Finite-time Control of a Class of Switched Nonlinear One-sided Lipschitz System

7668 words (31 pages) Essay in Physics

08/02/20 Physics Reference this

Disclaimer: This work has been submitted by a student. This is not an example of the work produced by our Essay Writing Service. You can view samples of our professional work here.

Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.

Finite-time control of switched nonlinear one-sided lipschitz systems; based on auxiliary matrices and average dwell time method.

Abstract: In this paper, the finite-time control of a class of switched nonlinear one-sided lipschitz systems with time-varying uncertainties and norm bounded disturbances is presented based on auxiliary matrices and the average dwell time method. First, the problem of finite-time boundedness is investigated for the mentioned switched systems. Then, a finite-time controller is designed, to make the closed-loop system be finite-time bounded in the presence of time varying uncertainties and external disturbances. Moreover the finite-time performance index of the closed-loop 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 one-sided 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: Finite-Time Boundedness, Switched Nonlinear Systems, One-Sided Lipschitz, Time-Varying Exogenous Disturbances, Auxiliary Matrices.

  1. 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 [1-3].

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 infinite-time convergence for these systems are unacceptable. hence the finite-time stability should be studied [4]. The finite-time stability (FTS) and finite-time boundedness (FTB) have been studied in a large number of papers [5-7]. Recently, plenty of researchers focus on the finite-time stability and finite-time boundedness of the switching systems [8-17]. The reference [8, 9, 12-14] has focused on finite-time stability of switched linear systems. Furthermore, finite time control for switched nonlinear systems that has the Lipschitz conditions include discrete-time [18-20] , 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 one-sided lipschitz [24]. One-sided lipschitz condition is Less conservative than the well-known Lipschitz condition, therefore many researchers have done a lot of works in this regard [25-27].

 Although, finite-time boundedness for switched linear systems has been extensively investigated, to the best of authors’ knowledge, there is no result available yet on finite-time control for switched nonlinear one-sided lipschitz systems; using auxiliary matrices and average dwell time method, which is the motivation of this research.

In this paper, a class of one-sided lipschitz switched nonlinear systems is considered and three theorems are proposed to guarantee the finite-time boundedness and stabilization of the switched system in the presence of time-varying uncertainties and norm bounded disturbances and performance index is satisfied. Using the Finsler’s lemma, auxiliary matrices, one-sided 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 finite-time 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.

  1. 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 one-sided 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 one-sided 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 inner-boundedness function on .

(5)  

where are quadratic inner-boundedness 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 finite-time 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 dwell-time.

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 i-th subsystem is active at the time t, i.e.,

(11)          

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

(12)          

where

  1. Finite-time boundedness

In this section, using the method of multiple Lyapunov-like functions and the average dwell-time technique and also the Finsler’s lemma, sufficient conditions of the finite-time boundedness of the nonlinear switched one-sided 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:

  1. such that and .
  2. 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 finite-time boundedness of the system (1) with :

Theorem 1: For given parameters, if there exist symmetric positive-definite matrices for and parameters such that the following inequality holds, the system (1) with is finite-time bounded with respect to parameters :

(13)          

(14)          

where

Moreover, the average dwell-time can be obtained as,

(15)          

Proof:consider a piecewise Lyapunov-like function as follows

(16)          

when . Define the following function:

(17)          

In the other hand, the function is one-sided Lipschitz and
also quadratic inner-boundedness, 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.       ■

  1. Finite-time stabilization

Considering the proposed finite-time boundedness analysis in the previous section, now, the finite-time 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 closed-loop system for will be as follows:

(43)          

where

In the sequel, the switching gain of the controller is designed using the sufficient conditions for finite-time stability of the system (43).

Theorem 2:Considering the closed-loop 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 closed-loop system (43) is finite-time bounded with respect to . Moreover, the average dwell-time of the switching signal will be calculated from (15).

Proof:similar to the proof of Theorem 1for the closed-loop 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 post-multiplying (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 closed-loop 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 closed-loop system is finite-time bounded with respect to :

(55)          

Where have already been presented.

In this case, the average dwell-time is calculated from (15). Furthermore, the controller gain is calculated as for.

  1. Finite-time performance index

In the following theorem, by using the piecewise Lyapunov-like function and average dwell time and also auxiliary matrices method, sufficient condition of the finite-time control for the switched nonlinear one-sided Lipschitz system is obtained.

Theorem 3: For given parameters, the closed loops system (43) is finite-time 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 post-multiplying (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. ■

  1. 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 one-sided lipschitz and quadratic inner boundedness can be selected as follow [33, 34]:

To verify the finite-time 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.1-Fig.3.

