- Email: [email protected]
Exponential Stability and Feasibility of Receding Horizon Control for Constrained Systems
Copyright co IFAC System Structure and Control, Nantes, France, 1998
EXPONENTIAL STABILITY AND FEASmILITY OF RECEDING HORIZON CONTROL FOR CONSTRAINED SYSTEMS Jae-Won Lee * Wook Hyun Kwon** Hyung S. Cho***
* System and Control Sector, Samsung Advanced Institute of
Technology, Email:[email protected] ** School of Electrical Engineering, Seoul National University ***
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology
Abstract: In this paper, we propose a new receding horizon controller for discrete linear systems with hard constraints on inputs and states. It is shown that exponential stability of the closed loop system with the proposed receding horizon controller, is guaranteed utilizing terminal ellipsoid constraints. In order to improve the feasibility, we also propose some implementable versions of the proposed controller where the system matrix is partitioned into the stable and unstable modes and hard state constraints are softened. We also propose some methods to penalize violations resulting from softened state constraints, and make comparison among them by some illustrative examples. Copyright © 1998 IFAC Reswne Dans cette communication, nous proposons des commandes a horizon fuyant par retour d'etat ou de sortie pour des systemes lineaires en temps discret soumis a des contraintes sur l'entree et sur l'etat. Les commandes a horizon fuyant proposees sont obtenues a partir d'un probleme d'optimisation a horizon fini ou la matrice de cOlIt final et la contrainte ellipsOlde invariante artificielle sont moins restrictives que la contrainte egalite finale habituelle. On considere les problemes avec contraintes strictes ou contraintes mixtes dans le cas de retour d'etat, et le probleme avec contraintes mixtes dans le cas de retour de sortie. On montre, en utilisant l'approche de Lyapunov, que toutes les lois de commande proposees guarantissent la stabilite exponentielle des systemes en boucle fermee pour tout les ensembles initiaux faisables.
1. INTRODUCTION
Originally, the terminal equality constraint has been utilized in order to guarantee the closed loop stability of the receding horizon controller for unconstrained systems (Kwon and Pearson, 1977; Kwon and Pearson, 1978) and for constrained systems (Rawlings and Muske, 1993; Zheng and Morari, 1995). This artificial constraint is satisfied by putting the state (or unstable mode) to the origin at the finite terminal time. However, this terminal equality constraint is restrictive, because the equality constraint rather than the inequality constraint should be satisfied. Moreover this approach may make the optimization problem infeasible under the hard state constraint. Hence,
The receding horizon control has been emerged as a powerful strategy for constrained systems with limitations on inputs, states, and outputs (Richalet, 1993; Kwon and Pearson, 1978; Rawlings and Muske, 1993; Muske and Rawlings, 1993; Meadows and Rawlings, 1995; Zheng and Morari, 1995; Lee et al. , 1997). Especially, the stability issue of the receding horizon control for constrained systems has been focused on in recent literatures (Rawlings and Muske, 1993; Muske and Rawlings, 1993; Zheng and Morari, 1995; Lee et al., 1997).
317
2. CONSTRAINED RECEDING HORIZON CONTROL
the horizon size may have to be made longer so as to make the problem feasible. Even though the mixed constraint has been introduced in order to relax the hard state constraint (Zheng and Morari, 1995), the feasibility and stability problems remain as issues, because the terminal equality constraint should be satisfied under the input constraint and only attractivity property has been shown in the existing results (Rawlings and Muske, 1993; Zheng and Morari, 1995) . Recently, the invariant ellipsoid constraint has been introduced in order to relax the conventional terminal equality constraint (Lee et al., 1997). This artificial constraint is satisfied by putting the state (or unstable mode) into an invariant ellipsoid. However, in (Lee et al., 1997), the exponential stability property is shown only for the initial states inside the invariant ellipsoid.
In this section, we consider the follOwing time-
invariant discrete linear system with input and state constraints : (1)
subject to U- ~ Uk ~ u+, k { g- ~ GXk ~ g+, k
= 0,1, ··· ,00 = 0,1, ... , 00,
(2)
where u-,u+ E Rm,G E Rngxn , and g- , g+ E Rng . It is assumed that Uk = 0 and GXk = 0 satisfy the constraiqt (2) . Note that the output constraint Y- ~ Yk ~y+ can be expressed as the state constraint in (2) with G = C . We denote U as the feasible set for the above input constraint and X as the feasible set for the above state constraint.
In (Scokaert and Rawlings, 1996), the infinite
horizon LQR for constrained systems is proposed. While finite horizon problems have been considered in most of results on the constrained receding horizon controller, the infinite horizon problem is also considered in (Scokaert and Rawlings, 1996). The idea is to put the states at a finite time step to a ball in which there exists the conventional state feedback LQR satisfying input and state constraints. However, it requires long horizon size to put the states into the ball and hence much computational burden.
