Assessment and Future Directions of Nonlinear Model Predictive ...

Techniques for Uniting Lyapunov-Based and Model Predictive Control 85where x =[x 1 ···x n ] ′ ∈ IR n denotes the vector of state variables, u =[u 1 ···u m ] ′∈ IR m denotes the vector of manipulated inputs, U ⊆ IR m ,X ⊆ IR n denote theconstraints on the manipulated inputs and the state variables, respectively, f(·)is a sufficiently smooth n × 1 nonlinear vector function, and g(·) is a sufficientlysmooth n×m nonlinear matrix function. Without loss of generality, it is assumedthat the origin is the equilibrium point of the unforced system (i.e., f(0) = 0).4.2 Lyapunov-Based Predictive Control DesignPreparatory to the characterization of the stability properties of the Lyapunovbasedpredictive controller, we first state the stability properties of the boundedcontroller of Eqs.5–6 in the presence of both state and input constraints. Forthe controller of Eqs.5–6, one can show, using a standard Lyapunov argument,that whenever the closed–loop state, x, evolves within the region described bythe set:Φ x,u = {x ∈ X : L ∗ f V (x) ≤ umax ‖(L g V ) ′ (x)‖} (14)then the controller satisfies both the state and input constraints, and the timederivativeof the Lyapunov function is negative-definite. To compute an estimateof the stability region we construct a subset of Φ x,u using a level set of V , i.e.,Ω x,u = {x ∈ IR n : V (x) ≤ c maxx,u } (15)where c maxx,u > 0 is the largest number for which Ω x,u ⊆ Φ x,u . Furthermore, thebounded controller of Eqs.5-6 possesses a robustness property (with respect tomeasurement errors) that preserves closed–loop stability when the control actionis implemented in a discrete (sample and hold) fashion with a sufficiently smallhold time (∆). Specifically, the control law ensures that, for all initial conditionsin Ω x,u , the closed–loop state remains in Ω x,u and eventually converges to someneighborhood of the origin (we will refer to this neighborhood as Ω b )whosesize depends on ∆. This property is exploited in the Lyapunov-based predictivecontroller design of Section 4.2 and is stated in Proposition 1 below (the proofcan be found in [25]). For further results on the analysis and control of sampleddatanonlinear systems, the reader may refer to [13, 14, 28, 31].Proposition 1. Consider the constrained system of Eq.1, under the boundedcontrol law of Eqs.5–6 with ρ>0 and let Ω x,u be the stability region estimateunder continuous implementation of the bounded controller. Let u(t) =u(j∆)for all j∆ ≤ t