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Frequency domain properties of uncertain rational functions depending on two independent interval polynomials are investigated. Several simplifications for efficient computation of the envelope of the Nyquist plots of the uncertain family are reported. Stronger computational reduction results are given for the Bode magnitude and phase plots of an interval plant-controller family of transfer functions. These results allow for a clear understanding of the extremality properties of several frequency domain performance specifications commonly used in control system design. >

Computation of all possible frequency responses of a linear system that depends on m uncertain parameters is the subject of this paper. Using a brute force approach, this computation could be carried out using an (m + l)-dimensional parameter sweep, i.e. there is the usual frequency sweep plus one sweep for each uncertain parameter. A much simpler computational alternative is possible if the uncertain system's transfer function has a linear dependence on the parameters and if the set of possible parameter values has a polytopic structure. For this class of uncertain systems, it has previously been shown that, for a single fixed frequency, all possible responses can be determined using several 1-dimensional sweeps. Thus, for a range of frequencies, the obvious extension would involve several 2-dimensional sweeps. However, this obvious extension is not the simplest way to carry out the computation. Indeed, this paper will show that all possible responses for a range of frequencies can be determined...

Determination of the steady state response of a system to an input composed of steps, ramps, and various other signals is considered. If the system and/or the input depend on uncertain parameters then a worst-case philosophy dictates that the analysis goal should be to determine the maximum and minimum possible equilibrium values of the output. It is shown that the system is robustly stable and if the dependence on the uncertain parameters is to an extent multiaffine, then the extreme values of the steady state response over the entire set of parameters is equal to the extreme values over just the vertices of the parameter set. This vertex result is doubly useful because it is valid in both continuous-time and discrete-time. For the transient response of stable systems, the vertices do not provide sufficient information for a complete analysis. Examples are presented which show that the maximum overshoot to the step response of a robustly stable uncertain system does not necessarily occur at a vertex. The systems in these examples have an affine dependence on a single uncertain parameter. >

A number of procedures exist for determining the stability of a linear system that is subject to parametric uncertainty provided the system order is constant over the range of parameter variations. This paper extends these tests to systems with parametric uncertainties that induce order change. The extended tests are also shown to apply to systems with nonparametric (unstructured) uncertainty through an equivalent parameterization of the nonparametric uncertainty class. The results are applied to the stability analysis of a crane model with a flexible cable.

When applying the parameter space design method of the preceding chapter, regions of controller coefficients are first determined, guaranteeing simultaneous Γ-stability of a finite plant family. The design method of [111,110], which will be presented in this chapter, also searches for simultaneously stabilizing controller coefficients. However, in contrast to the parameter space method no stability boundaries in the space of controller coefficients are generated. Rather, the coefficients are determined by optimizing a vector performance index, the components of which rate different design specifications.

With Chapter 11 we are entering into Part IV of this book, the part that deals with design. For an uncertain closed-loop characteristic polynomial p(s, q, k) we want to find a k = k0 such that the polynomial p(s, q, k0) is Γ-stable for all q ∈ Q. Generic situations are:1. There exists such a k0 and also a neighborhood KΓ of k0 has the same property, see Fig. 3.10. We want to pick the “best” k ∈ KΓ in consideration of further design requirements.

The examples of Chapter 1 have in common that the plant is insufficiently damped or even unstable in its operating domain Q. Thus, a primary task of a control system is stabilization with sufficient damping for all q ⋲ Q. Since eigenvalues cannot be shifted by a feedforward control system, we need a feedback structure. Fig. 2.1 shows an example of a control system with a plant family G(s,Q) = {g{s,q) |q ⋲ Q}, a feedback compensator (or controller) c(s), and a feedforward path (or prefilter) Ƒ(s).

It is well known that a linear time-invariant system is stable if the roots of its characteristic polynomial have a negative real part. In short we speak of “stability of a polynomial”.

In the previous chapter the stability test by zero exclusion from the value set at all frequencies was used as a concept for proofs of Kharitonov’s theorem and the edge theorem. For nonlinear parameter dependency there are no such simple results, it may be possible, however, to construct the value set and to use it for the stability test by zero exclusion. We will see that the construction of value sets can be performed extremely fast if the system has a so-called tree structure.

By assuming controller structures in Chapter 2 we have generated several examples of closed-loop characteristic polynomials p(s, q,k), where the vector k contains the free design parameters in the fixed controller structure and q contains the uncertain plant parameters in a given operating domain Q, i.e. q ∈Q.

