Multivariable control of a submarine using the LQG/LTR method. · 2016. 6. 12. ·...
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Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1985
Multivariable control of a submarine using the
LQG/LTR method.
Mette, James Allen Jr.
http://hdl.handle.net/10945/21457
DUDLEY KNO.
.
NAVAL POSTGRADUA1MONTEREY, CALIFORNI
DEPARTMENT OF OCEAN ENGINEERINGMASSACHUSETTS INSTITUTE OF TECHNOLOGY* CAMBRIDGE, MASSACHUSETTS 02139
MULTIVARIABLE CONTROL OF A SUBMARINE
USING THE L3G/LTR METHOD
BY
JAMES ALLEN METTE, JR.
OESM(ME)
COURSE XIIIAMAY: 1985
MULTIVARIABLE CONTROL OF A SUBMARINEUSING THE LOG/LTR METHOD
byJAMES ALLEN ME77E, JR.
BSME, Purdue University (1978)
SUBMITTED TO THE DEPARTMENTS OF OCEAN ENGINEERINGAND MECHANICAL ENGINEERING IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREES OF
OCEAN ENGINEERAND
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
10 MAY 1985
©James Allen Mette, Jr., 1985
Permission is hereby granted to the Charles Stark Draper Laboratory,to the Massachusetts Institute of Technology and to the United StatesGovernment and its agencies to reproduce and to distribute copies ofthis thesis document in whole or in part.
MULTIVARIABLE CONTROL OF A SUBMARINEUSING THE LQG/LTR METHOD
by
James Allen Mette, Jr.
Submitted to the Departments of Ocean Engineering and MechanicalEngineering in Partial Fulfillment of the Requirements for theDegrees of Ocean Engineer and Master of Science in MechanicalEngineering.
ABSTRACT
A multivariable control system for a deeply submerged submarinewith active roll control is designed using the Linear QuadraticGaussian with Loop Transfer Recovery (LQG/LTR) method. A linearmodel of the submarine is developed for a 1 degree rudder deflectionat a speed of 15 knots. The linear model is then scaled for unitsand input/output weightings and augmented with integral control in
all four input channels. Using the properties of the linear model, a
Model Based Compensator (MBC) is designed by shaping the singularvalues of a Kalman Filter to meet desired performance criteria andthen recovering the singular value shapes using the Kwaakernaakrecovery process. During extensive testing at speeds from 15 to 30
knots, the compensator performed well enough so that gain schedulingwas not employed. Next, the compensator is compared to one designedfor a 30 knot model. Finally, an Anti-Reset Windup (ARW) strategy isemployed to counter the effects of control surface saturation.
THESIS SUPERVISORS: Dr. Michael Athans, Professor of Systems Scienceand Engineering
Dr. Lena Valavani, Assistant Professor ofAeronautics and Astronautics
ACKNOWLEDGEMENTS
I would like express my thanks to my thesis supervisors, MichaelAthans and Lena Valavani, for their support, guidance and patience.I would also like to thank Mr. Petros Kapasouris for his ideas andthe interaction that helped in many ways.
The Charles S. Draper Laboratory deserves special recognitionfor providing the submarine simulation facilities. A special thanksto Mr. William Bonnice for his continual programming and technicalsupport.
Last, but not least, I want to express my thanks to Sue, Lizzieand Jimmy for the support they have given not only during my stay atMIT but throughout my naval career. A man could not ask for a moreloving or supportive family.
This research was performed at the MIT Laboratory for Informationand Decision Systems with partial support provided by the Office ofNaval Research under contract 0NR/N00014-82-K-0582(NR606-003)
.
TABLE OF CONTENTS
ABSTRACT 2
ACKNOWLEDGEMENTS 3
TABLE OF CONTENTS 4
LI5T OF FIGURES 6
LIST OF TABLES 7
CHAPTER l: INTRODUCTION AND SUMMARY
1 .
1
Background and Discussion of Prior Wori< 3
1 .
2
Contributions of the Thesis 10
1.3 Outline of the Thesis 10
CHAPTER 2: MODELING OF THE SUBMARINE
2.1 Introduction 12
2.2 Description of the Nonlinear Submarine Model 122.3 Description of the Linear Submarine Model 16
2.4 Selection of the Output Variables 19
2 .
5
Summary 27
CHAPTER 3: LINEAR MODEL ANALYSIS
3 .
1
Introduction 23
3 .
2
Reduction of the Model 283.3 Scaling 293 .
4
Eigenstructure 31
3.5 Performance Specifications 41
3 .
6
Summary 47
CHAPTER 4: LINEAR COMPENSATOR DESIGN
4 .
1
Introduction 43
4.2 The LQG/LTR Design Methodology 434.3 Design of the LQG/LTR Compensator 554.4 Preliminary Testing of the Compensator 654.5 Summary 81
CHAPTER 5: ADDITIONAL ANALYSIS OF THE COMPENSATOR
5.1 Introduction 82
5.2 Additional Analysis of the S15R1 Compensator 825.3 Comparison of the 15 and 30 Knot Compensators 91
5.4 Summary 118
CHAPTER 6: IMPLEMENTATION OF ANTI-RESET WINDUP FEED3ACX
6.1 Introduction 1196.2 ARW Feedback Loop 119
6.3 Simulations for a System With ARW Feedback.. 1236 .
4
Summary 136
CHAPTER 7: SUMMARY AND DIRECTIONS FOR FURTHER RESEARCH
7 .
1
Summary 137
7.2 Conclusions and Directions for Further Research 138
REFERENCES 141
APPENDIX A: MATRICES OF THE S15R1 LINEAR MODEL
Al Original Plant Matrices Prior to Reduction and Scaling.. 144
A2 Matrices Scaled for Units 145A3 Matrices Scaied for Units and Weights 149
APPENDIX B: PROPERTIES OF THE 15 KNOT PLANT AND COMPENSATOR
Bl Poles , Zeros and Eigenvectors of S15R1 152
B2 Observability and Controllability Matrices 153
B3 Model Based Compensator Matrices 154
B4 Poles and Zeros of the Open and Closed Loop Systems 156
APPENDIX C: MATRICES OF THE 30 KNOT PLANT AND COMPENSATOR
CI State Space Matrices of the 530R1 Linear Model 159C2 Matrices of the 30 Knot Comoensatcr 162
:!3P:
2.1 Body-Fixed Coordinate System for a Submarine 12
2.2 Comparison of Nonlinear and Linear Dynamics for Model 515R1 20
2.3 Comparison of Noniinear and Linear Response for 202
Perturbation Above Nominal Point. 23
3.1 Block Diagram Representation After Scaling 31
3 .
2
3reakdown of 515R1 Modes by State 333 .
3
Effect of Control Surfaces by Mode 36
3.4 Singular Value Decomposition of S15R1 39
3 .
5
Feedback Loop Structure 423.6 Singular Value Requirements for G(s)K(s) 42
3.7 Plant with Integrators in Input Channels 433.8 Singular Values of Original Plant Gp(s) 44
3.9 Singular Values of Augmented Plant G(s) 44
3.10 MIMO Feedback Loop with Multiplicative Error 45
4.
1
Feedback Loop Structure with G (s) 49
4.2 Structure of the Model Based Compensator in a
Feedback Configuration 50
4.3a Singular Values of G_f ]_(s) 534.3b Singular Values of Gj^f(s) 53
4.3c Singular Values of G(s)K(s) 584.4 Block Diagram Representation of the Closed Loop System 59
4.5a Singular Values of K(s) 61
4.5b Singular Values of K q ( s
)
61
4 .
6
Singular Values of the Closed Loop 624.7 Singular Values of Kr <s) 624.8 Singular Value Decomposition of K_r(s) 634.9 Symmetry Comparison for +/- 1 Degree/Second Yawrate 66
4.10 Symmetry Comparison for +/- 5 Degree Roil Angle 72
4.11 Simulation for 1 Foot/Second Depthrate 77
4.12 Simulation for 4.5 Feet/Second and -10 Degree Pitch Angle 79
5.1 Comparison of Singular Values of 515R1 and S30R1 Linear Models.. 845.2 Robustness Comparison of G(s)K(s) and the Error E(s) 855.3 Comparison of 15 Knot Compensator Performance for
Simultaneous Turn and Dive During Speed Change 875.4 Comparison of 15 Knot Compensator Performance for
Simultaneous Roil and Dive During Speed Change 925.5 Comparison of 15 Knot and 30 Knot Compensators
for -1 Degree/Second Yawrate at 15 Knots 975.6 Comparison of 15 Knot and 30 Knot Compensators
for -1 Degree/Second Yawrate at 30 Knots 1025.7 Comparison of 15 Knot and 30 Knot Compensators for
Simultaneous Turn and Dive (30 to 15 Knots) 1075.8 Comparison of 15 Knot and 30 Knot Compensators for
Simultaneous Turn and Dive (15 to 30 Knots) 113
6.1 Structure of the Ciosea Loop System with ARW 120
6.2 3cde Plot, of Simple Lag for Varying o< 121
6.3a Singular Values of GCs)K(s) for o< = 0.1 122
6.3b Singular Vaiues of GCs)K(s) for o< = 0.05 1226.4 Singular Vaiues of the Closed Loop for oc = 0.05 124
6.5 Comparison of Systems Without and With ARW (o<= 0.05)for 2 Feet/Second Depthrate 126
6.6 Comparison of Systems Without and With ARW (o<= 0.05)
for 4 Feet/Second DeDthrate 131
LIST OF TABLES
2.1 State and Control Variables 15
2.2 20* Perturbations Applied to Model 515R1 19
3.1 Control Surface Rate and Deflection Limits 30
CHAPTER 1
INTRODUCTION AND SUMMARY
1.1 Background and Discussion of Prior Work
The introduction of nuclear propulsion and hydrodynaraically efficient
hullforms on submarines have resulted in platforms capable of very high
submerged speed for prolonged periods of time. At these high speeds, ship
control is a complex problem due to the dynamic forces and moments acting
on the submarine. During a maneuvering situation, the cross-flow and
nonsymmetric flow over the hull may create a situation in which the
submarine can not be safely operated using our present method of ship
control. This requires that speed and depth restrictions be imposed on the
employment of the submarine.
Presently, submarine control systems are designed by decoupling the
vertical and horizontal planes of motion, looking at the individual
input/output transfer functions and "shaping" the response as desired.
This requires the use of an oversimplified model of a very complex dynamic
system. The recent development of the Linear-Quadratic-Gaussian with Loop
Transfer Recovery (LQG/LTR) methodology has provided a powerful tool for
the design of multivariable control systems. However, as LQG/LTR is
relatively new, research is required to investigate the inherent
limitations of the methodology and develop practical applications for its
use.
Within the last few years, several examples of the practical use of
LQG/LTR have been researched. Multiple Input Multiple Output (MIMO)
8
controllers for gas turbine engines have been developed CI] C2] [33
.
Several examples of submarine control systems have also been developed.
The submarine controller design by Harris C43 dealt with use of an
inverted-Y sternplane configuration. Of particular interest though are the
designs of Milliken C5] , Dreher C63 and Lively [7] . These designs all used
the standard cruciform stern configuration presently in use on U.S.
submarines. Milliken' s controller was sufficient for depth and course
keeping. Acceptable performance during depth and/or heading changes was
obtained only for small changes over long periods of time (i.e. small depth
and heading rates) . Dreher designed two rate controllers. One controlled
depthrate and heading rate directly and the other attempted to control
these rates by using pitch angle and yawrate. In both cases, acceptable
performance was obtained only for small rates.
The design by Lively provided the best example of using the
Linear-Quadratic-Gaussian with Loop Transfer Recovery (LQG/LTR) design
methodology for submarine control. In an attempt to capture the
maneuvering dynamics of the submarine, the linear model for this design
used a diving turn for a nominal point. By using linear compensators
designed at speeds of 5, 10, 15, 20 and 25 knots, a nonlinear compensator
was employed using gain scheduling. This provided for control of the
submarine over the speed range of 5 to 25 knots during speed changes as
well as at discrete intermediate speeds.
Additionally, Martin C8] has recently shown the merits of active roll
control over the conventional cruciform configuration for a submarine at 30
knots. By comparing LSG/LTR controllers with and without differential
sternplane deflection capability, better maneuvering characteristics were
realized especially in the area of depthkeeping during turns.
1.2 Contributions of the Thesis
The purpose of this thesis is twofold; to develop a practical example
of the use of LQG/LTR and to help open an avenue for advancement in the
area of submarine ship control systems. It is intended that this thesis be
considered a feasibility design showing the promise of using active roll
control through differential deflection of the sternplanes and not a final
detailed design ready for implementation aboard a submarine. A project of
that magnitude is obviously beyond the scope of this thesis due to
financial and time considerations.
In comparison to the work of Lively, this thesis shows that using
active roll control through differential deflection of the sternplanes
eliminates the need for a gain scheduled nonlinear compensator to provide
adequate control for speeds of 15 to 30 knots. Further research may even
permit this speed range to be expanded.
To show the eixects of crossover frequency on the response and
stability of the closed loop system, a comparison is made to the
compensator designed by Martin [8] . Following this comparison, an
Anti-Reset Windup (ARW) feedback loop is employed to counter the effects of
integrator windup during control surface saturation.
1.3 Outline of the Thesis
The nonlinear and linear models of the submarine are discussed in
Chapter 2. A brief description of the nonlinear equations of motion used
for submarine simulation at the Charles S. Draper Laboratory (CSDL) is
presented. This is followed by a discussion of the linearized model
employed in the design of the LQG/LTR compensator.
10
Chapter 3 contains an analysis of the linear model. The analysis
begins with a reduction in the order of the linear model. Scaling is then
applied to provide the designer with a more understandable system of units
and to weight the inputs and outputs. Next, the pole/zero structure and
eigenvectors of the open loop submarine model are presented. The chapter
ends with a discussion of performance requirements and specifications. In
this section, the concept of singular values is introduced.
Chapter 4 begins with a discussion of the Model Based Compensator
(MBC) concept and the LQG/LTR methodology. The design of the compensator
for the linear model is then presented. The chapter ends with a
description of the compensator and results of some of the maneuvering
simulations.
Chapter 5 begins with a discussion of additional maneuvering
simulations for the 15 knot compensator, concentrating on more complex
maneuvers. A direct comparison between the compensator designed in this
thesis and that designed by Martin C83 is also presented in Chapter 5.
A brief discussion on the use of Anti-Reset Windup feedback to reduce
the effects of integrator windup in the presence of saturating actuators is
contained in Chapter 6. This is followed by implementation of an
Anti -Reset Windup feedback loop in the 15 knot compensator and presentation
of some results which demonstrate its effectiveness.
Chapter 7 contains a summary of results, conclusions and some
recommendations for future research.
11
CHAPTER 2
MODELING AND ANALYSIS OF THE SUBMARINE
2.1 Introduction
The design of any control system begins with a mathematical
representation of the real world nonlinear system which we want to control.
The model used for this thesis was the analytic version of the SUBMODEL
submarine simulation computer programs resident on the computer systems at
the Charles Stark Draper Laboratory (CSDL)
.
The chapter begins with a brief description of the nonlinear model and
relevant computer programs. This is followed by a description of the 15
knot linear model using differential deflection of the sternplanes to
provide the necessary degree of freedom to actively control the roll angle
of the submarine.
