Abstract
In this paper, control for the uncertain switched nonlinear cascade systems with passive and nonpassive subsystems is investigated. Based on the average dwell time method, for any given passivity rate, average dwell time, and disturbance attenuation level, the feedback controllers of the subsystems by predetermined constants are designed to solve the exponential stability and -gain problems of control for switched nonlinear cascade systems. Two examples are provided to demonstrate the effectiveness of the proposed design method.
1. Introduction
With the development of scientific computing technology, the research on control problems of nonlinear systems has been greatly promoted, and the results of nonlinear control problems continue to emerge [1, 2]. However, these methods usually bring a difficulty that needs to solve the Hamilton–Jacobi equation.
The passivity, from the electrical network, becomes an extremely useful property in switched systems, and many results about the passivity of switched systems have been published [3–11]. Storage functions that characterize passivity can be used as Lyapunov functions to analyze stabilization problems [3]. And the passivity is closely related to the robust stability of systems under certain negative feedback disturbances [6]. Recently, the storage function method has been found to ensure a top limit of the minimum dwell time to keep the passivity of linear systems [5]. For switched nonlinear systems, stability was inferred from the passivity described by using multiple storage functions [10]. The necessary and sufficient conditions were obtained for the local passivity of discrete-time switched nonlinear systems which consisted of passive and nonpassive modes, and the passivity of the affine system was studied [9]. Using multiple barrier storage functions, sufficient conditions were derived for guaranteeing the regional passivity of the switched systems [8]. And literature [7] considered the stability of switched nonlinear systems with feedback incrementally passive subsystems via the average dwell time method.
With the systems becoming more complex in actual problems, the robustness caused by external disturbance becomes a source of trouble, and there are a few achievements on passivity of control problem of switched systems [12–15]. The stability of two types of passive control for discrete-time linear switched systems was considered by multiple storage functions [12, 13]. And combining the piecewise Lyapunov function and the average dwell time method, the literature [15] investigated the disturbance of time-controlled switched systems consisting of several linear time-invariant subsystems. The control of uncertain switched nonlinear systems with passivity was researched, and this research avoided solving the Hamilton–Jacobi inequality problem [14].
Complex nonlinear systems can be transformed into cascaded systems through certain conditions, and the stability of the cascaded system is studied by the stability and cascade properties of the subsystems [16]. This not only reduces the complexity of the controller but also reduces the difficulty of stability analysis [17–20]. A natural question is how to study the stability of switching nonlinear cascade robust control systems through passivity, and in this paper, we will work to solve these problems.
In this paper, based on the method of average dwell time, the robust control problem for a class of passive uncertain cascade switched systems with passiveness is considered. For passive subsystems and nonpassive subsystems, we design controllers and apply the multiple storage functions method to solve the stability and -gain of the nonlinear uncertain cascade switched system under the given conditions. Finally, two numerical simulations are illustrated to support our analytical results. Compared with the method of existing nonlinear cascade switched systems’ control problem, the advantages are that we adopt the parametric equation method to avoid the Lyapunov function construction and the Hamilton–Jacobi equation solution, which reduces the computational difficulty.
Notions: is the -dimensional real Euclidean space; denotes the matrix transposition; means the smallest eigenvalue of the matrices and , and is the largest; is the Euclidean norm of vector; stands for , where ; and means .
2. Problem Statement and Preliminaries
Consider the uncertain switched nonlinear cascade system with the formwhere , and , is the disturbance input and , is the control input, and is the output, defining the right continuous function is the switching law. For , , , , and are smooth functions of appropriate dimensions, , , and are bounded and smooth functions of appropriate dimensions, and is uncertain nonlinear functions of appropriate dimensions. Especially, , , and . In the ideal state, the subsystem switching signal is defined on the following switching sequence:where and are initial time and initial state, respectively, means that the th subsystem is activated at . Without loss of generality, we assume . In order to better understand the switching between subsystems, the block diagram of the switched system (1) is shown in Figure 1.
Assumption 1. (see [21]). For , the constants , and , there exist positive definite functions , such that the conditionshold.
For the subsystems of the switched system (1), we classify them into two groups: represents that the th closed-loop subsystem is passive; represents nonpassive. Then, and satisfy Assumption 2.
