Improved Distributed Model Predictive Control with Control Planning Set

We focus on distributed model predictive control algorithm. Each distributed model predictive controller communicates with the others in order to compute the control sequence. But there are not enough communication resources to exchange information between the subsystems because of the limited communication network. This paper presents an improved distributed model predictive control scheme with control planning set. Control planning set algorithm approximates the future control sequences by designed planning set, which can reduce the exchange information among the controllers and can also decrease the distributed MPC controller calculation demandwithout degrading the whole system performancemuch.The stability and system performance analysis for distributedmodel predictive control are given. Simulations of the four-tank control problem andmultirobotmultitarget tracking problem are illustrated to verify the effectiveness of the proposed control algorithm.


Introduction
Model predictive control (MPC), also referred to receding horizon control (RHC), is an attractive control strategy because of its ability to control systems with input and output constraints in the optimization problem.The input sequence is calculated by solving an optimization problem (minimization of a given performance index) over a prediction horizon.Once the optimization problem is solved, only the first input value is implemented into the system.In the next sampling time, a new optimization problem is solved repeatedly.MPC has been widely applied in various control areas over the past few decades [1][2][3].
Nowadays, systems are becoming more and more complex.In centralized MPC, all the inputs sequences are optimized with respect to one given performance index in a single optimization problem.However, when the number of the state variables and inputs of the system becomes larger and larger, the computation burden of the centralized optimization problem may increase significantly.Moreover, the entire system would be out of control if the centralized MPC controller fails.Therefore it is impractical to apply the centralized MPC to large-scale systems.In fact, a largescale system is composed by physically parted subsystems.Many decentralized and distributed model predictive control (DMPC) algorithms have been recently proposed [4][5][6][7], which are some feasible alternatives to overcome the computational burden of the centralized MPC.
In DMPC architecture, subsystems communicate with each other via networks and the inputs are computed by solving more than one optimization problem in each subsystem in a coordinated fashion.There are many achievements on DMPC strategy and a survey of major DMPC algorithms is presented in [8,9].The existing DMPC algorithms can be divided into different categories.
Based on the topology of the communication network, DMPC can be divided into fully connected algorithms and partially connected algorithms.In fully connected algorithms, DMPC is able to communicate with the rest of the local controllers [10,11].In partially connected algorithms, local optimization problems are solved by taking into account the neighboring (not the whole system) interaction and solution, which is suitable for loosely connected subsystems [12,13].However, it will deteriorate the whole system performance.
Based on the exchange times among the distributed controllers, DMPC can be divided into noniterative algorithms and iterative algorithms.In iterative algorithms, information is transmitted among the DMPC controllers many times in the sampling interval [14,15].On the contrary, in noniterative algorithms DMPC controller communicates with the other controllers only once in the sampling interval [16,17].
In this article, we consider that the DMPC controllers can exchange information only once while they are solving their local optimization problems at each sampling time and the connectivity of the communication is sufficient for the distributed controller to obtain information.This paper proposes an extension of the fully connected noniterative DMPC algorithm.However, the exchange information between subsystems is usually realized over a digital communication network.Thus, the local systems can only have limited communication resource.For example, in a networked environment, bandwidth limitations can restrict the amount of exchange information.Thus, it is necessary to restrict the distributed controllers to exchange information.The proposed DMPC in the paper reduces the communication information compared to the standard distributed MPC control scheme in complex large-scale systems and at the same time decreases computational burden of each controller.This algorithm also provides a reasonable trade-off between system performance and low communication requirements needed to reach a cooperative solution.
The rest of the paper is organized as follows.In Section 2, the centralized and distributed model predictive control problem is formulated.In Section 3, the improved distributed model predictive control with control planning set (CP-DMPC) is proposed.The stability and performance analysis is provided in Section 4. In Section 5, the simulations of the proposed controller to four-tank system and multirobot multitarget tracking system are presented.Finally, the conclusions of the work are given in Section 6.

Centralized and Distributed Model Predictive Control Formulation
Without loss of generality, suppose that the whole system is comprised of  interconnected subsystems.And consider that each subsystem only couples through the input [18].The discrete-time state-space model for th subsystem is as follows: , ( + 1) =  ,  , () +  ,   () where  = 1, . . ., .  , (),   (), and   () are the state vector, the control input vector, and the output vector of th subsystem at kth sampling time.The model (1a), (1b) is changed to suit the model predictive control design with an embedded integrator.The augmented model of the th subsystem state space model is where a new state variable vector is chosen to be and a new control variable vector is chosen to be and the difference of the state variable is denoted by The state interaction vector is given by The triplet   , [  ,   ],   is The model of the whole system (centralized model) can be expressed in compact way Figure 1: Centralized MPC control system architecture.
with state vector () ∈    , control input vector Δ() ∈    , and output vector () ∈    .A, B, and  are the whole system matrices.This implies that

Model of subsystem i
Setpoint

Controller information exchange Inherent interaction
Inherent interaction ] is applied to the whole system; after new measurements are available, a new optimization problem is solved in the next sampling time.
Many engineering applications such as power systems, unmanned aerial vehicles, sensor networks, economic system, transportation systems, and process control systems, have become larger and more complex.The overall number of inputs and states (outputs) is very large, and the optimized control sequence Δ * (,   | ) is highly dimensional.A single optimization problem may require computational resources (CPU time, memory, etc.).In view of the above consideration, it is natural to look for distributed MPC algorithms.

