Exponentially Convergent Receding Horizon Strategy for Constrained Optimal Control

Receding horizon control has been a widespread method in industrial control engineering as well as an extensively studied subject in control theory. In this work, we consider a lag L receding horizon strategy that applies the initial L optimal controls from each quadratic program to each receding horizon. We investigate a discrete-time and time-varying linear-quadratic optimal control problem that includes a nonzero reference trajectory and constraints on both state and control. We prove that, under boundedness and controllability conditions, the solution obtained by the receding horizon strategy converges to the solution of the full problem interval exponentially fast in the length of the receding horizon for some lag L. The exponential rate of convergence provides a systematic way of choosing the receding horizon length given a desired accuracy level. We illustrate our theoretical findings using a small, synthetic production cost model with real demand data.

on a finite horizon extending from the current time point k. The finite-horizon problem uses the current state of the system as its initial state, yields sequences of optimal controls and states, applies the optimal control at time point k to the system, and uses the optimal state at k + 1 as the initial state of the next receding horizon problem. This on-line feature of RHC makes the method adaptive to changing system parameters, since only a finite horizon extending into the future is required for the current control [18], and particularly attractive when off-line computation of the control policy is difficult [22].
Several results prove the stability of RHC for constrained linear and nonlinear systems. For example, reference [21] proves that RHC yields an asymptotically stable closed-loop system for continuous-time nonlinear systems. In addition, reference [15] establishes stability of RHC for discrete-time, time-varying, constrained nonlinear systems. Both references employ the value function as a Lyapunov function for the stability analysis. Stability results for linear systems can be found, for example, in [16,17,24,25]. More recently, studies [9][10][11]26] have shown stability conditions and performance estimates without stabilizing terminal constraints or terminal costs under suitable controllability conditions for both discrete and continuous-time systems. In addition, [5] extends the previous studies by taking into account the state and control constraints.
In this work, we consider a slight variation of the standard RHC described in the references above. In particular, on each receding horizon, instead of applying the optimal control at only the current time point k, we apply the optimal controls at the initial L time points for some lag L > 0, and the next receding horizon starts at time point k + L. The same receding horizon strategy is considered in other references for continuous time, in order to account for the sampling time; see, for example, [13,14]. Building on our recent analytical developments in [30], we prove that for an inequality-constrained time-varying linear system with a quadratic cost function and nonzero reference trajectory, the optimal states and controls obtained by this receding horizon strategy converge, for some lag L, to the solutions on the full problem interval exponentially fast in the length of the receding horizon. The appropriate lag L in the result is determined by the problem parameters and, in particular, by the controllability properties of the constrained system. Our analysis directly investigates the solutions of a related equality-constrained control problem and connects them to the original inequality-constrained problem through a sensitivity analysis. Specifically, we consider the following problem. min 1 2 In (1), we refer to x k , u k , and d k as the state, control, and reference trajectory, respectively. Problem (1) lacks nonlinear dynamics, which is studied in some stability analyses of RHC, for example, [13,21]. However, we include the inequality path constraint (1c) of state and control as considered in [24,25]. Furthermore, we allow a nonzero reference trajectory d k and prove the exponential convergence of RHC solutions to the solutions of problem (1) instead of a fixed equilibrium point [13,14].
In our proofs, we use two important results in optimal control theory. One is developed in [31], where the authors prove that for an unconstrained, switched-time, and discrete-time linear-quadratic optimal control problem, the optimal trajectory stabilizes exponentially under some mild conditions. They also give an estimate of the exponential rate, which we use here. The other one is established in [29], where the authors propose a Riccati-based approach for solving linear-quadratic optimal control problems subject to linear equality path constraints. They derive a solution procedure based on solving the KKT conditions via the Riccati recursion. We borrow similar manipulations and reductions of the KKT conditions here. Moreover, a few results regarding the Riccati recursion, closed-loop matrix, and sensitivity analysis are nearly the same as those in our previous work [30]. We present those proofs in the Appendix. We note that our previous work [30] had more complex algebra since it did not use the KKT-based ideas from [29], had bound constraints on control only, and did not investigate RHC convergence; these features are present in this work.
The rest of the article is organized as follows. In Section 2, we consider an equality constrained subproblem of (1) and investigate the dependences of solutions on the initial state and terminal reference. In Section 3, we define the lag L receding horizon strategy and prove the exponential convergence of RHC solutions based on results derived in Section 2. In Section 4, we demonstrate our theoretical findings using a synthetic production cost model with real demand data.

