International Journal of Automation and Computing
8(1), February 2011, 134-140 DOI: 10.1007/s11633-010-0565-x
An Improved Robust Model Predictive Control Approach to Systems with Linear Fractional Transformation Perturbations Peng-Yuan Zheng1,2 1 2
Yu-Geng Xi1,2
De-Wei Li1,2
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, PRC
Key Laboratory of System Control and Information Processing, Ministry of Education, PRC
Abstract: In this paper, a robust model predictive control approach is proposed for a class of uncertain systems with time-varying, linear fractional transformation perturbations. By adopting a sequence of feedback control laws instead of a single one, the control performance can be improved and the region of attraction can be enlarged compared with the existing model predictive control (MPC) approaches. Moreover, a synthesis approach of MPC is developed to achieve high performance with lower on-line computational burden. The effectiveness of the proposed approach is verified by simulation examples. Keywords: Robust model predictive control, linear fractional transformation (LFT) perturbations, linear matrix inequalities (LMIs), feedback model predictive control (MPC) framework, sequence of feedback control laws.
1
Introduction
Model predictive control (MPC), also known as receding horizon control, is a popular technique for industrial process control due to its capability of handing constraints explicitly[1,2] . Over the last decade, robust model predictive control has been greatly developed by using the control invariant set and linear matrix inequalities (LMIs). Kothare et al.[3] proposed an efficient approach for robust MPC synthesis that allows explicit incorporation of the description of the affine plant uncertainty. By using the control invariant set and LMIs, a broad class of model uncertainty descriptions can be addressed with guaranteed robust closed-loop stability. In order to improve the control performance and decrease the computational burden at the same time, many synthesis approaches of MPC were proposed[4−7] . Based on the concept of invariant set, [8–10] off-line constructed a sequence of explicit control laws corresponding to a sequence of invariant sets and on-line calculated the control input with low computational burden. Recently, the authors of [11–14] have proposed a systematic way to derive a sequence of state feedback control laws to enlarge the size of region of attraction, where system state is allowed not to be in an invariant set. As we know, the uncertain linear system described by time-varying, linear fractional transformation (LFT) perturbations includes affine uncertainty as a special case, and is often more appropriate for accurate modeling of nonlinear systems. However, all the techniques above only considered the robust MPC problem for affine plant uncertainty rather than the general class of LFT uncertain systems. Wu[15] extended the LMI-based robust MPC technique and adopted general block diagonal scaling matrices corresponding to Manuscript received April 13, 2010; revised June 21, 2010 This work was supported by National Natural Science Foundation of China (No. 60934007, No. 61074060), China Postdoctoral Science Foundation (No. 20090460627), Shanghai Postdoctoral Scientific Program (No. 10R21414600), and China Postdoctoral Science Foundation Special Support (No. 201003272).
the structured uncertainty to deal with the uncertain linear systems with LFT perturbations. In [15], the single state feedback policy was adopted, which leaves room for further improvement. As mentioned in [16], a sequence of feedback control laws can introduce more freedom to improve the control quality. Motivated by this consideration, this paper combines the procedure in [12] with the modeling procedure in [15] and introduces a sequence of state feedback control laws to improve the control performance and enlarge the region of attraction for the corresponding LFT uncertain systems. Moreover, a synthesis approach of MPC is further proposed to reduce the on-line computational burden. The approach off-line constructs a continuum of terminal constraint sets, and on-line computes a feedback control sequence by optimizing the combination coefficients. With the current feedback control law, this algorithm achieves high performance with lower on-line computation burden. Finally, the proposed robust MPC techniques are applied to the constrained control problem of an industrial continuous stirred tank reactor (CSTR). This paper is organized as follows. Section 2 gives the formulation of LFT uncertain systems. In Section 3, a novel approach is proposed by adopting a sequence of state feedback control laws. In Section 4, a synthesis approach of MPC with low on-line computation burden is developed, and an illustrative numerical example is given in Section 5 to show the merits of the proposed approaches. Finally, conclusions are drawn in Section 6. In this paper, R stands for the set of real numbers, and Rn denotes the n-dimensional space of real valued vectors. Rm×n represents the set of real m × n matrices and Sn×n is the set of real n × n symmetric matrices. In the matrix, the symbol ∗ induces a symmetric when L# and " structure, # e.g., " L ∗ L NT R are symmetric matrices, = . N R N R
