Abstract
In this paper, the H∞ control problem is investigated for a class of time-varying state-saturated systems with both stochastic parameters and nonlinearities under the stochastic communication protocol (SCP). The sensor-to-controller network is considered where only one sensor can obtain access to the communication network at each transmission instant. The SCP is adopted to mitigate the undesirable data collision phenomenon. The model transformation technique is employed to simplify the addressed problem, and then, the completing squares method is carried out to obtain a sufficient condition for the existence of the finite-horizon H∞ controller. The controller parameters are characterized by solving two coupled backward recursive Riccati-like difference equations. A simulation example is finally utilized to illustrate the effectiveness of the proposed controller design scheme.
1. Introduction
Due to its advantages of easy sharing, strong flexibility, and easy installation and maintenance, networked control systems (NCSs) have attracted a large number of scholars’ attention in recent years [1–4].
Most existing methods of the controller and the filter design are only for infinite time-invariant controlled object. In the actual NCSs, due to the operating point offset, equipment aging, and environmental factors, many systems are time varying. Therefore, it is of more practical significance to study the transient characteristics of time-varying systems in the finite horizon than the steady-state characteristics of time-invariant systems. The corresponding fuzzy control problem, finite-time domain filtering problem, and envelope constraint problem for continuous time-varying system, discrete time-varying system, and discrete time-varying nonlinear system are discussed in [5–7], respectively.
In NCSs, the bandwidth of communication network is limited, which tends to cause data conflicts of control or measurement signals in the process of network transmission, resulting in packet loss and network-induced delay [8, 9]. If multiple nodes in the network attempt to simultaneously transfer data, these phenomena caused by nonideal networks may be even more serious. Therefore, an effective way to avoid data conflicts is to join the communication protocol to manage the rights of each node to access the network. Random communication protocols have been widely studied and applied for their advantages of random access and strong expansibility [10, 11]. In [12], the filtering problem of a kind of linear time-invariant system under a random communication protocol was studied, which described the communication process between the sensor and the filter as a Markov chain. In [13], the auxiliary vector and a new state estimation structure were proposed to the problem of finite-time state estimation of delayed artificial neural network under the random communication protocol. In [14], a random pulse delay model was proposed under the consideration of packet loss, network-induced delay, and random communication protocol. Furthermore, an output feedback controller was designed to ensure the stability of the system in the mean square sense. In [15, 16], the sliding mode control and robust finite-time stability problem with time-varying delays were investigated.
It is important to note that the above literature implicitly assumes that the system state is not saturated. However, in practice, due to the physical limitations or protection measures of the equipment, state saturation often occurs. For example, mechanical systems with position and speed limits, digital filters implemented in finite word length formats, and neural networks with saturated transfer functions [17, 18]. In these cases, the state of the system is constrained within a bounded set. If state saturation is not considered in the comprehensive analysis, the desired performance of the closed-loop system cannot be guaranteed. Although the control and filtering of NCSs in infinite time domain and random communication protocol have been studied in depth, the control problem in finite-time domain of state-saturated time-varying system has not attracted attention under the influence of random communication protocol, which arouses the author’s research interest.
In summary, this paper is concerned with the control problem of the state saturation system in the finite-time domain based on the random communication protocol and considering the influence of random parameters and random nonlinearity. The main contributions in this paper are as follows: (1) both stochastic parameters and nonlinearities under the stochastic communication protocol are considered in the design of the output feedback controllers guaranteeing the desired performance, (2) a sufficient condition is provided for an auxiliary index which is closely related to the desired finite-horizon performance, and (3) a suboptimal controller design scheme is provided by solving two coupled backward recursive Riccati-like difference equations.
The rest of this paper is organized as follows. Section 2 introduces the problem description. Sufficient conditions for the performance analysis and controller design are given in Sections 3 and 4. In Section 5, a numerical example is presented. Finally, Section 6 concludes the paper.
Notations: the notations used in this paper are as follows. Rn denotes the n-dimensional Euclidean space. is the space of square summable vectors. stands for the mathematical expectation and means the occurrence probability of the event. In symmetric block matrices or complex matrix expressions, we use to represent a term that is induced by symmetry and denotes the block diagonal matrix. If their dimensions are not explicitly stated, they are assumed to be compatible for algebraic operations. The notation () means that is a symmetric and positive definite (semidefinite) matrix. and represent the zero matrix and identity matrix with compatible dimensions.
