Distributed model predictive control of discrete-time interconnected

2016 24th Iranian Conference on Electrical Engineering (ICEE)

Distributed Model Predictive Control of DiscreteTime Interconnected Systems with Time Delays

Iman Hosseini PhD Student School of Elect. and Comp. Eng. Shiraz University Shiraz, Iran Email: [email protected]

S. Vahid Naghavi Assistant Professor Zarghan Branch Islamic Azad University, Zarghan, Iran Email: [email protected]

Abstract— This paper investigates a distributed model predictive control strategy for discrete-time interconnected linear systems. The MPC controllers of subsystems communicate their information to handle the interconnection effects. It is known that distributed MPC relies on communication networks to transmit and share information among subsystems. Communication delays, resulting in delayed information exchange among subsystems, may inhibit the use of any distributed MPC approach. To address this problem, a delay handling mechanism is employed. Besides, it is showed that the proposed method guarantees input-to-state stability characterization for both local subsystems and the global system under some predetermined assumptions. The simulation results are exploited to illustrate the effectiveness of the proposed method.

Keywords-component; Distributed netwoked control system; MPC; time delay; ISS stability

I.

INTRODUCTION

Model predictive control (MPC) (or receding horizon control (RHC)) is a control strategy that offers attractive solutions for regulatory and tracking problem of constrained linear or nonlinear systems. MPC algorithms provide the intuitive way of addressing the control problem. In last decades, MPC has become an important research interest in control community within industry and academia. This is illustrated by its applications to a wide industrial problems and many theoretical developments. See for example [1-5]. Most processes in modern industries are physically distributed and generally composed of different subsystems,

Ali Akbar Safavi Professor School of Elect. and Comp. Eng. Shiraz University Shiraz, Iran Email: [email protected]

which are interconnected and characterized by significant interactions. Besides, the implementation of a control strategy for distributed systems can be achieved better in both economical and physical aspects via communication networks. In these networked systems, the information exchange through the communication networks unavoidably exhibits time delays. The study of these distributed networked control systems (DNCSs) is quite important yet a challenging issue. The MPC-based approach to study NCSs has appealing features in comparison to other approaches. First, the MPC strategy can predict the states and generate a sequence of future control signals by optimizing a control performance function at each time instant. The generated future control sequence is particularly effective in compensating for communication constraints in NCSs such as packet dropouts and delays. Second, the MPC is capable of handling various system constraints including input constraints and state constraints, which is also desirable in many NCS applications. Third, there have been a lot of applications of MPC in many industrial systems. Thus, the study of MPC for NCSs would facilitate the modification and advancement of the network-based control implementations. It is obvious that safe operation of plants relies, among other things, on controller designs that account for the inherently complex dynamics of the processes (manifested as nonlinearities), operational issues, such as constraints and uncertainties [6], [7]. These problems force control strategy to tackle the obstacles of constraints, nonlinearities, and uncertainties. In centralized architecture, it could be challenging to address these difficulties for interconnected systems since there are computational and communication