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2
Problem formulation
The problem formulation follows that of the discretetime uncertain linear system in [15] and is reproduced here for ease of reading. It is described by systems with timevarying, LFT perturbations: A x(k + 1) z(k) = C0 C2 y(k)
B2 x(k) D02 w(k) D22 u(k)
B0 D00 D20
w(k) = ∆(k)z(k)
(1)
(2)
where u(k) ∈ Rnu is the control input, x(k) ∈ Rn is the state of the plant, y(k) ∈ Rny is the plant output, z(k), w(k) ∈ Rnw are the regulated output and the perturbation, respectively. ∆ is a time-varying structured uncertainty which belongs to ∆ = {diag{δ1 Ir1 , · · · , δα Irα , Ξα+1 , · · · , Ξα+β } : mj
δi : l2 → l2 , kδi k 6 1, Ξα+j : l2
m
→ l2 j , kΞα+j k 6 1,
i = 1, · · · , α, j = 1, · · · , β} where the operator δi and Ξα+j is the induced Pα normPon β l2 norm and r + i i=1 j=1 mj = nw . In essence, the above uncertainty structure is a modified version of that described in [3]. Obviously, the above uncertainty structure could utilize more information if it is available for a practical application. As a result, the more information makes it helpful to improve the control design. Consequently, another scaling set is reproduced from [15] as follows: D = {diag{D1 , · · · , Dα , dα+1 Im1 , · · · , dα+β Imβ } : Di ∈ Sri ×ri , Di > 0, dα+j ∈ R, dα+j > 0, i = 1, · · · , α, j = 1, · · · , β}.
i>0
ξ = 1, · · · , p
(3)
with p defining a p-tuple of integers {l1 , · · · , lp } , which Pp satisfies ξ=1 lξ = nu , and possibly also on the output y(k + i|k): max kyη (k + i|k)k 6 yη,max , i>0
η = 1, · · · , q
(4)
with q defining a q-tuple of integers {m1 , · · · , mq }, which P satisfies pη=1 mη = ny . At each time k, the control objective of the robust MPC problem is to compute the control moves u(k + i|k) by minimizing the following robust performance index: min
max J∞ (k) =
u(k+i|k),i>0 ∆∈∆
∞ X [kx(k+i|k)k2Q +ku(k+i|k)k2R ]. i=0
3
Robust MPC based on a sequence of feedback control laws
The type of the above optimization problem has been studied in [15] based on the LMI framework. To distinguish the items appeared in this paper and in [15], in the following, the items in [15] will be added by symbol ?. Thus, the effective robust MPC approach Theorem 2 proposed in [15] will be denoted as Theorem 2? in the following. Theorem 2? assumes the use of a single state feedback policy and requires strict invariance in the definition of a positively invariant set. This leaves room for further improvement. In order to improve the control quality, the control inputs for the uncertain system can be replaced by a strategy π = {F0 , F1 , · · · , FN −1 } in feedback MPC framework suggested by [16], where Fi is the feedback control gain at the i-th step, and after the N -th step, the feedback control gain is always FN −1 . The goal of this paper is to find the control sequence π = {F0 , F1 , · · · , FN −1 } to minimize the cost function J∞ (k) subject to input and output constraints as described in (3) and (4). Consider the following quadratic function: V (i, k) = x(k + i|k)T P (i, k)x(k + i|k),
It is clear that for any ∆ ∈ ∆ and D ∈ D, ∆D = D∆ holds. Let x(k + i|k) be the state of the plant at time k + i predicted at k and u(k + i|k) the future control move at time k + i, respectively. In addition, consider the following constraints on the control input u(k + i|k): max kuξ (k + i|k)k 6 uξ,max ,
Before proposing the robust MPC approach, two assumptions, retained from [15], are given as follows: 1) State x is measurable in real-time. 2) The uncertain system (1) and (2) is robustly stable, i.e., there exist matrices X ∈ Sn×n , X > 0, F ∈ Rnu ×n , and S ∈ D, such that " # " # X −1 0 (A + B2 F )T (C0 + D02 F )T − × T 0 S −1 B0T D00 " # " # X −1 0 A + B2 F B0 × > 0. 0 S −1 C0 + D02 F D00
(5)
i = 0, 1, · · · , ∞
with P (i, k) = P (N −1, k), i > N , and impose the following robust stability constraint: h i V (i + 1, k) − V (i, k) < − kx(k + i|k)k2Q + ku(k + i|k)k2R which is equivalent to k(A+B2 Fi )x(k+i|k)+B0 w(k)k2P (i+1,k) − kx(k+i|k)k2P (i,k) < h i − kx(k+i|k)k2Q +ku(k+i|k)k2R .
