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Stability of Receding Horizon Control Part 2: Ingredients

Mar´ ıa M. Seron September 2004

Centre for Complex Dynamic Systems and Control

Outline

1

Stability of Receding Horizon Control The Receding Horizon Control Principle Ingredients for Stability Main Stability Result Linear Systems

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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:

PN(x) :

V

N (x) min VN({xk}, {uk}),

(1) subject to: xk+1 = f(xk, uk) for k = 0, . . . , N − 1, (2) x0 = x, (3) uk ∈ U for k = 0, . . . , N − 1, (4) xk ∈ X for k = 0, . . . , N, (5) xN ∈ Xf ⊂ X, (6) where VN({xk}, {uk}) F(xN) +

N−1

• k=0

L(xk, uk). (7)

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Stability of Receding Horizon Control

U ⊂ Rm, X ⊂ Rn, and Xf ⊂ Rn are constraint sets.

All sequences {uk} = {u0, . . . , uN−1} and {xk} = {x0, . . . , xN} 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 ⊂ Rm is a compact set, X ⊂ Rn and Xf ⊂ Rn 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.

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Stability of Receding Horizon Control

Denote the minimising control sequence, which is a function of the current state xi, by

U 

xi

{u

0 , u 1 , . . . , u N−1} ;

(8) then the control applied to the plant at time i is the first element of this sequence, that is, ui = u

0 .

(9) 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.

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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

• f the fixed horizon. This effectively turns the finite horizon

problem into an infi nite horizon one.

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Ingredients for Stability

Following this line of reasoning, we will define a terminal control law and an associated terminal state weighting in the

• bjective 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.

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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 (Xf, Kf, F) (i) a terminal constraint set Xf in the state space which is invariant under the terminal control law; (ii) a feasible terminal control law Kf 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.

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Main Stability Result

Let us define the set SN of feasible initial states. Definition (Set of Feasible Initial States) The set SN 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

xi+1 = f(xi, ui) for i ≥ 0, f(0, 0) = 0. (10) Definition (Positively Invariant Set) The set S ⊂ Rn is said to be positively invariant for the system (10) under the control ui = K(xi) (or positively invariant for the closed loop system xi+1 = f(xi, K(xi))) if f(x, K(x)) ∈ S for all x ∈ S.

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Main Stability Result

We make the following assumptions on the data of problem PN(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 ∈ SN, u ∈ U, where γ : [0, ∞) → [0, ∞) is continuous, γ(t) > 0 for all t > 0, and limt→∞ γ(t) = ∞. B2 The terminal state weighting F(x) in (7) satisfies F(0) = 0, F(x) ≥ 0 for all x ∈ Xf, and the following property: there exists a terminal control law Kf : Xf → U such that F(f(x, Kf(x))) − F(x) ≤ −L(x, Kf(x)) for all x ∈ Xf. B3 The set Xf is positively invariant for the system (10) under

Kf(x), that is, f(x, Kf(x)) ∈ Xf for all x ∈ Xf.

B4 The terminal control Kf(x) satisfies the control constraints in Xf, that is, Kf(x) ∈ U for all x ∈ Xf. B5 The sets U and Xf contain the origin of their respective spaces.

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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 SN of feasible initial states is positively invariant for the closed loop (CL) system. (ii) The origin is globally attractive in SN for the CL system. (iii) If, in addition to B1–B5, 0 ∈ int SN and V

N (·) in (1) is

continuous on some neighbourhood of the origin, then the

• rigin is asymptotically stable in SN for the CL system.

(iv) If, in addition to B1–B5, 0 ∈ int Xf, SN is compact, γ(t) ≥ atσ in B1, F(x) ≤ bxσ for all x ∈ Xf in B2, where a > 0, b > 0 and σ > 0 are some real constants, and V 

N (·) in (1) is

continuous on SN, then the origin is exponentially stable in SN for the CL system.

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Main Stability Result

Proof. (i) Positive invariance of SN. Let xi = x ∈ SN. At step i, and for the current state xi = x, the receding horizon algorithm solves the optimisation problem PN(x) in (1)–(7) to obtain the optimal control and state sequences

U 

x

{u

0 , u 1 , . . . , u N−1},

(11)

X 

x

{x

0 , x 1 , . . . , x N−1, x N } .

(12) Then the actual control applied to (10) at time i is the first element

• f (11), that is,

ui = KN(x) = u

0 .

