REAL CONTROLLERS USING NUMERICAL
LARGE FLEXIBLE SPACE STRUCTURES
James Richard Forbes V
A Thesis Submitted In Conformity With the Requirements For the Degree of Masters of Applied Science
Graduate Department of Aerospace Science and Engineering
University of Toronto
Numerical Optimization for the Control of Large Flexible Space
Structures
Masters of Applied Science Graduate Department of Aerospace Science and Engineering
University of Toronto
Abstract
The design of optimal strictly positive real (SPR) compensators using numerical opti-
mization is considered. The plants to be controlled are linear and nonlinear flexible ma-
nipulators. For the design of SISO and MIMO linear SPR controllers, the optimization
objective function is defined by reformulating the H2-optimal control problem subject
to the constraint that the controllers must be SPR. Various controller parameterizations
using transfer functions/matrices and state-space equations are considered. Depending
on the controller form, constraints are enforced (i) using simple inequalities guaranteeing
SPRness, (ii) in the frequency domain, or (iii) by implementing the Kalman-Yakubovich-
Popov lemma. The design of a gain-scheduled SPR controller using numerical optimiza-
tion is also considered. Using a family of linear SPR controllers, the time dependent
scheduling signals are parameterized, and the objective function of the optimizer seeks
to find the form of the scheduling signals which minimizes the manipulator tip tracking
error while minimizing the control effort.
ii
Acknowledgements
I would like to thank my parents James Wilfred and Ruth Evelyn Forbes for the
enormous amount of love and support they have given me. My father is by far the most
humble, kind and honest man I have ever known, a constant inspiration to me in all
aspects of my life. My mother is by far the most supportive woman I have ever known,
sacrificing more than ever expected for her children. Both my Mom and Dad have always
helped and encouraged me to pursue whatever life endeavor I chose, I am truly blessed
to have such wonderful parents.
My supervisor, Dr. Christopher Damaren is also a constant inspiration to me. His
brilliance is sometimes daunting, but at the same time his confidence in me, his patience,
his encouragement and our mutual excitement for our work has given me the courage to
continue without fear of failure. He is the best of the best, and I am extremely grateful
of his support.
Christopher Andre Beltempo has been more than a great friend while at UTIAS. Our
discussion and debates about engineering, politics, religion and life are wonderful. He too
has been very supportive and encouraging, challenging me to think and consider many
things I would most likely dismiss as insignificant or wrong. Again, I am blessed to have
such a great friend.
Lastly, I would like to thank society in general. I, by the luck of the draw, have been
born into a society filled with wealth and freedom. Every day I thank God that I have
been given this amazing opportunity to pursue my dream, unconstrained and without
prosecution. I am by no means the most deserving of this opportunity, but I do feel
obligated to do the best I possibly can. One day, I hope to give back to society what
society has so graciously given me.
James Richard Forbes V December 2007
iii
Contents
List of Figures vi
List of Tables viii
1 Introduction 1 1.1 Topics of Study In This Thesis . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Passivity-Based Control . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Multibody Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Thesis Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Passivity 6 2.1 Mathematical Background Pertaining to Passivity Concepts . . . . . . . 7
2.1.1 Lp-Norms and Lp-Spaces . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 System Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 L2-Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Passivity Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Passivity Associated with Linear Time-Invariant Systems . . . . . . . . . 11
2.3.1 Positive Real and Strictly Positive Real Transfer Functions . . . . 11 2.3.2 Positive Real and Strictly Positive Real Transfer Matrices . . . . 13 2.3.3 The Kalman-Yakubovich-Popov Lemma . . . . . . . . . . . . . . 14
2.4 The Passivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Flexible Multibody Dynamics 17 3.1 Dynamics of a Single-Link Flexible Manipulator . . . . . . . . . . . . . . 17
3.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 Kinetic and Potential Energy . . . . . . . . . . . . . . . . . . . . 19 3.1.3 Equations of Motion of a Single-Link Flexible Manipulator . . . . 20
3.2 Dynamics of a Two-Link Flexible Manipulator . . . . . . . . . . . . . . . 21 3.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.2 Kinetic and Potential Energy . . . . . . . . . . . . . . . . . . . . 26
iv
3.2.3 Equations of Motion of a Two-Link Flexible Manipulator . . . . . 28
4 Control of Flexible Multibody Systems 32 4.1 µ-Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1 Input-Output Mapping . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1.2 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.3 Motivation for Using the µ-Tip Rate . . . . . . . . . . . . . . . . 35
4.2 H2-Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2.1 General Control Formulation . . . . . . . . . . . . . . . . . . . . . 36 4.2.2 Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2.3 Optimal Regulator Problem . . . . . . . . . . . . . . . . . . . . . 39 4.2.4 Optimal Estimation Problem . . . . . . . . . . . . . . . . . . . . 39 4.2.5 The H2-Optimal Controller . . . . . . . . . . . . . . . . . . . . . 40
4.3 Passivity-Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3.1 Passivity of the Plant . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3.2 Passivity-Based Control of Flexible Manipulators . . . . . . . . . 41
4.4 Feedforward and Feedback Control . . . . . . . . . . . . . . . . . . . . . 43 4.4.1 Desired Tip Position and Velocity . . . . . . . . . . . . . . . . . . 43 4.4.2 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4.3 Closed-Loop Dynamics Utilizing Feedfoward and Feedback Control 44
5 Optimization Formulation 46 5.1 Gradient-Based Optimization Overview . . . . . . . . . . . . . . . . . . . 46 5.2 General Controller Optimization Formulation . . . . . . . . . . . . . . . 47
6 Optimization of SISO SPR Controllers 50 6.1 Third-Order Controller with Deterministic Inequality Constraints . . . . 51
6.1.1 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Variable-Order Controller with Frequency Domain Constraints . . . . . . 56
6.2.1 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.3 Controller Optimization Using State-Space Parameterization . . . . . . . 63
6.3.1 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.4 SISO Optimization Discussion . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Optimization of MIMO SPR Controllers 71 7.1 Decoupled-Diagonal Controller Optimization With Deterministic Constraints 72
7.1.1 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.2 Optimization of Composite Controller Utilizing Deterministic and Fre-
quency Domain Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.1 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.3 Full-State Controller Optimization Using State-Space Parameterization . 80 7.3.1 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.4 MIMO Optimization Discussion . . . . . . . . . . . . . . . . . . . . . . . 83
v
8 Optimal Gain Scheduling 84 8.1 Passivity Properties of Gain-Scheduled Very Strictly Passive Compensators 85 8.2 The Gain-Scheduling Optimization Problem . . . . . . . . . . . . . . . . 89
8.2.1 Optimal Scheduling Signals and The Design Variables . . . . . . . 90 8.2.2 The Gain-Scheduling Controller Optimization Objective Function 92 8.2.3 The Optimization Formulation . . . . . . . . . . . . . . . . . . . . 93
8.3 Gain-Scheduling Optimization Results . . . . . . . . . . . . . . . . . . . 93 8.4 Gain-Scheduled Control Optimization Discussion . . . . . . . . . . . . . 95
9 Conclusions 97 9.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
Bibliography 101
Appendices 102
vi
1.1 Steam Engine Governer Designed by James Watt, 1788 . . . . . . . . . . 1
2.1 System Response of Second-Order System . . . . . . . . . . . . . . . . . 8 2.2 y2 Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 A Negative Feedback Interconnection of Two Systems . . . . . . . . . . . 15
3.1 Single-Link Flexible Manipulator . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Two-Link Flexible Manipulator . . . . . . . . . . . . . . . . . . . . . . . 22
4.1 H2-Optimal Control Block Diagram . . . . . . . . . . . . . . . . . . . . . 37 4.2 Passivity-Based Control Block Diagram . . . . . . . . . . . . . . . . . . . 42
6.1 Nyquist Plot of SPR Controller Optimized with Deterministic Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 Bode Diagram of SPR Controller Optimized with Deterministic Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3 System Response Using Deterministic SPR Controller; [ρ(0)µ=1 ρ(0)µ=1] T
= [ π 4
1 ]T
(m,m/s), set point = [0 0]T (m,m/s) . . . . . . . . . . . . . . . 54 6.4 System Response With Increased Mass Using Deterministic SPR Con-
troller; [ρ(0)µ=1 ρ(0)µ=1] T =
[ π 4
1 ]T
(m,m/s), set point = [0 0]T (m,m/s) . 55 6.5 Design Space Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.6 Nyquist Plot of Iterative SPR Controller Optimized with Frequency Do-
main Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.7 Bode Diagram of Iterative SPR Controller Optimized with Frequency Do-
main Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.8 System Response Using Iterative SPR Controller; [ρ(0)µ=1 ρ(0)µ=1]
T =[ π 4
1 ]T
(m,m/s), set point = [0 0]T (m,m/s) . . . . . . . . . . . . . . . . . 60 6.9 Bode Diagrams of Iterative SPR Controllers n = 1 through n = 5 . . . . 61 6.10 System Responses Using Iterative SPR Controllers; [ρ(0)µ=1 ρ(0)µ=1]
T =[ π 4
1 ]T
(m,m/s), set point = [0 0]T (m,m/s) . . . . . . . . . . . . . . . . . 62 6.11 Nyquist Plot of State-Space SISO SPR Controller . . . . . . . . . . . . . 65 6.12 Bode Diagram of State-Space SISO SPR Controller . . . . . . . . . . . . 65 6.13 System Response Using State-Space SISO SPR Controller; [ρ(0)µ=1 ρ(0)µ=1]
T
= [ π 4
1 ]T
(m,m/s), set point = [0 0]T (m,m/s) . . . . . . . . . . . . . . . 66 6.14 Bode Diagram of State-Space SISO SPR Controller and H2-Optimal Con-
troller Designed With Plant Modeled With 2 Basis Functions . . . . . . . 67