(6)
By summing (6) from i = 0 to i = ∞, an upper bound on the robust performance objective can be obtained: max J∞ (k) < V (0, k) 6 γ
∆∈∆
(7)
where γ is a nonnegative parameter to be minimized in the optimization. Inequalities (6) and (7) can be transformed into LMIs by analogy to Theorem 2? . Thus, a preliminary lemma is proposed as follows. Lemma 1. For the uncertain system (1) and (2) without constraints on input and output, the policy π =
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International Journal of Automation and Computing 8(1), February 2011
{F0 , F1 , · · · , FN −1 } in feedback MPC framework which guarantees (6) and (7) is given by Fi = Yi Xi−1 with P (i, k) = γXi−1 , if there exists γ > 0 and matrices Xi ∈ Sn×n , Xi > 0, Yi ∈ Rnu ×n , Si ∈ D, i = 0, 1, · · · , N − 1, satisfy the following conditions:
Xi 0
∗ Si
∗ ∗
∗ ∗
∗ ∗
0
γI
∗
∗
0 B0 Si D00 Si
0 0 0
γI 0 0
∗ Xi+1 0
1
R 2 Yi 1 Q 2 Xi
AXi + B2 Yi C0 Xi + D02 Yi
∗ ∗ ∗ >0 ∗ ∗
XN −1 0
1 R 2 YN −1 1 Q 2 XN −1 Ψ Ξ
∗ SN −1
∗ ∗
∗ ∗
∗ ∗
∗ ∗
0
γI
∗
∗
∗
0 B0 SN −1 D00 SN −1
0 0 0
γI 0 0
∗
∗ ∗
XN −1 0
Fi = Yi Xi−1
Si
i = 0, 1, · · · , N − 2
(8a) >0
SN −1 (8b)
"
# 1 x(k)
∗ X0
>0
(9)
where Ψ = AXN −1 +B2 YN −1 and Ξ = C0 XN −1 +D02 YN −1 . Lemma 1 can be proven by taking a similar way to the proof of the upper bound in Theorem 2? . Thus, here it is omitted. In the following, we shall show that limits on both the plant inputs and outputs can also be incorporated into our robust MPC algorithm as sufficient LMI constraints. Lemma 2. The input and output constraints (3) and (4) are satisfied if there exists γ > 0 and matrices Xi ∈ Sn×n , Xi > 0, Yi ∈ Rnu ×n , Si ∈ D, Ti,η ∈ D, satisfy conditions (8) and (9), and also satisfy the following conditions: " # u2ξ,max Iξ ∗ >0 T Yi,ξ Xi i = 0, 1, · · · , N − 1,
2 yη,max Xi 0 C2,η Xi + D22,η Yi C0 Xi + D02 Yi
ξ = 1, · · · , p
∗ Ti,η D20,η Ti,η D00 Ti,η
i = 0, 1, · · · , N − 1,
∗ ∗ I 0
∗ ∗ ∗ Ti,η
(10)
>0
η = 1, · · · , q.
