Stability of Receding Horizon Control Part 2: Ingredients Mar´ ıa M. Seron September 2004 Centre for Complex Dynamic Systems and Control
Outline Stability of Receding Horizon Control 1 The Receding Horizon Control Principle Ingredients for Stability Main Stability Result Linear Systems Centre for Complex Dynamic Systems and Control
Stability of Receding Horizon Control We will now derive conditions that guarantee stability of receding horizon control, the principle underlying MPC. Recall the receding horizon control principle: At the current time, and for the current state x , solve: P N ( x ) : V N ( x ) � min V N ( { x k } , { u k } ) , (1) subject to: x k + 1 = f ( x k , u k ) for k = 0 , . . . , N − 1 , (2) x 0 = x , (3) u k ∈ U for k = 0 , . . . , N − 1 , (4) x k ∈ X for k = 0 , . . . , N , (5) x N ∈ X f ⊂ X , (6) where N − 1 � V N ( { x k } , { u k } ) � F ( x N ) + L ( x k , u k ) . (7) Centre for Complex Dynamic k = 0 Systems and Control
Stability of Receding Horizon Control U ⊂ R m , X ⊂ R n , and X f ⊂ R n are constraint sets. All sequences { u k } = { u 0 , . . . , u N − 1 } and { x k } = { x 0 , . . . , x N } satisfying the constraints (2)–(6) are called feasible sequences. The functions F and L in the objective function (7) are the terminal state weighting and the per-stage weighting , respectively. In the sequel we make the following assumptions: f , F and L are continuous functions of their arguments; U ⊂ R m is a compact set, X ⊂ R n and X f ⊂ R n are closed sets; there exists a feasible solution to problem (1)–(7). Because N is finite, these assumptions are sufficient to ensure the existence of a minimum by Weierstrass’ theorem. Centre for Complex Dynamic Systems and Control
Stability of Receding Horizon Control Denote the minimising control sequence, which is a function of the current state x i , by U � { u 0 , u 1 , . . . , u N − 1 } ; (8) x i then the control applied to the plant at time i is the first element of this sequence, that is, u i = u (9) 0 . Time is then stepped forward one instant, and the above procedure is repeated for another N -step-ahead optimisation horizon. The first element of the new N -step input sequence is then applied, and so on. Centre for Complex Dynamic Systems and Control
Ingredients for Stability To prove stability of MPC we can use the fact that the fi xed horizon control sequence is optimal. Actually, optimality can be turned into a notion of stability by utilising the value function (that is, the function V N ( x ) in (1)) as a Lyapunov function . However, the optimisation problem that we are solving is only defined over a fi nite future horizon, yet stability is a property that must hold over an infi nite future horizon. A trick to resolve this conflict is to add an appropriate weighting on the terminal state in the finite horizon problem so as to account for the impact of events that lie beyond the end of the fixed horizon. This effectively turns the finite horizon problem into an infi nite horizon one. Centre for Complex Dynamic Systems and Control
Ingredients for Stability Following this line of reasoning, we will define a terminal control law and an associated terminal state weighting in the objective function that captures the impact of using the terminal control law over infi nite time . Usually, the chosen terminal control laws are relatively simple and only “feasible” in a restricted (local) region. The above implies that one must be able to steer the system into this restricted terminal region over the finite time period available in the optimisation window. It is also important to ensure that the terminal region is invariant under the terminal control law, that is, once the state reaches the terminal set, it remains inside the set if the terminal control law is used. Centre for Complex Dynamic Systems and Control
Ingredients for Stability Thus, in summary, the ingredients typically employed to provide suffi cient conditions for stability are captured by the following terminal triple : Ingredients for Stability: The Terminal Triple ( X f , K f , F ) (i) a terminal constraint set X f in the state space which is invariant under the terminal control law; (ii) a feasible terminal control law K f that holds in the terminal constraint set; (iii) a terminal state weighting F on the finite horizon optimisation problem, which usually corresponds to the objective function value generated by the use of the terminal control law over infinite time. We will show below how, based on these “ingredients,” Lyapunov-like tests can be used to establish stability of RHC. Centre for Complex Dynamic Systems and Control
