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Controllers are usually implemented in a digital computer. Figure 11.1 shows a single-loop sampled-data control system with c z (z) representing the z-transfer function of the digital controller, and (1 - e - s T s as the transfer function of the hold element, T is the sampling interval.

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).

In Chapters 8 and 9, the stability of polynomial families $$ P\left( {s,Q} \right)\; = \;\left\{ {p\left( {s,q} \right)\;|\;q\; \in \;Q} \right\}$$ (1) with p(s, q) = ao(4) + ai(4)s +… + an(q)sn and q, E [4,; 4:], i = 1, 2,…, € was investigated. The primary interest was necessary and sufficient conditions for stability. These conditions give primarily a yes or no answer to the robust stability problem. If the answer is yes, then there is some kind of stability margin of the uncertain parameters, the domain Q of the parameters can be enlarged without loosing stability. The next question is: what is the smallest perturbation in the parameters q that destabilizes the system?

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.

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.

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.

A mathematical model for the plant dynamics is the basis for analysis and design of control systems. For linear time-invariant systems we commonly have a state space model of the form $$\eqalign{{\bf\it{\dot x}}(t)= {\bf\it Ax}(t)+ {\bf\it Bu}(t)\cr{\bf\it y}(t)= {\bf\it Cx}(t)}$$ (1.0.1) with u the vector x of input signals (manipulated variables), the state vector , and the vector of output signals y (performance variables). The output signals are often comprised of those variables that are measured and hence available for feedback.