vii
6.15 Bode Diagram of State-Space SISO SPR Controller Designed With Plant Modeled With 6 Basis Functions . . . . . . . . . . . . . . . . . . . . . . . 68
6.16 System Response Using State-Space SISO SPR Controller Designed With
Plant Modeled With 6 Basis Functions; [ρ(0)µ=1 ρ(0)µ=1] T =
[ π 4
1 ]T
(m,m/s), set point = [0 0]T (m,m/s) . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.1 Nyquist Plot and Bode Diagram of Optimized SPR Transfer Function G11(s) 73 7.2 Nyquist Plot and Bode Diagram of Optimized SPR Transfer Function G22(s) 73 7.3 Two-Link Flexible Manipulator System Response as Controlled via De-
coupled SPR Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.4 Nyquist Plot and Bode Diagram of Optimized SPR Controller G11(s) . . 77 7.5 Two-Link Flexible Manipulator System Response as Controlled via Com-
bined Third-Order and Iterative SPR Controller . . . . . . . . . . . . . . 78 7.6 Two-Link Flexible Manipulator System Response as Controlled via Com-
bined Third-Order and Iterative SPR Controller, Link Thickness Reduced to 2.5 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.7 Nyquist Plot and Bode Diagram of Optimized State-Space SPR Controller G11(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.8 Two-Link Flexible Manipulator System Response as Controlled via SPR Controller Parametrized Using State-Space Techniques . . . . . . . . . . 82
8.1 Gain-Scheduled Feedback Control System . . . . . . . . . . . . . . . . . 85 8.2 Linearization Points to Design Three Optimal SPR Controllers . . . . . . 89 8.3 Optimal Scheduling Signals . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.4 Optimal Gain-Scheduling Control System Response . . . . . . . . . . . . 95
A.1 A Negative Feedback Interconnection of Two Systems . . . . . . . . . . . 103 A.2 A Negative Feedback Interconnection of an Inductor and an Inductor-
Resistor Parallel Connection . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.3 Inductor Circuit and Inductor-Resistor Parallel Connection Circuit . . . 109 A.4 A Negative Feedback Interconnection of an Inductor and a Series Inductor-
Resistor Connection in Parallel With Another Resistor . . . . . . . . . . 112 A.5 Inductor and Inductor-Resistor Series Connection in Parallel With An-
other Resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.1 Single-Link Manipulator Physical Properties . . . . . . . . . . . . . . . . 50 6.2 Iterative SPR Controller Results . . . . . . . . . . . . . . . . . . . . . . . 59
7.1 Two-Link Manipulator Physical Properties . . . . . . . . . . . . . . . . . 71
ix
Introduction
Since the dawn of time, feedback has played a pivotal role in the output of dynamic
systems. Charles Darwin theorized that feedback over long periods of time contributes
to the evolution of species. The human body maintains a very specific temperature via a
biological feedback mechanism. More recently, ecologists have become more interested in
the feedback characteristics of CO2 in the atmosphere and its effect on global warming.
Feedback and in particular feedback control is all around us.
During the industrial revolution in Europe, man-made feedback control devices be-
came an integral part of industrial machinery. What is considered the first closed-loop
feedback control device is the governor implemented by Scottish Engineer James Watt,
shown in Figure 1.1. James Watt designed his first governor in 1788 to control the output
Figure 1.1: Steam Engine Governer Designed by James Watt, 1788
speed (and thus power) of a steam engine. As the speed of the output shaft increased, the
governor’s central spindle would also increase in speed. The governor’s rotating weights
1
would spin faster causing them to move outward and up. This motion, translated via
a series of connecting rods, would close the intake valve, regulating steam sent to the
engine. As a result, the output shaft speed would decrease. This type of feedback control
is what is now known as proportional feedback control.
The first governors implemented were not well understood from a theoretical stand
point; they just worked. Not until James C. Maxwell published his ever famous paper
On Governors in 1868 [1], did academics become intrigued with the mathematics and
physics behind their governing characteristics. What would follow would be a flood of
intellectual curiosity and development in the field that is now known as control theory.
In general feedback control systems are designed to regulate, or stabilize a dynamic
system. The implementation of feedback control can be found everywhere; in resistor-
inductor-capacitor (RLC) circuits, aircraft, chemical process, water treatment plants etc.
One of the major foci of modern control theory is determining stability in the presence of
disturbances, sensor noise or modeling errors. The famous Kalman Filter was invented
with the primary objective of acquiring an accurate estimate of a systems states from a
series of incomplete and noisy measurements, while ensuring the system remain stable.
1.1 Topics of Study In This Thesis
In the aerospace community, the successful flight of aircraft and satellites, or operation
of planetary rovers and robotic manipulators has hinged on control design. For example,
the unconventional NASA/Grumman X-29 aircraft, featuring forward swept wings, is
naturally unstable (unlike conventional aircraft). The X-29 would never fly without an
active control system.
Although the major focus of this thesis lies in the field of control theory, it is also
an amalgamation of three very interesting, exciting and challenging engineering disci-
plines related to aerospace engineering: passivity-based control, multibody dynamics
and numerical optimization. We will be using what is known as passivity-based control
to control a very complicated multibody dynamical system, while employing numerical
optimization techniques to find ‘the best controller’.
1.1.1 Passivity-Based Control
Before the digital age, the most common electrical engineering building blocks other than
copper wire and solder were resistors, capacitors and inductors. These circuit elements
are passive devices, that is, they do not generate energy. Circuits composed of only
2
passive elements may conserve or dissipate energy (i.e., Ein ≥ Eout) when measured from
the same point in the circuit (such as an input port). Circuits that conserve energy are
known as passive, and those that dissipate energy are known as strictly passive.
Before the introduction of the operational amplifier (the Op-Amp) and the digital
computer, feedback controllers where usually implemented in the form of simple RLC
circuits. As a result, feedback controllers in the form of RLC circuits have prompted
significant study in the academic community; a plethora of mature knowledge is available
in the literature. One of the most famous and influential feedback control laws developed
while studying passivity is the passivity theorem.
Although we will discuss at length the passivity theorem and the different classes of
passive systems, the passivity theorem guarantees the stability of the negative feedback
interconnection of two systems, one of which is passive, while the other is very strictly pas-
sive. It will be shown that very strictly passive systems, both single-input single-output
(SISO) and multi-input multi-output (MIMO), that are linear time-invariant (LTI) take
the form of strictly positive real (SPR) transfer functions and transfer matrices. It will be
these SPR transfer functions/matrices that we will focus on implementing as controllers.
1.1.2 Multibody Dynamics
The control of multibody systems is an interesting challenge for controls engineers. An
interesting class of multibody systems are flexible space structures. Large structures in
space are very light, somewhat flexible, have very little damping and must have active
feedback control to suppress structural vibrations. In the case of large flexible robot
manipulators such as the Canadarm and Canadarm2, their dynamics are configuration
dependent and thus nonlinear. The structural vibrations of the manipulator need to be
mitigated and controlled, as does the nonlinear motion of the end effector as it moves
through space following a particular trajectory. The controller must be able to control the
vibrations and nonlinear dynamics, while guaranteeing robust stability; robust stability
meaning stability will be guaranteed in the presence of disturbances and unmodeled
dynamics (such as unmodeled vibration modes).
1.1.3 Numerical Optimization
As first world nations have entered into the digital age, the computational horsepower
of the average desktop PC has increased exponentially as predicted by Moore’s Law.
Structural finite element analysis and computational fluid dynamics simulations that
would take days on an old 1970’s super computer now take minutes, even seconds, on a
3
desktop PC.
Not only has computational throughput increased but also the knowledge and ex-
pertise in the area of numerical optimization. Most engineering students first encounter
optimization while studying calculus; finding the minimum of a function by taking the
first derivative and setting the result to zero. The idea of ‘minimization of a function’ is
easily extended to N -dimensional problems, and can be used to find the optimal solution
of very complicated engineering problems. However, engineering problems are in gen-
eral huge and analytical solutions are infeasible, thus they are solved numerically using
computational methods.
Today, engineers can use computational tools (FEA, CFD etc.) to run simulations and
predict physical outcomes, but they can also employ numerical optimization techniques
to vary design parameters until a desired outcome is reached. Numerical optimization
has been a welcomed addition to the engineering community. The idea of having a set of
design variables coupled to an objective function, and finding the ‘best solution’ (i.e., a
minimum) via some optimization procedure is very powerful and being implemented in
all fields of engineering.
1.2 Thesis Purpose
The purpose of this thesis will be to investigate the optimal design of SPR transfer
functions and matrices to control flexible robot manipulators. The passivity theorem will
be the driving force behind the stability of the closed-loop systems we are controlling,
and numerical optimization will be the method by which we find optimal SPR controllers.
1.3 Thesis Overview
This thesis is very broad; topics related to passivity-based control, multibody dynamics
and numerical optimization will be discussed in detail.
