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Conclusions and Future Work
C H A P T E R
11 Conclusions and Future Work This chapter draws conclusions on the book and points out some possible research directions related to the work done in this book.
11.1 CONCLUSIONS The focus of the book has been placed on robust control and filtering for T-S fuzzy model, switched systems, and nonhomogeneous Markovian jump systems. Several research problems have been investigated in detail. First, we aim at T-S fuzzy models with modeling uncertainties. By properly constructing LF with nonmonotonic behavior, the criteria for stability analysis, stabilization, and filtering are presented and then are extended to the case of dynamic output feedback control of nonlinear systems. A nonmonotonic Lyapunov function (NLF) has been employed to design robust H∞ state feedback controllers for uncertain T-S fuzzy systems in Chapter 2. In the NLF approach, the monotonicity requirement of the LF is relaxed by allowing it to increase locally but go to zero ultimately. Based on the NLF approach, sufficient conditions for the existence of a robust state feedback H∞ controller that guarantee the stability and a prescribed H∞ performance have been provided in terms of LMIs. In Chapter 3, we have examined the robust H∞ filtering problem for T-S fuzzy models of nonlinear systems by using the idea of nonmonotonic approach. Our main contribution has further reduced the conservatism and improved the H∞ performance. Motivated by the results in Chapter 2, in Chapter 4, robust H∞ output feedback control for uncertain T-S fuzzy systems has been studied via NLF approaches. For comparison, the proposed design technique has been shown to be less conservative than the existing nonmonotonic approach, namely, N -sample variations of LFs. Second, we focus on arbitrary switched systems and average dwelltime switched systems. Methodologies that can effectively handle control and filtering problems with less conservatism are developed by allowing the LF to increase both at the switching instant and during the running time of each subsystem. Non-Monotonic Approach to Robust H∞ Control of Multi-Model Systems https://doi.org/10.1016/B978-0-12-814868-6.00017-4
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Copyright © 2019 Elsevier Inc. All rights reserved.
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11. CONCLUSIONS AND FUTURE WORK
Chapter 5 has been focused on the H∞ control for a class of discretetime switched systems by introducing the one-step ahead LF approach. The one-step ahead LF is a function of future states. The design objectives are to reduce the conservatism of the stability criterion developed for arbitrarily switched systems and further get a better disturbance attenuation capability. The distinguishing feature is that the one-step ahead LF has no structural constraint, such as a diagonal structure, and the resulting analysis and synthesis criteria can cover the nonmonotonic method considering two-sample variation as a particular case. In Chapter 6, we have developed an N-step ahead LF approach, which allows a nonmonotonic behavior both at the switching instants and during the running time of each subsystem. The asymptotic stability criterion has been improved as well as the capability of disturbance attenuation. By introducing a series of auxiliary variables the future knowledge of states and exogenous noises can be properly used to derive sufficient conditions for the existence of a robust H∞ controller in the form of a set of numerical testable conditions. Moreover, the relationship between the N-step time difference of LF and switching rate, i.e., the ADT constraint, is thoroughly discussed. In Chapter 7, the nonmonotonic Lyapunov function approach with N-step ahead predictive horizon has been developed for designing a robust H∞ filter for a discrete-time uncertain switched system. With increasing the number N, the filtering performance can be improved as well as the capability of disturbance attenuation. To further relax the restrictions on the switching law, the mode-dependent ADT switching has been introduced to reduce the ADT bound such that a trade-off between the switching frequency and filtering performance can be achieved. Chapter 8 developed dissipative dynamic output feedback (DOF) control for a class of average dwell-time switching systems via the multistep LF approach. First, a larger dissipative region with guaranteed stability and, specifically, smaller H∞ level can be achieved by increasing a predictive step N, which means that the monotonic requirement of LF is relaxed. Then, based on the results of dissipative analysis, a robust dissipative DOF controller is further designed. Finally, we deal with