AN INTRODUCTION TO THE STABILIZATION PROBLEM FOR PARAMETRIZED FAMILIES OF LINEAR SYSTEMS *

This talk provides an introduction to definitions and known facts relating to the stabilization of parametrized families of linear systems using static and dynamic controllers. New results are given in the rational and polynomial cases. 1. General discussion We shall consider a set of problems which have appeared in algebraic system theory and whose solutions involve tools of various different types. These problems, in their simplest form, deal with parametrized families of pairs ("systems") {(Aλ,Bλ), λ∈Λ}, where Aλ is an n×n and Bλ is an n×m matrix for each λ (with n,m fixed integers). To be found is a new parametrized family {Kλ, λ∈Λ} such that (1) a given design criterion is satisfied by the closed-loop matrix Aλ+BλKλ for all λ, and (2) the Kλ depend in a suitably ’nice’ form on the parameter. (For the purposes of this talk, the entries of all matrices take real values, for each λ.) For example, one design objective is that Aλ+BλKλ should (for each λ) have all its eigenvalues in the inside of the unit circle (discrete-time stabilization); one nice form of parameter dependence if, for instance, Λ is an Euclidean space Rr (or an algebraic variety), is that Kλ be required to have entries which are polynomials in λ. Many other design objectives and types of parametrizations will be mentioned below. As a general rule the former will always deal with stability-related properties. Regarding parametrizations, we’ll talk about the continuous, (real-)analytic, rational, or polynomial cases; when doing so, it will be implicit that the parameter set is respectively a topological space, a real-analytic manifold, or in the last two cases an Euclidean space, and that the given family {(Aλ,Bλ)} is parametrized in this way. One may also consider of course other situations, for example the smooth (=C∞) case, or polynomial and rational families over algebraic varieties; for simplicity, we restrict attention to the above. For any given type of parametrization for the family {(Aλ,Bλ)}, we shall search for families of controllers parametrized in the same way. This insistence on a ’nice’ parameter dependence for {Kλ} (or its dynamic versions described later) is what distinguishes the area from the more classical study of single systems. Most *1980 Mathematics Subject Classification numbers 93B25,93D15 **Research Supported in part by US Air Force Grant AFOSR-85-0196E results attempt in essence to establish local-global principles: does solvability for each individual λ imply the existence of a nicely parametrized solution? There are various motivations for studying the general type of problems mentioned here. From a purely mathematical point of view, these are a natural next step after the solution of their nonparametrized versions, which constitute what a great deal of linear control theory is about. The systemtheoretic interpretation of the present setup is as follows. The pairs (Aλ,Bλ) represent a discrete or continuous time system x(t+1) [or x⋅ (t)] = Aλx(t) + Bλu(t) (1) whose general structure is known in advance but where certain parameters are a priori undetermined. (There may also be an output or measurement specified, of the type y(t) = Cλx(t), in which case the transposed family {(Aλ ,Cλ T)} becomes also of interest, as in the example given below.) Parametrized equations 1 arise for instance in the case where the (Aλ,Bλ) correspond to linearizations of a given nonlinear system at many different operating points, and one is interested in the design of controllers for all the linear systems so obtained. Such a situation appears frequently in aircraft control ("gain scheduling"), where controllers are precomputed for a large variety of operating conditions, with an on-board computer choosing the appropiate controller to be used at any given time based on environmental, geometric, flight-mode, and other factors, like pitch angle, air speed, angle of attack, and so forth. An alternative approach to this precomputation and storage would be to try to apply the tools of parametrized families to achieve the simultaneous design of these controllers, in the form of a parametrized controller which regulates once its parameters are properly tuned. Thus, only the functional form of the controllers needs to be stored. Together with an on-line identification procedure, this becomes in effect a method for adaptive control; recent work ([E2]) makes this application more precise. The resulting families will be typically analytic or rational, but other situations may appear too; for example polynomially parametrized families appear when dealing with systems with finite Volterra series (see [RU]). The simplest control problem, that of stabilizing the above system by a static linear law u(t) = Kλx(t), consists mathematically of finding a (nicely parametrized) family {Kλ} such that, say in continuous-time, all eigenvalues of Aλ+BλKλ have negative real parts, for each λ in the parameter set Λ. A