Path-constrained Linear-quadratic Problem
In this section, we mainly consider a subproblem of the constrained linear-quadratic optimal control problem (1). For (1), we have that A k ∈ R n×n , B k ∈ R n×m , P k ∈ R r×n , C k ∈ R r×m , and Q k ∈ R n×n , R k ∈ R m×m are positive definite. We make the following uniform boundedness assumption. Assumption 1 For any n 1 , n 2 , and n 1 ≤ k ≤ n 2 , we assume that Note that we use symbols with a tilde for some upper bounds in Assumption 1 (a), since we reserve the corresponding straight symbols for the frequently used quantities defined later in Lemma 1. The subproblem of (1) we investigate is an equality-constrained problem obtained by considering some active subsets of the polyhedral path constraint (1c).

Equality-constrained Subproblem
To define the equality-constrained subproblem, we let I k ⊂ {1, . . . , r} be some index set of the constraint (1c) attaining the bound. Let P k = P k (I k , :) and C k = C k (I k , :) be the corresponding submatrices, and denote q k = q k (I k ). Then the equality constraint corresponding to the sequence of index sets I = {I k } is P k+1 x k+1 + C k u k = q k . The equality-constrained problem we consider is hence the following: where we denote Note that if I k = ∅, then we have P k = 0, C k = 0, q k = 0, and hence E k = 0, H k = 0.
If I is the active set of problem (1) at optimality, then problems (1) and (2) have the same solutions. Note that H k in (3) is determined by the sequence of index sets I encoding the equality constraints under consideration, and hence we define the following uniform boundedness property of H k in terms of the index sets.
Definition 1 Given a sequence of index sets, let H k be as in (3). With some λ H > 0, the sequence of index sets is uniformly bounded from below with respect to λ H , denoted as UDB(λ H ), if for any n 1 ≤ k ≤ n 2 and I k = ∅, H k has full row rank and From (3), the number of rows of H k equals to the number of equality constraints corresponding to the index set I k . Therefore, by requiring H k to have full row rank and be uniformly bounded from below, Definition 1 restricts the total number of equality path constraints for a sequence of index sets that is UDB(λ H ). In the rest of this subsection, we restrict our attention to the index sets that are UDB(λ H ). First, we define some matrices frequently used in the subsection.

Definition 2
For some sequence of index sets that is UDB(λ H ), we define the following matrices for n 1 ≤ k ≤ n 2 − 1: Note thatÂ k andB k are modifications, inspired from [29], of A k and B k , respectively, by taking into account the equality constraints (2c) as determined by the index sets. The rationale for Definition 2 will be made clear later in Lemma 4, which investigates the KKT conditions. To prepare for that, we first derive some properties for the matrices defined.
Lemma 1 Under Assumption 1, for any n 1 , n 2 , and n 1 ≤ k ≤ n 2 , if the sequence of index sets I is UDB(λ H ), then we have for some C A , C B , C Q , C H , C E , CĤ > 0 independent of n 1 , n 2 , and the particular choice of I. Here λ Q is the same as that in Assumption 1.
Proof From Assumption 1 and Definition 2, we have Note that throughout the article, we use the notations A B and A B to mean A − B is symmetric positive semidefinite and symmetric positive definite, respectively.
Note thatB k = B k X k R k from Definition 2, so we havê Since R k 0, we have thatR k 0.
To prove the results in this subsection, we employ the approach in [29] by considering the KKT conditions of problem (2). We first define and derive properties for the following matrices, some of which are similar to those in [29].

Definition 3
For some sequence of index sets that is UDB(λ H ), define the following backward recursions for n 1 ≤ k ≤ n 2 − 1: Proof We prove the statement by backward induction based on (5a)-(5f). To start, we have K n 2 = Q n 2 0 as the induction basis. Suppose K k+1 is positive definite. Then we have W k R k 0, and SinceR k 0 as shown in Lemma 2, we have that I +R k K k+1 is invertible and hence M k is well defined. Also we have so that K k is positive definite. By induction we have that M k is well defined, K k 0, W k 0, and K k+1 M k 0 for all n 1 ≤ k ≤ n 2 − 1.
Note that since K k is symmetric, Therefore (6) holds.
Note that (6) is also stated (without proof) in [29]. Now we derive a recursion of the optimal states of problem (2) by investigating the KKT conditions.

Lemma 4 Let u *
k and x * k be the optimal controls and states of problem (2), and let λ * k and η * k be the Lagrange multipliers associated with the dynamical constraint (2b) and the equality constraint (2c), respectively. For n 1 ≤ k ≤ n 2 − 1, if the sequence of index sets I is UDB(λ H ), then we have where K k and T k are given by the backward recursions (5a)-(5f).
In the following, we investigate the notion of controllability for the system (2b)-(2c). We define the following controllability matrix for the sequence of index sets I in terms of the sequence pair {Â k ,B k } in Definition 2.

