Assessment and Future Directions of Nonlinear Model Predictive ...

284 H. Chen et al.of the moving horizon system seems to play an important role. For the finitehorizonmoving horizon formulations, this is achieved by choosing the terminalpenalty function to satisfy the Hamilton-Jacobi-Isaacs inequality locally in apositive invariant terminal region defined as a value set of the terminal function.In [CSA97, MNS01], the positive invariance of the terminal region is sufficientlyguaranteed by restricting the amplitude of the disturbance to ‖w(t)‖ ≤δ‖z(t)‖(or the discrete form in [MNS03]), whereas in [Gyu02] it is included in primalassumptions. Removing the rather restrictive hypotheses on the size of disturbances,it is shown in [CS03] that closed-loop dissipation might fail, even ifdissipation inequalities are satisfied at each optimization step. This is first observedin [SCA02] with respect to switching between H ∞ controllers, where acondition is derived to recover dissipation. This condition is called dissipationcondition in [CS03, CS04] and introduced into the on-line optimization problemto enforce dissipation for the moving horizon system.This paper extends the results in [CS03, CS04] to address the disturbanceattenuation issue of nonlinear moving horizon control. Section 2 presents a conceptualminimax moving horizon control formulation for nonlinear constrainedsystems and the theoretical results on closed-loop dissipation, L 2 disturbance attenuationand stability. In Section 3, the implementation issue of the suggestedformulation is addressed with respect to tracking in the presence of disturbancesand control constraints. Simulation and comparison results of setpoint trackingcontrol of a CSTR are given in Section 4.2 Moving Horizon Control with Disturbance AttenuationConsider a nonlinear system described byẋ(t) =f (x(t),w(t),u(t)) ,x(t 0 )=x 0 ,z 1 (t) =h 1 (x(t),w(t),u(t)) , z 2 (t) =h 2 (x(t),u(t)) ,(1)with time-domain constraints|z 2j (t)| ≤z 2j, max ,j =1, 2, ··· ,p 2 ,t≥ t 0 , (2)where x ∈ R n is the state, w ∈ R m1 is the external disturbance, u ∈ R m2is the control input, z 1 ∈ R p1 is the performance output and z 2 ∈ R p2 is theconstrained output. It is assumed that the vector fields f : R n ×R m1 ×R m2 → R n ,h 1 : R n × R m1 × R m2 → R p1 and h 2 : R n × R m2 → R p2 are sufficiently smoothand satisfy f(0, 0, 0) = 0, h 1 (0, 0, 0) = 0 and h 2 (0, 0) = 0.With respect to disturbance attenuation, we strive to solve the following minimaxoptimization problem for the system (1) with the initial state x(t 0 )inmoving horizon fashion:∫ ∞min max ‖z 1 (t)‖ 2 − γ 2 ‖w(t)‖ 2 dt. (3)u∈U w∈Wt 0