978-1-4673-8789-7/16/$31.00 ©2016 IEEE 477

2016 24th Iranian Conference on Electrical Engineering (ICEE) constraints. With this in mind, researchers are interested in distributed or decentralized control strategies [8-10]. The MPC algorithms are generally based on optimization control methods and usually in the centralized form [11, 12]. For centralized MPC, the computation requirement is the main obstacle to expanding the areas of applications, especially for nonlinear large-scale systems. Utilizing the distributed MPC is a solution to reduce the computation time. Optimization in distributed MPC in comparison with the centralized MPC is totally decentralized into a number of small-scale optimizations. Because of limitation in communication bandwidth, the subsystems have not full information of the neighboring subsystems in order to solve the optimization problem. Consequently, performance in centralized MPC may be better than decentralized one from this aspect, but its advantages in general are more than the centralized one. There have been many papers on distributed MPC [14-16], but not addressing the issues considered in this paper. Subsystems are distributed in most cases; thus, it is required to share information between them. Nevertheless, if the amount of subsystems is enormous, the communication load will be huge. In [17] a distributed model predictive control is presented to improve the performance of the overall system. In this method, a local MPC is designed to control each subsystem and a reduced set of information is exchanged between these controllers via network. In [18] a framework for distributed model predictive control of discrete-time nonlinear systems with decoupled dynamics, subject to coupled constraints and a common cooperative task is presented but delay is not included. It is assumed in some papers that information transmission can be done successfully between subsystems [22, 23, 25]. In [19] the problem of stability analysis and stabilization of the network distributed control systems (which have both random delay and random packet loss in their communication networks) is investigated, yet the system is considered to be linear. In [20] for a class of continuous-time decoupled nonlinear systems the distributed MPC problem subject to communication delays is studied, in which a waiting mechanism is used and robustness constraint is exploited to handle the time delay. Note that a described large-scale system is associated with the communication topology/structure has been used for modeling many systems such as a group of vehicles [21], a team of robots[22, 23] and power generation systems[24]. In [26], we presented a novel approach to address the decentralized fault tolerant model predictive control of discrete-time interconnected nonlinear systems. The overall system is composed of a number of discrete-time interconnected nonlinear subsystems at the presence of multiple faults occurring at unknown time-instants. In order to deal with the unknown interconnection effects and changes in model dynamics due to multiple faults, both passive and active fault tolerant control design were considered. In this paper, we investigate a distributed model predictive strategy for discrete-time interconnected linear systems. The overall system is composed of a number of discrete-time interconnected linear subsystems. The MPC controllers

communicate their information to handle the interconnection effects. It is known that distributed MPC relies on communication networks to transmit and share information among subsystems. Communication delays, resulting in delayed information exchange among subsystems, may inhibit the use of the proposed distributed MPC approach. To address this problem, a delay handling mechanism is employed [20]. Besides, it is showed that the proposed method guarantees input-to-state stability characterization for both local subsystems and the global system under some predetermined assumptions. The simulation results are exploited to illustrate the effectiveness of the proposed method. This paper is organized as follows: In Section II preliminary definitions are presented. Section III describes the problem and formulations. In Section IV main results and delay handling mechanism are presented. Section V give the simulation results and we conclude the paper in Section VI.

II.

PRELIMINARY DEFINITIONS

Consider a discrete time system of the form x ( k + 1) ∈ ( Ax ( k ) + Bu ( k ) + w ) , k ∈ Z + (1)

where ‫ݔ‬ሺ݇ሻ ‫ܴ א‬௡ is the state, ‫ݑ‬ሺ݇ሻ ‫ܴ א‬௠ is the control input, ‫ݓ‬ሺ݇ሻ ‫ܴ  ؿ ܹ א‬௟  is an unknown input at discrete instant ݇ ‫ܼ  א‬ା . Definition 1: Regional ISS: The system (1) is said to be ISS in X ∈ R n if there exist a KL-function β , a K-function γ such that, for each x 0 ∈ X , all W, it holds that the corresponding state trajectory of (1) satisfies

(

)

§

·

x k ≤ β x 0 , k + γ ¨ w ªk −1º ¸ ∀k ∈ Z + ¨ ©

¬«

¼»

(2)

¸ ¹

Definition 2: CLF (Control Lyapunov Function) Consider system (1) and a function V: ܴ ௡  ՜  ܴା such that satisfies

α1 ( x ) ≤V ( x ) ≤ α2 ( x

)

(3)

for all ‫ ܺ א ݔ‬is called a control Lyapunov function (CLF) for system

x ( k +1) ∈Ax ( k ) + Bu ( k ) +w ; k ∈ Z +

If for all ‫ܺ א ݔ‬Ǣ‫ ܷ א ݑ׌‬such that the following holds V ( Ax + Bu ) −V ( x ) ≤ −α3 x (4)

( )

III.

PROBLEM DESCRIPTION

We consider a system composed of ݊ interconnected subsystems. The ݅‫ ݄ݐ‬subsystem ݅ ൌ ͳǡʹǡ ǥ ݊ is described by:

x i ( k +1) = Ai x i ( k ) + Bi ui ( k ) +w i

(5)

478

2016 24th Iranian Conference on Electrical Engineering (ICEE) where x i ( k ) = ª«x i 1 x i 2 " x i ρ ¬

i

T º » ¼

∈R

ρi

is the state of the T

݅‫ ݄ݐ‬subsystem, x ( k ) = ª x 1 x 2 " x n º ∈ R ρ where ¬ ¼ n

m ρ = ¦ ρi is the state of the system, ui ∈ R is the control i

i =1

action of the ݅‫ ݄ݐ‬subsystem, A i is the state matrix, B i is the input matrix and w i is nonlinear interconnection from other subsystems.

