Assessment and Future Directions of Nonlinear Model Predictive ...

600 A.N. Venkat, J.B. Rawlings, and S.J. WrightFig. 3. SFE process (Extractor). Closed-loop performance of different plantwide controlformulations.additional terminal constraint U ui ′ x i (k + N|k) = 0, that forces the unstabledecentralized modes to the origin at the end of the control horizon, is enforcedin the FC-MPC optimization problem (7). It is assumed that x(k) ∈X N ,theN-step constrained stabilizable set for the system. For initialization, a feasibleinput trajectory for each subsystem i ∈{1,M} can be computed by solvinga linear program. If Q i > 0,R i > 0 and the terminal penalty is chosen to beQ i = U si t lower Q i U si ′ ,inwhicht lower Q i is the solution to the Lyapunov equationA si ′ t lower Q i A si − t lower Q i = −U si ′ Q i U si , ∀ i ∈{1,M}, then the closed-loop systemx(k +1)=Ax(k)+Bu(x(k)) is exponentially stable under the distributedcontrol law defined by (8) for x(k) ∈X N and all p(k) > 0.Fig. 4. Behavior of the cooperation-based cost function with iteration number at k =6Example. We consider a 5 input (m), 5 output (y), 30 state (n) plantwith3 subsystems. Each subsystem has 1 unstable decentralized mode. Subsystem1 is represented by a CM consisting of 2 inputs, 2 outputs and 10 states. TheCMs for subsystems 2, 3havem =2,y =2,n =14andm =1,y =1,n =6.