION KP KI KD 126 -.316 -.0040 -9.00 ( 0.00 0.00 > ( -0.33 0.31 0 . 000 0.878 COMBINATION KP KI KD 0.00 12 7 -.316 • .0060 < 0.00 0.00 ) < 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 128 -.316 • .0060 -1.80 ( 0.00 0.00 ) < 0.01 0.73 0.000 0.730 COMBINATION KP KI KD 129 - . 3 1 6 • .0060 -3.60 < 0.00 0.00 ) < -0.12 0.77 0.000 0.783 KI COMBINATION KP KD -5.40 130 -.316 -.0060 0.82 ( 0.00 0.00 ) < -0.21 0.000 0.848 COMBINATION KP KI KD 131 -.316 • .0060 -7.20 -0.30 0.86 < 0.00 0.00 ) < 0.911 0.000  ) (  -0.31 -0.73 0.792  ) (  0.82 0.00 0.817  )  ) (  0.26 0.52 0.575  ) <  0.26 -0.52 0.575  )  ) <  0.04 -0.55 0.553  ) <  0.33 0.00 0.335  )  ) <  -0.09 -0.64 0.648  ) (  0.49 0.00 0.488  )  ) (  -0.18 -0.71 0.734  ) (  0.57 0.00 0.570  )  ) <  -0.26 -0.77 0.810  ) (  0.62 0.00 0.624  )  ) (  -0.33 -0.81 0.873  ) <  0.66 0.00 0.664  )  ) (  0.15 0.72 0.732  ) (  0.15 -0.72 0.732  )  ) (  0.01 -0.73 0.730  ) (  0.19 0.00 0.192  )  ) (  -0.12 -0.77 0.783  ) (  0.33 0.00 0.334  )  ) (  -0.21 -0.82 0.848  ) <  0.43 0.00 0.427  )  ; <  -0.30 -0.86 0.911  ) (  0.49 0.00 0.494  )  133.  COMBINATION KP KI KD 132 -.316 -.0060 -9.00 ( 0.00 0.00 ) < -0.37 0.90 0.000 0.971 COMBINATION KP KI KD 133 -.316 -.0080 0.00 ( 0.00 0.00 ) < 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 134 -.316 -.0080 -1.80 < 0.00 0.00 ) < -0.07 0.87 0.000 0.871 COMBINATION KP KI KD 135 -.316 -.0080 -3.60 < 0.00 0.00 ) < -0.18 0.89 0.000 0.910 COMBINATION KP KI KD 136 -.316 -.0080 -5.40 ( 0.00 0.00 ) ( -0.27 0.92 0.000 0.959 COMBINATION KP KI KD 137 -.316 -.0080 -7.20 < 0.00 0.00 ) ( -0.35 0.95 0.000 1.012 COMBINATION KP KI KD 133 -.316 -.0080 -9.00 ( 0.00 0.00 ) < -0.43 0.97 0.000 1.063 COMBINATION KP KI KD 139 -.316 -.0100 0.00 ( 0.00 0.00 ) ( 0.00 0.00 0.000 0.000 COMBINATION KP KI KD 140 -.316 -.0100 -1.30 ( 0.00 0.00 ) ( -0.15 0.98 0.000 0.989 COMBINATION KP KI KD 141 -.316 -.0100 -3.60 < 0.00 0.00 ) ( -0.25 0.99 0.000 1.022 COMBINATION KP KI KD 142 -.316 -.0100 -5.40 ( 0.00 0.00 ) ( -0.34 1.01 0.000 1.063 COMBINATION KP KI KD 143 -.316 -.0100 -7.20 < 0.00 0.00 ) < -0.42 1.02 0.000 1.108  > (  -0.37 -0.90 0.971  ) <  0.54 0.00 > 0.543  ) <  0.05 0.86 0.861  ) (  0.05 -0.86 ) 0.861  > (  -0.07 -0.87 0.871  ) <  0.13 0.00 0.135  )  ) (  -0.18 -0.89 0.910  ) <  0.25 0.00 0.248  )  ) <  -0.27 -0.92 0.959  ) (  0.33 0.00 0.334  )  > (  -0.35 -0.95 1.012  > <  0.40 0.00 0.400  )  ) (  -0.43 -0.97 1.063  > (  0.45 0.00 0.453  )  > (  -0.05 0.97 0.972  ) (  -0.05 -0.97 0.972  )  ) (  -0.15 -0.98 0.989  > (  0.10 0.00 0.105  )  ) (  -0.25 -0.99 1.022  ) (  0.20 0.00 0. 