Definition 4
For some n 1 ≤ q ≤ n 2 , t > 0, and some sequence of index sets I that is UDB(λ H ), define the controllability matrix associated with time steps [q, q + t − 1] as To see the relationship between C q,t (I) and the controllability of the equality constrained system (2b)-(2c), we start by defining the notion of controllability for the system (2b)-(2c). (3). At time step q, the system

Definition 5 Given a sequence of index sets, define E k and H k as in
is controllable in t steps if for any x 0 q and x, there exist admissible controls {u k } k=q:q+t−1 and corresponding states {x k } k=q+1:q+t satisfying (13) and x q+t = x. (2) is controllable at time point q in t steps, then C q,t (I) has full row rank.

Proposition 1 If the sequence of index sets I is UDB(λ H ) and the resulting constrained system (2b)-(2c) of problem
Proof The system (2b)-(2c) being controllable in t steps implies that there exist admissible controls {u k } k=q:q+t−1 and corresponding states {x k } k=q+1:q+t so that x q+t = x for any x. Then we have, for q ≤ k ≤ q + t − 1, which means that the same sequences {u k } and {x k } also satisfy the linear dynamics and that x q+t = x. In other words, (14) can be controlled in t steps to x. Since x is arbitrary, it follows that C q,t (I) has full row rank.
Proposition 1 connects the controllability of the equality-constrained system (2b)-(2c) to the full rank of a related controllability matrix C q,t (I). For our purpose, however, we need a uniform boundedness property of the controllability matrix, which is stronger than the standard assumption of merely full rankness.

Definition 6
For some sequence of index sets I that is UDB(λ H ), letÂ k ,B k be as in Definition 2. With some 0 < t < n 2 − n 1 and λ C > 0, the sequence of index sets is uniformly completely controllable with respect to λ C , denoted as UCC(λ C ), if the sequence pair {Â k ,B k } is uniformly completely controllable [15, Definition 3.1], i.e., for any n 1 ≤ q ≤ n 2 , The main purpose of this subsection is to investigate the dependencies of the solutions of problem (2) on the initial value x 0 n 1 and terminal reference d n 2 for some sequence of index sets that is UDB(λ H ) and UCC(λ C ). To start, we derive properties for the quantities defined in Definition 3. The proofs of the results regarding the Riccati matrix K k and closed-loop matrix D k are structurally the same as those in [30], and hence they are provided in the Appendix.
where M k , K k , and W k are from Definition 3.
Proof Definition 3 and Lemma 2 imply that Then we have which proves (15b).

Proposition 2 Under Assumption 1, if the sequence of index sets I is UDB(λ H ) and
UCC(λ C ), then for any n 1 ≤ q ≤ n 2 , we have K q 2 ≤ β for some β > 0 independent of n 1 , n 2 , and the particular choice of I.
Proof See Appendix A.1; also see [30, Proposition 2.7]. In the following, we show the dependencies of the solutions to problem (2) on the initial state and terminal reference decay exponentially. To start, we prove a short lemma about the recursion defined in (5e).

Proposition 3 Under Assumption 1, for any
for some C s > 0 independent of n 1 , n 2 , and the particular choice of I.
Proof Recursion (5e) gives The statement is proved by using Proposition 3 and taking C s = C Q C 1 for C Q defined in Assumption 1.
Proposition 4 Let x * k and u * k be the optimal states and controls of problem (2). Under Assumption 1, if the sequence of index sets I is UDB(λ H ) and UCC(λ C ), then

and the particular choice of I.
Proof From Assumption 1 and Lemma 1 we have, for X k andR k defined in Definition 2,

