Exponential Feedback Passivity of Switched Polynomial Nonlinear Systems

The exponential feedback passivity problem of switched polynomial nonlinear systems is studied. To obtain this aim, a method of parameterization of controller is presented and parameter solution algorithm is described. Then, the addressed method is utilized to solve the robust stabilization for a class of switched polynomial nonlinear systems with parameter uncertainties and external disturbance. This result extends the previous results on exponential feedback passivity from the case of nonlinear systems to switched nonlinear systems. A numerical example is given to demonstrate the effectiveness of the proposed result.


Introduction
Switched polynomial nonlinear systems (SPNSs) [1][2][3] are a class of dynamical systems consisting of nonlinear subsystems with polynomial structure and a switching law, which defines a specific subsystem being activated during a certain interval of time.Due to the theoretical development as well as practical applications, analysis and synthesis of switched systems have recently gained considerable attention.Many tools, such as single Lyapunov function [4], multiple Lyapunov functions [5], and average dwell time [6], have been proved effective in solving the problem of stability and stabilization for switched nonlinear systems.In addition, in practical control systems, the parameter uncertainty and external disturbance are common, which implies that it is meaningful to study the robust stabilization of switched polynomial nonlinear system with parameter uncertainties and external disturbance [6][7][8].
On the other hand, as a particular case of dissipativity, passivity was introduced in [9] and later generalized in [10].In virtue of the property of keeping a system internal stability, passivity theory is extensively used as an effective tool for the stability analysis and synthesis of nonlinear systems [11][12][13][14][15][16][17][18][19][20][21][22].Recently, the exponential passivity problem for nonlinear systems has attracted the attention of many scholars.In [23], the concept of exponential passivity was introduced.In this result, it needs to satisfy that there exists an -continuously differentiable storage function   (⋅) with supply rate (, ) =    meeting some conditions.Using exponentially weighted system storage functions with appropriate exponentially weighted supply rates, Chellaboina and Haddad [24] introduced the notion of exponential dissipativity, which was more general than the notion of exponential passivity introduced in [23].In [25], a delaydependent exponential passivity condition for memristive neural networks was proposed in terms of linear matrix inequalities.By constructing a proper Lyapunov-Krasovskii functional, utilizing free-weighting matrix method and some stochastic analysis techniques, some delay-dependent sufficient conditions that ensure the exponential passivity of stochastic neural networks were deduced in [26].In [27], by constructing an appropriate Lyapunov-Krasovskii functional and using the Wirtinger-based integral inequality to estimate its derivative, a delay-range-dependent and delay-ratedependent criterion was presented to ensure the augmented filtering dynamic system to be robustly exponentially passive with an expected dissipation.The exponential passivity of some systems can be ensured with the conditions of [25][26][27].Still and all, whether these conditions are true depend on the structure of the systems themselves.In addition, the nonlinear functions of the systems of [25][26][27] should satisfy the so-called Lipschitz condition, which restrict the application scope of these results.In [23], some feedback controllers were designed to obtain exponential passivity for a class of affine systems possessing globally defined normal form and factorisable high-frequency gain.In [28], a continuous output feedback controller was designed to solve the exponentially stability problem for a class of uncertain systems that contain a nominal part which is affine in the control and an uncertain part which is norm bounded by a known function.However, the results of [23,28] are only suitable for some special systems.Based on the above discussions, we known that the related research on the exponential feedback passivity of nonlinear systems is still rather limited.Especially to the author's knowledge, there exist no results on the feedback exponential passivity of switched nonlinear systems so far, which is the motivation of the research of this paper.
In this paper, the exponential feedback passivity problem and the exponential passivity property are investigated for a class of SPNSs.The focus is on designing parameterized controller and solving out the feasible interval solution of the parameter.First, a concept of   -parameterized feedback exponentially passive of switched systems is introduced.Second, some parameterized symmetric matrix inequality conditions are proposed to ensure the exponential passivity of the SPNSs.Then, a parameter solution algorithm is formulated to solve the matrix inequalities.At last, the exponential passivity property is utilized to solve the robust stabilization problem of the SPNSs with parameter uncertainties and external disturbance.
The remainder of this paper is organized as follows.Section 2 is the problem formulation.The main contributions of this paper are then given in Section 3, in which a method of parameterization of controller is produced and a parameter solution algorithm is provided.In Section 4, a numerical example to support the results of this paper is given, which is followed by the conclusion in Section 5.
Remark 1.For the SPNSs, all the terms of the matrices are polynomial functions about system states.The requirement of possessing polynomial structure is not very restrictive for system (1) since any continuous functions can be approximated by polynomials of sufficient large order.
In the following subsections, some definitions and lemma to be used in this paper are provided.
0 refers to the initial number of jitter.Without loss of generality, we make the value  0 = 0. Definition 3. Considering the following switched system: if, for ∀,  ∈ , any given average dwell time   , there exist positive constant    ,  ≥ 1, positive constants  1 ,  2 , positive definite continuous function   () called storage function with   (0) = 0, and parameterized controller   (,   ) =   (,   ) +    ]  () in which   (,   ) is the part to parameterize the controller, ]  () = −ℎ  () is the new input,   is a weighting matrix with full column rank,   is the parameter with serial number , and there exists feasible interval solution for   , such that, ∀ ≥ 0 (ii) (iii) (iv) hold.Then, under the parameterized controller   (,   ) and switching law with average dwell time   , the closedloop system (2) is said to be   -parameterized feedback exponentially passive.
Remark 5.The systems rendered to be   -exponentially passive by a parameterized feedback controller are called  parameterized feedback exponentially passive systems or said to be feedback equivalent to   -parameterized exponentially passive systems.In Section 3, the   -parameterized feedback exponentially passive property will be used to solve the robust stabilization problem for the SPNSs with parameter uncertainties and external disturbance.

