In this book, we present the reader with some of the latest developments in robust control theory for uncertain systems. In particular, we will consider the quadratic stabilizability approach to the control of uncertain systems with norm-bounded uncertainty and a more general absolute stabilizability approach to the control of uncertain systems described by integral quadratic constraints (IQCs). The reader will see that the integral quadratic constraint concept and its generalizations leads to an uncertain system framework in which the standard LQR and LQG optimal controller design methodologies can be extended into minimax optimal control and guaranteed cost control methodologies. Such an uncertain system framework allows the designer to retain existing methods of LQR/LQG design while the issue of robustness can be addressed by an appropriate choice of the uncertainty structure in the uncertain system model.
Note that for the most part, the robust control problems considered in this book are closely related to corresponding H¥ control problems. Furthermore, throughout the book, the solutions to these robust control problems will be formulated in terms of Riccati equations of the form arising in H¥ control theory. These facts motivate the title of this book.
Frequently Used Notation
1 Introduction
1.1 The concept of an uncertain system
1.2 Overview of the book
2 Uncertain Systems
2.1 Introduction
2.2 Uncertain systems with norm-bounded uncertainty
2.2.1 Special case:
sector-bounded nonlinearities
2.3 Uncertain systems with integral quadratic
constraints
2.3.1 Integral quadratic
constraints
2.3.2 Integral quadratic
constraints with weighting coefficients
2.3.3 Integral uncertainty
constraints for nonlinear uncertain systems
2.3.4 Averaged integral
uncertainty constraints
2.4 Stochastic uncertain systems
2.4.1 Stochastic uncertain
systems with multiplicative noise
2.4.2 Stochastic uncertain
systems with additive noise: Finite-horizon relative entropy constraints
2.4.3 Stochastic uncertain
systems with additive noise: Infinite-horizon relative entropy constraints
3 H¥ Control
and Related Preliminary Results
3.1 Riccati equations
3.2 H¥
control
3.2.1 The standard
H¥
control problem
3.2.2 H¥
control with transients
3.2.3 H¥
control of time-varying systems
3.3 Risk-sensitive control
3.3.1 Exponential-of-integral
cost analysis
3.3.2 Finite-horizon
risk-sensitive control
3.3.3 Infinite-horizon
risk-sensitive control
3.4 Quadratic stability
3.5 A connection between H¥
control and the absolute stabilizability of uncertain systems
3.5.1 Definitions
3.5.2 The equivalence
between absolute stabilization and H¥
control
4 The S-procedure
4.1 Introduction
4.2 An S-procedure result for a quadratic
functional and one quadratic constraint
4.2.1 Proof of Theorem
4.2.1
4.3 An S-procedure result for a quadratic
functional and k quadratic constraints
4.4 An S-procedure result for nonlinear functionals
4.5 An S-procedure result for averaged sequences
4.6 An S-procedure result for probability
measures with constrained relative entropies
5 Guaranteed Cost Control of Time-Invariant Uncertain Systems
5.1 Introduction
5.2 Optimal guaranteed cost control for uncertain
linear systems with norm-bounded uncertainty
5.2.1 Quadratic guaranteed
cost control
5.2.2 Optimal controller
design
5.2.3 Illustrative
example
5.3 State-feedback minimax optimal control
of uncertain systems with structured uncertainty
5.3.1 Definitions
5.3.2 Construction
of a guaranteed cost controller
5.3.3 Illustrative
example
5.4 Output-feedback minimax optimal control
of uncertain systems with unstructured uncertainty
5.4.1 Definitions
5.4.2 A necessary
and sufficient condition for guaranteed cost stabilizability
5.4.3 Optimizing the
guaranteed cost bound
5.4.4 Illustrative
example
5.5 Guaranteed cost control via a Lyapunov
function of the Lur'e-Postnikov form
5.5.1 Problem formulation
5.5.2 Controller synthesis
via a Lyapunov function of the Lur'e-Postnikov form
5.5.3 Illustrative
Example
5.6 Conclusions
6 Finite-Horizon Guaranteed Cost Control
6.1 Introduction