2.2 Description of the Nonlinear Submarine Model
Motion of the submarine is described in six degrees of freedom, three
translational and three rotational. Three force equations (X, Y and 2)
define the translational motions of surge (axial), sway (lateral) and heave
(normal). The rotational motions of roll, pitch and yaw are described by
three moment equations (K, M and N)
.
The complex hydrodynamic forces and moments which act on a submerged
submarine are best defined in a right-hand orthogonal coordinate system
fixed in the body. Figure 2.1 shows this coordinate system (and sign
12
STARBOARD VIEW
REFERENCE
TOP
Kj
HORIZONTALREFERENCE
'*
STERN VIEW
Figure 2.1 Body-Fixed Coordinate System for a Submarine
13
convention) with the origin at the center of gravity (CG) . The equations
of motion are more easily derived in and control of the submarine requires
knowledge of the submarine's motion (i.e. course, speed, depth and
orientation) with respect to an inertial or earth-fixed coordinate system.
The earth-fixed system is also a right-handed orthogonal system, but is
fixed at some geographic point on the surface of the water. The
orientation and motion of the submarine in these two reference frames are
related by Euler angles C9] , each representing a rotation about a
body-fixed axis. By convention, the rotations are applied in the following
order.
1. V (rotation about the z-axis)
2. 8 (rotation about the y-axis)
3. (rotation about the x-axis)
From the discussion above, it is evident that we are talking about
nine state variables, six representing the translational and rotational
velocities in the body-fixed reference frame and three representing the
orientation of the body-fixed axes with respect to earth reference. The
submarine's depth represents a tenth state variable. Control of the
submarine is accomplished by deflecting the rudder, stern planes, fairwater
planes and/or changing propeller speed.
The SUBMODEL simulation facilities at CSDL are based on the "2510
Equations" CIO] , updated with the cross-flow terms of the "Revised
Equations" [11] . Several additional features have been added to provide a
constant RPS constraint on the propeller and a differential deflection
option for the sternplanes. The constant RPS constraint reflects the
current operating policy used on submarines. The use of differential
14
sternplane deflection provides an avenue for active roil controi that is
not otherwise possible. Table 2.1 contains a summary of state and conr.ro!
variables.
State Variables
u = Xi, axial velocity (along x axis) - ft/sec
v = X2» lateral velocity (along y axis) - ft/sec
w = X3, heave velocity (along z axi3) - ft/sec
p = X4, roll rate (angular velocity about x axis) - rad/sec
q = X5, pitch rate (angular velocity about y axis) - rad/sec
r = Xfe, yaw rate (angular velocity about z axis) - rad/sec
= X7, roll (rotation about x axis) - radians
8 = X3, pitch (rotation about y axis) - radians
l|/ = Xg, yaw (rotation about z axis) - radians
z = X10, depth - feet
Control Variables
db = uj_, fairwater plane deflection - radians
dr = U2> rudder deflection - radians
dsi = U3, port stern plane deflection - radians
ds2 = U4, starboard stern plane deflection - radians
Table 2.1 State and Control Variables
It should be noted that these equations of motion do not account for
actuator dynamics. This will be addressed further in the discussion of
performance requirements in Section 3.5. Intuitively, one would expect the
dynamics of the submarine to be independent of the depth (z) and the
15
heading angle (IJJ) . This will prove true as we wiii see in the next.
section.
The SUBMODEL program provides the following analytic capabilities:
1. Integration of the nonlinear equations of motion
2. Determination of a local equilibrium point in
the nonlinear equations
3. Calculation of the linearized dynamics about
the equilibrium point
4. Integration of the linearized equations of motion
During integration of the equations of motion, the control surface
deflections may be set to initial values, varied as a function of time or
calculated using full state feedback or the dynamic compensation derived
using the LQG/LTR method. The analytical results of the integration may be
output graphically, in tabular form or both. A full description of the
submarine model may be found in [12] through [15]
.
2.3 Description of the Linear Submarine Model
The compensator design procedure is based on a Linear Time Invariant
(LTD model of the nonlinear system. The nonlinear equations described in
Section 2.2 can be generally put in the form
x(t) = f[x(t),u(t)] (2.1)
y_(t) = g_[x(t)] (2.2)
where
x(t) is the 3tate vector
u(t) i3 the control inDut vector
16
y_(t) is the output vector
These equations can then be linearized about aoie nominal point
(Xo,!^) by using a Taylor Series expansion. Neglecting the higher order
terms of the expansion, the linear dynamics may be expressed in the sLare
space form
x(t) = A x(t) + B u(t) (2.3)
y_(t) = C x(t) (2.4)
where
A = (al/dx.>|5Lo»lLo
C = (a£/dx.)|5Lo
The nominal point used here corresponds to a local equilibrium point.
This point is found by integrating the nonlinear equations of motion for a
specific set of initial conditions. At their steady-state value, each
state variable is then perturbed so that its local minimum is found.
Since the linear dynamics change with different nominal points, the
question arises as to what is the "correct" nominal or operating point
about which to linearize. Selection of a "benign" nominal point such as
that by Milliken C5] is probably satisfactory for an "autopilot" type of
controller. This controller however, will be used for maneuvering as well
as course and depth keeping. For this reason, a nominal point which
"captures" the maneuvering dynamics of the submarine is required. To
achieve this, the submarine must be oriented such that it experiences
crossflow over the hull much like it would encounter while simultaneously
turning and diving.
17
The nominal point used in this thesis was determined using an initial
forward velocity (surge) of 15 knots with a rudder deflection of 1 degree.
This model is named S15R1 to reflect the speed and rudder deflection. This
nominal point was selected so the linear model would have dynamics similar
to that used by Martin C83 and provide a basis for comparison. The nominal
point provides sufficient cross-coupling between the horizontal and
vertical planes of motion to capture the desired maneuvering
characteristics but is not too "radical". As a result, acceptable
performance was obtained for straight and level as well as maneuvering
trajectories. Figure 2.2 shows the excellent agreement between the time
response of the linear and nonlinear models. The A and B matrices and
values of the nominal states may be found in Appendix Al. Inspection of
the last two rows of these matrices indicate that our intuition about the
effect of z and ty on the other states was correct. This will be of vaiue
when we consider reducing the order of the system.
To determine the range of accuracy of the linear model, it can be
perturbed from its nominal point by a set of initial conditions, integrated
and then compared to the nonlinear system for the same set of initial
conditions. Figure 2.3 shows the results of the initial condition
perturbation of 20* above the nominal point. The perturbations are
summarized in Table 2.2.
18
u = 5.08 ft/aec
v = 0.150 ft/sec
w = 0.395E-02 ft/sec
p = -.115E-04 rad/sec
q = 0.352E-04 rad/sec
r = -.125E-02 rad/sec
= -.325 deg
8 = -.105 deg
Table 2.2 20% Perturbations Applied to S15R1
2.4 Selection of the Output Variables
Several factors must be considered in the selection of the output
variables. First and foremost is the intent of the controller and the
inherent capabilities of the system to be controlled. For example, an
attempt to control roll angle without using differential sternplane
deflection cannot possibly succeed.
Next, a constraint of the methodology is that the dimension of the
output vector must equal the dimension of the control input vector C16 ]
.
Since there are four control inputs (db, dr, dsl, ds2), there must be four
system outputs.
Finally, the effect of each control input "felt" at the output must be
considered. Two states may have influence on the results the designer is
trying to affect. The state which requires the least control surface
deflection to achieve the desired results is the better one to use since it
will require less energy, result in iower compensator gains and reduce the
chance of saturating the inputs.
19
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A submarine is presently maneuvered by deflecting the control surfaces
until a particular turning rate, depth rate or attitude is achieved.
Knowing the handling characteristics of the submarine, the operators
maintain the rates until a predetermined point when the surfaces are
deflected to counteract the rates which have developed. If done correctly
„
the submarine will "meet" the desired course and/or depth with little or no
overshoot. For this reason, the yaw rate ty and depth rare z will be used
as output variables.
The attitude of the submarine in pitch and roll strongly affects its
performance. As the pitch angle becomes excessive, it becomes increasingly
difficult to maintain depth or control depth rate because the fairwater
planes saturate. As roll angle increases, the forces generated by the
control surfaces act out of the plane they were meant to. To visualize
this problem imagine the submarine rolling 900. At this point, the rudder
will act as a sternplane and the sternplanes as rudders. Additionally, the
force and moment generated by crossflow over the sail is quite significant
and contributes to roll and pitch. These illustrations show the importance
of roll and pitch 8.
For this thesis then, the output vector y_(t) is given by
y_(t) = C0(t) 8(t) lj/(t) z(t)]T (2.5)
where
l//(t) = -u*sin8 v*cos8»sin0 w*cos8»cos0 (2.6)
z(t) = (r»cos0 * q*sin8)/cos8 (2.7)
The linearized equations for $ and z are easily derived from the state
space equation (2.3). The elements of the C matrix which represent ty and z
are simply the last 2 rows of the A matrix.
26
2.5 Summary
This chapter has briefly described the origin of the nonlinear model
and the linearization process. The state space form of the linear model
and the reasoning behind selection of the output variables were aiso
presented. Chapter 3 will cover model reduction, scaling, the
eigenstructure of the linear model and performance specifications.
27
CHAPTER 3
LINEAR MODEL ANALYSIS
3.1 Introduction
This chapter begins with a reduction of the tenth-order model and
discussion of the scaling. Next, the eigenstructure and pole/zero
structure of the reduced model will be presented. Finally, system
performance in the area of steady state error and crossover frequency will
be discussed. In the section on performance, the concept of singular
values and their use will be introduced. The reader already familiar with
the concept and its application to MIMO control system design may wish to
skip to the last paragraph of that section for a summary of the performance
specifications applicable to this design.
3.2 Reduction of the Model
As noted in Section 2.3, the depth and heading angle (yaw) have no
effect on the dynamic response of the other eight states. This may be
verified by inspection of the last two columns of the A matrix in Appendix
Al. Had the assumption of a deeply submerged submarine not been made, one
would expect depth and heading to have a significant effect on the states
due to near-surface suction forces and direction of the seas.
Additionally, inspection of the last four rows of the B matrix reveals that
the control surfaces have no direct effect on (J/, 9, or z.
28
Since this controller does not use W or z as outputs and the other
states are not functions of these variables, they may be deleted from the
model. This is accomplished by deleting the last two columns of the A
matrix and the last two rows of the A and B matrices. As we will see in
ChaDter 4, this will also reduce the order of the comoensator state vector.
3.3 Scaling
The use of scaling has been employed to change the state and control
variables of the linear system into units which are more easily understood
and to provide weighting on the inputs and outputs of the system. Kappos
C2] and Boettcher [17] have addressed scaling and it3 effects. Obviously,
the singular values of the system will be changed but the effect on system
robustness is not fully understood.
To provide the designer with a more easily understood system of units,
rotational or angular variables are expressed in units of degrees and
degrees per second vice radians and radians per second. Transiational
variables remain in units of feet and feet per second.
The unsealed state space system as expressed in equations (2.3) and
(2.4) is scaled by defining new state, control and output vectors where
x' = Sx x
u' = Sn u
y_' = Sy y_ (3.1)
The scaling matrices S_x, Sq and Sy are diagonal matrices whose elements
provide the desired transformation of units. Consequently, the system
matrices (scaled for units) become
29
A' = S_x A S.x-1
B' = S.x B Su"l
C' = Sy C S.x-1 (3.2)
Details of the unit transformation scaling matrices and scaled system
natrices may be found in Appendix A2.
Mow that the system has been scaled for units, we must consider
scaling the outputs and inputs to reflect their relative importance. These
weights were selected to coincide with those for a compensator being
concurrently developed by Martin C8] . Weights were selected for the
outputs such that yaw rate and depth rate were more important than roll and
pitch and are given by
Sy =
.1
.1
1
1
Input weights were selected by comparing the maximum deflection of each
control surface. Rate and deflection limits are listed in Table 3.1.
Control Rate Limit Deflection Limit
db 7 deg/sec 20 degrees
dr 4 deg/sec ±_ 30 degrees
dsl»dS2 7 deg/sec + 25 degrees
Table 3.1 Control Surface Rate and Deflection Limits
For this thesis, the control surface dynamics (deflection rates) will be
treated as high frequency modeling error. Considering the deflection
limits above, the selected input scaling is given by
30
0.S670.3
0.8
Figure 3.1 shows the block diagram representation of the system after
applying the scaling for units and input/output weights.
Sx — A
Figure 3.1 Block Diagram Representation After Scaling
The effect of scaling on the plant transfer function matrix is given by
G'(3) = Sy Sy G(s) Sjq"! Su"1 (3.3)
where G(s) is the plant transfer function matrix prior to any scaling. The
input and output scaling as well as the unit transformation is identical to
that used by Martin C8] so that a better comparison of the two compensators
may be made (presented in Chapter 5). A summary of the plant
matrices after scaling can be found in Appendix A3.
3.4 Eiqenatructure
To examine the modes of the state space system defined by (2.3) and
(2.4), the linear transformation (3.4) is applied to the state vector where
x(t) = P z(t) (3.4)
31
the matrix P is the constant and nonsmgular matrix of eigenvectors Pj.j
through P_ni as defined in (3.5). The new state space (3.6) representation
P =
I I
EliJ
P2i| 23 £niI
Z(t)
V_(t)
P-l A P z(t) P-1 B u(t)
C T z(t)
(3.5)
(3.6)
consists of n decoupled equations describing the modal response of the
system. The matrix P~l A P is a diagonal matrix whose eieraent3 are the
eigenvalues of the state space. Each eigenvector describes the motion of
its associated submarine mode along the coordinate axes of the
8-dimensional A matrix. Since the dynamic response of the submarine is a
linear combination of these modal responses, useful information may be
obtained by analyzing the contribution of each state to a particular mode.
Figure 3.2 shows the state breakdown of each normalized eigenvector
and it3 associated eigenvalue in bar chart form. The vertical scale of 0.0
to 1.0 has been selected to reflect the percent contribution of each state
to the overall magnitude of the eigenvector. Due to the difference in
variables and units (translational and rotational), cross-coupling between
terms and the complex nature of the dynamics involved, a clear cut physical
explanation of all the modes is not always possible. Several modes do
however exhibit clear physical meaning leading to the following
interpretation.
(1) Modes 1 and 2 are a complex conjugate pair representing
the oscillatory nature of pitch.
32
«J4-1
cn
>
<Q
0)
TJO
• pnijuSouj •pn||u6oui
-1 1 r« N «d d d
t 1 1 1 rn * •* «i ••
d d d d d
I
\
-
a. o
i
&
I"
(Xin•-i
CO
o
c3o-a
o0)
uCO
(N
aen
•pn||uSeui • pn||u6oiu
33
-»J
<0
•uen
>-Q
<0
uT3O
•pn||u6oui
&£
men
vio
c3OT3
"eo
(1)
UCO
<U
u3
• pr\||u6oui
34
(2) Mode 3 is heavily dependent on forward 3peed and is
therefore most likely related to drag.
(3) Modes 5 and 6 are a complex conjugate pair representing
the oscillatory nature of roll.
(4) The submarine is open loop 3table since all poles are
in the left half plane (LHP).
The eigenvalues, eigenvectors (modal matrix) and transmission zeros for the
linear model are presented in Appendix Bl.
