Abstract
The term double-fault networked control system means that sensor faults and actuator faults may occur simultaneously in networked control systems. The issues of modelling and an guaranteed cost fault-tolerant control in a piecewise delay method for double-fault networked control systems are investigated. The time-varying properties of sensor faults and actuator faults are modelled as two time-varying and bounded parameters. Based on the linear matrix inequality (LMI) approach, an guaranteed cost fault-tolerant controller in a piecewise delay method is proposed to guarantee the reliability and stability for the double-fault networked control systems. Simulations are included to demonstrate the theoretical results of the proposed method.
1. Introduction
Networked control systems (NCS) are frequently encountered in many fields of applications due to their suitable and flexible structure [1–9]. However, in practical NCS, there inevitably exists time delay and data packet dropouts because of the introduction of the communication network [10–13]. Sensor faults and actuator faults also occur easily at the device level because of its large scale and complicated structure [14], which can have a negative impact on the system, such as performance decline and instability. Thus, guaranteed cost and fault-tolerant control of networked control systems has become a new popular issue in the network control field.
To achieve stability requirements concerning sensor faults or actuator faults, fault-tolerant control has been investigated in many works [15–26]. Based on the Lyapunov stability theorem, a methodology for the design of fault-tolerant control systems for chemical plants with distributed interconnected processing units was presented by N. H. El-Farra and A. P. D. Gani [15]. Z. Qu and C. M. Ihlefeld devised a fault-tolerant robust controller for a class of nonlinear uncertain systems considering possible sensor faults and developed a robust measure to identify the stability- and performance-vulnerable failures [16]. Based on the integrity control theory, a robust fault-tolerant controller for NCS with actuator faults was discussed by Y. N. Guo [17]. The diagnosis of actuator component faults and fault-tolerant control for a class of networked control systems using adaptive observer techniques was investigated in [18]. A switched model based on probability for NCS was proposed in [19] to research issues of fault-tolerant control when actuators age or become partially disabled.
Recently, guaranteed cost control that can guarantee the stability of a system and make it meet a certain performance indicator has become popular in NCS [4–7, 27–29]. Stability guaranteed active fault-tolerant control against actuators failures in NCS was addressed by S. Li [27]. X. Y. Luo and M. J. Shang proposed the so-called guaranteed cost active fault-tolerant controller (AFTC) strategy in [28]. The issue of guaranteed cost reliable control with regional pole constraint against actuator failures was investigated by H. M. Soliman in [6]. In [7], X. Li investigated the issue of integrity against actuator faults for NCS under variable-period sampling, in which the existence conditions of a guaranteed cost fault-tolerant control law was tested in terms of the Lyapunov stability theory. The resilient reliable dissipativity performance index for systems including actuator faults and probabilistic time delay signals is investigated by authors in [30, 31]. Unfortunately, all the previously mentioned literature investigated the guaranteed cost fault-tolerant control for NCS of single-faults (just considering the condition that only actuator faults occur or sensor faults occur). Few articles examine the double-fault issue. In practical application, it is easy for the sensors and actuators to become faulty simultaneously when the NCS works in a poor environment and is affected by external disturbance. Improving the control performance of NCS when double-faults occur is important. This motivates us to investigate guaranteed cost fault-tolerant control of double-fault NCS, which is a necessary but challenge task.
This paper develops a guaranteed cost fault-tolerant control for a double-fault NCS to guarantee its stability. The time-varying properties of sensor faults and actuator faults are modelled as two time-varying and bounded parameter matrices, and the networked control system is built as a linear closed-loop system with transmission delays and data packet dropouts. Different from [30, 31], here it is necessary to deal with two dynamic matrices. One of them is located at the left hand side of the gain matrix, and the other one is located at the right hand side of the gain matrix. This brings challenges to searching a feasible control gain. To solve such problem, a piecewise delay method is proposed to analyse the delay-dependent faulty system for reducing the conservatism. The delay falling in each subinterval is treated as a case. For different cases, different weighted technology is used to derive the guaranteed cost fault-tolerant condition, and the controller parameter for this NCS is obtained by solving several sets of LMIs. Compared with [26, 32–34], the delay considered in this paper can be continuously changed with time.
Notation. denotes the n-dimensional Euclidean space. The superscript “” stands for matrix transposition. The notation means that the matrix is a real positive definite matrix. is the identity matrix of appropriate dimensions. denotes a symmetric matrix, where*denotes the entries implied by symmetry.