Distributed Model Predictive Control Formulation.
In the distributed model predictive control formulation, the large size optimization problem is replaced by  small ones that work cooperatively towards achieving the performance of centralized control system.And the following assumptions are made.The th subsystem minimizes the following local performance index, which is the th optimization problem [19]:

Improved Distributed Model Predictive Control with Control Planning Set (CP-DMPC)
Besides the computational advantages of DMPC, the amount of data needs to be exchanged among distributed controllers.
In the paper, fully connected noniterative DMPC algorithm is focused on.However, each system exchanges information with each other by both their initial state and their optimized input.And time delays exist in communication network.In Figure 3, we can see that time delay consists of three parts, sensor measurement delay, DMPC controller calculation delay, and controller information communication delay.
In this paper, a control planning set algorithm is combined with DMPC controller to reduce the controller information communication delay and meanwhile it also can decrease the DMPC controller calculation demand without degrading the whole system performance much.The control planning set method presented in the paper is inspired by the pulse-step control strategy [20].Suboptimal strategies can be obtained by restricting the future control sequence For specification and simplicity, we choose function  as a linear function: In the control planning set algorithm, the future control sequence is restricted by one possibility.The parameter  is chosen to plan the future control sequence increases or decreases in the same direction, which is suitable for the experience of control engineering.And it will prevent the frequent oscillation of the control input; see Figure 4.
In a traditional MPC scheme, the optimized control sequence is calculated via the performance index, which may oscillate during the control horizon.In CP MPC scheme, the optimized control sequence changes in one direction, which may not obtain the optimum solution but is suitable for the control engineering.In control engineering, in some time period control value does not change suddenly and frequently, and this is good for the control hardware device.
If  = 1, the control sequence is set in equal increase.If  > 1, the weight of the future control is larger than that of the current control.
Let one assume that Lemma 1.The interaction predictions of th subsystem at time  are given by and the compact predictions have the following form: Proof.With ( 6) and ( 13), the prediction of the interaction vectors of time  is given by . . .
Lemma 2. The state and output predictions of th subsystem at time  are expressed by and the compact predictions have the following form: Proof.With (2a), (2b), and ( 13), the state and output predictions of th subsystem at time  are expressed by By definitions (14a)-(14l), this implies the relations (18) where Proof.Using the local performance index (11a), the cost function can be written in the equivalent form Applying (18) into it, the local performance index   takes the form (21).
Theorem 5.For th subsystem, the explicit form of the control law is given by And the compact expression is where The distributed MPC algorithm with control planning set (CP-DMPC) can be summarized as shown in Algorithm 1.

Stability and Performance Analysis
4.1.Stability Analysis.We provide sufficient conditions that guarantee practical stability of the closed-loop system.

Theorem 6. The closed-loop system with 𝑁 subsystems is asymptotically stable if and only if
Proof.Combining the process (8a) and (8b) and control law (23), the closed-loop state-space representation is derived: ( Define the extended state Remark 8.However, the computation of the optimization problem is reduced greatly because of the dimension reduction of the optimal variables.The control value is calculated as In the traditional DMPC algorithm, when the number of subsystem inputs and the control horizon becomes large, the optimized control sequence Δ  ( | ) = [Δ  ( | ), Δ  ( + 1 | ), . . ., Δ  ( +   − 1 | )] is highly dimensional.The matrices Φ  have also high dimensions.The computation load of (10a) and (10b) is mainly to calculate the inverse of the matrix (Φ     Φ  +   ) −1 , which may require significant computational resources.
In CP-DMPC algorithm, Φ  is a vector not a matrix.Compared with (10a) and (10b), the computation load of ( 21) is lower because of no calculation of the matrix inverse.As a result, the CP-DMPC controller decreases the computation demand greatly.

Simulations and Results
In this section the theoretical results are illustrated using two different examples.The first example is focused on the process control system, four-tank system whose sampling time interval is about several seconds.The second example is focused on the motion control, multirobot target tracking scenario whose sampling time interval is about milliseconds.All the simulations are run in MATLAB on the same computer with Intel(R) Core (TM) 2.6 GHz processor and 8 GB RAM.