2. Problem Description
Consider the state-saturated random time-varying system in the finite horizon :where , , , and , respectively, represent system state, control input, controlled output, and measured output vector before transmission from the sensor-to-controller network. and are process noise and measurement interference of the system, respectively. The matrices , , , , and are time-varying parameter matrices with known appropriate dimensions. is a saturation function, is a random parameter matrix, is a zero-mean Gaussian white noise sequence with variance 1, and is a random nonlinear function of the system. The random parameter matrix and the random nonlinear function are independent of each other.
As [19] shows, the random parameter matrix has the following statistical characteristics:where is a known parameter and is th element of matrix , i.e., .
For all , the first moment and covariance of the random nonlinear function satisfywhere is a known nonnegative integer and and are known appropriate dimension matrices.
Define saturation function :where is the component of vector and is the saturation level.
In order to solve the problem of network congestion and data conflict caused by multiple transmission nodes using the communication network at the same time, random communication protocol is adopted in the network. Under the SCP scheduling, only one node is allowed to access the network and transmit data at each transport moment. is defined as the node that obtains access to the network at the instant k, and its value is determined by the Markov chain. The transition probability matrix is , specifically defined aswhere represents the transition probability of moment from state to state and satisfies .
Set the measurement output after transmission from the sensor-to-controller network as , and represents the measurement output of the sensor node received by the controller at the transmission time . Zero input processing strategy is adopted, and the updating rules of are
On the basis of formula (9), there arewhere and is the Kronecker Delta function.
For system (1), the following output feedback controller is designed:where is the controller gain to be designed.
According to formula (2), can be expressed as , and is defined for ease of analysis. The following closed-loop system is obtained by substituting equations (10) and (11) into system (1):
The purpose of this study is to design a finite-time domain output feedback controller (11) so that, for a given disturbance suppression level , positive definite matrix , and system initial state , the controlled output of the system meets the following performance requirements:
3. H∞ Performance Analysis
System (12) contains , so to facilitate the solution of controller gain , we define and . From the definition of , perturbation rewrites formulae (12) to (14). The positive real number is introduced to describe the scale of measurement interference and to provide more degrees of freedom [20] for the design of the controller:
Next, similar to equation (13), for system (14), the following performance indicators are defined:
Lemma 1. For performance indicators (13) and (14), if and are satisfied, then .
Proof. Considering and , thenSo, . Proving is completed.
Lemma 2. For , if the matrix is a positive definite symmetric matrix, is invariant, where represents the maximum eigenvalue of the matrix .
Proof. According to formula (7), can be obtained, and then, , or can be obtained. Because is a positive definite symmetric matrix, then . The proof is completed.
Lemma 3 (see [21]). For giving nonzero matrices , , and , the optimal solution to minimize problem is , where and represent the Moore–Penrose pseudoinverse of and , respectively.
Theorem 1. Consider the state-saturated system (1) over finite-time domain and its SCP model parameter . Given the controller gain , positive scalar , performance index , and positive definite matrix , the sufficient condition for the system to meet the performance requirements is that the existence of positive definite symmetric matrix makes the following recursive Riccati-like difference equations valid:and subjects to constraintswhere , , , and is the element of the symmetric matrix .
Proof. Define the following function:According to equations (14) and Lemma 2, it can be obtained thatAccording to properties of matrix covariance and trace, equations (2)∼(5) can be obtained:Therefore, equation (20) is equivalent toAdd zero item to the right side of equation (22), and use perfect plane method to change equation (22) intowhere .
According to the properties of conditional expectation, inequality (23) is accumulated from 0 to N, and the difference equation (17) such as Riccati is obtainedConsidering , , , and the matrix is positively definite symmetry, it is not difficult to obtainTherefore, ; the system meets the performance requirements. Proof is completed.