Denoting that xk+ilk and uk+ilk are predicted variables at the time k with xklk = Xk, we define the following finite horizon cost function which should be optimized at every current time k : N-l
J(xk , k) =
L(X~+ilkQXk+ilk + u~+ilkRuk+i lk) i=O
(3)
The motivation of this work is to show the exponential stability of a new receding horizon controller for constrained systems and consider how to increase feasibility while guaranteeing the exponential stability. It is noted that most conventional results on constrained receding horizon controllers showed only their attractivity property. In order to make optimization problems of the conventional results feasible, long horizon size is needed, since the terminal equality constraint and the terminal LQR constraint are restrictive. It is also noted that as horizon size increases, the computational burden also increases.
where Q > 0, R > 0,111 > 0, and N is a finite positive integer. We assume that the terminal weighting matrix 111 satisfies the following inequality condition : 'IT ~ (A
+ BH)''IT(A + BH) + Q + H'RH
,
(4)
where H is a free parameter and will be specified later. First, we introduce a state feedback receding horizon controller for unconstrained systems which was proposed in (Lee et al., 1997) and is defined as the first solution of the following optimization problem :
In this paper, we consider the finite horizon prob-
lem with finite terminal weighting matrices. For the artificial constraint to guarantee stability, we introduce the invariant ellipsoid constraint which is less restrictive than the conventional terminal equality constraint. It is shown that the proposed receding horizon controller guarantees exponential stability of the closed loop system for all feasible initial states sets. We also propose some implement able versions of the proposed controller, where feasibility is improved by introducing system partition and constraints softening. An illustrative example is included to compare some constraints softening methods.
u.,.,Minimize ···,U"+N_l,.
J(xk,k) .
(5)
Then, the closed loop stability of this state feedback receding horizon controller is guaranteed by the following theorem. Theorem 1. (Lee et al., 1997) Suppose that the inequality condition (4) is satisfied. Then the receding horizon control Uk = u klk where U k+1Ik ' i = 0, ... ,N - 1 is the optimal solution of the optimization problem (5), exponentially stabilizes the system (1).
318
Also, x" = 0 is the exponential stable equilibrium of the closed loop system with the state feedback receding horizon controller stemming from this optimization problem, for all initial states
Now, we propose a constrained receding horizon controller for hard constrained systems and investigate its exponential stability property. Consider the following optimization problem subject to the additional constraint of invariant ellipsoid :
Xo
Minimize
subject to
(6)
3. FEASIBILITY AND IMPLEMENTATION
U- ~Uk+ilk~U+, i=0,1, ··· N-1 g- ~ GXk+ilk ~ g+, i 0, 1, . .. , N (7) { x~+Nlk 1{Ix/o+NI.\: ~ 1
=
Generally, input constraints are imposed by physical limitations of actuators, valves, pumps, and etc., while state constraints are often desirable. There are often cases that state constraints cannot be satisfied all the time and hence some violations of state constraints are allowable. Especially, even if state constraints are satisfied in nominal operations, unexpected disturbances may put the states aside from the feasible region where state constraints are satisfied. In this case, it may happen that some violations of state constraints are unavoidable, while input constraints can be still satisfied. Hence, if some violations of state constraints are allowed, we can guess larger feasible initial sets. Moreover, if the terminal ellipsoid constraints are relaxed, we can guess much larger feasible initial sets. This strategy makes more sense in the output feedback case, because it is difficult to satisfy the hard state constraint. More details for the output feedback case will be followed in the next section. Hence we introduce the mixed constraints 1 which are given by
where we assume that the terminal weighting matrix '11 satisfies the following assumption. Assumption 1. The finite terminal weighting matrix '11 satisfies the following LMIs with X = '11- 1 > 0: X
(AX
+ BY) Ql/2X Rl/2y
(AX
[
E F(iJI,N).
+ BY)' (Q l /2 X)' (Rl/2Y)'] X 0 0
0
0 0
I 0
>0
I
where X, Y, Z, and V are variables which should be found. Then, the state feedback receding horizon controller with the invariant ellipsoid constraint is defined as the first optimal solution of the above constrained optimization problem with the terminal weighting matrix satisfying Assumption 1 . We also define the feasible initial states set of the optimization problem (6) : :F(1{I , N) = {xc E Rn
I there exists Ui such that
U- ~ U" { GXk ~ 9
E U, i = O, ·· ·N -1,
Xi+l
E X and XN E £'II}.
~ u+, k
+ ~", k
= 0, 1,· . . , 00 = 0,1,· .. , 00
(8)
where foie ~ 0 denotes tolerance for violation of state constraints. In order to enlarge feasibility, we partition the system matrix A into stable and unstable modes as follows :
It is noted that F(w, N) contains an open neighborhood of the origin for sufficiently large N, since X and U include the origin as an interior point. In order to show the exponential stability of the state feedback receding horizon controller stemming from the optimization problem (6), we need the following lemma :
A
= V JV- 1 = [Vu
Vs]
[~ J.]
[t:],
(9)
where Ju's eigenvalues are unstable and Js's eigenvalues are stable. Then the unstable mode ZU = Vux satisfies z:+l = Juz: + VuBu". Now, the idea is to drive only unstable mode into the ellipsoid instead of full states.