From Lemma 1, it follows that the index of the feedback control sequence is the upper bound of the control performance for the uncertain system (1)–(4), which means that minimizing the corresponding index approximately optimizes the control performance. Thus, the following robust MPC can be proposed as: Algorithm 1. Let x(k) = x(k|k) be the state of the uncertain system (1) and (2) measured at sampling time k, and constraints on the plant input and output are described as in (3) and (4). Then, the state feedback matrix Fi in the control law u(k + i|k) = Fi x(k + i|k), i > 0 that minimizes the upper bound V (0, k) on the robust performance objective function at sampling time k is given by
(11)
Lemma 2 can be proven in a similar way to the proof of constraints in Theorem 2? . Therefore, it is omitted here. Thus, we derive the sufficient LMI conditions which guarantee that both the manipulated variables and output constraints are satisfied.
where γ > 0, Xi ∈ Sn×n , Xi > 0, Yi ∈ Rnu ×n and Si ∈ D, Ti,η ∈ D, i = 0, 1, · · · , N − 1, η = 1, · · · , q, are obtained from the solution (if it exists) of the following linear objective minimization problem min
γ,Xi ,Yi ,Si ,Ti,η
γ
subject to inequalities (8)–(11). In order to simplify the presentation in the following, we → − − → denote Qi = (Xi , Yi , Si , Ti ), Ti = {Ti,1 , Ti,2 , · · · , Ti,q }, i = 0, 1, · · · , N −1, then the abbreviation (γ, Q 0 , Q 1 , · · · , Q N −1 ) can represent the solution of the optimization problem above. In the above equation, an approach is proposed which could minimize the cost function effectively through semidefinite programming. Next, the feasibility and stability property of the close-loop system can be asserted as follows. Theorem 1. If there is a feasible solution of the optimization Algorithm 1 at time k, there will also exist a feasible solution for all times t > k, and the feasible receding horizon state feedback control law obtained from Algorithm 1 robustly asymptotically stabilizes the closedloop system. Proof. Suppose that the optimization in Algorithm 1 is feasible at time k, and Γ∗ (k) = (γ ∗ (k), Q ∗0 , Q ∗1 , · · · , Q ∗N −1 ) is the optimal solution for the current state x(k), where − → Q ∗i = (Xi∗ , Yi∗ , Si∗ , Ti∗ ), i = 0, 1, · · · , N − 1. Thus, to prove the feasibility, we only need to prove that LMIs (8)–(11) in the optimization problem are feasible for all future measured states x(k + i) = x(k + i|k + i) for any i > 1. Now, the feasibility of the problem at time k implies satisfaction of (8)–(11). To prove feasibility, we shall construct Q∗1 , · · · , aQ Q∗N −1 , aQ Q∗N −1 ) a solution Γ(k + 1) = (aγ ∗ (k), aQ with a 6 1 where parameter a is defined by a = V (1, k)/γ ∗ (k). From Lemma 1, inequalities (8a) and (8b) establish V (1, k) 6 γ ∗ (k), so it must satisfies a 6 1. We observe that conditions (8a) and (8b) are affine in the ma→ − trices (γ, Q 0 , Q 1 , · · · , Q N −1 ), where Q i = (Xi , Yi , Si , Ti ), i = 0, 1, · · · , N − 1. By multiplying inequalities (8a) and (8b) by parameter a, it is obvious that (8a) and (8b) are satisfied with Γ(k + 1). Although (11) is not affine in the matrices (γ, Q 0 , Q 1 , · · · , Q N −1 ), it can also be proven that Q∗1 , the constructed solution Γ(k + 1) = (aγ ∗ (k), aQ ∗ ∗ QN −1 , aQ QN −1 ) with a 6 1 satisfies the following · · · , aQ
