Assigning closed-loop invariant polynomials over polytopes for linear periodic systems (Q1295106)

Pole assignment is a well-known and well-tried technique of feedback controller design. The main object of such a design is to locate the poles of a closed loop system in a specific region of the complex plane, without attempting to produce some exact locations for these poles. The specification of the desired region amounts to assigning a polytope to the coefficient space. An interesting class of designs consists of periodic, time-discretized systems. The authors approach these systems by dealing with solutions to certain diophantine equations. The authors proceed as follows: First, they describe some basic properties of a time-discretized system \(\Sigma\). Let \(k= 0,1,2,3,\dots\) be discrete instants of time. The state equation is of the form: \[ x(k+ 1)= A(k)x(k)+ B(k)u(k),\quad y(k)= C(k)x(k), \] where \(x\in\mathbb{R}^n\) is the state vector, \(u(k)\) is the control input, \(y\in\mathbb{R}^q\) is the observed output. \(A\), \(B\), \(C\) are periodic matrices, of period \(\omega\). The notion of associated system \(\Sigma^a\) also called lifted reformulation: \[ x_k(h+ 1)= E_kx_k(h)+ J_k u_k(h),\quad y_k(h)= L_k x_k(h)+ M_x u_x(h) \] has been introduced around 1975. Let \(z\) be the forward shift operator, and \(d\) denote the backward shift: \(d= z^{-1}\). The invariant polynomials of a polynomial matrix \(I_n- dE_k\) of rank \(r\) are the unique \(r\) non-zero monic polynomials forming the main diagonal of the Smith form of \(M(d)\). The product of these polynomials characterizes the stability of \(\Sigma\). As was shown by S. Bittanti in 1986 and by O. M. Grasselli and the second author of this article in 1988, the notion of associated system at time \(k\) allows the analysis of stability for the pole-zero structure of a periodic feedback system \(\Sigma\). O. M. Grasselli and S. Longhi have shown previously that products of invariant polynomials characterize stability of \(\Sigma\). The authors now define the polytopes of polynomials, and in particular discuss the minimal number of generators of such polytopes. This leads to theorems about minimal realizability of controllers solving the polytope problem, and the general form of the corresponding transfer matrix. A parametrized family of such controllers depends on a general solution of a parametrized diophantine equation. The actual numerical computations based on this approach are highly complex. The authors remark that a certain arbitrariness in choosing unimodular matrices in this process may help in specific cases to reduce this complexity.