Assumption 2. For ,For , there exists a constant satisfyingwhere and are smooth functions of appropriate dimensions.
Assumption 3. (see [6]). For uncertain function , it satisfies the bound , , where is a nonnegative function and satisfies for nonnegative constants and .
Definition 1. (see [22]). Let represent the number of switchings of in the interval for any switching signal and . Ifholds for . The constant is called average dwell time, and is the chatter bound. Without loss of generality, we choose .
The notion of average dwell time is often used for identifying switching signals which have certain desirable properties.
Definition 2. (see [23]). For any , let denote the total time when the passive subsystems are active on . Then, the passivity rate of the switched system is recorded as . Clearly, .
In this paper, we will study the following robust control problem for system (1). For any constant , define the control laws of each subsystems and . Under the switching signal , system (1) has the following properties [6, 19]:(i)The closed-loop system (1) with is globally robustly exponentially stable for all admissible uncertainties.(ii)The closed-loop system (1) has a weighted -gain level for some real-valued function with and , that is, there exist a constant and , such thatholds.
Definition 3. In the nonlinear system,for degree , it is exponentially small-time norm-observable if there exist positive constants and , such that when holds for and , , is established.
Remark 1. The small-time norm-observability has been proposed for ensuring the asymptotical stability of switched systems [24]. In this paper, the exponential small-time norm-observability with degree is exponential form, and it is used to research global robust exponential stability of system (1).
Remark 2. A method is given to verify that system (10) is exponentially small-time norm-observable. Assume that there exist positive constants and and positive definite matrices and , such that the following condition is satisfied:LetWe can getFrom (11), the time derivative of along the trajectory of system (10) isWhen holds for with length , we obtainBy (13) and (15), using the differential inequality theory, we obtainHence,which meansAccording to Definition 3, system (10) is exponentially small-time norm-observable.
Lemma 1. If system (10) is exponentially small-time norm-observable with degree , for any , it haswhere , , and .
Proof. If system (10) is exponentially small-time norm-observable with degree , there exists a constant , such thatholds. By (20), we havenamely,Apply the integral mean value theorem to the above formula, and there exists a constant , and , such asThen,Because system (10) is exponentially small-time norm-observable, if can be given with , we obtainwhich meansThen,Then, the sum of (24) and (27) iswhere .
3. Main Results
In this section, we will discuss system (1) in two parts. Part : when , we will analyze the globally robustly exponentially stable of system (1) for all admissible uncertainties. Part : when , the weighted -gain level will be researched.
3.1. Part I: The Stability Analysis of
Theorem 1. Under the conditions of Assumptions 1 and 2, let the positive constants and be any given average dwell time and passivity rate, respectively. For all admissible uncertainties, system (1) with is assumed to be exponentially small-time norm-observable with the positive constants , and satisfying , , and , wherefor a constant . Design the controllerswhere
Then, the switched system (1) with is globally robustly exponentially stable under any switching signals with the average dwell time and passivity rate .
Proof. Letwhere and are smooth functions of appropriate dimensions.
For , we make the set . Then, we divide the proof into two cases: one is , and the other is .
Case 1:. .
Assume that the th subsystem is active. For , the time derivative of along the trajectory of the switched system (1) isSubstituting controller (30) into (33), from (3), (4), and (7), for , we obtain thatwhere .
Similarly, for , it follows from (3), (4), and (6) thatFor , we apply the integral of (34) and (35) thatwhere , Definewhere From Assumption 3, thenChoose the piecewise function:where .
On the contrary, , for , and we obtainFrom , we have ; then,Taking (41) into (40), we obtainFrom (3), we get thatwhich means
Case 2:. .
In the case, for , we suppose that .
From Definition 3, we have . Then,From (3), we obtainSimilar to Case 1, we can achieveTherefore, the closed-loop switched system (1) is exponential stability for all admissible uncertainties. This completes the proof.
3.2. Part II: The -Gain Analysis of
In this section, we investigate the performance analysis of system (1) by the -gain .
