Assessment and Future Directions of Nonlinear Model Predictive ...

82 P. Mhaskar, N.H. El-Farra, and P.D. Christofidesswitching scheme (formalized in Theorem 1 below; for the proof, see [9]) thataddresses the above problem while focusing on achieving closed–loop stability.3.2 Controller Switching LogicTheorem 1. Consider the constrained nonlinear system of Eq.10, with any initialcondition x(0) ≡ x 0 ∈ Ω k (u max ),forsomek ∈K≡{1, ··· ,p}, whereΩ kwas defined in Eq.7, under the model predictive controller of Eqs.3-4. Also let¯T ≥ 0 be the earliest time for which either the closed–loop state, under MPC,satisfiesL f V k (x( ¯T )) + L g V k (x( ¯T ))M(x( ¯T )) ≥ 0 (11)or the MPC algorithm fails to prescribe any control move. Then, the switchingrule given by{ }1, 0 ≤ t< ¯Ti(t) =2, t ≥ ¯T(12)where i(t) = 1 ⇔ u i (x(t)) = M(x(t)) and i(t) = 2 ⇔ u i (x(t)) = b k (x(t)),guarantees that the origin of the switched closed–loop system is asymptoticallystable.Remark 1. Theorem 1 describes a stability-based switching strategy for controlof nonlinear systems with input constraints. The main components of this strategyinclude the predictive controller, a family of bounded nonlinear controllers,with their estimated regions of constrained stability, and a high–level supervisorthat orchestrates the switching between the controllers. A schematic representationof the hybrid control structure is shown in Figure 1. The implementationprocedure of this hybrid control strategy is outlined below:• Given the system model of Eq.1, the constraints on the input and the familyof CLFs, design the bounded controllers using Eqs.5-6. Given the performanceobjective, set up the MPC optimization problem.• Compute the stability region estimate for each of the bounded controllers,p⋃Ω k (u max ), using Eqs.7-8, for k =1,...,p,andΩ(u max )= Ω k (u max ).• Initialize the closed–loop system under MPC, at any initial condition, x 0within Ω, andidentifyaCLF,V k (x), for which the initial condition is withinthe corresponding stability region estimate, Ω k .• Monitor the temporal evolution of the closed–loop trajectory (by checkingEq.11 at each time) until the earliest time that either Eq.11 holds or theMPC algorithm prescribes no solution, ¯T .• If such a ¯T exists, discontinue MPC implementation, switch to the k-thbounded controller (whose stability region contains x 0 ) and implement itfor all future times.Remark 2. The main idea behind Theorem 1, and behind the hybrid predictivecontroller (including the designs that address issues of unavailability ofk=1