Interval plants are of interest in control theory as models of uncertain systems. They are useful because many worst-case analyses of these models are simple to carry out. For example, robust stability of an interval plant can be determined by investigating only the four Kharitonov vertices of the denominator polynomial. Also, the maximum peak of the Bode magnitude plot can be found using just 16 special plant vertices. These 16 vertices are connected by 32 special line segments. Most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out using just the segments. From these results, it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses. This paper presents examples showing that these conjectures are not true.

This paper investigates the frequency response of an uncertain system whose set of possible transfer functions is represented by a polytope of numerator-denominator polynomial pairs. It will be shown that the frequency response of the edge transfer functions provides most of the information required to determine all possible frequency responses of the system Once the edge response is computed, a simple procedure can be used to obtain the complete response. These results can be used to compute the frequency response at any point in the complex plane, so they are applicable to continuous-time and discrete-time systems. For an interval family of tansfer functions, the edge result in this paper can be combined with known properties of interval polynomials to give an even simpler method of computing the frequency response. This simplification is only valid for frequencies on the jw-axis, so its application is limited to continuous-time systems.

The stability version of the Edge Theorem states that a polytope of polynomials is D-stable if and only if its edge polynomials are all D-stable. Unfortunately, this statement is only true for restricted classes of polytopes and restricted classes of stability regions. If either of these restrictions is removed, the theorem will not be valid. In order to remove these constraints, this paper will present a simple precondition for the Edge Theorem. If this condition is not satisfied then the polytope is not D-stable. If this condition is satisfied then the stability version of the Edge Theorem is valid for all stability regions D and all polytopes of polynomials. The Generalized Edge Theorem of Soh and Berger is stated in a similar form. They consider stability regions D whose complements are simply connected in the complex plane. A special case of this theorem states that a polytope of polynomials is D-stable if and only if the edge polynomials are D-stable. It will be shown by counter example that this statement is not true. The precondition presented in this paper suggests ways of correcting this theorem.

The stability robustness of both continuous and discrete-time dynamical systems is tantamount to the stability of a whole family of (characteristic) polynomials. The authors treat such families as sets in coefficient space and examine some of their geometric properties. Specifically, they examine cases where the stability of the extreme points of a set implies the stability of the complete family. They are also concerned with the construction and convexity of the stability region in coefficient space.

Polytopes of (characteristic) polynomials arise when systems experience real parameter variations. A necessary and sufficient condition (requiring only a finite number of computations) is presented for determining whether all the roots of a polytope of polynomials are contained in a desired region. This can be any region that is the image of the open left-half plane under a real linear fractional transformation. Since the open unit cycle is such a region, this result establishes a finite algorithm for strict Schur invariance. Finite algorithms for generalized stability regions such as maximum bandwidth and maximum setting time are also established. >

The presence of uncertain parameters in a state space or frequency domain description of a linear, time-invariant system manifests itself as variability in the coefficients of the characteristic polynomial. If the family of all such polynomials is polytopic in coefficient space, we show that the root locations of the entire family can be completely determined by examining only the roots of the polynomials contained in the exposed edges of the polytope. These procedures are computationally tractable, and this criterion improves upon the presently available stability tests for uncertain systems, being less conservative and explicitly determining all root locations. Equally important is the fact that the results are also applicable to discrete-time systems.

In this paper we will examine how the spectrum (of eigenvalues) of an interval matrix family M depends on the spectrum of its extreme sets. We present three results: 1) We show that the roots of pairwise convex combinations of the characteristic polynomials of the vertices of M provide a bound for the spectrum of M. 2) We state conditions on M which guarantee that the spectrum of M can be determined from the spectrum of its relative boundary. 3) For the special case of M having a real spectrum, we show that this spectrum is completely determined by the eigenvalues of its vertices.

This paper presents a design methodology for the synthesis of a robust controller that accounts for both unmodelled dynamics and structured real parameter uncertainty for multiple-input, multiple-output systems. In a design aimed at constraining both the H ? norm of a certain disturbance transfer matrix and a quadratic gaussian performance index under their respective bounds, a surrogate system may be formed by modelling the structured real parameter uncertainty as additional noise inputs and additional weights at the existing noise inputs and measurement outputs of the system. Application of a Riccati equation approach to this surrogate system then yields a robust controller which, when used in the actual system, will result in a closed-loop system having the same H ? bound and quadratic gaussian performance index bound as the surrogate system, even in the presence of given real parameter variations.

In [1], Kharitonov gave an elegant and simple stability criterion for continuous-time systems. This note reports on similar results for discrete-time systems.