Main Results
In this section, the exponential feedback passivity of SPNSs is addressed with the method of parameterization of controller.In what follows, they will be accomplished in two subsections.

Exponential Feedback Passivity
Theorem 8.Under the switching law with the given average dwell time   , the closed-loop system (1) without parameter uncertainties and external disturbance is globally exponentially stable, if it is said to be feedback equivalent to   -parameterized exponentially passive system as Definition 3.
Then, under the switching law with average dwell time   and following parameterized controller system ( 1) is said to be feedback equivalent to   -parameterized exponentially passive systems.
Considering the th subsystem, according to the conditions   (0) = 0 and Lemma 6, system (1) can be transformed into the form of (12).
Choose the storage functions for each subsystem In view of (22), noting that ]  () = −ℎ  (), the differential of   () along the trajectory of the system (1) is obtained which means (3) holds.
By (23) and ( 24), we know that ( 4) and ( 7) hold.According to Definition 3, the closed-loop system (1) without parameter uncertainty and external disturbance is said to be   -parameterized exponentially passive.This completes the proof.
Remark 10.In order to render that (3) holds, condition (22) should be satisfied.However, the major difficulty is to solve out the interval value of   such that (22) holds over the entire system state space.According to Remark 1, expression   in the left side of condition ( 22) is polynomial function.By taking advantage of this feature, a symbolic computation based algorithm is produced to solve the interval value of   and α in Section 3.2.
Remark 11.The aims of the parameterization of controller include two aspects.Firstly, compensate for   and render it to be dissipative matrix: that is,   ()+  ()  (,   )   () ≥ 0. Secondly, render that the storage function   () has decay rate α along each subsystem.In Theorem 13 and Remark 14, we will know that the robust stabilization problem of system (1) can also be solved if the values of α are big enough.As a matter of fact, by adjusting the values of   contained in the parameterized controller (25), the values of α can be improved according to their feasible interval solutions.
In the following subsection, the   -parameterized feedback exponential passivity property of system (1) is utilized to solve its robust stabilization problem.Firstly, introduce an assumption as follows.
The uncertain term Δ  (, ) described by Assumption 12 is usual in nonlinear system [22].
In this paper, the robust stabilization problem of system (1) is summarized as follows: for a given disturbance attenuation level  > 0, given average dwell time   , parameterized controllers   (  , ),  ∈  in which   are the parameters with serial number  and there exist feasible interval solutions for   , such that (a) the closed-loop system (1) is said to be globally robustly exponentially stable when  = 0, (b) the closed-loop system (1) has  2 gain from  to  for all admissible uncertainties.That is to say, there exists a constant λ ≥ 0 and a real-valued function Ṽ() with Ṽ(0) = 0, such that holds.
Theorem 13.Suppose the conditions of Theorem 9 are satisfied and the uncertain terms of system (1) satisfy Assumption 12.For any given disturbance attenuation level , if there exist positive constants   ,  ∈ ,  3 such that ( 29), (30), and (31) where α ,  1 , , and   () are the same as Theorem 9 and   is a reassigned average dwell time.Then, the robust stabilization problem is solvable for system (1) under the parameterized controller (25) and switching law with average dwell time   .
Remark 14.Aside from the two aims described in Remark 11, the procedure of parameterization of controller has another effect that compensates some nonlinear actions for the external disturbance terms  1 ().Hence, if the external disturbance structure matrices  1 () contain state variables, the intervals of the parameters should be solved again by combining conditions ( 22) and (29).Namely, synthesize conditions ( 22) and ( 29) and obtain a new condition Solve conditions ( 31) and (47) with the PS algorithm described in Section 3.2 and obtain the interval values of the parameters   and   .Then the controller satisfying the requirements can be obtained as the same way of Theorem 9.
In addition, it is easy to know that whether conditions ( 29) and ( 31) hold depends on the values of α .Considering this property, we have the following three ways to confirm the values of the parameters when the external disturbance structure matrices  1 () do not contain state variables.
(1) If ( 29) and ( 31) are still established with the parameter values which are obtained according to Theorem 9, the controllers obtained in Theorem 9 are still effective in Theorem 13.
(2) If ( 29) or ( 31) is not established with the parameter values which is obtained according to Theorem 9, reselect the parameter values according to the interval value of the parameter (see Remark 10) obtained in Theorem 9.
(3) Resolve the interval values of the parameters according to (31) and (47) and reselect the parameter values.This way is always effective.
Remark 15.By the interval value of the parameter   , several specific values of   can be chosen and several controllers are achieved.Then, the optimized controller can be got by comparing with the performance of system (1) under the above obtained different controllers.Considering that the dissipation of the system is closely related to some performances of the system [11], it is reasonable to optimize the system with this way (see Remark 11).In Section 4, the disturbance attenuation capability of the system is improved with this method.

