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Time‐varying gain‐scheduling ‐error mean square stabilisation of semi‐Markov jump linear systems

2016; Institution of Engineering and Technology; Volume: 10; Issue: 11 Linguagem: Inglês

10.1049/iet-cta.2015.1327

1751-8652

Ting Yang, Lixian Zhang, Xunyuan Yin,

Advanced Control Systems Optimization

IET Control Theory & ApplicationsVolume 10, Issue 11 p. 1215-1223 Regular PapersFree Access Time-varying gain-scheduling -error mean square stabilisation of semi-Markov jump linear systems Ting Yang, Ting Yang School of Automation, Northwestern Polytechnical University, Xi'an, 710072 People's Republic of ChinaSearch for more papers by this authorLixian Zhang, Corresponding Author Lixian Zhang lixianzhang@hit.edu.cn State Key Laboratory of Robotics and System (HIT), Harbin, 150080 People's Republic of China School of Astronautics, Harbin Institute of Technology, Harbin, 150080 People's Republic of ChinaSearch for more papers by this authorXunyuan Yin, Xunyuan Yin School of Astronautics, Harbin Institute of Technology, Harbin, 150080 People's Republic of China Department of Chemical and Materials Engineering, University of Alberta, Edmonton, ABT6G2V4 CanadaSearch for more papers by this author Ting Yang, Ting Yang School of Automation, Northwestern Polytechnical University, Xi'an, 710072 People's Republic of ChinaSearch for more papers by this authorLixian Zhang, Corresponding Author Lixian Zhang lixianzhang@hit.edu.cn State Key Laboratory of Robotics and System (HIT), Harbin, 150080 People's Republic of China School of Astronautics, Harbin Institute of Technology, Harbin, 150080 People's Republic of ChinaSearch for more papers by this authorXunyuan Yin, Xunyuan Yin School of Astronautics, Harbin Institute of Technology, Harbin, 150080 People's Republic of China Department of Chemical and Materials Engineering, University of Alberta, Edmonton, ABT6G2V4 CanadaSearch for more papers by this author First published: 01 July 2016 https://doi.org/10.1049/iet-cta.2015.1327Citations: 13AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, a time-varying gain-scheduling approach is proposed to deal with the problem of stabilisation for a class of semi-Markov jump linear systems. A more general class of Lyapunov functions that depends not only on the system modes, but also on the staying time during the current system mode is constructed, which can cover the common time-invariant Lyapunov functions as special cases. In the sense of the σ-error mean-square stability proposed previously, the numerically testable sufficient criteria for the stability analysis are derived and certain techniques are employed such that the obtained conditions are linear in the system matrices. Both the time-invariant and time-varying control syntheses are investigated, and the results in a recent study can be deemed as extreme cases of the obtained criteria. Finally, the developed theoretical results are verified by three numerical examples, and it is demonstrated that the results based on the time-varying approach is less conservative than those based on the time-invariant method. 1 Introduction Markov jump systems, due to their capability of modelling the interactions between continuous-time and discrete-time dynamics, such as measurement sensors or control actuator faults, package dropouts or time-delays in networked control systems, colliding masses, electrical circuits with switches, and biological systems with impulsive behaviours, have drawn much attention from researchers throughout the world. In fact, Markov jump systems are a class of autonomous multi-model systems, within which each model corresponds to a system mode. At each time, only one of these system modes follows the real system dynamics. The transitions among these system modes are governed by a Markov stochastic variable. For the related results, we refer the readers to [1–3] for stability and stabilisation problems, [4–6] for filtering design, and the references therein. The Markov process demands that the sojourn time (the interval between two consecutive jumps) of each mode should be subject to an exponential distribution in the continuous-time domain or a geometric distribution in the discrete-time domain. As a result, the mode switching will be memoryless. If the sojourn time does not obey the exponential or geometric distribution, Markov jump systems would be incompetent in some scenarios to model this kind of stochastic switching systems. To overcome this difficulty, semi-Markov processes, in which the transition probabilities (TPs) among different modes are memory or history dependent, have been introduced to describe the stochastically switching signals. Since the semi-Markov process is not restricted by the momeoryless property, it is able to describe more general dynamic phenomena, while in the meantime the Markov process can be deemed as a special case of the semi-Markov processes [7, 8]. In addition, it is also worthwhile to mention that, because the switching among system modes now depends on the sojourn time, the results already obtained for Markov jump systems cannot be directly applied to semi-Markov jump systems. So far the studies of semi-Markov jump systems have been confined to the use of several special probability density functions (PDFs) of the sojourn time [9–13]. For instance, the results for the nearly exponential distribution can be found in [14], the PH-distribution in [15], and the Weibull distribution in [16]. It should be noted that in these representative studies, only a single type of distribution of the sojourn time is utilised to describe the switching among different modes, and moreover the parameters of the distribution are usually independent of the next mode. From the perspective of system modelling, it is highly possible that the statistic characteristics of the sojourn time depend on both the current and the next system mode. Therefore, in [17], based on the semi-Markov kernel (SMK) concept, different types of distributions of sojourn time and/or different parameters in the same type of distribution, depending on the target mode towards which the system jumps, were considered such that the studied semi-Markov jump linear system (s-MJLS) were more general. However, the results were obtained based on a conventional time-invariant Lyapunov function and the designed controller was also independent of time as is usually done in the literature. In fact, some efforts have been made to accomplish the time-varying purpose [18], and it has been verified that the time-varying scheme is more powerful and less conservative than the time-invariant approach. In light of the above observations, this paper proposed a time-varying approach to address the stability and stabilisation problems for a class of discrete-time stochastic switching linear systems. The semi-Markov chain (SMC), which covers more types of distribution of sojourn time than the Markov chain, is introduced to describe the system dynamics. Compared with the recently derived results, the time-varying Lyapunov function is explored and the corresponding control scheme is then investigated. It is finally verified that the developed stability and stabilisation criteria are less conservative than those derived via the time-invariant method. The remaining of this paper is organised as follows. Some preliminaries are given in Section 2. The stability and stabilisation criteria are established in Section 3. Numerical examples including an illustration via the vertical take oil and landing (VTOL) helicopter are provided in Section 4 and Section 5 gives the conclusion of the paper. Notation: The notation used throughout the paper is fairly standard. The superscript ' ' stands for matrix transposition; and denote the -dimensional Euclidean space and the set of real matrices, respectively; and represent the set of non-negative real numbers and the set of non-negative integers, respectively; the subscript of a set denotes an additional constraint upon the set, for example , ; the notation () means that is real symmetric and positive (semi-positive) definite and () means (). For the complete probability space , represents the sample space, is the -algebra of subsets of the sample space, and is the probability measure on . A function is said to be of class function if it is continuous, strictly increasing, unbounded, and . stands for the mathematical expectation. For symmetric block matrices, the asterisk () is employed to represent a term induced by symmetry. The stands for a block-diagonal matrix constituted by , and stands for an block-diagonal matrix with being the diagonal entries. and represent the identity matrix and zero matrix, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2 Preliminaries Consider the complete probability space , the discrete-time s-MJLS can be described in the following form (1)where , are the system state and control input, respectively; , are the system matrices; is a stochastic process and considered to be a semi-Markov chain which takes values in a finite set and which governs the switching among different modes of the system. For better understanding of later derivations, some relevant terminologies used in the proof of our main results are first introduced as below. Definition 1 [7].Consider the state space of a stochastic variable . The stochastic process is said to be a discrete-time homogeneous Markov renewal chain (MRC), where denotes the jump time with and is the associated state, if the TPs of the mode switches depending on the sojourn time satisfy , and , , where takes values in with and denotes the sojourn-time of mode between the th jump and th jump. The matrix is called the discrete-time SMK. Meanwhile, is named the embedded Markov chain (EMC) of MRC , and the TPs matrix of EMC is defined by (2)with . Based on the definition of SMK, it is easy to prove that has the following properties: (i) , , ; and (ii) , . Furthermore, can be rewritten as where is defined in (2), and , is defined as the PDF of the sojourn time, which depends not only on the current mode of the system, but also on the subsequent mode that the system will jump to. Definition 2 [7].Given an MRC , the stochastic process is said to be a SMC associated with the MRC if , where . Remark 1.For Markov jump systems, only one kind of TPs , defined in terms of the sampling time , is discussed. However, for semi-Markov jump systems, one can hardly obtain the same type of TPs since they now have memory (i.e. dependent on the past switching sequence). Therefore, another two kinds of TPs = = , , depending on the switching instant , are considered for use in later