2. Modelling of Double-Fault NCS
The linear control plant of NCS with uncertain parameters and external disturbance can be expressed aswhere , , , and represent state value, input, output, and external disturbance, respectively. Separately, , , , and are matrices with appropriate dimensions; and are matrices with uncertain time-varying parameters, satisfying ; is an unknown matrix function with Legesgue measurable properties, satisfying ; and , , and are constant matrices with appropriate dimensions.
For the convenience of the later formulation, two concepts can be initially introduced.
Definition 1. The sensor faults and actuator faults could occur simultaneously, that is, there may be two types of faults at one time; this kind of fault is called a double-fault.
Definition 2. The sensor faults and actuator faults do not occur simultaneously, that is, there is only one type of fault at one time; this kind of fault is called a single-fault.
The structure of a double-fault NCS is shown in Figure 1. Transmission delays induced by the network are the sensor-to-controller delay and the controller-to-actuator delay . These two delays can be combined when the feedback controller is static. The state of the system is assumed to be completely measurable. A piecewise continuous feedback controller, which is realized by a zero-order hold (ZOH), is employed: where is the state feedback gain matrix to be designed and is the sampling instant.
Considering that sensor faults may occur, is used to represent the data from the -th sensor. In this paper, we consider faults that include outage and loss of effectiveness. If the -th sensor is an outage, the corresponding sampling work is interrupted, and the sampling data keeps the default value . If the -th sensor loses effectiveness, the sampling data is inaccurate and nonzero. We denote the sensor fault model aswhere is the time-varying sensor efficiency factor, represents that the -th sensor is normal, represents that its fault is outage, and or represents that its fault is loss of effectiveness. The upper bound of sensor efficiency factor is denoted by a constant satisfying , while its lower bound is denoted by a constant satisfying .
Denoting , we havewhere is the sensor fault indicator matrix. Correspondingly, its upper bound matrix is , and its lower bound matrix is .
Data packet dropouts in NCS are also unavoidable because of limited bandwidth. Considering that data packet dropouts may occur, the network is modelled as a switch. When the switch is located in position, the data packet containing is transmitted, and the controller utilizes the updated data. When it is located in the position, the data packet dropouts occur, and the controller uses the old data. For a fixed sampling period , the dynamics of the switch can be expressed as follows:
The NCS with no packet dropout at time : The NCS with one packet dropout at time :The NCS with packet dropout at time :Because the feedback controller is static, (2) can be expressed as
Considering that actuator faults may also occur, is used to represent the signal from the -th actuator. Similarly, we denote the actuator fault model aswhere is the time-varying actuator efficiency factor, represents that the -th actuator is normal, represents that its fault is an outage, and or denotes that its fault is a loss of effectiveness. The upper bound of sensor efficiency factor is denoted by a constant satisfying , while its lower bound is denoted by a constant satisfying .
Denoting the faulty control signal , we can obtain the fault-tolerant control law aswhere is the actuator fault indicator matrix. Correspondingly, its upper bound matrix is , and its lower bound matrix is .
Let ; ; (10) can now be expressed as follows:Obviously, the delay part may vary with time , and it satisfies
From the upper bounds of fault indicator matrices and lower bounds of fault indicator matrices, the nonsingular mean-value matrices are separately obtained as
Moreover, the following two time-varying matrices are introduced:Obviously, we haveSimilarly, we have In expressions from (13) to (16), and meet , . Based on (15) and (16), we haveFrom on (14), the following can be obtained: Naturally, the time-varying fault indicator matrices can be rewritten asInserting (19) into (11), we haveThen, the new model of double-fault NCS can be obtained as follows:
Remark 3. The networked control systems in faulty case can be modelled as system (21) with the effects of time-varying delay whose upper bound and lower bound are described in (12). Unlike the previous models, [17, 21, 27, 35], this model is related to both sensor faults and actuator faults, the faults are time-varying, which are reflected by the time-varying parameters and . From (17), we undoubtedly know the time-varying parameter satisfies , while the parameter satisfies .
Remark 4. From the descriptions of faults matrices, we undoubtedly know if (), we can obtain and , and then model (21) is an actuator fault model. Similarly, if (), we can obtain and , and then model (21) represents a sensor fault model. Therefore, model (21) of a double-fault NCS contains cases of single-faults, and the single-faults are a special form of double-faults.
In the following section, a fundamental preliminary result is presented to guarantee the performance of a double-fault NCS based on the delay information.
3. Performance Analysis of Double-Fault NCS
For the system model (21) established in Section 2, the cost function is given as follows:where and are symmetric positive definite matrices.
Definition 5. For model (21) and its cost function (22), if there exists a control gain matrix satisfying the conditions
the closed-loop system is asymptotically stable when ;
for any zero initial condition and any nonzero vector , given , the output satisfies ;
there exists a constant , and the cost function defined as (22) satisfies ,
then matrix is the guaranteed cost control gain of double-faults NCS.