Lemmas 5 and 1 and Proposition 2 give
where the first inequality uses the relation W k R k , which is given by (5b) and Lemma 3. From Lemma 4 we have so which follows from Proposition 3. Also, from (16) we have and ∇ d n 2 x n 1 = 0. From Recursion (18) we have from which, using Proposition 3 and Lemma 6, we have, for Equation (7a) gives In this expression, for the term multiplying x * k and T k+1 , we have the following from Assumption 1 and Lemma 1, respectively: Note that from (5e), T k does not depend on the initial value x n 1 . Therefore we have and Considering (17), (19), (20), and (21) and letting prove the statement.
Proposition 4 is the main result of this subsection. It shows that the effect of the initial state (or terminal reference) on the solutions of problem (2) decays exponentially fast in the time distance between the solution and the initial (or terminal) time point. Moreover, under the uniform boundedness Assumption 1, the decay rate is independent of the problem interval [n 1 , n 2 ], and the particular choice of the index set given it is UDB(λ H ) and UCC(λ C ). This property is essential for proving that a receding horizon strategy approximates the solution on the full horizon in Section 3. We now conclude this subsection with a boundedness result of the solutions and adjoint variables of problem (2). Assumption 2 For any n 1 , n 2 and n 1 ≤ k ≤ n 2 , we have d k 2 ≤ m 0 and x 0 n 1 2 ≤ u 0 .
Lemma 7 Let x * k and λ * k be the optimal states and adjoint variables of problem (2), respectively. Under Assumptions 1 and 2, if the sequence of index sets I is UDB(λ H ) and UCC(λ C ), then we have

and the particular choice of I.
Proof In (5e), denote T n 2 = T n 2 and for k < n 2 , As a result, we have From Assumption 1 and Definition 2 we have From Lemmas 5 and 1 and Proposition 2 we have M k 2 ≤ 1 + C 2 B β/λ R = C M . Then using Assumption 1, Lemma 1, and Proposition 2, we have Combining the above with Proposition 3, we have Denote G k = −M kRk T k+1 + M kqk . Then, from Lemma 4, we have Next, we prove that the bound on λ * k . Lemma 4 gives Using Proposition 2, (22) and (23), we have This completes the proof.

Path-constrained Inequality Problem
In this subsection, we return to the inequality-constrained problem (1). Using the results established for (2), we investigate the solutions and adjoint variables of (1). We make the following controllability assumption of the active set of problem (1) at optimality.

Assumption 3 Let
A be the active set of problem (1) at optimality. Then (a) A is UDB(λ H ) as defined in Definition 1; (b) the equality-constrained system (2b)-(2c) corresponding to A is controllable in t steps as defined in Definition 5 for any n 1 ≤ q ≤ n 2 ; (c) under (b), C q,t (A) has full row rank by Proposition 1, and we further assume that A is UCC(λ C ) as defined in Definition 6.
Corollary 1 Let x * k and λ * k be respectively the optimal states and adjoint variables of problem (1). Under Assumptions 1, 2, and 3, we have Proof Note that when the sequence of index sets for problem (2) is the active set A of problem (1) at optimality, problems (1) and (2) have the same solutions and adjoint variables. Since A is UDB(λ H ) and UCC(λ C ) by Assumption 3, applying Lemma 7 gives the result.
Note that the constant upper bounds in Corollary 1 depend on λ H and λ C in Assumption 3.

Lag L Receding Horizon Strategy
In this section, we prove an exponentially decaying approximation error for a lag L receding horizon strategy. Let N > L be the length of each but the last receding horizon, and let n 0 = (n 2 −n 1 −N +1)/L +1 be the number of receding horizons. Then for i = 1, . . . , n 0 , define the ith receding horizon R i = [n 1 (i), n 2 (i)] as For simplicity, we denote m = n 1 (n 0 ) to be the starting index of the last receding horizon. Note that with (24), we have n 1 (i + 1) = n 1 (i) + L and that the length N 1 = n 2 − m + 1 of the last receding horizon satisfies N ≤ N 1 < N + L. On a receding horizon R i , we define the following parametrized problem whose parameters are the initial state and terminal reference.
The parametrized problem P i θ is essentially a subproblem of (1) restricted on the receding horizon R i with terminal reference parametrized by θ (d) and reinitialized with θ (h) . Denote x * k (P i θ ), u * k (P i θ ) as the optimal state and control, respectively, of problem P i θ at some time point k ∈ R i . Then the RHC policy (e.g., [13,18]) is the sequence { u k } n 2 k=n 1 defined as where θ 0 (i) = ( x n 1 (i) , d n 2 (i) ), and the state sequence { x k } n 2 k=n 1 is defined as In other words, the RHC policies u n 1 (i)+j −1 for 1 ≤ j ≤ L and 1 ≤ i ≤ n 0 − 1 are obtained by solving problem P i θ 0 (i) on R i initialized with x n 1 (i) = x n 1 (i−1)+L , which in turn is obtained by solving P i−1 θ 0 (i−1) on R i−1 . On the last receding horizon, u k for m ≤ k ≤ n 2 −1 are defined as the optimal controls of problem P n 0 θ 0 (n 0 ) on R n 0 . To bound the error of this RHC strategy, we need to relate the solutions of problems P i θ 0 (i) to those of problem (1). To start, we consider a different choice of the parameter θ 1 (i). The following result establishes a connection between the solutions of (1) and those of P i θ 1 (i) .
Proposition 5 shows that the solution of P i θ 1 (i) is identical to the solution, restricted to R i , of problem (1). However, problem P i θ 1 (i) is formal and cannot be defined without first solving problem (1). Hence, we need to investigate the relationship between solutions of problems P i θ 1 (i) and P i θ 0 (i) , the latter of which gives the RHC solutions. Since problem P i can be viewed as resulting from a perturbation of the parameters of problem P i θ 0 (i) , we employ the following parametric sensitivity results derived from [6].