Chapter 2 will define and discuss the various classes of passive systems, and the mon-
umental passivity theorem. First, we will present the required mathematical background
necessary to understand passivity concepts, followed by the specific definitions of passive
systems, very strictly passive systems, SPR transfer functions and matrices, and finally
the passivity theorem.
Chapter 3 will focus on the development of the plants we wish to control. The
dynamic equations of motion of a single-link flexible manipulator and a two-link flexible
manipulator will be developed.
4
The control of flexible systems is by no means simple, and in Chapter 4 we will discuss
the required plant modifications and different methods of control design available. H2-
Optimal control, passivity-based control and feedforward coupled with feedback control
will all be discussed with respect to the control of flexible multibody systems.
In Chapter 5, the means by which we will define our controller optimization problem
will be presented.
In Chapter 6 we will first investigate the optimization of SISO SPR controllers to
control the single-link flexible manipulator. Three different controller parameterizations
will be formulated, and the optimization results for each controller will be presented.
In Chapter 7 the various ways we will parameterize the MIMO SPR controllers to
control the two-link flexible manipulator will be presented, along with the optimization
results.
In Chapter 8 a new way of defining the optimization problem will be investigated.
Rather than designing and optimizing single SPR controllers, we will investigate the
optimization of a nonlinear SPR controller. The nonlinear SPR controller will utilize a
gain-scheduling algorithm coupled with a family of linear SPR controllers. A stability
proof of the gain-scheduled SPR controller presented, and the types of scheduling signals
possible will be discussed.
Finally, in Chapter 9 the conclusions of this thesis will be discussed, along with future
work.
5
Passivity
The foundation of this thesis is rooted in the well-developed passivity theorem. Originally
the passivity theorem was developed with regard to the stability of RLC circuits; how-
ever, it’s power is being exploited in other branches of control engineering. Briefly, the
passivity theorem states that a passive system when connected via a negative feedback
interconnection to a strictly passive system that has finite gain will be stable. 1
In order to be able to use the passivity theorem we need to understand what a
passive system and a strictly passive system with finite gain is. Additionally, because
linear time-invariant systems and their representation as transfer functions and transfer
matrices are so common, we naturally would like to know when transfer functions and
transfer matrices are passive or strictly passive. It will be shown that a positive real
transfer function corresponds to a passive system, and a strictly positive real transfer
function plus a small constant δ corresponds to a strictly passive system with finite gain.
1This definition of the passivity theorem is not rigorous and is stated for introduction purposes only. For a rigorous definition, see Section 2.4.
6
ity Concepts
u2 =
uTu dt (2.1)
The space of functions that have a finite L2-norm is the L2-space:
L2 = {u(t) | u2 <∞}
When a signal is truncated it is zero after time T :
uT (t) =
{ u(t) , t ≤ T
0 , t > T
Signals whose truncations belong to L2 are part of the extended L2-space:
L2e = {u(t) | uT (t) ∈ L2, 0 < T <∞}
The truncated L2-norm is simply the L2T -norm:
u2T =
uTu dt (2.2)
Note that the square of the L2T -norm as defined in Eq. (2.2) is:
u22T =
The L∞-Norm and L∞-Space
Assume that u(t) ∈ L∞, ∀t ≥ 0; the L∞-norm of any vector signal u ∈ L∞ is:
u∞ = max {sup|u1(t)|, sup|u2(t)|, · · · , sup|un(t)|} (2.3)
7
The space of functions that have a finite L∞-norm is the L∞-space:
L∞ = {u(t) | u∞ <∞}
Lp-Norm and Lp-Space Example
To clarify the difference between the L2-norm, L2-space and the L∞-norm, L∞-space
consider the following second-order SISO system:
y + 1 2 y + y = u ;
y(s)
u(s) =
1
s2 + 1 2 s+ 1
The system response to an impulse function is given in Figure 2.1. The maximum value
0 5 10 15 −0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Figure 2.1: System Response of Second-Order System
of y during the response time t ∈ [0, 15] is y = 0.7115. Clearly y(t) ∈ L∞, ∀t ≥ 0, where
y∞ = 0.7115. Any vector set of signals that are bounded by some upper bound are in
the L∞-space.
y2T =
y2 dt .
To better understand the L2T -norm of a signal (and by association the L2-norm), consider
the square of the output y in Figure 2.2. The L2T -norm of the system is simply the
8
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
A m
pl itu
te S
qu ar
Figure 2.2: y2 Response
square root of the area under the y2 curve between t = [0, 15]. For the above system,
y2T = 0.9997. Any signal (vector or otherwise) that has a finite L2T -norm is part of
the extended L2-space, L2e-space.
Norm Inequalities
The Cauchy - Schwarz inequality is a very useful inequality related to system norms [2]:∫ T
0
uTy dt
≤ u2T y2T ∀u ∈ L2e, ∀y ∈ L2e, 0 ≤ T ≤ ∞ (2.4)
The triangle inequality is another useful inequality:
u + v2 ≤ u2 + v2 ∀u ∈ L2, ∀v ∈ L2 (2.5)
2.1.2 System Operators
A system mapping or operator G : L2e → L2e maps a function u(t) into another function
y(t):
y = Gu
In controls engineering we are often interested in signals that are in the L2-space and are
mapped through a linear time-invariant system. For example, consider the differential
9
equation:
The solution to the above differential equation is:
x(t) = x0e −at +
e−a(t−τ)u(τ) dτ
If x0 = 0 and the output y(t) = x(t), the system operator is:
G : u(t)→ y(t) =
e−a(t−τ)u(τ) dτ
This is analogous to the convolution of the system input and the system operator, y(t) =
g(t) ∗ u(t). Using Laplace transforms this can be written as y(s) = g(s)u(s).
2.1.3 L2-Stability
Finite L2-gain refers to the maximum gain of the system at a particular frequency. The
L2-gain of a system is the ratio of the output L2-norm and the input L2-norm:
||G||∞ = sup 06=u∈L2
||Gu||2 ||u||2
= sup 06=u∈L2
||y||2 ||u||2
σmax(G(jω)) (2.6)
The L2-gain of a system is also known as the induced L2-norm or the H∞-norm. When
a system is said to have finite L2-gain, ||G||∞ <∞.
The system G is L2-stable if u ∈ L2 ⇒ y ∈ L2; that is to say: finite energy input
implies finite energy output [2]. Based on the results of Eq. (2.6) if ||G||∞ <∞ then
||Gu||2 ≤ ||G||∞||u||2
and the system G is L2-stable.
10
The following definitions apply to both linear and nonlinear systems.
Definition 2.2.1 (Passive System). A square system y = Gu, u ∈ Rm, y ∈ Rm and
G : L2e → L2e is passive if:∫ T
0
uTy dt ≥ β ∀u ∈ L2e, ∀T ≥ 0 (2.7)
Definition 2.2.2 (Very Strictly Passive System). A square system y = Gu, u ∈ Rm,
y ∈ Rm and G : L2e → L2e is very strictly passive (VSP) if there exists δ > 0 and ε > 0
such that∫ T
yTy dt ∀u ∈ L2e, ∀T ≥ 0 (2.8)
A system that is very strictly passive can also be called input-output strictly passive,
input strictly passive with finite gain, or simply strictly passive with finite gain.
In general, β ≤ 0 and corresponds to the initial energy of the system. If the initial
conditions of the system are relaxed such that the states of the system are all zero at
t = 0, then β = 0.
2.3 Passivity Associated with Linear Time-Invariant
Systems
If a system is linear time-invariant, the system operator H can be expressed as a transfer
function h(s), or as a transfer matrix H(s), corresponding to SISO and MIMO system,
respectively. Often, it is not convenient to determine if a system operator is passive or
very strictly passive based on evaluation of the passivity integrals given in Section 2.2.
This leads to the definition of positive real and strictly positive real transfer functions and
transfer matrices which are closely related to passive and very strictly passive systems.
2.3.1 Positive Real and Strictly Positive Real Transfer Func-
tions
The following definitions for positive real and strictly positive real transfer functions are
associated with a SISO LTI system operator H : L2e → L2e defined by y(t) = h(t) ∗u(t).
11
The operator H may be expressed as a rational transfer function, h(s) = p(s) q(s)
, of the
Definition 2.3.1 (Positive Real Transfer Function [3]). The rational transfer function
h(s) is positive real (PR) if:
1. h(s) is real for all real s (i.e., coefficients of p(s) and q(s) are all real numbers),
2. h(s) is analytic in Re[s] > 0 (i.e., the derivative of h(s) exists in the open right half
plane),
3. Re[h(s)] ≥ 0 for all Re[s] > 0 or equivalently:
(a) poles on the imaginary axis are simple, such that the associated residues are
non-negative definite,
(b) Re[h(jω)] ≥ 0 ∀ω ∈ [−∞,∞] such that jω 6= a pole of h(s).
A passive LTI SISO system will have a positive real transfer function h(s).
Definition 2.3.2 (Frequency Domain Conditions for Strictly Positive Real Transfer
Functions [3]). A rational transfer function h(s), that is strictly proper (i.e., relative
degree −1) is strictly positive real (SPR) if:
1. h(s) is real for all real s and h(s) is analytic Re[s] ≥ 0 (i.e., h(s) is Hurwitz),
2. Re[h(jω)] > 0 ∀ω ∈ (−∞,∞),
3. lim ω→∞
ω2Re[h(jω)] > 0
A strictly proper SPR transfer function on its own is NOT strictly passive. If the
transfer function h(s) is SPR then h(s) + δ (where δ is some small real number) corre-
sponds to a strictly passive system with finite gain.
