- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 973082, 18 pages

http://dx.doi.org/10.1155/2012/973082

## Fuzzy Control of DC-DC Converters with Input Constraint

^{1}LAMEL, Faculty of Science and Technology, University of Jijel, BP. 98, Ouled Aissa, 18000 Jijel, Algeria^{2}MIS (EA 4290), Université de Picardie Jules Verne, Rue du Moulin Neuf, 80000 Amiens, France^{3}Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 6 June 2012; Accepted 9 September 2012

Academic Editor: Peng Shi

Copyright © 2012 D. Saifia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a method for designing fuzzy control of DC-DC converters under actuator saturation. Because linear control design methods do not take into account the nonlinearity of the system, a T-S fuzzy model and a controller design approach is used. The designed control not only handles the external disturbance but also the saturation of duty cycle. The input constraint is first transformed into a symmetric saturation which is represented by a polytopic model. Stabilization conditions for the state feedback system of DC-DC converters under actuator saturation are established using the Lyapunov approach. The proposed method has been compared and verified with a simulation example.

#### 1. Introduction

The main task of DC-DC converters is the adaptation of the voltage and current levels between sources and loads while maintaining a low power loss in the conversion [1–3]. With the extensive use of DC-DC converters in different industry applications (e.g. power supplies for personal computers, DC-motor drive, telecommunications equipment, etc.), improving their performances has become an interesting problem in recent years [1–17]. Recently, different converter circuits (buck converter, boost converter, buck-boost converter, Cuk converter, etc.) are known. According to each application purpose (increase or decrease the magnitude of the DC voltage and/or invert its polarity), the converter circuit was chosen. Among them, we consider here, the control of the basic Pulse-Width-Modulation (PWM) buck converters, but it could be easily adapted for other converters.

The DC-DC switching power converters are highly nonlinear systems [4, 5, 15, 16]. Consequently, the conventional linear controls based on averaging and linearization techniques [12, 14] will result in poor dynamic performance or system instability. In order to resolve this problem, the fuzzy logic approach has been proposed as an alternative solution [4, 5, 7, 8, 11, 13, 15, 16]. Specifically, in [4, 5], the authors have reported very significant results on the modelling and control of DC-DC converters with T-S fuzzy systems. In [4, 5, 8, 11, 15, 16], authors have proposed methods for designing fuzzy control of DC-DC converters. Nevertheless, in the aforementioned papers, they have not taken into account the inherent nonlinearity of actuator saturation (duty cycle) and the external disturbance. Motivated by this observation, our aim is to use T-S fuzzy systems in order to control a nonlinear DC-DC converter subject to external disturbances and actuator saturation.

The actuator saturation can degrade the performance of the closed-loop system and often make the stable closed-loop system unstable [1, 18–21]. The T-S fuzzy system in the presence of saturation has been receiving increasing attention for control of nonlinear systems [18–29]. In these works, the saturation effect is considered as a symmetric function. This is not the case in DC-DC converter application where the saturation function constrained is between 1 and 0. The solution of this problem proposed in linear control case [2] has been leading to complexity analysis with the nonlinearity of DC-DC converter model. Here, we proposed a simple mathematical transformation to obtain a symmetric saturation.

The approach is used to analyze and to synthesize controllers/observers achieving an optimal level of disturbance attenuation and to guarantee the stability of the closed-loop system. To achieve this goal, the idea is to minimize the norm which represents the maximum value ratio between the output signal energy (controlled output) and the input signal energy (disturbance input). Recently, many researchers have used this approach for control design of T-S fuzzy systems (see for example [20, 21, 30, 31] and their references). In most cases, the quadratic Lyapunov function and linear matrix inequality (LMI) techniques are used to analyze and synthesize of stabilization of T-S fuzzy systems [17–21, 30, 31]. Based on these works, we address the control problem of DC-DC converters via PDC controller with actuator saturation and external disturbance. In this paper, we will use T-S fuzzy systems to represent the DC-DC converters. The control design is based on the parallel distributed compensation (PDC) scheme [32]. The idea is that for each local linear model, a linear feedback control is designed. The control problem is formulated and solved as a LMI optimization problem [33].