Lemma 2. Suppose that Xk E F(w,N). Then there exist", > 0 and uk+ilk E U, i = 0,··· , N-1 such that IUk+ilk 12 ~ "'IXk 12, Xk+ilk E X Vi = 0,· .. , N - 1 and Xk+Nlk E [.p.
We assume that there exist Xu > 0 and Yu which satisfy the following couple of LMIs and denote that iJl u = X;;l and Hu = YuiJI u.
Now, we are ready to state the result on the exponential stability of the state feedback receding horizon controller with the invariant ellipsoid constraint.
Assumption 2. There exists '11 u which satisfies the following LMIs :
Theorem 3. Suppose that Assumption 1 holds. Then the optimization problem minimizing J(Xk, k) subject to the constraint (7), is always feasible for all k ~ 0 and for all initial states Xo E F(iJI,N).
319
1 We will represent the hard state constraint 9- ~ GXk ~ g+ as Gx.\: ~ 9, which can be easily derived by modifying
G.
Before stating the theorem on the exponential stability, we made an assumption on the cost function S(e(k)).
Xu X~ X~ X~l Xl Xu 0 0 X 0 I 0 > 0 Xu > 0, (10) [ 2 X3 0 0 I
ZYu] [ Y~ Xu
Assumption 3. See) satisfies the following conditions :
2· :2: OZ , ii ~ Ui' J = 1,2,···,m (11)
where Xl = JuXu + BuYu , X 2 = Q~/2 X u , Xa R I / 2 y u , Xu = w;;I,Bu = V-IB, and Qu
• Define eOO(k) such that the jth element of eOO(k) denotes maxi=l,'" ,N.ejek+ilk , where ej = [0 · · · 1 .. ·oy is the unit vector with nonzero jth element. Then there exists b. > 0 such that S(e(k)) ~ b.leOO(kW· • Sh(k)) ~ S(e2(k)), if Ilel(k)11 ~ Ile2(k)ll·
= =
V~QVu.
Since J s is stable, there exists iI!' 8 > 0 satisfying the following inequality :
Theorem 4. Suppose that Assumption 2 is satisfied and the terminal weighting matrix iI!' is defined by (12). Then the optimization problem minimizing J.(Xk, k) subject to the constraint (16), is always feasible for all k :2: 0 and for all initial states Xo E j:u·(~,N). Also, Xk = 0 is the exponentially stable equilibrium of the closed loop system with the receding horizon controller stemming from this optimization problem, for all initial states Xo E Fu(~, N) .
iI!'s :2: J;iI!'sJs + V;QVs. Now denoting
= V [iI!'u 0] V-I o iI!'s H = [Hu OjV-I, iI!'
(12) (13)
it can be easily verified that iI!' and H satisfy the inequality condition (4). Using iI!' in (12) as the terminal weighting matrix and introducing a cost function for violations of state constraints, we modify the cost function and the corresponding optimization problem as
Now we consider how to choose See). Note that the additional cost See) is included so as to penalize a measure of violation of the state constraint. First, we consider an 00 norm of e( k). In this case, S (e) is represented as
(14)
where S > o. And then the second and third constraint in (16) can be represented as
(15)
J.(Xk, k)
Minimize
(17)
Uklk, " ,Uk+N_llk,.(k)
subject to U- ~ Uk+ilk ~ u+, i = 0,1, ... N - 1 GXk+ilk ~ 9 + e(k), i = 1" . . ,N _. 16 GAJXk+Nlk ~ 9 + e(k), j = 1"" ,00 ( ) { x~+Nlk ~Xk+Nlk ~ 1,
GXk+ilk ~ 9 + eOO(k), i G(A
i
= {xo
E Rn I there exists
Ui
~9
+ eOO(k),
In view of the optimization problem (15), the
decision variables are
where A = A + BH, ~ = V~iI!'u Vu, and e(k) :2: o. It is noted that the third constraint in (16) can be checked within a finite step because A + BH is stable (Rawlings and Muske, 1993). We can observe that the terminal ellipsoid constraint in this optimization is relaxed from that in the optimization in the previous section, because we only have to put the unstable modes instead of full states into the ellipsoid. We also define the feasible initial states set of the optimization problem (15)
Fu(~,N)
+ BH)ixk+Nlk
= 0,1,··· ,N j = 1,·" ,00.
Uklk,··· ,uk+N-llk, eOO(k).
Hence the size of decision variables are m x N +ng , which doesn't require much additional computation burden. However, this approach has some drawbacks such as mismatch between open-loop predictions and closed-loop behavior, which can lead to poor performance and tuning difficulties. One of alternatives to overcome these drawbacks is to introduce 2-norm cost for violations. A 2norm cost for violations can be chosen as :
E U,
N
= O,···N -1,such that XN E £",}.
See) =
It is noted that Fu(~,N) ;2 F(iI!',N) because there is only input constraint and the terminal ellipsoid constraint is much relaxed.
