Static output feedback: a survey
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paper_title: Control System Synthesis : A Factorization Approach

paper_content:

This book introduces the so-called "stable factorization approach" to the synthesis of feedback controllers for linear control systems. The key to this approach is to view the multi-input, multi-output (MIMO) plant for which one wishes to design a controller as a matrix over the fraction field F associated with a commutative ring with identity, denoted by R, which also has no divisors of zero. In this setting, the set of single-input, single-output (SISO) stable control systems is precisely the ring R, while the set of stable MIMO control systems is the set of matrices whose elements all belong to R. The set of unstable, meaning not necessarily stable, control systems is then taken to be the field of fractions F associated with R in the SISO case, and the set of matrices with elements in F in the MIMO case. The central notion introduced in the book is that, in most situations of practical interest, every matrix P whose elements belong to F can be "factored" as a "ratio" of two matrices N,D whose elements belong to R, in such a way that N,D are coprime. In the familiar case where the ring R corresponds to the set of bounded-input, bounded-output (BIBO)-stable rational transfer functions, coprimeness is equivalent to two functions not having any common zeros in the closed right half-plane including infinity. However, the notion of coprimeness extends readily to discrete-time systems, distributed-parameter systems in both the continuous- as well as discrete-time domains, and to multi-dimensional systems. Thus the stable factorization approach enables one to capture all these situations within a common framework. The key result in the stable factorization approach is the parametrization of all controllers that stabilize a given plant. It is shown that the set of all stabilizing controllers can be parametrized by a single parameter R, whose elements all belong to R. Moreover, every transfer matrix in the closed-loop system is an affine function of the design parameter R. Thus problems of reliable stabilization, disturbance rejection, robust stabilization etc. can all be formulated in terms of choosing an appropriate R. This is a reprint of the book Control System Synthesis: A Factorization Approach originally published by M.I.T. Press in 1985. Table of Contents: Introduction / Proper Stable Rational Functions / Scalar Systems: An Introduction / Matrix Rings / Stabilization
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paper_title: Algebraic tests for output stabilizability

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In this paper, we provide algebraic tests to determine whether a linear single-input-single-output (SISO) system, is stabilizable with a constant output feedback. >
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paper_title: Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint

paper_content:

Classical control theory is concerned with the topics in our title in the context of single-input/single-output systems. There is now a large and growing literature on the extension of these ideas to the multiinput/multioutput case. This development has posed certain difficulties, some due to the intrinsic nature of the problem and some, we would argue, due to an inadequate reflection on what the multivariable problem calls for. In this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory. A rather satisfactory generalization of the Nyquist criterion will be described, and a clear analog of the asymptotic properties of the root locus will be obtained in the "multiparameter" case. However, an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains. This calls for a new ingredient, a compactification of the space of gains, and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement. This problem is difficult in the multivariable case, but by establishing a correspondence with a classical set of problems in geometry, we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field.
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paper_title: Hermitian pencils and output feedback stabilization of scalar systems

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Abstract Necessary and sufficient semi-algebraic conditions for (a) the stabilization of scalar transfer functions, and (b) the assignability of real poles by static output feedback, are given in terms of the Weierstrass invariants of an associated hermitian matrix pencil. An explicit graphical test for output feedback stabilizability is derived which is equivalent to the Nyquist criterion.
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paper_title: Geometric methods for the classification of linear feedback systems

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Abstract The precise determination of the nature and form of system invariants under output feedback is of fundamental interest in classical and modern control theory. An algebraic solution of this problem would entail, for example, a description of the generators and relations of the ring of output feedback invariants. Geometrically, we seek to describe the basic properties of the quotient space whose points consist of feedback equivalence classes of the system, recovering the algebraic theory in terms of the function theory on this space. Naturally, any description should be made in system theoretic terms. In this paper we construct and study, in the scalar case, the quotient manifold for linear systems modulo the full output feedback group using classical algebraic geometry. As a corollary, we obtain explicit generators for the associated ring of invariants which we then interpret in terms of classical control theory, e.g. in terms of root-locus plots. Turning to the question of the existence of continuous canonical forms, we show that these exist globally, only when the system order is odd. In the even case, the obstruction to the existence of globally defined continuous forms is described in terms of the values of the system Cauchy index. This is illustrated for low order examples and certain questions concerning relations among the generators is treated using methods of the Schubert calculus. The complete set of invariants for the full output feedback group is obtained and our results concerning the existence and nonexistence of canonical forms is also compared to related results contained in the literature.
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paper_title: Parametrization of all stabilizing controllers via quadratic Lyapunov functions

paper_content:

