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1985 | Buch | 3. Auflage

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Linear Multivariable Control

A Geometric Approach

verfasst von: W. Murray Wonham

Verlag: Springer New York

Buchreihe : Stochastic Modelling and Applied Probability

Enthalten in: Professional Book Archive

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Über dieses Buch

In wntmg this monograph my aim has been to present a "geometric" approach to the structural synthesis of multivariable control systems that are linear, time-invariant and of finite dynamic order. The book is ad dressed to graduate students specializing in control, to engineering scientists involved in control systems research and development, and to mathemati cians interested in systems control theory. The label "geometric" in the title is applied for several reasons. First and obviously, the setting is linear state space and the mathematics chiefly linear algebra in abstract (geometric) style. The basic ideas are the familiar system concepts of controllability and observability, thought of as geometric prop erties of distinguished state subspaces. Indeed, the geometry was first brought in out of revulsion against the orgy of matrix manipulation which linear control theory mainly consisted of, around fifteen years ago. But secondly and of greater interest, the geometric setting rather quickly sug gested new methods of attacking synthesis which have proved to be intuitive and economical; they are also easily reduced to matrix arithmetic as soon as you want to compute. The essence of the "geometric" approach is just this: instead of looking directly for a feedback law (say u = Fx) which would solve your synthesis problem if a solution exists, first characterize solvability as a verifiable property of some constructible state subspace, say Y. Then, if all is well, you may calculate F from Y quite easily.

Inhaltsverzeichnis

Frontmatter

0. Mathematical Preliminaries

Abstract

For the reader’s convenience we shall quickly review linear algebra and the rudiments of linear dynamic systems. In keeping with the spirit of this book we emphasize the geometric content of the mathematical foundations, laying stress on the presentation of results in terms of vector spaces and their subspaces. As the material is standard, few proofs are offered; however, detailed developments can be found in the textbooks cited at the end of the chapter. For many of the simpler identities involving maps and subspaces, the reader is invited to supply his own proofs; an illustration and further hints are provided in the exercises. It is also recommended that the reader gain practice in translating geometric statements into matrix formalism, and vice versa; for this, guidance will also be found in the exercises.

W. Murray Wonham

1. Introduction to Controllability

Abstract

It is natural to say that a dynamic system is “controllable” if, by suitable manipulation of its inputs, the system outputs can be made to behave in some desirable way. In this chapter one version of this concept will be made precise, and some of its implications explored, for the system of Section 0.17:

$$\dot x(t) = Ax(t) + Bu(t),t \ge 0$$

(0.1)

We start by examining those states which, roughly speaking, the control u(·) in (0.1) is able to influence.

W. Murray Wonham

2. Controllability, Feedback and Pole Assignment

Abstract

Consider as usual the system

$$\dot x(t) = Ax(t) + Bu(t),t \ge 0$$

(0.1)

Suppose we are free to modify (0.1) by setting

$$u(t) = Fx(t) + v(t),t \ge 0$$

(0.2)

where υ(·) is a new external input, and F: X → Uis an arbitrary map. We refer to F as the state feedback. The obvious result of introducing state feedback is to change the pair (A, B) in (0.1) into the pair (A + BF, B). We shall explore the effect of such a transformation of pairs on controllability and on the spectrum of A + BF. Our main result is that if (A, B) is controllable then σ(A + BF) can be assigned arbitrarily by suitable choice of F, and this property in turn implies controllability.

W. Murray Wonham

3. Observability and Dynamic Observers

Abstract

Observability is a property of a dynamic system together with its observable inputs and outputs, according to which the latter alone suffice to determine exactly the state of the system. A data processor which performs state determination is called an “observer.” In an intuitive sense observability is a property dual to controllability: a system is controllable if any state can be reached by suitable choice of input; it is observable if (when the input is known) its state can be computed by suitable processing of the output. For linear time-invariant systems this intuitive duality translates into a precise algebraic duality.

W. Murray Wonham

4. Disturbance Decoupling and Output Stabilization

Abstract

In this chapter we first discuss a simple feedback synthesis problem, concerned with decoupling from the system output the effect of disturbances acting at the input. Examination of this problem leads naturally to the fundamental geometric concept of (A, B)-invariant subspace, which underlies many of our constructions and results in later chapters. As an immediate application, we show how state feedback may be utilized to stabilize outputs or, more generally, to realize a given set of characteristic exponents in the time response of the output.

W. Murray Wonham

5. Controllability Subspaces

Abstract

Given a system pair (A, B) we consider all pairs (A + BF, BG) which can be formed by means of state feedback F and the connection of a “gain” matrix G at the system input (Fig. 5.1). The controllable subspace of (A + BF, BG) is called a controllability subspace (c.s.) of the original pair (A, B). The family of c.s. of a fixed pair (A, B) is a subfamily, in general proper, of the (A, B)-invariant subspaces: the importance of c.s. derives from the fact that the restriction of A + BF to an (A + BF)-invariant c.s. can be assigned an arbitrary spectrum by suitable choice of F.

W. Murray Wonham

6. Tracking and Regulation I: Output Regulation

Abstract

A typical multivariable control problem requires the design of dynamic compensation to guarantee the following desirable behavior of the closed loop system.