The notion of weak SPR and strong SPR is sometimes overlooked. A weak SPR
transfer function is one that satisfies conditions 1 and 2 in Definition 2.3.2. A strong
SPR controller is one that satisfies conditions 1, 2 and 3 in Definition 2.3.2.
In the literature there exist SPR definitions for transfer functions that are of relative
degree 0 and +1. In controls applications we wish to implement a SPR transfer function
as a controller, and we are only concerned with transfer functions of relative degree −1.
These transfer functions have gain that rolls off at high frequencies, and therefore do not
amplify high frequency signal noise.
12
From the definitions of passive, PR, very strictly passive and modified SPR (i.e.,
hSPR(s) + δ) it can be inferred that the equivalent transfer function h(s) of the system
will have a phase angle that satisfies:
• −π 2 ≤ ∠ h(jω) ≤ π
• −π 2 < ∠ h(jω) < π
2 if the operator H is very strictly passive or modified SPR.
2.3.2 Positive Real and Strictly Positive Real Transfer Matrices
Definitions of positive real and strictly positive real transfer matrices for MIMO systems
naturally evolve from the SISO definitions. The following definitions are associated with
a MIMO LTI system operator H : L2e → L2e defined by y(t) = H(t)∗u(t). The operator
H may be expressed as a transfer matrix, H(s) ∈ Cm×m, that is composed of rational
elements, Hi,j(s) = pi,j(s)
i,jq(s) .
Definition 2.3.3 (Positive Real Transfer Matrix [4] [5]). The rational transfer matrix
H(s) is positive real (PR) if:
1. H(s) is real for all real s and all elements of H(s) are analytic in Re[s] > 0,
2. H(s) + HH(s) ≥ 0 for all Re[s] > 0 or equivalently:
(a) poles on the jω axis are simple and have non-negative definite residues,
(b) H(jω) + HH(jω) ≥ 0 ∀ω ∈ [−∞,∞] such that jω 6= a pole of H(s).
A passive LTI MIMO system will have a positive real transfer matrix H(s).
Definition 2.3.4 (Frequency Domain Conditions for SPR Transfer Matrices [6] [4]).
The rational transfer matrix H(s) composed of strictly proper transfer functions (i.e.,
Hi,j(s) = pi,j(s)
qi,j(s) are all strictly proper) is strictly positive real (SPR) if:
1. H(s) is real for all real s and all elements of H(s) are analytic in Re[s] ≥ 0,
2. H(jω) + HH(jω) > 0 ∀ω ∈ (−∞,∞),
3. lim ω→∞
ω2[H(jω) + HH(jω)] > 0
If the transfer matrix H(s) is SPR, then H(s) + δ1 corresponds to a very strictly
passive system.
Linear time-invariant systems can be expressed using transfer functions and matrices or
in state-space form as shown in Eq. (2.9):
x = Ax + Bu x ∈ Rn , u ∈ Rm
y = Cx + Du (2.9)
The Kalman-Yakubovich-Popov lemma can be used to determine if a LTI system is
positive real or strictly positive real using the state-space equations of the system.
Definition 2.3.5 (Positive Real Systems as Formulated Using the KYP Lemma [7] [8]).
Assume a system of the form presented in Eq. (2.9) with D = 0 is controllable and
observable. Said system as described by a m × m rational transfer matrix H(s) =
C(s1−A)−1B is positive real (PR) if there exist matrices P = PT > 0, P ∈ Rn×n and
Q ≥ 0, Q ∈ Rn×n such that:
PA + ATP = −Q
PB = CT (2.10)
Definition 2.3.6 (Strictly Positive Real Systems as Formulated Using the KYP Lemma
[7] [8]). Assume a system of the form presented in Eq. (2.9) with D = 0 is controllable,
observable and all eigenvalues of A are negative. Said system as described by a m ×m rational transfer matrix H(s) = C(s1 − A)−1B is strictly positive real (SPR) if there
exist matrices P = PT > 0, P ∈ Rn×n and Q > 0, Q ∈ Rn×n such that:
PA + ATP = −Q
PB = CT (2.11)
If the transfer matrix H(s) = C(s1−A)−1B is SPR, then H(s) + δ1 corresponds to
a very strictly passive system.
14
2.4 The Passivity Theorem
The motivation for the various definitions of passive, very strictly passive systems and
positive real, strictly positive real transfer functions/matrices is the calibrated passivity
theorem.
Theorem 2.4.1 (The Passivity Theorem [9]). Consider the negative feedback intercon-
nection of the systems G : L2e → L2e and H : L2e → L2e in Figure 2.3. If G is
passive and H is strictly passive with finite gain (i.e., input-output strictly passive) then
u1,u2 ∈ L2 implies y1,y2 ∈ L2.
G
H
u1
y2
y1
u2e2
e1
Figure 2.3: A Negative Feedback Interconnection of Two Systems
Proof. Refer to Appendix A for the proof of the passivity theorem.
Note that the passivity requirements of the systems G and H can be interchanged
and the passivity theorem will still hold; that is to say if G is strictly passive with finite
gain and H is passive, the closed loop system is L2-stable.
To clarify, let the operator H be a SISO LTI plant we wish to control with transfer
function h(s). Let the operator G be a SISO LTI controller with transfer function g(s).
The plant and controller are connected in a negative feedback interconnection as in Figure
2.3. Let us assume that the plant is passive, and therefore has a transfer function that
is positive real. If the controller is designed such that g(s) is strictly positive real, then
via the passivity theorem the controller g(s) + δ (which corresponds to a system that is
strictly passive with finite gain) is guaranteed to stabilize the closed loop system.
In order to give a more thorough understanding of the stability associated with the
negative feedback interconnection of a passive and very strictly passive system, we will
consider the open loop Nyquist plot of the system in Figure 2.3. We will start by assuming
there are no disturbances, u2 = 0. Recall the closed loop system can be written as
y2 = g(s)h(s) 1+g(s)h(s)
u1, and stability is dependent on g(jω)h(jω) 6= −1 when s = jω. The phase
15
angle of a passive system is always between −π 2 ≤ ∠ g(jω) ≤ π
2 , and the phase of a very
strictly passive system is always between −π 2 < ∠ h(jω) < π
2 . When constructing an open
loop Nyquist plot, the phase angle of the two systems sum, therefore the open loop phase
angle must remain within −π < ∠ g(jω)h(jω) < π. Hence, via the passivity theorem,
the open loop Nyquist plot can never encircle the −1 point, hence g(jω)h(jω) 6= −1 and
the system will always be stable. This result is known as phase stabilization.
An interesting feature of the passivity theorem is that a modified SPR transfer func-
tion (h(s)+δ) which is equivalent to a very strictly passive system can be used to stabilize
a nonlinear passive plant. This is a very powerful feature of the passivity theorem that
will be exploited in this thesis.
16
Flexible Multibody Dynamics
This thesis is concerned with the design of controllers to control large flexible structures
with very little damping. A very interesting class of flexible structures are flexible robot
manipulators such as the Canadarm and Canadarm2. These robots are not only flexible,
but have dynamics that are nonlinear due to their continually morphing configuration.
In this thesis, two flexible robots will be modeled and used for controller synthesis.
For simplicity, a single-link flexible manipulator will be specified first. This will enable us
to become familiar with the various techniques and pitfalls associated with modeling such
complicated systems. Next, a two-link flexible manipulator will be modeled. Throughout
this thesis we adopt the very powerful Vectrix notation developed by Peter C. Hughes [10].
3.1 Dynamics of a Single-Link Flexible Manipulator
Consider the planar single-link flexible manipulator in Figure 3.1. It is composed of a
link mass, a hub mass and a tip mass representing the manipulator payload. The link
has width w, height h and length L. A motor applies a torque τ at the hub, causing the
link to rotate.
In order to formulate the dynamic equations of motion associated with a single-link
flexible manipulator, we will utilize Lagrange’s equation:
d
dt
( ∂L
∂qi
) − ∂L
∂qi = Qi (3.1)
To be able to employ Lagrange’s equation we need to first develop the kinematic equations
of motion and define the generalized coordinates qi. Next, kinetic and potential energy
expressions must be formulated in order to define the Lagrangian, L = T −V . With qi, L
and the generalized forces, Qi, the equations of motion for the system can be determined
17
θ(t)
ue(t)
using Eq. (3.1).