derivations. It should be kept in mind that the TPs matrix of EMC is different from that for Markov jump systems. In order to clearly present the purpose of this paper, the following stability definition is needed. Definition 3 [17].Given an upper bound of the sojourn time , , system (1) is said to be -error mean square stable ( -MSS) if for and any initial condition , the following equality holds (3)with where , , is the cumulative density function (CDF) of the sojourn time for system mode and it is assumed that . For the semi-Markov stochastic jump system (1), both the time-invariant and time-varying state-feedback control schemes are considered in designing the controller in this paper. Then, supposing the knowledge on mode variations is accessible, the two kinds of controllers are of the following forms (4)and (5)respectively, where with is a scheduler for the activated control gain. The operation of the scheduler is illustrated in Fig. 1. In (5), the scheduler is the time staying in mode at the sampling time . The controller of the above structure is dependent on the sampling time. The closed-loop system can be therefore obtained as (6)where is for the time-invariant feedback scheme and for the time-varying feedback scheme. Fig. 1Open in figure viewerPowerPoint Illustration of the time-varying Lyapunov function and control strategy The objectives of this paper are to establish the stability criteria for the underlying s-MJLSs in the sense of -MSS, and then to design the stabilising controller in the forms of (4) and (5), respectively, such that the closed-loop system (6) is -MSS. 3 Main results This section will present the sufficient conditions of -MSS stability and stabilisation for the s-MJLS. For ease in later developments, we first recall the MSS stability conditions for a general stochastic switching system. In order to reduce the conservatism of the criteria, the time-varying approach is used and the corresponding Lyapunov function is defined as (7)where , , , and . See Fig. 1 for the illustration. Lemma 1.Consider a discrete-time switching system where is the mode index associated with the switching instants and . For a given constant , the system is MSS if there exist a set of functions and three class functions , such that (8) (9) (10) Proof.Choosing the Lyapunov function in the form of (7), we can obtain (11)where . Define (12)and then one has the following inequality hold (13)where . When , we have (14)Further, one has (15) (16)As trends to infinity, it can be proved that . Setting , one has . Similar to the proof of Lemma 1 in [17], taking mathematical expectations at both sides of (8) and (9), one has (17)As , , and therefore , which implies . That is to say, . The proof is completed.□ Based on Lemma 1, the following theorem gives a set of sufficient conditions ensuring the -MSS of system (1). For notational simplicity, we denote as in what follows. Theorem 1.Let be a set of given constants. Then, the s-MJLS (1) is -MSS if for any , there exist and a set of symmetric matrices such that the following inequalities are satisfied (18) (19)where with . Proof.Construct a stochastic Lyapunov function candidate as expressed in (7). Since satisfies the constraints (18) and (19), it is straightforward to show that which guarantees (8) in Lemma 1. Suppose that the system mode does not switch at the sampling time , i.e. , , which can be described as , , . Then, (9) in Lemma 1 with can be rewritten as According to (18), holds. On the other hand, between two consecutive switching instants, i.e. , , we have (20)where , and is defined in (19). Since satisfies (19), one has which then guarantees (10) in Lemma 1. The proof is completed. □ Note that if is independent of the time staying on the current mode at the current sampling time , i.e. , , the derived theorem can be easily converted to the result (given in the following corollary) based on the time-invariant Lyapunov function approach. Corollary 1 [17].Let be a set of given constants. Then, the s-MJLS (1) is -MSS if for any , there exist and a set of symmetric matrices such that the following inequalities are satisfied (21) (22)where with being defined in (19). Remark 2.If the criteria in either Theorem 1 or Corollary 1 are guaranteed when , then the s-MJLS (1) will be stable in the conventional sense of MSS. However, the derived conditions (18), (19) and (21), (22) are generally difficult to be numerically checked because of the uncountable number of inequalities involved therein. It can be observed that conditions (18), (19) and (21), (22) contain the matrix power term . Since the system matrices are given a prior, the matrix power term can be computed, and thus is essentially known and will not bring any difficulty in checking the solvability of these criteria. However, the results cannot be directly extended for a solution of the stabilisation problem. To overcome this obstacle, certain techniques are needed to eliminate the matrix power terms, for which the following theorem is established. Theorem 2.Let , , be a set of given finite constants. Then, the s-MJLS (1) is -MSS if for any , there exist and a set of symmetric matrices , , , , , and , , , such that , , the following inequalities are satisfied (23) (24) (25) (26)where with being defined in (19). Proof.From (23), one can infer that (27)which is equivalent