This paper is organized as follows. Section 2 gives the averaged model of basic PWM buck converter. Section 3 presents the T-S fuzzy model of the DC-DC converter. Section 4 formulates the conditions for the stabilization of fuzzy control with actuator saturation problem in terms of LMI. The simulation results to show the effectiveness of the proposed method are given in Section 4.

*Notation 1. * denotes the set , denotes the set of real number and the set of all real matrix. is used to denote a symmetric positive definite (positive semidefinite, negative definite, negative semidefinite, resp.) matrix. * denotes the symmetric bloc matrix, denotes , denotes the multiplication, co denotes the convex hull, and is used to denote the intersection of sets.

#### 2. Averaged Model of Basic PWM Buck Converter

Figure 1 shows the basic circuit of the nonlinear PWM buck converter proposed in [4, 5] with an external disturbance as proposed in [1, 2]. is the on-state resistance of the MOSFET transistor, is the winding resistance of inductor, is the threshold voltage of the diode, and is the equivalent series resistance of the filter capacitor.

By applying the Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) in on-state of the MOSFET transistor case, we obtain Now, in off-state of the MOSFET transistor case and by applying of KVL and KCL, we get Using averaging method for one-time scale discontinuous system (AM-OTS-Ds) [9], the global dynamical behavior of the DC-DC converter is modeled as follows: where is the duty cycle.

#### 3. T-S Fuzzy Model of DC-DC Converter

Our control approach is based on T-S fuzzy models and the robustness of this control depends on the ability of the fuzzy model to represent the real system. In this section our objective is to show the effectiveness of T-S fuzzy model to represent the DC-DC converter.

The T-S fuzzy model has been successfully used to approximate nonlinear systems by interpolation of numerous local linear models [34].

: If is about and is about then in which is the fuzzy set of in rule , is the number of if-then fuzzy rules and are the decision variable assumed measurable, is the system state vector, is the saturate control input, is the measurable output, is the controlled output variable, and is the disturbance variable with .

The global dynamic system is inferred as follows: where is the grade of membership of in . The normalized activation function in relation with the th submodel is such that Among different methods (identification method, linearization around different operating points, or by transformation in nonlinear sector), the method by transformation in nonlinear sector is based on bounded function and gives a minimum number of local models. Here we will use this method to represent the DC-DC converter.

The following Lemma will be used in the sequel of the paper.

Lemma 3.1. *Let is a bounded function, it always exist tow functions, and and two scalars and such that:
**
with
*

In this case, the obtained T-S fuzzy model represents exactly the nonlinear system for with locals models where represents the number of local models.

Assuming that, and by Lemma 3.1, the system described by (2.3) is modeled with a two-rules T-S fuzzy system as follows: Rule 1: if is about then: . Rule 2: if is about then: .

The overall model of T-S fuzzy system can be given by the following: with and The membership function is such that To show the effectiveness of the fuzzy model to represent the nonlinear system, we simulate the fuzzy model and the nonlinear system for the same inputs with different initial conditions. The simulation parameters used in this work are as follows [4]: with and the input current .

Figure 2 shows the membership functions of the fuzzy model. The DC-DC converter responses of nonlinear system (2.3) and T-S fuzzy system (3.7) with initial conditions and are shown in Figures 3 and 4, respectively.

These figures demonstrate that, with different initial conditions, the T-S fuzzy system (2.3) has the same behaviour as the nonlinear system (3.7). This means the satisfactory approximation ability of the fuzzy model.

#### 4. Fuzzy Control of DC-DC Converter

In this section, we present an fuzzy approach to the control design of DC-DC converter.

##### 4.1. Saturated Control Analysis

In order to control the output voltage of the DC-DC converter, we define the following variables: where is the reference voltage of .

