Robust Model Predictive Control Using a Discrete ... - Semantic Scholar
Robust Model Predictive Control Using a Discrete-Time Recurrent Neural Network Yunpeng Pan and Jun Wang Department of Mechanical and Automation Engineering The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong {yppan,jwang}@mae.cuhk.edu.hk
Abstract. Robust model predictive control (MPC) has been investigated widely in the literature. However, for industrial applications, current robust MPC methods are too complex to employ. In this paper, a discrete-time recurrent neural network model is presented to solve the minimax optimization problem involved in robust MPC. The neural network has global exponential convergence property and can be easily implemented using simple hardware. A numerical example is provided to illustrate the effectiveness and efficiency of the proposed approach. Keywords: Robust model predictive control, recurrent neural network, minimax optimization.
1 Introduction Model predictive control (MPC) is a powerful technique for optimizing the performance of control systems, with several advantages over other control systems [1]. MPC applies on-line optimization to the model of a system, by taking the current state as an initial state, a optimization problem is solved at each sample time, and at the next computation time interval, the calculation repeated with a new state. MPC that take consideration of uncertainties in the process model is called robust MPC. One way to deal with uncertainties in MPC is the worst case approach, which obtains a sequence of feedback control laws that minimizes the worst case cost. In industrial processes, it required the real-time solution to a minimax optimization problem. Although the robustness of MPC has been studied and is now well understood, the research outcomes are conceptual controllers that can work in principle but not suitable for hardware implementation [2]. As a result, further investigations on a more implementable controller are needed. One very promising approach to dynamic optimization is to apply recurrent neural networks. Recurrent neural networks are brain-like computational models for solving optimization problems in real time. Compared with traditional numerical methods for constrained optimization, neural networks have several advantages: first, they can solve many optimization problems with time-varying parameters; second, they can handle large-scale problems with their parallelizable ability; third, they can be implemented effectively using VLSI or optical technologies. Neural networks for optimization and their engineering applications have been widely investigated in the past two decades. Many neural network models have been proposed F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 883–892, 2008. c Springer-Verlag Berlin Heidelberg 2008
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for solving both linear and nonlinear programming problems. In this paper, we present a discrete-time recurrent neural network model for solving the quadratic minimax optimization problem associated with robust MPC. The neural network is globally exponentially convergent to the saddle point of the objective function. The rest of this paper is organized as follows. In Section 2, we formulate robust MPC as a quadratic minimax optimization problem. In Section 3, we proposed a recurrent neural network model for minimax optimization, and prove its global exponential convergence property, a control scheme for robust MPC is also presented. In Section 4, we provide a example in industrial application to illustrate the performance of the proposed approach. Furthermore, a comparision is made between the proposed approach and linear matrix inequalities approach. Finally, Section 5 conclude this paper.
2 Problem Formulation 2.1 Process Model Consider the following discrete-time linear system with global bounded uncertainties: x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k) + Dw(k),
(1)
with the constraints umin ≤u(k) ≤ umax , Δumin ≤Δu(k) ≤ Δumax , wmin ≤w(k) ≤ wmax ,
(2)
ymin ≤y(k) ≤ ymax , where k ≥ 0, x(k) ∈ n is the state vector, u(k) ∈ m is the input vector, and y(k) ∈ p is the output vector. w(k) ∈ q denotes the vector of bounded uncertainties. umin ≤ umax , wmin ≤ wmax , ymin ≤ ymax are vectors of upper and lower bounds. 2.2 Robust MPC Design MPC is a step-by-step optimization technique: at each sampling time k, measure of estimate the current state, obtain the optimal input vector by solving a optimization problem. When bounded uncertainties are considered explicitly, a robust MPC law can be derived by minimizing the maximum cost within the model described by the uncertainty set. The optimal control action is obtained by solving a minimax optimization problem: (3) min max J(Δu, w), Δu
subjected to the constraints in (2).
w
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The objective function J(Δu, w) can be with an infinite or finite, linear or quadratic norm criterion. In this paper, we consider an objective function with a finite horizon quadratic criterion: J(Δu, w) =
N
[r(k + j|k) − y(k + j|k)]T Φ[r(k + j|k) − y(k + j|k)]+
j=1
(4)
N u −1
T
[Δu(k + j|k)] Ψ [Δu(k + j|k)]
j=0
where k is the current time step, y(k + j|k) denotes the predicted output, r(k + j|k) denotes the reference trajectory of output signal (desired output), and Δu(k + j|k) denotes the input increment, where Δu(k + j|k) = u(k + j|k) − u(k − 1 + j|k). Φ ∈ p×p , Ψ ∈ m×m are appropriate weighting matrices. N denotes the predictive horizon (1 ≤ N ). Nu denotes the control horizon (0 < Nu ≤ N ). After Nu control moves, Δu(k + j|k) becomes zero. According to the process model (1): y(k + j) = CAj x(k) + C
j−1
Ai Bu(k + j − i − 1) + Dw(k + j),
j = 1, ..., N (5)
i=0
Define following vectors: ···
y¯(k) = [y(k + 1|k)
u(k + Nu − 1|k)]T ∈ Nu m ,
···
u ¯(k) = [u(k|k)
Δu(k + Nu − 1|k)]T ∈ Nu m ,
···
Δ¯ u(k) = [Δu(k|k)
y(k + N |k)]T ∈ N p ,
r(k + N |k)]T ∈ N p ,
···
r¯(k) = [r(k + 1|k)
(6)
where the reference trajectory r¯(k) is known in advance. The predicted output y¯(k) is expressed in the following form: y¯(k) = Sx(k) + M u ¯(k) + Ew(k) = Sx(k) + M Δ¯ u(k) + V u(k − 1) + Ew(k), where S = [CA ⎡
CA2
E = [D
D
···
CAN ]T ∈ N p×n ,
···
D]T ∈ N p×q , ⎤
CB ⎢ ⎥ C(A + I)B ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ N p×m Nu −1 ⎢ + · · · + A + I)B ⎥ , V = ⎢ C(A ⎥∈ ⎢ C(ANu + · · · + A + I)B ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ . C(AN −1 + · · · + A + I)B
(7)
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⎡
CB C(A + I)B .. .