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inequalities: ∗ 2 yη,max (aXi ) ∗ ∗ 0 aTi,η ∗ ∗ ∗ C2,η (aXi ) + D22,η (aYi ) D20,η (aTi,η ) ∗ ∗ ∗ C0 (aXi ) + D02 (aYi ) D00 (aTi,η )
∗ ∗ ∗ ∗ I ∗ ∗ 0 aTi,η
>0
i = 0, 1, · · · , N − 1, η = 1, · · · , q i.e., (11) is also satisfied with Γ(k + 1). Similarly, it can be proved that (10) also holds with Γ(k + 1). According to the definition of a , i.e., V (1, k) = aγ ∗ (k), it leads to x(k + 1|k)T (aX1∗ )−1 x(k + 1|k) = 1, which means the constructed solution satisfies (9). Hence, Γ(k + 1) is a feasible solution at time k + 1. This argument can be repeated for k + 2, k + 3, · · · to prove the recursive feasibility. Q∗1 , · · · , aQ Q∗N −1 , aQ Q∗N −1 ) is a Since Γ(k + 1) = (aγ ∗ (k), aQ feasible solution at time k + 1 , and a = V (1, k)/γ ∗ (k) 6 1, it can be obtained that γ ∗ (k + 1) 6 aγ ∗ (k) 6 γ ∗ (k). This proves the asymptotically stability for the uncertain system. ¤ Remark 1. Note that Wu[15] requires strict invariance in the definition of a positively invariant set, which will result in somewhat conservativeness. For the proposed approach, the states are allowed to move from one ellipsoid to another and finally into the terminal invariant set. The conditions in Wu0 s work can be recovered by imposing Q i = Q 0 , → − i = 0, 1, · · · , N − 1, with Q i = (Xi , Yi , Si , Ti ) in the proposed approach, and included as its special case (N = 1). Therefore, the proposed approach has more freedom and less conservativeness of design compared with Wu0 s. Consequently, it would enlarge the region of attraction and improve the control performance. The optimization problem of minimizing a linear objective subject to LMI constraints can be solved efficiently by using interior point methods. The fastest interior point algorithms show O(ρθ3 ) growth in computation, where ρ is the total row size of the LMIs and θ is the total number of scalar decision variables[17] . The parameters to be searched in Algorithm 1 are γ > 0, Xi ∈ Sn×n , Xi > 0, Yi ∈ Rnu ×n , and Si ∈ D, Ti,η ∈ D, i = 0, 1, · · · , N − 1, η = 1, · · · , q. Compared with Wu0 s technique, the proposed approach has more decision variables and LMI constraints, and introduces a much heavier online computational burden.
4
Reduction of computation burden
Although Algorithm 1 could achieve high control performance and a large region of attraction, the online computational burden for solving the optimization problem might be excessive with great N or n. Based on Algorithm 1, a synthesis approach of MPC would be developed in this section to reduce the on-line computational burden. In the following, we shall propose two preliminary lemmas, based on which the synthesis approach of MPC could be established. Lemma 3. For the uncertain system (1)–(4), suppose that the two groups (γ1 , Q 1,0 , Q 1,1 , · · · , Q 1,N −1 ) and (γ2 , Q 2,0 , Q 2,1 , · · · , Q 2,N −1 ) satisfy conditions (8a), (8b), (10), and (11), respectively, and that the conˆ ,Q ˆ , · · · ,Q ˆ vex combination (ˆ γ, Q 0 1 N −1 ) would also satisfy