Theorem 2. Assume the positive constants , , and are average dwell time, passivity rate, and disturbance attenuation level, respectively. , , and still satisfy (3)–(7). For all admissible uncertainties and disturbance inputs, assume the passive subsystems are exponentially small-time norm-observability with positive constants , , , , and . All of these satisfy and . Then, we design the controllers:whereand is given by Theorem 1 with and . Then, the switched system (1) achieves a weighted -gain from to for all admissible uncertainties.
Proof. On the basis of Theorem 1, the time derivative of along the trajectory of switched system (1) isWhen , , similarly the proof of Theorem 1, we havewhere .
When , on the interval , we haveFrom (3), we obtainDue to , we get . Substituting into (53), we knowFor , which means ; then,Similarly, when ,Let , where . For (51), (55), and (56), the differential equation theory and the constant variable formula are used, respectively. When and ,When ,And when ,whereCombining (57)–(59), we obtainwhere . Define the piecewise function and . When the time satisfies , by the property , , and from (61), we haveDue to , we have .
Then,which meansWe multiply both sides of the above formula by :Obviously, , then , and putting , into (65), we obtainFor the trivial case of and , we obtainNext, we consider the nontrivial case of . Rearranging the double-integral area leadsHence, the switched system (1) achieves a weighted -gain from to for all admissible uncertainties. This completes the proof.
Remark 3. Under zero initial condition, we have , and from (68), we can get the weighted -gain level . The smaller the weighted -gain level is, the better the performance of robust control of system (1) is [25, 26].
Remark 4. The system in [19] is similar to system (1) in this paper, and it needs to satisfy these conditions in [19]: (i) ; (ii) for unbounded positive definite differentiable functions , constants and , such that holds. In our paper, we just need , and the positive definite differentiable functions do not need to satisfy condition (ii). So, this paper gets less conservative.
4. Numerical Example
In this section, we give two examples to demonstrate the effectiveness of the proposed method.
Example 1. Consider a switched continuous stirred tank reactor system with two modes feed stream [27, 28]:In this paper, we ignore the influence of temperature on reaction speed, and only consider the disturbance on the concentration. And the physical meaning of the parameters in system (69) can be found in [28].
The control objective is to make the temperature to some constant reference . And is a steady-state control corresponding to the temperature set points . Let . System (69) can be expressed in the form with equilibrium point at the origin:where and .
Then, let the steady-state point , , and parameters for the simulation , . And defining the output , we get two subsystems as follows:where , are controllers and , are unknown constants.
It is not difficult to know that and . So, , where .
Let and . For system (71),For system (72),Then,It is easy to verify that system (71) is passive and system (72) is nonpassive.
A simple calculation shows that , , , , , and . In addition, we acquire that and . Let the average dwell time , the passivity rate , the disturbance attenuation level , and . Then, , , , and . According to Theorem 1, when , we construct the controllersFigure 2 gives the control input of systems (71) and (72) and the switching signal . Figure 3 is the simulation result with the initial states and .
When , we construct the controllersFrom Remark 3, we define the functionIn this example, we can get . Figure 4 gives the -gain level with the initial states and . And we can easily see the -gain less than .
Example 2. Consider the uncertain switched nonlinear cascade systems with two systems:where and are generated constants by random numbers.
Let , and . For system (79),For system (80),Then,It is easy to see that system (79) is passive and system (80) is nonpassive. A simple calculation shows that , , , , and . In addition, we acquire that and . Let the average dwell time , the passivity rate , the disturbance attenuation level , and . Hence, , , , and .
According to Theorem 1, we construct the controllers; when ,Figure 5 shows the control input of systems (79) and (80) and the switching signal . Figure 6 is the simulation result with the initial states , , , and .
When , we construct the control functions:In this example, we can get . Figure 7 is the -gain level with the initial states , , , and . And we can easily see the -gain less than .
5. Conclusion
Based on the method of average dwell time, we give sufficient conditions to ensure the solvability of the problem avoiding the Lyapunov function construction by the storage functions and reducing the computational complexity of the solution. For any switching signal, the system can achieve stability and have the weighted -gain property under the action of the feedback controller designed by the given passivity rate, average dwell time, and interference attenuation level. The proposed scheme supplements the research methods of robust control for the nonlinear cascade systems. In the future, we will extend the results of this paper to global stabilization of switched stochastic nonlinear robust control systems.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.