ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics Meng Zhao, Baocang Ding n College of Automation, Chongqing University, Chongqing 400044, China

art ic l e i nf o

a b s t r a c t

Article history: Received 14 February 2014 Received in revised form 17 June 2014 Accepted 15 July 2014 This paper was recommended for publication by Dr. Rickey Dubay

This paper considers the distributed model predictive control (MPC) of nonlinear large-scale systems with dynamically decoupled subsystems. According to the coupled state in the overall cost function of centralized MPC, the neighbors are confirmed and fixed for each subsystem, and the overall objective function is disassembled into each local optimization. In order to guarantee the closed-loop stability of distributed MPC algorithm, the overall compatibility constraint for centralized MPC algorithm is decomposed into each local controller. The communication between each subsystem and its neighbors is relatively low, only the current states before optimization and the optimized input variables after optimization are being transferred. For each local controller, the quasi-infinite horizon MPC algorithm is adopted, and the global closed-loop system is proven to be exponentially stable. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Distributed control Model predictive control (MPC) Compatibility condition Exponentially stability

1. Introduction Model predictive control (MPC) has received much attention in recent years due to its capability to control the constrained systems [1,2]. The MPC algorithms are optimization-based control strategies and generally in centralized form [3,4]. For centralized MPC, the computation requirement [5] is the main barrier to enlarge the scopes of applications, especially for nonlinear largescale systems. A viable way to reduce the computation time is by utilizing the distributed MPC. Compared to the centralized MPC, the optimization in distributed MPC is totally decentralized into a number of small-scale optimizations. Since the communication bandwidth is often limited, each subsystem cannot get the full information of other subsystems when solving its optimization. As a result, the control performance may be worse than centralized MPC, but the advantages outweigh its disadvantages. Up to now, there are many works on distributed MPC [6–8]. In [9], an overall introduction to all existing important works on distributed MPC is given. In [8], the distributed MPC algorithms are divided into 2 kinds: the cooperative distributed MPC [10–13] and noncooperative distributed MPC [14]. In a cooperative distributed MPC, each local controller optimizes a global objective function,

n

Corresponding author. Tel.: þ 86 23 6534 1593. E-mail addresses: [email protected] (M. Zhao), [email protected] (B. Ding).

while in a non-cooperative MPC, a local one. In this paper, we consider distributed MPC of a set of dynamically decoupled subsystems with a separable common objective function. The common objective function, in which the states and inputs of subsystems are coupled, is separated into each local controller. In order to guarantee stability, the overall compatibility constraints are also disassembled. Since the common objective function is totally divided, the proposed distributed MPC is the noncooperative one. Distributed MPC has been widely applied to control systems, such as multi-agent systems [35], four-tank systems [15,16], building temperature regulation systems [17], and supply chain management systems [18]. Most of these systems are physically distributed and can be divided into 2 categories: system with coupled dynamic [15,16,19] and that with decoupled one [35,17,20,21]. In this paper, the latter system is considered. Since the subsystems are spatially distributed in most cases, the communication between them is required to share the information. Nevertheless, the communication burden will be huge if the amount of subsystems is large and all the local information is required to exchange. In [22], a method which requires no communication is proposed, and the algorithm utilized is called the decentralized MPC. In [23], the information of each subsystem is transmitted iteratively between samplings. However, in [32,34–36], the information is transmitted in a non-iterative way. In [24], the communication delay in the

http://dx.doi.org/10.1016/j.isatra.2014.07.012 0019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

information network is considered. In [25], the data losses is considered. In the present paper, the communication between each subsystem and its neighbors is required. By comparison with procedures for finding neighbors in [35,36], each subsystem in our proposed distributed MPC has less neighbors because the relations are non-mutual. In order to guarantee closed-loop stability, a special form of compatibility constraint [34,32] is adopted in this paper, which is a sufficient condition for stability. Unlike the compatibility constraint utilized in [35,36,19], the proposed compatibility constraint makes the controlled system exponentially stable. Other techniques include the Lyapunov-based approach [26–28] and the game approach [29,30]. The main contribution of this paper is that the distributed MPC for nonlinear systems with decoupled dynamics is considered. The local controller in each subsystem is designed by using the quasi-infinite horizon MPC algorithm. The proposed distributed MPC decomposes a centralized objective function into each local controller. In order to guarantee the overall stability, the compatibility constraint is deduced and disassembled. Moreover, new procedures for finding neighbors are presented. Unlike any other methods, the proposed method is available not only for the isomorphic subsystems, but also for the heterogeneous subsystems. Previous results of this paper are given in [34] for linear nominal systems and in [32] for uncertain linear systems. This paper is organized as follows. Section 2 describes the nonlinear system with dynamically decoupled subsystems, and the corresponding centralized MPC. Section 3 introduces the naive distributed MPC, and the compatibility condition for stability. In Section 4, by decoupling the compatibility condition, the synthesis approach of distributed MPC is presented. In Section 5, a multivehicle formation control example and the corresponding simulation results are given. In the Appendix, new procedures for finding neighbors are proposed. Notations: I is the identity matrix with appropriate dimension. In is the identity matrix of n-th order. For a vector x and positivedefinite matrix W, J x J 2W ¼ xT Wx. xðjjkÞ is the value of vector x at a future time k þ j predicted at time k. For any integer Na, Na ≔f1; 2; …; N a g. For any integer N 4 0, N≔f0; 1; …; N  1g, N1 ≔f0; 1; …; N 2g and N2 ≔f1; 2; …; N  1g. A variable with n as superscript indicates that it is the optimal solution of the optimization problem. The time-dependence of the MPC decision variables is often omitted for simplicity.

Define the local linearization of (1) at origin

Ai ¼

∂f i ð0; 0Þ; ∂xi

Let us consider the following Na physically distributed local systems:

∂f i ð0; 0Þ ∂ui

ð3Þ

Assumption 3. ðAi ; Bi Þ is stabilizable. At each time k, the control objective is to minimize i 1 h JðkÞ ¼ ∑ J xðjjkÞ J 2Q þ J uðjjkÞ J 2R

ð4Þ

j¼0

h iT with respect to uðjjkÞ, j Z0, where x ¼ xT1 ; xT2 ; …; xTNa , h iT u ¼ uT1 ; uT2 ; …; uTNa ; xi ðj þ 1jkÞ ¼ f i ðxi ðjjkÞ; ui ðjjkÞÞ, xi ð0jkÞ ¼ xi ðkÞ; Q ; R 4 0 are symmetric weighting matrices. The input and state constraints, under Assumption 2, are supposed to have the following form in the minimization: X i ≔fxi j  ψ r Ψ i xi ðj þ 1jkÞ r ψ i g; i

U i ≔fui j  u i r ui ðjjkÞ r u i g;

jZ0

ð5Þ

where i A Na , Ψ i A Rqi ni , qi is the number of rows in matrix Ψi, h iT h iT ψ i ≔ ψ i1 ; ψ i2 ; …; ψ iq and ψ i ≔ ψ i1 ; ψ i2 ; …; ψ iqi with ψ i ; ψ il 4 0, l i h iT h iT   i i i i i i and u i ≔ u 1 ; u 2 ; …; u mi with l A 1; …; qi , u i ≔ u 1 ; u 2 ; …; u m i

u il ; u il 4 0, l A f1; …; mi g . In order to implement the control in a distributed manner, J(k) is divided as Ji(k)'s with 1

J i ðkÞ ¼ ∑

j¼0

h

i J zi ðjjkÞ J 2Q þ J vi ðjjkÞ J 2R ; i

i

Na

JðkÞ ¼ ∑ J i ðkÞ;

ð6Þ

i¼1

 T  T where zi ¼ xTi ; xT i , vi ¼ uTi ; uT i ; x  i ðu  i Þ includes the states (inputs) of the neighbors of i; Q i Z 0 and R i Z0 are symmetric weighting matrices, 2 i 3 2 i 3 i i Q 1 Q 12 R 1 R 12 5; R i ¼ 4 5: ð7Þ Q i ¼ 4 iT i iT i R 12 R 3 Q 12 Q 3 In this paper, the neighbors of i are defined as those local systems that are related to i via Ji(k). Denote  T n o x  i ¼ xTνi ; xTνi ; …; xTνi ; νi1 ¼ i; νi1 ; νi2 ; …; νiN i DNa 2

2. Problem statement

Bi ¼

3

Ni

where N i  1 Z 0 is the number of neighbors of local system i, and νij ; j A f2; 3; …; N i g is the sequence number of the (j  1)-th neighbor of i. For each local system, the control objective is to minimize Ji(k) satisfying its input and state constraints. Remark 1. Given Q, R, a procedure is proposed in [35] (see its Section 4.3) for finding Q i , R i . With the procedure in [35] applied,

xi ðk þ 1Þ ¼ f i ðxi ðkÞ; ui ðkÞÞ;

i A Na

ð1Þ

where xi A Rni and ui A Rmi are measurable state and input, respectively. The state and control are confined by xi A X i D Rni ;

xi A U i DRmi

ð2Þ

a JðkÞ ¼ ∑N i ¼ 1 J i ðkÞ. A procedure different from that in [35] is given in a Appendices A and B. JðkÞ ¼ ∑N i ¼ 1 J i ðkÞ if and only if J xðÞ J Q ¼

Na a ∑N i ¼ 1 J zi ðÞ J Q and J uðÞ J R ¼ ∑i ¼ 1 J vi ðÞ J R . If dimfxi g adimfzi g or i

i

dimfui g a dimfvi g, then i has neighbors. The neighbors of i include any local system whose state is a part of zi, or whose input is a part of vi. Since the neighbors of i are defined through Ji(k), which is equivalent to through Q i and R i , finding the neighbors of i clearly depends on Q, R.