196  )  > (  -0.34 -1.01 1.063  ) (  0.27 0.00 0.272  )  -0.42 -1.02 1.103  ) (  0.33 0.00 0.334  )  ) <  134.  COMBINATION KP KI KB 144 -.31G -.0100 -9.00 < 0.00 0.00 ) < - 0 . 5 0 1.04 0.000 1.153  ) <  -0.50 -1.04 1.153  ) <  0.38 0.00 ) 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 7-Sep-1986 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  14:06 14:05  PROGRAM NUUAVE C. C SIEVE HODGE 31712771 C. C C . ..PROGRAM TO GRAPHICALLY DEP1CI WAVEBOARD MOTION AND C . . . T H E RESULTING WAVEHEIGHI PROFILE. THE WAVEBOARD STROKE C. ..AND THE WAVEHEIGHI ARE ALSO PLOTTED VERSUS TIME. C 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)  100  A  0PEN<UNIT=5,FILE='WAVDAT.DAT;2',STATUS='OLD' ) READ(5,A) U,EM1,MM(1),DT,TEND,DX,XEND,STROKE,TIME FORMAT(II,F5.3.T9,F5.3,T17,F5.3,T25,F5.3.T33,12,137,F5.3,145,12, I49,F5.3> DATA X / 0 . / . A L 0 T / 2 0 / EM<1)=EM1 C=(WAA2)A0.249  C C . ..CALCULATE WAVE COEFFICIENTS BY NEWTON'S METHOD C 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 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  7-Sep-1986 14:06 7-Sep-1986 14:05:  C . . . C A L C U L A T E ALL UAVEHEIGHTS S BOARD POSITIONS FOR X t TIME C DO 40 1=1,TEND T IME = T IME+DT ! !WAVEBOARD INPUT " 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 CONTINUE 50 !! OUTPUT CONTINUE 40 C C . ,DRAW WAVEBOARD POSITIONS t WAVEHEIGHT PROFILES C . •ALSO PLOT THE STROKE AND WAVEHEIGHT VERSUS TIME C CALL GRSTRT(4027,5) !CHECK THIS CALL M0VE<5.,85.) CALL DRAU(5.,55.) CALL DRAW(145 . , 5 5 . ) 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. , 4 5 . ) 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 0115 0116 0117 0118 0119 0120 0121 0122 0123 0124 0125 0126 0127 0128 0129 0130 0131 0132 0133 0134 0135 0136 0137 0138  7-Sep-1986 7-Sep-1986 TIME=TIME*DT CALL M0VE<15.,68.5) CALL APPEAR CALL DRAU(B0ARDX(I) B0ARDY<I)) DO 900 L = l , 2 X=0. DO 90 J=1,XEND X=X+DX H0RZ=15.+XA9.84 VERT=78.7+NU<I,J)A9.84 IF(XO<I).GT.X) GOIO 90 I F ( L . E Q . l ) GOIO 89 CALL REMOVE CALL MARKER(HORZ,VERT 42> CONTINUE I F ( ( L . E Q . 2 ) . O R . < I . E Q . T E N D ) ) GOTO 30 CALL M 0 V E ( 1 5 . , 6 8 . 5 ) CALL REMOVE CALL DRAU(BOARDX<I),BOARDY(I)) CONTINUE CONTINUE CALL GRSTOP STOP END f  89 90  900 30  r  14:06 14:05  

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