System Description.
The four-tank problem used in the section is described by [21][22][23] and the description of the system is shown in Figure 5.It is a multivariable system with two manipulates variables and four state variables.The differential equations that model the nonlinear dynamics of the system can be expressed as where the parameters in (29) can be found in Table 2.
The set-point levels of tank 1 and tank 2 are as follows: (1) From 0 s to 1000 s, the set-point of tank 1 is 0.65 m and the set-point of tank 2 is 0.65 m.
(2) From 1001 s to 3000 s, the set-point of tank 1 is 0.3 m and the set-point of tank 2 is 0.3 m.
(3) From 1001 s to 3000 s, the set-point of tank 1 is 0.5 m and the set-point of tank 2 is 0.75 m.
From Figures 6, 7 and 8, we can conclude that CMPC has the best control and that CP-DMPC can also have similar control performance as traditional DMPC (noniterative).But from Figure 9, we can see that the CMPC and traditional DMPC provide a higher optimization time than CP-DMPC algorithm.[

Multirobot Target Tracking
where The objective of the whole system is to track a target with  robots and to keep the distant between the robots and the target.Meanwhile there will not be a collision among the robots during tracking the target.As a result, the local performance index of the ith robot can be selected as  We simulate the scenario from time  = 1, . . ., 30 s.The target moves according to the dynamic (25) with the sampling time   = 1 in the area collectively monitored by the three robots states above.The initial positions of three robots is (20 m, 0 m), (−20 m, 0 m), and (0 m, −10 m).The maximum velocities of robots are 2 m/s of the  and  coordinates.The initial positions of the target is (−10 m, −10 m) and the target motion trajectory is illustrated in Figure 10.

Simulations with CP-DMPC Algorithm.
In this section, the system performance of two control algorithms is compared, which are DMPC (noniterative) and CP-DMPC.Both of these strategies have the same input constraints, input and output weights, prediction, and control horizon.The parameters used in the simulations are ).The parameter used in CP-DMPC is  = 1.The traditional DMPC and CP-DMPC algorithms are applied to the scenario by the same parameters.
The trajectories of three robots and target and four typical snapshots at time = 1, 10, 20, 30 are depicted in Figure 11.The simulation results demonstrate that the multirobot system with the CP-DMPC controller can track the target well.

Comparisons between Traditional MPC and CP-DMPC
Algorithm.In the section, we compare the computational  1).
In [24], communication energy is made up of transmitting energy   and receiving energy   :   (, ) = ( 1 +  2  (, )  ) , where (, ) is the distance between the two robots,  is the path loss index,  is a transmitting data rate, and  1 ,  2 are constants (45 nJ/bit and 10 pJ/bit).And the receiving energy   is constant, which is 135 nJ/bit.The computational complexity corresponds to the number of operations required to complete the task, where an operation is defined as a combination of one addition and one multiplication.And model predictive control requires the solution of an open-loop optimal control problem at every sampling instant.In the paper, we use fast gradient method which has low implementation calculation and numerical robustness.
The two optimization problems between traditional DMPC and CP-DMPC algorithm are evaluated 50 times.The simulation results are shown in Figure 11.From Figure 12(a), the traditional DMPC provides a lower performance cost (better system performance) than CP-DMPC algorithm.

➁
(a) Predictive horizons   and control horizons   are the same for each subsystem.(b) Controllers are synchronous.(c) Controllers communicate with each other only once within a sampling time interval.(d) Controllers are interconnected and can obtain information which the controllers need.And the DMPC control system architecture diagram is shown in Figure2.DMPC-i (i = 1, . . ., N) controller calculation delay

Figure 3 :
Figure 3: Delay time analysis per sampling interval.

9 Figure 4 :
Figure 4: The comparison between traditional MPC and CP MPC.

Figure 5 :
Figure 5: Description of the four-tank system.

Figure 6 :
Figure 6: Dynamic response of the four-tank system of centralized MPC for tracking.

Figure 7 :
Figure 7: Dynamic response of the four-tank system of DMPC for tracking.

Figure 8 :
Figure 8: Dynamic response of the four-tank system of CP-DMPC for tracking.
Here   is the prediction horizons and   is the control horizons.And   ≥   .  and   are penalties on the output variables and control variables, respectively.   is the output set point.And because the central controller can handle all the information of the system, the interaction predictions   ( +  | ) are known at time .(,   | ) = [Δ * ( | ), * ( +  | ) −    ( + ) ( +  + 1) =     ( + ) +   Δ  ( + ) +   ( + −1) ,   ( + ) =     ( + ) .(11b)It can be seen that the global performance index can be decomposed into a number of local performance indexes, but the output of each agent is still related to all the input variables due to the input coupling.Because controllers communicate with each other only once within a sampling time interval, the interaction predictions   ( +  | ) are unknown for the th subsystem.And only the prediction   ( +  |  − 1) based on the information broadcasted at time  − 1 is available.A noniterative algorithm is developed to seek the distributed solution at each sampling time.Based .In CP-DMPC algorithm, the optimal variable is Δ  ( | ) and the dimension of variable is   , which decreases greatly.As a result, exchange information among the CP-DMPC controllers reduces from  *    to   .

Table 1 :
Metrics comparisons among different algorithms.

Table 2 :
Parameters of the four-tank system.