⎢ ⎢ ⎢ ⎢ ⎢ Nu −1 M =⎢ ⎢ C(A N + · · · I)B ⎢ C(A u + · · · I)B ⎢ ⎢ .. ⎣ . C(AN −1 + · · · I)B
··· ··· .. .
0 0 .. .
··· ··· .. .
CB C(A + I)B .. .
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∈ N p×Nu m , ⎥ ⎥ ⎥ ⎥ ⎦
· · · C(AN −Nu + · · · I)B
I denotes the identity matrix. Define vectors: Δ¯ umin = [Δumin · · · Δumin ]T ∈ Nu m , Δ¯ umax = [Δumax · · · Δumax ]T ∈ Nu m u ¯min = [umin · · · umin]T ∈ Nu m , u ¯max = [umax · · · umax ]T ∈ Nu m , y¯min = [ymin · · · ymin ]T ∈ Nu p , y¯max = [ymax · · · ymax ]T ∈ Nu p , ⎡ ⎤ I 0 ··· 0 ⎢I I ··· 0⎥ ⎢ ⎥ I˜ = ⎢ . . . . ⎥ ∈ Nu m×Nu m . ⎣ .. .. . . .. ⎦ II I I Thus, the original minimax optimization can be expressed in the following form: min max Δu
w
s.t.
[¯ r(k) − Sx(k) − M Δ¯ u(k) − V u(k − 1) − Ew(k)]T Φ[¯ r (k) − Sx(k) − M Δ¯ u(k) − V u(k − 1) − Ew(k)] + Δ¯ uT (k)Ψ Δ¯ u(k) ˜ u(k) ≤ u u ¯min ≤ u ¯(k) + IΔ¯ ¯max Δ¯ umin ≤ Δ¯ u(k) ≤ Δ¯ umax w ¯min ≤ w(k) ¯ ≤w ¯max y¯min ≤ y¯(k) + M (k)Δ¯ u(k) ≤ y¯max (8)
By defining the variable vectors u = Δ¯ u(k) ∈ Nu m , w = w(k) ∈ q . By neglecting the constraints on u(k) and y(k), the problem (8) can be rewritten as a minimax quadratic programming problem: min max u
w
s.t.
1 T 1 u Qu + cT u − uT Hw − wT Rw − bT w 2 2 u ∈ U, w ∈ W
(9)
where U and W are two box set defined as U = {u ∈ Nu m |Δ¯ umin ≤ u ≤ Δ¯ umax }, ¯min ≤ w ≤ w ¯max }. The coefficient matrices and vectors are W = {w ∈ q |w Q = 2(M T ΦM +Ψ ) ∈ Nu m×Nu m , c = −2M T Φ(¯ r (k)−Sx(k)−V u(k−1)) ∈ Nu m , R = 2E T ΦE ∈ q×q , b = Φ(¯ r (k) − Sx(k) − V u(k − 1)) ∈ q , H = 2M T ΦE ∈ Nu m×q .
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The solution to the minimax quadratic programming problem (9) gives the vector of control action Δ¯ u(k). The control law is given by u ¯(k) = f (Δ¯ u(k) + u ¯(k − 1)), where f (·) is defined as εi , (Gε)i ≤ li , (10) f (εi ) = li , (Gε)i > li . and G and l are defined as G = [−I˜ I˜
−M
M ]T ∈ (2Nu m+2N p)×Nu m ,
⎡
⎤ −¯ umin + u¯(k) ⎢ u ¯max − u¯(k) ⎥ ⎥ ∈ 2Nu m+2N p . l=⎢ ⎣ −¯ ymin + y¯(k) ⎦ y¯max − y¯(k) The first element u(k|k) is used as the control signal. In industrial control processes, to solve large-scale minimax optimization problems in real-time is a major obstacle for robust MPC. In the next section, we will propose a recurrent neural network for solving (9).
3 Recurrent Neural Network Approach 3.1 Neural Network Model In recent years, many neural network models have been proposed for solving optimization problems [3,4,5,6,7]. In particular, continuous-time neural networks for solving minimax problems has been investigated in [8,9,10]. However, in view of the availability of the digital hardware and the compatibility to the digital computers, discrete-time neural network is more desirable in practical implementation. In this section, we proposed a discrete-time recurrent neural network for minimax problem (9). By the saddle point condition [11], (9) can be formulated as a linear variational inequality (LVI): (s − s∗ )T (M s∗ + q) ≥ 0, where
M=
Q −H , HT R
q=
c , b
∀s ∈ Ω,
(11)
Ω = U × W.
(12)
According to the well-known saddle point theorem [11], s∗ = (u∗ , w∗ ) is a saddle point of J(u, w) if satisfying J(u∗ , w) ≤ J(u∗ , w∗ ) ≤ J(u, w∗ ),
∀(u, w) ∈ Ω.