the conditions (8a), (8b), (10), and (11), where Q 1,i = − − → − − → ˆ (X1,i , Y1,i , S1,i , T1,i ), Q 2,i = (X2,i , Y2,i , S2,i , T2,i ), Q i = → − ˆ i , Yˆi , Sˆi , Tˆi ) = λ1Q + λ2Q , γˆ = λ1 γ1 + λ2 γ2 , λ1 > 0, (X 1,i 2,i λ2 > 0, λ1 + λ2 6 1, i = 0, 1, · · · , N − 1. Proof. Since both (γ1 , Q 1,0 , Q 1,1 , · · · , Q 1,N −1 ) and (γ2 , Q 2,0 , Q 2,1 , · · · , Q 2,N −1 ) are assumed to be the solutions of conditions (8a), (8b), (10), and (11), for each i, we multiply (8a) and (8b) by λ1 and λ2 , where (γ, Q 0 , Q 1 , · · · , Q N −1 ) are replaced with (γ1 , Q 1,0 , Q 1,1 , · · · , Q 1,N −1 ) and (γ2 , Q 2,0 , Q 2,1 , · · · , Q 2,N −1 ), respectively, and sum the resulting inequalities to get
ˆi X 0 1 R 2 Yˆi 1 ˆi Q2 X
ˆ i + B2 Yˆi AX ˆ i + D02 Yˆi C0 X
0 ˆ Si
∗ ∗
∗ ∗
∗ ∗
0
γˆ I
∗
∗
0 B0 Sˆi D00 Sˆi
0 0 0
γˆ I 0 0
∗ ˆ i+1 X 0
∗ ∗ ∗ >0 ∗ ∗ Sˆi
i = 0, 1, · · · , N − 2
ˆ N −1 X 0 1 ˆ 2 R YN −1
1 ˆ N −1 Q2 X ˆ Ψ ˆ Ξ
∗ SˆN −1
∗ ∗
∗ ∗
∗ ∗
∗ ∗
0
γˆ I
∗
∗
∗
0 B0 SˆN −1 D00 SˆN −1
0 0 0
γˆ I 0 0
∗
∗ ∗
ˆ N −1 X 0
>0
SˆN −1
ˆ = AX ˆ N −1 + B2 YˆN −1 , Ξ ˆ = C0 X ˆ N −1 + D02 YˆN −1 Ψ this means that the convex combination also satisfy (8a) and (8b), where λ1 > 0, λ2 > 0, λ1 + λ2 6 − − → 1, γˆ = λ1 γ1 + λ2 γ2 , Q 1,i = (X1,i , Y1,i , S1,i , T1,i ), − − → ˆ Q 2,i = (X2,i , Y2,i , S2,i , T2,i ), Q i = λ1Q 1,i + λ2Q 2,i , i = 0, 1, · · · , N − 1. Similarly, (10) and (11) can be established with the convex combination solution. ¤ Lemma 4. For the uncertain system (1)–(4), if a group (γ, Q 0 , Q 1 , · · · , Q N −1 ) satisfies conditions (8a), (8b), (10), and (11), then the convex combination ¯ ,Q ¯ , · · · ,Q ¯ (¯ γ, Q 0 1 N −1 ) would also satisfy the conditions → − (8a), (8b), (10), and (11), where Q i = (Xi , Yi , Si , Ti ), → − ¯ = (X ¯ i , Y¯i , S¯i , T¯i ) = λ0Q +λ1Q +· · ·+λN −i−1Q Q i i+1 N −1 + i · · · + λN −1Q N −1 , γ¯ = (λ0 + λ1 + · · · + λN −1 )γ, λi > 0, λ0 + λ1 + · · · + λN −1 6 1, i = 0, 1, · · · , N − 1. Proof. Since the group (γ, Q 0 , Q 1 , · · · , Q N −1 ) satisfies conditions (8a), (8b), (10), and (11), then (γ, Q i , Q i+1 , · · · , Q N −1 , Q N −1 , Q N −1 , · · · , Q N −1 ) would | {z } | {z } N −i
i
also be the solution of conditions (8a), (8b), (10), and → − (11) with Q i = (Xi , Yi , Si , Ti ), i = 0, 1, · · · , N − 1. Using Lemma 3, it can be proved that the convex combination ¯ ,Q ¯ , · · · ,Q ¯ (¯ γ, Q 0 1 N −1 ) would also be the solution of conditions (8a), (8b), (10), and (11). ¤ Remark 2. Lemmas 3 and 4 reveal that if two groups satisfy the robust stability constraints, the convex combina-
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tion of all the ellipsoid sets within the two groups would also satisfy these LMI constraints, so does the convex combination of all the ellipsoid sets within each group. Therefore, the two lemmas above drop a hint that we could develop a synthesis approach of MPC to make an off-line design of two groups which satisfy the robust stability conditions and online optimize the combination coefficients. In the following, we will present the off-line design, which transfers part of the online computational burden to offline and completes the main computation of constructing the ellipsoid sets. The off-line design algorithm is given as follows. Algorithm 2 (Off-line design). Step 1. Choose N . Step 2. Choose γ1 and solve the following optimization problem: max
~1,i γ1 ,X1,i ,Y1,i ,S1,i ,T
s.t.
log(det(X1,0 )),
(8a), (8b), (10), and (11)
(12) − → − − → with γ = γ1 , Xi = X1,i , Yi = Y1,i , Si = S1,i , Ti = T1,i , i = 0, 1, · · · , N − 1. Step 3. Choose γ0 (γ0 < γ1 ), and solve the following optimization problem: max
~0,i γ0 ,X0,i ,Y0,i ,S0,i ,T
s.t.