System stabilizablility (unstable modes are controllable) and
detectability (unstable modes are observable) can also be addressed by
simple inspection of the T" 1 B and C T matrices in Appendix B2. The
elements of these matrices represent the link between the original state
space system and the decoupled system and a zero element in one of these
matrices would indicate a that the link is not present. The breakdown of
the controllability matrix in Figure 3.3 shows the relative impact of the
control inputs on each mode. Similarly, the observability matrix
represents the relative contribution of each mode to the system output
variables. When used in conjunction with the eigenvector breakdown,
additional physical information can be gained about the system dynamics.
Inspection of Figure 3.3 reveals that Modes 1 and 2 which represent
pitch of the submarine are mostly affected by the fairwater planes and
sternplanes. The dominance of Mode 3 by rudder deflections explains the
reason for the large dependence of Mode 3 on forward velocity. All control
surface deflections cause a loss in forward speed due to added drag. The
rudder, being the largest control surface, has significant effect even for
small deflections. Mode 4 which has no real dominant state is also most
35
a3
o<2
0.9 -
o.a -
0.7 -
o.s -
0.5 -
0.4 -
\\
\Z7\ <*B IXNl dRMODEE2 dsi
S 7
S3 dS2
Fiaure 3.3 Effect of Control Surfacss by 3ode
36
affected by rudder deflection. Since roil angle is a major contributor to
this mode followed cioseiy by forward speed, this probably represents the
speed loss and roil angle induced by the crossflow over the sail of a
turning submarine. Modes 5 and 6 which represent the roll characteristics
of the submarine are dominated by the sternplanes with a smaller
contribution from the rudder. Obviously, the differential deflection of
the sternplanes will affect roll. The effect of the rudder on roll has
already been mentioned. Mode 7 which is dominated by pitch with lesser
contributions from pitch-related states is mostly controlled by the
fairwater planes. No clear interpretation of the rudder deflections which
control Mode 8 is possible.
To provide an indication of the coupling between control inputs and
output variables, a singular value decomposition of the plant transfer
function G(s) is presented. Details of the theory behind singular value
decomposition may be found in Athans CIS] and Lehtomaki [18] . The plant
transfer function matrix ia given by
G(s) = C (si - A)'l B = U E V.H
where
U is the matrix of left singular vectors
E is a diagonal matrix of the singular values
\/H is the matrix of right singular vectors
and at s = by
G(0) = C (-A)-l B
37
Therefore, since
y_(t) = G(0) u<t) = U E VH u(t)
we can define
y/ (t) = E u ' (t)
where
y_' <t> = U _1 y_<t)
u' (t) = V.H u(t)
Since E is a diagonal matrix, a direct coaparison of the left and right
singular vectors for each singular value will yield the desired information
about input/output coupling. The left and right singular vectors for each
singular value are presented in Figure 3.4. For Oil, we see that
deflection of the 3ternplanes results in the coupled response in pitch
angle and depthrate. Intuitively, one would think that sternplanes would
affect pitch more that depthrate. One explanation is that these variables
are strongly coupled but have different units. As is obvious from o"22>
deflection of the rudder affects primarily z but does exhibit some effect
on roll angle. This is a reflection of the effect of rudder movement and
should not be interpreted as a feasible way to control roll angle. The
effect of differential deflection of the sternplanes is exhibited by O33.
As shown by 0*44, deflection of the fairwater planes affects primarily pitch
and depthrate to a somewhat lesser extent. Again, the difference in
variables and coupling between pitch and depthrate is the probable reason
for this behavior.
38
a m6 6
i r« - o6 6
OO
CM
IT)
en
H
— oc
a:in
en
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09
oQeooa>
QCD
•HO>
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3CT
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09
3
39
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5
o
in
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-
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1 1 1 1 1 1 1 r
666666666
O
5
5
<N
'OvvvVNN \\\\ v\\\\\ W\\ V '
OS,
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u3
40
3.5 Performance Specifications
The performance requirements used in this thesis represent the
choices of the designer and may not necessarily meet any established
criteria set by legal authority. The specifications here will deal with
steady-state error, disturbance rejection and stability robustness. First,
the concept of singular values will be briefly discussed.
A logical extension of the Bode plot can be made from the SISO case to
the MIMO system by consideration of the singular values of the system. The
singular values of a complex matrix M are defined by
where
oVW) = / *i CM.HM]'
(3.4)
o~i is the i tn singular value of M
M.H is the complex conjugate transpose of M
Aj_ is the ith eigenvalue of CM^M]
Referring to (3.4), a matrix M is considered "large" when its minimum
singular value is large and "small" when its maximum singular value is
small. Using these definitions, we can now address command following,
disturbance rejection, reduction in sensitivity to modeling errors and
response to sensor noise.
Our ultimate goal is to design a robust, dynamic compensator, K(s) to
control the Linear Time Invariant (LTD plant, Gp(s). Figure 3.5 3hows the
feedback loop structure with unity negative feedback, command input r(s),
plant input u(a), system output y_<3), output disturbance vector d(s) and
measurement noise n(s) . From Figure 3.5, the following frequency domain
*U
d
^O
Figure 3.5 Feedback Loop Structure
relation can be easily derived
y_(s) s £1 + G(3)K(a)]-lG(3)K(3) r(s)
CI G(s)K(s)]-l d(s)
« CI G(s)K(3)]-i n<3) (3.5)
From (3.5), we see that to have "good" command following and
disturbance rejection, the Loop Transfer Matrix G(s)K(s) and therefore
o"a i n [G(3)K(s) } must be "large" in the frequency range of the reference
commands. To reduce the sensitivity to modeling errors, o,
max CG(s)K(3)
}
must be "small" in the frequency range of the modeling errors. Similarly,
for "good" noi3e rejection, 0"max {G(3)K(s) } must be "small" in the frequency
range of the noise. Figure 3.8 shows the Bode-like plot interpretation of
L09 crtjwl
7777771777//low frequency//performance //BARRIER m „ /
a_-JG <N»> <(iv»H
•ma.ISIi-'Sf-'l
logjwj
777777777'
UNMO0EL2DOYNAMICS/SENSOR '
NOISE
Figure 3.6 Singular Value Requirements for G(3)K(3)
h2
these singular value requirements. More detail on thi3 subject nay be
found in Athans [16]
.
With respect to command following, it is desired to have zero steady
state error to step inputs. This coincides with the singular value
requirement discussed above. As such, integrators were placed in all four
input channels. Figure 3.7 shows the new plant configuration where the
Figure 3.7 Plant With Integrators in Input Channels
state space representation of the augmentation is defined by
G-(a) = (I/-) (3.6)
and the resultant plant transfer function matrix is
G(s) = Gp(3)G_a(s) (3.7)
The augmented plant is now a 12-dimensional system, however the 4
integrator states are included as part of the plant only during the L2G/LTR
compensator design. During implementation, the integrators become part of
the compensator. The singular values of the original plant Gp(s) and the
augmented plant G(s) are shown in Figures 3.8 and 3.9 respectively. Note
that augmenting the plant has increased the singular values at .001
radians/second by approximately 60 db. The matrices of the augmented plant
nay be found in Appendix A3.
43
o3
o
o
90 -
ao -
70 -
so -
50 -
AO -
30 -
20 -
10 -
n — ^-10 -
-20 - ^-30 -
-AO -
-SO -
-60 -
-70 -
-80 -
-90 -
-100 - 1— " T" " -"-"-""
.001 •01 .1 1 10omega (radians/second)
100
Figure 3.8 Singular Values of Original Plant Gp (s)
100
cm
O
o
.001 .1 1
omega (radians/second)100
'igure 3.9 Singular Valuea of Augmented Plant G(a)
kk
The next area which needs to be addressed is crossover frequency. As
crossover frequency is increased, the system responds faster and one would
expect better performance. At slower speeds however, the control surfaces
have less effect for a given deflection since the forces and moments
generated are proportional to speed squared. Since this compensator will
be compared to the one being concurrently developed by Martin C8] , a
comparable crossover frequency is desired. To avoid frequent control
surface saturation and in some cases limit cycling due to the decrease in
effectiveness, a compromise must be reached. Additionally, actuator
dynamics which exist at approximately 2 to 3 radians/second C5] C6] , have
not been included in this model and must be treated as modeling error. For
this thesis, a crossover frequency of 0.1 radian per second was found to be
suitable.
Finally, a brief discussion of robustness is in order. According to
Lehtomaki C183 , the error E(3) between the real system G(s) and the linear
model G(s> can provide a measure of stability robustness. These errors are
characterized as additive, subtractive, multiplicative or division in
nature. In particular, the multiplicative and additive errors can be used
to provide a measure of the relative and absolute errors, respectively. A
block diagram representation of multiplicative error as defined by (3.8) is
shown in Figure 3.10. Additive error is defined by (3.9). Using these
r(s) * ">.
£(s)
+ <r
K(s) G(s)Ji
vfe)
Figure 3.10 MIMQ Feedback Loop With Multiplicative Error
^5
G(s) s CI E(s)] G(s) = L(s)G(s)
E(s) = CG(s) - G(s)] G(s)-1 (3.8)
G(s) = E(s) G(s)
E(s) = Z(.a) - G(s) (3.9)
relationships, it can be shown that if the singular value inequalities
(3.10) and (3.11) hold, we are guaranteed a closed loop 3table 3V3tem .
o"*ax<k(s) - 13 < o-min {I. CG(s)K(s)] "-} (3.10)
o-max (L-l(s) -I) < <r«infl G(s)K(s)} (3.11)
The modeling errors may be difficult to quantify but looking at it from
another aspect, if we calculate the right hand aide of the inequalities,
then the maximum tolerable error is known. These relationships provide
conservative stability bounds so that a closed loop system may still be
stable if the bounds are crossed. System stability would then have to be
determined through extensive testing and evaluation. Additional
information on stability robustness may be found in C19] through C223
.
Performance specifications may therefore be quickly summarized aa
(1) The singular value requirements in Figure 3.6 must
be met
(2) Integrators in all input channels will insure zero
steady-state error to step inputs
(3) Maximum crossover frequency is 0.1 radian/3econd
46
3.6 Summary
In Chapter 3, the order of the iinear model has been reduced by 2
states and scaled for units and input/output weightings. The poie/zero
structure and eigenvectors have been presented. In the section on modal
analysis, the relative contribution of the states to each mode was
considered and the issues of stabilizability and detectability were
addressed. The coupling between inputs and outputs was also presented.
Finally, the performance specifications in terms of singular value
requirements and crossover frequency were detailed. In Chapter 4, the
LQG/LTR methodology and a summary of the compensator design are presented,
^7
CHAPTER 4
LINEAR COMPENSATOR DESIGN
4. i Introduction
This chapter covers the design of linear compensator for the S15R1
model discussed in Chapters 2 and 3. It begins with a general discussion
of the Model Based Compensator (MBC) concept and the Linear Quadratic
Gaussian with Loop Transfer Recovery (LQG/LTR) methodology tailored for
performance and robustness at the plant output. This is followed by a
summary of the linear compensator design for the S15R1 model. Finally, the
results of several simulations with the compensator integrated in the
nonlinear analytic model at CSDL are presented. The compensator is further
analyzed and critiqued in Chapter 5 where it is compared to the S30R1
compensator designed by Martin C8j .
4.2 The LQG/LTR Desicn Methodology
Figure 4.1 again shows the feedback loop structure with unity negative
feedback. Note that the original plant Gp(s) has been replaced with the
augmented plant G(s) as defined by (3.7). The compensator K(s) must
provide closed loop performance commensurate with the criteria noted in
Chapter 3. This will be accomplished through the design of a Model Based
Compensator (M8C) [163 using the Linear Quadratic Gaussian with Loop
Transfer Recovery (LQG/LTR) methodology [23] through [25]
.
48
uyuT
G(s)
+ 'F
^O
Figure 4.1 Feedback Loop Structure with G(s)
The LQG/LTR methodology guarantees the designer a MIMO compensator
K(3) that is closed loop stable while providing the necessary degrees of
freedom for "loop shaping" (shaping of the singular values of G(s)K(s)) to
achieve desired performance while maintaining stability robustness. Figure
4.2 shows the structure of the feedback loop using the MBC. The state
vector of the MBC is z(t)ERn (i.e. the same dimension as x(t), the plant
state vector) , hence we ultimately wind up with an open loop system
G(s)K(s) which is 2n-dimensional. The reduction of the model described in
Section 3.2 has therefore reduced the resultant closed loop system by a
total of 4 states.
The MBC derives its name from the fact that the A, 3 and C matrices
defined in the state-space representation of the plant appear in
the dynamics of the compensator. The open-loop plant dynamics are
x(t) = A x(t) + B u(t)
y_(t) = C x(t)
(4. 1)
(4.2)
where x(t)ERn, u(t)ER"» and y_(t)ER». Note that u(t) and y_(t) are the same
dimension as constrained by the methodology. The frequency domain
representation for the plant transfer function G(s) is given by
^9
ss
>.l
ol
xt
<!
col
ai
z<-Ia.
ol xl
.- ai Ol>l
XJn XI
++
t-i OlJS i cal31 +ml *l j-. <l+ Ol ^^ u*
|a>i
N|w» (A
I<I >-t *i Ola 1 ii
•mi >l 31 31 ^\
CQl31
ail wt1
+ 31 <l1
•^ ** —i
xl XI to u*
<l Ol Ol olu u 1 u
mm*** ^i «• ^.•XI >>l >-l Ol
<0
3a
coo-Xo«0
j3T3OJ
<u
u.
to
c
uo*J<B
<D
cai
a£Oo-oai
(0
(0
CO
ai
ao=ai
ai
3Uo3
4-1
cn
CM
41
U3cn
50
G(s) = C (si - A)"i 3 (4.3)
Dynamics of the MBC in the time domain are
z(t) = CA - B G - H C] z(t) - H e(t)
u(t) = -G z(t)
(4.4)
(4.5)
and the coraoensator transfer function K(s) 13
K(s) =G(sI-A+BG+H C)~- K (4.6)
where the Filter Gain Matrix H and Control Gam Matrix G are design
parameters for the compensator.
As derived in C16] , by defining a state estimation vector (4.7), the
dynamics of the closed loop system can be represented by (4.3).
x(t)
w(t)
w(t) = x(t) - z(t)
(A - B G) -3 G
(A - H Z)
x(t)+
w(t) -H
r(t)
(4.7)
(4.8)
By examining the closed loop A matrix above, it is evident that the roots
of (4.9) yield the 2n eigenvalues of the closed loop system and for closed
loop stability, (4.10) and (4.11) must hold.
detCsI - A + B G ] • dettal - A H C3 =
ReCXiCA - B G ] } < ; i=l,2, . . . ,n
ReOiCA - HC11 < ; i = l,2, . . .,n
(4.9)
(4.10)
(4.11)
It now becomes clear that the compensator design decomposes into
finding G and H 3uch that (4.10) and (4.11) hold. The Linear Quadratic
Gaussian (LOG) methodology provides the necessary tools to make
51
"intelligent" choices for the compensator gain matrices thus the L3G
controller is a special type of Model Based Compensator. A review of
equations (4.4) and (4.5) reveai the LOG controller to be a cascaded
combination of a Kalman-Bucy Filter and a full-state feedback Linear
Quadratic Regulator.