Definition 8
For θ ∈ R q , define the one-sided directional derivative of y(θ) along a direction p ∈ R q at θ 0 as given that the limit exists.
Note that problem P i θ has the same structure as that defined by (31), and Lemma 9 connects the dependence on parameters of the solutions for the inequality-constrained problem (31) with that of a related equality-constrained problem (32), whose equality constraints are subsetted from the active constraints of (31) at optimality. The equality-constrained problem has smooth and regular KKT conditions which facilitate the derivation for the dependence of solutions on parameters as shown in Section 2.1. Now we are ready to investigate the effect on solutions of perturbing the parameters of P i θ . Since the proof for each receding horizon is the same, for notational simplicity we suppress the dependence of n 1 (i), n 2 (i) and P i θ on i whenever the index of the receding horizon under consideration is clear. To connect the solutions of P θ 1 and P θ 0 , we consider a continuously indexed family of problems P θ s for θ s = θ 0 + s(θ 1 − θ 0 ) and s ∈ [0, 1]. Let P k+1 (s)h k+1 + C k (s)w k = q k (s) be the active constraints of problem P θ s at optimality. We let E k (s) = P k+1 (s)A k , H k (s) = P k+1 (s)B k + C k (s).
Thus, the active constraints of P θ s are where θ s = (θ (h) s , θ (d) s ). As in Assumption 3 (a), we make the following uniform boundedness assumption about the active constraints of P θ s .
In particular, Assumption 4 implies that H k (s) defined in (33) has full row rank, with which we can now apply Lemma 9 to problem P θ s .

Lemma 10
Denote θ 0 = ( x n 1 , d n 2 ) and θ 1 = (ĥ n 1 ,d n 2 ) as defined in (28). For θ = (θ (h) , θ (d) ), let x(θ) be the solution of problem P θ . Under Assumption 4, for s ∈ [0, 1] and θ s = θ 0 + s(θ 1 − θ 0 ), we have and y s (θ) is the solution of the following equality-constrained problem: where rows of E k (s) and H k (s) are respectively subsets of rows of E k (s) and H k (s) defined by the active constraints of P θ s at optimality as in (33). In other words, E k (s)h k + H k (s)w k = q k (s) is a subset of the equality constraints (34b).
Proof Problem P θ s is an instance of problem (31) with the following parameters: Here Ax ≤ r and Bx = d(θ) correspond respectively to the inequality constraints (25c) and the dynamical constraints (25b). Note that A and B have the same number of columns. G and F are positive definite from Assumption 1. The quantities c 1 , c 2 , and C of problem (31) do not enter in the proof, so their definitions are not shown. Let A(s) be the matrix whose rows are subsets of rows of A corresponding to the active constraints P k+1 (s)h k+1 + C k (s)w k = q k (s) at optimality for problem P θ s . Then we define In the following we show that rows of A(s) and B are linearly independent. Denote Then

Now we let α T A(s) + β T B = 0.
We show that α, β = 0 by backward induction. We have Suppose α k , β k−1 = 0 for some n 1 + 1 < k ≤ n 2 . Then we have So LICQ holds for problem P θ s at optimality. Directly applying Lemma 9 concludes the proof.
Problem (35) is an equality-constrained problem for which the results derived in Section 2, especially the exponential decay property of the dependence of solutions on the initial state and terminal reference, can be applied under certain assumptions. In the following, we investigate the controllability conditions for problem (35). Denote the active set of problem P i θ s (i) and the index set for the corresponding equality constraints of problem (35) as A s (i) and I s (i), respectively.

