A perturbat ion approach to the stabil ization of cascade systems with t ime-delay nonlinear

The effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay differential equations is analyzed. We investigate under which conditions global stability is preserved and if not, whether semi-global stabilization is possible by controlling the size or shape of the perturbation. These results are used to study the stabilization of partially linear cascade systems with partial state feedback. 1 I n t r o d u c t i o n The stability analysis of the series (cascade) interconnection of two stable nonlinear systems described by ordinary differential equations is a classical subject in system theory [11, 12, 15]. > NL > NL stable stable Contrary to the linear case, the zero-input global asymptotic stability of each subsystem does not imply the zero-input global asymptotic stability of the interconnection. The output of the first subsystem acts as a transient input disturbance which can be sufficient to destabilize the second subsystem. In the ODE case, such destabilizing mechanisms are well understood, since the seminal work by Sussmann and Kokotovic [13]. They can be subtle but are almost invariably associated to a finite escape time in the second subsystem (some states blow up to infinity in a finite time). The present paper explores similar instability mechanisms generated by the series interconnection of nonlinear DDEs. In particular we consider the situation where the destabilizing effect of the interconnection is delayed and examine the difference with the ODE situation. *Department of Computer Science, Katholieke Universiteit Leuven, Belgium, {Wim. Michiels, Dirk. Roose}©cs. kuleuven, ac. be tlnstitut Montefiore, University of Liege, Belgium, r. sepulchre©ulg, ac. be ~M.A.E. Department, Princeton University, USA, imoreau©princet on. edu Instrumental to the stability analysis of cascades, we first study the effect of external (affine) perturbations w on the stability of nonlinear time-delay systems = f ( z , z ( t 7 ) )+ ~ ( z , z ( t 7))w, z 6 ~ n , w 6 ~, (1) where we assume that the equilibrum z = 0 of i = f(z, z(t T)) is globally asymptotically stable. We consider perturbations w = ~(t) which belong to both £1([0, oc)) and £~([0, oc)) and investigate the set of initial conditions which give rise to bounded solutions, under various assumptions on the system and the perturbations. These results are strengthened to asymptotic stability results when the perturbations are generated by a globally asymptotically stable DDE. We consider both global and semi-global results. In the ODE case, an obstruction to global stability is formed by the fact that arbitrarily small input perturbations can cause the state to escape to infinity in a finite time, for instance when the interconnection term ~(z) is nonlinear in z. This is studied extensively in the literature in the context of stability of cascades, see e.g. [13, 11] and the references therein. Even though delayed perturbations do not cause a finite escape time we explain a similar mechanism giving rise to unbounded solutions, caused by nonlinear delayed interconnection terms. In situations where unbounded solutions are inevitable for large initial conditions, we investigate under which conditions trajectories can be bounded semi-globally in the space of initial conditions, in case the perturbation is parametrized, i.e. ~ = ~(t, a). Here we let the parameter a control the £1 or £ ~ norm of the perturbation. We also consider the effect of concentrating the perturbation in an arbitrarily small time-interval. The study of controlled perturbations is motivated by the situation where the perturbation is the output of a controlled system, see Figure 1.1. In particular, we will be interested in the stabilization of the following cascade, { ~ f ( z , z ( t v)) + ~ ( z , z ( t v))y A~ + Bu y C ~ , ~ 6 ~ ~, u, y 6 ~ (2) > N stal L 'controlled per turbat ion '> NL )le stable