Selection of the Filter Gain Matrix H is made through the solution of
a Kalman-Bucy Filter (KBF) problem based on equation (4.4). The Filter
Gain Matrix is defined by
H = (1/^j) E C_T (4.12)
where E = E* >_ is the unique solution of the Filter Algebraic Riccati
Equation (FARE)
= AE*E_aJ + LLJ- (l/^i) E CT C E (4.13)
subject to the constraints
» CA,C_] detectable
* CA,L] stabilizable
• p >
The Control Gain Matrix G is defined by (4.14) as determined through
the solution of the "cheap" LQR (Linear Quadratic Regulator) control
problem
u(t) = -G z(t) = -R-l B_T K (4.14)
where K = Xj >_ is the unique solution of the Control Algebraic Riccati
Equation (CARE)
= -K A - Aj K - q C? C K 3 R/l B~ K (4.15)
52
subject, to the constraints
* CA,B] stabilizabie
« CA,C] detectable
8.s I
» q >
The compensator gains could new be determined if the design parameters
q, u and L were known. To determine the "correct" values for these design
parameters, we now turn our attention to Loop Transfer Recovery (LTR)
.
Refering to Figure 4.2, performance and stability robustness can be
measured at the plant input or plant output by breaking the loop at the
corresponding point. For the design of the submarine controller, physical
significance at the plant output has been retained (point (2) on Figure 4.2)
and therefore the latter approach will be used. The reader is referred to
[23] and [24] for an explanation of the methodology with respect to
performance and robustness at the plant input.
The design of the controller can now be performed in the following
fashion.
(1) Design a Kalman Filter with appropriate singular values
using the KBF equations (4.12) and (4.13)
(2) Recover the singular value shapes using the LQR equations
(4.14) and (4.15)
The following definitions of the filter open loop and Kalman Filter
transfer functions are useful in the development of the design procedure
Gf l(s) = C (si - A)*l L (4.16)
G^f(s) = C (sl_ - A)"l H (4.17)
53
Note that (4.17) represents the ioop transfer function when the LQG
loop in Figure 4.2 is broken at point (l) . The relationship between these
transfer functions is given by the following Return Difference Identity
(4.18) also known as the Kalman Frequency Domain Equality (FDE) for the
filter.
CI+Gjcf Cs)] CI+Gfcf <s)]H = I d/u)Gfol (s)GHfol ( S ) (4.18)
By looking at the singular values of both sides of the FDE (4.19), the
approximation (4.20) holds for 0"i(G_^f(s)} >> 1, i.e. at low frequency.
0-j.CI Gj<f(s)} = /l (l/^)o-i2(Gf i(s))1
(4.19)
0-i{G_i<f(s)} = 0"i((l/y7i)Gfo i(s)} (4.20)
This means the singular values of the KBF can be "shaped" to meet the
desired performance, crossover and robustness requirements by selecting
1. L for the desired loop shape
2. u for the desired crossover frequency
Using the value of }i and the matrix L from above, H is calculated
using the KBF equations and the following properties are guaranteed.
* Closed-loop stability (for the filter)
» -6db < Gain Margin < oo
* -60° < Phase Margin < 60°
Now that the Filter Gain Matrix has been determined, the zeros of the
plant tranfer function G(s) are checked. If the plant is minimum phase, we
are guaranteed recovery of the "nice" shape and properties of the Kalman
Filter using the Kwakernaak "sensitivity recovery" C26] via the LSR
5^
equations. At this time, the effects of non-minimua phase (NM?) zeros are
not. clearly understood, but suffice it to say that some degracaticn in the
recovery process and resulting cicsed loop performance is certain if one or
more NMP zeros are within the bandwidth of the controller. If the plant is
non-minimum phase, all that can presently be done is to proceed and use
extensive testing and evaluation to detect instabilities or other problems.
The LQR equations are now solved (as q
—
go) to determine the Control
Gain Matrix G. Using the G and H matrices, the LQG compensator matrix KCs)
is determined using (4.6). The singular values of G(s)K(s) are then
compared to the singular values of Gj<f(s). If the Cm (G(s)K(s) } are
sufficiently different from o"i (G^f (s) 1 , q must be increased and the LGR
equations solved for a new Control Gain Matrix. Recovery of the Kalman
Filter loop shape is considered satisfactory when there is good agreement
for at least 1 decade past crossover. At that point, the roiloff of the
singular values is at least -40 db/decade (two pole roiloff) . The Loop
Transfer Matrix G(s)K(s) is then checked for robustness as discussed in
Section 3.5.
4.3 Design of the LQG/LTR Compensator for the S15R1 Model
As discussed in the previous section, the design of the compensator is
performed as follows.
(1) Design a Kalman Filter loop with appropriate singular
value shapes
(2) Attempt to recover these singular value shapes using
the Kwaakernaak recovery method
55
The reader is once again reminded that during the LQG/L7R design process,
the plant matrices which are used are that for the augmented scaiec plant
G(s) . Any augmentation is separated from the plant model and included in
the compensator prior to implementation for testing.
Design of the Kalraan Filter loop hinges on selection of the parameter
)i (magnitude of the measurement noise intensity) and the matrix L so that
the singular values of Gf l<s) as defined by (4.16), have a "nice" shape
like that shown in Figure 3.5 and meet the performance specifications set
in Section 3.5. To enhance chances of a good recovery of these shapes in
the LQR loop, it is desired to have the minimum and maximum singular vaiues
natch. This is accomplished by selecting L such that the singular values
match at high and low frequencies.
The augmented plant transfer function G(s) = QB (.a)Gp(a) is defined by
G(s) = C (si - A)"l B (4.21)
whereA =
0.
Bp Cp
C = " Cp]
and therefore
[si - A]si
-Bp (si. - Ap) (4.22)
Looking at (4.22) at low frequency, (si. - Ap) —*> -Ap and a good
approximation for (si. - A) "I is
(si - A)"l £<i/sn
-(l/ 3 )Ap-l|p -Ap" 1 (4.23)
provided Ap~l exists. Therefore by (4.16), Gfol ( s) is approximately
56
Gf i(s) ^ CpJ
(1/S)I
-(l/ s )Ap~lBp -Ap-1
(l/ s )CpAp-lBpLi - CpAp _1 L2
kl
k2
(4.24)
Note that the matrix L has been partitioned into L± and L2 associated with
the low frequency and high frequency behavior respectively. From (4.24),
low frequency matching of singular values occurs if
Li [CoAo-lBol-1 (4.25)
Similarly, at high frequency, (sl_ - Ap) — sl_ and
(si - A)"l £3
(l/a)IS'±.
(l/ 32)Bp (i/s)!
(1/S )I
-(l/s2)Bp (1 / S >I
(!/s2)CpBpLi (l/ 3 )CpL2
(4.26)
Gf l(s) & Cp 1k2
(4.27)
As s —* 00, the second term in (4.27) dominates and the singular values are
matched if
L2 = CpT<CpCpT)-l (4.28)
The L matrix (listed in Appendix B3) has now been selected for low and high
frequency matching, however the behavior of the singular values at middle
frequencies has not been considered. Unacceptable differences at these
frequencies are dealt with by appropriate scaling of the plant inputs and
outputs such as that described in Section 3.3.
Once an appropriate L matrix has been determined, the singular values
are shifted up or down by varying the scalar parameter p until the desired
crossover frequency is obtained. The value of p. = 132 produced the desired
57
ioo
.001 .1 1 10
oaeqa (radians/second)
Figure 4.3a Singular Values of G-foi (a)
100
ICO
.001 .1 1 10
omega (radians/second)
Figure 4.3b Singular Values of Gir-f(s)
100
.001 .1 1
omega (radians/second)
Figure 4.3c Singular Values of G(s)K(s)
100
58
maximum crossover frequency of 0.1 radians/second. Figure 4.3a shows the
plot of o"i { (l//JJ)Gf i (s) ) for these parameters.
Now, using the p. and L such that meet the desired specifications, the
FARE is solved and the Filter Gain Matrix H determined from (4.12). Next,
the singular values of the Kalman Filter o"i(Gj<f(s)} are calculated as shown
in Figure 4.3b and compared to the performance specifications. Thi3 is the
"loop shape" of G(s)K(s) which results from the Kwaakernaak recovery.
The next step in the LOG/LTR process is solution of the CARE (the MIMO
transmission zeros have already been checked in Section 3.4) and
calculation of the Control Gain Matrix using (4.14). Using the H and G
matrices, the compensator K(s) is calculated using (4.6) and the singular
values of the Loop Transfer Matrix G(s)K(s) are compared to the singular
values of the Kalman Filter loop. As stated in Section 4.2, satisfactory
recovery is said to occur when o~i (G(s)K(s) ) match o"i£Gj<f(s)) for at least
one decade past crossover. Figure 4.3c shows the recovered singular value
loop shape for q = 105. The crossover bandwidth for this design was from
approximately 0.05 to 0.1 radians/second. The G and H matrices are listed
in Appendix B3.
As stated in Section 3.5, the integrators are actually part of the
final compensator matrix Ka(s). The final compensator must also include
the effects of scaling on the inputs and outputs. Figure 4.4 shows the
block diagram representation of the resultant closed loop system. The
Figure 4.4 Block Diagram Representation of Closed Loop Sv3tam
59
singular values for the LOG compensator K(s) and the augmented and scaled
compensator |£a(s) are shown in Figures 4.5a and 4.5b respectively. From
these singular vaiue plots, the lead-lag characteristics, particularly the
effects of the integrators at low frequency, are evident. The spread in
singular values at frequencies below crossover reflects the different
amplification requirements for certain directions so that the singular
values of the recovered Loop Transfer Matrix G(s)K(s) match.
The singular values of the closed loop plant as defined by (4.3) are
shown in Figure 4.6. They reflect the desired chararacteristic of db
gain until crossover or break frequency and at least 2 pole rolloff
(-40 db/decade) after crossover. The poles and zeros of the open loop and
closed loop systems are found in Appendix B4.
By examining the transfer function matrix for the loop broken at the
plant input, information can be obtained on how the control inputs vary
with respect to reference commands. The transfer function relating r(s) to
u(s) is given by
u(s) = CI + K(s)G(s))-l K(s) r(s) = Kr(s)r(s) (4.29)
Figure 4.7 shows the singular values for the frequency range of interest.
The spread in singular values indicates that certain directions have higher
gains than others. The singular value decomposition of (4.29) shown in
Figure 4.3 allows us to examine the coupling of reference commands to
control inputs. The singular value decomposition was accomplished in the
manner discussed in Section 3.4 on plant eigenstructure. From Figure 4.8,
we see that for <T\\, the controller responds to a depthrate command with
deflection of the fairwater planes which makes physical sense. For 0*22 » it
is evident that if a yawrate command causes deflection of the rudder and
6o
100
3
C
o
o
.001 .1 1 10
omega (radians/second)
Figure 4.5a Singular Values of K(a)
100
100
o"3
c
a
o
.001 .1 1
omega (radians/second)100
Figure 4.5b Singular Values of Ka (3)
61
>uO3
c
O
O
100 -r-
90 -
SO -
70 -
SO -
50 -
40 -
30 -
20 -
10 -
O --lO -
-20 -
-30 -
-40 -
-50 -
-SO -
-70 -
-30 -
-90 -
-100 -
.001
—
i
r-
.01 .1 1
omega (radians/second)
100
Figure 4.6 Singular Values of the Closed Loop
TOO
o3
a
©
.001 .01 .1 1
omega (radians/second)
100
Figure 4.7 Singular Values of Kr <3)
62
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the sternpianes. The rudder obviously deflects to induce the yawrate.
Movement of the sternpianes can be attributed to differential deflection to
counter the initial roll angle caused by the rudder deflection and the
steady state roll angle induced by crossflow over the sail. Furthermore,
the 0*33 information indicates that roll commands will cause the rudder and
sternpianes to respond. Differential sternplane deflection is obviously
the only feasible way to control roll so the question arises as to why the
rudder deflects so much. This is most likely due to the coupling between
the rudder and sternpianes. Finally, 044 3hows that a pitch command
results in deflection of the sternpianes. In this case we would expect
deflection in the same general direction with possible slight differences
due to induced roll angles.
4.4 Preliminary Testing of the Compensator
As a preiude to further analysis, preliminary testing of the
compensator was performed to gain some insight on possible limitations of
the design. Simulations were performed for single command inputs and
multiple command inputs using step and square commands.
First, a aeries of simulations were made to examine the effects of
symmetry on the response of the submarine. Figure 4.9 shows the results of
the 15 knot nonlinear simulations for ^1 degree/second yawrate step inputs
applied at t = 5 seconds and removed at t = 75 seconds. As the submarine
enters into the turn, it initially rolls slightly outward and then due to
the large force generated by crossflow over the sail, snap rolls inward.
To counter this effect, the sternpianes deflect differentially to generate
a righting moment which drives the roll angle to zero. To counter the
effect of the initial rudder deflection on pitch and depthrate, there is
65
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70
slight movement of the fairwater planes. The slight differences which do
exist are most likely due to propeller torque with a smaller contribution
probably due to the -1 degree/second command (turn to port.) being in the
direction of the nominal point whereas the +1 degree/second (turn to
starboard) is in the opposite direction.
Another maneuver which demonstrates motion about a plane of symmetry
is displayed in Figure 4.10. At t = 5 seconds, a ^_5 roll angle step input
is applied. The sternpianes deflect differentially to a steady-state valu
of approximately +7.5 degrees to achieve the desired roll angle. The
rudder and fairwater piane3 deflect only slightly in reaction to the
initial transients and then go to zero.
Various other maneuvers were simulated for depthrate with and without
pitch to ensure control surface deflections for motion in the vertical
plane were practical. The results of the simulation for a 1 foot/second
depthrate are presented in Figure 4.11. The singular value decomposition
in Figure 4.8 predicts that the fairwater planes should deflect in response
to this reference command. Thi3 does indeed occur a3 shown in Figure 4.11
and is accompanied by a sraail deflection of the sternpianes which produce a
moment to counter the slight trimming moment induced by the fairwater
planes.
Figure 4.12 shows the results of the simulation for a 4.5 feet/second
at -10 degree pitch. The initial decrease is attributed to the additional
drag generated by movement of the control surfaces. A3 the control surface
movements decrease, forward speed is regained. The sraail transient
disturbances in the states are a result of of control surface deflection
and to a lesser degree, mild cross-coupling. Referring again to the
singular value decomposition in Figure 4.8 (d*n and 044), one would predict
71
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deflection of the fairwater planes in response to the ciepthrate command,
aternpianes in response to the ordered pitch angle and minor deflection of
the rudder. Figure 4.12 confirms this to be the case.
Subsequently, simulations which flexed the compensated system in the
vertical and horizontal planes of motion simultaneously were made at speeds
from 15 to 30 knot3 as a prelude to gain scheduling. These simulations and
justification for not employing gain scheduling are discussed in Chapter 5.
4.5 Summary
The first section of this chapter covered the LQG/LTR methodology for
robustness and performance at the piant output. Next, the design and
analysis of the linear compensator for the S15R1 model were presented.
Finally, results of some simulations at 15 knots were presented. These
simulations demonstrated the symmetry of yawrate and roll angle and
confirmed the predictions of the singular value decomposition for the
transfer function relating the control inputs to the reference commands.
Chapter 5 will cover some additional simulations which were performed for
speeds from 15 to 30 knots. A comparison to the compensator designed by
Martin C8] is also presented.
