Stabilization of discrete‐time bilinear systems

The problem of feedback stabilizing a bilinear discrete‐time system is studied. Under an accessibility condition on an associated nonlinear system on the projective space it can be shown that null controllability is equivalent to feedback stabilizability. We present a way in which a stabilizing feedback may be computed using some ideas from discounted optimal control.


Introduction
We consider systems on IRd of the form where Ao, . . . , A , E Rdxd do not span a subspace of non-invertible matrices, and the set of admissible control values U = cl int U c Ern is compact, with connected interior and satisfies 0 E int U. The associated system on the projective space Pd-' is given by Here only those control sequences u E U" are admissible for which the corresponding solution @,,(t, 0)xo # 0 for all t E IN, where zO is such that it spans ( 0 . These control values or sequences will be denoted by U([),U"((). The solution of (2) corresponding to an initial value [ and a control sequence u is denoted by cp (., 4, u). Let us note that the setup is a particular c a e of the systems studied in [3], [4]. Proofs for the statements of the theorems below can be found in [3].
We call system (1) asymptotically null-controllable if for every x E IRd there exists a control sequence ti E U" such that limt,, @,,(t, 0)x = 0. System (1) is called (state) feedback stabilizable if there exists a map F : IRd + U such that the system is globally asymptotically stable. If F can be chosen such that (3) is exponentially stable, then we call (1) exponentially (state) feedback stabilizable and F is called exponentially stabilizing.
Recall that a nonlinear system is called forward accessible, if for each point the interior of the forward orbit is nonempty. The following theorem states the main result on exponential stabilizability of (1).
T h e o r e m 1. If (2) is forward accessible, then the following statements are equivalent.

Construction of the feedback
In t,his section we will give a constructive approach for the calculation of the exponentially stabilizing feedback for system (1). For this purpose we will base our construction on a dynamic programming technique using the optimal value function of a discounted optimal control problem. Note that this procedure yields an existence result for stabilizing feedbacks and in addition makes the problem numerically feasible, cp. Remark 3.
From the application point of view one reason why feedback stabilization is preferred t o open loop asymptotic null controllability lies in the fact that one expects robustness of the closed loop system against small perturbations.
Since in this paper we follow an optimal control approach the resulting feedback will in general be discontinuous, and the desired robustness property cannot be obtained as a simple conclusion from the continuous dependency on the initial value. However, the continuity of the associated value function may be used in order to obtain this property.
The construction of the feedback is related to the following optimal control problem: Define the function q : Pd-' x U + RU (-00) by The associated value function on projective space is given by V6(<) := infuEUm Ja (C,u). For small 6 the function 0 6 may be interpreted as an approximation of the smallest exponential growth rate that may be attained via an arbitrary control sequence from [. is indeed an optimal control strategy for V6. Furthermore it holds that