The time-derivative of is as follows: Using the converter’s model defined in (2.3), we have By Lemma 3.1, this system can be modeled with a two-rule T-S fuzzy system as follows: Rule 1: if is about then: . Rule 2: if is about then: .

and the global T-S fuzzy model can be given by the following: with: The output voltage of the DC-DC converter can be controlled by means of variation of duty cycle (Figure 1). The duty cycle is defined by the ratio , where is the frequency of the PWM circuit and is the time when the MOSFET is on (Figure 5). A transistor gate driver (Figure 1) switches the MOSFET between the conducting (on) and blocking (off) states using a binary signal . This signal is produced by the modulator. In this operation, the duty cycle is compared with a sawtooth signal of amplitude equals to 1. Consequently, the duty-cycle is constrained in amplitude between 0 and 1.

Thus, in this application, the control input is subject to actuator saturation with the following saturation: which is not a symmetric saturation. However, most results reported in open literature have treated the controller design analysis with symmetric saturation [18–21]. To transform this saturation into a symmetric saturation, authors in [2] have proposed a change of variables to the linear converter model. This idea cannot be used in our case (nonlinear converter model) because it increases the complexity of the control analysis. Here, we will propose another idea that allows a simple analysis thereafter.

The system described in (4.4) can be written as follows: with is the external disturbance, , and is the new control input.

This last system can be represented with the following T-S fuzzy model: Thus, the new control input is constrained as follows:

In this work, the controller is a nonlinear state feedback which shares the same activation functions as the T-S fuzzy model (3.2) of the following form [32]: where and is the local controller matrix to be determined.

The control input is subject to actuator saturation the saturation function means: denotes the saturation level and and denote the th element of and . In our case: The saturation obtained in (3.6) is a symmetric saturation and the following lemma can be used to the stability analysis.

Lemma 4.1 (see [18]). *Let be the set of diagonal matrices whose diagonal elements are 1 or 0. Suppose that for all , where and denote the th element of and , respectively. If for , then:
**
where denotes all elements of , is matrix and is the th row of the matrix . *

In our case , is the saturation level and is the number of local models. Consequently, the saturation function in (4.10) can be rewritten as follows: with and and .

##### 4.2. Quadratic Lyapunov Stability

For a constant and a symmetric positive matrix , define an ellipsoid as follows: This ellipsoid can be rewritten as follows: , this implied that is a symmetric positive matrix.

We define the Lyapunov function as follows: An ellipsoid is said to be contractively invariant set if , for all . Therefore, if an ellipsoid is contractively invariant, it is inside the domain of attraction.

An ellipsoid is inside if and only if for all : In order to design a DC-DC converter to perform adequately in the presence of external disturbances, the attenuation performance is chosen as the performance measure, which is defined as follows: where is a positive scalar and small as possible, and .

In our problem, the controlled output is the error , that is: where . The closed-loop system composed of (4.9), (4.22), and (4.15) is given by the following:

Theorem 4.2. * The ellipsoid is contractively invariant set of the closed-loop system (4.15) and achieves in a disturbance rejection level , if there exist a symmetric positive definite matrix and matrices , solutions of the following LMI problem:
**
Then one gets
**
Using Lemma 3.1 for inequality (4.20), one has:
**
Let the following change of variables:
**
Consequently, the inequality (4.28) can be rewritten as follows
**
where is the th row of the matrix . By Schur complement [33], this last inequality can be transformed as LMI (4.25).*

Proposition 4.3 (see [30]). *If the Lyapunov function defined in (4.19) satisfies the following Hamilton-Jacobi-Bellman inequality:
**
along (4.23) for all and , then (4.21) is verified. Using (4.23), one has:
**
By Schur complement [13], the condition (4.32) holds if
**
Pre- and postmultiplying this last inequality by one has
**
Taking account (4.27) and (4.30), one gets the following LMI:
**
which ends the proof.*

##### 4.3. The Initial Conditions Constraint

The main purpose of the DC-DC converter control is to maintain the voltage level equal to the desired level . The initial condition for the state vector is as follows: However, it is not possible to assure that any initial condition will belong to the ellipsoid of stability given in Theorem 4.2. In this section, the objective is to guarantee the start up stability with maximum elimination of external disturbances in the presence of saturation.