81
CHAPTER 5
ADDITIONAL ANALYSIS OF THE COMPENSATOR
5.1 Introduction
This chapter covers additional analysis performed on the compensator
designed for the S15R1 linear model at speeds from 15 to 30 knots. This is
followed by a comparison to the S30R1 compensator designed by Martin C8]
.
As a result of thi3 analysis, it was determined that while gain scheduling
would offer improved performance, it was not required to assure a closed
loop stable system and therefore would not be implemented.
5.2 Additional Analysis of S15R1 Compensator
It has been shown that using active roll control through differential
deflection of the sternplanes reduces the depth excursion problem
experienced by a high speed submarine in a turn. In this section, it
becomes obvious that active roll control also provides improved performance
for depth changing and coordinated maneuvers over a wide speed range.
To implement a gain scheduling scheme, linear compensators must be
designed for various discrete speeds in the speed range of interest. A
question arises as to how to select those discrete speeds. In this case,
quantifying the modelling error between the linear model and the actual
plant is difficult. We can use the robustness singular value inequalities
(3.10) and (3.11) in a similar manner by assuming the linear model at one
speed is the real system G(s), the linear model at another speed is the
82
model G(s) and using the relationships (3.8) and (3.9) as a gauge of the
errors between the models. This is not. strictly accurate representation of
the errors and the following comments should be kept in mind. Due to this
assumption, the error between the S15R1 model and the actual 30 knot system
is probably larger in some directions and smaller in others. The
inequalities provide a conservative estimate of stability robustness 30
that the system may be stable even if the relations (3.10) and (3.11) do
not hold. These two conditions tend to offset one another and
determination of system stability must be confirmed through extensive
simulation.
A comparison of the singular values of the S15R1 and S30R1 linear
models is shown in Figure 5.1. At low frequency, the singular values of
S15R1 are generally lower than those of S30R1 with the exception of the
minimum singular values which are identical up to approximately 0.05
radians/second. We see that U3e of integral augmentation has increased the
gain of all singular values by approximately 60 dB at 0.001 radians/second
and has shifted the maximum crossover frequency somewhat higher. The state
space matrices for S30R1 can be found in Appendix CI.
Preliminary simulations showed that the compensator designed for the
15 knot model adequately controlled the nonlinear model even at 30 knots.
Assuming the S30R1 model as representative of the real or nominal system
G(s) and the S15R1 model as the system model G(s), we can get a feel for
the error of the S15R1 model at 30 knot3. Figure 5.2 shows the comparison
of the singular values of CI (G(s)K(s) ) "1] and [I G(s)K(s)] with the
singular values of CL(s) - JJ and CL~l(s) - I_] and reveals that the
singular value inequalities (3.10) and (3.11) do not hold. As previously
stated, the relations provide a conservative estimate of stability and the
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85
direction of the instability (if it does exist.) must be determined through
testing. The results of some nonlinear simulations for the 15 knot
compensator are now presented to support this claim.
The maneuvers presented in Chapter 4 were rerun for various speed
between 15 and 30 knots with favorable results. Since these maneuvers were
not very stressful, two maneuvers which flexed the system were simulated.
The first maneuver consisted of a combined dive and turn performed at 15
knots, at 30 knots, as speed increased from 15 to 30 knots and finally as
speed decreased from 30 to 15 knots. The maneuver begins with a 4.5
feet/second depthrate, -10 degree pitch angle and -2 degree/second yawrate
initially applied at t = 5 seconds. At t = 80 seconds, the yawrate command
is zeroed, followed at t = 130 by the depthrate and pitch commands.
Results of these simulations were satisfactory. For the sake of brevity,
only the speed change maneuvers are presented. Figure 5.3 shows the
results with the 15 to 30 knot run on the left. Both simulations were
completed without encountering instability however, there were notable
differences in the yawrate command following and minor differences in depth
change. The roll angle for both simulations was adequately controlled
after the initial transients due to control surface movement died out.
Maximum pitch errors of approximately 3 degrees and 2 degrees respectively
were experienced. The bottom line of this maneuver though was to effect a
course and depth change through appropriate rate commands during a speed
change to stress the system. Considering the time the rate commands were
in effect, the depth should have increased to approximately 1060 feet and
the course change have been 150 degrees. The 15 to 30 knot simulation
displayed some overshoot in depth but both converged to within a few
percent of the intended depth as did the 30 to 15 knot simulation. Even
86
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though the yawrate command following for the 30 to 15 knot was not felt to
be adequate, both simulations converged on a 150 decree course change.
The second maneuver consisted of a dive with an ordered roll angle
during speed changes 3uch as that in the previous maneuver. At t = 5
seconds, a 20 degree roll angle, 4.5 feet/second depthrate and -10 degree
pitch angle were ordered. The 20 degree roil angle remains for the entire
simulation however, the depthrate and pitch angle orders are removed at
t = 130 seconds. Figure 5.4 shows the results with the 15 to 30 knot
simulation on the left again. The simulation for 15 to 30 knots showed
favorable results by maintaining the ordered 20 degree roll angle. The
depthrate and pitch angle responses exhibited some overshoot but were
converging on the commanded values as the fairwater planes came out of
saturation. Throughout the maneuver only slight disturbances to the
yawrate were present. The 30 to 15 knot simulation showed that the system
was stable, but due to the speed decrease, the roil angle could not be
maintained as is evident by sternpiane saturation at t = 95 seconds.
5.3 Comparison of 15 and 30 Knot Compensators
Since the 15 knot compensator performed adequately, the next logical
step was to determine how it's performance at various speeds compared to
the compensator designed by Martin for the 30 knot model. For details of
the design as well as a comparison to a system without active roil control
the reader is referred to C83 . The matrices for the 30 knot compensator
are found in Appendix C2.
First, several benign maneuvers were performed to observe how the
control inputs and outputs of the two systems responded. The first set of
91
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simulations shown in Figures 5.5 and 5.6 were for a -I degree/second
yawrate step input applied at t s 5 seconds at speeds of 15 and 30 knots
respectively. The response for the 15 knot compensator is on the left and
the 30 knot on the right. In general, both exhibited good command
following with the 30 knot compensator reaching the commanded yawrate
quicker. It is evident that this occurs because the control surfaces move
ituch faster than the 15 knot compensator. It should also be noted that
this causes major disturbances to the states and other outputs as compared
to the 15 knot compensator. Of particular interest is that the 15 knot
compensator showed more oscillation at 30 knots and the 30 knot compensator
showed more oscillation at 15 knots. This is due to the particular
simulation being far from the nominal point. The oscillations for the
system using the 30 knot compensator were generally of larger amplitude
than those of the system with the 15 knot compensator. This set of
simulations was followed by a similar set for a 1 foot/second depthrate
with similar results.
The two maneuvers presented in Section 5.2 above were then repeated
for the 30 knot compensator. The results of the simulations are again
shown with the 15 knot compensator on the left and the 30 knot compensator
on the right. The 30 to 15 knot simulation for the turn and dive is
presented in Figures 5.7. The results are generally the same a3 previously
stated. The 30 knot compensator responds quicker, is more oscillatory and
exhibits more overshoot. Of particular interest though is the saturation
of the fairwater planes at t = 80 seconds due to the sternplanes deflecting
quicker than those of the 15 knot compensator as the yawrate command is
removed. This results in growing errors in pitch and depthrate until these
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commands are removed. The results of the roil and dive maneuver provided
similar results and are therefore not presented.
As shown in Figure 5.8, very interesting results were obtained when
the simulation was repeated for 15 to 30 knots. The results are similar
until the yawrate command is removed. At that time, the system with the 30
knot compensator experiences a large 3nap roll, almost twice that of the 15
knot compensator. At t = 110 seconds, the fairwater planes of the 30 knot
compensator saturate which the system also handles but with a larger error
in pitch and depthrate. At t = 130 however, the pitch and depthrate
commands are zeroed and the system goes unstable in the vertical plane as
evidenced by the growing depthrates and pitch angles and repeated
saturation of the fairwater planes and the oncoming saturation of the
sternplanes.
As shown by Martin, the 30 knot compensator performs well for large
yawrates. It appears then that the problem arises only for motion in the
vertical plane. The reason or reasons why the system with the 30 knot
compensator goes unstable and the system with the 15 knot compensator does
not are unknown but the following is offered as a possible explanation.
(1) The higher crossover frequency for the 30 knot compensator
causes the control surfaces to move faster. These quick
movements result in quicker control surface saturations resulting
in large intergated errors.
(2) Since we are operating away from the nominal design point, the
model is no longer accurate for motion in the vertical plane.
(3) The drag terms associated with the 15 knot compensator
simulation are actually 4 times larger at 30 knot3 than at
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15 knots (drag is proportional to velocity squared) where the
compensator was designed. This has sufficiently slowed the
system response at 30 knots, particularly in the directions
related to motion in the vertical plane.
Determination of the exact cause is a subject which requires further
research. In any case, the 30 knot does exhibit better performance in mild
maneuvers so to implement a global control system for the submarine at
various speed and for variouys maneuvers, there may be 3ome merit to using
a quick controller for mild maneuvers and a slower controller for more
radical maneuvers.
As a result of this analysis, it was determined that implementing a
gain scheduling algorithm wouid not be feasible without either redesigning
the 30 knot compensator for a lower crossover frequency or limiting the
maneuvers at higher speeds. As shown by Lively C7] , implementing a gain
scheduling scheme is rather straightforward and in this case, it was felt
that nothing new would be learned from the exercise.
5.4 Summary
This chapter has covered some of the more stressful maneuvers which
the 15 knot compensator performed. The 30 knot compensator exhibited
better behavior for less stressful maneuvers and at lower speeds where the
quick control surface movements did not excite any instabilities like it
did at higher speeds. As a result of this comparison, gain scheduling was
not used as it was felt that more could be learned from investigating the
effect of control surface saturation. The use of an Antireset Windup (ARW)
feedback loop is investigated in Chapter 6.
118
CHAPTER 6
IMPLEMENTATION OF ANTI-RESET WINDUP FEEDBACK
6. 1 Introduction
In Chapter 5, it became apparent that as the submarine is ordered to
do more difficult maneuvers, control surface saturation enters into the
picture, While the 15 knot compensator remained stable throughout the
testing, the 30 knot compensator did not. As was evident in the
simulations presented in Chapter 5, control surface saturation of the
fairwater planes occurred frequently and in some cases for long periods of
time. To decrease the time required for a control surface to come out of
saturation, an Anti-Reset Windup (ARW) feedback loop is installed. The
chapter begins with a discussion of the ARW feedback loop. This is
followed by an analysis of the impact of the ARW loop on the open loop and
closed loop systems. Finally, several simulations which demonstrate the
use of ARW are presented.
6.2 ARW Feedback Loop
As a result of prolonged control surface saturation, the system
experiences a phenomena known as integrator windup or reset windup. The
delay in the control surface or surfaces coming out of saturation after the
saturating condition is no longer in effect is caused by the integration of
the error between the ordered deflection and the saturation level. If the
119
error is large and/or the period of saturation is long, the integrated
error may also become large and quite significant delays can be
encountered. To counter this problem, a nonlinear feedback loop is
employed whose effect is to "turn off" the integrator for the saturating
actuator.
The use of ARW strategy in a MIMG control system is not fully
developed and the method employed here is based on C27] and personal
interactions with it3 authors, Xapasouns and Athan3. Figure 5.1 shows the
closed loop system with the ARW feedback loop installed around the
:Q—Ka-Q -ffll
:I 7L
bfl- G»
Figure 6.1 Structure of the Closed Loop System With ARW
integrators. The nonlinearity prior to the plant represents the saturating
control surfaces of the submarine, with the saturation limits as listed In
Table 3.1. When the control surfaces are not in saturation, the feedback
loop is "deenergized" through the "dead-zone" as shown. As a control
surface saturates, the feedback loop for that control channel energizes.
As a result of this feedback, the integrator is replaced by a lag network
as shown by (6.1). The amount of feedback used is a function of the
uCs) = (I/ 3 )CI +«(i/ s )3" 1 K(s) e(s)
= K(a) CI/(a «)] e<a) (6.1)
120
variable oc in the loop but the nee effect of the feedback is to reduce the
gain at low frequency. Figure 6.2 shows the affect of the parameter °< on
the Bode plot of a lag network. A3 <X i3 decreased, the behavior of the
100 o
Figure 6.2 Bode Plot of Siiaole Lag for Varying o<
loop approaches that for the integrator. As <* is increased, the gain at
low frequency decreases. At present, there is no known straighforward
method for selecting °<, especially for the MIMO case. To gain some insight
on a method of selection, several areas were examined.
Figure 6.3 shows the effect the ARW method has on the singular values
of the loop transfer matrix G(s)K(s) where G(s) now contains the dynamics
of the ARW feedback loop instead of the integrators as presented in Section
3.5. At low frequencies, the singular values have spread with two 3howmg
slight increases, one a slight decrease and the one associated with the
saturated control surface a drastic decrease. When o< was decreased by a
factor of 1/2 from 0.1 to 0.05, the minimum singular value increased by a
factor of 2 or 6 dB. There was little or no effect on the transfer
function relating reference commands to control inputs (4.30) or its
singular value decomposition. Finally, the effect on the singular values
121
3
C
t3»
O
©
lOO(alpha = 0.1)
.001 .01 .1 1
omega (radians/second)
igure 6.3a Singular Value3 of G(s)K(s) for <* = 0.1
100
100(alpha = 0.05)
O>l.
c
o
o
.1 1 10omega (radians/second)
'igure 6.3b Singular Values of G(3)K(s) for o< = 0.05
100
122
of the cicsed loop system for c* = 0.05 is shown in Figure 6.4. Again,
spreading of the singular values occurs at low frequency and one would
expect problems in the channel with the -10 dB minimum singular value. For
this design then, the only noticabie effect is on the singular values and
not the singular vectors. Therefore, selection of the parameter o< appears
to be based on how much the minimum singular value of the unaugmented
system needs to be increased at low frequency to assure acceptable command
performance. In the next section, several simulations which shew the merit
of using an ARW feedback loop are presented.
6.3 Simulations for System With ARW Feedback
Using the ARW feedback loop as described in Section 6.2, several
simulations were made for various values of c<. Because the minimum
singular value of G(s)K(s) exibits such a large dependence on <X, it was
thought that to have adequate performance and robustness, c< would have to
be selected such that the value of the minimum singular value was at least
20 dB at low frequency, consistent with the low frequency barrier in Figure
3.6. 7hi3 proved not to be the case as an c< of 0.05 seemed to provide
adequate command following. This may only be sufficient for the series of
tests performed by the author. Additional testing may indeed indicate that
command following is not satisfactory for some combination of command
inputs. But that is a subject for further research.
Since saturation of the fairwater planes occurs most frequently, the
first two simulations deal with an ordered depthrate with zero commanded
pitch as this would easily cause saturation. A comparison of the closed
loop system response for a 2 feet/second commanded depthrate with and
123
100
a3o»
"5?
o»o
o
.001 .1 1
omega (radians/second)
100
Figure 6.4 Singular Values of the Closed Loop for <* = 0.05
12^
without ARW feedback is shown in Figure 6.5. A "mild" saturation is
created by application of the reference command at t = 5 seconds. Noting
the change in scale on the graphs, we see that the responses remain
identical until t = 130 seconds when the reference command is zeroed. At
that time, sternplane and rudder deflections are the same but a slight
difference in the fairwater piane deflection occurs with the ARW simulation
displaying a slight overshoot. Review of the outputs show that the
oscillations have been effectively decreased and the output errors slightly
reduced in magnitude.