W. Murray Wonham

7. Tracking and Regulation II: Output Regulation with Internal Stability

Abstract

In this chapter, we continue the investigation begun in Chapter 6 on output regulation for the system {fy151-1|(0.1a)} {fy151-2|(0.1b)} {fy151-3|(0.1c)} As in Chapter 6, we shall explore the basic geometric structure, while in Chapter 8, we turn to simple and direct procedures that lend themselves to computation.

W. Murray Wonham

8. Tracking and Regulation III: Structurally Stable Synthesis

Abstract

In this chapter we investigate the regulator problem with internal stability (RPIS) discussed in Chapter 7, from the viewpoint of well-posedness and genericity in the sense of Section 0.16, and of structurally stable implementation. Subject to mild restrictions it is shown that, if and only if RPIS is well-posed, a controller can be synthesized which preserves output regulation and loop stability in the presence of small parameter variations, of a specified type, in controller and plant. Synthesis is achieved by means of a feedback configuration which, in general, incorporates an invariant, and suitably redundant, copy of the exosystem, namely the dynamic model adopted for the exogenous reference and disturbance signals which the system is required to process. The geometric idea underlying these results is transversality, or the intersection of subspaces in general position.

W. Murray Wonham

9. Noninteracting Control I: Basic Principles

Abstract

Consider a multivariable system whose scalar outputs z_ij have been grouped in disjoint subsets, each having a physical significance to distinguish it from the remaining subsets. Represent the output subsets by vectors

$${z_i} = col(zi1,...,{z_{ipi}}),i \in k$$

For instance, with k = 3 and each p_i = 2, z_i could represent angular position and velocity of a rigid body relative to the ith axis of rotation. Next, suppose the system is controlled by scalar inputs u₁, …, u_m, where m ≥ k. In many applications it is desirable to partition the input set into k disjoint subsets U₁, …, U_k, such that for each i ∊ k the inputs of U_i control the output vector z_i completely, without affecting the behavior of the remaining z_j, j ≠ i. Such a control action is noninteracting, and the system is decoupled. From an input-output viewpoint decoupling splits the system into k independent subsystems. Considerable advantages may result of simplicity and reliability, especially if control is partially to be executed by a human operator.

W. Murray Wonham

10. Noninteracting Control II: Efficient Compensation

Abstract

In this chapter we continue the discussion in Chapter 9 on solution of EDP by dynamic compensation. A refinement of the construction used to prove Theorem 9.4 permits a further reduction of the bound (9.6.2) on dynamic order. The reduced bound turns out to be strictly minimal if the number of independent control inputs is equal to the number of output blocks to be decoupled. As these results are somewhat specialized and their proofs are intricate, the reader interested only in the main features of the theory is advised to skip to Chapter 11.

W. Murray Wonham

11. Noninteracting Control III: Generic Solvability

Abstract

In this chapter we discuss solvability of the noninteraction problem from the viewpoint of genericity, in the parameter space of the matrices A, B, and the D_i (i ∊ k). It turns out that noninteraction is possible for almost all data sets (A, B, D₁, …, D_k) if and only if the array dimensions of the given matrices satisfy appropriate, rather mild, constraints. When these conditions fail decoupling is possible, if at all, only for system structures which are rather special. Finally, in the generically solvable case we determine the generic bounds on dynamic order of a decoupling compensator, corresponding to the “naive” and “efficient” extension procedures of Chapters 9 and 10, respectively.

W. Murray Wonham

12. Quadratic Optimization I: Existence and Uniqueness

Abstract

In previous chapters our objectives in system synthesis have been almost entirely qualitative: we have indeed imposed requirements like stability on the system spectrum, but in the main have sought to realize very general properties of signal flow, as in tracking or noninteraction. By contrast, in this chapter and the next we take a somewhat more quantitative approach to realizing good dynamic response. We describe a systematic way of computing linear state feedback which ensures “optimal” recovery from an impulsive disturbance acting at the system input. It will later be clear how to incorporate the method into the framework of synthesis techniques already presented.

W. Murray Wonham

13. Quadratic Optimization II: Dynamic Response

Abstract

The approach to state feedback optimization described in the previous chapter has been widely advertised as a systematic technique to achieve good transient response with reasonable computational effort. While not generally disputed, this claim is based more on numerical experience than compelling theoretical arguments. Nevertheless, some precise information is available about the qualitative behavior of the closed loop system as a function of the weighting matrices M and N of the cost functional (12.1.3). In this chapter we present a selection of the simpler results, referring the reader to the literature for supplementary developments.

W. Murray Wonham

Backmatter

Titel: Linear Multivariable Control
verfasst von: W. Murray Wonham
Verlag: Springer New York
Electronic ISBN: 978-1-4612-1082-5
Print ISBN: 978-1-4612-7005-8
DOI: https://doi.org/10.1007/978-1-4612-1082-5

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

0. Mathematical Preliminaries

1. Introduction to Controllability

2. Controllability, Feedback and Pole Assignment

3. Observability and Dynamic Observers

4. Disturbance Decoupling and Output Stabilization

5. Controllability Subspaces

6. Tracking and Regulation I: Output Regulation

7. Tracking and Regulation II: Output Regulation with Internal Stability

8. Tracking and Regulation III: Structurally Stable Synthesis

9. Noninteracting Control I: Basic Principles

10. Noninteracting Control II: Efficient Compensation

11. Noninteracting Control III: Generic Solvability

12. Quadratic Optimization I: Existence and Uniqueness

13. Quadratic Optimization II: Dynamic Response

Backmatter