2015 American Control Conference Palmer House Hilton July 1-3, 2015. Chicago, IL, USA
On Average Performance of Economic Model Predictive Control with Time-Varying Cost and Terminal Constraints David Angeli, Alessandro Casavola and Francesco Tedesco Abstract— This note extends from a theoretical perspective recent results in Economic Model Predictive Control to the case of time-varying cost functionals and terminal constraints. Formal conditions are provided in order to achieve average performance bounds of the closed-loop system under suitable technical assumptions guaranteeing existence of an asymptotic average stage cost (for frozen control and state variables).
I. I NTRODUCTION Model predictive control (MPC) has been one of the most widely investigated control methods over the last decades. This is mainly due to its capability of delivering a control action that takes into account all relevant constraints on input and output variables, addressing robustness issues with respect to model uncertainty and disturbances as well as satisfying optimality conditions related to tracking performance. In particular, in the context of tracking problems addressed via MPC, the cost to be minimized is usually assumed to be a positive definite function of the state, command and set-point or reference samples. In recent years, however, the standard MPC framework has been extended to a more general setup, referred to as Economic MPC, where the above assumption on the positiveness of the cost function is relaxed. In fact, in this new formulation, more general cost functions (usually related to some economic variables of the considered process) are taken into account. In this respect, several variants have been proposed, see e.g. [1], [2], [3], [4], [5], [6]. Among them, different assumptions or additional (terminal) constraints have been adopted in order to address the critical challenges arising from formulating a stabilizing control scheme with guaranteed recursive feasibility and asymptotic performance while minimizing a general cost function. For an extensive review on the topic please refer to [7]. Most of the existing Economic MPC schemes are formulated with time-invariant economic cost functions and are oriented to force the state of the process to converge to an optimal constant operating regime. However, in the case where the time constants of both time-varying economic D. Angeli is with the Department of Electrical and Electronic Engineering, Imperial College London, SW7 2AZ, U.K., and also with the Department of Information Engineering, University of Florence, Italy
variables and dominant process dynamics are comparable, the use of constant optimal regimes may be inadequate to profitably and efficiently manage the plant. Examples of such a situation involve the management of energy in buildings ([8], [9]), the control of chemical plants ([10]) and the supervision of distribution networks, such as water networks ([11]), power grids ([12], [13], [14],) gas networks, etc ([15]), where the price of the utilities (e.g. water, energy, gas) to be delivered often varies hourly or daily, by reflecting their availability against the current demand. Hence, the notion of time-varying (yet a-priori known) stage cost appears of interest in Economic MPC. While this has been already used in practice, there are few works ([16], [17]) addressing such a topic from a theoretical perspective and important aspects such as stability, robustness and asymptotic performance still need to be investigated in a formal way. This paper aims at presenting a theoretical study on Economic MPC in the general case when both time-varying stage costs and terminal constraints are considered. Its main contribution relies upon extending the asymptotic average performance analysis carried out in [5] for the Economic MPC framework involving a time-invariant cost function, to the more general scheme considered here where the cost is either periodic or fulfills certain average assumptions. It is shown that an average performance can be guaranteed by introducing time-varying terminal constraints where the state is forced to reach a periodic solution instead of converging to an optimal steady-state equilibrium. The remainder of the paper is organized as follows. In Section II the problem setup is introduced. Then, an Economic MPC strategy with a time-varying periodic stage cost and terminal constraints of same period is proposed in Section III where the related asymptotic performance is rigorously analyzed. Section IV presents a further Economic MPC scheme with a time-varying (not necessarily periodic) cost endowed with a terminal constraint that forces the state to reach the precomputed best periodic solution with respect to the related averaged cost. Finally, an illustrative example and conclusions of this work are presented in Section V and VI respectively.
[email protected] A. Casavola and F. Tedesco and versity of Calabria, DIMES, 87036
are with the UniRende (CS), Italy. {ftedesco,casavola}@dimes.unical.it. This work has been partially supported by the European Commission, the European Social Fund and the Calabria Region. The authors are solely responsible for the content of this paper and the European Commission and Calabria Region disclaim any responsibility for the use that may be made of the information contained therein.