Assumption 1. The functions fi's are twice continuously differentiable, with f i ð0; 0Þ ¼ 0.

3. Stability of the naive distributed MPC

Assumption 2. X i is a closed set and U i a compact, convex set, both of them containing the origin as interior point.

We will discuss the stability of distributed MPC by simply applying, in each local controller, the procedure as in the centralized MPC.

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Step 1: Solve a control problem based on the local linearization to get a locally stabilizing state feedback gain Fi. Step 2: Choose a constant ζ i A ½0; 1Þ satisfying ζ i o 1  ρðAi þ Bi F i Þ2 , where ρðÞ defines the spectral radius. Step 3: Solve the Lyapunov equation (13) to get a positivedefinite Pi. Step 4: Find the largest possible α0i such that  u i rF i xi ru i and  ψ r Ψ i f i ðxi ; F i xi Þ r ψ i , for all xTi P i xi r α0i . i Step 5: Find the largest possible αi A ð0; α0i  such that θi ðxi ÞT P i θi ðxi Þ þ 2θi ðxi ÞT P i ðAi þ Bi F i Þxi r ζ i xTi P i xi , for all xi A Ωi , where θi ðxi Þ ¼ f i ðxi ; F i xi Þ  ðAi þBi F i Þxi .

3.1. Formulation of the naive distributed MPC Each Ji(k) is infinite-dimensional and couplings exist between different Ji(k) and, for the neighbors, apply the assumed control trajectories as in [35,36]. The MPC with N free control moves (N is named as switching horizon) will be applied. Consider i i N 1 h 1 h J
i ðkÞ ¼ ∑ J z
i ðjjkÞ J 2Q þ J v
i ðjjkÞ J 2R þ ∑ J xi ðjjkÞ J 2Q i þ J ui ðjjkÞ J 2Ri i

i

j¼0

3

j¼N

ð8Þ where h iT z
i ðjjkÞ ¼ xi ðjjkÞT ; x^  i ðjjkÞT ; x^ t ð0jkÞ ¼ xt ðkÞ;

x^ t ðj þ 1jkÞ ¼ f t ðx^ t ðjjkÞ; u^ t ðjjkÞÞ;

t A fνi2 ; νi3 ; …; νiN i g;

h iT v
i ðjjkÞ ¼ ui ðjjkÞT ; u^  i ðjjkÞT ;

ð9Þ

u^  i ðjj0Þ ¼ 0;

ð10Þ

u^ is the assumed value of u; x^ is the assumed value of x dependent ^ Q i 40 and Ri 4 0 are symmetric weighting matrices, satisfyon u; ing diagfQ 1 ; Q 2 ; …; Q Na g ZQ ;

diagfR1 ; R2 ; …; RNa g Z R:

ð11Þ

Define ui ðjjkÞ ¼ F i xi ðjjkÞ;

8 j ZN;

ð12Þ

where Fi is determined such that Ai þBi F i is Schur stable. For each i A Na , appropriate ζi and Pi should be chosen such that ðAi þ Bi F i ÞT P i ðAi þ Bi F i Þ P i ¼  ζ i P i  Q i  F Ti Ri F i :

ð13Þ

The following lemma is directly obtained by referring to “Lemma 1” in [39]. Lemma 1. Let Assumptions 1 and 2 hold, and P i 4 0 satisfy the Lyapunov equation (13) for some   ζ i 4 0. Suppose (12) is applied. Denote Ωi ¼ xi A Rni jxTi P i xi r αi . Then, there exists a constant αi 4 0 such that, for all xi ðNjkÞ A Ωi , (i) the constraints in (5) are satisfied for all j Z N, and xi ðjþ 1jkÞ ¼ f i ðxi ðjjkÞ; F i xi ðjjkÞÞ, j ZN is asymptotically stable; (ii) J xi ðj þ 1jkÞ J 2P i  J xi ðjjkÞ J 2P i r  J xi ðjjkÞ J 2Q i  J ui ðjjkÞ J 2Ri for all j ZN; (iii) the cost function (8) satisfies i N1 h J
i ðkÞ rJ i ðkÞ ¼ ∑ J z
i ðjjkÞ J 2Q þ J v
i ðjjkÞ J 2R þ J xi ðNjkÞ J 2P i : ð14Þ i

j¼0

i

Proof. By adapting the proof of “Lemma 1” in [39] and letting t Z N; κ ¼ ζ i , and Wðxi Þ ¼ xTi P i xi , the claims (i) and (ii) can be deducted directly for each subsystem i. The main difference between Lemma 1 in this paper and “Lemma 1” in [39] is that [39] only considers the centralized MPC. Since the claim (ii) holds for j Z N, by summing the equation J xi ðjþ 1jkÞ J 2P i  J xi ðjjkÞ J 2P i r  J xi ðjjkÞ J 2Q i  J ui ðjjkÞ J 2Ri from j¼N to j ¼ 1, we have o 1 n J xi ð1jkÞ J 2P i  J xi ðNjkÞ J 2P i r  ∑ J xi ðjjkÞ J 2Q i þ J ui ðjjkÞ J 2Ri j¼N

"

Xi

Y Ti

Yi

Zi

"

# Z 0;

Z i;jj r u 2j;inf ;

j A f1; 2; …; mi g;

ðX i ATi þ Y Ti BTi ÞΨ i

T

Xi

Ψ i ðAi X i þ Bi Y i Þ

Γi

# Z 0;

Γ i;ss r ψ 2s;inf ; s A f1; 2; …; qi g;

where Z i;jj ðΓ i;ss Þ is the j-th (s-th) diagonal element of Z i ðΓ i Þ, u j;inf ¼ minfu j ; u j g, ψ s;inf ¼ minfψ ; ψ s g. Various criteria can be used s to find Xi, Γi, Zi, Yi, such as maxfX i ;Γ i ;Z i ;Y i g log detðX i Þ and maxfX i ;Γ i ;Z i ;Y i g trðX i Þ. As a result, let P i ¼ X i 1 and F i ¼ Y i X i 1 . The obtained J i ðkÞ is a finite-horizon cost function. At each time k, the distributed MPC considers the following optimization problem: min J i ðkÞ u~ i ðkÞ

s:t:

ð5Þ for jA N;

ð9Þ–ð10Þ and xi ðNjkÞ A Ωi :

ð15Þ

h iT n Of the optimal u~ i ðkÞ ¼ uni ð0jkÞT ; uni ð1jkÞT ; …; uni ðN 1jkÞT , only ui ðkÞ ¼ uni ð0jkÞ is implemented, and problem (15) is solved again at time k þ1. Correspondingly, the centralized MPC considers the following optimization problem: ~ uðkÞ

s:t:

ð5Þ for j A N; xðNjkÞ A Ω;

ð16Þ

h i 2 2 2 1 where J ðkÞ ¼ ∑N j ¼ 0 J xðjjkÞ J Q þ J uðjjkÞ J R þ J xðNjkÞ J P , Ω and P are

j¼N

A procedure for selecting Pi, follows:

Remark 3. If the subsystem i is linear, one can choose ζ i ¼ 0 and achieve the LQ optimal control around the equilibrium point. However, it would be better to use the procedure in [40], to determine Xi, Γi, Zi, Yi satisfying the following linear matrix inequalities (LMIs): 3 2 1=2 1=2 X i ATi þ Y Ti BTi X i Q i Y Ti Ri Xi 6 7 6 Ai X i þ Bi Y i Xi 0 0 7 6 7 6 7 Z 0; 1=2 0 I 0 7 6 Q i Xi 4 5 1=2 Ri Y i 0 0 I

min J ðkÞ

From claim (i), we have J xi ð1jkÞ J 2Pi ¼ 0, then o 1 n J xi ðNjkÞ J 2Pi Z ∑ J xi ðjjkÞ J 2Q i þ J ui ðjjkÞ J 2Ri Eq. (14) can be easily derived, and the claim (iii) is proven.

Remark 2. In Step 2, the principle of choosing ζi, which satisfies the ζ i o 1  ρðAi þ Bi F i Þ2 , is that the discrete generalized Lyapunov equation (13) must have positive solution Pi [38]. In the above procedures, the most crucial steps are to fix the value of Fi and Pi. In [39], Fi will be set as the LQ optimal gain matrix associated 2 2 with the performance cost ∑1 j ¼ 0 ½ J xi ðjjkÞ J Q i þ J ui ðjjkÞ J Ri . In this 0 case, one obtains a positive-definite P i satisfying ðAi þ Bi F i ÞT P 0i ðAi þ Bi F i Þ  P 0i ¼  Q i  F Ti Ri F i . The same conclusion can be drawn from [33]. By choosing any P i 4 P 0i , ζi exists. However, the procedure for finding Pi is not given in [39,33]. By adapting from [31] which is for continuous systems, Fi is not necessarily specified as the LQ optimal gain matrix.