L e~+ilkSek+ilk i=O
(18)
where S > O. In this approach, there is no such a big mismatch between open-loop predictions and
320
(8 = 1,20,50,100). In this figure, we also observe that the peak of violation reduces as violation weighting increases as in the case of 00 norm cost for violation. However, the responses in this case are better than those in the first case. Especially, when the weighting is large (8 = 50,100), the responses are much better than those of Figure 1. Figure 4 shows the mismatch between finite horizon (open-loop) predictions and the receding horizon (closed-loop) responses. In this figure, we observe that the mismatch is much smaller than that of Figure 2.
closed-loop behavior. Therefore, it is somewhat easy to tune intuitively. However, in this case, the decision variables are
and therefore the size of decision variables increases to mN + ngN., which requires much computational burden. Hence, we conclude that there is a trade-off between two approaches. Some comparisons between two approaches will be presented through an example in the following section.
From these figures, we observe that the violation peak is small in the case of 00 norm, while duration of violation is small in the case of 2-norm. Figure 5 makes this observation clear.
4. ILLUSTRATIVE EXAMPLE
We can also observe that overall performances including mismatch between finite horizon and receding horizon, are better when the 2-norm cost for violations is adopted. However, the case of 2-norm cost needs much computational burden, because the size of the optimization problems becomes large.
In this example, we present the effects of constraints softening which was introduced in the previous section. Consider the output regulation problem of the following discrete linear system :
5. CONCLUSIONS In this paper, we proposed state feedback receding horizon controllers for discrete linear systems with hard and mixed constraints. The proposed controllers are obtained from finite horizon optimization problems with the artificial invariant ellipsoid constraints. We showed that the proposed controllers guarantee exponential stability in the Lyapunov sense, for all feasible initial states.
where we assume that there is no input constraints and the output constraint is given by -1:$ Yk :$1, i
= 1,2,· · · ,00 . = C'C, R = I
We also assume that Q and the initial state is given by [1.5 1.5 1.5]'. We construct two constrained receding horizon controller with the 00 norm cost and the 2-norm cost for constraint violations, respectively.
It is noted that in the state feedback case, only attractivity property has been shown (Rawlings and Muske, 1993; Zheng and Morari, 1995) or exponential stability shown only for the initial sets inside an invariant ellipsoid (Lee et al., 1997). Since proposed controllers utilize the invariant ellipsoid constraint which is less restrictive than the conventional terminal equality constraint, we can guess larger feasible initial sets or less computational burden than the conventional receding horizon controllers with the terminal equality constraint.
First, we construct the constrained receding horizon controller with the 00 norm cost for violation defined by (17). Figure 1 shows the corresponding output responses according to several violation weightings (8 = 1,20,50,100). In this figure, we observe that the peak of violation reduces as violation weighting increases, while duration of violations increases. In other words, increased weighting makes an effect of hardening the state constraint. Hence, we conclude that there is a tradeoff between peak and duration of violation. In this figure, we also observe that the responses are not so good when violations appear. Figure 2 shows the mismatch between finite horizon (open-loop) predictions and the receding horizon (closed-loop) responses. We observe that the result is not so satisfactory, because the mismatch is so big.
6. REFERENCES Kwon, W.H. and A.E. Pearson (1977) . A modified quadratic cost problem and feedback stabilization of a linear system . IEEE 1hms. Automat. Contr. AC-22(5), 838 - 842. Kwon, W.H. and A.E. Pearson (1978). On feedback stabilization of time-varying discrete linear systems. IEEE 7hlns. Automat. Contr. 23(3), 479 - 481.
Second, we construct the receding horizon control with the 2-norm cost for violations defined by (18). Figure 3 shows the corresponding output responses according to several violation weightings
321
Lee, J.-W., W.H. Kwon and J. Choi (1997). On stability of constrained receding horizon control with finite terminal weighting matrix. In: Proceedings of the 1997 European Control Conference. Brussels, Belgium. Meadows, E.S. and J.B. Rawlings (1995). Topics in model predictive control. In: Methods of Model Based Process Control. Kluwer. R. Berber, Eds. Muske, K.R. and J .B. Rawlings (1993). Linear model predictive control of unstable processes. J. Proc. Cont. 3, 85-96. Rawlings, J.B. and K.R. Muske (1993). The stability of constrained receding horizon control . IEEE 7rans. Automat. Contr. 38, 1512 1516. Richalet, J . (1993) . Industrial applications of model based predictive control. A utomatica 29(5) , 1251-1274. Scokaert, P.O.M. and J .B. Rawlings (1996). Constrained linear quadratic regulation. To appear in IEEE 7rans. Automat. Contr. Zheng, A. and M. Morari (1995). Stability of model predictive control with mixed constraints . IEEE Trans. Automat. Contr. 40, 1818 - 1823.