This paper presents a parametrization of all finite-dimensional, linear time-invariant controllers which asymptotically stabilize a given finite-dimensional, linear time-invariant system. Both continuous-time and discrete-time systems are considered. A potential advantage over existing parametrization schemes in the frequency domain is that the controller order can be fixed. Consequently, necessary and sufficient conditions for stabilizability via static output feedback controller are obtained and stated by the existence of a quadratic Lyapunov functionV(x):=x T Px such thatP satisfies a linear matrix inequality (LMI), whileP−1 satisfies another LMI. If the controller order is not fixed a priori, then the resulting computational problem can be made convex, and a controller of order less than or equal to the plant order may always be constructed.
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paper_title: Output feedback stabilization and related problems-solution via decision methods

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Given an unstable finite-dimensional linear system, the output feedback problem is, first, to decide whether it is possible by memoryless linear feedback of the output to stabilize the system, and, second, to determine a stabilizing feedback law if such exists. This paper shows how this and a number of other linear system theory problems can be simply reformulated so as to allow application of known algorithms for solution of the existence question, with the construction problem being solved by some extension of these known algorithms. The first part of the output feedback problem is solvable with a finite number of rational operations, and the second with a finite number of polynomial factorizations. Other areas of application of the algorithm are described: Stability and positivity tests, low-order observer and controller design, and problems related to output feedback. Alternative computational procedures more or less divorced from the known algorithms are also proposed.
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paper_title: Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint

paper_content:

Classical control theory is concerned with the topics in our title in the context of single-input/single-output systems. There is now a large and growing literature on the extension of these ideas to the multiinput/multioutput case. This development has posed certain difficulties, some due to the intrinsic nature of the problem and some, we would argue, due to an inadequate reflection on what the multivariable problem calls for. In this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory. A rather satisfactory generalization of the Nyquist criterion will be described, and a clear analog of the asymptotic properties of the root locus will be obtained in the "multiparameter" case. However, an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains. This calls for a new ingredient, a compactification of the space of gains, and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement. This problem is difficult in the multivariable case, but by establishing a correspondence with a classical set of problems in geometry, we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field.
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paper_title: On pole assignment by output feedback

paper_content:

A new result of the problem of pole assignment by output feedback is presented. A simple condition is given under which min (n, m + r − 1) closed-loop poles can be assigned arbitrarily by a gain output feedback, where n, mand r are the dimensions of the state, output and input vectors, respectively. Under the same condition, a new lower bound for the number of poles assignable by a fixed-order dynamic compensator is given which improves considerably on the results of Breach and Pearson (1970) and Ahmari and Vacroux (1973). A design algorithm and illustrative examples are also given.
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paper_title: A further result on the problem of pole assignment by output feedback

paper_content:

In this paper a new result in the problem of pole assignment by gain output feedback is given. Roughly speaking, this result says that arbitrary pole assignment is possible for almost all systems if n \mu, m \geq \nu . Here n, r and m are the number of states, of inputs and of outputs, respectively, and ν and μ are the so-called controllability index and the observability index of the system, respectively. This result extends the author's previous one in [1]. The basic idea in [1] is developed further and its geometrical meaning is amplified. The proof of the theorem itself gives a method of constructing a desired gain matrix. An example is given to show the feasibility of the algorithm.
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Title: Static Output Feedback: A Survey

Section 1: Introduction
Description 1: This section introduces the static output feedback problem, its importance in control engineering, and outlines the structure of the paper.

Section 2: Necessary Conditions
Description 2: This section identifies the cases and theorems where static output feedback cannot stabilize an open-loop unstable system.

Section 3: Sufficient Conditions
Description 3: This section discusses conditions under which static output feedback can stabilize systems, particularly focusing on SISO systems and simple root-locus arguments.

Section 4: Design Approaches and Limitations
Description 4: This section reviews various graphical and algebraic methods used for designing stabilizing static output controllers and highlights their limitations in both SISO and MIMO cases.

Section 5: Parameterization Methods
Description 5: This section lists parameterization results that are useful in solving the static output feedback problem and discusses some non-testable conditions.

Section 6: Covariance Assignment and Stabilizability
Description 6: This section explains how covariance assignment methods can be used to find static output feedback controllers that assign a given matrix value to the steady-state covariance matrix.

Section 7: Decision Methods
Description 7: This section introduces the usage of decision methods in finding stabilizing compensators and outlines a basic approach to solve the output feedback problem using these methods.

Section 8: Pole Placement With Output Feedback
Description 8: This section covers methods to select the gain K for placing poles in the closed-loop system at desired symmetric locations.

Section 9: LQR With Static Output Feedback
Description 9: This section discusses the selection of K to minimize a performance index subject to the closed-loop system constraints.

Section 10: Conclusion
Description 10: This section summarizes the survey findings, discusses the complexity and limitations of existing methods, and suggests directions for future research.