□

ζi, αi can be adapted from [31], as

as defined in Lemma 1 removing the subscript i, and uðjjkÞ ¼ FxðjjkÞ h n for all j Z N. Of the optimal u~ ðkÞ ¼ un ð0jkÞT ; un ð1jkÞT ; …; un ðN  1jkÞT T , only uðkÞ ¼ un ð0jkÞ is implemented, and problem (16) is solved again at time k þ1.

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

By substituting a centralized MPC with a distributed one, loss of optimality may be incurred. The bounds for this loss of optimality are discussed in Appendix C.

n Substituting (7) and (17)–(19) into J~ ðk þ 1Þ  J~ ðkÞ yields n J~ ðk þ 1Þ  J~ ðkÞ

n h i h i o Na r  ∑ ð1  σ i ðkÞÞ J xi ðkÞT ; x  i ðkÞT J 2Q þ J uni ð0jkÞT ; u^  i ð0jkÞT J 2R

r ð1  σ ðkÞÞ J xðkÞ J 2Q

Define Na N  1 n

H nx ðkÞ ¼  ∑ ∑

i¼1j¼1

ð17Þ

Na N  1 n  i  H nu ðkÞ ¼  ∑ ∑ 2uni ðjjkÞT R 12 u^  i ðjjkÞ  un i ðjjkÞ i¼1j¼1

o i i þ u^  i ðjjkÞT R 3 u^  i ðjjkÞ  un i ðjjkÞT R 3 un i ðjjkÞ :

ð18Þ

The significance of H nx ðkÞ and H nu ðkÞ will be clear in the following. For centralized MPC, the stability is usually guaranteed by enforcing that the optimal cost decreases at each time k, by at least a stage cost (stage j¼0) value (see [39]). In distributed MPC, the stability becomes more involved. In the following, it is allowed that the optimal global performance cost can decrease, at each time k, by less than the stage cost of j¼0. Lemma 2. Suppose (11) holds and there exist σ i ðkÞ, i A Na such that, for all k 4 0, 0 r σ i ðkÞ o1; i A Na ; h i i o n h Na      ∑ σ i ðkÞ  xi ðkÞT ; x  i ðkÞT 2Q þ  uni ð0jkÞT ; u^  i ð0jkÞT 2R n

n

þ H x ðkÞ þ H u ðkÞ r 0

ð19Þ

Then, by solving the receding-horizon problem (15), the implementation of uni ðkÞ ¼ uni ð0jkÞ, iA Na guarantees exponential stability of the global closed-loop system, once a feasible solution at time k ¼0 is found. Proof. Compared with [34], the only difference for this proof is a that Pi's are not the decision variables. Define J~ ðkÞ ¼ ∑N i ¼ 1 J i ðkÞ. Suppose, at time k, there is optimal solution u~ ni ðkÞ yielding Na

i¼1

h h i2 i2

     xi ðkÞT ; x  i ðkÞT  þ  uni ð0jkÞT ; u^  i ð0jkÞT  Qi

Ri

Qi

Ri

    þ ∑ xni ðNjkÞ2P i Na

At time k þ1, u~ i ðk þ 1Þ ¼ funi ðjjkÞ; j A N2 ; F i xni ðNjkÞg is feasible which, according to Lemma 1, yields N

i¼1j¼1

h h i2 i2

 n     xi ðjjkÞT ; xn i ðjjkÞT  þ  uni ðjjkÞT ; un i ðjjkÞT  Qi

Ri

Na

h h i2 i2

    ¼ ∑ ∑  xni ðjjkÞT ; xn i ðjjkÞT  þ  uni ðjjkÞT ; un i ðjjkÞT  Na N  1

Qi

i¼1j¼1

Ri

i

) h i2 h i 2   r ∑ ∑  xni ðjjkÞT ; xn i ðjjkÞT  þ J uni ðjjkÞT ; un i ðjjkÞT   Na N  1

Na

i¼1

In the synthesis approach, it is appropriate to incorporate the compatibility condition (19) into the optimization problem. Since uni ðjjkÞ for all iA Na couple with each other through (19), it is necessary to assign constraints to the individual i's so as to satisfy (19) along the optimization. The continued discussion on stability depends on handling (19).

ð21Þ

1 2 Na where ε ¼ x^  xn . Choose symmetric matrices Q~ 3 ; Q~ 3 ; …; Q~ 3 such that n 1 2 o Na T diag Q~ 3 ; Q~ 3 ; …; Q~ 3 Z ðQ~ 3 þ Q~ 3 Þ=2: h i Denote Q~ 2 ¼ Q~ 2;1 Q~ 2;2 ⋯Q~ 2;Na conformal with the partition of x. Then, n

Qi

i¼1

n

N1 n

H x;i ðkÞ≔ ∑

j¼1

o i i i x^ i ðjjkÞT Q~ 20 εi ðjjkÞ þ z^ þ i ðjjkÞT Q~ 2 þ εi ðjjkÞ þ εi ðjjkÞT Q~ 3 εi ðjjkÞ ;

ð22Þ

i N a h 2 þ ∑ xni ðNjkÞQ þ J uni ðNjkÞ J 2Ri þ J xni ðN þ 1jkÞ J 2P i

þ ∑ J xni ðNjkÞ J 2P i

4. Synthesis approach of distributed MPC

H nx ðkÞ r ∑ H x;i ðkÞ;

i¼1

i¼1j¼1

Remark 4. For N ¼ 1, by solving the receding-horizon problem (15), the implementation of uni ðkÞ, i A Na guarantees exponential stability of the global closed-loop system, once a feasible solution at time k ¼0 is found.

Na

þ ∑ J xni ðN þ 1jkÞ J 2Pi

i¼1

Satisfaction of (19) indicates that all (at least some) xi ðjjkÞ's and ui ðjjkÞ's (j A N2 ) should not deviate too far from their assumed values (x^ i ðjjkÞ's and u^ i ðjjkÞ's, respectively). Hence, (19) can be named as compatibility condition. In the synthesis approach that follows, this compatibility condition will be disassembled to each local system, which leads to compatibility constraint. Our compatibility constraint is derived from a single compatibility condition that collects all the states and inputs (whether predicted or assumed) within the switching horizon, which is different from those of [35,36].

j¼1

i¼1

Na

where λmin ðQ Þ is the minimum eigenvalue of Q. Then, we can see n that J~ ðkÞ is the Lyapunov function for proving exponential stability. □

Define εi ¼ x^ i  xni . There exist Q~ 2 , Q~ 3 such that o N 1 n T ~ ^ Q 2 εðjjkÞ þ εðjjkÞT Q~ 3 εðjjkÞ ; H nx ðkÞ ¼ ∑ xðjjkÞ

h h i2 i2

    þ ∑ ∑  xni ðjjkÞT ; x^  i ðjjkÞT  þ  uni ðjjkÞT ; u^  i ðjjkÞT 

J~ ðk þ 1Þ ¼ ∑ ∑

n n J~ ðk þ1Þ  J~ ðkÞ r ð1  σ ðkÞÞ J xðkÞ J 2Q r ð1  σ ðkÞÞλmin ðQ Þ J xðkÞ J 2Q

4.1. Decoupling of the compatibility condition

Na N  1

i¼1j¼1

ð20Þ

i

i

i¼1

i

where σ ðkÞ ¼ maxi A Na σ i ðkÞ o 1. At time k þ1, by re-optimization, n J~ ðk þ1Þ r J~ ðk þ 1Þ. Hence, (20) leads to

 i  2xni ðjjkÞT Q 12 x^  i ðjjkÞ xn i ðjjkÞ

o i i þ x^  i ðjjkÞT Q 3 x^  i ðjjkÞ  xn i ðjjkÞT Q 3 xn i ðjjkÞ ;

n J~ ðkÞ ¼ ∑

i

i¼1

3.2. Compatibility condition for stability

Ri

where

2 3 h i Q~ i20 T T T x^ i z^ þ i 4 i 5 ¼ x^ Q~ 2;i Q~ 2 þ

ð23Þ

is satisfied and z^ þ i includes the assumed states of those local T systems that take i as their neighbor. It is easy to verify that x^ Q~ 2;i only relates to x^ i and z^ þ i .