The point is equivalent to the following: Or in LMI form by the following: The optimization problem proposed in Theorem 4.2 can be formulated as follows:

##### 4.4. Simulation Results

In order to demonstrate the effectiveness of the proposed method, the DC-DC converter is controlled by the proposed -based state feedback and the PDC controller proposed in [4]. In this simulation, the current perturbation step equals to 1 A and the PWM frequency equals to 5 kHz.

Solving the optimization problem described in (4.40) for initial condition , we have: The control is defined by .

Figures 6, 7, and 8 show the DC-DC converter responses (output voltage and inductance current, resp.). Figure 9 shows the trajectory of the saturated control input signal (duty cycle ). Figure 8 shows, with the presence of saturation, the control can guarantee the stability of all initial conditions in the interval in presence of actuator saturation and external disturbances (see initial time and 0.1 s time).

Simulation results (Figures 6–8) demonstrate that our controller guarantees better stabilization performance, better perturbation rejection (see initial time in Figure 6) and better time response (see Figures 6–8) Moreover, the proposed controller is robust with respect to frequency change. Figure 10 shows the simulation results for a frequency of 1 kHz. It shows that our controller achieves better performance even with different frequency values.

Despite that we take into account not only the actuators saturation but also the external disturbance in the designing control, our controller gives better stabilization performance (perturbation rejection and better time response) of the system as reported in the open literature [4]. Moreover, with different frequency values, our saturated control gives robust control signal (guarantees the same performances despite the change of frequency). These results demonstrate the effectiveness of proposed method.

#### 5. Conclusion

This work has presented an controller design for DC-DC converters via state feedback under actuator saturation and external disturbances. The T-S fuzzy system is first used to describe the DC-DC converter. Then, the state controller is designed to guarantee the stability of the closed loop system with performance. The saturation effect is represented by a polytopic model. Based on Lyapunov approach, the problem of stabilization in the presence of actuator saturation was formulated as an LMI optimization and solved easily by using existing numerical tools. Finally, the simulation results on a DC-DC converter have demonstrated the effeteness of the proposed control.

The analysis given here is applied to the buck converter; we can extend this work to other converters (boost converter, buck-boost converter, Cuk converter, etc.). Finally, the results developed in the paper can be extended to the case that the underlying systems are involved with any switching dynamics such as a nondeterministic or stochastic switching system (see [35–37]).