Fig. 3. The case of 2-norm cost for violations
I
j
j 0 .' 0.2
"RttltI'iOtltOn..,..,.. :"ac..crh . 10
'"
30
Fig. 4. Mismatch between finite horizon (openloop) prediction and receding horizon (closed-loop) response (the case of 2-norm)
5-1
10
"
'"
Fig. 1. The case of oo-norm cost for violations
Fig. 2. Mismatch between finite horizon (openloop) prediction and receding horizon (closed-loop) response (the case of oo-norm)
Fig. 5. Response comparison between both cases of costs for violations
322
EXPONENTIAL STABILITY AND FEASmILITY OF RECEDING HORIZON CONTROL FOR CONSTRAINED SYSTEMS Jae-Won Lee * Wook Hyun Kwon** Hyung S. Cho***
* System and Control Sector, Samsung Advanced Institute of
Technology, Email:[email protected] ** School of Electrical Engineering, Seoul National University ***
Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology
Abstract: In this paper, we propose a new receding horizon controller for discrete linear systems with hard constraints on inputs and states. It is shown that exponential stability of the closed loop system with the proposed receding horizon controller, is guaranteed utilizing terminal ellipsoid constraints. In order to improve the feasibility, we also propose some implementable versions of the proposed controller where the system matrix is partitioned into the stable and unstable modes and hard state constraints are softened. We also propose some methods to penalize violations resulting from softened state constraints, and make comparison among them by some illustrative examples. Copyright © 1998 IFAC Reswne Dans cette communication, nous proposons des commandes a horizon fuyant par retour d'etat ou de sortie pour des systemes lineaires en temps discret soumis a des contraintes sur l'entree et sur l'etat. Les commandes a horizon fuyant proposees sont obtenues a partir d'un probleme d'optimisation a horizon fini ou la matrice de cOlIt final et la contrainte ellipsOlde invariante artificielle sont moins restrictives que la contrainte egalite finale habituelle. On considere les problemes avec contraintes strictes ou contraintes mixtes dans le cas de retour d'etat, et le probleme avec contraintes mixtes dans le cas de retour de sortie. On montre, en utilisant l'approche de Lyapunov, que toutes les lois de commande proposees guarantissent la stabilite exponentielle des systemes en boucle fermee pour tout les ensembles initiaux faisables.
1. INTRODUCTION
Originally, the terminal equality constraint has been utilized in order to guarantee the closed loop stability of the receding horizon controller for unconstrained systems (Kwon and Pearson, 1977; Kwon and Pearson, 1978) and for constrained systems (Rawlings and Muske, 1993; Zheng and Morari, 1995). This artificial constraint is satisfied by putting the state (or unstable mode) to the origin at the finite terminal time. However, this terminal equality constraint is restrictive, because the equality constraint rather than the inequality constraint should be satisfied. Moreover this approach may make the optimization problem infeasible under the hard state constraint. Hence,
The receding horizon control has been emerged as a powerful strategy for constrained systems with limitations on inputs, states, and outputs (Richalet, 1993; Kwon and Pearson, 1978; Rawlings and Muske, 1993; Muske and Rawlings, 1993; Meadows and Rawlings, 1995; Zheng and Morari, 1995; Lee et al. , 1997). Especially, the stability issue of the receding horizon control for constrained systems has been focused on in recent literatures (Rawlings and Muske, 1993; Muske and Rawlings, 1993; Zheng and Morari, 1995; Lee et al., 1997).
317
2. CONSTRAINED RECEDING HORIZON CONTROL
the horizon size may have to be made longer so as to make the problem feasible. Even though the mixed constraint has been introduced in order to relax the hard state constraint (Zheng and Morari, 1995), the feasibility and stability problems remain as issues, because the terminal equality constraint should be satisfied under the input constraint and only attractivity property has been shown in the existing results (Rawlings and Muske, 1993; Zheng and Morari, 1995) . Recently, the invariant ellipsoid constraint has been introduced in order to relax the conventional terminal equality constraint (Lee et al., 1997). This artificial constraint is satisfied by putting the state (or unstable mode) into an invariant ellipsoid. However, in (Lee et al., 1997), the exponential stability property is shown only for the initial states inside the invariant ellipsoid.
In this section, we consider the follOwing time-
invariant discrete linear system with input and state constraints : (1)
subject to U- ~ Uk ~ u+, k { g- ~ GXk ~ g+, k
= 0,1, ··· ,00 = 0,1, ... , 00,
(2)
where u-,u+ E Rm,G E Rngxn , and g- , g+ E Rng . It is assumed that Uk = 0 and GXk = 0 satisfy the constraiqt (2) . Note that the output constraint Y- ~ Yk ~y+ can be expressed as the state constraint in (2) with G = C . We denote U as the feasible set for the above input constraint and X as the feasible set for the above state constraint.
In (Scokaert and Rawlings, 1996), the infinite
horizon LQR for constrained systems is proposed. While finite horizon problems have been considered in most of results on the constrained receding horizon controller, the infinite horizon problem is also considered in (Scokaert and Rawlings, 1996). The idea is to put the states at a finite time step to a ball in which there exists the conventional state feedback LQR satisfying input and state constraints. However, it requires long horizon size to put the states into the ball and hence much computational burden.