To demonstrate the effects of a prolonged "hard" saturation, another
simulation was made with a 4 feet/second commanded depthrate as shown in
Figure 6.6. There is a slight difference in the rudder and sternplane
deflections but an obvious improvement in fairwater plane response is
noted. At t = 130 seconds when the reference command is zeroed, the
fairwater planes for the system with ARW immediately deflect to reduce the
depthrate to zero while the fairwater planes for the 3y3tera without ARW do
not come out of saturation for another 110 seconds! A comparison of output
variables again shows that the ARW feedback loop has decreased oscillations
and more importantly, decreased the error by as much as a factor of 2 in
the case of depthrate and yawrate.
These simulations have 3hown that for a system which experiences
prolonged control surface saturation, the use of an ARW feedback loop will
decrease oscillations in the outputs and sometimes significantly decrease
the magnitude of the maximum error. Additionally, the use of the fairwater
planes as an effective dynamic control input is regained almost immediately
instead of waiting until the integrated error is nulled.
125
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6.4 Summary
This chapter has covered the use o£ an Anti Reset Windup feedback loop
to improve performance of the submarine in the presence of saturating
control surfaces. In particular, the effect on fairwater plane saturation
was demonstrated with notable improvements by reduction in the oscillation
and maximum error of the output variables and the time lag until dynamic
control of the saturating surface is regained.
136
CHAPTER 7
SUMMARY AND DIRECTIONS FOR FURTHER RESEARCH
7. 1 Summary
This thesis has presented the design of a multivariate control system
for a submarine using the Linear Quadratic Gaussian with Loop Transfer
Recovery (LQG/LTR) methodology. The submarine model was based on the NSRDC
2510 equations of motion modified to include the crossfiow terms of the
updated equations, differential control of the sternplanes and a constant
RPS constraint on, the propeller.
The 10th order linear model based on a nominal point of 15 knots
forward speed and 1 degree rudder deflection was reduced to an 8th order
system and then scaled for units and input/output weightings. Using
LQG/LTR, a Model Based Compensator was designed for a crossover frequency
of 0.1 radians/second for the plant augmented with integrators to reduce
steady state error to step inputs. The design was extensively tested at
speeds from 15 to 30 knots and then compared to the compensator design by
Martin C83 . That design is based on a 30 knot model of the same submarine
but has a crossover frequency of 0.5 radians/second.
Subsequently, an Anti-Reset Windup (ARW) feedback loop was
incorporated to reduce the effects of saturating control surfaces on the
dynamic response of the submarine. This is accomplished by replacing the
integrator in the input channel with a simple lag when that channel issues
a command above the saturation level of the control surface.
137
7.2 Conclusions and Directions for Further Research
The LQG/LTR multivariable control system design methodology has been
used to design a controller which considers rather than neglects the cross
coupling present in the dynamics of a submarine. This provides for better
control over a wider variety of maneuvers since the modei is vaiid for
larger perturbations than one which is based on decoupled vertical and
horizontal planes of motion. Active roil control through the differentia-
deflection of the sternplanes also helps by limiting perturbations in roll
angle during maneuvers, thereby limiting cross-coupling.
The use of modal analysis and singular vaiue decomposition have proven
to be an excellent means "of providing the designer with insight on the
states and control inputs which dominate the response of the submarine.
Additionally, the singular value decomposition of the transfer function
relating the reference commands to the control inputs Kr-(s), provides
important information on their coupling, i.e. what control inputs are
generated in the controller for a given reference command.
Comparisons of the 15 knot compensator designed in this thesi3 and the
30 knot compensator designed by Martin [8] indicate that a higher crossover
frequency is desirable for improved performance during mild maneuvers of
the submarine, but a lower crossover design is preferable for more radical
maneuvers. This was evident by the instability experienced during the turn
and dive maneuver by the system using the 30 knot compensator.
The effectiveness of an ARW feedback loop was demonstrated in a system
that experienced control surface saturation. It was determined that the
feedback parameter alpha must be selected such that the gain of G(s)K(s)
remains high enough to guarantee adequate command following. There was no
138
effect on the singular value decomposition of Kt-(s) and only a slight
effect on its singular values.
This thesis has provided an avenue for advancement in the area of
submarine control system design but prior to any actual system being built
and put to sea, much more research must be performed. This thesis and
Martin'3 have demonstrated the advantage of using differential deflection
of the sternpianes to provide active roll control for the submarine.
Additional research is required however, in the area of casualty control in
the event of a sternplane or other control surface failure. In Chapter 5,
it was shown that the 15 knot and 30 knot compensators could be operated at
speeds and operating points far away from nominal but thatthe severity of
the maneuver had to be restricted for the 30 knot compensator due to
instabilities. The reason for this unstable condition is mostly attributed
to the higher crossover frequency, but the effect of the higher nominal
speed must be investigated. The use of gain scheduling will obviously be
required for an actual controller design which effectively controls the
submarine at all speeds.
The singular value decomposition of the plant and the reference to
command input transfer function Kj-Cs), provided useful information about
the coupling of the system at low frequency, but the singular values and
singular vectors change as a function of frequency. Additional research in
this area will provide insight into how the coupling of the system changes
as a function of frequency.
The last area which requires further research is the use of Anti -Reset
Windup feedback. The improvement in the submarine's performance was
evident, but the method of selecting the feedback parameter o< needs
further development. Additionally, the effect of prolonged multiple
139
saturations while using as ARW feedbackloop and the effect, of the ARW loop
on closed loop stability must be addressed.
1^-0
REFERENCES
1. Kapasouris, P., "Gain-Scheduled Multivariable Control for the GE-21
Turbofan Engine Using the LGR and LQG/LTR Methodologies", Master'sThesis, MIT, Cambridge, MA, May 1984.
2. Kappos, P., "Robust Multivariable Control for the F100 Engine",
Master's Thesis, MIT, Cambridge, MA, September 1983.
3. Pfeil, W., "Multivariable Control for the GE T700 Engine Using theLQG/LTR Design Methodology", Master's Thesis, MIT, Cambridge. MA,
August 1984.
4. Harris, K., "Automatic Control of a Submersible", Masters' Thesis,
MIT, Cambridge, Ma, May 1984.
5. Milliken, L., "Multivariable Contron of an Underwater Vehicle",Engineer's Thesis, MIT, Cambridge, MA, May 1984.
6. Dreher, L., "Robust Rate Control System Designs for a
Submersible", Engineer's Thesis, MIT, Cambridge, MA, May 1984.
7. Lively, K., "Multivariable Control System Design for a Submarine",Engineer's Thesis, MIT, Cambridge, MA, May 1984.
8. Martin, R., "Multivariable Control System Design for a SubmarineUsing Active Roll Control", Engineer's Thesis, MIT, Cambridge, MA, May1985.
9. Abkowitz, M., Stability and Motion Control of Ocean Vehicles ,
MIT Press, Cambridge, MA, 1969.
10. Gertler, M. and Hagen, G., "Standard Equations of Motion forSubmarine Simulation", NSRDC Report 2510, Washington, DC, June 1967.
11. Feldman, J., "DTNSRDC Revised Standard Submarine Equations ofMotion", DTNSRDC/SPD-0393-09, Bethesda, MD, June 1979.
12. Bonnice, W. and Valavani, L., "Submarine Configuration andControl", Charles S. Draper Laboratory Memo SUB 1-1083, October 1983.
13. Bonnice, W. and Valavani, L., "Submarine Configuration andControl", Charles S. Draper Laboratory Memo SUB 2-1183, November 1983.
14. Bonnice, W., "Propulsion and Drag Models for the SubmodelSubmarine Simulation", Charles S. Draper Laboratory Memo SUB 3-984,September 1984.
15. Bonnice, W., CONSTRPS Fortran and Command List Computer Programs,Charles S. Draper Laboratory, May 1984.
lM
16. Athans, M., Class Notes for Course 6.232. MIT, Cambridge, MA,
Spring 1984.
17. Boettcher, K., "Analysis of Multivariable Control Systems withStructural Uncertainty", Doctoral Dissertation, MIT, Cambridge, MA,
1983.
18. Lehtomaki, N., "Practical Robustness Measures in MultivariableControl System Analysis", Doctoral Thesis, MIT, Cambridge, MA, May
1981.
19. Lehtomaki, N., Sandell, N. and Athans, M., "Robustness Results in
Linear-Quadratic-Gaussian Based Multivariable Control Designs",IEEE Transactions on Automatic Control , Vol.AC-26. No.l,
February 1981.
20. Safanov, M., "Robustness and Stability Aspects of StochasticMultivariable Feedback System Design", Doctoral Thesis, MIT,
Cambridge, MA, 1979.
21. Lehtomaki, N., et al, "Robustness Tests Utilizing the Structure ofModelling Error", LIDS Report P-1151, MIT, Cambridge, Ma, October1981.
22. Lehtomaki, N., et al, "Robustness and Modelling Error
Characterization", LIDS Report P-1311, MIT, Cambridge, MA, June 1983.
23. Doyle, J. and Stein, G., "Multivariable Feedback Design: Conceptsfor a Classical/Modern Synthesis", IEEE Transactions on AutomaticControl . Vol. AC-26, No.l, February 1981.
24. Stein, G., "LCG-Based Multivariable Design: Frequency DomainInterpretation", AGARD-LS-117, 1981.
25. Stein, G. and Athans, M., "The LQG/LTR Procedure for MultivariableFeedback Control Design", LIDS Report P-1384, MIT, Cambridge, MA, May1984.
26. Kwakernaak, H. and Si van, R. , Linear Optimal Control Systems,Wiiey, New York, NY, 1972.
27. Kapasouris, P., and Athans, M., "Multivariable Control Systemswith Saturating Actuators Antireset Windup Strategies", Draft of Paperfor American Control Conference, Boston, MA, June 1985.
1>2
APPENDIX A
MATRICES 0? THE S15R1 LINEAR MCOEL
143
APPENDIX Al
ORIGINAL PLANT MATRICES PRIOR TO SCALING AND 1QDEL REDUCTION
A MATRIX (10 x 10)
-1.9012E-02 -1.1096E-02 -4.2241E-04 -9.594tE-03 -1.6035E-02 1.5933E+00 O.OOOOE+00 2.9550E-04 O.OOOOE+OO O.OOOOE+00