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Define ϵi ¼ u^ i  uni . Similarly, there exist R~ 2 , R~ 3 such that o N1 n T~ ^ H nu ðkÞ ¼ ∑ uðjjkÞ R 2 ϵðjjkÞ þ ϵðjjkÞT R~ 3 ϵðjjkÞ ;

ð24Þ

j¼1

1 2 Na where ϵ ¼ u^  un . Choose symmetric matrices R~ 3 ; R~ 3 ; …; R~ 3 such that n 1 2 o Na T diag R~ 3 ; R~ 3 ; …; R~ 2 Z ðR~ 3 þ R~ 3 Þ=2: h i Denote R~ 2 ¼ R~ 2;1 R~ 2;2 ⋯R~ 2;Na conformal with the partition of u. Then,

sufficiently small neighborhood of x ¼ 0, when κ is fixed. However, the compatibility constraint (28) remains effective in the neighborhood of x ¼ 0. For more details, readers can refer to [34]. 4.3. Final synthesis approach Based on the compatibility constraint (28), for all k 4 0 the optimization problem (15) is finally approximated: min J i ðkÞ u~ i ðkÞ

Na

5

s:t:

ð5Þ for jA N;

ð9Þ; ð10Þ; ð28Þ and xi ðNjkÞ A Ωi :

n

H nu ðkÞ r ∑ H u;i ðkÞ; i¼1

N1 n

n

H u;i ðkÞ≔ ∑

j¼1

ð29Þ For k ¼0, it is not required to satisfy (19). Hence, for k ¼0 (15) should be solved instead of (29). Define

o i i i u^ i ðjjkÞT R~ 20 ϵi ðjjkÞ þ v^ þ i ðjjkÞT R~ 2 þ ϵi ðjjkÞ þ ϵi ðjjkÞT R~ 3 ϵi ðjjkÞ ;

ð25Þ where h

T u^ i

ð30Þ

For practical implementation, distributed MPC is formulated in the following algorithm:

2 3 i R~ i20 T T v^ þ i 4 i 5 ¼ u^ R~ 2;i R~

ð26Þ

2þ

is satisfied and v^ þ i includes the assumed controls of those local systems that take i as their neighbor. By substituting (22) and (25) into (19), it is shown that (19) is guaranteed by assigning 0 r σ i ðkÞ o 1; n h i h i o 2 2  σ i ðkÞ : xi ðkÞT ; x  i ðkÞT :Q þ J uni ð0jkÞT ; u^  i ð0jkÞT :R i i

n

S i ¼ fνi2 ; νi3 ; …; νiN i g [ fμi1 ; μi2 ; …; μiMi g:

n

þH x;i ðkÞ þ H u;i ðkÞ r 0

ð27Þ i Q~ 3

i R~ 3 ,

and conservativeness is for each iA Na . By selecting introduced and, hence, (27) is more stringent than (19). 4.2. Allocation of the compatibility constraints

Algorithm 1 (The synthesis approach of distributed MPC given in (1)–(5)). Off-line stage: (i) For all i A Na , find fQ i ; R i g according to (6), partition fQ i ; R i g as in (7), and find fQ i ; Ri g satisfying (11). (ii) For all i A Na , find Fi by solving the LQ optimal gain matrix associated with fQ i ; Ri g, find Pi according to (13) and find Ωi according to Lemma 1. (iii) Find fQ~ 2 ; Q~ 3 g such that (17) and (21) hold. Find fR~ 2 ; R~ 3 g such that (18) and (24) hold. (iv) Choosing symmetric matrices Q~ 1 ; Q~ 2 ; …; Q~ Na such that 3 3 3 n 1 2 o Na T diag Q~ 3 ; Q~ 3 ; …; Q~ 3 Z ðQ~ 3 þ Q~ 3 Þ=2. Choosing symmetric n 1 2 o 1 2 Na Na such that diag R~ 3 ; R~ 3 ; …; R~ 3 Z matrices R~ 3 ; R~ 3 ; …; R~ 3 ðR~ 3 þ R~ 3 Þ=2. h i h i (v) Denote Q~ ¼ Q~ ; Q~ ; …; Q~ ~ ~ ~ ~ 2 2;1 2;2 2;Na , R 2 ¼ R 2;1 ; R 2;2 ; …; R 2;N a . T

Denote h i i z^ þ i ¼ x^ μi ; x^ μi ; …; x^ μM ; 1

2

i

n

o

μi1 ; μi2 ; …; μiMi D Na ;

where Mi Z0 is the number of local systems that take i as their neighbor. Define   N 1 1 H x;i ðkÞ ¼ H x;i ðkÞ þ ∑ J x^ i ðjjkÞ  xi ðjjkÞ J 2~ i ; 1

j¼1

   i  i x^ i ðjjkÞT Q~ 20 x^ i ðjjkÞ  xi ðjjkÞ þ z^ þ i ðjjkÞT Q~ 2 þ x^ i ðjjkÞ  xi ðjjkÞ

1

N 1

H u;i ðkÞ ¼ H u;i ðkÞ þ ∑

j¼1

1

N1 n

H u;i ðkÞ ¼ ∑

j¼1

On-line stage: For each local system i, perform the following steps at time k ¼0:

Q3

j¼1

N1 n

H x;i ðkÞ ¼ ∑

i i i i Determine fQ~ 20 ; Q~ 2 þ g according to (23), fR~ 20 ; R~ 2 þ g according to (26). (vi) Choose fR i;i ; R i;  i g such that R i Z diagfR i;i ; R i;  i g, for all i A Na .



o ;

 J u^ i ðjjkÞ  ui ðjjkÞ J 2~ i ; R3

  o i  i u^ i ðjjkÞT R~ 20 u^ i ðjjkÞ  ui ðjjkÞ þ v^ þ i ðjjkÞT R~ 2 þ u^ i ðjjkÞ  ui ðjjkÞ :

For each i, (27) is satisfied by imposing the following constraint in the optimization: 0 r σ i ðkÞ o 1;

n h i h i o 2 2  σ i ðkÞ  : xi ðkÞT ; x  i ðkÞT :Q i þ J ui ð0jkÞT ; u^  i ð0jkÞT :R i þH x;i ðkÞ þ H u;i ðkÞ r 0

ð28Þ

for optimization, xi ðjjkÞ should be properly expressed as functions of ui ðjjkÞ; j A N. Remark 5. In [35,36], the state compatibility constraints J xi ðjjkÞ  x^ i ðjjkÞ J r κ ; j A N2 for i A Na are adopted. Compared with the compatibility constraints proposed in this paper, the compatibility constraints in [35,36] may become ineffective in a

(i) Measurement: Take the measurement of xi ð0Þ. (ii) Communication: Send xi ð0Þ to local systems fμi1 ; μi2 ; …; μiMi g. Receive xt ð0Þ, t A fνi2 ; νi3 ; …; νiN i g. (iii) Initialization: Set x^ t ð0j0Þ ¼ xt ð0Þ, t A fνi2 ; νi3 ; …; νiN i g. (iv) Initialization: Set u^ t ðjj0Þ ¼ 0 for all j A N and all t A fνi2 ; νi3 ; …; νiN i g. (v) Optimization: Solve problem (15). (vi) Control: Implement ui ð0Þ ¼ uni ð0j0Þ. (vii) Communication: Send uni ð1j0Þ; uni ð2j0Þ; …; uni ðN  1j0Þ; F ni ð0Þ to local systems in S i . Receive unr ð1j0Þ; unr ð2j0Þ; …; unr ðN  1j0Þ; F nr ð0Þ; r A S i . For each local system i, perform the following steps at each time k 4 0: (i) Measurement: Take the measurement of xi(k). (ii) Communication: Send xi(k) to local systems S i . Receive xr ðkÞ; r A S i . (iii) Initialization: Set x^ r ð0jkÞ ¼ xr ðkÞ; r A fi; S i g. (iv) Initialization: Set u^ r ðjjkÞ ¼ unr ðj þ 1jk  1Þ; u^ r ðN  1jkÞ ¼ n n F r ðk  1Þxr ðNjk 1Þ, for all j A N1 and all r A fi; S i g.

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

(v) Optimization: Solve problem (29). (vi) Control: Implement ui ðkÞ ¼ uni ð0jkÞ. (vii) Communication: Send uni ð1jkÞ; uni ð2jkÞ; …; uni ðN  1jkÞ; F ni ðkÞ to local systems in S i . Receive unr ð1jkÞ; unr ð2jkÞ; …; unr ðN  1jkÞ; F nr ðkÞ; r A S i . Remark 6. Since the algorithm proposed in the paper is noniterative, there is no communication when solving the local optimization problems. Moreover, the information which needs to be transferred is only between each local system i and its neighbors, so the communication burden in the algorithm is relatively low. At time k ¼0, subsystem i sends its state xi ð0Þ to the local systems fμi1 ; μi2 ; …; μiMi g and receives xt ð0Þ, t A fνi2 ; νi3 ; …; νiN i g before solving (15). However, at time k 4 0, subsystem i sends its state xi(k) to local systems S i and receives xr ðkÞ; r A S i before solving (29). After the local optimization problems have been solved, each subsystem i sends uni ð1jkÞ; uni ð2jkÞ; …; uni ðN 1jkÞ; F ni ðkÞ to local systems in S i and receives unr ð1jkÞ; unr ð2jkÞ; …; unr ðN  1jkÞ; F nr ðkÞ; r A S i , both at k¼0 and k 40. Take the Example A.1 in Appendix A for example. At k ¼0, subsystem 1 sends its local state x1 to subsystem 4, and receive the local state x2 of subsystem 2. However, at k 4 0, subsystem 1 needs to send x1 to subsystems 2 and 4, and receive x2 and x4 from the subsystems 2 and 4, respectively. Remark 7. In [37], the distributed MPC for nominal nonlinear large-scale system with decoupled dynamics is studied. The local controller i optimizes not only u~ i ðkÞ, but also u~  i ðkÞ. Also in [37], there is no compatibility constraint, so the local controller is implemented as in the centralized MPC manner and the related stability conditions are given. The stability conditions in [37] cannot be disassembled in each local optimizer. Remark 8. Since the subsystems do not cooperate with each other when solving the local optimization problems, the solution is not globally optimal. As indicated in [8], the solution we get is the Nash equilibrium because the cost function of each local controller is selfish. Theorem 1. By applying Algorithm 1, the implementation of u(k) guarantees exponential stability of the global closed-loop system, once a feasible solution at time k ¼0 is found. Proof. By imposing (28) in the optimization for each local system i and for all k 4 0, (19) is always satisfied. Hence, the recursive feasibility of the optimization and the closed-loop stability are easily obtained by referring to Lemma 2. □