to (28)Then according to (24) and (28), it can be verified that (29)Further, letting and in (18), it is straightforward to see that (18) is guaranteed by (29). On the other hand, it follows from (25) that (30)By changing the order of summation, (30) can be rewritten as which yields (31)Substituting (26) into (31), one obtains Note that if we set , and , the inequality holds. Thus, based on Theorem 1, it can be concluded that system (1) is -MSS. □ A result extended from Theorem 2 is given as below, by setting , , and paralleling the extension from Theorem 1 to Corollary 1. Corollary 2 [17].Let be a set of given finite constants. Then, the s-MJLS (1) is -MSS if for any , there exist and a set of symmetric matrices , , , , , and , , , such that , , the following inequalities hold (32) (33) (34) (35)where with being defined in (19). Then, based on Theorem 2 and Corollary 2, the stabilisation problem of the underlying systems via both time-varying and time-invariant control schemes is addressed in the following two theorems. Theorem 3.Let be a set of given finite constants. Then, the closed-loop s-MJLS (6) is -MSS if for any , there exist and a set of symmetric matrices , , , , , , , and , , such that , , , we have (36) (37) (38) (39) (40)with where , , , , , , , , , , , , , and with and . Further, the admissible controller gain is given by Proof.The stability condition in Theorem 2 can be extended to the closed-loop system by replacing with . Therefore, (23)–(26) can be rewritten as (41) (42) (43) (44)By Schur complement, (43) is equivalent to (45)with where , , , , , .Define , and pre- and post-multiplies the right-hand side of the inequality (45) by and , respectively.Since implies that , it follows that (45) is guaranteed by (46)with .Performing the congruence transformation by , , and setting , , , , , , , , , , , , one can see that (46) is equivalent to (38).Besides, because when , (46) can be modified as (47)which is equivalent to (39). Meanwhile, it follows that (44) holds if (40) is satisfied by pre- and post-multiplying and on (44). Using similar techniques, it yields that (41) and (42) can be guaranteed by (36) and (37), respectively. This ends the proof. □ In comparison with the time-varying state-feedback controller, the following theorem summarises the result of the time-invariant state-feedback controller. Theorem 4.Let be a set of given finite constants. Then, the closed-loop s-MJLS (6) is -MSS if for any , there exist and a set of symmetric matrices , , , , , and , such that , , , we have (48) (49) (50) (51) (52)with where , , and the other parameters are defined in Theorem 3. Further, the admissible time-invariant controller gain is given by . Proof.Let with be defined as Then (25) and (26) can be rewritten as follows (53) (54)By Schur complement, (53) gives the following inequality (55)where . Pre- and post-multiplying , , on the inequality (55), respectively, one can obtain that (55) is guaranteed by Performing the congruence transformation by on (50), setting , , and replacing with , the above inequality is found to be equivalent to (50). Using similar techniques in Theorem 3, we have (51) when and it is ready to prove that (48) and (49) guarantee (23) and (24) in Theorem 2, respectively. □ In an analogous way, one can also obtain a feasible time-invariant controller using the following corollary which is derived by the time-invariant Lyapunov approach. Note that the corollary can be covered by Theorem 4 via setting , . Corollary 3 [17].Let be a set of given finite constants. Then, the closed-loop s-MJLS (6) with the controller (4) is -MSS if for any , there exist and a set of symmetric matrices , , , , and , such that , , (56) (57) (58) (59) (60)with are satisfied, where , and the other parameters are defined in Theorem 4. Further, the admissible controller gain is given by . Remark 3.A noteworthy observation is that when , we have , which implies that is the inherent requirement of Lemma 1 and the established corollaries and theorems in Section 3. Accordingly, when , the constraint (9) will become invalid in Lemma 1, and moreover the constraints deduced based on (9) in the related theorems and corollaries, such as (18) in Theorem 1, (23) and (24) in Theorem 2, (36) and (37) in Theorem 3, and (48) and (49) in Theorem 4, should be ignored. Remark 4.Although in the theorems derived based on the time-invariant Lyapunov function the number of the matrix variables is less than that for the time-varying Lyapunov function approach, the results could be more conservative, which will be shown in the examples in the following section. 4 Numerical examples This section presents three examples which illustrate the effectiveness and potential of the theoretical results developed previously. 