Design of H ∞ Predictive Controller for Networked Control System

: For a class of network control systems with both data packet dropout and network communication delay problems, a new robust model predictive control method with compensation function is proposed. Considering that the system has interference problems, in the two cases of long-delay and short-delay, the packet loss problem is established as a Bernoulli sequence, and then a discrete NCS model based on the state observer is obtained. The state observer in the model can deal with the data packet dropout compensate and predict the state of the long-delay problem. Through linear matrix inequality and Lyapunov method, the controller is designed to obtain sufficient conditions for the closed-loop system to be exponentially stable and meet the specified performance indicators. Finally, compared with the method without any compensation measures, the method in this paper can get better control effect.

system, and a better control effect is obtained. The problem of data packet dropout in the actual field is a random process, so it is very necessary to establish a random model of the network control system. Literature [11][12] regards the problem of data packet dropout as a random problem, and describes it as a random effort. Sequences and random processes based on Markov. At the same time, the actual engineering systems are all disturbing, and many current articles are studying pure linear systems. In [13] considering the data packet dropout and long network communication delay at the same time, the controller design with disturbing network control system is carried out. Literature [14] compensates the control signal according to the operating parameters of the random long-delay system, so as to achieve the stable control of the random long-delay system. Literature [15] studied the problem of nonlinear networked quantitative control with network communication delay. Literature [16] describes the application of the truetime toolkit in the network control system. The simulation of numerical examples in the article can visually see the effect of the control system. In this paper, the data packet dropout problem is established as a Bernoulli model, considering the short-delay and long-delay conditions respectively. For the state unobservable problem, a linear system model based on the state observer is constructed to compensate for the data packet dropout and predict the state. Using Lyapunov's theorem and linear matrix inequality and other methods, a controller design method that satisfies the specified performance index is given, so that the closedloop system is exponentially stable and can withstand certain external disturbances.

System Modeling
The network control system structure considered in this article is shown in Figure 2. Suppose the state space of the continuous controlled object is described as is the input of the controlled are constant matrices with corresponding dimensions.
, the discrete time model of equation (1) can be obtained as Due to the packet loss phenomenon in the actual system, the controller can be described as , where α are known constants, and P is the value random Bernoulli sequence of 0 and 1.

Short network communication delay
When m=1, a short-delay event occurs. This event is marked as 1 E , and the probability of occurrence is 1 r . When the network communication delay is less than one sampling period, that is, can be written in the following form.
. So the discrete model of the controlled object can be described as the following form: Since the state of the controlled object in the network control system is not measurable, a full state observer is designed to reconstruct its state. The steps are as follows: Step 1: Use the output value k x to correct the value: Step 2: Calculate the state value τ + k x and control quantity k u at the time of delay τ + T : Step 3: Calculate the state value of the observer at time (K+1)T: Step 4: Define the error variable

Long network communication delay
When a long-delay event occurs at 1 > m , the mark is 2 E , and the probability is 2 r . Assuming that the network-induced delay is greater than one sampling period and the part exceeding one period is equal to the value of the short-delay. Taking 2 = m as an example below, the method can be extended to the case of 2 > m . The same can be obtained . Since the state of the controlled object in the network control system is not measurable, a full state observer is designed to reconstruct its state. The steps are as follows: Step 1: Use the output value a to correct the value: Step 2: Calculate the state value τ + k x and the control quantity k u at the time of delay τ + T : Step 3: Calculate the state value of the observer at time (K+1)T: Step 4: Define the error variable Thus, we can obtain the following derivation, we can get that the method has generality and is suitable for the long time delay of 1 > m .

Stability Analysis
The feedback controller based on the state observer is designed for the above closed-loop system, so that the system meets the following requirements: (1) When external disturbance 0 = k G , the closed-loop system gradually stabilizes.
(2) Under zero initial conditions, the controlled output k z of the closed-loop system satisfies the Where γ is a given scalar. Lemma 1: For the asynchronous state system Where * represents the symmetric item in the symmetric matrix, the system is asymptotically stable.
Proof: By simplifying the formula (3) of Lemma 1, we can get: ( ) ( ) According to Lemma 2, we can get According to Lemma 3: 0 < Ω is equivalent to: there is a constant 0 < ε such that  I  I  I  I  R  Q  P  diag , , , , , , is combined, the conclusion of Theorem 1 can be obtained. Remark 2: , , , , , and a controller coefficient K matrix, the following inequality holds Then the closed-loop system meets the above ∞ H performance index. Proof: Choose the Lyapunov function in Remark 1 as follows: The Bernoulli probability condition satisfies:  I  I  I  I  I  I  Q  R  R  P  diag  ,  ,  ,  , , the controller parameter can be obtained by Easy to get when 0 = can be obtained: to the above formula can be obtained: , and the closed-loop system is gradually stable,

Example Simulation
The matrix of known coefficients is   The network scheduling situation of the system is shown in Figure 3, where the low level of the curve represents the idle state, the high level represents the sending state, and the medium represents the waiting state.
The curve can indicate that the controller node, sensor/actuator node, and interference node all have packet loss and long delay, but the system state can still be stable, which further verifies that the method proposed in this paper is effective.
, the state response curve and control output curve can be obtained through MATLAB/Truetime simulation as shown in Figure 4 and Figure 5, respectively. Through the simulation curve, the system state variables and control output can be gradually stabilized in a small range. Meet the given ∞     By comparison, adopting the design method of this paper, the curve oscillation amplitude is small, the overshoot is small, and better control performance can be obtained.

Conclusions
In order to solve the problems of network communication delay, data packet dropout and interference in the network control system, a new robust model predictive control method with compensation function is proposed. Data packet dropout has the characteristics of Bernoulli sequence random binary distribution; at the same time, the network communication delay in the network control system conforms to the asynchronous dynamic theorem, and the delay less than one sampling period and the delay greater than one sampling period are classified into short-delay and long-delay. Because the state in the system is not measurable, the state observer is used for predictive control, and the ∞ H controller strategy is designed based on Lyapunov's theorem and linear matrix inequality. Finally, the effectiveness of the method is verified by simulation. The curve shows that the system can maintain stability in the presence of external interference. Therefore, the ∞ H controller designed in this paper is feasible.