Denoting that xk+ilk and uk+ilk are predicted variables at the time k with xklk = Xk, we define the following finite horizon cost function which should be optimized at every current time k : N-l
J(xk , k) =
L(X~+ilkQXk+ilk + u~+ilkRuk+i lk) i=O
(3)
The motivation of this work is to show the exponential stability of a new receding horizon controller for constrained systems and consider how to increase feasibility while guaranteeing the exponential stability. It is noted that most conventional results on constrained receding horizon controllers showed only their attractivity property. In order to make optimization problems of the conventional results feasible, long horizon size is needed, since the terminal equality constraint and the terminal LQR constraint are restrictive. It is also noted that as horizon size increases, the computational burden also increases.
where Q > 0, R > 0,111 > 0, and N is a finite positive integer. We assume that the terminal weighting matrix 111 satisfies the following inequality condition : 'IT ~ (A
+ BH)''IT(A + BH) + Q + H'RH
,
(4)
where H is a free parameter and will be specified later. First, we introduce a state feedback receding horizon controller for unconstrained systems which was proposed in (Lee et al., 1997) and is defined as the first solution of the following optimization problem :
In this paper, we consider the finite horizon prob-
lem with finite terminal weighting matrices. For the artificial constraint to guarantee stability, we introduce the invariant ellipsoid constraint which is less restrictive than the conventional terminal equality constraint. It is shown that the proposed receding horizon controller guarantees exponential stability of the closed loop system for all feasible initial states sets. We also propose some implement able versions of the proposed controller, where feasibility is improved by introducing system partition and constraints softening. An illustrative example is included to compare some constraints softening methods.
u.,.,Minimize ···,U"+N_l,.
J(xk,k) .
(5)
Then, the closed loop stability of this state feedback receding horizon controller is guaranteed by the following theorem. Theorem 1. (Lee et al., 1997) Suppose that the inequality condition (4) is satisfied. Then the receding horizon control Uk = u klk where U k+1Ik ' i = 0, ... ,N - 1 is the optimal solution of the optimization problem (5), exponentially stabilizes the system (1).
318
Also, x" = 0 is the exponential stable equilibrium of the closed loop system with the state feedback receding horizon controller stemming from this optimization problem, for all initial states
Now, we propose a constrained receding horizon controller for hard constrained systems and investigate its exponential stability property. Consider the following optimization problem subject to the additional constraint of invariant ellipsoid :
Xo
Minimize
subject to
(6)
3. FEASIBILITY AND IMPLEMENTATION
U- ~Uk+ilk~U+, i=0,1, ··· N-1 g- ~ GXk+ilk ~ g+, i 0, 1, . .. , N (7) { x~+Nlk 1{Ix/o+NI.\: ~ 1
=
Generally, input constraints are imposed by physical limitations of actuators, valves, pumps, and etc., while state constraints are often desirable. There are often cases that state constraints cannot be satisfied all the time and hence some violations of state constraints are allowable. Especially, even if state constraints are satisfied in nominal operations, unexpected disturbances may put the states aside from the feasible region where state constraints are satisfied. In this case, it may happen that some violations of state constraints are unavoidable, while input constraints can be still satisfied. Hence, if some violations of state constraints are allowed, we can guess larger feasible initial sets. Moreover, if the terminal ellipsoid constraints are relaxed, we can guess much larger feasible initial sets. This strategy makes more sense in the output feedback case, because it is difficult to satisfy the hard state constraint. More details for the output feedback case will be followed in the next section. Hence we introduce the mixed constraints 1 which are given by
where we assume that the terminal weighting matrix '11 satisfies the following assumption. Assumption 1. The finite terminal weighting matrix '11 satisfies the following LMIs with X = '11- 1 > 0: X
(AX
+ BY) Ql/2X Rl/2y
(AX
[
E F(iJI,N).
+ BY)' (Q l /2 X)' (Rl/2Y)'] X 0 0
0
0 0
I 0
>0
I
where X, Y, Z, and V are variables which should be found. Then, the state feedback receding horizon controller with the invariant ellipsoid constraint is defined as the first optimal solution of the above constrained optimization problem with the terminal weighting matrix satisfying Assumption 1 . We also define the feasible initial states set of the optimization problem (6) : :F(1{I , N) = {xc E Rn
I there exists Ui such that
U- ~ U" { GXk ~ 9
E U, i = O, ·· ·N -1,
Xi+l
E X and XN E £'II}.
~ u+, k
+ ~", k
= 0, 1,· . . , 00 = 0,1,· .. , 00
(8)
where foie ~ 0 denotes tolerance for violation of state constraints. In order to enlarge feasibility, we partition the system matrix A into stable and unstable modes as follows :
It is noted that F(w, N) contains an open neighborhood of the origin for sufficiently large N, since X and U include the origin as an interior point. In order to show the exponential stability of the state feedback receding horizon controller stemming from the optimization problem (6), we need the following lemma :
A
= V JV- 1 = [Vu
Vs]
[~ J.]
[t:],
(9)
where Ju's eigenvalues are unstable and Js's eigenvalues are stable. Then the unstable mode ZU = Vux satisfies z:+l = Juz: + VuBu". Now, the idea is to drive only unstable mode into the ellipsoid instead of full states.