5.3364E-04 -6.0771E-02 3.4469E-04 -6.46B2E-01 9.48B2E-02 -7.8164E+00 1.3L76E-01 -3.4021E-05 O.OOOOE+00 O.OOOOE+00
7.2953E-06 -7.3932E-05 -5.3862E-02 -B.0323E-01 6.1384E+00 9.7422E-03 O.OOOOE+OO 7.6175E-03 O.OOOOE+OO O.OOOOE+OO
1.1097E-04 -5.8726E-03 -7.1679E-04 -2.2072E-01 -1.2851E-01 -1.4504E-02 -1.6206E-01 4.1B45E-05 O.OOOOE+OO O.OOOOE+OO
-6.5774E-07 -2.2004E-06 6.7287E-04 -5.7594E-03 -2.0682E-01 1.4001E-04 O.OOOOE+00 -2.5124E-03 O.OOOOE+00 O.OOOOE+00
1.3843E-05 -1.0317E-03 3.7870E-06 -3.5813E-03 4.7420E-04 -1.9379E-01 2.6177E-04 -6.7592E-08 O.OOOOE+OO O.OOOOE+00
O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO l.OOOOE+OO 2.5811E-04 -9.1543E-03 -1.3327E-12 -6.2735E-03 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 9.9960E-01 2.8184E-02 6.2730E-03 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO -2.8186E-02 9.9965E-01 1.4553E-10 5.7451E-05 O.OOOOE+00 O.OOOOE+00
9.1576E-03 -2.B183E-02 9.9956E-01 O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO 7.9107E-01 -2.4B18E+01 O.OOOOE+OO O.OOOOE+OO
B MATRIX 14 x 10)
-4.2090E-04 -1.5065E-02 7.2244E-04 7.2244E-04
O.OOOOE+OO 5.9604E-01 -4.3727E-02 4.3727E-02
-3.7257E-01 -3.8218E-07 -2.5391E-01 -2.5391E-01
O.OOOOE+OO 1.0988E-02 5.3783E-02 -5.3783E-02
3.5786E-03 1.2605E-07 -6.1424E-03 -6.1424E-03
O.OOOOE+OO -1.5105E-02 -8.6874E-05 8.6874E-05
O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+00
O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00
O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO O.OOOOE+00
1*44
APPENDIX A2
MATRICES TO PERFORM UNIT TRANSFORMATIONS
Matrix used to premiltiply the A and B aatrices:
1.0O0OE+M O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 0.0OOOE+O0 O.OOOOE+OO 0.0OOOE+OO O.OOOOE+OO
O.OOOOE+OO l.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 5.7300E+O1 O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO
Matrix used to postaultiply the A latrix:
l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.74S2E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO- O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO
145
APPENDIX A2
Matrix used to postaultiply the B aatrix:
1.7452E-02 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00
0.0000E+00 1.74S2S-02 O.OOOOE+00 O.OOOOE+00
O.OOOOE+00 O.OOOOE+00 1.7452E-02 O.OOOOE+00
O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO 1.7452E-02
Matrix used to pree-ultiply the C aatrix:
5.7300E+01 O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO 5.7300E+01 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO 5.7300E+01 O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO
Matrix used to postaultiply the C utrix:
l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO l.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.7452E-02 O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO
146
a?pe:;d:x A2
PLANT MATRICES SCALED FOR UNIT TRANSFORMATION ONLY
A MATRIX (8 x 8)
-1.9012E-02
5.8364E-04
7.2953E-06
6.35B4E-03
-3.7689E-05
-7.9321E-04
O.OOOOE+OO
O.OOOOE+OO
-1.1096E-02
-8.0771E-02
-7.8932E-05
-3.3650E-01
-1.2608E-04
-5.9116E-02
O.OOOOE+OO
O.OOOOE+OO
•4.2241E-04
3.4469E-04
5.3862E-02
4.1072E-02
3.8555E-02
2.1699E-04
O.OOOOE+OO
O.OOOOE+OO
1.6744E-04
1.1288E-02
1.4018E-02
2.2072E-01
5.7594E-03
3.5813E-03
l.OOOOE+OO
O.OOOOE+OO
2.7984E-04
1.6559E-03
1.0713E-01
1.2B51E-01
•2.0682E-01
4.7420E-04
2.581 tE-04
9.9960E-01
2.7806E-02
1.3641E-01
1.7002E-04
•1.4504E-02
1.4001E-04
1.9379E-01
•9.1543E-03
2.8184E-02
O.OOOOE+OO
2.2994E-03
O.OOOOE+OO
1.6206E-01
O.OOOOE+OO
2.6177E-04
•1.3327E-12
6.2730E-03
5. 1570E-06
-5.9374E-07
1.3294E-04
4.1845E-05
-2.5124E-03
-6.7592E-08
-6.2735E-03
O.OOOOE+OO
B MATRIX (8 x 4)
7.3455E-06
O.OOOOE+OO
6.5021E-03
O.OOOOE+OO
3.5786E-03
O.OOOOE+OO
O.OOOOE+OO
O.OOOOE+OO
2.6291E-04
1.0402E-02
6.6698E-09
1.0988E-02
1.2605E-07
•1.5105E-02
O.OOOOE+OO
O.OOOOE+OO
1.2608E-05
•7.6312E-04
4.4313E-03
5.3783E-02
•6.1424E-03
•8.6874E-05
O.OOOOE+OO
O.OOOOE+OO
1.2608E-05
7.6312E-04
-4.4313E-03
-5.3783E-02
-6.1424E-03
8.6874E-05
O.OOOOE+OO
O.OOOOE+OO
1^+7
APPENDIX A2
C MATRIX (4 x 8)
O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO l.OOOOE+00 O.OOOOE+00
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE+00
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO -2.3185E-02 9.9964E-01 1.4552E-10 5.7450E-05
9.1576E-03 -2.8183E-02 9.9956E-01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.3806E-02 -4.3313E-01
148
APPENDIX A3
PLANT MATRICES SCALED FOR UNIT TRANSFORMATION AND WEIGHTINGS
ON INPUTS AND OUTPUTS
A MATRIX (8 x 8)
-1.9012E-02 -1.1096E-02 -4.2241E-04 -1.6744E-04 -2.7984E-04 2.7806E-02 O.OOOOE+OO 5.1570E-06
5.8364E-04 -8.0771E-02 3.4469E-04 -1.1288E-02 1 . 6559E-03 -1.3641E-01 2.2994E-03 -5.9374E-07
7.2953E-06 -7.8932E-05 -5.3862E-02 -1.4018E-02 1.0713E-01 1.7002E-04 O.OOOOE+OO 1.3294E-04
6.3584E-03 -3.3650E-01 -4.1072E-02 -2.2072E-01 -1.2851E-01 -1.4504E-02 -1.6206E-01 4.1845E-05
-3.7689E-05 -1.2608E-04 3.8555E-02 -5.7594E-03 -2.0682E-01 1.4001E-04 O.OOOOE+OO -2.5124E-03
-7.9321E-04 -5.9116E-02 2.1&99E-04 -3.5813E-03 4.7420E-04 -1.9379E-01 2.6177E-04 -6.7592E-08
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO l.OOOOE+OO 2.5811E-04 -9.1543E-03 -1.3327E-12 -6.2735E-03
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 9.9960E-01 2.8184E-02 6.2730E-03 O.OOOOE+OO
B MATRIX (8 x 4)
-7.3455E-06 -3.9417E-04 1.5760E-05 1.5760E-05
O.OOOOE+OO 1.5595E-02 -9.5390E-04 9.5390E-04
-6.5021E-03 -9.9997E-09 -5.5391E-03 -5.5391E-03
O.OOOOE+OO 1.6473E-02 6.7228E-02 -6.7228E-02
3.5786E-03 1.8898E-07 -7.6780E-03 -7.6780E-03
O.OOOOE+OO -2.2646E-02 -1.0859E-04 1.0859E-04
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
1^+9
APPENDIX A3
C MATRIX (4 x 8)
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 l.OOOOE-Oi O.OOOOE+OO
O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE-Ot
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -2.3185E-02 9.9964E-01 1.4552E-10 5.7450E-05
9.1576E-03 -2.8183E-02 9.9956E-01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.3806E-02 -4.3313E-01
150
APPENDIX 3
MATRICES OF THE 15 KNOT PLANT AND COMPENSATOR
151
APPENDIX 3:
POLES, ZEROS AND EIGENVECTORS
EIGENVALUES
2.0413E-02 -2.0413E-02 -2.0618E-02 -3.3461E-02 -1.0603E-01 -1.0603E-01 -2.2040E-01 -2.4761E-01
1.4066E-02 -1.4066E-02 O.OOOOE+OO 0.0000E+OO 3.8084E-01 -3.8084E-01 0.0OOOE+OO O.OOOOE+OO
TRANS?1ISSI0N ZEROS
3.1712E+08 3.3004E+06 5.8381E+03 -1.9007E-02 -1.2729E-01 -5.3321E+03 -1.2904E+07 -7.9657E+08
EI6ENVECT0RS (MODAL MATRIX)
5.4585E-03 5.4585E-03 9.7809E-01 5.2178E-01 -4.3995E-05 -4.3995E-05 -5.4014E-04 3.3478E-02
2.1218E-03 -2.1218E-03 O.OOOOE+OO O.OOOOE+OO 2.9056E-04 -2.9056E-O4 O.OOOOE+OO O.OOOOE+OO
2.0006E-03 2.0006E-03 6.9651E-02 3.4612E-01 -7.5258E-04 -7.5258E-04 2.7841E-03 -3.7199E-01
1.8330E-03 1.8330E-03 O.OOOOE+OO O.OOOOE+OO 1.2856E-02 -1.2856E-02 O.OOOOE+OO O.OOOOE+OO
-1.1406E-02 -1.1406E-02 1.4151E-02 6.0006E-02 7.784AE-03 7.7846E-03 1.3399E-01 -4.6995E-03
7.1516E-02 7.1516E-02 O.OOOOE+OO O.OOOOE+OO 1.1036E-02 -1.1036E-02 O.OOOOE+OO O.OOOOE+OO
-5.2527E-03 -5.2527E-03 1.0682E-03 2.2280E-02 3.4260E-01 3.4260E-01 -2.2220E-02 -1.9978E-01
3.3729E-03 -3.3729E-03 O.OOOOE+OO O.OOOOE+OO -1.3309E-01 1.3309E-01 O.OOOOE+OO O.OOOOE+OO
6.0737E-03 6.0737E-03 4.7564E-03 1.5018E-02 1.8075E-03 1.8075E-03 -2.1241E-01 -1.7366E-02
2.4203E-02 2.4203E-02 O.OOOOE+OO O.OOOOE+OO 4.8844E-03 -4.8844E-03 O.OOOOE+OO O.OOOOE+OO
-5.5494E-04 -5.5494E-04 -2.8418E-02 -1.3178E-01 -1.9431E-03 -1.9431E-03 4.6092E-03 -4.2513E-01
4.9147E-04 -4.9147E-04 O.OOOOE+OO O.OOOOE+OO 3.0331E-03 -3.0331E-03 O.OOOOE+OO O.OOOOE+OO
7.7598E-04 7.7598E-04 -1.1242E-01 -7.3935E-01 -5.5690E-01 -5.5690E-01 1.2856E-01 7.9363E-01
3.7620E-02 -3.7620E-02 O.OOOOE+OO O.OOOOE+OO -7.4456E-01 7.4456E-01 O.OOOOE+OO O.OOOOE+OO
•7.4935E-01 -7.4935E-01 -1.5755E-01 -1.9905E-01 1.9061E-03 1.9061E-03 9.5910E-01 9.8394E-02
6.5659E-01 -6.5659E-01 O.OOOOE+OO O.OOOOE+OO 4.0419E-03 -4.0419E-03 O.OOOOE+OO O.OOOOE+OO
152
APPENDIX 3:
CONTROLLABILITY MATRIX
1.6552E-01 1.0504E-01 3.5334E-01 3.6637E-01
-5.7600E-01 -1.8233E-01 -2.9672E-01 -3.0245E-01
1.6552E-01 1.0504E-01 3.5334E-01 3.6637E-01
5.7600E-01 1.8233E-01 2.9672E-01 3.0245E-O1
-6.8062E-03 -3.9264E-01 -5.6056E-03 -4.4330E-03
2.0B19E-10 -1.5146E-09 -3.3261E-09 3.2745E-09
4.0734E-03 7.1184E-01 1.7878E-03 -2.5278E-03
3.1612E-10 2.7899E-09 6.1340E-09 -6.1256E-09
2.3352E-02 1.2539E-01 4.3242E-01 -3.9234E-01
3.4091E-03 2.8794E-01 2.9887E-01 -3.2909E-01
-2.3352E-02 1.2539E-01 4.3242E-01 -3.9234E-01
-3.4091E-03 -2.3794E-01 -2.98B7E-01 3.2909E-01
-5.2958E-01 -2.3894E-02 1.4670E-01 1.5495E-01
-2.0413E-11 3.4676E-10 7.9478E-10 6.7265E-10
-5.4850E-03 2.2961E-01 -5.6294E-03 9.4034E-03
-9.9081E-12 2.1548E-11 4.4051E-11 -2.7408E-11
OBSERVABILITY MATRIX
7.7598E-04 7.7598E-04 -1.1242E-01 -7.3935E-01 -5.5690E-01 -5.5690E-01 1.2856E-01 7.9363E-01
3.7620E-02 -3.7620E-02 O.OOOOE+OO O.OOOOE+OO -7.4456E-01 7.4456E-01 O.OOOOE+OO O.OOOOE+OO
7.4935E-01 -7.4935E-01 -1.5755E-01 -1.9905E-01 1.9061E-Q3 1.9061E-03 9.5910E-01 9.8394E-02
6.5659E-01 -6.5659E-01 O.OOOOE+OO O.OOOOE+OO 4.0419E-O3 -4.0419E-03 O.OOOOE+OO O.OOOOE+OO
7.6899E-04 -7.6899E-04 -2.8551E-02 -1.3217E-01 -1.9932E-03 -1.9932E-03 1.0649E-02 -4.2448E-01
1.2112E-03 -1.2U2E-03 O.OOOOE+OO O.OOOOE+OO 2.8946E-03 -2.8946E-03 O.OOOOE+OO O.OOOOE+OO
3.1317E-01 3.1317E-01 8.7825E-02 1.3101E-01 -7.1196E-04 -7.1196E-04 -2.7979E-01 -2.5567E-02
-3.5528E-01 3.5528E-01 O.OOOOE+OO O.OOOOE+OO -1.3537E-03 1.3587E-03 O.OOOOE+OO O.OOOOE+OO
153
APPENDIX E3
MODEL BASED COMPENSATOR MATRICES
L MATRIX (12 x 4)
-5.2859E-01 -3.6891E+01 -1.9800E-02 -8.9967E+00
3.1664E-03 5.6612E-02 -3.0290E+00 1.0072E-02
1.2335E+01 2.0142E+00 -4.8602E+00 7.1824E-01
-1. 1585E+01 -7.66B1E-01 5.2128E+00 1.1306E-01
-1.2643E-03 3.9664E-02 -4.5390E-16 9.1576E-03
3.B909E-03 -1.2207E-01 1.3969E-15 -2.8183E-02
-1.3800E-01 4.3294E+00 -4.9544E-14 9.9956E-01
0.0000E+00 0.0000E+00 0.0000E+00 O.OOOOE+00
4.1013E-11 1.M91E-05 -2.B183E-02 2.8272E-14
-1.4546E-09 -5.7425E-04 9.9956E-01 -1.0027E-12
1.0000E+01 6.4753E-11 1.0100E-18 4.3422E-U
2.9472E-10 1.0000E+01 8.5B22E-14 2.9325E-09
154
APPENDIX B3
MODEL BASED COMPENSATOR MATRICES
CONTROL 6AIKI MATRIX - G (4 x 12)
1.9075E+00 1.7309E-02 3.3175E-01 3.3927E-01 -2.2603E+00 6.3586E+00
2.3374E+02 6.0251E-02 1.1518E+02 2.9814E+O0 -2.4295E+00 1.2755E+02
1.7309E-02 3.6408E+00 2.7069E-02 -5.3234E-02 8.8669E-02 9.1610E+00
•4.3139E+00 1.4955E+00 7.0839E+00 -2.8537E+02 1.26&7E+00 1.3071E+00
3.3175E-01 2.7069E-02 2.2008E+00 -4.1713E-01 -1.1099E+00 -3.4200E+00
1.3371E+02 2.5322E+01 -1.5522E+01 1.1943E+01 1.736BE+01 4.0942E+01
3.3927E-01 -5.3234E-02 -4.1713E-01 2. 1780E+00 -1.3532E+00 1.0440E+01
1.3407E+02 -2.5060E+01 -1.0521E+01 -3.B677E+00 -2.1052E+01 4.189BE+01
FILTER 6AIN HATRIX - H (12 x 4)
-5.9202E-02 -3.2113E+00 -3.B250E-02 -7.79B3E-01
3.2510E-02 8.4010E-03 -2.6152E-01 -3.6334E-03
1.1180E+00 1.8450E-01 -2.8643E-01 1.646BE-02