more interesting problem is that of "pole-assignment", which consists of finding a family {Kλ} such that Aλ+BλKλ has specified eigenvalues for each λ. (The terminology "poles" is due to the fact that the eigenstructure of this matrix gives rise to the poles of the transfer function of the closed loop system.) A specific, though somewhat artificial, example of where a parametrized control problem appears is the following one. A cart can move horizontally, in one dimension, controlled by a motor. To its top is attached an inverted pendulum. The objective is to keep the pendulum in an upright position, using suitable controls (horizontal forces on the car). This is a standard textbook problem in control theory, and is analogous to a rocket stabilization problem. We shall assume that the mass M of the pendulum is concentrated at its top, and will disregard friction effects. The available observations will be the displacement of the cart and the angular velocity of the pendulum. Linearizing around the position corresponding to a static cart and a static vertical pendulum, the equations can be written as x⋅ (t) = Ax(t)+Bu(t), y(t) = Cx(t), with A,B parametrized matrices as above, C a p×n matrix, m=1, n=4, p=2, and 0 1 0 0 0 (2) 0 0 -λ 0 1 1 0 0 0 A= 0 0 0 1 B= 0 C= 0 0 0 1 0 0 g+λ 0 -1 (see [FH,p.47]), where the state coordinates are respectively the position and velocity of the cart, and the angle (with respect to the vertical) and the angular velocity of the pendulum. We are assuming that the pendulum has unit length, the cart has unit mass, g is the acceleration of gravity, and the parameter λ=Mg is scalar, Λ=R. The linearized model is controllable and observable for any λ (see below), so most design techniques can be easily applied, yielding in particular dynamic stabilizers for the above system. Here, however, we wish to consider the mass M of the pendulum, or equivalently λ, as a parameter, and we ask if it is possible to design a family of control systems {∑λ} for 2, such that each ∑λ results in a closed loop system with appropiate dynamic characteristics, and such that these controllers depend again nicely on λ. The results to be reviewed later insure that pole assignment (in particular, stabilization with arbitrary degrees of convergence,) can be achieved, by regulators ∑λ of fixed dimension and depending polynomially on λ, if and only if 2 is controllable and observable for all values of λ, real and complex. The controllability constraint means that the controllability matrix (B,AB,A2B,A3B) has full rank; since its determinant = g2 = nonzero constant, this is clearly satisfied. The observability property deals with the analogous matrix for the dual pair (AT,CT). This observability matrix (CT,ATCT,(AT)2CT,(AT)3CT)T has in particular minors ∆1 = -λ 2 (rows 1,3,5,7) and ∆2 = g+λ (rows 1,2,3,4); since these two polynomials are relatively prime, the observability condition holds too. In fact, it is absolutely trivial (because m=1) to design a polynomially parametrized state feedback that achieves any desired pole location: if k=k(λ) is desired so that A+BkT has a characteristic polynomial z+α3z +α2z +α1z+α0, one may choose k1:=α0/g, k2:=α1/g, k4:=α3+α1/g, and k3:=α2+g+(α0/g)+λ, so that k is a polynomial of degree 1 in λ. In order to obtain a regulator that uses only the available measurements, we may proceed by constructing first a "Luenberger observer" for (A,C), which reduces to the solution of a pole-assignment problem for the transposed pair (AT,CT), and then combining this with the above feedback law. The usual methods for obtaining an observer, however, will not result in a polynomially parametrized observer. For example, since A(λ) is cyclic for all λ, one may try to find a v=v(λ) such that, for each λ, CTv is cyclic for AT, reducing the construction to the single-output case ([WO]). But a simple calculation shows that no such v exists in this case, and the usual generalization, "Heymann’s lemma," is not available over the polynomial ring R[λ]. Since Λ=R, R[λ] is a principal ideal domain, however, and the construction of an observer can be completed, as discussed later. We omit the calculations here, but remark that they involve a certain amount of linear algebra over R[λ], at the level of computations with Smith forms. Another, more theoretical example, related to the "gain scheduling" idea mentioned earlier, of families of systems appearing as linearizations of nonlinear systems around different operating points (in the style of [BR]), is as follows. Consider the problem of obtaining observers for the linear systems that result when linearizing the 2-dimensional system x⋅ 1=u, x ⋅ 2=x2+v, y1=x1x2+x1, y2=x1 x2-x2, at the natural equilibrium states (λ,0), λ∈R. The duals of these systems have 0 0 1 0 (3) A = B= . 0 1 λ λ2-1 It turns out that this is the example that we shall mention later in order to illustrate the fact that it is in general impossible to obtain arbitrary characteristic polynomials for A+BK, with polynomially parametrized K. Final