#### References

- H.-S. Kim, J. H. Kim, B. D. Min, D. W. Yoo, and H. J. Kim, “A highly efficient PV system using a series connection of DC-DC converter output with a photovoltaic panel,”
*Renewable Energy*, vol. 34, no. 11, pp. 2432–2436, 2009. View at Publisher · View at Google Scholar · View at Scopus - C. Olalla, I. Queinnec, R. Leyva, and A. El Aroudi, “Robust optimal control of bilinear DC-DC converters,”
*Control Engineering Practice*, vol. 19, no. 7, pp. 688–699, 2011. View at Publisher · View at Google Scholar · View at Scopus - J.-K. Kaldellis, “The wind potential impact on the maximum wind energy penetration in autonomous electrical grids,”
*Renewable Energy*, vol. 33, no. 7, pp. 1665–1677, 2008. View at Publisher · View at Google Scholar · View at Scopus - Y. Li and Z. Ji, “T-S modeling, simulation and control of the Buck converter,” in
*Proceedings of the 5th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2008*, pp. 663–667, Schandong, China, October 2008. View at Publisher · View at Google Scholar · View at Scopus - K. Y. Lian, J. J. Liou, and C. Y. Huang, “LMI-based integral fuzzy control of DC-DC converters,”
*IEEE Transactions on Fuzzy Systems*, vol. 14, no. 1, pp. 71–80, 2006. View at Publisher · View at Google Scholar · View at Scopus - I. Gadoura, T. Suntio, and K. Zenger, “Model uncertainty and robust control of paralleled DC/DC converters,” in
*Proceedings of the 15th World IFAC Congress on Power Electronics, Machines and Drives*, pp. 74–79, Barcelona, Spain, April 2002. View at Scopus - D. Alejo, P. Maussion, and J. Faucher, “Multiple model control of a Buck dc/dc converter,”
*Mathematics and Computers in Simulation*, vol. 63, no. 3–5, pp. 249–260, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Guesmi, N. Essounbouli, A. Hamzaoui, J. Zaytoon, and N. Manamanni, “Shifting nonlinear phenomena in a DC-DC converter using a fuzzy logic controller,”
*Mathematics and Computers in Simulation*, vol. 76, no. 5-6, pp. 398–409, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Sun and H. Grotstollen, “Averaged modelling of switching power converters: reformulation and theoretical basis,” in
*Proceedings of the Power electronics specialists Conference*, pp. 1165–1172, Taledo, Spain, 1992. - A. Shahin, B. Huang, J. P. Martin, S. Pierfederici, and B. Davat, “New non-linear control strategy for non-isolated DC/DC converter with high voltage ratio,”
*Energy Conversion and Management*, vol. 51, no. 1, pp. 56–63, 2010. View at Publisher · View at Google Scholar · View at Scopus - K. Guesmi, N. Essounbouli, and A. Hamzaoui, “Systematic design approach of fuzzy PID stabilizer for DC-DC converters,”
*Energy Conversion and Management*, vol. 49, no. 10, pp. 2880–2889, 2008. View at Publisher · View at Google Scholar · View at Scopus - F. D. Tan and R. D. Middlebrook, “Unified model for current-programmed converters,”
*IEEE Transactions on Power Electronics*, vol. 10, no. 4, pp. 397–408, 1995. View at Publisher · View at Google Scholar · View at Scopus - C. Elmas, O. Deperlioglu, and H. H. Sayan, “Adaptive fuzzy logic controller for DC-DC converters,”
*Expert Systems with Applications*, vol. 36, no. 2, pp. 1540–1548, 2009. View at Publisher · View at Google Scholar · View at Scopus - P. R. K. Chetty, “Modelling and design of switching regulators,”
*IEEE Transactions on Aerospace and Electronic Systems*, vol. 18, no. 3, pp. 333–344, 1982. View at Scopus - H. K. Lam, T. H. Lee, F. H. E. Leung, and P. K. S. Tam, “Fuzzy control of DC-DC switching converters: stability and robustness analysis,” in
*Proceedings of the 27th Annual Conference of the IEEE Industrial Electronics Society*, Denver, Colo, USA, December 2001. - H. K. Lam and S. C. Tan, “Stability analysis of fuzzy-model-based control systems: application on regulation of switching DC-DC converter,”
*IET Control Theory & Applications*, vol. 3, no. 8, pp. 1093–1106, 2009. View at Publisher · View at Google Scholar - S. Bououden, M. Chadli, S. Filali, and A. El Hajjaji, “Fuzzy model based multivariable predictive control of a variable speed wind turbine: LMI approach,”
*Renewable Energy*, vol. 37, no. 1, pp. 434–439, 2012. View at Publisher · View at Google Scholar · View at Scopus - Y. Y. Cao and Z. Lin, “Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation,”