The trajectory of the for open-loop and closed-loop systems shown in Fig.1. It is easy to see that for any switching signal with , the switched nonlinear system is finite-time 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 open-loop and closed-loop 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 norm-bounded 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.4-Fig.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 open-loop system is unstable while the closed-loop system is finite-time 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 open-loop and closed-loop systems

Fig.5: state variable for closed loop switched system

Fig.6: Curves of the switching signal

Fig.7: the trajectory of signal control.

  1. Conclusion

In this paper, we study the finite-time boundedness of switched nonlinear one-sided lipschitz systems with time-varying uncertain and norm bounded disturbance, using auxiliary matrices and average dwell time method. Based on the average dwell time method and multiple Lyapunov-like 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 closed-loop switched system. The finite-time 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.

References

[1] N. Baleghi, and M. Shafiei, “Stability analysis and stabilization of a class of discrete-time nonlinear switched systems with time-delay and affine parametric uncertainty,” Journal of Vibration and Control, pp. 1077546318819737, 2019.

[2] N. A. Baleghi, and M. H. Shafiei, “Design of static and dynamic output feedback controllers for a discrete-time switched non-linear system with time-varying delay and parametric uncertainty,” IET Control Theory and Applications, 2018.

[3] Y. Chen, C. Wu, J. Yang, L. Cui, and W. Qian, “Finite-time state estimation and active mode identification for uncertain switched linear systems,” Transactions of the Institute of Measurement and Control, pp. 0142331219827049, 2019.

[4] J. Song, Y. Niu, and Y. Zou, “A parameter-dependent sliding mode approach for finite-time bounded control of uncertain stochastic systems with randomly varying actuator faults and its application to a parallel active suspension system,” IEEE Transactions on Industrial Electronics, vol. 65, no. 10, pp. 8124-8132, 2018.

[5] F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459-1463, 2001.

[6] L. Weiss, and E. Infante, “Finite time stability under perturbing forces and on product spaces,” IEEE Transactions on Automatic Control, vol. 12, no. 1, pp. 54-59, 1967.

[7] P. Dorato, Short-time stability in linear time-varying systems, POLYTECHNIC INST OF BROOKLYN NY MICROWAVE RESEARCH INST, 1961.

[8] H. Du, X. Lin, and S. Li, “Finite-time boundedness and stabilization of switched linear systems,” Kybernetika, vol. 46, no. 5, pp. 870-889, 2010.

[9] H. Liu, Y. Shen, and X. Zhao, “Finite-time stabilization and boundedness of switched linear system under state-dependent switching,” Journal of the Franklin Institute, vol. 350, no. 3, pp. 541-555, 2013.

[10] Y. Wu, J. Cao, A. Alofi, A.-M. Abdullah, and A. Elaiw, “Finite-time boundedness and stabilization of uncertain switched neural networks with time-varying delay,” Neural Networks, vol. 69, pp. 135-143, 2015.

[11] H. Liu, Y. Shen, and X. Zhao, “Asynchronous finite-time H∞ control for switched linear systems via mode-dependent dynamic state-feedback,” Nonlinear Analysis: Hybrid Systems, vol. 8, pp. 109-120, 2013.

[12] W. Xiang, and J. Xiao, “Finite-time stability and stabilisation for switched linear systems,” International Journal of Systems Science, vol. 44, no. 2, pp. 384-400, 2013.

[13] M. S. Ali, S. Saravanan, and J. Cao, “Finite-time boundedness, L2-gain analysis and control of Markovian jump switched neural networks with additive time-varying delays,” Nonlinear Analysis: Hybrid Systems, vol. 23, pp. 27-43, 2017.

[14] X. Lin, H. Du, S. Li, and Y. Zou, “Finite-time boundedness and finite-time l2 gain analysis of discrete-time switched linear systems with average dwell time,” Journal of the Franklin Institute, vol. 350, no. 4, pp. 911-928, 2013.

[15] Y. Wang, X. Shi, G. Wang, and Z. Zuo, “Finite-time stability for continuous-time switched systems in the presence of impulse effects,” IET control theory and applications, vol. 6, no. 11, pp. 1741-1744, 2012.

[16] G. Zong, H. Ren, and L. Hou, “Finite-time stability of interconnected impulsive switched systems,” IET Control Theory and Applications, vol. 10, no. 6, pp. 648-654, 2016.

[17] J. Ban, W. Kwon, S. Won, and S. Kim, “Robust H∞ finite-time control for discrete-time polytopic uncertain switched linear systems,” Nonlinear Analysis: Hybrid Systems, vol. 29, pp. 348-362, 2018.

[18] W. Xiang, and J. Xiao, “H∞ finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance,” Journal of the Franklin Institute, vol. 348, no. 2, pp. 331-352, 2011.