Assumption 1: The state and the control variables are restricted to fulfill the following constraints xi ∈X i

, u i ∈U i

where X i and U i are compact sets, both containing the T

origin as an interior point. Let us denote X  ª¬X 1T , X 2T ,..., X nT º¼ the overall constraint set on the state.

( )

Assumption 2: The state of the plant x i k at each sample time.

can be measured

Assumption 3: The term w i is such that w i ∈W i

where W i is a compact set containing the origin as an interior T

point. Let us denote W  ª¬W1T ,W 2T ,...,W nT º¼ interconnection set.

the overall

As was stated before, information which are exchanged between the subsystems and controllers are subject to the delay that is a challenging problem in distributed systems and may result in instability of the overall system. Hence, the control goal is to design a control law ui ( k ) for each

subsystem, which satisfies the constraints on the input and the state along the system evolution for any possible interconnections and overcome the delay problem so that the overall system moves to the origin or a neighbor of it.

FIGURE 1: EXAMPLE OF APPLYING THE CONTROL ACTIONS ACCORDING TO THE COMMUNICATION DELAYS[20]

Suppose at time ‫ݐ‬௞ , the distributed system is synchronized and the optimal control inputs are applied for subsystems ݅ and ݆ , and the assumed state information begins to be transmitted from their neighbors to subsystems ݅ and ݆. The system states are measured at time instant ‫ݐ‬௞  ൅ ߜ . For subsystem ݅ , the time delays from its neighbors to itself are ߬௞௜௜ଵ , ߬௞௜௜ଶ , and , respectively. Since ߬௞௜௜ଵ < ߬௞௜௜ଶ < ߬௞௜௜ଷ , it has ߬௞௜  ൌ  ߬௞௜௜ଷ , and all the neighbors’ information has been ௜  ൌ  ‫ݐ‬௞  ൅ ߜ ൅  ߬௞௜ . For subsystem ݆, the received at time ‫ݐ‬ǁ௞ାଵ ௝௝ଵ ௜ ൌ time delay from its neighbor to itself is ߬௞ . Thus, ‫ݐ‬ǁ௞ାଵ ௝ ௝௝ଵ ௝ ௜ ‫ݐ‬௞  ൅ ߜ ൅  ߬௞ = ‫ݐ‬௞  ൅ ߜ ൅  ߬௞ . Since ߬௞  ൐  ߬௞ , the synchronized time of subsystems ݅ and ݆ will be at ‫ݐ‬௞ାଵ  ൌ ௝௝ଵ ‫ݐ‬௞  ൅ ߜ ൅  ߬௞ , and subsystems ݅ and ݆ will generate and apply the new control signals at time ‫ݐ‬௞ାଵ ሾʹͲሿǤ Moreover, ISS approach is used to show the stability of the overall system at the presence of time delay.

B. Distributed Model Predictive Control In order to implement the distributed MPC algorithm a cost function is defined for the subsystem as Np

IV.

J MPCi ( k ) = ¦x iT ( k + j |k )Q i x i ( k + j |k ) j =0

MAIN RESULTS

N c −1

+ ¦u

A. The Waiting Mechanism Transmitting the information between subsystems and controllers is subject to delay. To handle distributed NCS subject to time delay and interconnections, the waiting mechanism of [20] is employed. A simple example of delay mechanism is depicted in Figure 1. In this example, there are two subsystems. One is subsystem ݅ , neighbors ݅ଵ ǡ ݅ଶ ǡ and ݅ଷ , and the other is subsystem݆, neighbor ݆ଵ .