978-1-4799-8684-2/$31.00 ©2015 AACC
II. P RELIMINARIES AND P ROBLEM S ETUP A. Notation and Definitions Let I≥0 denote the set of nonnegative integers, Ip:q , p < q the integer sequence {p, p + 1, p + 2, ..., q} and R the set of real numbers. Let | · |Q denote the weighted Euclidean norm
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of a vector (i.e. |x|Q = xT Qx where Q is a positive definite matrix). B. Problem setup Consider a discrete-time nonlinear system of the form x(t + 1) = f (x(t), u(t)),
x(0) = x0
(3)
which is assumed to be continuous in x and u. The cost functional one wants to minimize at each sampling time is defined over a sufficiently long yet finite prediction horizon (N steps) given by VN (t, x, u) :=
N −1 X
`(t + k, x(k), u(k))
(4)
k=0
where u = [u(0), ..., u(N −1)] and x(k+1) = f (x(k), u(k)) for all k in I0:N −1 , while we denote x(0) = x. Accordingly, one is interested in investigating Economic MPC schemes characterized by both a time-varying cost and terminal constraints that can be described in the following general form min u subject to
III. P ERIODIC C OST AND T ERMINAL C ONSTRAINTS A typical situation complying with Assumption 1 occurs when ` itself is periodic in t. More formally Assumption 2: (` periodicity) There exists T ∈ I≥0 such that:
(2)
for some compact set Z ⊆ X × U. In order to compute a control input to system (1) within a model predictive control framework, system (1) is equipped with the following time-varying stage cost related to its economic performance `(t, x, u) : I≥0 × X × U → R
The above assumption is crucial for the investigation of the asymptotic performance of system (1) under the control actions computed by means of algorithm (5).
(1)
with t ∈ I≥0 , state x ∈ X ⊂ Rn , input u ∈ U ⊂ Rm and state-transition map f : X × U → X which is assumed to be continuous in (x, u). The system is subject to (possibly coupled) pointwise-in-time constraints (x(t), u(t)) ∈ Z
VN (t, x, u) x(k + 1) = f (x(k), u(k)), k ∈ I0:N −1 (x(k), u(k)) ∈ Z, k ∈ I0:N −1 (5) x(N ) = xF (t) x(0) = x
where the term xF (t) appearing in the terminal equality constraint is going to be defined in next Sections, according to several possible design strategies. Moreover, as customary in receding horizon control, it is assumed that only the first element of the minimizing control sequence computed at time t, is actually applied to the plant and the whole optimization program reruns at the subsequent sampling time. A typical situation in addressing tracking problems via standard MPC is that of using a time-invariant stage cost. On the contrary, in the present context, the following assumption appears more suitable in a number of Economic MPC applications: Assumption 1: (Existence of averaged stage cost) The ¯ i.e. there function ` admits an asymptotic time average `, ¯ exists ` : X × U → R such that PL−1 t=0 `(t, x, u) ¯ u) := lim . (6) `(x, L→+∞ L
`(t + T, x, u) = `(t, x, u)
∀ (t, x, u) ∈ I≥0 × X × U (7)
In this context, one can define the best T -periodic performance as the solution of the following optimization problem (which, for the sake of simplicity, we assume unique) PT −1 `(k, x(k), u(k)) `?T := min k=0 x,u T x(k + 1) = f (x(k), u(k)) k ∈ I0:T −1 (x(k), u(k)) ∈ Z k ∈ I0:T −1 subject to x(T ) = x(0) (8) By denoting with x? = [x? (0), ..., x? (T − 1), x? (T )] and u? = [u? (0), ..., u? (T − 1)] the optimal state and, respectively, input sequence for this problem, one can obtain a time-varying state feedback control law u0 (t, x) by solving online the following optimization problem over the set of T -periodic terminal constraints VN0 (t, x)
subject to
:= min VN (t, x, u) u z(k + 1) = f (z(k), u)(k), k ∈ I0:N −1 (z(k), u(k)) ∈ Z, k ∈ I0:N −1 (9) z(N ) = xF (t) = x? ((t + N ) mod T ) z(0) = x
Let (z0 (t, x), u0 (t, x)) denote the optimal state and input sequences of (9) for the initial state x. Then, according to the Receding Horizon (RH) approach, the command to be applied is given by u(t) = u0 (0; t, x(t))
(10)
leading to the following closed-loop system x(t + 1) = f (x(t), u0 (0; t, x(t))).
