State-space solutions to standard H 2 and H ∞ control problems
15 Jun 1988-Iss: 25, pp 1691-1696
TL;DR: In this article, simple state-space formulas are presented for a controller solving a standard H∞-problem, where 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 theory.
Abstract: 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.
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TL;DR: In this article, 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.
Abstract: 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. >
5,272 citations
TL;DR: In this paper, the continuous and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI), and two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H ∞-suboptimal controllers, including reduced-order controllers.
Abstract: The continuous- and discrete-time H∞ control problems are solved via elementary manipulations on linear matrix inequalities (LMI). Two interesting new features emerge through this approach: solvability conditions valid for both regular and singular problems, and an LMI-based parametrization of all H∞-suboptimal controllers, including reduced-order controllers.
The solvability conditions involve Riccati inequalities rather than the usual indefinite Riccati equations. Alternatively, these conditions can be expressed as a system of three LMIs. Efficient convex optimization techniques are available to solve this system. Moreover, its solutions parametrize the set of H∞ controllers and bear important connections with the controller order and the closed-loop Lyapunov functions.
Thanks to such connections, the LMI-based characterization of H∞ controllers opens new perspectives for the refinement of H∞ design. Applications to cancellation-free design and controller order reduction are discussed and illustrated by examples.
3,200 citations
Book•
21 Apr 2008
TL;DR: 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.
Abstract: 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
1,927 citations
Cites background from "State-space solutions to standard H..."
...A major breakthrough was made by Doyle, Glover, Khargonekar and Francis [65], who showed that the solution to the problem could be obtained using Riccati equations and that a controller of low order could be found....
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...Much later it was shown that solutions to robust control problems also had a similar structure but with different ways of computing observer and state feedback gains [65]....
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Proceedings Article•
01 Aug 2008
TL;DR: Insights into inherent properties of robust systems will provide a better understanding of complex diseases and a guiding principle for therapy design.
Abstract: Robustness is a ubiquitously observed property of biological systems. It is considered to be a fundamental feature of complex evolvable systems. It is attained by several underlying principles that are universal to both biological organisms and sophisticated engineering systems. Robustness facilitates evolvability and robust traits are often selected by evolution. Such a mutually beneficial process is made possible by specific architectural features observed in robust systems. But there are trade-offs between robustness, fragility, performance and resource demands, which explain system behaviour, including the patterns of failure. Insights into inherent properties of robust systems will provide us with a better understanding of complex diseases and a guiding principle for therapy design.
1,875 citations
Book•
01 Aug 19941,655 citations
References
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TL;DR: In this paper, a complete characterization of all rational functions that minimize the Hankel-norm is derived, and the solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations.
Abstract: The problem of approximating a multivariable transfer function G(s) of McMillan degree n, by Ĝ(s) of McMillan degree k is considered. A complete characterization of all approximations that minimize the Hankel-norm is derived. The solution involves a characterization of all rational functions Ĝ(s) + F(s) that minimize where Ĝ(s) has McMillan degree k, and F(s) is anticavisal. The solution to the latter problem is via results on balanced realizations, all-pass functions and the inertia of matrices, all in terms of the solutions to Lyapunov equations. It is then shown that where σ k+1(G(s)) is the (k+l)st Hankel singular value of G(s) and for one class of optimal Hankel-norm approximations. The method is not computationally demanding and is applied to a 12-state model.
2,980 citations
TL;DR: In this article, the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problems of stabilization, and the feedback schemes obtainable from a given plant are parameterized.
Abstract: In this paper, the problem of sensitivity, reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization. Stable feedback schemes obtainable from a given plant are parameterized. Salient properties of sensitivity reducing schemes are derived, and it is shown that plant uncertainty reduces the ability, of feedback to reduce sensitivity. The theory is developed for input-output systems in a general setting of Banach algebras, and then specialized to a class of multivariable, time-invariant systems characterized by n \times n matrices of H^{\infty} frequency response functions, either with or without zeros in the right half-plane. The approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity, and on the concept of an approximate inverse. Approximate invertibility, of the plant is shown to be a necessary and sufficient condition for sensitivity reduction. An indicator of approximate invertibility, called a measure of singularity, is introduced. The measure of singularity of a linear time-invariant plant is shown to be determined by the location of its right half-plane zeros. In the absence of plant uncertainty, the sensitivity, to output disturbances can be reduced to an optimal value approaching the singularity, measure. In particular, if there are no right half-plane zeros, sensitivity can be made arbitrarily small. The feedback schemes used in the optimization of sensitivity resemble the lead-lag networks of classical control design. Some of their properties, and methods of constructing them in special cases are presented.
2,203 citations
TL;DR: In this paper, the optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated, and the integrand of the performance criterion is allowed to be fully quadratically in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem.
Abstract: The optimal control of linear systems with respect to quadratic performance criteria over an infinite time interval is treated. Both the case in which the terminal state is free and that in which the terminal state is constrained to be zero are treated. The integrand of the performance criterion is allowed to be fully quadratic in the control and the state without necessarily satisfying the definiteness conditions which are usually assumed in the standard regulator problem. Frequency-domain and time-domain conditions for the existence of solutions are derived. The algebraic Riccati equation is then examined, and a complete classification of all its solutions is presented. It is finally shown how the optimal control problems introduced in the beginning of the paper may be solved analytically via the algebraic Riccati equation.
1,436 citations
TL;DR: In this article, the problem of selecting rational weighting matrices to ensure required feedback system properties is reduced to a pair of matrix polynomial equations and the optimal solution is highly non-unique, special solutions with additional optimality properties are considered.
Abstract: The properties of a feedback system where the plant has rational transfer matrix H and the compensator has transfer matrix G can be characterized through the system functions S:=(I +HG)− and T:= G(I+HG)−1. Good disturbance attenuation, robustness, limited bandwidth and compensator roll-off may be obtained by minimizing a criterion of the form ||Z||∞, where Z:= V∗(S∗T∗W1∗WlS + T∗W2∗W2T)V, with respect to the compensator transfer matrix G. Here V, W1, and W2 are suitable rational weighting matrices. The solution of the problem can be reduced to a pair of matrix polynomial equations. Since the optimal solution is highly non-unique, special solutions with additional optimality properties are considered as well. The paper includes a discussion of the numerical solution of the polynomial equations and of the question how to choose the weighting matrices to ensure required feedback system properties. An example illustrates the results.
208 citations
TL;DR: In this article, the pole-zero cancellations which occur in a class of H-optimal control problems which may be embedded in the configuration of Fig. 1 are studied, and a general bound on the McMillan degree of all controllers which are stabilizing and lead.
Abstract: The aim of this paper is to study the pole-zero cancellations which occur in a class of H-optimal control problems which may be embedded in the configuration of Fig. 1. H control problems are said to be of the first kind if both P12(s) and P21(s) are square but not necessarily of the same size. It is primarily this class Of problems which will concern us here. A general bound on the McMillan degree of all controllers which are stabilizing and lead.to a closed loop which satisfies (((s)((o -< p (p need not be optimal in the L-norm sense) is derived. As illustrated in Fig. 1, (s) is the transfer function relating yl(s) to Ul(S). If the McMillan degree of P(s) in Fig. is n, we show that in the single-loop (SISO) case the corresponding (unique) H-optimal controller never requires more than n states. In the multivariable case, there is a continuum of optimal controllers whose McMillan degree satisfies this same bound, although other controllers with higher McMillan degree also exist. The derivation of these bounds require several steps, each of which is of independent system theoretic interest.
83 citations