(13)
We define the saddle point set Ω ∗ = {(u∗ , w∗ ) ∈ Ω|(u∗ , w∗ ) satisfy (13)} and assume Ω ∗ is not empty. It is obvious that if (u∗ , w∗ ) ∈ Ω ∗ , then (u∗ , w∗ ) is the optimal solution to the minimax problem (9).
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According to inequalities (13), we can get that v ∗ is a global minimizer of the objective function J(v, w∗ ) with respect to U, while w∗ is the global minimizer of J(v ∗ , w) with respect to W. As a result, the following LVIs hold: (u − u∗ )T (Qu∗ + c − Hw∗ ) ≥ 0, (w − w∗ )T (Rw∗ + b + H T u∗ ) ≥ 0,
∀u ∈ U,
(14)
∀w ∈ W.
(15)
According to the basic property of the projection mapping on a closed convex set: [z − PΩ (z)]T [PΩ (z) − v] ≥ 0,
∀z ∈ , v ∈ Ω.
(16)
Based on (14)-(16) and lemma 1 in [9], we can get that (u∗ , w∗ ) ∈ Ω ∗ if and only if the following equations hold: u∗ = PU [u∗ − α(Qu∗ + c − Hw∗ )]
(17)
w∗ = PW [w∗ − α(Rw∗ + b + H T u∗ )]
(18)
where α > 0 is a scaling constant, PU (·) and PW (·) are piecewise activation functions defined as: ⎧ ⎧ ⎨ Δumin , εi < Δumin ; ⎨ wmin , εi < wmin ; wmin ≤ εi ≤ wmax ; PU (εi ) = εi , Δumin ≤ εi ≤ Δumax ; PW (εi ) = εi , ⎩ ⎩ Δumax , εi > Δumax . wmax , εi > wmax . (19) Based on the equations (17) and (18), we propose a recurrent neural network for solving (9) as follow: u(t + 1) = PU [u(t) − α(Qu(t) + c − Hw(t))] (20) w(t + 1) = PW [w(t) − α(Rw(t) + b + H T u(t))] The proposed recurrent neural network has a simple structure, and can be easily implemented using digital hardware. In the next section, we will prove that the proposed neural network has global exponential convergence property under some mild conditions. 3.2 Convergence Analysis Definition 1. Neural network (20) is said to be globally exponentially convergent to the equilibrium point (ue , we ) if both ue and we satisfy u(t) − ue ≤ c0 u(0) − ue e−ηt , w(t) − w ≤ b0 w(0) − w e e
e
−ηt
∀t ≥ 1; ,
∀t ≥ 1;
(21)
where η is a positive constant independent of the initial point, c0 and b0 are positive constant dependent on the initial point.
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Lemma 1. The neural network (20) has a unique equilibrium point, which is the saddle point of J(u, w). Proof. Similar to the proof in [12], we can establish that the neural network (20) has a unique equilibrium point (ue , we ). Define a equilibrium point set Ω e = {(ue , we ) ∈ Ω|(ue , we ) satisfy (17)and(18))}. According to the above derivation, it is obvious that the equations (17) and (18) is equivalent to (13) for all (u, w) ∈ Ω, from the definition of Ω ∗ , we can get that Ω e = Ω ∗ , which means the equilibrium point of (20) is the saddle point of J(u, w). Lemma 2. For all z ∈ n , PU (v) − PU (z)2 ≤ v − z2 ,
PW (v) − PW (z)2 ≤ v − z2 .
Proof. From the inequality (16) we can easily prove that PU (v) − PU (z)2 ≤ (v − z)T [PU (v) − PU (z)] ≤ v − z2 , PW (v) − PW (z)2 ≤ (v − z)T [PW (v) − PW (z)] ≤ v − z2 ,
∀v, z ∈ n . (22)
R Define λQ i > 0(i = 1, ..., Nu m), λj > 0(j = 1, ..., N q) as the eigenvalues of Q, R Q R R respectively, let λQ min , λmax , λmin , λmax be the smallest and largest eigenvalues of Q and R. Define two functions
Q
ψ (α) = ψ R (α) =
1 − λQ min α,
Q 0 < α ≤ 2/(λQ min + λmax )
λQ max α − 1,
Q 2/(λQ min + λmax ) ≤ α < +∞
1 − λR min α,
R 0 < α ≤ 2/(λR min + λmax )
λR max α − 1,
R 2/(λR min + λmax ) ≤ α < +∞
(23)
(24)
Then we give the following lemma: Lemma 3. ψ Q (α) < 1 and ψ R (α) < 1
if and only if
R 0 < α < min{2/λQ max , 2/λmax }.
(25)
Proof. From the Theorem 2 in [13], we can get that ψ Q (α) < 1 if and only if α ∈ R R (0, 2/λQ max ), similarly, ψ (α) < 1 if and only if α ∈ (0, 2/λmax ). We can easily verify Q that the sufficient and necessary condition for both ψ (α) < 1 and ψ R (α) < 1 is R 0 < α < min{2/λQ max , 2/λmax }. Theorem 1. With any α that satisfies (25), the neural network (20) is globally exponentially convergent to the saddle point of J(u, w). 2 Proof. From (23) and (24), we can obtain that ψ Q (α) = max{(1 − αλQ 1 ) , ..., (1 − Q 2 R R 2 R 2 αλNu m ) }, ψ (α) = max{(1 − αλ1 ) , ..., (1 − αλN q ) }.
