Finitetime Control of a Class of Switched Nonlinear Onesided Lipschitz System
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Finitetime control of switched nonlinear onesided lipschitz systems; based on auxiliary matrices and average dwell time method.
Abstract: In this paper, the finitetime control of a class of switched nonlinear onesided lipschitz systems with timevarying uncertainties and norm bounded disturbances is presented based on auxiliary matrices and the average dwell time method. First, the problem of finitetime boundedness is investigated for the mentioned switched systems. Then, a finitetime controller is designed, to make the closedloop system be finitetime bounded in the presence of time varying uncertainties and external disturbances. Moreover the finitetime performance index of the closedloop system is guaranteed based on with the auxiliary matrices and average dwell time method. The proposed theorems are less conservative compared to existing methods, due to separation of the Lyapunov matrix from the system matrix using auxiliary matrices. Moreover in this paper the lipschitz condition is not necessary and the onesided lipschitz condition is considered for nonlinear terms. Finaiy, Two examples are given to illustrate the effectiveness of the proposed method and to show its less conservatism.
Keywords: FiniteTime Boundedness, Switched Nonlinear Systems, OneSided Lipschitz, TimeVarying Exogenous Disturbances, Auxiliary Matrices.
 Introduction
Switched systems compose a special classe of hybrid systems. Over the ago years, the switching systems have a lot of attentions by researchers, due to their wide applications. The focus of most of these papers is on the stability of the system based on lyapunov analysis in a infinite time interval [13].
On the other hand, a large number of practical applications, behavior of system is important in a finite time interval and the asymptotic stability which concerns on an infinitetime convergence for these systems are unacceptable. hence the finitetime stability should be studied [4]. The finitetime stability (FTS) and finitetime boundedness (FTB) have been studied in a large number of papers [57]. Recently, plenty of researchers focus on the finitetime stability and finitetime boundedness of the switching systems [817]. The reference [8, 9, 1214] has focused on finitetime stability of switched linear systems. Furthermore, finite time control for switched nonlinear systems that has the Lipschitz conditions include discretetime [1820] , impulsive switched systems[21], stochastic hybrid systems[22]. In fact, the Lipschitz condition may lose when large Lipschitz constants have to be selected [23]. For this purpose, the definition of the lipschitz generalized condition has been to onesided lipschitz [24]. Onesided lipschitz condition is Less conservative than the wellknown Lipschitz condition, therefore many researchers have done a lot of works in this regard [2527].
Although, finitetime boundedness for switched linear systems has been extensively investigated, to the best of authors’ knowledge, there is no result available yet on finitetime control for switched nonlinear onesided lipschitz systems; using auxiliary matrices and average dwell time method, which is the motivation of this research.
In this paper, a class of onesided lipschitz switched nonlinear systems is considered and three theorems are proposed to guarantee the finitetime boundedness and stabilization of the switched system in the presence of timevarying uncertainties and norm bounded disturbances and performance index is satisfied. Using the Finsler’s lemma, auxiliary matrices, onesided lipschitz condition and average dwell time method, have led to less conservative results, since the auxiliary matrices make the Lyapunov matrix separate from the system matrix. Finally, two examples is given to illustrate the effectiveness of the method proposed.
The paper is organized as follows. In Section 2, some important definitions and assumptions are given. the analysis of finitetime boundedness is presented in Section 3. Then, in Section 4, a state feedback controller based on the presented analysis is designed. Finite time performance index is presented for the switched system in section 5 and in order to show the effectiveness of the proposed results two examples are presented in Section 6. Finally, the paper is concluded in Section 7.
 Problem formulation
Consider a class of switched nonlinear systems in the following form:

(1) 
where is the state variable, is the control input and is an exogenous disturbance. Also, is the switching signal which is a piecewise constant function, is a real nonlinear vector field for, satisfying onesided lipschitz condition and , and are constant real matrices and and are unknown matrices satisfying the following condition:

(2) 
where are know matrices and for are unknown time varying matrices that satisfying,

(3) 
Definition 1[24]: If there exist constant , such that following condition hold, then the nonlinear function for is onesided lipschitz on .

(4) 
where can be positive, negative or even zero.
Definition 2[24]: If there exists such that the following inequality hold then the nonlinear function is a quadratic innerboundedness function on .