Lemma 12
If the equality-constrained system (34) corresponding to the active sets of P θ s is controllable at q in t steps, then the controllability matrix C q,t (I s ) defined by the subsetted system (35b)-(35c) has full row rank.
Proof If system (34) can be controlled to an arbitrary state x in t steps with some admissible controls and corresponding states, then the subsetted system (35b)-(35c) can also be controlled to x in t steps with the same controls and states, because the feasible set defined by (35b)-(35c) contains that defined by (34). As a result, from Proposition 1 C q,t (I s ) has full row rank.
Assumption 5(a) assumes controllability of problem P θ s only at optimality. Proposition 1 and Lemma 12 imply that C q,t (A s ) and C q,t (I s ) are bounded below, and Assumption 5 (b) in addition assumes that the lower bounds are uniform for all time points and receding horizons. Now we are ready to bound the distance between solutions of P θ 0 and P θ 1 , which by Proposition 5, is also the distance between solutions of RHC and problem (1). The assumptions needed in the proof are summarized as follows. Assumptions 1 and 2 give uniform bounds for the problem parameters. Assumption 3 assumes uniform boundedness and controllability for the active set of problem (1) at optimality. Lastly, Assumptions 4 and 5 give the uniform boundedness and controllability conditions, respectively, for the active sets of problems P i θ s (i) for all receding horizons i = 1, . . . , n 0 and s ∈ [0, 1].

Theorem 6
For 1 ≤ i ≤ n 0 , let u * n 1 (i):n 2 (i)−1 , x * n 1 (i)+1:n 2 (i) be the solution of problem P i θ 1 (i) , which from Proposition 5 is exactly the solution of problem (1) restricted to R i . Let u k and x k be the receding horizon control and state defined in (26) and (27), respectively. Under Assumptions 1-5, we have, for some lag L so that Z 1 ρ L < 1 and receding horizon length N > L, Here C d > 0 is independent of N , n 1 , and n 2 ; Z 1 , Z 2 , and ρ are as in Proposition 4.
Proof Assumption 1 and Lemma 8 give, for i = 1, . . . , n 0 , Let θ 0 (i) = ( x n 1 (i) , d n 2 (i) ), θ 1 (i) = (ĥ n 1 (i) ,d n 2 (i) ) as defined in (28), and θ s (i) = θ 0 (i) + s(θ 1 (i) − θ 0 (i)) for s ∈ [0, 1]. Note that for 1 ≤ j ≤ L, x * n 1 (i)+j is the optimal state of problem (1), which by Proposition 5 is also that of problem P i θ 1 (i) ; while x n 1 (i)+j is the optimal state of problem P i θ 0 (i) by (27). Denote s * k (θ s (i)) and p * k (θ s (i)) for n 1 (i) ≤ k ≤ n 2 (i) as the optimal control and state of problem P i θ s (i) , and s * k (θ s (i)) and p * k (θ s (i)) as those of the corresponding subsetted equality-constrained problem (35). Then for 1 ≤ i ≤ n 0 − 1, we have from Proposition 5 and Lemma 10 Lemma 11 states that the index set of the corresponding problem (35) is UDB(λ H ), and Assumption 5(b) further states it is UCC(λ C ). Note that the exponential bounds obtained in Proposition 4 are independent of the problem interval and the particular choice of the equality constraint index set, which is UDB(λ H ) and UCC(λ C ). Therefore, applying Proposition 4, we have, for 1 ≤ i ≤ n 0 − 1 and 1 ≤ j ≤ L, When j = L, note that n 1 (i) + L = n 1 (i + 1), so (37) becomes From this recursion, at the starting index of R i for 1 ≤ i ≤ n 0 we have Note that L is chosen so that Z 1 ρ L < 1. Substituting (38) into (37), we have, for 1 ≤ i ≤ n 0 − 1 and 1 ≤ j ≤ L, Now we prove the approximation error bound for the RHC policies. For 1 ≤ i ≤ n 0 − 1 and 1 ≤ j ≤ L, u * n 1 (i)+j −1 is the optimal control of problem (1), which by Proposition 5 is also that of problem P i θ 1 (i) , whereas u n 1 (i)+j −1 is the optimal control of problem P i θ 0 (i) by (26). Therefore, by Proposition 5 and Lemma 10 we have
Note that since the quantities Z 1 and ρ are independent of the problem interval [n 1 , n 2 ], so is the choice of L. Therefore, Theorem 6 proves the approximation error of the RHC solution with an appropriate lag L decays exponentially in the length N of the receding horizon regardless of the full problem interval under the uniform boundedness and controllability conditions. The exponential decay rate provides systematic means to choose the length of the receding horizon given a desired accuracy level. We also note that it is a bit surprising that the choice L = 1 may sometimes not satisfy our assumptions; that case occurs when Z 1 ρ > 1. In particular, we note that in reference [11] it was shown that, for nonlinear RHC with zero reference, under controllability conditions similar to ours [11, Assumption 3.1] stabilization does occur for any L (though note that the exponential decay of the initial value effect is assumed in that reference [11, (3.2)] as opposed to proved, as we do in Proposition 4, assuming the state-space form definition of controllability; moreover, the problem described in [11] is unconstrained, though nonlinear). That result was shown by analyzing the descent properties of the Lyapunov function induced by the objective of the RHC problem, given that the system had the zero state vector as a reference. Therefore the objective function value acted as a metric of state size, and reducing it meant approaching optimality and the stable solution. In the case with nonzero reference signal that we treat here, however, only the difference between the current value of the objective and the global minimum can at best play that role, and the latter value is not known apriori. While for analytical results, this may be less of an issue, several of the assumptions used in [11] are not obviously extensible to the case of non-zero reference signal, as is the case with, for example, of [11,Assumption 3.1]. Further difficulties are created by the fact that the infinite horizon objective function [11, (2.3)] for N = ∞ may be ∞ in the nonzero reference case. While it is certainly unsatisfying that, when the reference signal d k goes to zero, we do not recover the results from [11], we do not see an obvious way to extend our technique or combine it with the one in [11].