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By Lemma 2: u(k) − u∗ 2 =PU [u(t − 1) − α(Qu(t − 1) + c − Hw(t − 1))]− PU [u∗ − α(Qu∗ + c − Hw∗ )]2 ≤(I − αQ)(u(t − 1) − u∗ )2 Q 2 2 ∗ 2 ≤ max{(1 − αλQ 1 ) , ..., (1 − αλNu m ) }u(t − 1) − u
=ψ Q (α)2 u(t − 1) − u∗ 2
(26)
=⇒ u(t) − u∗ ≤ ψ Q (α)u(t − 1) − u∗ ≤ ψ Q (α)t u(0) − u∗ ≤ e−η
Q
(α)t
u(0) − u∗
Similarly, w(t)−w∗ ≤ e−η (α)t w(0)−w∗ . From Lemma 3, η Q (α) > 0 (ψ Q (α) < 1) and η R (α) > 0 (ψ R (α) < 1) for all α that satisfy (25). From the above proof and lemma 1, we can obtain that for any α that satisfies (25), the neural network (20) is globally exponentially convergent to the unique equilibrium point (u∗ , w∗ ), which is the saddle point of J(u, w). R
3.3 Control Scheme The control scheme based on proposed recurrent neural network can be summarized as follows: 1. Let k = 1. Set terminal time T , sample time t, predictive horizon N , control horizon Nu , weighting matrices Φ and Ψ . 2. Calculate process model matrices S, E, V , M , neural network parameters Q, R, H, c, b. 3. Solve the quadratic minimax problems (9) using the proposed recurrent neural network, obtaining the optimal control action Δ¯ u(k). 4. Calculate the optimal input vector u¯(k) = f (Δ¯ u(k) + u¯(k − 1)), the first element u(k|k) is sent to the process. 5. If k < T , set k = k + 1, return to step 2; otherwise, end.
4 Numerical Example Consider a two-tank system described in [14], which is a two-input, two-output system, with the flow rates of the two inlet streams as the two inputs, and the liquid level in each tank as the two output variables. By sampling at 0.2 min using a zero-order holder, the following discrete-time statespace model can be obtained:
x(k + 1) =
− 0.5 3 0.5 2
0.2 3 − 0.5 2
10 y(k) = x(k) 01
1 x(k) +
3
0
0 1 2
u(k) (27)
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The set-point for the liquid levels (output) of tanks 1 and 2 are 0.8 and 0.7, respectively; the prediction and control horizons are N = 10 and N u = 4; weighting matrices Φ = I, Ψ = 5I; scaling constant α = 0.2; an uncertainty −0.02 ≤ w ≤ 0.02 is considered to affect both liquid levels of tanks 1 and 2; moreover, the following constraints are considered:
0 0.5 0 0.6 ≤u(k) ≤ ≤ y(k) ≤ 0 0.5 0 0.7
(28) −0.05 0.05 ≤ Δu(k) ≤ −0.05 0.05 Input u1
Input u2
0.5
0.4 RNN LMI
0.45
RNN LMI 0.35
0.4 0.3 0.35 0.25
0.3 0.25
0.2
0.2
0.15
0.15 0.1 0.1 0.05
0.05 0
0
10
20
30
40
50 60 Samples k
70
80
90
100
0
0
10
20
30
40
50 60 Samples k
70
80
90
100
Fig. 1. Input signals of tanks 1 and 2 using the proposed RNN approach and LMI approach
Output y1
Output y2
0.9
0.8
0.8
0.7
0.7
0.6
0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2
0.2 0.1 0
0.1
RNN LMI 0
10
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30
40
50 60 Samples k
70
80
90
100
0
RNN LMI 0
10
20
30
40
50 60 Samples k
70
80
90
100
Fig. 2. Output responses of tanks 1 and 2 using the proposed RNN approach and LMI approach
In order to compare the effectiveness and efficiency of the proposed approach, a linear matrix inequalities (LMI) approach [1] is also applied to the process. The simulation results are showed in Figs. 1 - 2. We can see that the proposed neural network approach gives a better set-point tracking performance with faster stable output responses.
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5 Conclusion This paper presents a new approach to robust MPC based on a discrete-time recurrent neural network by solving a minimax optimization problem. The neural network is proved to have global exponential convergent property. Simulation results show the superior performance of the neural network approach. Compared with a linear matrix inequalities approach, the proposed neural network approach gives a better performance in set-point tracking.
