This paper will very briefly review the history of the relationship between modern optimal control and robust control. The latter is commonly viewed as having arisen in reaction to certain perceived inadequacies of the former. More recently, the distinction has effectively disappeared. Once-controversial notions of robust control have become thoroughly mainstream, and optimal control methods permeate robust control theory. This has been especially true in H-infinity theory, the primary focus of this paper.

http://www.gbv.de/dms/goettingen/18606599X.pdf

Robust and Optimal Control

Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >

State-space solutions to standard H/sub 2/ and H/sub infinity / control problems

1. Introduction. 2. Linear Algebra. 3. Linear Systems. 4. H2 and Ha Spaces. 5. Internal Stability. 6. Performance Specifications and Limitations. 7. Balanced Model Reduction. 8. Uncertainty and Robustness. 9. Linear Fractional Transformation. 10. m and m- Synthesis. 11. Controller Parameterization. 12. Algebraic Riccati Equations. 13. H2 Optimal Control. 14. Ha Control. 15. Controller Reduction. 16. Ha Loop Shaping. 17. Gap Metric and ...u- Gap Metric. 18. Miscellaneous Topics. Bibliography. Index.

/pdf/essentials-of-robust-control-3kaks6yk3h.pdf

Essentials of Robust Control

Simple state-space formulas are presented for a controller solving a standard H∞-problem. The controller has the same state-dimension as the plant, its computation involves only two Riccati equations, and it has a separation structure reminiscent of classical LQG (i.e., H2) theory. This paper is also intended to be of tutorial value, so a standard H2-solution is developed in parallel.

/pdf/state-space-solutions-to-standard-h-2-and-h-control-problems-54ciqky6pt.pdf

State-space solutions to standard H 2 and H ∞ control problems

This book provides an introduction to the mathematics needed to model, analyze, and design feedback systems. It is an ideal textbook for undergraduate and graduate students, and is indispensable for researchers seeking a self-contained reference on control theory. Unlike most books on the subject, Feedback Systems develops transfer functions through the exponential response of a system, and is accessible across a range of disciplines that utilize feedback in physical, biological, information, and economic systems. Karl strm and Richard Murray use techniques from physics, computer science, and operations research to introduce control-oriented modeling. They begin with state space tools for analysis and design, including stability of solutions, Lyapunov functions, reachability, state feedback observability, and estimators. The matrix exponential plays a central role in the analysis of linear control systems, allowing a concise development of many of the key concepts for this class of models. strm and Murray then develop and explain tools in the frequency domain, including transfer functions, Nyquist analysis, PID control, frequency domain design, and robustness. They provide exercises at the end of every chapter, and an accompanying electronic solutions manual is available. Feedback Systems is a complete one-volume resource for students and researchers in mathematics, engineering, and the sciences.Covers the mathematics needed to model, analyze, and design feedback systems Serves as an introductory textbook for students and a self-contained resource for researchers Includes exercises at the end of every chapter Features an electronic solutions manual Offers techniques applicable across a range of disciplines

/pdf/feedback-systems-an-introduction-for-scientists-and-596yj6j5hm.pdf

Feedback Systems: An Introduction for Scientists and Engineers

Linear Robust Control

The 2-Riccati H∞ controller formulae and derivations are simplified via various ‘loop-shifting’ transformations that are naturally expressed in terms of a degree-one polynomial system matrix (PSM) closely related to the Luenberger descriptor form of a system. The technique enables one without loss of generality to restrict attention to the simple case in which D 11 =0, D 22 = 0, D T 12= [0 l], D 21 = [0 l],D l 12 C1=0 and B 1 D T 21 =0. Matrix-fraction descriptions (MFDs) for the algebraic Riccati equation solutions afford another change of variables, which brings the 2-Riccati H∞ controller formulae into a cleaner, more symmetric descriptor form having the important practical advantage that it eliminates the numerical difficulties that can occur in cases where one or both of the Riccati solutions, Pi and Q, blow up and in cases where l— QP is nearly singular. Numerical difficulties previously associated with verifying the existence conditions P ≥ 0, Q ≥ 0, and λmaxlpar;QP) < 1 are largely eliminated by e...

Simplifying the H∞ theory via loop-shifting, matrix-pencil and descriptor concepts

The established theory of nonzero sum games is used to solve a mixed H/sub 2//H/sub /spl infin//, control problem. Our idea is to use the two pay-off functions associated with a two-player Nash game to represent the H/sub 2/ and H/sub /spl infin// criteria separately. We treat the state-feedback problem and we find necessary and sufficient conditions for the existence of a solution. Both the finite and infinite time problems are considered. In the infinite horizon case we present a full stability analysis. The resulting controller is a constant state-feedback law, characterized by the solution to a pair of cross-coupled Riccati equations, which may be solved using a standard numerical integration procedure. We begin our development by considering strategy sets containing linear controllers only. At the end of the paper we broaden the strategy sets to include a class of nonlinear controls. It turns out that this extension has no effect on the necessary and sufficient conditions for the existence of a solution or on the nature of the controllers. >

A Nash game approach to mixed H/sub 2//H/sub /spl infin// control

Starting from an existing advanced motorcycle dynamics model, which allows simulation of reasonably general motions and stability, modal and response computations for small perturbations from any trim condition, improvements are described. These concern (a) tyre/road contact geometry, (b) tyre shear force and moment descriptions, as functions of load, slip and camber, (c) tyre relax- ation properties, (d) a new analytic treatment of the monoshock rear suspension mechanism with sample results, (e) parameter values describing a contemporary high performance machine and rider, (f) steady-state equilibrium and power checking and (g) steering control. In particular, the "Magic Formula" motorcycle tyre model is utilised and complete sets of parameter values for contemporary tyres are derived by identification methods. The new model is used for steady turning, stability, design parameter sensitivity and response to road forcing calculations. The results show the predictions of the model to be in general agreement with observations of motorcycle behaviour from the field and they suggest that frame flexibility remains an important design and analysis area, despite improvements in frame designs over recent years. Motorcycle rider parameters have significant influences on the behaviour, with results consistent with a commonly held view, that lightweight riders are more likely to suffer oscillation problems than heavyweight ones.

/pdf/advances-in-the-modelling-of-motorcycle-dynamics-2426qmelhy.pdf

Advances in the modelling of motorcycle dynamics

It is shown that necessary and sufficient conditions for the existence of sub-optimal solutions to the model matching problem associated with H∞ control are that two coupled J-spectral factorizations exist. The second J-spectral factor provides a parameterization of all solutions to the model matching problem. The existence of the J-spectral factors is then shown to be equivalent to the existence of stabilizing, non-negative definite solutions to two algebraic Riccati equations, allowing a state-space formula for a linear fractional representation of all controllers to be given.

David J. N. Limebeer

Papers

Linear Robust Control

Simplifying the H∞ theory via loop-shifting, matrix-pencil and descriptor concepts

A Nash game approach to mixed H/sub 2//H/sub /spl infin// control

Advances in the modelling of motorcycle dynamics

A J-Spectral Factorization Approach to H∞ Control