Dieter Kaesbauer

Analysis and Design

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.

Robust gamma-stability analysis in a plant parameter space

Abstract Given a characteristic polynomial whose coefficients depend polynomially on l uncertain parameters, the following robustness problem arises: Determine whether all the roots of the polynomial are located in a prescribed region Г in the complex plane for all admissible parameter values. To this end, the boundary ∂Г of Г is mapped into the parameter space. A necessary and sufficient condition for Г-stability of an operating domain in parameter space is that it contains at least one Г-stable point and is not intersected by the image of ∂Г. This condition may be tested graphically by gridding l − 2 parameters and projecting all boundaries into a two-dimensional subspace of the parameter space. Finally the method is applied to a track-guided bus with uncertain mass and velocity.

Robustness analysis: a case study

Linear systems with uncertain parameters q are analyzed for stability for all q in a box Q. An example with multilinear dependency of the characteristic polynomial P(s, q) on the parameters is used as a test case for the composition of four methods: (1) eigenvalues (s-plane), (2) zero exclusion from the complex value set (P-plane), (3) algebraic test of Hurwitz conditions, and (4) parameter space (stability boundary exclusion from Q in projected q-space). The methods are also discussed with respect to gridding, graphical warning for singular cases, extension to many parameters, extension to polynomic coefficient functions, extension to Gamma -stability, and insight beyond a yes-no answer to the analysis problem.<<ETX>>

On robust stability of polynomials with polynomial parameter dependency: two/three parameter cases

We consider uncertain polynomials whose coefficients depend polynomially on the elements of the parameter vector. The size of perturbation is characterized by the weighted norm of the perturbed parameter vector. The maximal perturbation defines the stability radius of the set of uncertain polynomials. It is shown that determining this radius is equivalent to solving a finite set of systems of algebraic equations and picking out the smallest solution. The number of systems depend crucially on the dimension of the parameter vector, whereas the complexity of systems increases mainly with the kind of polynomial dependency and the degree of the polynomial. This method also yields the critical parameter combination and the corresponding critical frequency. For a small number of parameters, this transfomed problem can be solved by symbolic computations. For a large number of parameters, numerical methods must be used.

Design by Search

Abstract Consider a world in which computers are infinitely fast. How would you do design with such computers? One of the possibilities is an on-line search in a space of free design parameters. The bottlenecks are then the man-machine interfaces. in particular keyboard input to the computer and interpretation of displays showing attributes like performance of a control system to be designed. This paper deals with the question: Is it possible to do design by search in sensomotoric interaction between designer and design machine? for design of a robust controller for a simple two mass system with two uncertain parameters the answer is “yes”

Single-Loop Feedback Structures

With Chapter 8 we are entering into Prt III of this book.It deals with specific feedback structures and with additional robustness requirements on the closed-loop system,like sector uncertainty of a nonlinearity,Gamma-stability and discrete-time implementation of a robust controller.

Parameter Space Design

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.

Design by Optimizing a Vector Performance Index

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.

Control System Structures

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).

Classical Stability Tests Applied to Uncertain Polynomials

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”.