References 1. Camacho, E., Bordons, C.: Model Predictive Control. Springer, Heidelberg (2004) 2. Mayne, D., Rawlings, J., Rao, C., Scokaert, P.: Constrained model predictive control: Stability and optimality. Automatica 36, 789–814 (2000) 3. Zhang, Y., Wang, J.: A dual neural network for convex quadratic programming subject to linear equality and inequality constraints. Physics Letters A 298, 271–278 (2002) 4. Xia, Y., Feng, G., Wang, J.: A recurrent neural network with exponential convergence for solving convex quadratic program and related linear piecewise equations. Neural Networks 17, 1003–1015 (2004) 5. Liu, S., Wang, J.: A simplified dual neural network for quadratic programming with its KWTA application. IEEE Trans. Neural Netw. 17, 1500–1510 (2006) 6. Hu, X., Wang, J.: Solving pseudomonotone variational inequalities and pseudoconvex optimization problems using the projection neural network. IEEE Trans. Neural Netw. 17, 1487– 1499 (2006) 7. Liu, Q., Wang, J.: A one-layer recurrent neural network with a discontinuous hard-limiting activation function for quadratic programming. IEEE Trans. Neural Netw. 19, 558–570 (2008) 8. Tao, Q., Fang, T.: The neural network model for solving minimax problems with constraints. Control Theory Applicat. 17, 82–84 (2000) 9. Gao, X., Liao, L., Xue, W.: A neural network for a class of convex quadratic minimax problems with constraints. IEEE Trans. Neural Netw. 15, 622–628 (2004) 10. Gao, X., Liao, L.: A novel neural network for a class of convex quadratic minimax problems. Neural Computation 18, 1818–1846 (2006) 11. Bazaraa, M., Sherali, H., Shetty, C.: Nonlinear programming: theory and algorithms (1993) 12. Perez-Ilzarbe, M.: Convergence analysis of a discrete-time recurrent neural network toperform quadratic real optimization with bound constraints. IEEE Trans. Neural Netw. 9, 1344– 1351 (1998) 13. Tan, K., Tang, H., Yi, Z.: Global exponential stability of discrete-time neural networks for constrained quadratic optimization. Neurocomputing 56, 399–406 (2004) 14. Alamo, T., Ramırez, D., Camacho, E.: Efficient implementation of constrained min–max model predictive control with bounded uncertainties: a vertex rejection approach. Journal of Process Control 15, 149–158 (2005)
Abstract. Robust model predictive control (MPC) has been investigated widely in the literature. However, for industrial applications, current robust MPC methods are too complex to employ. In this paper, a discrete-time recurrent neural network model is presented to solve the minimax optimization problem involved in robust MPC. The neural network has global exponential convergence property and can be easily implemented using simple hardware. A numerical example is provided to illustrate the effectiveness and efficiency of the proposed approach. Keywords: Robust model predictive control, recurrent neural network, minimax optimization.
1 Introduction Model predictive control (MPC) is a powerful technique for optimizing the performance of control systems, with several advantages over other control systems [1]. MPC applies on-line optimization to the model of a system, by taking the current state as an initial state, a optimization problem is solved at each sample time, and at the next computation time interval, the calculation repeated with a new state. MPC that take consideration of uncertainties in the process model is called robust MPC. One way to deal with uncertainties in MPC is the worst case approach, which obtains a sequence of feedback control laws that minimizes the worst case cost. In industrial processes, it required the real-time solution to a minimax optimization problem. Although the robustness of MPC has been studied and is now well understood, the research outcomes are conceptual controllers that can work in principle but not suitable for hardware implementation [2]. As a result, further investigations on a more implementable controller are needed. One very promising approach to dynamic optimization is to apply recurrent neural networks. Recurrent neural networks are brain-like computational models for solving optimization problems in real time. Compared with traditional numerical methods for constrained optimization, neural networks have several advantages: first, they can solve many optimization problems with time-varying parameters; second, they can handle large-scale problems with their parallelizable ability; third, they can be implemented effectively using VLSI or optical technologies. Neural networks for optimization and their engineering applications have been widely investigated in the past two decades. Many neural network models have been proposed F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 883–892, 2008. c Springer-Verlag Berlin Heidelberg 2008
884
Y. Pan and J. Wang
for solving both linear and nonlinear programming problems. In this paper, we present a discrete-time recurrent neural network model for solving the quadratic minimax optimization problem associated with robust MPC. The neural network is globally exponentially convergent to the saddle point of the objective function. The rest of this paper is organized as follows. In Section 2, we formulate robust MPC as a quadratic minimax optimization problem. In Section 3, we proposed a recurrent neural network model for minimax optimization, and prove its global exponential convergence property, a control scheme for robust MPC is also presented. In Section 4, we provide a example in industrial application to illustrate the performance of the proposed approach. Furthermore, a comparision is made between the proposed approach and linear matrix inequalities approach. Finally, Section 5 conclude this paper.
2 Problem Formulation 2.1 Process Model Consider the following discrete-time linear system with global bounded uncertainties: x(k + 1) = Ax(k) + Bu(k), y(k) = Cx(k) + Dw(k),
(1)
with the constraints umin ≤u(k) ≤ umax , Δumin ≤Δu(k) ≤ Δumax , wmin ≤w(k) ≤ wmax ,
(2)
ymin ≤y(k) ≤ ymax , where k ≥ 0, x(k) ∈ n is the state vector, u(k) ∈ m is the input vector, and y(k) ∈ p is the output vector. w(k) ∈ q denotes the vector of bounded uncertainties. umin ≤ umax , wmin ≤ wmax , ymin ≤ ymax are vectors of upper and lower bounds. 2.2 Robust MPC Design MPC is a step-by-step optimization technique: at each sampling time k, measure of estimate the current state, obtain the optimal input vector by solving a optimization problem. When bounded uncertainties are considered explicitly, a robust MPC law can be derived by minimizing the maximum cost within the model described by the uncertainty set. The optimal control action is obtained by solving a minimax optimization problem: (3) min max J(Δu, w), Δu
subjected to the constraints in (2).