T i

j =0

+¦

j ∈N i

where

Np

(k

+ j | k ) R i u i ( k + j |k )

(6)

ª r ( x ( k + j |k ) − x a ( k + j |k ) )T j « ij i «Q x k + j |k − x a k + j |k ) j( )) ¬ ij ( i (

is the prediction horizon,

Nc

∗º » » ¼

is the control horizon

( k + j |k ) and u ( k + j |k ) denote the predicted state and control action of the ith subsystems at step k , Q i n ×n and R i m ×m are positive definite states control weights respectively and x aj ( k + j |k ) are the assumed state trajectories of the neighbors of the subsystem i , Qij > 0 (N c < N p )

,

x

T i

T i

479

2016 24th Iranian Conference on Electrical Engineering (ICEE)

݆

∈ Ni

are the given cooperation matrices;

rij , j ∈ N i

are the

cooperation weight. Suppose that each subsystem ݅ can receive information from its neighboring subsystems (neighbors) whose indices are denoted by N i with N i ⊆ {M } and N i ≠ φ . The cooperation among the subsystems is achieved by designing the cooperation weights and the control objective function of (6).

Problem 1: Let α3i ∈Κ∞ , J MPCi (.) in (5.7) and a CLF Vi (.) be given for the i th subsystem. At time k measure the state

x i (k )

and

minimize

the

cost

J MPCi (.)

over

U i ( k ):= (u i ( k | k ),u i ( k + 1| k ),...,u i ( k + N c − 1 | k )) subject to the following constraints

for all ‫ ܺ א ݔ‬and ‫ ܹ א ݓ‬. From this property and the second inequality in (7) we have V i (A i x i + B i u i + w i ) −V i ( A i x i + B i u i ) ≤ Li

(

)

(

)

(

©

V i ( Ai x i + Bi ui +w i ) −V i ( x i ) ≤ −α3i x i + Li w i

Thus, one can find σi ∈ K ,σi w i := Li w i such that

( )

V i ( Ai x i + B i u i + w i ) −V i ( x i ) ≤ −α 3i x i + σ i w i

For all x i ∈ X i . From V i ( x i ) ≤ α 2i (

xi

) or

α 2−i1 ( x i ) .V i ( x i ) ≤ 1

−α 2−i1

( x )V. ( x )α ( x ) + σ i

i

i

3i

i

i

i

( ) (w )

(7)

where λi = 1 −

i

(

ci ∈[0,1) . Then we have bi

)

k

V i x i ( k + 1) ≤ λik +1V i ( x i 0 ) + ¦ λi j σ i §¨ w ik − j

( )

(14)

©

· ¸ ¹

j =0

(15)

For all ‫ݔ‬௜଴ ‫ܺ א‬௜ ǡ ‫ݓ‬௜ ‫ܹ א‬௜ and ܼ݇߳ା . The following inequalities hold:

(

(

)

)

k

V i x i ( k + 1) ≤ λik +1α 2i x i 0 + ¦ λi j σ i §¨ w ik − j

for some

CLFs V i (.) and costs J MPCi (.) be given for all systems

i = 1,2,..., n . Suppose Problem 1 is feasible for each subsystem i = 1,2,..., n and for all states x i ∈X i . Then

indexed by

each ith subsystem is ISS ( X i , W i ) and the global interconnected dynamically coupled nonlinear system described by the collection of difference inclusions

x i (k + 1)∈f i (x i (k ), π i ( x i (k ))) +w i ( k )

§

k +1 · i ¸

(

)

(

) ¨¨© 11−−λλ

(

)

(

) ¨¨© 1−1λ ¸¸¹

i

§

(16)

¸ ¹

·

i

From α1i ( x i ( k + 1) ) ≤ V i ( x i ( k + 1) ) and inequality (16) one can yield that §

(

)

(

x i ( k + 1) ≤α1−i 1 ¨ λik +1α 2i x i 0 + σ i w ik ¨ ©

··

§

) ¨¨©1−1λ ¸¸¹ ¸¸ i

( )¹

x i ( k + 1) ≤α1−i 1 §¨ 2λik +1α 2i x i 0 ·¸ (18)

§ § 1 ·· ¸¸ +α1−i 1 ¨ 2σ i w ik ¨ ¨ ¨1− λ ¸ ¸ i ¹ © ©

is ISS(X,W).