(11)
where u0 (k; t, x(t)) denotes the k-th sample of the sequence u0 (t, x). Equation (10) is defined on the set of x for which the optimization problem (9) is solvable. Notice that, in the case of both periodic cost and terminal constraints, the resulting closed-loop system (11) is also a T -periodic nonlinear system. It is shown in the next Theorem 1 that the resulting feedback law (10) induces an asymptotic average cost that is not worse than that corresponding to the optimal periodic solution. Theorem 1: Let x(0) be a feasible initial condition for problem (9). Then, x(t) as defined through iteration (11) is
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feasible for all t and the average asymptotic performance of the closed-loop system fulfills PL `(k, x(k), u(k)) ≤ `?T lim sup k=0 (12) L+1 L→+∞ For space reasons the proof is missing. However it will be available in a journal version of the paper as soon as possible. IV. O PTIMAL P ERIODIC C ONSTRAINTS BASED ON AVERAGED S TAGE C OST In this section the main properties of an Economic MPC scheme based on the best Q-periodic solution related to the averaged stage cost `¯ are investigated. Such a solution can be computed by solving the following optimization problem PQ−1 ¯ `(k, x(k), u(k)) `¯?Q := min k=0 x,u Q x(k + 1) = f (x(k), u(k)) k ∈ I0:Q−1 (x(k), u(k)) ∈ Z k ∈ I0:Q−1 subject to x(Q) = x(0). (13) ˜? = Again, let ˜ x? = [˜ x? (0), ..., x ˜? (Q − 1), x ˜? (Q)] and u [˜ u? (0), ..., u ˜? (Q−1)] denote respectively the best Q-periodic solution and the corresponding control sequence. In this respect, it is interesting to observe how `?T in (8) and `¯?Q in (13) are related each other. In the case where Q = T , the following formal result can be proved: Proposition 1: The best T -periodic solution computed in (8) is never worse than the T -periodic solution based on the averaged cost `¯ as in (13), i.e. `?T ≤ `¯?T . (14) Proof : Because ` is T -periodic one has that PT −1 `(t + i, x? (t + i), u? (t + i)) ? `T = t=0 , ∀i ∈ I0,T −1 T Then PT −1 PT −1 ? ? ? i=0 t=0 `(t + i, x (t + i), u (t + i)) T `T = T PT −1 PT −1 ˜? (i), u ˜? (i)) i=0 t=0 `(t + i, x ≤ T T −1 X ¯ x? (i), u `(˜ ˜? (i)) = T `¯? = T
i=0
where the last equality follows by taking into account the fact that the average cost `¯ can be computed over a single period as: PT −1 ¯ u) = t=0 `(t, x, u) `(x, (15) T In the cases where Q 6= T , and especially if the time constants of the system are comparable with the period T of the cost being considered, the inequality `?T ≤ `¯?Q
(16)
need not hold true in general. In fact, it is not hard to build up a counter-example that refutes it:
Example 1: Consider a system without inputs x(t + 1) = −x(t) with related stage cost x4 x2 − + [x − (t mod 3) − 1]2 4 2 In this case the averaged cost is `(t, x) =
(17)
2 4 ¯ = x − x − 4x + 14 `(x) 4 2 3 The best and unique 3-periodic solution according to (8) is x(0) = 0 leading to the optimal sequence x? = [0, 0, 0] and optimal cost `?3 = 32 . On the contrary, the best 2-periodic ˜ ? = [1, −1] sequence obtained by solving (13) can be either x 5 ? ? ¯ ˜ = [−1, 1] with related cost `2 = 12 . It can be observed or x that `¯?2 < `?3 The latter observation further motivates the introduction of Economic MPC schemes which adopt Q-periodic terminal constraints based on the optimal solutions arising from the average cost (13). In particular, algorithms of the following form are considered
V˜N0 (t, x)
subject to
:= min VN (t, x, u) u z(k + 1) = f (z(k), u(k)), k ∈ I0:N −1 (x(k), u(k)) ∈ Z, k ∈ I0:N −1 (18) z(N ) = x ˜? (t mod Q) z(0) = x
˜0 (t, x(t))) denote the optimal state and Here (˜z0 (t, x(t)), u input sequences of (18) for the initial state x. As in the case of Algorithm (9), the RH approach still applies and the following command u(t) = u ˜0 (0; t, x(t))
(19)
is applied to the plant, leading to the closed-loop system x(t + 1) = f (x(t), u ˜0 (0; t, x(t)))
(20)
with u ˜0 (k; t, x(t)) denoting the k-th sample of the sequence 0 ˜ (t, x). As in the previous Section, a similar analysis on the u asymptotic performance of the resulting closed-loop system can be undertaken and an analogous result of Theorem 1 results. Nevertheless, in this case, the following additional assumption involving the stage cost ` is required Assumption 3: PL−1 k=0 `(t + kQ, x, u) ¯ `(x, u) = lim , ∀t ∈ I≥0 (21) L→+∞ L Notice that Assumption 3 holds if, for instance, ` is T periodic, and Q and T are relatively prime numbers. Another possibility is when ` is periodic and continuous for some irrational T > 0. Also in this case Assumption 3 holds. Finally, the following Theorem 2 can be stated: Theorem 2: : Let x(0) be a feasible initial condition for problem (18). Then, x(t) as defined through iteration (20) is
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10
9
u1
feasible for all t and the average asymptotic performance of the closed-loop system fulfills PL `(t, x(t), u(t)) lim sup t=0 ≤ `¯?Q (22) L+1 L→+∞
8
7
The proof is not included for space reasons. Remark 1: It is worth remarking that Theorem 2 still holds in the case where the T -periodicity of the stage cost ` (Assumption 2) is relaxed.