(5) 
where are quadratic innerboundedness constants and can be positive, negative or zero.
Assumption 1 [28]: is the external disturbance input and satisfies the following condition:

(6) 
where is the finite time of the study.
Assumption 2 [8]: the trajectory is continuous and does not jump and the switching signal has a finite number of switchings.
Definition 3 [29]. Consider the following switched system,

(7) 
if the following condition holds,

(8) 
then system (7) is said to be finitetime bounded (FTB) with respect to.
Definition 4 [30]. For any , The function denotes the number of switching of over , if

(9) 
holds, then is called the average dwelltime.
Definition 5 [29]: considerthe system (1) with the zero initial condition. The performance index is satisfies if the following condition hold,

(10) 
For simplicity, suppose that , which denotes that the ith subsystem is active at the time t, i.e.,

(11) 
Thus, the equation of the system at this time will be as

(12) 
where
 Finitetime boundedness
In this section, using the method of multiple Lyapunovlike functions and the average dwelltime technique and also the Finsler’s lemma, sufficient conditions of the finitetime boundedness of the nonlinear switched onesided lipschitz system (1) with are investigated. First, the following lemmas are introduced which will be used to develop main results of the paper:
Lemma 1(Finsler’s lemma )[31]: Let such that . The following conditions are equivalent:
 such that and .
 such that .
Lemma 2 [32]:For any scalar, one has

where and are constant and real matrices with appropriate dimensions and is time varying matrix with appropriate dimensions that satisfied .
The following theorem presents, sufficient conditions for finitetime boundedness of the system (1) with :
Theorem 1: For given parameters, if there exist symmetric positivedefinite matrices for and parameters such that the following inequality holds, the system (1) with is finitetime bounded with respect to parameters :

(13) 


(14) 

where

Moreover, the average dwelltime can be obtained as,

(15) 
Proof:consider a piecewise Lyapunovlike function as follows

(16) 
when . Define the following function:

(17) 
In the other hand, the function is onesided Lipschitz and
also quadratic innerboundedness, considering definitions 1 and 2 for and , the following inequality will be obtained:

(18) 
Based on (17) and (18), we can define

(19) 
According to the inequality (19), if is negative, then will be negative, too. The inequality can be written in the following matrix form:

(20) 
where

Since,

(21) 
according to Lemma 1, if the following inequality holds, then based on the dynamic of the system is negative:

(22) 
Consequently,

(23) 
The above inequality can be rewritten as follow:

(24) 
where
From (2), we can obtain

(25) 
where have already been defined in theorem 1.
Using Lemma 2, it yield that

(26) 
The inequality (26) is not in the form of LMI. Thus according to Schur complement lemma, it will be converted to the inequality (13) .
As mentioned before, the result of the inequality , leads to , that is,

(27) 
which leads to,

(28) 
Integrating (28) from to gives

(29) 
Then, according to the definitions of and, at the switching instant , we have

(30) 
Since , then

(31) 
That means that,and based on Assumption 2, . Without loss of generality, at the switching instant , assume that , . According to (31), we can get

(32) 
According to (29) and (32), we have

(33) 
For any , let be the number of switching of over , which implies that. Since ( ), and using an iterative method, we have

(34) 
where .
Now, the relation leads to

(35) 
From (16) we have

(36) 
The inequalities (35) and (36), leads to,

(37) 
On the other hand,

(38) 
Putting together (37) and (38), we get

(39) 
Now, consider the following two cases:
Case 1: , which is a trivial case, we have

(40) 
And is equivalent with the following matrix inequality:


Case 2: , from (39) , we have

(41) 
The proof is completed. ■
 Finitetime stabilization
Considering the proposed finitetime boundedness analysis in the previous section, now, the finitetime stabilization issue is investigated. In this paper, the following state feedback controller is designed to stabilize the system (1).

(42) 
Substituting (42) into the system (1), the dynamic of the closedloop system for will be as follows:

(43) 
where
In the sequel, the switching gain of the controller is designed using the sufficient conditions for finitetime stability of the system (43).
Theorem 2:Considering the closedloop switching system (43), and for given parameters, if there exist positive scalars and symmetric positive definite matrices with appropriate dimensions and real matrices , for, such that the condition (14) and the following inequalities hold:

(44) 


(45) 


(46) 


(47) 

where
Then, with the control law (42) where the switching gains are as for, the corresponding closedloop system (43) is finitetime bounded with respect to . Moreover, the average dwelltime of the switching signal will be calculated from (15).
Proof:similar to the proof of Theorem 1for the closedloop system (43), one has

(48) 
where
have already been defined.
Using the Schur’s complement lemma and Substituting (2) and , we have

(49) 
Where
Presented in theorem 1.
The above inequality can be rewritten as follows:

(50) 
According to Lemma 2, we can get

(51) 
Using the Schur complement lemma, we have

(52) 
have already been defined.
Pre and postmultiplying (52) by , leads to,

(53) 
where
By changing variables as , the LMI (44) will be achieved.
On the other hand, If it is assumed that , and according to , It can be concluded that

