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Although distributed model predictive control has caused significant attention and received many good results, the results are mostly under the assumption that the system states can be observed. However, the states are difficult to be observed in practice. In this paper, a novel distributed model predictive control is proposed based on state observer for a kind of linear discrete-time systems where states are not measured. Firstly, an output feedback control law is designed based on Lyapunov function and state observer. And the stability domain is described. Furthermore, the stability domain as a terminal constraint is added into the constraint conditions of the algorithm to make systems stable outside the stability domain. The simulation results show the effectiveness of the proposed method.

In industrial processes, there exists a class of hybrid systems which are comprised of some subsystems which couple each other through energy, quality, etc. For example, urban drainage network system, transportation system, energy power, Net system and irrigation system. These systems have many components, wide space dis- tribution, many constraints and many targets. We can obtain good control performance if the centralized control is used to control this kind of systems. But its flexibility and fault tolerance are relatively weak. If the distributed control is adopted, its flexibility and fault tolerance are better [

Model predictive control (MPC) is receding horizon control which can deal with the constraints of systems states and inputs during the design of optimization control [

In recent years, the research on the distributed predictive control method has developed greatly. There have been many beneficial results about it. In Literature [

However, in the actual application, the limitation of measuring equipment in economy makes the state feedback hard to realize. In reference [

In this paper, a distributed predictive control method based on Lyapunov function and state observer is designed to optimize the overall system’s performance. This algorithm adds the quadratic function of the decoy system's input variables to the performance index of the subsystem, expands the coordination degree, and optimizes the performance of the system.

This paper is arranged as follows. In the second section, the control problem for distributed system under network mode is described in detail. The output feedback controller based on Lyapunov functions and state observers is designed in the third section, and the stability domain is given. The fourth section designs distributed predictive controller. In the fifth section, the distributed prediction controllers performance is analyzed, and the steps of the algorithm design are given. The simulation results verify the effectiveness of the method proposed in this paper in the sixth section. Conclusion is given in Section 7.

Consider the distributed system S which is comprised of m related subsystems

where

satisfies

stant matrices with corresponding dimension, respectively. The distributed structure of the system under the network mode is shown in

Synthesizing all subsystems, we can get the system model as:

where,

The control objective is to design an output feedback control law for the linear discrete-time distributed system (3) (4) based on Lyapunov function and state observer under the premise that network connectivity and fault tolerance of the system are not added. And then the stability domain is described. Furthermore, taking the stability domain as a terminal constraint to design output feedback model predictive controller in order to make the system stable outside the stability domain. Make sure that under the premise of initial feasibility, the system is successive feasible.

This section shows the controller design based on Lypunov function under the states are available at first to get the stability domain description. Then it shows the output feedback controller design under the states are not available.

Consider the subsystem (1) (2), and structure the state feedback controller as follows:

where

Assumption (i). For the subsystem

Define the following matrices:

satisfy:

where

and

where

Lemma 1. If the Assumption (i) is satisfied, there exists a non-empty set

proof. Select a Lyapunov function candidate

The difference of

Since

Since the input constraint

And since

The proof is completed, and the set

Therefore, all states from

Thus, the stability domain of the subsystem

Suppose that at

Design the state observer as follows [

where

Therefore, the error dynamic equation of the state observer is regarded as a new autonomous system. That is to say if the new system (7) is stable, the estimation states can track the real states well.

Define a quadratic function on the observe error as follows:

where,

Theorem 1. Consider the error dynamic equation of the state observer (7), if there exist matrices

is satisfied, then the inequality

is satisfied, where

Proof. By the Schur complement lemma, the inequality (8) is equivalent to

Substituting

Multiply

By (8), we have

the inequality is satisfied.

Therefore,

Remark 1. By Theorem 1, the state observer gain F can be computed off line through the feasibility of the linear matrix inequality (8).

