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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. >

This paper presents a design methodology for the synthesis of a robust controller that accounts for both unmodeled dynamics and structured real-parameter uncertainty for multiple-input/multiple-output systems. The unmodeled dynamics are assumed to be characterized as a single block dynamic uncertainty at a point in the closed-loop system. 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 modeling 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 that, when used in the actual system, will result in a closed-loop system that has the same H^ bound and quadratic Gaussian performance index bound as the surrogate system, even in the presence of given real-parameter variations.

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.

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.

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