A new sufficient condition for generic pole assignment by output feedback

We show that for strictly proper systems withmoutputs,linputs (m geq l), McMillan degreen, and controllability indexeslambda_{1} geq lambda_{2} geq ...geq lambda_{1} >0, one can in the generic case arbitrarily assign min((q + 1)m + q + b(l - 1), n + q)closed-loop poles with a proper orderqcompensator. This represents an improvement over results reported earlier in the literature.     

Robust observers for systems with parameters

In many cases linear multivariable models which describe physical systems contain structured parametric type uncertainties. Specifically it may be that the state space model is expressed in a form which involves parameters. We first consider circumstances in which a parameter independent observer can be constructed for the system, and suggest conditions that guarantee its existence. We also give conditions under which such a robust observer can be subsequently used with additional dynamics in a parameter free configuration for regulation.     

The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

Stability preserving maps and robust design

In this paper we present the concept of a matrix stability preserving map and show its impact on the problem of robust controller design. We develop a number of tests for checking whether a given matrix is a stability preserving map. We show that the concept of a stability preserving map can be used to provide a different characterization of the existence of a fixed order controller that simultaneously stabilizes a finite number of plants. We also demonstrate how it can be used to state conditions for the robust stabilization of families of plants with real parameter uncertainty. In addition, we show how stability preserving map tests lead to robust stabilization techniques and apply the methodology to a number of examples.     

To stabilize a k real parameter affine family of plants it suffices to simultaneously stabilize 4 polynomials

A controller stabilizes an entire family of plants with affine uncertainty, if it simultaneously stabilizes a finite number of polynomials. An upper bound for this number is 4^k.     

Controller synthesis via the 4-polynomial approach

In some recent work it was shown that to stabilize systems with real parameter uncertainty it suffices to find a controller that simultaneously stabilizes a finite number of polynomials. These polynomials include those generated from the ‘vertex’ plants as well as some generated by some ‘fictitious’ vertex plants that involve the controller. This paper deals with the issues of existence of such a controller, controller synthesis, and conservativeness of the design. It is shown how this approach can ‘enhance’ the stability robustness of an H_∞ design.     

A note on robust stabilization for systems with parameters

We consider the problem of stabilizing a linear time invariant multivariable system whose transfer function admits a factorization which contains parametric type uncertainties. The parameters are known to take values in some region Ω_a. We first give a sufficient condition for robust stability when a proper feedback compensator is used. Here it is important to note that no assumption is made about the number of unstable plant poles. Following this we focus attention on how to compute an appropriate robustly stabilizing compensator. This is done in a two step procedure. Initially the solutions of a Diophantine equation generate a class of proper compensators, each of which stabilizes the ‘nominal’ closed loop system. This is followed by a Nyquist Theorem inspired minimization which yields the final choice. An illustrative physical example is given. The compensator thus constructed guarantees stability over . A discussion about how to compute the size of the attainable robust stability region is also included.     

A new parameterization of stable polynomials

In this note, we develop a new characterization of stable polynomials. Specifically, given n positive, ordered numbers (frequencies), we develop a procedure for constructing a stable degree n monic polynomial with real coefficients. This construction can be viewed as a mapping from the space of n ordered frequencies to the space of stable degree n monic polynomials. The mapping is one-one and onto, thereby giving a complete parameterization of all stable, degree n monic polynomials. We show how the result can be used to generate parameterizations of stabilizing fixed-order proper controllers for unity feedback systems. We apply these results in the development of stability margin lower bounds for systems with parameter uncertainty.     

Achieving robust stability by sensing additional system outputs

In this note we investigate the problem of robust stability in the context of linear time-invariant multivariable systems which are affected by structured parametric type uncertainty. We demonstrate that by generating additional outputs which are then made available for feedback, and thus using sensors for more measurements, robust closed-loop system stability can be achieved. We state and prove results that suggest when this is possible, and show how it can be done.     

Representations of controllers that achieve robust performance for systems with real parameter uncertainty

Consider a family of single input single output plants described by transfer functions that involve real parameter uncertainty. Parameter values are known to lie in a hypercube. Assume that a class of available controllers has been prescribed, along with a bound for the sensitivity transfer function to ensure tracking. It is of interest to determine whether a controller from the given class exists that guarantees robust stability and robust asymptotic tracking. In this paper we present a problem formulation and then provide a solution based on it. Not only do we address the existence question but also give representation of controllers from the class that meet the robustness requirements.