Proof: We have that each continuous and convex CLF ܸ௜ ሺ‫ݔ‬௜ ሺ݇ሻሻ is globally Lipchitz on ܺ௜ . Hence, assuming ‫ܮ‬௜ ‫א‬ ܴା is a Lipschitz constant of ܸ௜ ሺǤ ሻ in ܺ௜ , one may write (8)

(17)

¹

Then it is trivial to show that ©

CL

· ¸ ¹

©

j =0

≤ λik +1α 2i x i 0 + σ i w ik

ai , bi , c i , δ i ‰ R , bi p c i , continuous and convex

V i (Ai x i + B i u i + w i −V i ( Ai x i + B i u i ) ≤ Li w i

(13)

i

≤ λik +1α 2i x i 0 + σ i w ik

Let α1i (s )  ai s ,α2i (s )  bi s ,α3i (s )  c i s

(12)

and inequality

V i ( Ai x i + B i u i +w i ) −V i ( x i ) ≤ λiV i ( x i ) + σ i w i

instant k ∈ Z + . Theorem 1: δi

(11)

Now let α 2i ( s ) = bi s δ and α3i = c i s δ then one can obtain

denote the difference inclusion corresponding to the system without considering the disturbance in closed loop with the set of feasible solutions obtained by solving Problem 1 at each

δi

(10)

Adding inequalities (9) and (10) yields

≤

x i ( k + 1) ∈ A i x i ( k ) + B i π o ( x i ( k ))) := {f i ( x i ( k ), u i ( k )) u i ∈ π o ( x i ( k ))}

δi

· ¸ ¹

V i ( Ai x i + B i u i +w i ) −V i ( x i ) ≤ −α3i x i + σ i w i

)

Note that u i ( k |k ) = u i ( k ). Let π o ( x i ( k )):= {u i ( k ) Ψ holds } and

(9)

V i A i x i ( k ) + B i u i ( k ) −V i x i ( k ) ≤ −α 3i §¨ x i ( k )

(12) we have

­ x i ∈ X i , u i ∈U i ° °V i ( Ai x i ( k ) + B i u i ( k ) ) −V i ( x i ( k ) ) ≤ −α 3i x i ( k ) ° °x i ( k + j + 1) = Ai x i ( k + j ) + B i u i ( k + j ) ; j = 0,1,..., N p Ψ® °x i ( k | k ) = x i ( k ) ° °u i ( k + j ) = u i ( k + N c − 1) ; j ≥ N c − 1 °x a ( k + j ) = A x ( k + j ) + B u ( k + j − τ ) ; j = 0,1,..., N j j j i k p ¯ j

wi

(

)

¹

Let us consider

(

)

§ 2σ

βi (s , k ) := α1−i 1 2λik α 2i (s ) , γ i (s ) :=α1−i 1 ¨ ¨ ©

i

(s ) ·¸

1 − λi

¸ ¹

where ߚ௜ and ߛ௜ are KL-function and K-function respectively. Consequently each ith subsystem is ISS(ܺ௜ ǡ ܹ௜ ) according to definition 1.

480

2016 24th Iranian Conference on Electrical Engineering (ICEE) In order to prove the ISS characterization of the global system, one can consider

x (k + 1)∈f

( x ( k ), π ( x ( k ))) +w (k )

CL

(19)

which denotes the difference inclusion for the global interconnected system in closed loop with the set of feasible solutions of each ith subsystem. n

Now let V

(x (k )) = ¦V i (x i (k )) be a CLF for the

which is decreasing in time. The distributed MPC control effort and state responses are shown in Figure 3 and Figure 4. As one sees, the state responses of different subsystems are stabilized at presence of the interconnection effects. Moreover, the proposed distributed MPC is compared with the state feedback control method. One could see that this approach cannot even stabilize the system. Figure 5 shows that the control effort grows unbounded and Figure 6 shows state responses which diverge to infinity.

i =1

8 6 1

J MPC

global interconnected system in (19). Each continuous and convex CLF V i (x i (k )) implies

2

Lipschitz continuity on X i . Thus V ( x ( k ) is also a

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J MPC

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2

Lipschitz continuous CLF for the global interconnected system. Hence, assuming L ∈ R + is a Lipschitz constant of

4

10 0 0

V (.) in X, one can work out that J MPC

10

3

5

V ( f (x , u ) +w ) −V (f (x , u )) ≤ L w (20) for all x ∈ X and w ∈W . Then the ISS(X,W) of the difference inclusion (5.21) follows similar to i th subsystem.