*IEEE Transactions on Fuzzy Systems*, vol. 11, no. 1, pp. 57–67, 2003. View at Publisher · View at Google Scholar · View at Scopus - D. Saifia, M. Chadli, and S. Labiod, “Static output feedback stabilization of multiple models subject to actuators saturation,” in
*Proceedings of the 12th IFAC Symposium on Large Scale Systems*, Villeneuve d'Ascq, France, 2011. - D. Saifia, M. Chadli, and S. Labiod, “H∞ control of multiple model subject to actuator saturation: application to quarter-car suspension system,”
*Analog Integrated Circuits and Signal Processing*, pp. 1–10, 2011. View at Publisher · View at Google Scholar · View at Scopus - T. Zhang, G. Feng, H. Liu, and J. Lu, “Piecewise fuzzy anti-windup dynamic output feedback control of nonlinear processes with amplitude and rate actuator saturations,”
*IEEE Transactions on Fuzzy Systems*, vol. 17, no. 2, pp. 253–264, 2009. View at Publisher · View at Google Scholar · View at Scopus - C. P. Huang, “Model based fuzzy control with affine T-S delayed models applied to nonlinear systems,”
*International Journal of Innovative Computing*, vol. 8, no. 5, pp. 2979–2993, 2012. - C.-H. Sun, Y.-T. Wang, C.-C. Chang, and T.-S. Switching, “Fuzzy model-based guaranteed cost control for two-wheeled mobile robots,”
*International Journal of Innovative Computing A*, vol. 8, no. 5, pp. 3015–3028, 2012. - L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 41, no. 1, pp. 273–286, 2011. View at Publisher · View at Google Scholar · View at Scopus - L. Wu, X. Su, P. Shi, and J. A. Qiu, “Novel approach to filter design for T-S fuzzy discrete-time systems with time-varying delay,”
*IEEE Transactions on Fuzzy Systems*, vol. PP, no. 99, p. 1, 2012. - C.-H. Sun, S.-W. Lin, and Y.-T. Wang, “Relaxed stabilization conditions for switching T-S fuzzy systems with practical constraints,”
*International Journal of Innovative Computing*, vol. 8, no. 6, pp. 4133–4145, 2012. - X. Su, L. Wu, P. Shi, and Y. D. . Song, “${\text{H}}_{\infty}$ model reduction of T-S fuzzy stochastic systems,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. PP, no. 99, pp. 1–12, 2012. - L. Wu, X. Su, P. Shi, and J. Qiu, “Model approximation for discrete-time state-delay systems in the TS fuzzy framework,”
*IEEE Transactions on Fuzzy Systems*, vol. 19, no. 2, pp. 366–378, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. H. Chang and G. H. Yang, “Relaxed results on stabilization and state feedback H∞ control conditions for T-S fuzzy systems,”
*International Journal of Innovative Computing, Information and Control*, vol. 7, no. 4, pp. 1753–1764, 2011. View at Scopus - H. J. Lee and D. W. Kim, “Robust stabilization of T-S fuzzy systems: fuzzy static output feedback under parametric uncertainty,”
*International Journal of Control, Automation and Systems*, vol. 7, no. 5, pp. 731–736, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. Chadli and P. Borne, “Design of robust static output controller for uncertain fuzzy model: an LMI formulation,”
*Studies in Informatics and Control*, vol. 16, pp. 421–430, 2007. - K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs,”
*IEEE Transactions on Fuzzy Systems*, vol. 6, no. 2, pp. 250–265, 1998. View at Scopus - S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequalities in System and Control Theory*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar - T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,”
*IEEE Transactions on Systems, Man and Cybernetics*, vol. 15, no. 1, pp. 116–132, 1985. View at Scopus - L. Zhang and J. Lam, “Necessary and sufficient conditions for analysis and synthesis of Markov jump linear systems with incomplete transition descriptions,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 7, pp. 1695–1701, 2010. View at Publisher · View at Google Scholar - L. Zhang and P. Shi, “Stability, L2-gain and asynchronous ${\text{H}}_{\infty}$ control of discrete-time switched systems with average dwell time,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 9, pp. 2192–2199, 2009. View at Publisher · View at Google Scholar - L. Zhang, N. Cui, M. Liu, and Y. Zhao, “Asynchronous filtering of discrete-time switched linear systems with average dwell time,”
*IEEE Transactions on Circuits and Systems*, vol. 58, no. 5, pp. 1109–1118, 2011. View at Publisher · View at Google Scholar