Markovian jump systems governed by timevarying transition probabilities. The concept of N -step ahead approach is proposed such that the stability problem can be solved via a finite number of conditions. The systems involved with time-delay, which are handled by the Lyapunov–Krasovskii function approach, are also investigated. In Chapter 9, the robust H∞ control problem for a class of discretetime nonhomogenous Markovian jump linear systems (NMJLSs) has been investigated by a multistep LF approach. The proposed multistep LF is allowed to increase during the period of several sampling time steps ahead of the current time within the jump mode. First, a less conservative stability criterion has been derived based on this multistep LF approach. Second, an H∞ performance has been analyzed under the multistep case
11.2 FUTURE WORK
203
by properly dealing with the knowledge of future states and extraneous noises. These two results have then been employed to facilitate a robust H∞ control design for NMJLSs. Chapter 10 studied the robust H∞ filtering problem for a class of nonhomogeneous Markovian jump delay systems with the N -step ahead Lyapunov–Krasovskii function approach. In this chapter, we aim to design filters such that, for all possible time-varying transition probabilities and all admissible parameter uncertainties and time-delays, the filtering error system is mean-square stable with a smaller estimated error and a lower dissipative level. In terms of LMIs, sufficient conditions for the solvability of the addressed problem are developed via a moving-horizon method to avoid essential difficulties introduced by future noise.
11.2 FUTURE WORK Some of future works are listed as follows: • A possible future research direction is to investigate multiobjective H2 –H∞ control and filtering problems for multimodel systems. • It would be interesting to investigate the problems of fault detection and fault tolerant control for multimodel systems via the nonmonotonic Lyapunov function approach. • A trend for future research is to generalize the methods obtained in the book to the finite-time or finite-frequency control and filtering problems. • A challenging work is to employ the nonmonotonic approach to the model predictive control. A possible difficulty is to make a trade-off between the conservatism reduction and online computational burden. • The nonlinearities considered in the book have some condition constraints that bring conservativeness. The analysis and synthesis of general nonlinear systems could be one of the future research works.
11 Conclusions and Future Work This chapter draws conclusions on the book and points out some possible research directions related to the work done in this book.
11.1 CONCLUSIONS The focus of the book has been placed on robust control and filtering for T-S fuzzy model, switched systems, and nonhomogeneous Markovian jump systems. Several research problems have been investigated in detail. First, we aim at T-S fuzzy models with modeling uncertainties. By properly constructing LF with nonmonotonic behavior, the criteria for stability analysis, stabilization, and filtering are presented and then are extended to the case of dynamic output feedback control of nonlinear systems. A nonmonotonic Lyapunov function (NLF) has been employed to design robust H∞ state feedback controllers for uncertain T-S fuzzy systems in Chapter 2. In the NLF approach, the monotonicity requirement of the LF is relaxed by allowing it to increase locally but go to zero ultimately. Based on the NLF approach, sufficient conditions for the existence of a robust state feedback H∞ controller that guarantee the stability and a prescribed H∞ performance have been provided in terms of LMIs. In Chapter 3, we have examined the robust H∞ filtering problem for T-S fuzzy models of nonlinear systems by using the idea of nonmonotonic approach. Our main contribution has further reduced the conservatism and improved the H∞ performance. Motivated by the results in Chapter 2, in Chapter 4, robust H∞ output feedback control for uncertain T-S fuzzy systems has been studied via NLF approaches. For comparison, the proposed design technique has been shown to be less conservative than the existing nonmonotonic approach, namely, N -sample variations of LFs. Second, we focus on arbitrary switched systems and average dwelltime switched systems. Methodologies that can effectively handle control and filtering problems with less conservatism are developed by allowing the LF to increase both at the switching instant and during the running time of each subsystem. Non-Monotonic Approach to Robust H∞ Control of Multi-Model Systems https://doi.org/10.1016/B978-0-12-814868-6.00017-4
201
Copyright © 2019 Elsevier Inc. All rights reserved.