4.1 Example 1 To demonstrate that the time-varying Lyapunov function approach can lead to less conservative stability criteria, let us consider the three-mode semi-Markov stochastic system (s-MJLS) with the following data The TPs matrix of EMC is given as The sojourn time follows the Bernoulli distribution , , and the Weibull distribution , , , and . The purpose here is to carry out the stability analysis of the system and compare the conservatism of the obtained Corollary 2 and Theorem 2. The verification results are shown in Fig. 2, where the parameters and are chosen in the range of 0.01–0.1, and are set as 6. From Fig. 2, we can see that the feasibility when using the time-invariant approach fails if is smaller than 0.06, 0.05, 0.05, respectively. On the contrary, the time-varying approach still yields feasible solutions for every pair of , thereby demonstrating that Theorem 2 is less conservative. Fig. 2Open in figure viewerPowerPoint Feasible sets of parameters for Corollary 2 and Theorem 2 4.2 Example 2 The numerical example discussed above has illustrated the advantage of the time-varying Lyapunov function for stability analysis. To further show the advantage of the time-varying controller over the time-invariant controller in the sense of conservatism, let us consider the s-MJLS as follows Without loss of generality, the stationary distributions of the system, such as TPs matrix of EMC, are assumed to be the same as those in Example 1. By solving (48)–(52) and (56)–(60) in Theorems 3 and 4 with different pairs of parameters, respectively, an estimation of the feasible region for the two theorems can be obtained. Fig. 3 lists the feasible sets of parameters for Theorems 3 and 4, where the parameters and are chosen from 0.05 to 0.15 and are set as 3, 4, and 5, respectively. From Fig. 3, we can observe that the feasible sets of parameters for Theorem 4 are all suitable for Theorem 3, but not vice versa (it will be unfeasible if is smaller than 0.14). Therefore, it can be concluded that Theorem 3 has less conservatism than Theorem 4. Fig. 3Open in figure viewerPowerPoint Feasible sets of parameters for Theorems 3 and 4 4.3 Example 3 A short example of stabilisation of the VTOL helicopter is presented below in order to demonstrate the usefulness and applicability of the developed theoretical results. The motion of VTOL helicopter can be described as [19, 20] where are the system matrices; are the state variables, where denotes the horizontal velocity (kt), is the vertical velocity (kt), is the pitch rate (deg/s) and is the pitch angle (deg); is the force applied to the helicopter, where denotes the collective pitch input and denotes the longitudinal cyclic pitch control input. The correspond to different air speeds, 135 knots, 60 knots and 170 knots, respectively. The parameters , and are supposed to take values from Table 1. Table 1. Parameters for the VTOL processes Airspeed (knots) 135 0.37 1.42 3.55 60 0.07 0.12 1.0 170 0.51 2.52 5.11 By Euler's first-order approximation approach, the system matrices for the corresponding discrete-time system can be expressed as where is the sampling time and is set as . We assume the dynamics of airspeed satisfies the semi-Markov stochastic process and the TPs matrix of EMC is given as The sojourn time follows the Bernoulli distribution , and the Weibull distribution , , , . By solving (48)–(52) in Theorem 3, the corresponding control gains (omitted due to space limit) can be derived with . Under these control gains, the state responses of the system with the initial condition are given in Fig. 4 a, and the corresponding system mode and the active indices of the control gains are shown in Fig. 4 b. It can be seen from Fig. 4 b that even under the same mode, the control inputs are calculated by different control gains that are switched with the staying time of the active mode, which is the main difference between the time-varying and the time-invariant control schemes. In addition, the error which depends on the parameters can be computed as listed in Table 2. It is observed from Fig. 4 a that the designed controller is valid against the system switching, which demonstrates the applicability of the obtained theoretical results. Table 2. Error for different , 2 4 6 8 10 (2, 2) 5.7037 2.8297 1.1663 0.4394 0.3109 (4, 4) 5.4117 2.5378 0.8744 0.1474 0.0190 (6, 6) 5.3937 2.5197 0.8563 0.1294 (8, 8) 5.3928 2.5189 0.8555 0.1285 (10, 10) 5.3928 2.5189 0.8555 0.1285 (15, 15) 5.3928 2.5188 0.8555 0.1285 Fig. 4Open in figure viewerPowerPoint State responses, system modes and active control gains for the VTOL helicopter a State response of the system for with random jumping sequences subject to b System modes and the indices of the active control gains 5 Conclusion This paper is concerned with exploring a time-varying gain-scheduling approach to tackle the stability and stabilisation problems for a class of s-MJLSs. By introducing a time-varying Lyapunov function which depends also on the staying time at the current system mode, a set of numerically testable conditions is obtained in the sense of the -MSS concept. The advantages of the proposed method in terms of conservatism are illustrated by comparing the time-varying with time-invariant Lyapunov function approaches and the time-varying with time-invariant control schemes, respectively. The developed theoretical results are finally verified by three examples. The future work will be devoted to the control and filter problems of the underlying systems. 6 Acknowledgments The work was partially supported by the National Natural Science Foundation of China (61322301) and Heilongjiang (F201417, JC2015015), the Fundamental Research Funds for Central Universities, China (grant nos. HIT.BRETIII.201211, HIT.BRETIV.201306, HIT.MKSTISP.2016.32), and the Self- Planned Task (grant no. SKLRS201617B) and grant no. 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