[19] G. Zong, R. Wang, W. Zheng, and L. Hou, “Finite‐time H∞ control for discrete‐time switched nonlinear systems with time delay,” International Journal of Robust and Nonlinear Control, vol. 25, no. 6, pp. 914-936, 2015.

[20] Y. Mao, H. Zhang, and Z. Zhang, “Finite-time stabilization of discrete-time switched nonlinear systems without stable subsystems via optimal switching signal design,” IEEE Transactions on Fuzzy Systems, vol. 25, no. 1, pp. 172-180, 2017.

[21] Y. Wang, X. Shi, Z. Zuo, M. Z. Chen, and Y. Shao, “On finite-time stability for nonlinear impulsive switched systems,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 807-814, 2013.

[22] Y. Yang, J. Li, and G. Chen, “Finite-time stability and stabilization of nonlinear stochastic hybrid systems,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 338-345, 2009.

[23] G.-D. Hu, “A note on observer for one-sided Lipschitz non-linear systems,” IMA Journal of Mathematical Control and Information, vol. 25, no. 3, pp. 297-303, 2007.

[24] M. Abbaszadeh, and H. J. Marquez, “Nonlinear observer design for one-sided Lipschitz systems.” pp. 5284-5289.

[25] M. A. Razaq, M. Rehan, C. K. Ahn, A. Q. Khan, and M. Tufail, “Consensus of One-Sided Lipschitz Multiagents Under Switching Topologies,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019.

[26] Y. Yang, C. Lin, and B. Chen, “Nonlinear H∞ observer design for one‐sided Lipschitz discrete‐time singular systems with time‐varying delay,” International Journal of Robust and Nonlinear Control, vol. 29, no. 1, pp. 252-267, 2019.

[27] C. M. Nguyen, P. N. Pathirana, and H. Trinh, “Robust observer and observer-based control designs for discrete one-sided Lipschitz systems subject to uncertainties and disturbances,” Applied Mathematics and Computation, vol. 353, pp. 42-53, 2019.

[28] H. Gholami, and T. Binazadeh, “Sliding-mode observer design and finite-time control of one-sided Lipschitz nonlinear systems with time-delay,” Soft Computing, pp. 1-12, 2018.

[29] H. Gholami, and T. Binazadeh, “Observer-based H∞ finite-time controller for time-delay nonlinear one-sided Lipschitz systems with exogenous disturbances,” Journal of Vibration and Control, pp. 1077546318802422, 2018.

[30] H. Zhao, Y. Niu, and J. Song, “Finite-time output feedback control of uncertain switched systems via sliding mode design,” International Journal of Systems Science, vol. 49, no. 5, pp. 984-996, 2018.

[31] Y. Huang, S. Fu, and Y. Shen, “Finite-time H∞ control for one-sided Lipschitz systems with auxiliary matrices,” Neurocomputing, vol. 194, pp. 207-217, 2016.

[32] H. Gholami, and T. Binazadeh, “Robust Finite-Time H∞ Controller Design for Uncertain One-Sided Lipschitz Systems with Time-Delay and Input Amplitude Constraints,” Circuits, Systems, and Signal Processing, pp. 1-21, 2019.

[33] J. Song, and S. He, “Robust finite-time H∞ control for one-sided Lipschitz nonlinear systems via state feedback and output feedback,” Journal of the Franklin Institute, vol. 352, no. 8, pp. 3250-3266, 2015.

[34] R. Wu, W. Zhang, F. Song, Z. Wu, and W. Guo, “Observer-based stabilization of one-sided Lipschitz systems with application to flexible link manipulator,” Advances in Mechanical Engineering, vol. 7, no. 12, pp. 1687814015619555, 2015.

[35] L. Zhang, S. Wang, H. R. Karimi, and A. Jasra, “Robust finite-time control of switched linear systems and application to a class of servomechanism systems,” IEEE/ASME Transactions on Mechatronics, vol. 20, no. 5, pp. 2476-2485, 2015.

Get Help With Your Essay

If you need assistance with writing your essay, our professional essay writing service is here to help!

Find out more

Cite This Work

To export a reference to this article please select a referencing stye below:

Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.
Reference Copied to Clipboard.

Related Services

View all

DMCA / Removal Request

If you are the original writer of this essay and no longer wish to have the essay published on the UK Essays website then please:

McAfee SECURE sites help keep you safe from identity theft, credit card fraud, spyware, spam, viruses and online scams Prices from
£124

Undergraduate 2:2 • 1000 words • 7 day delivery

Order now

Delivered on-time or your money back

Rated 4.6 out of 5 by
Reviews.co.uk Logo (188 Reviews)