Numerical Results
In this section, we apply the receding horizon strategy to the following production cost model and verify some of the theoretical results.
In this model, d k is the hourly electricity demand to be satisfied, for which we employ the estimated hourly demand data in the northern Illinois region for year 2016 provided by PJM Interconnection [12]. The demand can be satisfied by two generators: one with a high quadratic cost c 1 = 10 and the other one with a low quadratic cost c 2 = 5. The low-cost generator has a limited generation level, modeled by the upper bound (42c) on the generation x k . The generator with a high cost serves the remaining loads d k − x k . We initialize problem (42) by setting the initial state x 0 1 to be the average demand of year 2015 on the same hour as the initial time point. We note that problem (42) has the form of problem (1). Moreover, the active set of problem (42) at optimality is UDB as in Definition 1 with We implement the receding horizon strategy described in Section 3 with lag L = 1. Specifically, we solve a short version of problem (42) on a receding horizon R i = [n 1 (i), n 2 (i)] with length N T and initial value x n 1 (i) , obtain the optimal control u n 1 (i) and state x n 1 (i)+1 , then reinitialize at time point n 1 (i) + 1 with x n 1 (i)+1 to solve the problem on the next receding horizon R i+1 = [n 1 (i) + 1, n 2 (i) + 1]. Problem (42) is solved on the full horizon and each receding horizon using the Ipopt software [4]. The model was defined by using the Julia/JuMP interface [20].
We investigate the solution accuracy of the receding horizon strategy with different choices of the generation upper bound G to verify our theoretical findings. Denote x * = {x * k } and u * = {u * k } as the optimal state and control of problem (42) on the full horizon, and x = { x k } and u = { u k } as those obtained by the receding horizon strategy. Figures 1 and 2 show the relative approximation errors of the optimal states and controls, respectively, for different choices of G. We observe exponential decay of the approximation error in the length of the receding horizon for all cases tested.
Assumption 3 requires problem (42) being controllable in t steps at optimality. We thus investigate numerically the longest period t for which problem (42) is not controllable. Since the problem is one-dimensional, controllability holds when no constraint is active. Therefore, we calculate the longest contiguous period for which (42c) is active at optimality. Table 1 shows t in hours for different choices of G. The longest periods of uncontrollability are all less than two days, which is well covered by the one-year problem horizon (42). At small G, though, the uncontrollability period may be longer than the small horizons used by RHC for which the error decay is displayed in Figs. 1 and 2. In summary, the example problem (42) satisfies Assumptions 1 and 2, the boundedness of the data, by direct inspection, Assumption 3, the UBD and UCC (controllability) properties by the discussion above, and Assumption 4, the UDB property (Definition 1) for the short horizon RHC problems. Assumption 5 is satisfied only for large values of G. The numerical experiments satisfy the exponential decay of the approximation errors for RHC as proved in Theorem 6, and while our assumptions do not all appear necessary, they are certainly sufficient as implied by that theorem.  We show in the following example that even with further moderate violation of the conditions, we may still have exponential decay of the approximation errors for the RHC solutions.
Problem (43) has one more box constraint (43d) on the controls u k to model a limited capacity for the low-cost generator to change its output. One example of such a situation is the combination of a fast but expensive gas plant and a cheap but slow coal plant. If both constraints (43c) and (43d) are active at optimality for some time point k, the corresponding H k does not have full row rank and hence the UDB condition is not satisfied. In fact, Table 2 shows that a small proportion p of the optimal solutions of problem (43) does have both constraints active for different choices of G and U , hence violating Assumption 3(a) and  Assumption 4. Nonetheless, the longest periods t of uncontrollability when at least one constraint is active are less than two days even under the tightest bounds of U = 200 or U = 400. Therefore, the controllability condition, Assumption 3(b) still holds for this example, in addition to the boundedness Assumptions 1 and 2. Figures 3 and 4 show the relative approximation errors of the optimal states and controls, respectively, for a fixed G. We observe exponential decay of the approximation error in the length of the receding horizon for different choices of U . Moreover, the rate of decay is faster for a larger bound U on the control. The decay rate ρ in Theorem 6 depends on the  quantity β defined in Proposition 2, which in turn depends on the uniform lower bound λ C of the controllability matrix defined in Definitions 4 and 6. A larger bound on the control improves the controllability of the problem and should lead to a faster rate of convergence, as is indeed observed here. Figures 5 and 6 plot the relative errors for different upper bound G on the generation. Similarly, we observe exponential rate of decay for the approximation error in the length of receding horizon. Furthermore, in this case the decay rate is larger for a smaller choice of G. Recall that G is the generation upper limit of the slow plant and our model (43) assumes the remaining load will be satisfied fully by a high-cost fast plant. A smaller G indicates that more demand is met by the fast plant, and therefore the system is more controllable, resulting in a faster rate of convergence (for example, if G = 0, then we get the optimal solution to be x k = u k = 0, ∀k and the convergence occurs in one step for any horizon).