3.1.1 Kinematics
The motion of any mass element dm within the manipulator is modeled using a combi-
nation of rigid and elastic displacements. The rigid motion component of a mass element
at point x along the length of the manipulator is the arc-length of the point, θ(t)x. The
angle θ(t) is the joint angle measured between 01 and 11. The elastic displacement of a
mass element at any point x along the link is modeled using a Rayleigh-Ritz discretiza-
tion. The Rayleigh-Ritz method expresses the elastic deflection of the link as a sum of
basis functions which satisfy the geometric boundary conditions. We will assume there
is only deflection in the 12 direction in the base frame F ~
1. The link deflection, ue, using
two basis functions is defined as:
ue = 2∑ i=1
ψi(x)qi(t) = x2q1(t) + x3q2(t) (3.2)
Let R ~
1 be the position of any infinitesimal mass element dm within the manipulator:
R ~
x
ue
0
(3.3)
The velocity of any point within the manipulator is composed of the rotation rate of
the link, and the deflection rate of the link itself. We will assume that any kinematics
18
V ~
3.1.2 Kinetic and Potential Energy
In order to use Lagrange’s equation to develop the dynamic equations of motion for the
single-link flexible manipulator, we must express the kinetic and potential energy of the
system as a function of the generalized coordinates. This will enable us to define the
Lagrangian, L = T − V . The generalized coordinates are θ(t), q1(t) and q2(t). The
generalized coordinate vector is defined as:
q = [θ q1 q2] T
Kinetic Energy
T = 1 2
VT 1 V1 dm (3.5)
Recall Eq. (3.4) which is the expression for the velocity of any point mass dm within the
link. Substitution of Eq. (3.4) into Eq. (3.5) yields the kinetic energy of the link due to
rotation and deflection of the link:
T link = 1 2
∫ B u2 e dm
= 1 2 qTMlinkq (3.6)
In a similar fashion, the kinetic energy of the payload mass and hub is found to be:
Ttip = 1 2 mtip(Lθ + ue(L, t))
2 = 1 2 qTMtipq (3.7)
Thub = 1 2 Jhubθ
19
Summing Eqs. (3.6), (3.7) and (3.8) yields the total kinetic energy of the system:
T = Tlink + Ttip + Thub = 1 2 qT (Mlink + Mtip + Mhub)q = 1
2 qTMq (3.9)
Potential Energy
The potential energy of the single-link flexible manipulator system is due to the elastic
strain energy associated with the link deflection. The amount of strain is a function of
the modulus of elasticity E of the material, and the second moment of area of the link,
Iyy. The potential energy of the flexible link is defined as:
V = 1 2 EIyy
tor
Employing Lagrange’s equation in Eq. (3.1) along with the Lagrangian (L = T −V ), the
dynamic equations of motion can be developed:
Mq + Kq = Bτ (3.11)
The column matrix B is simply [1 0 0]T and τ is the control torque (via the motor at
the joint). In order to solve the coupled differential equations in Eq. (3.11), we must
rearrange as follows:
q = −M−1Kq + M−1Bτ (3.12)
Note that the single-link manipulator system has differential equations that are SISO
and linear. Therefore the differential equations can be placed into state-space form.
Output Equations for The Deflected Tip Position and Rate
Solving the coupled differential equations in Eq. (3.12) will provide the generalized coor-
dinates as a function of time. We are concerned with the deflected tip position and tip
velocity of the single-link manipulator. Therefore, we need to state output equations for
tip position and tip velocity.
20
Starting with Eq. (3.12), we can express the system in state-space form:
x =
[ q
q
]T
y = Cx + Du (3.13)
Note that the control input u is the joint torque τ . Eq. (3.12) will yield the A and B
matrices. The C and D matrices are a function of the output, the deflected tip position
and tip rate:
ρµ=1 = Lθ(t) + ue(L, t) = Lθ(t) + L2q1(t) + L3q2(t) = [ L L2 L3
] q
ρµ=1 = Lθ(t) + ue(L, t) = Lθ(t) + L2q1(t) + L3q2(t) = [ L L2 L3
] q
Therefore the A, B, C and D matrices of a single-link flexible manipulator written, in
state-space form, with the output being the deflected tip position and rate are:
A =
] D =
[ 0
0
] (3.14)
For reasons that will become clear in Chapter 4, we have defined the deflected tip position
and rate as ρµ=1 and ρµ=1.
3.2 Dynamics of a Two-Link Flexible Manipulator
Consider the planar two-link flexible manipulator in Figure 3.2. Each link has a width
wn, height hn and length Ln when n = 1, 2. There are motors located at the joints of the
robot. Each motor applies a torque, τn. There is a payload to be moved at the end of
the second link, and the second motor at joint 2 can be considered a small intermediate
payload at the end of the first link. We will first derive the kinematic equations of motion.
The kinetic and potential energy of the system will then be defined in order to define the
Lagrangian, L = T−V , which will then be employed along with Lagrange’s equation (Eq.
(3.1)) to provide the differential equations of motion of the two-link flexible manipulator
21
θ1(t)
u1e(t)
F ~
0
F ~
1
θ2(t)
u2e(t)
F ~
2
F ~
system.
3.2.1 Kinematics
Base Frames
Each link has an in-board and out-board base frame. The in-board base frames, F ~
1 and
F ~
2, which are the coordinate origin for both links are located at the joint center. The
out-board base frames, F ~
1 and F ~
2, are located at the very end of each link. The F ~
1
and F ~
2 origins are at the same spatial point and the 13 and 23 axes are collinear. The
11 and 21 axes are along the length of the links and the 13 and 23 axes are through the
joint axes (the motors apply a positive torque about the positive 13 and 23 axes).
Position Kinematics; Rigid and Elastic Position
Let the absolute position of the base frames F ~
1 and F ~
r ~
1 and F ~
2 as if the links were rigid.
Both links have a lateral deflection; the deflections are modeled using a Rayleigh-Ritz
22
u ~
~ T 1
~ T 2
[ q21
q22
] (3.16)
Using some foresight we will also define the deflection and slope at the end of each link
as shown in Eqs. (3.17), (3.18) and (3.19), (3.20). Note that the slope at the end of each
link is in essence a rotation about the 13 or 23 axis of the corresponding base frame.
Thus in Eqs. (3.19), (3.20) the matrices Φ1 and Φ2 are sparse save the last rows.
u ~
0 0
u ~
F ~ T 1∇×u1e(x1, t)
x1=L1
x2=L2
23
The angular component of the manipulator motion is measured by the angles θ1(t)
and θ2(t). Angle θ1(t) is the angle between frames F ~
0 and F ~
angle between frames F ~
1 to frame F ~
(3.21)
We will assume that the link deflections are small and that C21 ≈ C21.
The position kinematics of links 1 and 2 are as follows. First define the rigid-elastic
interbody position vectors r ~
2e
Thus, the absolute position of any point within links 1 and 2 is:
R ~
Velocity Kinematics
We will now proceed to define the absolute velocity associated with any point mass dm
within body Bn (i.e., links 1 and 2):
R ~
R ~
⇒V ~ n = r
~ ne (3.22)
The variable r ~ n is the absolute velocity of the base frame F
~ n. The velocity of dm in
Bn relative to the base frame F ~ n as if the body were rigid is r
~ n,n+1. To incorporate the
flexibility of the body, u ~ ne is the rate of elastic deflection within the body. Expanding
Eq. (3.22) we find:
~ ne + ω
~ n ×u
~ ne (3.23)
We will assume ω ~ × u ~ ne is approximately zero and will be ignored. Eq. (3.23) above
simplifies:
24
We need explicit expressions for the absolute velocity of the tip masses at the ends
of links 1 and 2 in order to define the kinetic energy of said masses:
V ~ nt = r
Vnt = vn − r×n,n+1(Ln)ωn + Ψnqne (3.26)
Before continuing, let us define the linear and angular velocity of each link base frame
in terms of the generalized coordinates, θ1, θ2, q11, q12, q21 and q22. To do so, we will use
an augmented matrix form. First let us define the following variables:
z1 = z2 =
q11
q12
q21
q22
(3.27)
Note that the column matrices z1, z2 are the link rotation axes. The linear and angular
velocity of each link base frame can be consolidated into:
ν ~ n+1,n = F
0 Cn+1,n
(3.28)
It follows that ν1 and ν2 as a function of the generalized coordinates are:
ν1 =
[ v1
ω1
= J1θθ + J1eqe (3.29)
= J2θθ + J2eqe (3.30)
As with the base frames of link 1 and 2, we will define the linear and angular velocity
of the link tip frames F ~
1 and F ~
2 in an augmented matrix form (similar to Eq. (3.28)):
ν ~ nt = F
~ T nνnt = F
qne (3.31)
It follows that ν1t can be expressed as a function of the generalized coordinates:
ν1t =
= J1θtθ + J1etqe (3.32)
Similarly, ν2t can be expressed as a function of the generalized coordinates:
ν2t =
= J2θtθ + J2etqe (3.33)
3.2.2 Kinetic and Potential Energy
Lagrange’s equation will be used to develop the dynamic equations of motion for the two-
link flexible manipulator. In order to do so, we need to define the Lagrangian L = T −V .
Hence, expressions for the global kinetic and potential energy of the system are required.
26
Kinetic Energy
Referring to Eq. (3.5), the kinetic energy of one of the two flexible bodies being considered
is:
)T ( vn − r×n,n+1ωn + une
) dm
Mn,ee = matrix α,β
} (3.35)
In a similar manner, the kinetic energy of the tip mass at the end of link 1 or 2 can be
described by:
2 νTntMn,tνnt (3.36)
Utilizing Eqs. (3.29), (3.29), (3.32) and (3.33) the total kinetic energy of the two-link
flexible manipulator as a function of the generalized coordinates can be written as:
T = T1 + T2 + T1t + T1t
= [
2 qT1eM1,eeq1e
2 qT2eM2,eeq2e
T Mθeqe + 1
2 qTe Meeqe
Potential Energy
The potential or strain energy of the links is a function of the deformation une (Eqs.