w
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The objective function J(Δu, w) can be with an infinite or finite, linear or quadratic norm criterion. In this paper, we consider an objective function with a finite horizon quadratic criterion: J(Δu, w) =
N
[r(k + j|k) − y(k + j|k)]T Φ[r(k + j|k) − y(k + j|k)]+
j=1
(4)
N u −1
T
[Δu(k + j|k)] Ψ [Δu(k + j|k)]
j=0
where k is the current time step, y(k + j|k) denotes the predicted output, r(k + j|k) denotes the reference trajectory of output signal (desired output), and Δu(k + j|k) denotes the input increment, where Δu(k + j|k) = u(k + j|k) − u(k − 1 + j|k). Φ ∈ p×p , Ψ ∈ m×m are appropriate weighting matrices. N denotes the predictive horizon (1 ≤ N ). Nu denotes the control horizon (0 < Nu ≤ N ). After Nu control moves, Δu(k + j|k) becomes zero. According to the process model (1): y(k + j) = CAj x(k) + C
j−1
Ai Bu(k + j − i − 1) + Dw(k + j),
j = 1, ..., N (5)
i=0
Define following vectors: ···
y¯(k) = [y(k + 1|k)
u(k + Nu − 1|k)]T ∈ Nu m ,
···
u ¯(k) = [u(k|k)
Δu(k + Nu − 1|k)]T ∈ Nu m ,
···
Δ¯ u(k) = [Δu(k|k)
y(k + N |k)]T ∈ N p ,
r(k + N |k)]T ∈ N p ,
···
r¯(k) = [r(k + 1|k)
(6)
where the reference trajectory r¯(k) is known in advance. The predicted output y¯(k) is expressed in the following form: y¯(k) = Sx(k) + M u ¯(k) + Ew(k) = Sx(k) + M Δ¯ u(k) + V u(k − 1) + Ew(k), where S = [CA ⎡
CA2
E = [D
D
···
CAN ]T ∈ N p×n ,
···
D]T ∈ N p×q , ⎤
CB ⎢ ⎥ C(A + I)B ⎢ ⎥ ⎢ ⎥ .. ⎢ ⎥ . ⎢ ⎥ N p×m Nu −1 ⎢ + · · · + A + I)B ⎥ , V = ⎢ C(A ⎥∈ ⎢ C(ANu + · · · + A + I)B ⎥ ⎢ ⎥ ⎢ ⎥ .. ⎣ ⎦ . C(AN −1 + · · · + A + I)B
(7)
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⎡
CB C(A + I)B .. .
⎢ ⎢ ⎢ ⎢ ⎢ Nu −1 M =⎢ ⎢ C(A N + · · · I)B ⎢ C(A u + · · · I)B ⎢ ⎢ .. ⎣ . C(AN −1 + · · · I)B
··· ··· .. .
0 0 .. .
··· ··· .. .
CB C(A + I)B .. .
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ∈ N p×Nu m , ⎥ ⎥ ⎥ ⎥ ⎦
· · · C(AN −Nu + · · · I)B
I denotes the identity matrix. Define vectors: Δ¯ umin = [Δumin · · · Δumin ]T ∈ Nu m , Δ¯ umax = [Δumax · · · Δumax ]T ∈ Nu m u ¯min = [umin · · · umin]T ∈ Nu m , u ¯max = [umax · · · umax ]T ∈ Nu m , y¯min = [ymin · · · ymin ]T ∈ Nu p , y¯max = [ymax · · · ymax ]T ∈ Nu p , ⎡ ⎤ I 0 ··· 0 ⎢I I ··· 0⎥ ⎢ ⎥ I˜ = ⎢ . . . . ⎥ ∈ Nu m×Nu m . ⎣ .. .. . . .. ⎦ II I I Thus, the original minimax optimization can be expressed in the following form: min max Δu
w
s.t.
[¯ r(k) − Sx(k) − M Δ¯ u(k) − V u(k − 1) − Ew(k)]T Φ[¯ r (k) − Sx(k) − M Δ¯ u(k) − V u(k − 1) − Ew(k)] + Δ¯ uT (k)Ψ Δ¯ u(k) ˜ u(k) ≤ u u ¯min ≤ u ¯(k) + IΔ¯ ¯max Δ¯ umin ≤ Δ¯ u(k) ≤ Δ¯ umax w ¯min ≤ w(k) ¯ ≤w ¯max y¯min ≤ y¯(k) + M (k)Δ¯ u(k) ≤ y¯max (8)
By defining the variable vectors u = Δ¯ u(k) ∈ Nu m , w = w(k) ∈ q . By neglecting the constraints on u(k) and y(k), the problem (8) can be rewritten as a minimax quadratic programming problem: min max u
w
s.t.
1 T 1 u Qu + cT u − uT Hw − wT Rw − bT w 2 2 u ∈ U, w ∈ W
(9)
where U and W are two box set defined as U = {u ∈ Nu m |Δ¯ umin ≤ u ≤ Δ¯ umax }, ¯min ≤ w ≤ w ¯max }. The coefficient matrices and vectors are W = {w ∈ q |w Q = 2(M T ΦM +Ψ ) ∈ Nu m×Nu m , c = −2M T Φ(¯ r (k)−Sx(k)−V u(k−1)) ∈ Nu m , R = 2E T ΦE ∈ q×q , b = Φ(¯ r (k) − Sx(k) − V u(k − 1)) ∈ q , H = 2M T ΦE ∈ Nu m×q .
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The solution to the minimax quadratic programming problem (9) gives the vector of control action Δ¯ u(k). The control law is given by u ¯(k) = f (Δ¯ u(k) + u ¯(k − 1)), where f (·) is defined as εi , (Gε)i ≤ li , (10) f (εi ) = li , (Gε)i > li . and G and l are defined as G = [−I˜ I˜
−M
M ]T ∈ (2Nu m+2N p)×Nu m ,
⎡
⎤ −¯ umin + u¯(k) ⎢ u ¯max − u¯(k) ⎥ ⎥ ∈ 2Nu m+2N p . l=⎢ ⎣ −¯ ymin + y¯(k) ⎦ y¯max − y¯(k) The first element u(k|k) is used as the control signal. In industrial control processes, to solve large-scale minimax optimization problems in real-time is a major obstacle for robust MPC. In the next section, we will propose a recurrent neural network for solving (9).