Thus, for the given

tically stable at the origin. There exists

real number

Lemma 2 [

This section studies the design of model predictive controller when states are not measured. Since the input constraint is related to the observer states, states constraint is constraints of the real states, and there are some errors between real states and observer states, the observer errors have influence on the future input and states. So the observer states are used in the performance index directly to design the controller. In order to keep the system stable, we adopt infinite horizon model predictive control strategy. Therefore, the optimization problem at time k is as follows:

where

The optimization problem decomposes into two parts as

Suppose the Lyapunov function

satisfies the stability constraint

When the closed loop system is stable,

Superpose (16) from

That is to say

Therefore, the optimization problem (15) transforms into minimizing

We have the following predictive model based on state observer of the subsystem

Its partial derivative is

Because the control law of the subsystem affects not only the performance of its own subsystem, but also that of its downstream subsystem, controller

where

And in order to improve the convergence of optimization problem, the weighting coefficients are added.

Next, the model predictive control optimization problem of all subsystems in the distributed model predictive control algorithm is shown as:

Problem 1. For subsystem

s.t.

where,

constraint set as

Give the following assumption:

Assumption (ii). At initial moment

bounded.

The distributed model predictive controller based on Lyapunov function and state observer is designed on the condition of initial feasibility, so the main content in this section is to ensure successive feasibility and stability.

This part mainly studies: if the system is feasible at time

By

which satisfies the stability condition (20).

Lemma 3. If the Assumption (i) and (ii) are satisfied, and the problem (17)-(22) have feasible solution at any time

we have

Proof. Since the problem (17)-(22) have feasible solution at any time

Therefore,

From

thus

Lemma 4. If the Assumption (i) and (ii) are satisfied, and the problem (17)-(22) have feasible solution at any time

then for all

and

Proof. When

The above two formulas subtract, we have

By Theorem 1, there always exists a

Obviously, (23) is satisfied.

Furthermore,

where,

Therefore,

When

The above two formulas subtract, we have

Therefore, (23) is satisfied.

Next we can derive that

Lemma 5. If the Assumption (i) and (ii) are satisfied, and the problem (17)-(22) has feasible solution at any time

Proof. Since the problem (17)~ (22) has feasible solution at any time

then,

Therefore,

Lemma 6. If the Assumption (i) and (ii) are satisfied, and the problem (17)-(22) has feasible solution at any

time

Proof. By triangle inequality, we have

then,

Remark 2. According to Lemma 2 to 6, if the assumption (i) and (ii) are satisfied, then

Theorem 2. If the Assumption (i) and (ii) are satisfied, the control law satisfies the constraint condition (18)- (22), and design parameters

then the system asymptotically stable at the origin.

Proof. When

Define

By the constraint (20), we have

so

For

Since

By Theorem 2, we have

By Lemma 4, we have

Substitute (25)-(27) into (24), we derive

therefore , when

the system is asymptotically stable.

We give the distributed model predictive control algorithm based on Lyapunov function and state observer.

Algorithm Off-line part:

1. Give decay coefficient

2. By Theorem 1, we obtain observer gain

On-line part:

1. Choose the appropriate parameter

2. Initialize

3. Receive

4. Let

Consider the distributed system under networked control as follows:

where:

that is this system has two subsystems.

Subsystem 1:

subsystem 2:

where,

Let the subsystem control constraint as

We use the Matlab simulationtools to simulate the algorithm proposed in this paper:

By the algorithm above, we can obtain that the stability domain of the subsystem 1 and 2 shown in

From the simulation results, we can see the algorithm can guarantee that estimation stats track the real states well, and asymptotically stable to the origin. We can also see that the control low satisfied the constraint and stable eventually.

For a kind of the distributed systems with input and state constraint and unavailable states under networked con-

trol patten, we consider the design and stability problem of the output feedback predictive controller based on Lyapunov function and state observer. The main idea is: For the considered system, use Lyapunov function and

states reconstruction to design output feedback controller in order to get the stability domain. Furthermore, the stability domain as a terminal constraint, the distributed model predictive controller is designed. The controller is successive feasibility under the condition of initial feasibility. The simulation results verify the effectiveness of the method proposed in this paper.

Baili Su,Yanan Zhao,Jinming Huang, (2016) Networked Cooperative Distributed Model Predictive Control Based on State Observer. Applied Mathematics,07,1148-1164. doi: 10.4236/am.2016.710103