5. Numerical simulation We consider the multi-vehicle formation example in [41]. The model of robot i is 2

3 2 h 3 2 p hi ðk þ 1Þ p ðkÞ 0:5 cos 6 v 7 6 i 7 6 p i ðk þ 1Þ 7 ¼ 6 p vi ðkÞ 7 þ 6 4 5 4 5 4 0:5 sin θ ðk þ 1Þ θ ðkÞ 0 i

i

θ i ðkÞ θ i ðkÞ

3 v i ðkÞ 7 76  0 54 w i ðkÞ 5 0:5 0

32

where p hi and p vi are, respectively, positions in the horizontal and  i are, vertical directions, θ i is the orientation, and v i and w respectively, the linear speed and angular speed. The state and control vector of robot i are, respectively, defined as  i T . x i ¼ ½p hi ; p vi ; θ i T and u i ¼ ½v i ; w

The reference trajectory for all vehicles 3 2 3 2 r p ðk þ 1Þ p h;r ðkÞ 0:5 cos θ ðkÞ 6 v;r 7 6 v;r 7 6 6 p ðk þ 1Þ 7 ¼ 6 p ðkÞ 7 þ 4 0:5 sin θ r ðkÞ 4 5 4 5 θ r ðk þ 1Þ θ r ðkÞ 0 2

 h;r

is described as 3 # 0 " r 7 v ðkÞ 5 0  r ðkÞ w 0:5

The state and control vectors, of the reference trajectory, are h   r iT  r T , respectively. and u r ¼ v r ; w denoted as x r ¼ p h;r ; p v;r ; θ  T Defining x~ i ¼ x i  x r ¼ phi ; pvi ; θi and u~ i ¼ u i  u r ¼ ½vi ; wi T yields 2 h 3 2 h 3 pi ðk þ 1Þ pi ðkÞ 6 pv ðk þ 1Þ 7 6 pv ðkÞ 7 4 i 5¼4 i 5

θi ðk þ 1Þ



θi ðkÞ

3 r r 0:5 cos θi ðkÞ þ θ ðkÞ ðvi ðkÞ þ v r ðkÞ  0:5 cos θ ðkÞv r ðkÞ 6 7  6 7 þ 6 0:5 sin θi ðkÞ þ θ r ðkÞ ðvi ðkÞ þ v r ðkÞÞ  0:5 sin θ r ðkÞv r ðkÞ 7 4 5 0:5wi ðkÞ 2

For the formation problem, it is required that limk-1 x~ i ðkÞ ¼ v;d T T ~d ~ ~ x~ di ¼ ½ph;d i ; pi ; 0 and limk-1 u i ðkÞ ¼ u i ¼ ½0; 0 . Defining xi ¼ x i  d d T x~ i ¼ ½xi;1 ; xi;2 ; xi;3  and ui ¼ u~ i  u~ i ¼ ½ui;1 ; ui;2 T yields  8 r r > xi;1 ðk þ 1Þ ¼ xi;1 ðkÞ þ 0:5 cos xi;3 ðkÞ þ θ ðkÞ ðui;1 ðkÞ þ v r ðkÞÞ  0:5 cos θ ðkÞv r ðkÞ > > <  r  r r r > xi;1 ðk þ 1Þ ¼ xi;1 ðkÞ þ 0:5 sin xi;3 ðkÞ þ θ ðkÞ ðui;1 ðkÞ þ v ðkÞÞ  0:5 sin θ ðkÞv ðkÞÞ > > : x ðk þ 1Þ ¼ x ðkÞ þ 0:5u ðkÞ i;3

i;3

i;2

Construction of the performance cost is shown in Appendix D. For this numerical example, there are four vehicles. We v;d h;d v;d h;d v;d choose ½ph;d 1 ; p1  ¼ ½  1; 0, ½p2 ; p2  ¼ ½0;  0:5, ½p3 ; p3  ¼ ½1; 0, v;d ½ph;d ; p  ¼ ½0; 0:5, designate the 4th, 1st and 3rd robots as the 4 4 core robots, and let E 0 ¼ fð4; 1Þ; ð4; 2Þ; ð4; 3Þg. It is easy to show that d14 ¼ ½1; 0:5T , d24 ¼ ½0; 1T and d34 ¼ ½  1; 0:5T , and pd ¼ ½0; 0T . The global performance cost is obtained as  1 h JðkÞ ¼ ∑ J x~ 4;1 ðjjkÞ  x~ 1;1 ðjjkÞ þ x~ d1;1  x~ d4;1 J 2 j¼0

 þ J x~ 4;1 ðjjkÞ  x~ 2;1 ðjjkÞ þ x~ d2;1  x~ d4;1 J 2  þ J x~ 4;1 ðjjkÞ  x~ 3;1 ðjjkÞ þ x~ d3;1  x~ d4;1 J 2  1 þ J ðx~ 4;1 ðjjkÞ þ x~ 1;1 ðjjkÞ þ x~ 3;1 ðjjkÞÞ  x~ d4;1 þ x~ d1;1 þ x~ d3;1 J 2 9



~ þ J x~ 1;2 ðjjkÞ J 2 þ J x~ 2;2 ðjjkÞ J 2 þ J x~ 3;2 ðjjkÞ J 2 þ J x~ 4;2 ðjjkÞ J 2 þ J uðjjkÞ J2 :

Then, 2

0

0

0

1 9I 2

0

 89I 2

I 0

0 I2

0 0

0 0

0 0

0  I2

0

0

I

0

0

0

0

0

0

119I 2

0

 89I 2

0

0

0

0

I

0

9

0

I 2

0

 89I 2

0

319I 2

0

0

0

0

0

0

0

119I 2 6 0 6 6 6 0 6 6 0 6 Q ¼6 1 6 9I 2 6 6 0 6 6 8 4  I2

With the algorithm in Appendix x  1 ¼ x3 ; x  2 ¼ x4 ; x  3 ¼ x4 ; x  4 ¼ x1 . It is choose 2 3 2 2 0 19I 2 0 0 I2 9I 2 6 7 6 0 6 0 1 0 07 I 6 2 6 7 Q1 ¼61 7; Q 2 ¼ 6 6  I2 0 6 9I 2 0 29I 2 0 7 4 4 5 0 0 0 0 0 1 2

0

3

07 7 7 07 7 07 7 7; 07 7 07 7 7 05

R ¼ I8 :

I A applied, we have easily seen that we can  I2 0 119I 2 0

0

3

07 7 7; 07 5 1 3

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

8 9I 2

6 6 0 6 Q3 ¼6 8 6  9I 2 4 0

0

 89I 2

1 2

0

0

I2

0

0

3

2

7 07 7 7; 07 5

6 6 0 6 Q4 ¼6 8 6  9I 2 4 0

0

1 3

I2

0

 89I 2

1 3

0

0

8 9I 2

0

0

0

3

2

7 07 7 7 07 5 1 2

and R i ¼ diagfI 2 ; 0g; iA f1; 2; 3; 4g. Then, 2

119I2

6 0 6 6 6 0 6 h i 6 0 6 ~ ~ ~ ~ ~ Q 2 ¼ Q 2;1 ; Q 2;2 ; Q 2;3 ; Q 2;4 ¼ 6 1 6 9I 2 6 6 0 6 6 8 4  I2

0

0

0

1 9I 2

0

 89I2

I

0

0

0

0

0

0

I2

0

0

0

 I2

0 0

0 0

I 0

0 119I 2

0 0

0  89I2

0

0

0

0

I

0

9

0

 I2

0

 89I 2

0

319I 2

0

0

0

0

0

0

0

0

3

07 7 7 07 7 07 7 7 07 7 07 7 7 05

6 6 0 6 6 1I 6 27 2 Q3 ¼6 6 0 6 6 8 6  I2 4 27 0 2 65 I2 6 54 6 0 6 6 8 6  27I 2 6 6 0 6 Q4 ¼6 6  1I 2 6 2 6 6 0 6 6 8 6  27I 2 4 0

2

0

1 3

0

0

0

0

10 27I 2

0

8  27 I2

0

0

1 3

0

0

8  27 I2

0

19 27I 2

0

0

0

0

1 2I 2

6 6 0 6 Q2 ¼6 1 6  2I 2 4 0

0

 12I 2

1 2

0

0

1 2I 2

0

0

0

8  27 I2

1 3

0

0

0

0

10 27I 2

0

8  27 I2

0

0

1 3

0

0

8  27 I2

0

19 27I 2

0

0

0

0

0

8  27 I2

0

 12I 2

0

8  27 I2

1 4

0

0

0

0

0

0

10 27I 2

0

0

0

1 27I 2

0

0

1 3

0

0

0

0

0

0

1 2I 2

0

0 0

0

7 07 7 07 7 7; 07 7 7 07 5 1 4

0

0

0

0

1 2

0

1 27I 2

0

0

0

10 27I 2

0

0

0

0

0

0

0

3

7 07 7 7 07 7 07 7 7 07 7 7 07 7 7 07 5 1 3

0

3

0 7 07 7 07 7 7; 07 7 7 07 5

-1

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

1 vi

0

8  27 I2

0

1

1 4

0 -1 0.5

3

7 07 7 7; 07 5

0

vi

6 6 0 6 6 1I 6 27 2 Q1 ¼6 6 0 6 6 8 6  I2 4 27 0

1 27I 2

1 27I 2

and R i ¼ diagðI 2 ; 0Þ; i A f1; 2; 3; 4g. Further, choose Q 1 ¼ diag  1    29I 2 ; I , Q 2 ¼ diagf2I 2 ; Ig, Q 3 ¼ Q 1 , Q 4 ¼ diag 589I 2 ; I , Ri ¼ I 2 ; iA f1; 2; 3; 4g. In addition, κ ¼ 0:5, which is the parameter in comparability constraints of distributed MPC [35]. In the following, the reference trajectory is specified by  r ðkÞ ¼ 0 and x r ¼ ½0:5k  0:5; 0:3; 0T . At initial time v r ðkÞ ¼ 1, w

vi

i R~ 3 ¼ 0, R i;i ¼ I 2 , R i;  i ¼ 0; iA f1; 2; 3; 4g. For comparison, the distributed MPC in [35] is applied, where x  1 ¼ ½xT3 ; xT4 T ; x  2 ¼ x4 ; x  3 ¼ ½xT1 ; xT4 T ; x  4 ¼ ½xT1 ; xT2 ; xT3 T . Then, we can choose 10 27I 2