3 Recurrent Neural Network Approach 3.1 Neural Network Model In recent years, many neural network models have been proposed for solving optimization problems [3,4,5,6,7]. In particular, continuous-time neural networks for solving minimax problems has been investigated in [8,9,10]. However, in view of the availability of the digital hardware and the compatibility to the digital computers, discrete-time neural network is more desirable in practical implementation. In this section, we proposed a discrete-time recurrent neural network for minimax problem (9). By the saddle point condition [11], (9) can be formulated as a linear variational inequality (LVI): (s − s∗ )T (M s∗ + q) ≥ 0, where
M=
Q −H , HT R
q=
c , b
∀s ∈ Ω,
(11)
Ω = U × W.
(12)
According to the well-known saddle point theorem [11], s∗ = (u∗ , w∗ ) is a saddle point of J(u, w) if satisfying J(u∗ , w) ≤ J(u∗ , w∗ ) ≤ J(u, w∗ ),
∀(u, w) ∈ Ω.
(13)
We define the saddle point set Ω ∗ = {(u∗ , w∗ ) ∈ Ω|(u∗ , w∗ ) satisfy (13)} and assume Ω ∗ is not empty. It is obvious that if (u∗ , w∗ ) ∈ Ω ∗ , then (u∗ , w∗ ) is the optimal solution to the minimax problem (9).
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According to inequalities (13), we can get that v ∗ is a global minimizer of the objective function J(v, w∗ ) with respect to U, while w∗ is the global minimizer of J(v ∗ , w) with respect to W. As a result, the following LVIs hold: (u − u∗ )T (Qu∗ + c − Hw∗ ) ≥ 0, (w − w∗ )T (Rw∗ + b + H T u∗ ) ≥ 0,
∀u ∈ U,
(14)
∀w ∈ W.
(15)
According to the basic property of the projection mapping on a closed convex set: [z − PΩ (z)]T [PΩ (z) − v] ≥ 0,
∀z ∈ , v ∈ Ω.
(16)
Based on (14)-(16) and lemma 1 in [9], we can get that (u∗ , w∗ ) ∈ Ω ∗ if and only if the following equations hold: u∗ = PU [u∗ − α(Qu∗ + c − Hw∗ )]
(17)
w∗ = PW [w∗ − α(Rw∗ + b + H T u∗ )]
(18)
where α > 0 is a scaling constant, PU (·) and PW (·) are piecewise activation functions defined as: ⎧ ⎧ ⎨ Δumin , εi < Δumin ; ⎨ wmin , εi < wmin ; wmin ≤ εi ≤ wmax ; PU (εi ) = εi , Δumin ≤ εi ≤ Δumax ; PW (εi ) = εi , ⎩ ⎩ Δumax , εi > Δumax . wmax , εi > wmax . (19) Based on the equations (17) and (18), we propose a recurrent neural network for solving (9) as follow: u(t + 1) = PU [u(t) − α(Qu(t) + c − Hw(t))] (20) w(t + 1) = PW [w(t) − α(Rw(t) + b + H T u(t))] The proposed recurrent neural network has a simple structure, and can be easily implemented using digital hardware. In the next section, we will prove that the proposed neural network has global exponential convergence property under some mild conditions. 3.2 Convergence Analysis Definition 1. Neural network (20) is said to be globally exponentially convergent to the equilibrium point (ue , we ) if both ue and we satisfy u(t) − ue ≤ c0 u(0) − ue e−ηt , w(t) − w ≤ b0 w(0) − w e e
e
−ηt
∀t ≥ 1; ,
∀t ≥ 1;
(21)
where η is a positive constant independent of the initial point, c0 and b0 are positive constant dependent on the initial point.
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Lemma 1. The neural network (20) has a unique equilibrium point, which is the saddle point of J(u, w). Proof. Similar to the proof in [12], we can establish that the neural network (20) has a unique equilibrium point (ue , we ). Define a equilibrium point set Ω e = {(ue , we ) ∈ Ω|(ue , we ) satisfy (17)and(18))}. According to the above derivation, it is obvious that the equations (17) and (18) is equivalent to (13) for all (u, w) ∈ Ω, from the definition of Ω ∗ , we can get that Ω e = Ω ∗ , which means the equilibrium point of (20) is the saddle point of J(u, w). Lemma 2. For all z ∈ n , PU (v) − PU (z)2 ≤ v − z2 ,
PW (v) − PW (z)2 ≤ v − z2 .
Proof. From the inequality (16) we can easily prove that PU (v) − PU (z)2 ≤ (v − z)T [PU (v) − PU (z)] ≤ v − z2 , PW (v) − PW (z)2 ≤ (v − z)T [PW (v) − PW (z)] ≤ v − z2 ,
∀v, z ∈ n . (22)
R Define λQ i > 0(i = 1, ..., Nu m), λj > 0(j = 1, ..., N q) as the eigenvalues of Q, R Q R R respectively, let λQ min , λmax , λmin , λmax be the smallest and largest eigenvalues of Q and R. Define two functions
Q
ψ (α) = ψ R (α) =
1 − λQ min α,
Q 0 < α ≤ 2/(λQ min + λmax )
λQ max α − 1,
Q 2/(λQ min + λmax ) ≤ α < +∞
1 − λR min α,
R 0 < α ≤ 2/(λR min + λmax )
λR max α − 1,
R 2/(λR min + λmax ) ≤ α < +∞
(23)
(24)
Then we give the following lemma: Lemma 3. ψ Q (α) < 1 and ψ R (α) < 1
if and only if
R 0 < α < min{2/λQ max , 2/λmax }.