6 0
1
2
3
4
5 Time [s]
6
7
8
9
10
40 u( t ) u*( t ) 30
0.2 0.15
u2
~ Q (t) ~ R (t)
20
10
0.1
0
0.05
0
1
2
3
4
5 Time [s]
6
7
8
9
10
0 2
4
6
8
10
Time [s]
Fig. 1.
Fig. 3. Input signals under the action of Algorithm (9) and best T -periodic solution for problem (8).
Time-varying weights of cost (24).
x1
−1
−1.5
5
4
4
3
x2
0
3
2
2
1
1
0 0
2
4 6 Time [s]
8
10
0
2
4 6 Time [s]
8
10
−2 5.5 *
-*T
Q
5
6
7
8
9
4.5
10
x4
4
3
3
x
2
x(t) x* ( t ) 2.4
−2.5 1
2.6
5
4
2.2
Q 3.5 2
Fig. 2. Comparison between `?T , T = 10 (problem (8)) and `¯?Q with Q ∈ I1:10 (problem (13)).
V. I LLUSTRATIVE E XAMPLE : C ONSECUTIVE -C OMPETITIVE R EACTIONS In this section the control of a nonlinear isothermal chemical reactor with consecutive-competitive reactions [18] has been dealt with. Such networks arise in many chemical and biological applications. In the simple case of two reactions the following structure should be taken into account P0 + B = P1 P1 + B = P2 The dimensionless mass balances for this problem are x˙ 1 x˙ 2 x˙ 3 x˙ 4
= = = =
u1 − x1 − σ1 x1 x2 u2 − x2 − σ1 x1 x2 − σ2 x2 x3 −x3 + σ1 x1 x2 − σ2 x2 x3 −x4 + σ2 x2 x3
(23)
where x1 , x2 , x3 and x4 are the concentrations of P0 , B, P1 and P2 respectively, while u1 and u2 are the inflow rates of P0 and B and represents the manipulated variables. The parameters σ1 and σ2 have values 1 and 0.4 respectively.
3 2.5
1.8 0
2
4 6 Time [s]
8
10
0
2
4 6 Time [s]
8
10
Fig. 4. State profiles under the action of Algorithm (9) and best T -periodic solution for problem (8).
An upper bound of 10 is imposed on u1 . The simultaneous approach [19] has been used to solve the dynamic regulation problem. The state space is divided into a fixed number of finite elements. The input is parameterized according to zero-order hold with the input value constant across a finite element. A sample time Ts = 0.1 s has been chosen. In order to manage the system, the following stage cost has been considered 1 `(t, x, u) = −x3 + |x − xs |2Q(t) + |u − us |2R(t) 2 (24) where the linear term is included in order to maximize the average amount of P1 in the effluent flow while the convex term, whose penalties are periodically varied, is added in order to keep the system trajectories close to the steady state
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−2
−2.5
4
2.5
3.8
2 x2
x1
3.6
1.5
−3
3.4 −3.5
1
3.2
0.5 0
2
4 6 Time [s]
−4 k i =0
(k , x ( i ), u ( i ) ) k +1
50
100
150 steps k
200
250
300
4 6 Time [s]
8
10
2.1
3.3
2.05 2
3.2
Fig. 5.
2
2.15
3.4 x3
0
0
3.5
* T
−5
10
x4
Σ
−4.5
8
x(t) x ˜* (t)
1.95
3.1
Asymptotic performance of Algorithm (9).
0
2
4 6 Time [s]
8
10
0
2
4 6 Time [s]
8
10
10
Fig. 7. State profile under the action of Algorithm (18) and best Q-periodic solution for problem (13).
u1
8
6
1
Pk
i=0
0.5
4 0
2
4
6
8
(k,x(i),u(i)) k+1
*Q
10
0
Time [s]
−0.5
20 u(t) u ˜ *(t)
15 u2
-
−1 −1.5
10
−2 5
−2.5 0 0
2
4
6
8
0
50
10
100
150 steps k
200
250
300
Time [s]
Fig. 8.