Lemma 2. Suppose that Xk E F(w,N). Then there exist", > 0 and uk+ilk E U, i = 0,··· , N-1 such that IUk+ilk 12 ~ "'IXk 12, Xk+ilk E X Vi = 0,· .. , N - 1 and Xk+Nlk E [.p.
We assume that there exist Xu > 0 and Yu which satisfy the following couple of LMIs and denote that iJl u = X;;l and Hu = YuiJI u.
Now, we are ready to state the result on the exponential stability of the state feedback receding horizon controller with the invariant ellipsoid constraint.
Assumption 2. There exists '11 u which satisfies the following LMIs :
Theorem 3. Suppose that Assumption 1 holds. Then the optimization problem minimizing J(Xk, k) subject to the constraint (7), is always feasible for all k ~ 0 and for all initial states Xo E F(iJI,N).
319
1 We will represent the hard state constraint 9- ~ GXk ~ g+ as Gx.\: ~ 9, which can be easily derived by modifying
G.
Before stating the theorem on the exponential stability, we made an assumption on the cost function S(e(k)).
Xu X~ X~ X~l Xl Xu 0 0 X 0 I 0 > 0 Xu > 0, (10) [ 2 X3 0 0 I
ZYu] [ Y~ Xu
Assumption 3. See) satisfies the following conditions :
2· :2: OZ , ii ~ Ui' J = 1,2,···,m (11)
where Xl = JuXu + BuYu , X 2 = Q~/2 X u , Xa R I / 2 y u , Xu = w;;I,Bu = V-IB, and Qu
• Define eOO(k) such that the jth element of eOO(k) denotes maxi=l,'" ,N.ejek+ilk , where ej = [0 · · · 1 .. ·oy is the unit vector with nonzero jth element. Then there exists b. > 0 such that S(e(k)) ~ b.leOO(kW· • Sh(k)) ~ S(e2(k)), if Ilel(k)11 ~ Ile2(k)ll·
= =
V~QVu.
Since J s is stable, there exists iI!' 8 > 0 satisfying the following inequality :
Theorem 4. Suppose that Assumption 2 is satisfied and the terminal weighting matrix iI!' is defined by (12). Then the optimization problem minimizing J.(Xk, k) subject to the constraint (16), is always feasible for all k :2: 0 and for all initial states Xo E j:u·(~,N). Also, Xk = 0 is the exponentially stable equilibrium of the closed loop system with the receding horizon controller stemming from this optimization problem, for all initial states Xo E Fu(~, N) .
iI!'s :2: J;iI!'sJs + V;QVs. Now denoting
= V [iI!'u 0] V-I o iI!'s H = [Hu OjV-I, iI!'
(12) (13)
it can be easily verified that iI!' and H satisfy the inequality condition (4). Using iI!' in (12) as the terminal weighting matrix and introducing a cost function for violations of state constraints, we modify the cost function and the corresponding optimization problem as
Now we consider how to choose See). Note that the additional cost See) is included so as to penalize a measure of violation of the state constraint. First, we consider an 00 norm of e( k). In this case, S (e) is represented as
(14)
where S > o. And then the second and third constraint in (16) can be represented as
(15)
J.(Xk, k)
Minimize
(17)
Uklk, " ,Uk+N_llk,.(k)
subject to U- ~ Uk+ilk ~ u+, i = 0,1, ... N - 1 GXk+ilk ~ 9 + e(k), i = 1" . . ,N _. 16 GAJXk+Nlk ~ 9 + e(k), j = 1"" ,00 ( ) { x~+Nlk ~Xk+Nlk ~ 1,
GXk+ilk ~ 9 + eOO(k), i G(A
i
= {xo
E Rn I there exists
Ui
~9
+ eOO(k),
In view of the optimization problem (15), the
decision variables are
where A = A + BH, ~ = V~iI!'u Vu, and e(k) :2: o. It is noted that the third constraint in (16) can be checked within a finite step because A + BH is stable (Rawlings and Muske, 1993). We can observe that the terminal ellipsoid constraint in this optimization is relaxed from that in the optimization in the previous section, because we only have to put the unstable modes instead of full states into the ellipsoid. We also define the feasible initial states set of the optimization problem (15)
Fu(~,N)
+ BH)ixk+Nlk
= 0,1,··· ,N j = 1,·" ,00.
Uklk,··· ,uk+N-llk, eOO(k).
Hence the size of decision variables are m x N +ng , which doesn't require much additional computation burden. However, this approach has some drawbacks such as mismatch between open-loop predictions and closed-loop behavior, which can lead to poor performance and tuning difficulties. One of alternatives to overcome these drawbacks is to introduce 2-norm cost for violations. A 2norm cost for violations can be chosen as :
E U,
N
= O,···N -1,such that XN E £",}.
See) =
It is noted that Fu(~,N) ;2 F(iI!',N) because there is only input constraint and the terminal ellipsoid constraint is much relaxed.
