Assessment and Future Directions of Nonlinear Model Predictive ...

Robust Model Predictive Control for Obstacle Avoidance 619Proposition 1 (Robust Constraint Satisfaction). Suppose that the tube Xand the associated policy π satisfy the constraints (7)–(11). Then the state of thecontrolled system satisfies φ(i; x, π, w) ∈ X i ⊆ X O for all i ∈ N N−1 ,thecontrolsatisfies µ i (φ(i; x, π, w)) ∈ U(X i ,µ i ) ⊆ U for all i ∈ N N−1 , and the terminalstate satisfies φ(N; x, π, w) ∈ X f ⊆ T ⊆ X O for every initial state x ∈ X 0 andevery admissible disturbance sequence w ∈ ¯W.Let θ {X,π} and let, for a given state x ∈ X O , Θ(x) (the set of admissible θ)be defined by:Θ(x) {θ | x ∈ X 0 ,X i ⊆ X O ,U(X i ,µ i ) ⊆ U,F(X i ,µ i , W) ⊆ X i+1 , ∀i ∈ N N−1 ,X N ⊆ X f ⊆ T} (12)Consider the cost function defined by:V N (x, θ) N−1∑i=0l(X i ,µ i (·)) + V f (X N ) (13)where l(·) is path cost and V f (·) is terminal cost and consider the following,finite horizon, robust optimal control problem P N (·):P N (x) :VN 0 (x) =arginf N (x, θ) | θ ∈ Θ(x)}θ(14)θ 0 (x) ∈ arg inf N (x, θ) | θ ∈ Θ(x)}θ(15)The set of states for which there exists an admissible tube–control policy pair θis clearly given by:X N {x | Θ(x) ≠ ∅} (16)The robust optimal control problem P N (x) is highly complex in general case,since it requires optimization over control policies and sets. We focus attentionon the case when the system being controlled is linear and constraints specifiedby (2) are polytopic while obstacle avoidance constraint (3) are polygonic so thatthe overall state constraints (4) are polygonic.3 Linear – Polygonic CaseHere we consider the linear discrete-time, time invariant, system:x + = f(x, u, w) Ax + Bu + w (17)where, as before, x ∈ R n is the current state, u ∈ R m is the current controlaction, x + is the successor state, w ∈ R n is an unknown disturbance and(A, B) ∈ R n×n × R n×m . The disturbance w is persistent, but contained in a convexand compact set W ⊂ R n that contains the origin. We make the standing