-1.0548E+00 -7.6073E-02 3.2537E-01 5.3966E-02
9.5416E-04 2.0790E-03 3.5B01E-02 7.4520E-04
1.9485E-03 -5.3268E-03 -B.6929E-02 -3.5640E-03
-1.0162E-02 3.76B9E-01 1.0342E-03 8.7100E-02
5.2396E-03 6.0975E-03 -1.4220E-02 4.6202E-04
-3.9137E-03 -3.B591E-05 -1.9355E-03 B.6603E-05
-6.6970E-03 3. 1A29E-04 6.5711E-02 1.3147E-03
9.2545E-01 1.3114E-02 -6.5B35E-02 -3.1072E-02
1.3114E-02 B.7089E-01 3.6730E-03 -1.2924E-03
155
APPENDIX B4
POLES AND ZEROS OF THE OPEN AND CLOSED LOOP SYS7EKS
OPEN LOOP POLE:
(1 x 24)
PE£L PART
7. 1012E-09 i.3i4QE-09 e.0203E-iO a.020SE-10 -1.P007E-02 -2.0413E-42
-2.0*13E-02 -2.0ai=E-v2 -3.3461E-02 -L.0603E-01 -:.0cv3E-vt -i.272?E-01
-2.2040E-01 -2.47elE-01 -3.sOS3EH)i -3.a083c-0i -7.0970E-0! -7.0970E—)1
-7. ?225E-01 -1.3002E-00 -i.3002rX>O -i.cOOOE-OO -i.?273E-H>0 -1.P273E-00
IIWSI.NARY PART
O.OOOOE-00 O.OGOOE-HJO 5.ai4S£-09 -5.al4fiE-39 Q.3000&MJ0 1.4CSdE-02
-i.406iE-v2 0.0000=^)0 Q.QOOOS-HX) 3.3084SHJI -3.3vS4E-)i O.OOOOE^OO
0.0OOOE-00 O.OOOOE-rOO B.0017E-01 -a.OOlTE—>I 1.323SE-00 -L323SE-00
tOOE+OQ -1.3200E-HJ0 0.QCOQE+O0 i.'?220E-00 -t.?22OE~0O
OPEN LflO? ZEROS
(i : 24)
9.2849E+02 9.224PE-02 I.2343E*02 1.2343E-02 i.3i37E*)C L.5099E+O0
•i.?0O7E-32 -i.?IaiE-$2 -2.04O4E-02 -2.0404E-O2 -5.7I43E-02 -l.MME-M-1.M19EHU -1.2725E-01 -2.0323E-O1 -2.2075E-01 -1.2357E-HJ2 -i.2357E*<S
•i.44a3Er03 -i.*663c*03 -2.-0371Ei03 -4.?i9SE*03 -4.?!<?SE-03 -7.2S51E+Q4
ItlASI.NASY PART
1.7100E-03 -i.7I00E*03 1.2349E-02 -1.2349E+02 7.6mE*M -T.ii?lE-04
O.OOOOE-KW 0.3000S-00 1.4944E-02 -1.4044EH52 0.0O00E-KJ0 3. 74?0£-^H
-3.7490E-01 0.9000E-00 O.OOOOErOO O.OOOGE^OO 1.234PE-02 -i.234=E-02
2.3463S-H)3 -2.34a3E*03 O.OOOOE-00 i.a495£«)4 -l.a4?5E*04 0.0OOCE*)O
156
APPENDIX B4
POLES AKD ZEROS OF THE OPEN A KD CLOSED LOOP SYSTEMS
pi ncrr, t rmo s™ -5
(1 : 24)
REAL PART
L.S917E-02 -J.9007E-Q2 -2.Q299E-02 -2.0299E-02 -4.2237E-02 -4.2237E-02
-B.5722-02 -5.7075E-02 -9.3104E-02 -i.07A0E-0i -1.0740E-01 -I.2729E-01
-2.2005E-01 -2.e290E-01 -3.542SE-01 -3.342SE-01 -7.07I4E-Q1 -7.0714E-01
-7.1471E-0! -1.2584E+00 -1.2584E+0O -1.4I26E+0C -1.S944E+00 -1.B944E+00
0.0000E+O0
0.000GE+00
O.OOOOE^OO
O.OOOOE---00
Q.0000S+00
O.OOOOE+00
O.OOOOE+00
1.2795E+C0
IHAelMABY PART
I.4076E-02 -1.407&E-02 3.2491E-02 -3.2491E-02
O.OOOOE+00 3.62C2E-01 -3.8202-01 0.0000E+00
5.5033E-01_c r
033E-01 1.2751E+00 -1.2751E-C0
1.2795E+00 O.OOOOE+00 00 -1.BS98E+00
CLEEED LAPP ZEROS
(1 s 24)
REAL PART
1.9711E+03 9.7852E+01 9.7852E+01 -1.9007E-02 -1.9161E-02 -2.0403E-02
-2.0403E-02 -5.7143E-02 -l.MME-01 -1.0619E-01 -1.2729E-01 -2.0323E-01
-2.2C75E-01 -3.1904E+01 -3.1904E+01 -9.9733E+01 -9.9733E+01 -1.JC79E+02
-1.1079E+02 -1.799EE+03 -1.799SE+03 -1.2192+04 -1. 21921+04 -B.623SE+C4
MASHtftgY PART
-9.9187E+C2 9.8791E+01 -9.B791E+01 O.OOOOE+00 O.OOOOE+00 1.4O43E-02
-1.4O43E-02 0.0OO0E+00 3.7490E-01 -3.7490E-01 O.OOOOE+00 0.000OE+O0
O.OOOOE+00 2.1934£t03 -2.1984E+03 9.S791E+01 -9.6791E+01 2.7447E+04
-2.7447E+04 1.1325E+03 -1.1325E+03 2.3498E+04 -2.3498E+04 O.OOOOE+00
157
APPENDIX C
MATRICES OF THE 30 KNOT PLANT AND COMPENSATOR
158
APPENDIX CI
S
T
A7Z SPACE MATRICES OF THE S30R1 LINEAR MODEL
ORIGINAL MATRICES PRIOR TO SCALIN6
A BATRII
-3.3245E-02 -2.1911E-02 -2.7720E-03 -1.3964E-02 -2.9363E-01 3.1674E+00 0.0OOOE+OO 2.9326E-04 O.OOOOE+OO O.OOOOE+OO
1.1461E-03 -1.3919E-01 -1.933SE-03 -1.1464E+00 1.1276E-01 -1.5397E+01 1.3004E-01 -1.7564E-03 0.0000E+OO O.OOOOE+OO
2.4222E-05 4.4499E-04 -1.0631E-01 -1. 39846*00 1.2070E+01 3.0t94E-02 O.OOOOE+00 7.5597E-03 O.OOQOE+00 0.00OOE+OO
2.4614E-04 -I.1680E-02 -1.3226E-03 -4.344SE-01 -2.3879E-01 -7.1773E-03 -1.5995E-01 2.1603E-O3 O.OOOOE+OO O.OOOOE+00
-5.3732E-06 -1.3585E-05 1.3207E-O3 -1.1380E-02 -4.0755E-01 1.0074E-04 O.OOOOE+00 -2. 4934E-03 O.OOOOE+OO O.OOOOE+00
-2.7564E-03 -2.0277E-03 2.4063E-05 -8.1034E-03 3.4042E-03 -3.3180E-01 2.583AE-04 -3.4895E-06 O.OOOOE+00 O.OOOOE+00
O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO 1.0000E+00 1.3427E-02 -1.2348E-01 -2.0244E-10 -1.26A0E-02 O.OOOOE+OO O.OOOOE+00
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 9.9414E-01 1.08IOE-01 1.2467E-02 O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -1.0893E-01 I.0018E+00 1.4423E-09 1.5405E-03 O.OOOOE+00 O.OOOOE+00
1.2326E-01 -1.0728E-01 9.36S4E-01 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.5702E+00 -4.3493E+01 O.OOOOE+00 O.OOOOE+OO
8 BATRII
-1.4315E-03 -5.3396E-02 2.3022E-03 2.3022E-03
O.OOOOE+00 2.3119E+00 -1.4950E-01 1.4950E-01
-1.4442E+00 -1.4813E-0A -9.3474E-01 -9.3476E-01
O.OOOOE+OO 4.2586E-02 2.0848E-01 -2.0848E-01
1.3872E-02 4.8862E-07 -2.382SE-02 -2.3823E-02
O.OOOOE+OO -5.3593E-02 -3.3476E-04 3.2676E-04
O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 O.OOOOE+00
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00
O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 O.OOOOE+00
159
APPENDIX CI
STATE SPACE MATRICES OF THE S2QR1 LINEAR MODEL
REDUCED AND SCALED PLANT MATRICES WITH APPROPRIATE C MATRIX
A MATRIX
•3.3269E-02 -2.1964E-02 -2.7533E-03 -3.3173E-04 2.0734E-03 5.5394E-02 O.OOOOE+00 5. 12S5E-0&
1.1417E-03 -1.5939E-01 -3.3786E-05 -2.3578E-02 2. S353E-03 -2.6860E-0L 2.2745E-03 -2.5914E-05
-4.7476E-04 1.3910E-03 -9.4526E-02 -2.7949E-02 2.1163E-01 7.&140E-04 O.OOOOE+00 1.3221E-04
1.3945E-02 -6.6430E-01 -8.0931E-02 -4.3452E-01 -2.S262E-01 -2.1920E-O2 -1.6030E-01 1.3264E-03
7.1418E-05 -2.5929E-04 7.3U7E-02 -1.1406E-02 -4.0815E-01 -7.7327E-04 O.OOOOE+00 -2.4985E-03
-1.5782E-03 -1.1622E-01 3.4035E-04 -8.00IIE-03 2.2809E-03 -3.3201EH31 2.5893E-04 -2.9501E-06
O.OOOOE+00 O.OOOOE+OO O.OOOOE+00 1.0000E+OO 1.1328EH32 -1.0538E-OI -4.93S2E-10 -1.2635E-02
0.0O00E+0O O.OOOOE+00 O.OOOOE+00 O.OOOOE+OO 9.9427E-01 1.0689E-01 1.2494E-02 O.OOOOE+OO
3 HATRII
-I.2666E-03 -1.3279E-03 9.3625E-05 9.3625E-05
O.OOOOE+OO &.0491E-02 -3.6976E-03 3.6976E-03
-2.5204E-02 -3.3763E-08 -2.1483E-02 -2.1483E-02
O.OOOOE+OO 6.3847E-02 2.6060E-01 -2.6060E-01
1.3873E-02 7.3256E-07 -2.9781E-02 -2.9781E-02
O.OOOOE+00 -3.784AE-02 -4.2094E-04 4.2094E-04
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+00
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
160
APPENDIX CI
SPACE MATRICES 07 THE S30R1
C HATCH
O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE-Ol O.OOOOE+00
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 l.OOOOE-Ol
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -1.0749E-01 9.9984E-01 4.4827E-09 1.3316E-03
1.0539E-01 -1.0629E-01 9.3873E-01 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 2.7U9E-02 -8.4843E-01
161
APPENDIX C2
MATRICES 07 THE 30 KN'OT CC.M.PSSSAT33
FILTH SAIN MATRIX
-1.02B2E+00 -1.3198E+01 -3.1019E-01 -2.1680E+00
1.I923E-01 7.9327E-02 -7.5141E-01 2.3994E-02
1.9031E+00 1.76ME+Q0 -3.6529E-01 1.5857E-01
-1.4904E+00 4.2619E-01 1.2014E+00 1.141SE-01
9.3005E-03 4.4870E-01 9.2493E-HJ2 5.4981E-42
6.3777E-03 -3.4783E-0I -2.43546-01 -3.3983E-42
2.4814E-02 4.1892E+O0 2.1197E-01 4.9849Eh)1
LS704E-01 2.4765E-01 -3.5220EHJI 4.9200E-02
-3.0473E-02 -4.2343E-03 -3.1907EHJ2 3.5337E-44
-4.2073E-02 2.1331E-02 3.3510EHJI -4.2349E-03
5.343IE+00 2.1792E-01 -3.3301E-01 -1.5210E-01
2.1792E-01 4.9887E+00 2.3426E-01 -4.34I4Eh)3
CONTROL SAIN MATRIX
1.3487E+00 L4722EHH -I.8437E-03 3.2330E-03 -2.3935E+00 2.1865E+00
-2.3081E+01 4.6679E-02 2.3366E+0I 2.0238E+00 -2.3789E-01 2.5951E+01
1.4722E-03 2.0983E+00 -1.1733E-02 -4.2&71E-02 -4.7100E-02 2.5212E+00
•7.4947E-01 1.90S3E-01 L7016E+00 -2.3I96E+01 3.3982E-02 3.4430E-01
1.3437E-03 -1.I733E-02 1.3132E+00 -3.5292E-01 -7.9515E-01 -1.3192E+00
•3.6155E+00 2.2982E+00 -1.9369E+00 1.4596E+00 1.4934E+00 4.3954E+00
3.2330E-03 -4.2471E-02 -3.5292E-01 1.2835E+00 -9.4105E-01 2.7612E+00
-3.7112E+00 -2.5731E+00 -5.3521E-01 6.3046E-01 -1.9771E+00 5.4309E+00
162
APPENDIX C2
MATRICES OF THE 30 KNOT COMPENSATOR
L HATRIX
-a.0839E-Ol -3.M79E+01 5.i433E-0l -4.3555E+00
1.5610E-02 B.7114E-02 -1.5284E+00 &. 4870E-03
3.3627E+00 3.5527E+00 -2. 4355E+00 4.4871E-01
-2.6540E+00 8.S502E-01 2.7545E+00 1.5325E-01
-2.3532E-02 8.9419E-01 S.1547E-14 1.0S39EHH
2.8826E-02 -9.0182E-01 -8.2242E-14 -1.0629E-01
-2.6814E-01 8.3888E+00 7.4502E-13 9.3S73E-01
O.OOOOEtOO O.OOOOE^OO O.OOOOE+00 O.OOOOE+00
4.9774E-09 1.4154E-03 -1.0629E-01 -4.4369E-13
-4.6300E-08 -1.3166EHJ2 9.3873E-01 4.1272E-12
i.OOOOE+Ol -1.4691E-09 -3.3444E-17 1.2488E-11
-1.3286E-10 l.OOOOE+Ol 6.9211E-12 -5.5841E-09
OPEN LOOP EIGENVALUES
9.5843E-09 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 -1.4176E-02 -3.3412E-02
•4.4084E-01 -4.4084E-OI -4.5U4E-01 -5.0484EhJI -5.3965E-01 -3.8965E-01
-4.0661E-02 -4.2B86E-02 -7.1364E-02 -1.9689E-01 -1.96B9E-01 -2.5114E-01
9.5148E-01 -9.S148E-01 -1.01O4E+OO -1.3887E+00 -1.3887E+00 -1.4462E+00
O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00
4.7901E-01 -4.7901E-01 O.OOOOE+00 O.OOOOE+00 1.1366E+00 -i.i34aE+00
O.OOOOE+OO O.OOOOE+00 O.OOOOE+OO 3.1300E-01 -3.1300E-01 O.OOOOE+OO
1.0148E+00 -1.0148E+00 O.OOOOE+00 1.3186E+00 -1.31S6E+00 O.OOOOE+00
163
APPENDIX C2
MATRICES OF THE 20 KNOT COMPENSATOR
OPEN LOOP TRANSMISSION ZEROS
1.0000E+30 1.0000E+30 1.0000E+30 1.0000E+30 1.1625E+08 1.0192E+08
•3.9767E-02 -3.9767E-02 -1.9869E-01 -2.0469E-01 -2.0469E-01 -2.5097E-01
1.2816E+05 1.4260E+04 1.2133E+00 1.2202E+00 -1.425AE-02 -3.8414E-02
2.3357E-01 -4.5944E-01 -1.4263E+04 -1.2816E+05 -8.4484E+07 -5.303BE+09
O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00 O.OOOOE+00
1.6S40E-03 -1.M40E-03 O.OOOOE+00 2.3343E-01 -2.3343E-01 O.OOOOE+00
O.OOOOE+00 O.OOOOE+00 1.0851E+04 -1.0851E+04 O.OOOOE+00 O.OOOOE+00
O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00
. CLOSED LOOP EIGENVALUES
-1.4220E-02 -3.3414E-02 -3.8479E-02 -4.1399E-02 -1.0317EHJI -1.0317E-01
5.1064E-01 -5.10ME-01 -5.15UE-01 -5.2355E-01 -5.2355E-01 -7.1513E-01
-2.3970E-01 -2.3970E-01 -2.5106E-01 -2.7292E-01 -2.7292E-01 -4.1228E-01
-7.4533E-01 -7.5232E-01 -7.5232E-01 -1.0388E+00 -1.1957E+00 -1.1957E+00
O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 1.0A59E-01 -1.0659E-01
7.04OIE-O2 -7.0401E-02 O.OOOOE+OO 9.2146E-01 -9.2144E-01 O.OOOOE+00
3.3649E-01 -3.3449E-01 O.OOOOE+OO 2.7737E-01 -2.7737E-01 O.OOOOE+OO
O.OOOOE+00 8.5953E-01 -3.5953E-01 O.OOOOE+00 1.1439E+00 -1.1639E+00
CLOSED LOOP TRANSHISSION ZEROS
7.5203E+10 I.1350E+04 8.i830E+03 6.7422E+03 2.5300E+02 2.5300E+02
-3.9768E-02 -3.9768E-02 -1.98&9E-01 -2.0469E-01 -2.0469E-01 -2.5097E-01
L5900E+00 U4499E+00 2.3446E-01 7.5507E-0I -1.4253E-02 -3.84I4E-02
2.3357E-01 -4.5944E-01 -5.0698E+02 -6.7412E+03 -8.4823E+03 -1.1349E+04
O.OOOOE+OO O.OOOOE+OO O.OOOOE+00 O.OOOOE+00 4.3878E+02 -4.3878E+02
1.A527E-03 -1.M27E-03 O.OOOOE+00 2.3343E-01 -2.3343E-01 O.OOOOE+00
2.4470E+05 -2.4470E+05 1.5450E+04 -1.5450E+04 O.OOOOE+OO O.OOOOE+00
O.OOOOE+00 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
164
! ?
224690
Mette
,Multivariahltro1 of a «\
le c°n-
method. LQR/lTR
*$ «*r s?Q 03 9s
Thesis
M5622
c.l
214690
Mette
Multivariable con-
trol of a submarineusing the LQR/LTRmethod
.