4. H∞ Controller Design
Theorem 1 gives a sufficient condition for the closed-loop system (1) to satisfy the performance index. Next, the solution of controller gain matrix under the worst disturbance case of the system is . To do this, define , , and the cost function . Rewrite the closed-loop system (14) as
Theorem 2. Considering the state saturation system (1) and its SCP model parameter on finite-time domain and given the positive scalar , performance index , and positive definite matrix , the sufficient condition for the system to meet the performance requirements is that there are and positive definite symmetric matrices and , so the Riccati-like difference equation recursively can be established as follows:
And, it subjects to constraints
At this point, the gain matrix of the controller can be calculated by the following formula:where , , , and .
Proof. Theorem 2 ensures that Theorem 1 is true; it can be known that the closed-loop system (1) meets the performance requirements. Furthermore, under the worst interference , the following functions are defined:According to equation (26) and Lemma 2, it can be obtained asAdd the zero term to the right of the cost function , then define , use the complete square for , and combine equation (27) and constraint (28), and we obtainIt is noted that , in order to reduce the cost function, the gain matrix of the controller is selected as follows:Therefore, according to Lemma 3, equation (29) is the optimal solution to the minimization problem (33). Proving is complete.
Based on Theorem 2, the following controller design algorithm on finite-time domain is obtained (Algorithm 1).
|
5. Numerical Example
A state-saturated time-varying system (1) with random parameters and random nonlinearity is considered in finite-time domain , and its system matrix parameters are shown as below [22]:
The random nonlinear function iswhere , , and are three Gaussian white noise sequences whose mean value is 0 and variance is 1, and it is not difficult to verify that the random nonlinear function satisfies
In addition, it is known that the initial state , , , and of the system is 1, the state transition matrix of the random communication protocol , and the process noise and measurement interference of the system are and , respectively.
On the premise that the algorithm in this paper has a solution and the performance index is as small as possible, the given values of , positive definite matrix , and and are given as . By using the algorithm proposed in this paper, the gain matrix of the controller is obtained, as shown in Table 1. The corresponding simulation results of the system are shown in Figures 1–3. Figure 1 depicts the changes of sensor nodes that obtain network permissions under the influence of random communication protocol with the state transition matrix. Figures 2 and 3, respectively, show the state response curves of the open-loop system and the closed-loop system. The simulation results demonstrate the effectiveness of the proposed algorithm for the finite-time domain controller design.
In order to highlight the advantages of the controller design algorithm proposed in this paper in the aspect of state saturation system, the algorithm in [22] is compared with the algorithm proposed in this paper. Regardless of random parameters and random nonlinearity, only for the state saturation system under the influence of random communication protocol, the parameters set are exactly the same as those in the above simulation. It is worth noting that the algorithm in [22] is not applicable to networked control systems under the influence of random communication protocols, so and in [22] need to be adjusted. The controller gain matrix obtained according to the algorithm proposed in this paper and the controller gain matrix obtained according to the algorithm in [22] are shown in Table 2. The state response curves of the corresponding closed-loop system are shown in Figures 4 and 5, respectively. It can be seen from Figure 4 that the controller gain matrix obtained based on the algorithm proposed in this paper makes the state-saturated system enter a stable state at the 6th transmission moment. It can be seen from Figure 5 that the controller gain matrix obtained from [22] makes the system enter a stable state at the 20th transmission moment. For this system, other parameters remain unchanged. On the premise that the controller has a solution, in this algorithm can be taken as the minimum , and in [22], the minimum can be taken as . To sum up, for the state saturation system under the influence of random communication protocol, the algorithm proposed in this paper takes into account many system factors, and the value of is relatively large, but the dynamic performance of the system is better.
6. Conclusion
In this paper, the finite-time domain control problem of state-saturated time-varying systems with random parameters and random nonlinearity under the influence of random communication protocol is studied. Firstly, a state saturation system model with random parameters and random nonlinearity is constructed. Then, in the sensor-controller transmission network, random communication protocol is added to avoid data conflict. Then, the sufficient conditions for the system to meet the performance requirements are obtained by using the model transformation technology and matching method. On this basis, the controller design algorithm based on the reverse-recursive Riccati difference equation technology is presented. Finally, numerical simulation shows the effectiveness of the proposed algorithm.
Data Availability
The data used to support the findings of this study are available within the article as references and from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (no. 61773182).