log(det(X0,0 ))
(13) − → − − → with γ = γ0 , Xi = X0,i , Yi = Y0,i , Si = S0,i , Ti = T0,i , i = 0, 1, · · · , N − 1. Subsequently, the task is to design the on-line algorithm to optimize the control performance. According to the properties of Lemmas 3 and 4, the on-line optimization algorithm can be developed to minimize the cost upper bound, which adopts the convex combinations of all the ellipsoid sets within the two groups constructed from the offline design. Thus, it is formulated as follows. Algorithm 3 (On-line optimization). min
λ0 ,··· ,λ2N −1
(8a), (8b), (10), and (11)
(λ0 +λ2 +· · ·+λ2N −2 )γ0 +(λ1 +λ3 +· · ·+λ2N −1 )γ1 (14)
s.t. (9) with X0 =
λ0 X0,0 + λ2 X0,1 + · · · + λ2N −2 X0,N −1 + | {z } N
λ1 X1,0 + λ3 X1,1 + · · · + λ2N −1 X1,N −1 | {z } N
subject to λi > 0,
λ0 + λ1 + · · · + λ2N −1 6 1,
i = 0, 1, · · · , 2N − 1.
The feedback control gain is given by F = Y0 X0−1 where Y0 =
λ0 Y0,0 + λ2 Y0,1 + · · · + λ2N −2 Y0,N −1 + | {z } N
λ1 Y1,0 + λ3 Y1,1 + · · · + λ2N −1 Y1,N −1 . | {z } N
(15)
Since Algorithm 2 completes the design of two groups offline, the parameters to be searched on-line in Algorithm 3 are just λi , i = 0, 1, · · · , 2N − 1, and the related LMI constraints reserve only (9). Moreover, the number of the online optimization parameters does not depend on the system order n. Therefore, Algorithm 3 would reduce the computational burden dramatically, especially for high order systems. Remark 3. Algorithms 2 and 3 provide us a synthesis approach of MPC, which is a simplified version of Algorithm 1 and reduces the on-line computational burden greatly at the cost of control performance. For this simplified method, the increase of N could improve the control performance but lead to a little increase of on-line computational burden. Therefore, the parameter N should be chosen appropriately according to the requirement of practical applications. For the feasibility and stability property for the synthesis approach, we give the following theorem: Theorem 2. For the uncertain system (1)–(4), if optimization problem (14) is feasible at time k for the current plant state x(k), the close-loop system is asymptotically stable by the feedback control law (15). We can prove Theorem 2 in a similar way to the proof of Theorem 1, thus it is omitted.
5 5.1
Case study Modeling of CSTR
The continuous stirred tank reactor (CSTR) is a common ideal reactor type in chemical engineering. Assuming constant liquid volume and using the component balance and energy balance principle, the CSTR for an exothermic, irreversible reaction is described by the following dynamic model (e.g., [15,18,19]). q E C˙ A = (CAf −CA )−k0 exp(− )CA V RT −∆H E UA q k0 exp(− )CA + (Tc −T ) T˙ = (Tf −T )+ V ρCP RT V ρCP where CA is the concentration in the reactor, T is the reactor temperature, and Tc is the temperature of the coolant stream. The constraints are 280 K 6 Tc 6 370 K, 280 K 6 T 6 370 K, 0 6 CA 6 1 mol/l. The nominal operating conditions, which correspond to an unstable equieq librium CA = 0.5 mol/l, Teq = 350 K, Tceq = 300 K are q = 100 l/min, Tf = 350 K, V = 100 l, ρ = 1000 g/l, CP = 0.239 J/g·K, ∆H = −5 × 104 J/mol, E/R = 8750 K, k0 = 7.2 × 1010 min−1 , and U A = 5 × 104 J/min·K. To ease of comparison between the proposed approach and Wu0 s, we use the same uncertain linear model for the nonlinear CSTR plant under conic sector bounded uncertainties, adopt the same parameter settings, and assume δ1 = δ2 for the case study.