(25)
Proof. From the Theorem 2 in [13], we can get that ψ Q (α) < 1 if and only if α ∈ R R (0, 2/λQ max ), similarly, ψ (α) < 1 if and only if α ∈ (0, 2/λmax ). We can easily verify Q that the sufficient and necessary condition for both ψ (α) < 1 and ψ R (α) < 1 is R 0 < α < min{2/λQ max , 2/λmax }. Theorem 1. With any α that satisfies (25), the neural network (20) is globally exponentially convergent to the saddle point of J(u, w). 2 Proof. From (23) and (24), we can obtain that ψ Q (α) = max{(1 − αλQ 1 ) , ..., (1 − Q 2 R R 2 R 2 αλNu m ) }, ψ (α) = max{(1 − αλ1 ) , ..., (1 − αλN q ) }.
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By Lemma 2: u(k) − u∗ 2 =PU [u(t − 1) − α(Qu(t − 1) + c − Hw(t − 1))]− PU [u∗ − α(Qu∗ + c − Hw∗ )]2 ≤(I − αQ)(u(t − 1) − u∗ )2 Q 2 2 ∗ 2 ≤ max{(1 − αλQ 1 ) , ..., (1 − αλNu m ) }u(t − 1) − u
=ψ Q (α)2 u(t − 1) − u∗ 2
(26)
=⇒ u(t) − u∗ ≤ ψ Q (α)u(t − 1) − u∗ ≤ ψ Q (α)t u(0) − u∗ ≤ e−η
Q
(α)t
u(0) − u∗
Similarly, w(t)−w∗ ≤ e−η (α)t w(0)−w∗ . From Lemma 3, η Q (α) > 0 (ψ Q (α) < 1) and η R (α) > 0 (ψ R (α) < 1) for all α that satisfy (25). From the above proof and lemma 1, we can obtain that for any α that satisfies (25), the neural network (20) is globally exponentially convergent to the unique equilibrium point (u∗ , w∗ ), which is the saddle point of J(u, w). R
3.3 Control Scheme The control scheme based on proposed recurrent neural network can be summarized as follows: 1. Let k = 1. Set terminal time T , sample time t, predictive horizon N , control horizon Nu , weighting matrices Φ and Ψ . 2. Calculate process model matrices S, E, V , M , neural network parameters Q, R, H, c, b. 3. Solve the quadratic minimax problems (9) using the proposed recurrent neural network, obtaining the optimal control action Δ¯ u(k). 4. Calculate the optimal input vector u¯(k) = f (Δ¯ u(k) + u¯(k − 1)), the first element u(k|k) is sent to the process. 5. If k < T , set k = k + 1, return to step 2; otherwise, end.
4 Numerical Example Consider a two-tank system described in [14], which is a two-input, two-output system, with the flow rates of the two inlet streams as the two inputs, and the liquid level in each tank as the two output variables. By sampling at 0.2 min using a zero-order holder, the following discrete-time statespace model can be obtained:
x(k + 1) =
− 0.5 3 0.5 2
0.2 3 − 0.5 2
10 y(k) = x(k) 01
1 x(k) +
3
0
0 1 2
u(k) (27)
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The set-point for the liquid levels (output) of tanks 1 and 2 are 0.8 and 0.7, respectively; the prediction and control horizons are N = 10 and N u = 4; weighting matrices Φ = I, Ψ = 5I; scaling constant α = 0.2; an uncertainty −0.02 ≤ w ≤ 0.02 is considered to affect both liquid levels of tanks 1 and 2; moreover, the following constraints are considered:
0 0.5 0 0.6 ≤u(k) ≤ ≤ y(k) ≤ 0 0.5 0 0.7
(28) −0.05 0.05 ≤ Δu(k) ≤ −0.05 0.05 Input u1
Input u2
0.5
0.4 RNN LMI
0.45
RNN LMI 0.35
0.4 0.3 0.35 0.25
0.3 0.25
0.2
0.2
0.15
0.15 0.1 0.1 0.05
0.05 0
0
10
20
30
40
50 60 Samples k
70
80
90
100
0
0
10
20
30
40
50 60 Samples k
70
80
90
100
Fig. 1. Input signals of tanks 1 and 2 using the proposed RNN approach and LMI approach
Output y1
Output y2
0.9
0.8
0.8
0.7
0.7
0.6
0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2
0.2 0.1 0
0.1
RNN LMI 0
10
20
30
40
50 60 Samples k
70
80
90
100
0
RNN LMI 0
10
20
30
40
50 60 Samples k
70
80
90
100
Fig. 2. Output responses of tanks 1 and 2 using the proposed RNN approach and LMI approach
In order to compare the effectiveness and efficiency of the proposed approach, a linear matrix inequalities (LMI) approach [1] is also applied to the process. The simulation results are showed in Figs. 1 - 2. We can see that the proposed neural network approach gives a better set-point tracking performance with faster stable output responses.
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5 Conclusion This paper presents a new approach to robust MPC based on a discrete-time recurrent neural network by solving a minimax optimization problem. The neural network is proved to have global exponential convergent property. Simulation results show the superior performance of the neural network approach. Compared with a linear matrix inequalities approach, the proposed neural network approach gives a better performance in set-point tracking.
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