(3.15) and (3.16)):
Vn = 1 2
Kn,ee = matrix α,β
} (3.39)
The total potential energy of the system can be formulated as follows:
V = V1 + V2
3.2.3 Equations of Motion of a Two-Link Flexible Manipulator
The Lagrangian is defined as L = T − V ; employing Lagrange’s equation the dynamic
equations of motion can be developed for the two-link flexible manipulator:
d
dt
( ∂L
∂q
) − ∂L
∂q =
[ 1
0
][ τ1
τ2
]} ,
then the differential equations of motion for the two-link manipulator can be written as:[ Mθθ Mθe
MT θe Mee
] Mq + Kq = Bτ + fnon (3.41)
In order to solve for the states of the system (i.e., the generalized coordinates), we
must rearrange Eq. (3.41):
Note that the two-link flexible manipulator system has differential equations that are
MIMO and nonlinear. The equations can not be expressed in a linear state-space form
unless they are linearized about an equilibrium position qT = [θ T
0T ].
Output Equations for The Deflected Tip Position and Rate
While the above equation will yield the generalized coordinates as a function of time, in
order to control the end effector position of the two-link manipulator we need to generate
expressions for the tip position and tip velocity. As in the single link case we will define
the actual tip position and tip rate as ρµ=1 and ρµ=1.
We will start with tip position; the tip position is a function of both the rigid and
elastic motion of the links:
ρ ~ µ=1 = r
2:
The rotation between F ~
1 to F ~
1 is a result of link deflection only. Assuming small angles
29
C11 = 1− (Φ1q1e) ×
0. Starting with Eq. (3.43) and multiplying by
(C21C11C10) T yields the tip position in frame F
~ 0:
( 1− (Φ1q1e)
×)T C12 (r23(L2) + Ψ2q2e) (3.44)
Note that in the case of the two-link planar manipulator being considered, we are only
concerned with the first two rows of ρµ=1 which represent the true x and y position of
the end effector. For future use we will also define the end tip position in F ~
0 as if the
links were completely rigid:
~ T 0
0
(3.45)
Again, we are only concerned with the first two rows of rt which represents the assumed
x and y end-tip position of an equivalent rigid robot manipulator.
To define the tip velocity, we will start with Eq. (3.33) which is the velocity of the
tip in frame F ~
0 is: 1
) (3.46)
As with the tip position equation, we are only concerned with the top two rows of Eq.
(3.46) which represents the x and y velocity (vx and vy) of the tip:
ρµ=1 = Jθθ + Jeqe (3.47)
Thus, ρµ=1 is a 2× 1 column matrix. Ignoring singularities at θ2 = 0, π, Jθ is square and
invertible. The Jθ matrix is really an approximation; for it to be the true map between
the joint rates and tip velocity, it should be a funtion of both the joint coordinates and
1Note that the rotation sequence C01C12 does not include the small rotation C11 due to the link deflection.
30
elastic coordinates:
Jθ = Jθ(θ,qe)
However, by deliberately ignoring the rotation matrix C11 and the deformation of the
rigid-elastic interbody position vector while deriving the velocity expressions of the links, 2 Jθ is only an approximate map:
Jθ = Jθ(θ)
This approximation then yields a relationship between the rigid manipulator end-tip
velocity and the joint velocity:
rt = Jθθ
We can think of ρµ=1 as the rigid end-tip velocity combined with the elastic rates of the
links:
ρµ=1 = rt + Jeqe
Also note that Eq. (3.47) and the derivative of Eq. (3.44) are not equivalent; the tip rate,
ρµ=1, does not incorporate the rotation C11, but ρµ=1 does.
2For example, in Eq. (3.28), the r×n+1,n within Tn+1,n would be r×n+1,n if we did not ignore the deformation of the rigid-elastic interbody position vector.
31
Systems
In the previous chapter we developed the dynamic equations of motion for the systems
we wish to control. The purpose of this chapter is to discuss what it is we wish to
control, and the methods we will employ to execute the control. As with most mechanical
systems, the main control objective is to track some desired trajectory while rejecting
plant disturbances.
4.1 µ-Output
In this thesis we wish to control flexible manipulators carrying large payloads. These
plants have dynamics that are significantly more complicated than their rigid cousins.
In order to employ any type of control technique we must first establish if all the states
of the system (the angular rates of the joints and the elastic rates of the links) are
controllable/observable between the system input-output map.
4.1.1 Input-Output Mapping
From Eq. (3.47) the tip velocity of the two-link flexible manipulator is: ρµ=1 = Jθθ+Jeqe.
Let us now introduce the µ-tip rate which will be the output we wish to control:
y(t) = ρµ = Jθθ + µJeqe (4.1)
where µ = 1 captures the true tip velocity and µ = 0 captures tip velocity of the
manipulator as if it were rigid.
32
Let us now establish the control input u. Let us first define the system output to
be the rigid two-link manipulator tip rate, ρµ=0 = Jθθ. Given this output, the system
input can be derived by considering the passive input-output map (I/O map) between
the joint torques and the angular velocity of the joints:∫ T
0
∫ T
0
∫ T
0
∫ T
0
uTy dt ≥ 0
It is well known that any collocated I/O map is passive. The input τ and output ρµ=0
are non-collocated, and thus their mapping is not passive. Via the above transformation,
we have shown that the modified I/O mapping between J−Tθ τ and ρµ=0 is passive.
In the context of flexible manipulators this I/O map is still appropriate because as
the payload becomes massive, the modes of the structure appear clamped at the end
effector [11]. Thus, the I/O map for a flexible manipulator with a massive payload
emulates that of a rigid manipulator.
4.1.2 Modal Analysis
Recall the motion equations for the two-link manipulator we wish to study (Eq. (3.41)):
Mq + Kq = Bτ + fnon
where M = MT > 0 and Kee = KT ee > 0. Now linearize the above system and the output
y(t) about a specific configuration qT = [θ T
0T ]:
[ 1
0
y = C δq = [Jθ(q) µJe(q)] δq (4.3)
Assume the joints are not locked in place, but momentarily ignore the rigid body
modes of vibration. As is customary with mass-spring vibration problems, let the solution
to Eq. (4.2) be of the form δq = qαe −jωαt , α = 1 . . . 4 1 and solve the corresponding
eigenvalue problem:
−ω2 αMqα + Kqα = 0
where ωα are the joint-unlocked vibration modes and qα are the mode shapes (eigenvec-
1There are two basis functions for each link, thus the number of elastic coordinates, Ne, is 4.
33
qα =
[ θα
qeα
]
where θα are the mode slopes at the base of each link. The eigenvectors are normalized
with respect to M, and are orthogonal with respect to M and K:
qTαMqβ = δαβ qTαKqβ = ω2 αδαβ
Recall that δαβ is the Kronecker delta. Now let’s consider the rigid-body modes; because
K is positive semidefinite, there are two zero-frequency rigid-body modes which satisfy
KQr = 0, Qr =
] .
As given previously, these two eigenvectors which are combined in Qr, but not normalized
with respect to M, are orthogonal to M and K:
qTαMQr = 0 qTαKQr = 0
We can now express δq in terms of modal coordinates:
δq(t) = Qrηr(t) + 4∑
α=1
qαηα(t) (4.4)
Substituting Eq. (4.4) into Eq. (4.2), premultiplying by QT r and employing the orthonor-
mal relationships will yield the rigid motion equations for the two-link manipulator:
M δq + K δq = Bu
QT r M
Similarly, premultiplying Eq. (4.2) by qTβ and employing the orthonormal relationships
yields the flexible motion equations:
M δq + K δq = Bu
34
qTβM
( Qrηr +
ηα + ω2 αηα = θTαJTθ u = (Jθθα)Tu (4.6)
Finally, starting with Eq. (4.3) and substitution of Eq. (4.4) will yield the output y(t) in
terms of the modal coordinates:
y(t) = C δq = [Jθ µJe] δq
= [Jθ µJe]
(Jθθα + µJeqeα) ηα (4.7)
Taking Laplace transforms of Eqs. (4.5), (4.6) and (4.7) and substituting appropriately
yields the dynamics of the linearized system:
y(s) =
[ 1
4.1.3 Motivation for Using the µ-Tip Rate
In [12] it was shown that as the payload becomes massive, the base and tip mass will act
as fixed nodes and the manipulator will vibrate internally between said nodes. This fact
can be expressed in the following equality:
Jθθα + Jeqeα = 0
Substituting Eq. (4.9) into Eq. (4.8) yields:
y(s) = G(s)u(s)
] u(s) (4.10)
From Eq. (4.10) it is clear why we must employ the µ-tip rate as our output when the
payload is large; when µ = 1 the output is the true tip rate and the modes of vibration
35
become unobservable. When µ < 1, the modes of vibration are observable and become
more observable as µ approaches zero. Throughout this thesis, all single-link and two-link
manipulator simulations use µ = 0.8.
Although we will show that the full nonlinear two-link flexible manipulator model is
indeed passive in Section 4.3.1, it is noteworthy to observe the positive realness of the
linearized plant in Eq. (4.10). All the poles of the system are in the closed left half
complex plane, and G(jω) + GH(jω) ≥ 0 for ∀ω ∈ [−∞,∞], except at the poles of G(s)
(i.e., when jω equals a pole of G(s)).