(13)

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Main Stability Result

Let x+ xi+1 = f(x, KN(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 PN(x+) are then

˜ U = {u

1 , . . . , u N−1, Kf(x N )},

(14)

˜ X = {x

1 , . . . , x N−1, x N , f(x N , Kf(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

N

∈ Xf. Finally, by B3, the

terminal state f(x

N , Kf(x N )) in (15) also lies in Xf.

Thus, there exist feasible sequences (14) and (15) for x+ = f(x, KN(x)) and hence x+ ∈ SN. This shows that SN is positively invariant for the CL system x+ = f(x, KN(x)).

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Optimal and Feasible Sequences

Xf SN

x = x

0 x 1

x1 x0 x2 x

N

f(x

N , Kf(x N ))

X 

x

= {x

0 , x 1 , . . . , x N−1, x N }

(optimal at time 0)

˜ X = {x

1 , . . . , x N−1, x N , f(x N , Kf(x N ))}

(feasible at time 1) .

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Main Stability Result

(ii) Attractivity. It is easy to show that the origin is an equilibrium point for the closed loop system x+ = f(x, KN(x)). We will next use the value function V 

N (·) in (1) as a Lyapunov

function. We first show that V

N (·) satisfies property (i) in Theorem

(Attractivity in S). Let x ∈ SN. The increment of the Lyapunov function, upon using the control (13) and moving from x to x+ = f(x, KN(x)), satisfies V

N (x+) − V N (x) = VN(X  x+ , U  x+ ) − VN(X  x

, U 

x

) ≤ VN( ˜ X , ˜ U ) − VN(X 

x

, U 

x

) ,

since, by optimality we know that VN(X 

x+ , U  x+ ) ≤ VN( ˜

X , ˜ U ).

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Main Stability Result

Substituting (11), (12), (14) and (15) in the objective function expression (7), we obtain V

N (x+) − V N (x) ≤ VN( ˜

X , ˜ U ) − VN(X 

x

, U 

x

) = −L(x, KN(x)) + L(x

N , Kf(x N ))

+ F(f(x

N , Kf(x N ))) − F(x N ) .

From B2, and since x

N

∈ Xf, the sum of the last three terms on

the right hand side above is less than or equal to zero. Thus, V

N (x+) − V N (x) ≤ −L(x, KN(x)) ≤ −γ(x) for all x ∈ SN,

(16) where, in the last inequality, we have used the bound in Condition B1. Thus V

N (·) satisfies property (i) of Theorem (Attractivity in S).

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Main Stability Result

In addition, from Assumptions B1 and B2, V 

N (·) satisfies

V

N (x) ≥ L(x, u 0 ) ≥ γ(x) for all x ∈ SN .

(17) Hence, from the assumption on γ, V 

N (x) → ∞ when x → ∞,

and therefore V

N (·) satisfies property (ii) in Theorem (Attractivity

in S). It then follows from Theorem (Attractivity in S) that the origin is globally attractive in SN for the closed loop system.

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Main Stability Result

(iii) Asymptotic stability. To show asymptotic stability of the origin, note first that V 

N (0) = 0

(since the optimal sequences in (1)–(7) corresponding to x = 0 have all their elements equal to zero). Next, note from (17) and the properties of γ that V 

N (·) satisfies

property (ii) in Theorem (Lyapunov Stability) with S = SN. If, in addition, the origin is in the interior of SN and V

N (·) is

continuous on a neighbourhood of the origin, then Theorem (Lyapunov Stability) shows that the origin is a stable equilibrium point for the closed loop system, and hence, combined with attractivity in SN, it is asymptotically stable in SN.

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Main Stability Result

(iv) Exponential stability. By assumption, F(x) ≤ bxσ for all x ∈ Xf. It is easily shown that V 

N (x) ≤ F(x) for all x ∈ Xf.

To see this, let x ∈ Xf and denote by {xf

k(x) : k = 0, 1, . . .},

xf

0(x) x, the state sequence resulting from initial state x and

controller Kf(x) in (10). Then, by B2, F(x) ≥

N−1

• k=0

L(xf

k(x), Kf(xf k(x))) + F(xf N(x)),

where, by B3, xf

k(x) ∈ Xf, k = 0, . . . , N and, by B4, Kf(xf k(x)) ∈ U,

k = 0, . . . , N − 1 (that is, they are feasible sequences).

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Main Stability Result

Hence, by optimality, V

N (x) ≤ N−1

• k=0

L(xf

k(x), Kf(xf k(x))) + F(xf N(x)).

Thus, V

N (x) ≤ N−1

• k=0

L(xf

k(x), Kf(xf k(x))) + F(xf N(x)) ≤ F(x),

that is, V

N (x) ≤ F(x) ≤ bxσ for all x ∈ Xf.