Conclusions
RHC has made a significant impact on industrial control engineering and received extensive study of its theoretical characteristics. We investigate the convergence of its solution with respect to the length of the receding horizon for a linear-quadratic path-constrained optimal control problem.
The version of RHC considered in this work applies the model predictive control every L steps. Our theoretical result, Theorem 6, shows that, under some boundedness and controllability conditions, the RHC solution converges to the full horizon solution exponentially fast in the length of receding horizon for a certain choice of L. The exponential rate of convergence allows a principled way of choosing the length of the receding horizon and the control frequency, both important parameters for applications, to achieve a desired accuracy. Our problem admits a nonzero reference trajectory, which to the best of our knowledge is not assumed in the existing stability analysis of RHC. The inclusion of a reference trajectory makes the analysis different from previous approaches since now the convergence is with respect to the solution of the full horizon problem instead of a fixed equilibrium point. Therefore our proofs do not rely on the value function, as most RHC stability analyses do, but instead expose the solution properties of an equality-constrained subproblem and then use sensitivity analysis to connect it to the solution of the original problem. We verify numerically the exponential rate of convergence for a small, synthetic production cost model under various parameter settings. In this example, a lag L = 1 is sufficient to observe the exponential decay for the approximation error of the RHC solutions.
The class of optimal control problems investigated here is only one instance of the problems to which RHC can be applied. In particular, although we consider state and control constraints that are common in RHC literature, we do not include other intricate but practical features such as nonlinear dynamics or time delay. Moreover, our theory certifies only that an L, which is computable in terms of the problem data, exists; but L = 1 may not always satisfy our conditions. In future work, we will investigate extending the results to other complicating features as well as investigate whether we can obtain similar results with weaker assumptions, as our numerical results seem to indicate is possible.
The index set being UCC(λ C ) implies that C q,t is uniformly completely controllable and in particular that C q,t has full row rank. Therefore, there existsû = (û T q , . . . ,û T q+t−1 ) T so that Denote the corresponding states generated withû q:q+t−1 asx q:q+t . Thenx q+t = 0 by (46 Then from Definition 6 and Lemma 1, we have From (45), we have, for 1 ≤ j ≤ t − 1, Now we letû k = 0 for k ≥ q + t. Then it follows thatx k = 0 for k ≥ q + t. Also note that since (44) is a standard linear-quadratic regulator problem, the optimal value is given by x T q K q x q [2]. As a result, we have the following.
x T q K q x q = min u k n 2 −1 k=q x T kQ k x k + u T k R k u k + x T n 2 Q n 2 x n 2 ≤ Letting completes the proof. Note that β depends only on the quantities in Assumption 1, Definitions 1 and 6, and Lemma 1, which are independent of n 1 , n 2 , and the particular choice of I given it is UDB(λ H ) and UCC(λ C ).

A.2 Proof of Proposition 3
Define L k = −W −1 kB T k K k+1Âk . Then from Lemma 5 and (5d) we have D k =Â k +B k L k . In [2] the recursion (15b) is shown to be equivalent to For q ≤ j ≤ n 2 − 1, define x j +1 = D j x j . Then (49) and Proposition 2 imply that As a result, we have which proves the claim.