3

0

I

      1 2 3 4 Q~ 3 ¼ 189I 2 ; 12I 2 , Q~ 3 ¼ fI 2 ; 0g, Q~ 3 ¼ 129I 2 ; 12I 2 , Q~ 3 ¼ 489I 2 ; 23I 2 . h i R~ 2 ¼ R~ 2;1 ; R~ 2;2 ; R~ 2;3 ; R~ 2;4 ¼ R~ 3 ¼ 0. Further, choose Q 1 ¼ diag  1    29I 2 ; I , Q 2 ¼ diagf2I 2 ; Ig, Q 3 ¼ Q 1 , Q 4 ¼ diag 589I 2 ; I , Ri ¼ I 2 ,

2

10 27I 2

7

-0.5 -1

1 4

k

Fig. 2. Control input v i for the centralized MPC. 2

wi

piv

1 0 -2 -2

0

2

4

6

8

10

12

14

-1

0

0

2

4

6

8

10

12

-1

14

1

0

0

wi

piv

10

15

20

25

5

10

15

20

25

5

10

15

20

25

0

2

-2 -2

5

1

wi

piv

2

-2 -2

0

0

2

4

6

8

10

pih

Fig. 1. Movements of the vehicles by centralized MPC.

12

14

-1

k  i for the centralized MPC. Fig. 3. Control input w

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

k ¼0, p_ hi ¼ p_ vi ¼ 0. For input constraints, u i1 ¼ u i1 ¼ 1:5, u i2 ¼ u i2 ¼ π =4; i A f1; 2; 3; 4g. Choose N ¼12. For the terminal set, αi ¼ 0:25; 2 2 i A f1; 2; 3; 4g. Denote J true ¼ ∑1 k ¼ 0 ½ J xðkÞ J Q þ J uðkÞ J R .

With the centralized MPC in (5) being applied, for three sets of initial positions of the vehicles:

 the movements of the vehicles and the control input signals are shown in Figs. 1–3;

0

2

4

6

8

10

12

14

i

-2 -2

0

2

4

6

8

10

12

14

i

0

0

2

4

6

8

10

12

14

i

piv

2

piv

2 0 -2 -2

piv

2 0 -2 -2

pih

Fig. 4. Movements of the vehicles by the proposed distributed MPC. k

Fig. 7. Parameter σ i ðkÞ for the proposed distributed MPC.

1.5 vi

1

0

piv

0.5 5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

1.5 vi

1

0

piv

0.5

1.5 vi

1

0

piv

0.5

k

Fig. 5. Control input v i for the proposed distributed MPC.

pih Fig. 8. Movements of the vehicles by distributed MPC in [35].

0 -1

vi

wi

1

5

10

15

20

25

5

10

15

20

25

5

10

15

20

25

0 -1

vi

wi

1

0 -1

vi

wi

1

k  i for the proposed distributed MPC. Fig. 6. Control input w

k

Fig. 9. Control input v i for the distributed MPC in [35].

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

 the obtained Jtrue's are 37.8016, 42.6630 and 9.4801;  it takes about 1.6 min to obtain xð25Þ (for each set of the initial positions).

With the distributed MPC in (29) being applied, for three sets of initial positions of the vehicles same as in the centralized MPC:

 the movements of the vehicles and the control input signals are shown in Figs. 4–6. Moreover, the evolutions of σ i ðkÞ are plotted in Fig. 7;

 the obtained Jtrue's are 43.5503, 48.7345 and 10.6292;  it takes about 32 s to obtain xð25Þ (for each set of the initial positions).

With the distributed MPC in [35] being applied, for three sets of initial positions of the vehicles same as in the centralized MPC:

 the movements of the vehicles and the control input signals are

9

6. Conclusions and future work In this paper, a distributed MPC algorithm with compatibility constraint is proposed for nonlinear discrete-time systems with decoupled dynamics. Technically, this paper generalizes the centralized MPC in [39] to several dynamically decoupled local systems with global cost function. According to the coupling relation of subsystems in the overall objective function, a new procedure for finding the neighbors of each subsystem is given. After the neighbors of each subsystem are fixed, the overall optimization of centralized MPC algorithm is decomposed into the local controllers. For each subsystem, the local quasi-infinite horizon MPC algorithm is designed. The simulation example verifies the effectiveness of the proposed algorithm. Future works will focus on incorporating the coupling constraints in the neighborhood of subsystems, and the coupled dynamics if possible.

Acknowledgment

shown in Figs. 8–10;

This work is supported by the Fundamental Research Funds for the Central Universities (CDJZR12175501).

positions).

Appendix A. New procedures for finding neighbors of i and Q i , Ri

 the obtained Jtrue's are 43.0890, 49.3458 and 10.1969;  it takes about 34 s to obtain xð25Þ (for each set of the initial

As seen from Figs. 4–7, by applying the proposed distributed MPC, the overall stability is guaranteed. From the value of Jtrue, the performance of the proposed distributed MPC is worse than the centralized MPC. Note that the total time for all the sub-controllers is larger than the centralized MPC, but the real time they taken is much shorter because the distributed optimizations are solved simultaneously. Comparing the proposed distributed MPC with distributed MPC in [35], the performances of both algorithms are close, but the number of neighbors for each subsystem in [35] are much larger than our method. The communication load is increased when the number of neighbors is expanded. The marks ‘○’, ‘▿’, ‘⋆’ and ‘□’ in Figs. 2, 3, 5–7, 9 and 10 represent the 1st, 2nd, 3rd and 4th subsystem, respectively. Note that each figure contains three sub-figures, the top for the first set of initial positions, the middle for the second, and the bottom for the third. s The solver for optimization is the Matlab package FMINCOM. We did the simulation via Matlab2008a on our PC with a 2.6 GHz Intel Pentium Dual-Core processor and 2 GB RAM.

According to the dimensions of xi's and ui's, Q and R are partitioned into block matrices as     Q ¼ Q ij i A Na ;j A Na ; R ¼ Rij i A Na ;j A Na ; with Q ij A Rni nj and Rij A Rmi mj . For each i A Na , search through jA Na ; if Q ij a 0 or Rij a 0, then put xj into z~ i . Algorithm A.1 (The procedure for finding the neighbors of each local system). (i) For each i A Na , rearrange the states in z~ i such that h iT z~ i ¼ xTi ; xToi ; xToi ; xToi ; ⋯ ; dimfxoi g r dimfxoi g r dimfxoi g r ⋯ 2

where,

if

3

2

4

there

oia ,

exist

3

oib

such

4

dimfxoia g ¼ h z~ i ¼ xTi ; …; xToi ;

that

g ¼ ⋯ ¼ dimfxoi g, then let b iT with oia o oia þ 1 o ⋯ ooib xToi ; …; xToi ; ⋯ aþ1 b  T z~ i ¼ xTi ; …; xToi ; xToi ; …; xToi ; xToi ; …; xToi ; … dimfxoi

aþ1

a

aþ1

τab

τab þ 1

a

or with

b

oia o oia þ 1 o ⋯ o oiτab 4 oiτab þ 1 o⋯ o oib o oia , where

τab is cho-

wi

sen to make oiτab as the largest number in ½oia ; oib . (ii) Let z~ ¼ fz~ 1 ; z~ 2 ; …; z~ Na g. (iii) In z~ , with priority from left to right (the left has higher priority), find a z~ t with the largest dimension. Denote h iT z~ t ¼ xTt ; x~ T t . x~  t is a collection of a number of xs's. (iv) In x~ , with priority from right to left (the right has higher

wi

t

wi

priority), find a xs such that the corresponding z~ s is included in z~ , and that x~  s includes xt. Remove this xs from x~  t and z~ t . If, in x~  t , there is no such xs, then denote zt ¼ z~ t , x  t ¼ x~  t and remove z~ t from z~ . (v) If there are more than one z~ t in z~ , then go T to step (iii); else, then there is t A Na such that zt ¼ xTt ; xT t ¼ z~ .

k  i for the distributed MPC in [35]. Fig. 10. Control input w

The order of xs's in z~ t and priorities in steps (iii) and (iv) guarantee the uniqueness of the result. The dimensional order in z~ t and choice of the largest dimension in step (iii) are to balance