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Main Stability Result

From the above bound, it is easy to show that there exists a constant ¯ b > 0 such that V

N (x) ≤ ¯

bxσ for all x ∈ SN, since SN is compact. Combining the above inequality with (16) and (17): V

N (x+) − V N (x) ≤ −L(x, KN(x)) ≤ −γ(x) for all x ∈ SN,

V

N (x) ≥ L(x, u 0 ) ≥ γ(x) for all x ∈ SN ,

and using the assumption that γ(t) ≥ atσ for some constant a > 0, it follows from Theorem (Exponential Stability) that the closed loop system has an exponentially stable equilibrium point at the origin. This concludes the proof of the theorem.

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Linear Systems

For linear systems with convex constraints it is easy to find ingredients to guarantee stability of MPC. Consider the following optimisation problem:

PN(x) :

V

N (x) min

       F(xN) + 1

2

N−1

• k=0

(x

kQxk + u kRuk)

        ,

(18) subject to: xk+1 = Axk + Buk for k = 0, . . . , N − 1, (19) x0 = x, (20) uk ∈ U for k = 0, . . . , N − 1, (21) xk ∈ X for k = 0, . . . , N, (22) xN ∈ Xf ⊂ X, (23) as the underlying fixed horizon optimisation problem for the receding horizon algorithm.

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Linear Systems

We assume: Q > 0 and R > 0 in (18); hence, condition B1 of the main stability Theorem is satisfied with γ(t) = λmin(Q) t2, where

λmin(Q) is the minimum eigenvalue of the matrix Q.

F(x) = 1

2xPx with P ≥ 0 in (18).

The sets U, X and Xf are convex and U and Xf contain the

• rigin of their respective spaces (that is, condition B5 is

satisfied). The above assumptions guarantee that: The set of feasible initial states SN is convex. The value function V 

N ( · ) in (18) is convex in SN, and hence

continuous in the interior of SN.

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Linear Systems

Choices for the Terminal Triple: The terminal state weighting is usually F(x) = 1 2xPx, where P is the positive definite solution of the algebraic Riccati equation P = APA + Q − K  ¯ RK, (24) K ¯ R−1BPA,

¯

R R + BPB. (25) The terminal controller Kf(x) is chosen to be the optimal linear controller Kf(x) = −Kx, where K is given by (25).

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Linear Systems

The terminal set Xf is usually taken to be the maximal output admissible set O∞ for the closed loop system using the local controller Kf(x), defined as

O∞ {x : K(A−BK)kx ∈ U and (A−BK)kx ∈ X for k = 0, 1, . . .}. O∞ is the maximal positively invariant set for the system

xk+1 = (A − BK)xk in which constraints are satisfied.

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Linear Systems

With the above choice for the terminal triple (Xf, Kf, F), conditions B1–B5 of the main stability theorem are readily established. Recall that, as discussed before, conditions B1 and B5 of Theorem 2.1 are satisfied from the assumptions on problem PN(x) in (18)–(23). Clearly conditions B3 (the set Xf = O∞ is positively invariant for the system xk+1 = (A − BK)xk) and B4 (the terminal control satisfies the control constraints in Xf) hold with the above choices for the terminal triple.

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Linear Systems

Using (24)–(25), direct calculation yields that F(x) = xPx satisfies F(f(x, Kf(x))) − F(x) = 1 2x(A + BK)P(A + BK)x − 1 2xPx

= 1

2x(APA − P + 2K  ¯ RK + K ( ¯ R − R)K)x

= −1

2xQx − 1 2xK RKx

= −L(x, Kf(x)),

so that condition B2 is also satisfied. Thus far, we have verified conditions B1–B5 of the main stability theorem, which establishes global attractivity of the origin.

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Linear Systems

To prove exponential stability, we further need to show that the conditions in part (iv) of the theorem are also fulfilled. Note that F(x) ≤ λmax(P)x2. Also, as shown above, V 

N ( · ) is continuous on int SN.

Hence, exponential stability holds in any arbitrarily large compact subset contained in the interior of SN.

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Recapitulation

For linear systems with convex constraints and quadratic objective function VN = F(xN) + 1 2

N−1

• k=0

(x

kQxk + u kRuk),

the choices: Q > 0, R > 0 and F(x) = 1

2xPx, where P is the positive

definite solution of the algebraic Riccati equation P = APA + Q − K  ¯ RK, K ¯ R−1BPA,

¯

R R + BPB;

Xf = O∞;

ensure exponential stability of the closed loop system in any arbitrarily large compact subset contained in the interior of SN

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