Asymptotic performance of Algorithm (18).
Fig. 6. Input signals under the action of Algorithm (18) and best Q-periodic solution for problem (13).
solution xs = [0.3874, 1.5811, 0.3752, 0.2373]T and us = [1, 2.4310]T that we would obtain if `(t, x, u) = −x3 , with additional average constraint Av[u1 ] ⊆ (0, 1) [5]. Moreover ˜ ˜ Q(t) = Q(t)I 4 with Q(t) = 0.01[(T −t) mod T ] and R(t) = ˜ ˜ R(t)I where R(t) = 0.001[t mod T ] (see Figure 1). A 2 period T = 10 has been chosen for the cost (24). As a consequence, the averaged cost `¯ takes the following form 2 ¯l(x, u) = −x3 + 1 |x − xs |2 0.09I4 + |u − us |0.009I2 2 (25) In order to simulate the behavior of the plant under the actions of Algorithms (9) and (18) respectively we have first computed the best T -periodic solutions for problem (8) and obtained the minimum cost `?T = −2.0358 with solutions depicted in Figures 3-4 (red dashed line). Secondly, several Q-periodic solutions of problem (13) have been evaluated for different values of Q ∈ I1:10 obtaining the same minimum averaged cost `¯?Q = −1.8839. In this respect, as depicted in Figure 2, the inequality `?10 ≤ `¯?10 holds true and Proposition 1 is verified.
Next in order to show the effectiveness of Algorithm (9) we have initialized the system with a random non equilibrium state and simulated the plant under the action of Algorithm (9), with a prediction horizon N = 20, solved by means of the f mincon routine of Matlab. Figures 3,4 and 5 illustrate the achieved results. As expected, after a brief transient the trajectory presents a T -periodic behavior. In particular, it is possible to observe how the input actions u(t) are in phase ˜ in the sense that it is more with the time-varying weight R(t) ˜ active when R(t) is low, while, on the contrary, it decreases ˜ its action when R(t) starts increasing toward its maximum value. As far as the asymptotic performance is concerned, Figure 5 shows the results stated in Theorem 1. Further simulations have been undertaken involving Algorithm (18), with control horizon N = 20 and based on the best Q-periodic solution obtained by solving problem (13) with Q = 9. Please notice that with the above choices Assumption 3 is fulfilled. The simulation results concerning such a case are depicted in Figures 6-8. Also in this case, although the best Q-periodic averaged solution is actually constant, (red dashed lines in Figures 6-7) the trajectory of the plant shows a time-varying T -periodic behavior.
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−2.12
Σ
−2.14 −2.16
k i =0
(k , x ( i ), u ( i ) ) k +1
¯Q
−2.18 −2.2 −2.22 −2.24 −2.26 −2.28 0
100
200
300
400
500 steps k
600
700
800
900
1000
Fig. 9. Fig: Asymptotic performance of Algorithm (18) in the case of stage cost with irrational period.
Furthermore, Figure 8 shows that the performance obtained by the closed-loop law is clearly better with respect to that pertaining to the optimal Q-periodic averaged solution (see Theorem 2). Finally, in order to verify the observation stated in Remark 1, we have performed a simulation in the case where the stage cost ` has an irrational period T . In this respect we have considered 7 `(t, x, u) = −x3 + (1 + sin(t))(x1 − 1)2 ; 5 having period T = 2π and whose average is given by ¯ u) = −x3 + 7 (x1 − 1)2 `(x, 5 The system has been simulated under the action of Algorithm (18) with Q = 7. Even in this case, the results of Theorem 2 holds true. In fact, as depicted in Figure 9, the asymptotic performance obtained by the closed-loop system is the same of that achieved by the best Q-periodic solution. VI. C ONCLUSIONS In this paper, we investigated Economic MPC schemes in the general case of time-varying cost functions and terminal constraints. We first analyzed an Economic MPC strategy with a time-varying periodic stage cost and terminal constraints steering the state to the optimal solution having the same period. Then, we considered a further Economic MPC scheme with a time-varying (not necessarily periodic) cost endowed with a terminal constraint that forces the state to reach the precomputed best periodic solution with respect to the related averaged cost. In both cases it has been formally shown that the control laws induce an asymptotic average cost that is not worse than that corresponding to the respective optimal periodic solution computed offline. A final example has been provided where theoretical outcomes have been verified and discussed.
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