To analyse the stability of the system expediently, the following lemmas are introduced.
Lemma 6 (see [36, 37]). For any matrices , , , with , and any scalar , the inequality holds as
Lemma 7 (see [38]). If , for any matrices , , and , the following inequalities are equivalent:
;
.
The fundamental preliminary result is presented in the following theorem.
Theorem 8. Given symmetric positive definite matrices and , a set of constant , , , , and . If there exists a set of symmetric positive definite matrices and matrix , as well as matrices , , , , (), , and a set of constants, and , satisfying the LMIswherethen model (21) is asymptotically stable with the norm bound . In addition, the upper bound of cost function is given as
Proof. First, with the definition of and , the interval of delay is distributed into two subintervals as follows:Then, we consider the Lyapunov-Krasovskii functional as follows:where matrix satisfies and () are symmetric positive definite matrices with appropriate dimensions. For the convenience of writing, we denote and in the following expressions. Calculating the derivative of Lyapunov-Krasovskii function and based on (21), we havewhere [39].
Based on Jessen’s inequality, we havewhere . For the convenience of the following discussion, we define Case 1. If , weighted technology based on the principle of Newton-Leibniz is introduced as follows:andBecause , we haveandTherefore,In addition, both on the left and right sides of equality (29) and on the right of the equal sign “=”, and then inserting (30), (32), (34), (35), (38) to the obtained inequality, we havewhereifNext, we need to acquire the inequality (24) through a transformation based on inequality (41), which is equivalent to the following inequalities by applying the theory given in Lemma 7 and the Schur complement, also used by D. Yue [38] in the previous study:Premultiplying and postmultiplying the inequalities above by , we have Applying the theory of the Schur complement to inequality (44), we havewhereInequality (44) can be written as whereWith the definition of scalar , we apply the theory given in Lemma 6 to (47), also used by Y. Wang and L. Xie [37], in which the uncertain matrix can be eliminated, and a sufficient condition of (47) is obtained. Applying the Schur complement to inequality (49), we haveThere exists , . Based on (17) and Remark 3 in Section 2, we know that and . According to expressions (47), (49), and (50), we obviously know that . Using Lemma 6 again, we haveandBased on inequalities (51), (52) and the Schur complement, we know inequality (24) is a sufficient condition of inequality (50), while inequality (50) is equivalent to inequality (41). Therefore, we can undoubtedly obtain inequality (24) as a sufficient condition of inequality (41). Thus, based on inequality (39) and (41), we know
if , obviously, we have , so system (21) is asymptotically stable;
if , we know . In addition, it can be obtained that .
Therefore, Thus, Because , we haveIt is known that model (21) is asymptotically stable with the norm bound .
Moreover, according to (39) and (41), we haveThrough the integral operation, it can be determined that . In addition, by inserting into the Lyapunov-Krasovskii function shown as expression (28), the upper bound of cost function can be obtained and shown as expression (26). Therefore, the theorem is verified if .
Case 2. If , weighted technology based on the principle of Newton-Leibniz is introduced as follows:Because , we haveTherefore,In addition, both on the left and right sides of equality (29) and on the right side of equal sign “=”. By inserting (30), (31), (57), (58), and (61) into the obtained inequality, the item can be offset, while the item is offset in Case 1. Then, in the same methods of transformation as Case 1, the inequality (24) when can be obtained. Therefore, the proof is complete.
Remark 9. In the two different cases, a variational weighting matrix and Jessen’s inequalities are used to derive the guaranteed cost fault-tolerant condition of the system, in which more delay information is employed to reduce the conservatism.
The next section will provide sufficient conditions for designing the guaranteed cost fault-tolerant control for a double-fault NCS.
4. Guaranteed Cost Fault-Tolerant Control of Double-Fault NCS
Inequality (24) is not linear with respect to the gain matrices of the controller, so it is needs to be reformulated into LMIs via a change of variables.
Theorem 10. Given symmetric positive definite matrices and , a set of constant , , , , ( = from 1 to 7), and . If there exists a set of symmetric positive definite matrices , , and matrix , as well as matrices , , , , (from 1 to 8), , and a set of constants, and , satisfying the LMIswherethen the guaranteed cost control gain can render model (21) to be asymptotically stable with the norm bound . The upper bound of cost function is given as
Proof. The proof is based on a suitable transformation and a change of variables allowing us to obtain inequality (24) in Theorem 8. First, we define , ( from 1 to 4), in (24). Because we consider the dimension of stateis equal to that of outside disturbance in this paper, we can also define , . Obviously, (24) implies , so is nonsingular. Then, using the analysis method of D. Yue [40] and Z. Wang [41], pre- and postmultiplying both sides of inequality (24) with and its transpose, where and , introducing new variables ; (); , (); and . From the definition of , we know is invertible, so can be obtained by calculating . It is easy to see that (62) and (64) respectively imply (24) and (26). Therefore, from Theorem 8, we can complete the proof.