L e~+ilkSek+ilk i=O
(18)
where S > O. In this approach, there is no such a big mismatch between open-loop predictions and
320
(8 = 1,20,50,100). In this figure, we also observe that the peak of violation reduces as violation weighting increases as in the case of 00 norm cost for violation. However, the responses in this case are better than those in the first case. Especially, when the weighting is large (8 = 50,100), the responses are much better than those of Figure 1. Figure 4 shows the mismatch between finite horizon (open-loop) predictions and the receding horizon (closed-loop) responses. In this figure, we observe that the mismatch is much smaller than that of Figure 2.
closed-loop behavior. Therefore, it is somewhat easy to tune intuitively. However, in this case, the decision variables are
and therefore the size of decision variables increases to mN + ngN., which requires much computational burden. Hence, we conclude that there is a trade-off between two approaches. Some comparisons between two approaches will be presented through an example in the following section.
From these figures, we observe that the violation peak is small in the case of 00 norm, while duration of violation is small in the case of 2-norm. Figure 5 makes this observation clear.
4. ILLUSTRATIVE EXAMPLE
We can also observe that overall performances including mismatch between finite horizon and receding horizon, are better when the 2-norm cost for violations is adopted. However, the case of 2-norm cost needs much computational burden, because the size of the optimization problems becomes large.
In this example, we present the effects of constraints softening which was introduced in the previous section. Consider the output regulation problem of the following discrete linear system :
5. CONCLUSIONS In this paper, we proposed state feedback receding horizon controllers for discrete linear systems with hard and mixed constraints. The proposed controllers are obtained from finite horizon optimization problems with the artificial invariant ellipsoid constraints. We showed that the proposed controllers guarantee exponential stability in the Lyapunov sense, for all feasible initial states.
where we assume that there is no input constraints and the output constraint is given by -1:$ Yk :$1, i
= 1,2,· · · ,00 . = C'C, R = I
We also assume that Q and the initial state is given by [1.5 1.5 1.5]'. We construct two constrained receding horizon controller with the 00 norm cost and the 2-norm cost for constraint violations, respectively.
It is noted that in the state feedback case, only attractivity property has been shown (Rawlings and Muske, 1993; Zheng and Morari, 1995) or exponential stability shown only for the initial sets inside an invariant ellipsoid (Lee et al., 1997). Since proposed controllers utilize the invariant ellipsoid constraint which is less restrictive than the conventional terminal equality constraint, we can guess larger feasible initial sets or less computational burden than the conventional receding horizon controllers with the terminal equality constraint.
First, we construct the constrained receding horizon controller with the 00 norm cost for violation defined by (17). Figure 1 shows the corresponding output responses according to several violation weightings (8 = 1,20,50,100). In this figure, we observe that the peak of violation reduces as violation weighting increases, while duration of violations increases. In other words, increased weighting makes an effect of hardening the state constraint. Hence, we conclude that there is a tradeoff between peak and duration of violation. In this figure, we also observe that the responses are not so good when violations appear. Figure 2 shows the mismatch between finite horizon (open-loop) predictions and the receding horizon (closed-loop) responses. We observe that the result is not so satisfactory, because the mismatch is so big.
6. REFERENCES Kwon, W.H. and A.E. Pearson (1977) . A modified quadratic cost problem and feedback stabilization of a linear system . IEEE 1hms. Automat. Contr. AC-22(5), 838 - 842. Kwon, W.H. and A.E. Pearson (1978). On feedback stabilization of time-varying discrete linear systems. IEEE 7hlns. Automat. Contr. 23(3), 479 - 481.
Second, we construct the receding horizon control with the 2-norm cost for violations defined by (18). Figure 3 shows the corresponding output responses according to several violation weightings
321
Lee, J.-W., W.H. Kwon and J. Choi (1997). On stability of constrained receding horizon control with finite terminal weighting matrix. In: Proceedings of the 1997 European Control Conference. Brussels, Belgium. Meadows, E.S. and J.B. Rawlings (1995). Topics in model predictive control. In: Methods of Model Based Process Control. Kluwer. R. Berber, Eds. Muske, K.R. and J .B. Rawlings (1993). Linear model predictive control of unstable processes. J. Proc. Cont. 3, 85-96. Rawlings, J.B. and K.R. Muske (1993). The stability of constrained receding horizon control . IEEE 7rans. Automat. Contr. 38, 1512 1516. Richalet, J . (1993) . Industrial applications of model based predictive control. A utomatica 29(5) , 1251-1274. Scokaert, P.O.M. and J .B. Rawlings (1996). Constrained linear quadratic regulation. To appear in IEEE 7rans. Automat. Contr. Zheng, A. and M. Morari (1995). Stability of model predictive control with mixed constraints . IEEE Trans. Automat. Contr. 40, 1818 - 1823.
Fig. 3. The case of 2-norm cost for violations
I
j
j 0 .' 0.2
"RttltI'iOtltOn..,..,.. :"ac..crh . 10
'"
30
Fig. 4. Mismatch between finite horizon (openloop) prediction and receding horizon (closed-loop) response (the case of 2-norm)
5-1
10
"
'"
Fig. 1. The case of oo-norm cost for violations
Fig. 2. Mismatch between finite horizon (openloop) prediction and receding horizon (closed-loop) response (the case of oo-norm)
Fig. 5. Response comparison between both cases of costs for violations
322