(54) 
where , are chosen positive Scalars. The proof is completed. ■
Corollary 1: Consider the closedloop switching system (43) with . for given parameters, if there exist positive scalars and symmetric positive definite matrices with appropriate dimensions and real matrices , for, such that conditions (45)(47), (14) and the following inequality hold, Then, the closedloop system is finitetime bounded with respect to :

(55) 
Where have already been presented.
In this case, the average dwelltime is calculated from (15). Furthermore, the controller gain is calculated as for.
 Finitetime performance index
In the following theorem, by using the piecewise Lyapunovlike function and average dwell time and also auxiliary matrices method, sufficient condition of the finitetime control for the switched nonlinear onesided Lipschitz system is obtained.
Theorem 3: For given parameters, the closed loops system (43) is finitetime boundedness (FTB) and the performance index (10) is satisfied for , if there exist positive scalars and symmetric positive definite matrices and real matrices , for with appropriate dimensions, such that conditions (45)(47), (14) and also the following inequalities hold:

(56) 
where
where before defined.
Then, the average dwell time can be calculated from
, 
(57) 
and the controller gain is calculated as for.
Proof: consider the same Lyapunov function in Theorem 1 and the following auxiliary function:

(58) 
According to (18) , (21) and similar to the procedure in theorems 1and 2, we can get

(59) 
presented in (49).
The above inequality can be rewritten as follows:

(60) 
According to Lemma 2, we have

(61) 
Using Schur complement lemma, we can get

(62) 
where have already been defined.
Pre and postmultiplying (62) by and change of variables , lead to (56) is achieved.
In the other hand, , is equivalent with the following inequality:

(63) 
Integrating (63) from to and similar to (34) we can get

(64) 
From (64), with considering zero initial condition , thus

(65) 
On the other hand,

(66) 
Considering (66), (67) , we can get

(67) 
Since , then

(68) 
The proof is completed. ■
 Simulation results
In this section, two examples are presented to verify the theoretical results and to illustrate the effectiveness of the proposed method.
Example 1:Consider the switching system (1) by the following parameters
Model 1:

Model 2:

We assume the external disturbances is

From (4) and (5) parameters onesided lipschitz and quadratic inner boundedness can be selected as follow [33, 34]:

To verify the finitetime conditions on , we consider some of the parameters as follows

If there exist a such that the conditions of theorem 3 are satisfied then closed loop switching system is FTB on. The conditions theorem 3 are satisfied for and we can obtain the controller gain matrices as

Here, and average dwell time is . The computer simulation results are shown in Fig.1Fig.3.
The trajectory of the for openloop and closedloop systems shown in Fig.1. It is easy to see that for any switching signal with , the switched nonlinear system is finitetime boundedness with respect to while the open loop system is unstable. Fig. 2 shows the state trajectories under switching signal from the initial condition . The switching signal is shown in Fig.3.
Fig.1. Time history of for openloop and closedloop system
Fig.2. States of switched system
Fig.3. the switching signal
Example 2: consider a servomechanism system in [35]:

(69) 
Where the parameters are given as follow:
.
For different values , the subsystems will be shown as follows

(70) 
In order to show that the expressed procedures have a acceptable efficiency to the all admissible uncertainties, assume that there exist parameter uncertainties (2) and normbounded disturbance input in the servomechanism system with the matrices

(71) 
The corresponding parameters are specified as follows:

The closed loop system (1) with above values is finite time boundedness (FTB) with respect to , if conditions Corollary 1 is satisfied. By chosen and solving Corollary 1 by MATLAB Toolbox , we can obtain the following solutions

(72) 
The average dwell time is. Applying the obtained state feedback controllers above, under switching signal with , the simulation results are shown in Fig.4Fig.7.
The trajectory of state variables and signal control for closed loop switched system are shown in Fig.5 and Fig.7, respectively. As shown in Fig. 4, the openloop system is unstable while the closedloop system is finitetime boundedness with respect to . In other words, it means that for . Theorem presented in this article is an feasible solution for while In the reference [35] these values are. this shows that the theorems of this paper have less conservatism.
Fig.4: the trajectory of the for openloop and closedloop systems
Fig.5: state variable for closed loop switched system
Fig.6: Curves of the switching signal
Fig.7: the trajectory of signal control.
 Conclusion
In this paper, we study the finitetime boundedness of switched nonlinear onesided lipschitz systems with timevarying uncertain and norm bounded disturbance, using auxiliary matrices and average dwell time method. Based on the average dwell time method and multiple Lyapunovlike function approach, some sufficient conditions were derived to guarantee the FTB property of the switched linear and nonlinear system. Then, a switching state feedback controller was designed to stabilize the closedloop switched system. The finitetime controller was designed by static state feedback. In this regard, three theorem were presented which are less conservative with respect to existing methods. Finally, two simulation examples were provided to demonstrate the effectiveness of the proposed approach.
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