Main Stability Result Let us define the set S N of feasible initial states . Definition (Set of Feasible Initial States) The set S N of feasible initial states is the set of initial states x ∈ X for which there exist feasible state and control sequences for the fi xed horizon optimal control problem P N ( x ) in (1) – (7) . ◦ We also require the following definition. Consider the system x i + 1 = f ( x i , u i ) for i ≥ 0 , f ( 0 , 0 ) = 0 . (10) Definition (Positively Invariant Set) The set S ⊂ R n is said to be positively invariant for the system (10) under the control u i = K ( x i ) (or positively invariant for the closed loop system x i + 1 = f ( x i , K ( x i )) ) if f ( x , K ( x )) ∈ S for all x ∈ S . ◦ Centre for Complex Dynamic Systems and Control
Main Stability Result We make the following assumptions on the data of problem P N ( x ) . Conditions for Stability: B1 The per-stage weighting L ( x , u ) in (7) satisfies L ( 0 , 0 ) = 0 and L ( x , u ) ≥ γ ( � x � ) for all x ∈ S N , u ∈ U , where γ : [ 0 , ∞ ) → [ 0 , ∞ ) is continuous, γ ( t ) > 0 for all t > 0, and lim t →∞ γ ( t ) = ∞ . B2 The terminal state weighting F ( x ) in (7) satisfies F ( 0 ) = 0, F ( x ) ≥ 0 for all x ∈ X f , and the following property: there exists a terminal control law K f : X f → U such that F ( f ( x , K f ( x ))) − F ( x ) ≤ − L ( x , K f ( x )) for all x ∈ X f . B3 The set X f is positively invariant for the system (10) under K f ( x ) , that is, f ( x , K f ( x )) ∈ X f for all x ∈ X f . B4 The terminal control K f ( x ) satisfies the control constraints in X f , that is, K f ( x ) ∈ U for all x ∈ X f . B5 The sets U and X f contain the origin of their respective spaces. Centre for Complex Dynamic Systems and Control
Main Stability Result Theorem (Stability of Receding Horizon Control) Consider the closed loop system formed by system (10) , controlled by the receding horizon algorithm (1) – (9) , and suppose that Conditions B1 to B5 are satisfi ed. Then: (i) The set S N of feasible initial states is positively invariant for the closed loop (CL) system. (ii) The origin is globally attractive in S N for the CL system. (iii) If, in addition to B1 – B5 , 0 ∈ int S N and V N ( · ) in (1) is continuous on some neighbourhood of the origin, then the origin is asymptotically stable in S N for the CL system. (iv) If, in addition to B1 – B5 , 0 ∈ int X f , S N is compact, γ ( t ) ≥ at σ in B1 , F ( x ) ≤ b � x � σ for all x ∈ X f in B2 , where a > 0 , b > 0 and σ > 0 are some real constants, and V N ( · ) in (1) is continuous on S N , then the origin is exponentially stable in S N for the CL system. Centre for Complex Dynamic Systems and Control
Main Stability Result Proof. (i) Positive invariance of S N . Let x i = x ∈ S N . At step i , and for the current state x i = x , the receding horizon algorithm solves the optimisation problem P N ( x ) in (1)–(7) to obtain the optimal control and state sequences U � { u 0 , u 1 , . . . , u N − 1 } , (11) x X � { x 0 , x 1 , . . . , x N − 1 , x N } . (12) x Then the actual control applied to (10) at time i is the first element of (11), that is, u i = K N ( x ) = u 0 . (13) Centre for Complex Dynamic Systems and Control
Main Stability Result Let x + � x i + 1 = f ( x , K N ( x )) = f ( x , u 0 ) be the successor state. A feasible (but not necessarily optimal) control sequence, and corresponding feasible state sequence for the next step i + 1 in the receding horizon computation P N ( x + ) are then ˜ U = { u 1 , . . . , u N − 1 , K f ( x N ) } , (14) ˜ X = { x 1 , . . . , x N − 1 , x N , f ( x N , K f ( x N )) } . (15) Indeed, the first N − 1 elements of (14) lie in U (see the control constraint (4)) since they are elements of (11); also, by B4 , the last element of (14) lies in U since x ∈ X f . Finally, by B3 , the N terminal state f ( x N , K f ( x N )) in (15) also lies in X f . Thus, there exist feasible sequences (14) and (15) for x + = f ( x , K N ( x )) and hence x + ∈ S N . This shows that S N is positively invariant for the CL system x + = f ( x , K N ( x )) . Centre for Complex Dynamic Systems and Control
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