202
11. CONCLUSIONS AND FUTURE WORK
Chapter 5 has been focused on the H∞ control for a class of discretetime switched systems by introducing the one-step ahead LF approach. The one-step ahead LF is a function of future states. The design objectives are to reduce the conservatism of the stability criterion developed for arbitrarily switched systems and further get a better disturbance attenuation capability. The distinguishing feature is that the one-step ahead LF has no structural constraint, such as a diagonal structure, and the resulting analysis and synthesis criteria can cover the nonmonotonic method considering two-sample variation as a particular case. In Chapter 6, we have developed an N-step ahead LF approach, which allows a nonmonotonic behavior both at the switching instants and during the running time of each subsystem. The asymptotic stability criterion has been improved as well as the capability of disturbance attenuation. By introducing a series of auxiliary variables the future knowledge of states and exogenous noises can be properly used to derive sufficient conditions for the existence of a robust H∞ controller in the form of a set of numerical testable conditions. Moreover, the relationship between the N-step time difference of LF and switching rate, i.e., the ADT constraint, is thoroughly discussed. In Chapter 7, the nonmonotonic Lyapunov function approach with N-step ahead predictive horizon has been developed for designing a robust H∞ filter for a discrete-time uncertain switched system. With increasing the number N, the filtering performance can be improved as well as the capability of disturbance attenuation. To further relax the restrictions on the switching law, the mode-dependent ADT switching has been introduced to reduce the ADT bound such that a trade-off between the switching frequency and filtering performance can be achieved. Chapter 8 developed dissipative dynamic output feedback (DOF) control for a class of average dwell-time switching systems via the multistep LF approach. First, a larger dissipative region with guaranteed stability and, specifically, smaller H∞ level can be achieved by increasing a predictive step N, which means that the monotonic requirement of LF is relaxed. Then, based on the results of dissipative analysis, a robust dissipative DOF controller is further designed. Finally, we deal with Markovian jump systems governed by timevarying transition probabilities. The concept of N -step ahead approach is proposed such that the stability problem can be solved via a finite number of conditions. The systems involved with time-delay, which are handled by the Lyapunov–Krasovskii function approach, are also investigated. In Chapter 9, the robust H∞ control problem for a class of discretetime nonhomogenous Markovian jump linear systems (NMJLSs) has been investigated by a multistep LF approach. The proposed multistep LF is allowed to increase during the period of several sampling time steps ahead of the current time within the jump mode. First, a less conservative stability criterion has been derived based on this multistep LF approach. Second, an H∞ performance has been analyzed under the multistep case
11.2 FUTURE WORK
203
by properly dealing with the knowledge of future states and extraneous noises. These two results have then been employed to facilitate a robust H∞ control design for NMJLSs. Chapter 10 studied the robust H∞ filtering problem for a class of nonhomogeneous Markovian jump delay systems with the N -step ahead Lyapunov–Krasovskii function approach. In this chapter, we aim to design filters such that, for all possible time-varying transition probabilities and all admissible parameter uncertainties and time-delays, the filtering error system is mean-square stable with a smaller estimated error and a lower dissipative level. In terms of LMIs, sufficient conditions for the solvability of the addressed problem are developed via a moving-horizon method to avoid essential difficulties introduced by future noise.
11.2 FUTURE WORK Some of future works are listed as follows: • A possible future research direction is to investigate multiobjective H2 –H∞ control and filtering problems for multimodel systems. • It would be interesting to investigate the problems of fault detection and fault tolerant control for multimodel systems via the nonmonotonic Lyapunov function approach. • A trend for future research is to generalize the methods obtained in the book to the finite-time or finite-frequency control and filtering problems. • A challenging work is to employ the nonmonotonic approach to the model predictive control. A possible difficulty is to make a trade-off between the conservatism reduction and online computational burden. • The nonlinearities considered in the book have some condition constraints that bring conservativeness. The analysis and synthesis of general nonlinear systems could be one of the future research works.