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

the complexity of all the local optimal control problems. By applying Algorithm A.1, we obtain z ¼ fz1 ; z2 ; …; zNa g with zi ¼ ½xTi ; xT i T , iA Na . The neighbors of i are those local systems whose states are included in x  i . Example A.1. Suppose we have four local states x1 , x2 , x3 , x4 with n T  T  T equal dimensions, and z~ ¼ xT1 ; xT2 ; xT4 ; xT2 ; xT3 ; xT1 ; xT3 ; xT4 ; xT2 ; n  T T T T o T . Then by applying Algorithm A.1, z ¼ xT1 ; xT2 ; x4 ; x1 ; x3  T T T  T T T  T T T o . x2 ; x3 ; x3 ; x4 ; x4 ; x1 Example

A.2. Suppose we have four local states with n T dimfx1 g ¼ dimfx2 g ¼ dimfx3 g o dimfx4 g, and z~ ¼ xT1 ; xT2 ; xT4 ;  T T T T  T T T T  T T T T x2 ; x3 ; x1 ; x3 ; x2 ; x4 ; x4 ; x1 ; x3 g. Then by applying Algorithm n T  T  T o . A.1, z ¼ xT1 ; xT2 ; xT2 ; xT3 ; xT2 ; xT4 ; xT1 ; xT3 The procedure for finding Q i , R i for each local system i is simple. Write Qii and Rii as μi

μiM

μi

μi

Q ii ¼ Q iii þ Q ii 1 þ Q ii 2 þ ⋯ þ Q ii i ;

μiM

μi

Rii ¼ Riii þ Rii 1 þ Rii 2 þ ⋯ þ Rii

i

ðA:1Þ where μδi , δi A f1; 2; …; Mi g are defined in Section 4.2. μδi ’s are those local systems that take i as their neighbor. Then i

i

2

Qi i i 6 ν1 ;ν1 6 i 6Q i i 6 ν2 ;ν1 Qi ¼6 6 ⋮ 6 4 i Q νi ; νi Ni

Q iνi ;νi

⋯

Q iνi ;νi

Q iνi ;νi

⋯

Q iνi ;νi

⋮ Q iνi ;νi

⋱ ⋯

⋮ Q iνi ;νi N N

1

2

1

2

2

Ni

2

3 7 7 7 7 7; 7 7 5

Ni

1

Ni

2

i

i

2

Ri i i 6 ν1 ;ν1 6 i 6R i i 6 ν ;ν Ri ¼ 6 2 1 6 ⋮ 6 4 i Rνi ;νi Ni

1

Riνi ;νi

⋯

Riνi ;νi

Riνi ;νi

⋯

Riνi ;νi

⋮

⋱

⋮

Riνi ;νi 2 N

⋯

Riνi ;νi N N

1

2

2

2

i

3 7 7 7 7 7: 7 7 5

Ni

1

2

i

Ni

Appendix C. The bounds of the loss of optimality by distributed implementation

i

It is required that Q i Z 0, R i Z 0. μiδ

Define

μiδ

Usually, there are enumerable choices for Q ii i and Rii i in (A.1). μiδ

In case it is not easy to find Q ii following optimization problems:

μiδ

and Rii i , one can solve the

N1 h

i ‖zi ðjjkÞ‖2Q þ ‖vi ðjjkÞ‖2R þ ‖xi ðNjkÞ‖2P i ;

J i ðkÞ ¼ ∑

i

i

j¼0

N1 h

Na

J ðkÞ ¼ ∑ J i ðkÞ ¼ ∑

ηQ ;

max

μi δ

i

In the context of [35], the above three manners are equivalent. In Appendix A, we can find neighbors by observing z obtained from Algorithm A.1, or by observing Ji(k). If one finds neighbors through Algorithm A.1, then the results are different from those by [35]. In [35], the neighbor relationship between local systems is mutual, i.e., j is a neighbor of i if and only if i is a neighbor of j, By applying Algorithm A.1, the neighbor relationship is single-sided, i.e., j is a neighbor of i means that i is not a neighbor of j. The advantage by applying Algorithm A.1 is that the computational burden can be greatly reduced since each local system has less number of neighbors. The deductions in Sections 2–4 are suitable for the neighbors both by [35] and by Algorithm A.1. Moreover, the deductions in Sections 2–4 do not reply on the order of xj's in x-i. In [35], it is assumed that R is decomposed. Then, given Q, finding Q i can be achieved by handling G. The procedure obtains Q i which defines mutual neighbor relationship. By applying the procedure in [35], one should first find G. By applying the new procedure in Appendix A, one can obtain smaller dimensional Q i . According to “Remark 6.3” of [35], Q i affects the control performance, and applying Q i obtained in Appendix A has a leader– follower effect. The deductions in Sections 2–4 are suitable for the obtained fQ i ; R i g both by [35] and by Appendix A. If the procedures in [35] are applied for obtaining the neighbors and Q i ; R i , then for the technique in Sections 2–4, x  i ¼ z þ i .

j¼0

i¼1

i Na ‖xðjjkÞ‖2Q þ ‖uðjjkÞ‖2R þ ∑ ‖xi ðNjkÞ‖2Pi i¼1

ηQ ;Q ii i ;i A f1;…;N a g;δi A f1;…;Mi g

Q i Z ηQ I;

s:t: μi δi

μi

μiM

μi

Q iνi ;νi ¼ Q ii  Q ii 1  Q ii 2 ⋯  Q ii 1

1

i

; iA f1; …; N a g;

min J ðkÞ

ηR ;

max

~ uðkÞ

ηR ;Rii ;i A f1;…;N a g;δi A f1;…;Mi g

s:t:

R i Z ηR I;

μi2

μiM

Riνi ;νi ¼ Rii  Rii Rii  ⋯  Rii 1

s:t:

ð5Þ; 8 jA N; xi ðNjkÞ A Ωi ; 8 iA Na :

i

; i A f1; …; Na g: μiδ

μiδ

By applying the above optimizations, the solutions to Q ii i and Rii are unique.

i

Appendix B. A comparison with the procedures in [35]

Let J i ðkÞ be the optimal cost value of the i-th local MPC (that n achieved by (15)), J ðkÞ be the optimal cost value of the centralized n MPC (that is achieved by (16)), and J ðkÞ be the optimal cost value by solving (C.1). Suppose solving (15) yields optimal xni ðjjk; J i Þ and uni ðjjk; J i Þ. Substituting the optimal solutions of (15) into problem (C.1), and n n a  calculating ℓ1l ðJ i ðkÞÞ ¼ ∑N i ¼ 1 J i ðkÞ  J ðkÞ, yield Na N  1 h

n

ℓ1l ðJ i ðkÞÞ ¼ ∑ ∑ The procedures in Appendix A are different from those in [35]. In [35], finding neighbors of i can be achieved at least in three different manners:

 Suppose R is decomposed and Q ¼ GT G. Denote G ¼

 

ðC:1Þ

n

μi1

1

Consider the following optimization problem:

½Glj l A f1;…;Mg;j A Na , where the dimension of Glj is conformal with the local system dimension. For any j A Na , if there exists l A f1; …; Mg such that Gli a 0 and Glj a 0, then j is a neighbor of i. If xj or uj is in the local performance cost of i, then j is a neighbor of i. For multi-vehicle formation, the formation can be described as a graph. Two vehicles i and j are called as neighbors if the ordered pair (i, j) constitutes an edge of the graph.

i¼1j¼0

Na N  1 h

 ∑ ∑

n

i

‖zni ðjjk; J i Þ‖2Q þ‖vni ðjjk; J i Þ‖2R i

i¼1j¼0

i

n

‖z
i ðjjk; J i Þ‖2Q þ ‖v
i ðjjk; J i Þ‖2R

i

i i

h iT h iT n T T n n and v
i ¼ uinT ; u^  i . Since J ðkÞ Z J ðkÞ, where z
i ¼ xni T ; x^  i n

n

n

1 a  ∑N i ¼ 1 J i ðkÞ  J ðkÞ Z ℓl ðJ i ðkÞÞ. Further, suppose solving (C.1) yields xn ðjjk; J Þ and un ðjjk; J Þ. Substituting the optimal solutions of (C.1) n n into problem (16), and calculating ℓ2l ðJ ðkÞÞ ¼ J ðkÞ  J ðkÞ, yield n ℓ2l ðJ ðkÞÞ ¼ ‖xn ðNjk; J Þ‖2P  P ; n n n where P ¼ diagfP 1 ; P 2 ; …; P Na g. Since J ðkÞ Z J ðkÞ, J ðkÞ J ðkÞ Z