Parameter Solution Algorithm.
In order to obtain the interval values of the parameters restricted by ( 22), ( 24) or (31), (47), a symbolic computation based parameter solution (PS) algorithm is produced in this subsection.First, design the parameter matrices and obtain the parameterized controllers; secondly, according to conditions (22), (24) or ( 31), (47), obtain equivalent inequality set; thirdly, decompose in the parameter space and search out the required cell.At last, choose several values for each   and obtain the optimized controller.In this subsection, we will just consider the case of (31) and (47).The controller for the case of ( 22) and ( 24) can be obtained with the same way, which will not be considered.The algorithm now proceeds as follows.
In the following subsection, the parameterized controllers will be designed and the interval value of their parameters will be solved with the PS algorithm described in Section 3.2.
By Step 1 of the PS algorithm, define the following parameter matrix: Then, by ( 25) and (51), the parameterized controllers for the subsystems of Example 18 are obtained as Hence, by the values of , where " * " means the symmetric terms in the symmetric matrix P .
According to the symmetric positive definiteness of P , the constraint conditions   > 0,  ∈ {1, 2, 3} are obtained with the way of Step (3.4) of the PS algorithm.
Decompose the  1 - 1 plane into cells by drawing all the curves of  1 = 0,  ∈ {0, 1, 2, 3} in the plane and search out the required cell RC 1 with the method of Step (4.2) of the PS algorithm.The required cell RC 1 marked by gray background is surrounded by the curves of  10 = 0 and  13 = 0 in Figure 1.Following the above similar procedures, obtain the RC 2 that is partly surrounded by the curves of  20 = 0 and  23 = 0 in Figure 2.Then, considering RC  and (54) and (55), we know that  1 ∈ [3.61, k1 ], k1 ≥ 15 and  2 ∈ [2.54, k2 ], k2 ≥ 15.Choose By   = 1, for ,  ∈ , design the following switching law: Obtain a group of controllers by substituting (56) into (52).to describe the  2 disturbance attenuation gain level of the system from () to ().Simulate the system with the switching law (57) and the parameterized controller (52) with parameter (56).The simulation results are showed by Figure 5 (dotted line).In order to verify the tuning characteristic of the parameter containing in the parameterized controller, choose under controller (61) and switching law (57).Figure 3 (solid lines) shows the state responses and Figure 4 (solid lines) shows the partial enlargement of the state responses.Summary 3.According to Figures 3 and 4, the system controlled with the method of this paper became stable for no more than 1.5 s (see the dotted lines in Figures 3 and  4), while the system controlled with the method of [22] became stable for over 2.5 s (see the solid lines in Figures 3  and 4).Overall, the controller designed using Theorem 13 of this paper possesses a better robustness compared with the existing approach [22].
Summary 4. With the same external disturbance and  2 gain disturbance attenuation level , Figure 6 illustrates that the controllers designed using Theorem 13 of this paper provide better disturbance attenuation capability compared with the controller using the existing approach described in [22].

Conclusions
In this paper, the exponential feedback passivity of switched polynomial nonlinear systems is presented.Comparing with the existing results of [25][26][27], the nonlinear functions of the systems do not need to satisfy the so-called Lipschitz condition.In addition, the performance of the SPNSs can be improved by adjusting the value of the parameter contained in the parameterized controller according to the obtained feasible interval solution of the parameters.

Figure 4 :
Figure 4: Partial enlargement of the state response.