4.2 H2-Optimal Control
There are many controller algorithms such as PI, PD, PID, Lead-Lag etc. that can be
used to control the tip velocity of a flexible manipulator. A more modern control method
is H2-optimal control. In this thesis the ideas that follow from the H2-optimal control
formulation will be used in conjunction with numerical optimization, so it is appropriate
to discuss the H2-optimal control formulation here.
A H2-optimal controller can be applied to both the single- and two-link manipulators.
However, we will present it’s development with the objective of controlling the tip rate
of the more complicated two-link manipulator.
Before continuing, it should be understood that when we discuss controller design
we are referring to the design of a rate controller, not a position and rate controller.
Position control is handled by simple proportional control (P control). As a result, while
designing any rate controller such as a H2-optimal controller we are assuming the plant
(the flexible robot) has been pre-wrapped with proportional control.
4.2.1 General Control Formulation
Consider the block digram in Figure 4.1. H is the plant we wish to control, and G is
the controller we wish to design. The input and output signals are:
• z - regulated output; ρµ=1
• w - exogenous inputs or disturbances
• y - measurements; ρµ
• u - control output
Figure 4.1: H2-Optimal Control Block Diagram
In our specific case, the plant we wish to control is that of a nonlinear two-link flexible
manipulator. The H2-optimal control formulation is for linear time-invariant systems;
therefore, we must linearize the two-link manipulator about an equilibrium point (usu-
ally a specific manipulator configuration). The plant H can then be expressed by the
following state-space equations:
[ B1
D21
] DT
21 =
[ 0
Rw
]
The controller G is also linear and time invariant with the same dimensions as the plant:
xc = Acxc + Bcy
37
The plant and controller can be combined to create the closed loop system Tz :
Tz ≡
4.2.2 Control Objective
The controller G is designed to minimize the effect of the disturbance w or the initial
conditions of the system x(0) on the output z. One way of interpreting the H2-optimal
control objective is designing the controller to minimize the H2-norm 2 of the transfer
matrix Tz(s). It can be shown that this can be determined using the L2-norm of each
output z(i)(t) in response to a disturbance input w(i)(t) which contains a Dirac Delta
function in the ith input:
J2 = Tz(s)2 =
where
w(i)(t) =
0
0 ...
δ(t) ...
0
0
= 1iδ(t)
The square, symmetric, positive definite matrix P is found by solving the Lyapunov
equation
z Cz .
2The notation H2- rather then L2-optimal control is a result of the system operators being bounded and analytic in the closed right half plane. These operators are in turn stable and part of the Hardy Space named after the British mathematician G.H. Hardy (1877-1947) [13].
38
It is clear that the solution to the Lyapunov equation above and the minimum closed-loop
H2-norm in Eq. (4.12) is a function of the Az, Bz and Cz matrices, which are in turn a
function of the plant and the controller state-space models. We cannot change the plant,
but we do have control over the design of the controller. The controller which yields the
minimum closed-loop H2-norm, has an optimal regulator and optimal observer form.
4.2.3 Optimal Regulator Problem
The optimal regulator has the form of negative state feedback within the controller:
u = −Kcxc = Ccxc
It is found by solving the ever famous Algebraic Riccati Equation for Pc:
PcA + ATPc −PcB2R −1BT
where
12D12 .
Kc = RBT 2 Pc
4.2.4 Optimal Estimation Problem
While implementing controllers, some of the internal states of the system cannot be
measured. Therefore we must estimate what the states of the system should be. This is
the role of an observer, to drive the estimated value of the states to the true value of the
states. This is done via a gain matrix Ke:
xc = Axc + B2u + Ke(C2xc − y)
The solution to the optimal state estimation problem is found by solving the Algebraic
Riccati Equation for Pe:
PeA T + APe −PeC
T 2 R−1
where
T 21 .
w
of the optimal regulator and optimal estimator:
xc = (A−B2Kc −KeC2)xc + Key
u = −Kcxc
This controller will minimize the closed-loop H2-norm of the system in the presence of
the disturbances prescribed. The above state-space form of the controller can also be
placed in the form of a transfer matrix:
u = G(s)y = −Kc (s1− (A−B2Kc −KeC2)) −1 Key
4.3 Passivity-Based Control
This thesis is concerned with the optimal design of very strictly passive controllers to
control passive plants such as a flexible robot manipulators. Before presenting the main
formulations used to design said controllers, it is important to discuss passivity-based
control in general as applied to flexible systems.
As with the H2-optimal control method, we will assume the plant has been pre-
wrapped with proportional control.
4.3.1 Passivity of the Plant
From Section 2.4, for the passivity theorem to be employed we must have a passive plant.
Recall the input to our system is u = J−Tθ τ , while the output is y = ρµ. For µ = 0 it is
relatively easy to show that∫ T
0
(J−Tθ τ )T ρµ=0 dt = H(T )−H(0) ≥ −H(0)
40
H(q(t),q(t), t) = 1 2 qTM(q)q + 1
2 qTKq .
Thus the mapping from u = J−Tθ τ to y = ρµ=0 is passive. Now let us evaluate the
passivity integral when 0 < µ ≤ 1. The following derivation is from Damaren, 1995 [12].
Assuming H(0) = 0, the passivity integral yields:∫ T
0
∫ T
0
(J−Tθ τ )TJeqe dt (4.15)
By employing the method of virtual work and making a quasi-static approximation for
qe (i.e., qe ≈ 0), it can be shown that
Keeqe = −JTe J−Tθ τ (t) . (4.16)
If we premultiply Eq. (4.16) by qTe we find
qTe Keeqe = −qTe JTe J−Tθ τ = V
which is the time rate of change of the elastic potential energy of the links. By making
the appropriate substitution of V and assuming H(0) = 0, Eq. (4.15) becomes:∫ T
0
uTy dt = H(T )− µV (T ) = T (T ) + (1− µ)V (T ) ≥ 0
Clearly, when µ = 0 the system is passive as was previously stated. When 0 < µ < 1
the system is passive and the elastic deflections of the links are observable as previously
discussed in Section 4.1. When µ = 1, the system is passive but the elastic modes are
not observable.
4.3.2 Passivity-Based Control of Flexible Manipulators
Briefly, the passivity theorem in Section 2.4 states that the negative feedback interconnec-
tion of a passive plant H and a strictly passive controller with finite gain G is L2-stable.
This is an extremely powerful statement in the context of controlling flexible robot ma-
nipulators. Consider the two-link manipulator previously presented; the dynamics are
nonlinear due to the configuration dependence of the links, yet the system is always pas-
sive. Via the passivity theorem we are guaranteed to stabilize this complicated nonlinear
41
system with any linear time-invariant controller that is strictly passive with finite gain.
H
G(s)
While controlling flexible structures, spillover instability may be an issue. Spillover
is a consequence of unmodeled dynamics (modes of vibration). Usually the controller
is designed based on a finite number of modes, and inadvertently may destabilize un-
modeled modes. However, in the context of the flexible robots we control, the passivity
characteristics of the plant are independent of the number of vibration modes in the
system model, as well as the chosen basis functions and natural frequencies. As a result,
very strictly passive feedback control yields robust stability, as spillover is avoided.
As an example of passivity-based control, assume our control objective is to have
the tip of a two-link flexible robot with a large payload follow a prescribed trajectory,
ρµ=1(t)→ ρd(t), with prescribed velocity, ρµ=1(t)→ ρd(t). 3 We are concerned with the
rate portion of the control objective. 4 The controller I/O map would be:
u(s) = G(s)y(s) = G(s) ρµ(s)− ρd(s)
s
Via the passivity theorem, any SPR transfer matrix will suffice. Consider the transfer
matrix G(s):
0 s+d (s+a)(s+b)
] Provided a + b > c and a + b > d, the decoupled MIMO controller is SPR. Therefore
G(s)+δ1 is strictly passive with finite gain and is guaranteed to stabilize the closed-loop
system via the passivity theorem.
3This will be discussed in more detail in Section 4.4. 4Recall, we are assuming the system has been pre-wrapped with proportional control.
42
Previously, feedback control strategies such as H2-optimal and passivity-based control
were introduced. These strategies can be used for set-point regulation or tracking. In
general, feedback control strategies render the closed-loop system stable via some pole
placement procedure. That is to say, the closed-loop eigenvalues of a system will all have
negative real parts.
Usually in robotics the objective is to execute some sort of complicated maneuver
while following a desired trajectory. While performing a tracking task a popular form of
control is feedforward control. Feedforward control in conjunction with feedback control
usually results in better end-effector tracking. The objective of feedforward control is to
essentially cancel out plant dynamics.
4.4.1 Desired Tip Position and Velocity
Assume a rigid two-link manipulator is to move from an initial position with joint config-
uration θi to a final position with joint configuration θf in time tf > 0. The manipulator
must move from start to finish in a smooth fashion with a gradual increase and then de-
crease in both velocity and acceleration. Consider the following desired joint trajectory:
θd(t) =
[ 10
( t
tf
)3
− 15
( t
tf
)4
+ 6
( t
tf
)5 ]
(θf − θi) + θi (4.17)
The desired joint velocities θd(t) and joint accelerations θd(t) can be found by taking the
appropriate time derivatives of θd(t), in addition to being zero at t = 0 and t = tf (i.e.,
θd(0) = θd(tf ) = 0, θd(0) = θd(tf ) = 0).