6.2 The uncertainty averaging approach to
state-feedback minimax optimal control
6.2.1 Problem statement
6.2.2 A necessary
and sufficient condition for the existence of a state-feedback guaranteed
cost controller
6.3 The uncertainty averaging approach to
output-feedback optimal guaranteed cost control
6.3.1 Problem statement
6.3.2 A necessary
and sufficient condition for the existence of a guaranteed cost controller
6.4 Robust control with a terminal state constraint
6.4.1 Problem statement
6.4.2 A criterion
for robust controllability with respect to a terminal state constraint
6.4.3 Illustrative
example
6.5 Robust control with rejection of harmonic
disturbances
6.5.1 Problem statement
6.5.2 Design of a
robust controller with harmonic disturbance rejection
6.6 Conclusions
7 Absolute Stability, Absolute Stabilization and Structured
Dissipativity
7.1 Introduction
7.2 Robust stabilization with a Lyapunov function
of the Lur'e-Postnikov form
7.2.1 Problem statement
7.2.2 Design of a
robustly stabilizing controller
7.3 Structured dissipativity and absolute
stability for nonlinear uncertain systems
7.3.1 Preliminary
remarks
7.3.2 Definitions
7.3.3 A connection
between dissipativity and structured dissipativity
7.3.4 Absolute stability
for nonlinear uncertain systems
7.4 Conclusions
8 Robust Control of Stochastic Uncertain Systems
8.1 Introduction
8.2 H¥control
of stochastic systems with multiplicative noise
8.2.1 A stochastic
differential game
8.2.2 Stochastic H¥
control with complete state measurements
8.2.3 Illustrative
example
8.3 Absolute stabilization and minimax optimal
control of stochastic uncertain systems with multiplicative noise
8.3.1 The stochastic
guaranteed cost control problem
8.3.2 Stochastic absolute
stabilization
8.3.3 State-feedback
minimax optimal control
8.4 Output-feedback finite-horizon minimax optimal
control of stochastic uncertain systems with additive noise
8.4.1 Definitions
8.4.2 Finite-horizon
minimax optimal control with stochastic uncertainty constraints
8.4.3 Design of a
finite-horizon minimax optimal controller
8.5 Output-feedback infinite-horizon minimax
optimal control of stochastic uncertain systems with additive noise
8.5.1 Definitions
8.5.2 Absolute stability
and absolute stabilizability
8.5.3 A connection
between risk-sensitive optimal control and minimax optimal control
8.5.4 Design of the
infinite-horizon minimax optimal controller
8.5.5 Connection to
H¥
control
8.5.6 Illustrative
example
8.6 Conclusions
9 Nonlinear versus Linear Control
9.1 Introduction
9.2 Nonlinear versus linear control in the
absolute stabilizability of uncertain systems with structured uncertainty
9.2.1 Problem statement
9.2.2 Output-feedback
nonlinear versus linear control
9.2.3 State-feedback
nonlinear versus linear control
9.3 Decentralized robust state-feedback H¥
control for uncertain large-scale systems
9.3.1 Preliminary
remarks
9.3.2 Uncertain large-scale
systems
9.3.3 Decentralized
controller design
9.4 Nonlinear versus linear control in the
robust stabilizability of linear uncertain systems via a fixed-order output-feedback
controller
9.4.1 Definitions
9.4.2 Design of a
fixed-order output-feedback controller
9.5 Simultaneous H¥control
of a finite collection of linear plants with a single nonlinear digital
controller
9.5.1 Problem statement
9.5.2 The design
of a digital output-feedback controller
9.6 Conclusions
10 Missile Autopilot Design via Minimax Optimal Control of
Stochastic Uncertain Systems
10.1 Introduction
10.2 Missile autopilot model
10.2.1 Uncertain system
model
10.3 Robust controller design
10.3.1 State-feedback
controller design
10.3.2 Output-feedback
controller design
10.4 Conclusions
11 Robust Control of Acoustic Noise in a Duct via Minimax Optimal
LQG Control
11.1 Introduction
11.2 Experimental Setup and Modeling
11.2.1 Experimental
Setup
11.2.2 System Identification
and Nominal Modelling
11.2.3 Uncertainty
Modelling
11.3 Controller Design
11.4 Experimental Results
11.5 Conclusions
Appendix A Basic Duality Relationships for Relative Entropy
Appendix B Metrically Transitive Transformations
References
Index