To obtain the optimal bound shown in (26), the commonly used method is to consider it as an optimization problem like [42], in which the expression of initial state () needs to be known. The expression of initial state is not given in this paper. Therefore, the optimization method used in [42] cannot be used here. A practical method to obtain , also used by D. Yue [43], is employed as follows.
Suppose is bounded if and satisfying . In addition, suppose that there exists , satisfyingInserting this into (64), we haveApplying the Schur complement to the inequalities above, we haveThen, combining (62) and (67), (), , exist simultaneously. We cannot directly use the LMI tools to solve the problem. Defining (), , and using the idea of the cone complementary linearization algorithm, the guaranteed cost fault-tolerant controller of system (21) and the value of optimal performance indicator can be obtained in the following method:
5. Simulations
Example 11. Consider inverted pendulum model that can usually be modelled as (1), and the system parameters are given as follows:Therefore, we have For this simulation, the initial state of system is assumed , and the external disturbance is considered as . Here, we take the upper bound of time-varying delay as and its lower bound as , namely, . In addition, the fault bounds of the system are given in Table 1.
We choose the parameters as follows:By taking advantage of the LMI tool box and inserting the above parameters into inequalities (62) and (68), we can obtain the guaranteed cost control gainwith .
The corresponding optimal performance indicator (the upper bound value of guaranteed cost function) is .
The time-varying delay is shown in Figure 2. In Figure 3, (a) is the actuator fault, which is a piecewise-linear function. It keeps the minimum value from to , while it keeps the maximum value from to . The sensor faults are shown as (b), which is sinusoidal. It should be noted that the green dotted line represents the fault of sensor 1, while the blue solid line represents the fault of sensor 2. Through the state response of the double-fault NCS shown in Figure 4 and corresponding control signal shown in Figure 5, we know the guaranteed cost controller designed in this paper is able to make the double-fault NCS asymptotically stable. The system gets preliminarily steady at 2, and its state can return to the equilibrium position in a certain period of time when the NCS is affected by external disturbance. Compared with the state response of worse stability shown in Figure 6 when the method proposed in [28] is used for this double-fault problem, it sufficiently proves the effectiveness and feasibility of the method proposed in this paper.
To better illustrate the effectiveness of the method proposed in this paper, the following example is presented.
(a) Time-varying actuator fault
(b) Time-varying sensor fault
Example 12. Consider the parameters of system (1) as follows:Therefore, we have For this simulation, the initial state of system is assumed as , and the external disturbance is considered asThe uncertain time-varying delay satisfies . In addition, the fault bounds of the system are given in Table 2. Other parameters are selected as follows:By taking advantage of the LMI tool box and submitting these parameters above into inequalities (62) and (68), we can obtain the guaranteed cost control gainThe corresponding optimal performance indicator is . From the state response shown in Figure 7 and control signal shown in Figure 8, we undoubtedly know the double-fault NCS is asymptotically stable when the guaranteed cost controller is used. This further demonstrates the feasibility and effectiveness of the method proposed in this paper.
6. Conclusions
The issues of modelling and guaranteed cost fault-tolerant control of double-fault networked control systems have been addressed. The closed-loop model of a double-fault NCS is set up with regard to the influences of transmission delay, packet dropout, uncertain parameters, and external disturbance. In addition, the piecewise delay method is proposed to reduce the conservatism when analysing the delay-dependent faulty system. With the help of Lee Y S’s lemma, the sufficient condition of guaranteed cost fault-tolerant for time-varying double-fault NCS is introduced using the Lyapunov-Krasovskii theory and weighted technology. The method of designing a guaranteed cost fault-tolerant controller for this NCS is given based on LMI. Our next research task will be choosing more reasonable values of parameters () to reduce the conservatism further. Of course, the study on scheduling policy of double-fault NCS is also a challenge but indispensable work.
Data Availability
The data used in this paper can be got by simulations.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was partly supported by National Nature Science Foundation of China under Grants 51875380, 51375323, and 61563022, Cooperative Innovation Fund-Prospective of Jiangsu Province under Grant BY2016044-01, Major Program of Natural Science Foundation of Jiangxi Province, China, under Grant 20152ACB20009, High Level Talents of “Six Talent Peaks” in Jiangsu Province, China, under Grant DZXX-046.