Let us now return to consideration of a flexible two-link robot; in preparation to use
both feedforward and feedback control while performing tip trajectory tracking, we will
specify the desired tip trajectory based on the desired joint trajectory of an equivalent
rigid robot. Because we have a massive payload it is appropriate to design the end-tip
trajectory as if the manipulator were rigid. Recall Eq. (3.45), the rigid tip position of a
two-link robot. We can use Eq. (3.45) to specify the desired end-effector position as a
function of θd:
] (4.18)
ρd can be found by taking the time derivative of ρd.
43
4.4.2 Feedforward Control
Recall Eq. (3.41), the differential equations of a flexible two link manipulator with no
control:
Mq + Kq = Bτ + fnon
As previously stated, the objective of feedforward control is to cancel out all, or at least
some portion of the plant dynamics. Let us look at just the rigid component of the
differential equations:
Mθθθ = τ + fnon,θ
For feedforward control to cancel out the rigid motion dynamics it would have the form:
τ FF = Mθdθdθd − fnon,θd (4.19)
where Mθdθd is the rigid mass matrix evaluated at the desired joint position and fnon,θd is fnon,θ evaluated at the desired joint position.
4.4.3 Closed-Loop Dynamics Utilizing Feedfoward and Feed-
back Control
If our objective is to have the tip of the two-link flexible manipulator follow a desired
trajectory as prescribed by Eq. (4.18), the feedback control required would be composed
of position and rate control. This can be expressed in joint coordinates as:
τ FB = −JTθ [ Kp
)] (4.20)
Note that the term ρµ is not equivalent to ρµ=1, as presented in Eq. (3.44). The µ-tip
position can be derived in the following manner:
ρµ = Jθθ + µJeqe
44
where ρµ=1 is the actual tip position taking into account the rotation of the tip due to
link deflection (C11) and the deflected rigid-elastic interbody position vector, as derived
in Eq. (3.44), and rt(θ) is the end-tip position as if the links were rigid, as derived in
Eq. (3.45). Defining ρµ as the combination of rt(θ) and ρµ=1 is primarily motivated
by the practical measurements available; usually link deflections will be measured by a
series of strain gauges. Joint angles can be measured via encoders, and can be used to
determine rt(θ). The strain gauge and encoder measurements can be used to determine
the actual tip position, ρµ=1. Therefore, ρµ can easily be approximated with the available
measurements.
Augmenting the feedforward and negative feedback control from Eqs. (4.19) and
(4.20) (τ = τ FF + τ FB) we have the following closed-loop differential equations of the
two-link flexible manipulator system:
( Kp
))] + fnon
Here G is some MIMO rate controller such as a H2-optimal controller designed via
linearization at the end position θf , or a SPR controller designed via algorithms to be
discussed in the following sections.
45
Chapter 5
Optimization Formulation
This thesis is concerned with the optimal design of SPR controllers to control large
flexible structures. The approach we will take to design these SPR controllers will employ
numerical optimization techniques. In this chapter we will introduce how to pose a
numerical optimization problem, then specific characteristics of the SPR controller design
optimization problem will be introduced.
5.1 Gradient-Based Optimization Overview
The focus of this thesis is not to develop a new numerical optimization procedure, but
rather to use numerical optimization to solve a particular set of problems. Because of the
extensive use of numerical optimization in this thesis, a brief overview of how a numerical
optimization problem is posed is in order.
Numerical Optimization Problem Statement
In general, the numerical optimization problem statement is as follows:
Minimize a given objective function, f(x), by varying any or all design variables in
the vector x subject to a set of equality and inequality constraints of the form h(x) = 0
and g(x) ≤ 0 respectively.
The objective function is one numerical value, f(x) : Rn → R. The optimal solution
x? corresponds to the smallest value of the objective function (f(x?)) among all the
vectors that satisfy the constraints [14].
46
A Small Example
An objective function is something we want to minimize or maximize. We may do
so by varying design variables which are directly or indirectly subject to some sort of
constraints. As an example, while designing an aircraft we may want to maximize the
lift of the wings, or minimize the weight of the wing structure, or both. The design
variables associated with maximizing lift may be wing span, the root, mean and tip cord,
the angle of twist, etc. The design variables associated with minimizing the weight of the
wing may be the thickness and spacing of the wing spars. The constraints may be direct
(that is directly applied to the design variables), such as: the wing span must be less than
10 m, or the thickness of the spars can be no greater than 50 mm. The constraints may
also be indirect, a function of some other physical phenomena: the stress in the spar can
be no greater than σyield, and therefore the minimum thickness of the spar is affected.
Numerical Optimization Using MATLAB
There are various numerical optimization packages available, however we have elected
to use the function fmincon within MATLAB’s Optimization Toolbox. The function
fmincon uses a Sequential Quadratic Programming (SQP) method to solve constrained
optimization problems. It is a gradient-based optimization procedure that utilizes the
Karush-Kuhn-Tucker (KKT) conditions to ensure that both the optimum solution has
been found, and the constraints have been satisfied [15].
5.2 General Controller Optimization Formulation
The first objective of this thesis is to use a gradient-based optimizer to design optimal
SPR controllers to control the flexible manipulators developed in Chapter 3. As with
all optimization problems we need to define an objective function, design variables and
constraints.
Objective Function
Keeping with the optimal control tradition, the closed-loop H2-norm seems an appro-
priate objective function. Recall Section 4.2, the presentation of H2-optimal control.
The H2 control objective can be interpreted as designing the controller to minimize the
closed-loop L2-norm of the output z when the inputs w contain Dirac Delta functions:
J2 = √
47
The matrix P = PT > 0 is found by solving the Lyapunov equation
PAz + AT z P = −CT
z Cz .
Referring to Eq. (4.11), the Az, Bz and Cz matrices are the closed-loop state-space equa-
tion matrices. Each are a function of the plant and the controller to be optimized. While
designing optimal SPR controllers we will use the same H2-optimal control objective
function (Eq. (5.1)), but replace the controller with an SPR controller of the form:
xc = Acxc + Bcy
u(s) = G(s)y(s)
The controller form and parameterization will be independently specified, and may or
may not have the same number of states as the plant, such as the traditional H2-optimal
controller.
Design Variables
The design variables are the parameters we will be able to vary during the optimization
procedure. Depending on how we specify the form of the SPR controller (a transfer
function/matrix of specific order, or state-space equations of a specific dimension), the
design variables will change. The controller form and the design variables chosen is
termed the controller parameterization. The parameterization plays a significant role in
the optimization procedure, and one of the many objectives of this thesis is to determine
how different controller forms and parameterizations effect the optimization results. Sig-
nificant detail will be paid to each controller form/parameterization in future chapters.
As an example, the design variables are usually parameters such as transfer function
numerator and denominator polynomial coefficients.
Recall that the feedback controller is composed of proportional control and rate con-
trol. Not only will the rate control be optimized (the SPR controller), but also the pro-
portional control. For the single-link manipulator system the design variable associated
with proportional control is simply Kp. For the two-link manipulator, the proportional
48
control is a 2× 2 positive definite symmetric matrix Kp; design variables associated with
MIMO proportional control are diag {Kp11 , Kp12 = Kp21 , Kp22}.
Constraints
In order to pose the design of SPR controllers using numerical optimization properly,
we must also specify some sort of constraints on the design variables of the system. In
Section 2.3, the conditions for transfer functions, matrices and state-space representations
to be SPR are discussed in detail. Obviously, the main constraint while executing the
optimization is that the controller must be SPR. The way in which we enforce this SPR
constraint may be simplified depending on the controller form and parameterization. As
with the design variables, the constraints on the SPR controller may change depending
on the form/parameterization of the controller. Additional detail will be paid to each
constraint associated with each unique controller form and parameterization in future
chapters.
49
Controllers
Previously, in Chapter 5 the SPR controller optimization objective function was clearly
defined as the system closed-loop H2-norm. In this chapter, we will explore the dif-
ferent approaches to controller parameterization; designing the form of the controller,
specifying the design variables and enforcing the constraints. This chapter will only be
concerned with the optimization of SISO SPR controllers to control the single-link flexible
manipulator previously developed.
The physical values of the single-link manipulator length, mass, tip-mass, hub inertia
etc. used during optimization and simulation are given in Table 6.1.
Length L 1 m Link Mass m 1 kg
Modulus of Elasticity E 70× 109 Pa Link Height h 75 mm
Link Base Width w 2 mm Link Second Moment of Area I = 1
12hw 3 5× 10−11 m4
Hub Mass mhub 1 kg Hub Mass Moment of Inertia Jhub 3.125× 10−4 Kg ·m2
Tip Mass mtip 1.5 kg
Table 6.1: Single-Link Manipulator Physical Properties
50
equality Constraints
Recall Definition 2.3.2, which specifies the frequency domain requirements for a transfer
function to be SPR. These conditions can be somewhat complicated to enforce directly,
thus, we will first seek a simple way to define and constrain a fixed-order transfer function
to be SPR.
The authors Marquez and Damaren [16] have developed a formulation such that
given a strictly proper transfer function with a denominator polynomial that is Hurwitz,
necessary and sufficient conditions are formulated which when met will always yield a
numerator polynomial that creates a SPR transfer function. The formulation allows
the numerator polynomial to be expressed as a function of t