5.2
Result comparison
For the CSTR problem, it is possible to improve the control quality associated with the synthesis condition in [15] through a sequence of feedback control laws.
P. Y. Zheng et al. / An Improved Robust Model Predictive Control Approach to Systems with · · ·
In order to simplify the following presentation, we denote the controller designed by Algorithms 1 and 3 as C 1 and C 2 , respectively. Given initial state x(0) = [0.5 0.31], the one step optimization index for our approach is γ = 266.60 and that for Wu0 s is γ = 537.11, which indicates that our proposed approach can be expected a smaller value of γ than Wu0 s approach. The tracking performance from " initial state # δ1 x(0) = [0.5 0.31] with structured uncertainty = δ2 # " 0.5 is simulated in Fig. 1, and the region of at0.5 traction for both techniques are plotted in Fig. 2. Table 1 shows the comparison of computation burden for Wu0 s technique and our approaches.
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From Table 1, it can be concluded that the computational burden in C 1 is heavier than Wu0 s, and C 2 is the lowest among the three controllers. Moreover, denote P J= ∞ [kx(k + i)k2Q +ku(k + i)k2R ], then J ∗ = 117.35 for i=0 controller C 1 (N = 3), J ∗ = 126.58 for controller C 2 (N = 5, γ1 = 600, γ2 = 200) and J ∗ = 139.46 for Wu0 s technique. Fig. 1 shows that plant states x1 and x2 converge to their set point [0.3 1.96] (the equilibrium of the CSTR model). Fig. 2 demonstrates that the proposed approach also enlarges the size of the region of attraction. These clearly demonstrate the advantage of the feedback MPC framework application to uncertain systems with LFT perturbations.
6
Conclusions
This paper develops a new approach to robust constrained MPC for systems with uncertainty in linear fractional transformation form. The design conservativeness can be reduced by substituting a feedback control sequence for the single state feedback law. The effectiveness of the proposed approaches is demonstrated by means of a numerical example.
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Fig. 1 Set point tracking (Solid line for controller C 1 (N = 3), dash for controller C 2 (N = 5), and dash dot for Wu0 s technique)
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Fig. 2 Comparison of the region of attraction for two techniques (Solid line for controller C 1 (N = 3), dash for C 1 (N = 2), and dash dot for Wu0 s technique) Table 1
Computation comparison for Wu0 s and our approaches
Computation burden 0
ρ
θ
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30
16
122 880
C 1 (N =3)
82
46
7 981 552
C 2 (N =5)
16
10
16 000
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[18] A. Uppal, W. H. Ray, A. B. Poore. On the dynamic behavior of continuous stirred tank reactors. Chemical Engineering Science, vol. 29, no. 4, pp. 967–985, 1974. [19] L. Magni, G. D. Nicolao, L. Magnani, R. Scattolini. A stabilizing model-based predictive control algorithm for nonlinear systems. Automatica, vol. 37, no. 9, pp. 1351–1362, 2001. Peng-Yuan Zheng received his B. Sc. degree in electrical engineering and automation from the North University of China, PRC in 2000, the M. Sc. degree in measurement technology and instrumentation from University of Shanghai for Science and Technology, Shanghai, PRC in 2005. He is currently a Ph. D. candidate in the Department of Automation at Shanghai Jiao Tong University, PRC. His research interests include predictive control and robust control. E-mail:
[email protected] (Corresponding author) Yu-Geng Xi received the Dr.-Ing. degree in automatic control from the Technical University Munich, Germany in 1984. Since then, he has been with the Department of Automation, Shanghai Jiao Tong University, and as a professor since 1988. His research interests include predictive control, large scale and complex systems, and intelligent robotic systems. E-mail:
[email protected] De-Wei Li received the B. Sc. degree in automation from Shanghai Jiao Tong University, Shanghai, PRC in 1993, the Ph. D. degree in control theory and control engineering from Shanghai Jiao Tong University, Shanghai, PRC in 2009. He is currently a postdoctoral research fellow in Shanghai Jiao Tong University. His research interests include predictive control and robust control. E-mail:
[email protected]