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Figure 2: Cost function of subsystems in distributed MPC

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Remark: The stability proof approach can be generally used for nonlinear systems [26].

2 0 -2 0 2

u3

0

SIMULATION RESULTS

-2 -4 0

Figure 3: Distributed MPC control effort 0.2

0

-0.2

x 22 x 31 x 32

21

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31

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( k ) + u1 ( k ) )

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( k ) + u 2 ( k ))

3

The proposed control method is used to show the effectiveness of MPC-based controller. Figure 2 shows the cost function of the distributed interconnected subsystems

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x 11 ( k + 1) = x 11 ( k ) +T x 12 ( k ) x 12

0

0

-0.5

x32

( ) ( k + 1) = x ( k ) +T ( x ( k ) − x ( k ) − x ( k ) + x ( k + 1) = x ( k ) +T ( x ( k )) ( k ) = x ( k ) +T ( x ( k ) + 0.5x ( k ) + x ( k ) + x ( k + 1) = x ( k ) +T ( x ( k ) ) ( k + 1) = x ( k ) +T ( x ( k ) + 0.5x ( k ) + u ( k ))

0.5

x12

The linearized discrete-time version of the interconnected system presented in [12] is considered to illustrate the design methodology for the distributed model predictive control. Thus, to study the behavior of this system via numerical simulation, this dynamic is discretized by Euler backward method and sampling time is assumed to be 10 milliseconds (i.e. T = 0.01):

x11

V.

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-0.5

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Figure 4: State response in Distributed MPC

Now consider the delay between controllers information exchange. The simulation results show that the delay effect can be handled if it is bounded. Simulation results show that the upper bound of delay is about 30 samples. Two different time delay in the case of 10 and 30 sample delay in controller information exchange is studied.

481

2016 24th Iranian Conference on Electrical Engineering (ICEE) 20 10

17

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u1

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Figure 8: Distributed MPC control effort with 10 sample delay

Figure 5: control effort in state feedback control

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Figure 6: State response in state feedback control

In Figure 7 cost function of subsystems subject to 10 samples delay is shown. The cost functions are decreasing so that the overall system can be stabilized. Figure 8 indicates that the control efforts corresponding to each subsystem remain bounded and the control inputs are satisfied the constraint -2Ͳ ൑ ‫ ݑ‬൑ ʹͲ. The state responses are shown in Figure 9 where all states converges to desired values. In other hand, a delay in the information exchange of each subsystem causes the control input to adapt itself so as to accommodate the delay.

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Figure 9: State response in Distributed MPC with 10 sample delay

It can be deduced from Figures 7 to 9 that the system subject to 10 samples delay is stable and also the states converge to zero. To show the effectiveness of the proposed distributed MPC strategy, the delay bound is increased. Simulation results show that the closed loop system can tolerate the delay up to 30 samples which cause marginal stability. Figure 10 shows that state responses do not converge to zero and the responses are oscillatory. This oscillatory responses originate from nondecreasing cost function, and oscillatory control effort due to time delay in information exchange. From Figures 10 it can be seen that the upper bound of delay is about 30 samples. Consequently, system will be unstable with more than 30 samples delay.

4

1

J MPC

6

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2016 24th Iranian Conference on Electrical Engineering (ICEE) VI.

CONCOLUSION

This paper presented an approach for distributed model predictive control with time delay. The overall system is consists of a number of discrete-time interconnected subsystems at the presence of time delay. In order to deal with problem of time delay in exchanging the information between controllers a waiting mechanism is employed, and an upper bound is considered for the delay. Furthermore, the distributed MPC strategy was implemented for each subsystem with the MPC algorithm subject to some constraints. It was showed that the proposed method guarantees input-to-state stability characterization for both local subsystems and the global system under some predetermined assumptions. The proposed approach was investigated for a linear discrete-time system to make a practical sense.

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