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ n

ℓ2l ðJ ðkÞÞ. Hence, the lower bound is expressed as Na

n

n

n

n

∑ J i ðkÞ  J ðkÞ Zℓ1l ðJ i ðkÞÞ þ ℓ2l ðJ ðkÞÞ:

i¼1

Suppose solving (C.1) yields optimal xn ðjjk; J Þ and un ðjjk; J Þ. Substituting the optimal solutions of (C.1) into problem (15), and n a n calculating ℓ1u ðJ ðkÞÞ ¼ ∑N i ¼ 1 J i ðkÞ  J ðkÞ, yield Na N  1 h

n ℓ1u ðJ ðkÞÞ ¼ ∑ ∑

i¼1j¼0

n n ‖z
i ðjjk; J Þ‖2Q þ‖v
i ðjjk; J Þ‖2R

Na N  1 h

 ∑ ∑

i¼1j¼0

i

i i

‖zni ðjjk; J Þ‖2Q þ ‖vni ðjjk; J Þ‖2R i

i i

n n 1 n a n Since J i ðkÞ ZJ i ðkÞ, ∑N i ¼ 1 J i ðkÞ  J ðkÞ Z ℓu ðJ ðkÞÞ. Moreover, suppose n n solving (16) obtains x ðjjk; J Þ and u ðjjk; J Þ. Substituting the optimal n solutions of (16) into problem (C.1), and calculating ℓ2u ðJ ðkÞÞ ¼ J ðkÞ  J n ðkÞ, yield n

ℓ2u ðJ ðkÞÞ ¼ ‖xn ðNjk; J Þ‖2P  P : n

n

n

n

Since J ðkÞ Z J ðkÞ, J ðkÞ J ðkÞ r ℓ2u ðJ ðkÞÞ. Hence, the upper bound is expressed as Na

n n n n ∑ J i ðkÞ  J ðkÞ rℓ1u ðJ i ðkÞÞ þ ℓ2u ðJ ðkÞÞ:

i¼1

Appendix D. Deductions of the performance cost of the vehicles According to [35], we can choose the performance cost for the centralized MPC as " 1

JðkÞ ¼ ∑

j¼0

∑ ‖pi ðjjkÞ  pj ðjjkÞ þ dij ‖2 þ‖p∑ ðjjkÞ  pd ‖2

ði;jÞ A E 0 2

# 2

þ ‖vel ðjjkÞ‖ þ‖uðjjkÞ‖

;

ðD:1Þ

 T where pi ¼ phi ; pvi ; dij is the desired relative vector (relative h iT v;d þ distance) between two vehicles i and j, i.e., ph;d i ; pi h iT v;d ; P ∑ is the center of geometry of the core vehicles; dij ¼ ph;d j ; pj pd is the desired center of geometry of the core vehicles; vel is the velocity of all vehicles; E 0 is an orientation of the set E, with E being the set of relative vectors between vehicles. More details for the vector formation graph are referred to [35]. In [35], the term “formation” is explained in three manners:

 A formation associates a precise location for every vehicle.  A multi-vehicle formation is defined by a set of relative vectors that connect the desired locations of the vehicles.

 A formation can be described as a graph, being called as vector formation graph. In the context of [35], the above manners are equivalent. In our present paper, the term “formation” is defined same as in [35].

References [1] Mayne DQ, Rawlings JB, Rao CV, Scokaert POM. Constrained model predictive control: stability and optimality. Automatica 2000;36(6):789–814. [2] Findeisen R, Imsland L, Allgöwer F, Foss BA. State and output feedback nonlinear model predictive control: an overview. Eur J Control 2003;9(2–3): 190–206.

11

[3] Gu B, Gupta YP. Control of nonlinear processes by using linear model predictive control algorithms. ISA Trans 2008;47(2):211–6. [4] Zhang B, Yang W, Zong H, Wu Z, Zhang W. A novel predictive control algorithm and robust stability criteria for integrating processes. ISA Trans 2011;50(3):454–60. [5] Abu-Ayyad M, Dubay R. Real-time comparison of a number of predictive controllers. ISA Trans 2007;46(3):411–8. [6] Camponogara E, Jia D, Krogh BH, Talukdar S. Distributed predictive control. IEEE Control Syst Mag 2002;22(1):44–52. [7] Scattolini R. Architectures for distributed and hierarchical model predictive control—a review. J Process Control 2009;19(5):723–31. [8] Christofides PD, Scattolini R, de la Peña DM, Liu JF. Distributed model predictive control: a tutorial review and future research directions. Comput Chem Eng 2013;51:21–41. [9] De Schutter B, Scattolini R. Introduction to the special issue on hierarchical and distributed model predictive control. J Process Control 2011;21(5): 683–4. [10] Stewart BT, Venkat AN, Rawlings JB, Wright SJ, Pannocchia G. Cooperative distributed model predictive control. Syst Control Lett 2010;59(8): 460–9. [11] Stewart BT, Wright SJ, Venkat AN, Rawlings JB. Cooperative distributed model predictive control for nonlinear systems. J Process Control 2011;21 (5):698–704. [12] Ferramosca A, Limon D, Alvarado I, Camacho EF. Cooperative distributed MPC for tracking. Automatica 2013;49(4):906–14. [13] Trodden P, Richards A. Cooperative distributed MPC of linear systems with coupled constraints. Automatica 2013;49(2):479–87. [14] Farina M, Scattolini R. Distributed predictive control: a non-cooperative algorithm with neighbor-to-neighbor communication for linear systems. Automatica 2012;48(6):1088–96. [15] Mercangöz M, Doyle FJ. Distributed model predictive control of an experimental four-tank system. J Process Control 2007;17(3):297–308. [16] Alvarado I, Limon D, de la Peña DM, Maestre JM, Ridao MA, Scheu H, et al. A comparative analysis of distributed MPC techniques applied to the HD-MPC four-tank benchmark. J Process Control 2011;21(5):800–15. [17] Morosan PD, Bourdais R, Dumur D, Buisson J. Building temperature regulation using a distributed model predictive control. Energy Build 2010;42 (9):1445–52. [18] Maestre JM, de la Peña DM, Camacho EF. Distributed MPC: a supply chain case study. In: Joint 48th IEEE conference on decision and control and 28th Chinese control conference, Shanghai, 2009. [19] Dunbar WB. Distributed receding horizon control of dynamically coupled nonlinear systems. IEEE Trans Autom Control 2007;52(7):1249–63. [20] Wang C, Ong CJ. Distributed model predictive control of dynamically decoupled systems with coupled cost. Automatica 2010;46(12):2053–8. [21] Müller MA, Reble M, Allgöwer F. Cooperative control of dynamically decoupled systems via distributed model predictive control. Int J Robust Nonlinear Control 2012;22(12):1376–97. [22] Magni L, Scattolini R. Stabilizing decentralized model predictive control of nonlinear systems. Automatica 2006;42(7):1231–6. [23] Liu JF, Chen XZ, de la Peña DM, Christofides PD. Iterative distributed model predictive control of nonlinear systems: handling asynchronous, delayed measurements. IEEE Trans Autom Control 2012;57(2):528–34. [24] Franco E, Magni L, Parisini T, Polycarpou MM, Raimondo DM. Cooperative constrained control of distributed agents with nonlinear dynamics and delayed information exchange: a stabilizing receding-horizon approach. IEEE Trans Autom Control 2008;53(1):324–38. [25] Heidarinejad M, Liu JF, de la Peña DM, Davis JF, Charistofides PD. Handling communication disruptions in distributed model predictive control. J Process Control 2011;21(1):173–81. [26] Liu JF, de la Peña DM, Christofides PD. Distributed model predictive control of nonlinear process systems. AIChE J 2009;55(5):1171–84. [27] Liu JF, de la Peña DM, Christofides PD. Distributed model predictive control of nonlinear systems subject to delayed measurements. In: Proceedings of the 48th IEEE conference on decision and control, held jointly with the 28th Chinese control conference; 2009. p. 7105–12. [28] Liu JF, de la Peña DM, Christofides PD. Distributed model predictive control of nonlinear systems subject to asynchronous and delayed measurements. Automatica 2010;46(1):52–61. [29] Li SY, Zhang Y, Zhu QM. Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem. Inf Sci 2005;170 (2–4):329–49. [30] Giovanini L. Game approach to distributed model predictive control. IET Control Theory Appl 2011;5(15):1729–39. [31] Chen H, Allgöwer F. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 1998;34(10): 1205–17. [32] Ding BC. Distributed robust MPC for constrained systems with polytopic description. Asian J Control 2011;13(1):198–212. [33] Ding BC, Li SY. Design and analysis of constrained nonlinear quadratic regulator. ISA Trans 2003;42(2):251–8. [34] Ding BC, Xie LH, Cai WJ. Distributed model predictive control for constrained linear systems. Int J Robust Nonlinear Control 2010;20(11): 1285–98. [35] Dunbar WB. Distributed receding horizon control of multiagent system [Ph.D. thesis]. California Institute of Technology; 2004.

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i

12

M. Zhao, B. Ding / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

[36] Dunbar WB, Murray RM. Distributed receding horizon control for multivehicle formation stabilization. Automatica 2006;42(4):549–58. [37] Keviczky T, Borrelli F, Balas GJ. Decentralized receding horizon control for large scale dynamically decoupled systems. Automatica 2006;42 (12):2105–15. [38] Penzl T. Numerical solution of generalized Lyapunov equations. Adv Comput Math 1998;8(1–2):33–48.

[39] Johansen TA. Approximate explicit receding horizon control of constrained nonlinear system. Automatica 2004;40(2):293–300. [40] Lee JW. Exponential stability of constrained receding horizon control with terminal ellipsoidal constraints. IEEE Trans Autom Control 2000;45(1):83–8. [41] Gu DB, Hu HS. Receding horizon tracking control of wheeled mobile robots. IEEE Trans Control Syst Technol 2006;14(4):743–8.

Please cite this article as: Zhao M, Ding B. Distributed model predictive control for constrained nonlinear systems with decoupled local dynamics. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.07.012i