Title: Method for computing state transformations of time‐delay systems
Abstract: IET Control Theory & ApplicationsVolume 9, Issue 16 p. 2405-2413 Research ArticlesFree Access Method for computing state transformations of time-delay systems Dinh Cong Huong, Dinh Cong Huong Department of Mathematics, Quynhon University, Binh Dinh, VietnamSearch for more papers by this authorHieu Trinh, Corresponding Author Hieu Trinh [email protected] School of Engineering, Deakin University, Geelong, 3217 AustraliaSearch for more papers by this author Dinh Cong Huong, Dinh Cong Huong Department of Mathematics, Quynhon University, Binh Dinh, VietnamSearch for more papers by this authorHieu Trinh, Corresponding Author Hieu Trinh [email protected] School of Engineering, Deakin University, Geelong, 3217 AustraliaSearch for more papers by this author First published: 01 October 2015 https://doi.org/10.1049/iet-cta.2015.0108Citations: 14AboutSectionsPDF 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 This study provides a systematic method for deriving state transformations of a class of time-delay systems with multiple output. The significance of this study is that such state transformations can be used to transform time-delay systems into new coordinates where all the time-delay terms in the system description are associated with the output and input only. Therefore, in the new coordinate system, a Luenberger-type state observer can be readily designed. Subsequently, of the three possible versions of the original state vector, namely, instantaneous, delayed, and a mixed of instantaneous and delayed, a state observer which estimates one of these versions can be obtained. This new finding allows the authors to design state observers for a wider class of time-delay systems. Conditions for the existence of such coordinate changes and an effective algorithm for computing them are provided in this study. A numerical example and simulation results are given to illustrate the simplicity and effectiveness of the proposed method. 1 Introduction Since time delay is often encountered in many practical control systems [1, 2], the problem of designing a state observer to estimate the state vector of a time-delay system is an important research topic and it has received considerable research attention in the literature. In particular, state observers have important applications in realisation of state-feedback control, system supervision, fault diagnosis of dynamic processes, and general control and diagnosis issues from available information [3–7]. There are various state observer design methods and observer structures available in the literature for continuous time-delay systems (see, e.g. [8–16], and the references therein) and for discrete time-delay systems (see, e.g. [17–22], and the references therein). Nevertheless, there are instances where existing state observer design methods cannot be applied. To support this statement, in the following, we will present a motivated example of a fifth-order time-delay system with two outputs. For this example, some existing state observer design methods [8–15] cannot be used to design a stable state observer since their reported existence conditions [8–15] are not satisfied. This motivated example affirms that there is indeed an urgent need to develop some new state observer design methods in order to deal with a wider class of time-delay systems. In this study, we consider the following time-delay system: (1) (2) (3)where ϕ (θ) is a continuous initial function, τ > 0 is a known constant time delay, is the state vector, is the control input vector, is the measurement output vector, matrices A, Ad, B and C are constant and of appropriate dimensions. To motivate the discussion, let us now consider an example of a fifth-order time-delay system with two outputs, i.e. n = 5 and p = 2, where and For this example, the reported existence conditions [8–13] are not satisfied. Whereas, the method recently reported in [14] is delay dependent and does not always work for an arbitrary large time delay, τ. Note also that the matrix pair (A, C) is not observable and the condition for the existence of a coordinate change (i.e. a state transformation) according to [15] (i.e. column unimodular) is not satisfied. In this study, we draw our inspiration from the work of [15], and instead let us now consider the following new state transformation: (4) (5) (6) (7) (8) (9)By taking the derivatives of the change of the coordinates as given in (4)–(9) and letting zT (t) = [z1 (t) z2 (t) z3 (t) z4 (t) z5 (t) z6 (t)], we obtain (10) (11)From (10) and (11), it is straightforward to design a Luenberger-type state observer (see, e.g. [13]) to estimate z3 (t), z4 (t), z5 (t) and z6 (t). Upon a satisfactory state observer to estimate z3 (t), z4 (t), z5 (t) and z6 (t) has been designed, then based on the state transformation (4)–(9), and also note that, instead of an instantaneous state observer, a delayed state observer to estimate x3 (t − τ), x4 (t − τ) and x5 (t − τ) can be constructed (12) (13) (14)where , and are the estimations of x3 (t − τ), x4 (t − τ) and x5 (t − τ), respectively. Even though instantaneous estimation of the state variables x3 (t), x4 (t) and x5 (t) cannot be achieved, this new finding enables us to estimate the state variables after a finite time delay τ, i.e. can estimate x3 (t − τ), x4 (t − τ) and x5 (t − τ). Thus, we have demonstrated that a possible key to overcome some shortcomings of some existing state observer design methods [8–15] lays in our innovative development of some new state transformations [see, (4)–(9)] that allow us to transform a given time-delay system into a new canonical form [see, (10) and (11)] where all the time-delay terms in the system description are associated with the output and input only. In this study, inspired by the early work of [15], we develop a systematic method for deriving some generalised state transformations of a class of time-delay systems with multiple output. Through a numerical example we demonstrate the effectiveness of the proposed method. Due to the complexity of the problem and the constraint imposed on the page limit of a paper, this study only focuses on the case of general order with two outputs, (i.e. n > 2 and p = 2). The case with more than two outputs (p > 2) and multiple time delays, (i.e. τi, i = 1, 2, …, q) will be the subject for future research. Notations: For p = 2, and an arbitrary matrix , the following notations will be used throughout this paper: MT denotes the transpose of M. 0m, n denotes the m × n zero matrix. , where and are sub-matrices of M. , where and are the two elements of [M]L. 2 Main results 2.1 State transformation Without loss of generality, we assume that matrix C takes the canonical form, where Let us denote the following matrices: (15) (16) (17) (18) (19)where , , , , , , , , , , and are scalars to be determined later. We now define a new change of coordinates as follows: (20)where for i = 1, 2, 3, 4 and , (i = 5, 6, …, n + 1) are scalars to be determined later. The following theorem provides a coordinate state transformation (20) which enables the delay terms in the system to be injected into the system's output and input. Theorem 1.For some scalars , , , , (j = 3, 4, …, n + 1), , (k = 5, 6, …, n + 1), if the following equations hold: (21) (22) (23) (24) (25) (26)then the change of coordinate (20) transforms systems (1)–(3) into the following form: (27) (28)where and Proof.For i = 1, taking the derivatives of (20), using (15), (16) and (20), we obtain (29)For i = 2, taking the derivatives of (20), using (15), (17) and (20), we obtain (30)Similarly, for i = 3, taking the derivatives of (20), using (21)–(23), we obtain (31)For i = 4, 5, …, n, taking the derivatives of (20), and using (24), we obtain (32)From (18) and (19), we have for i = 4, 5, …, n (33) (34)Substituting (33) and (34) into (32), using (20) and (23), we obtain (35)for all i = 4, 5, …, n.On the other hand, when i = n + 1 and from (32), we have (36)Now, subject to the satisfaction of (25) and (26), (36) can be expressed as (37)Finally, (29)–(31), (35) and (37) can now be expressed in the form (27) and (28). This completes the proof of Theorem 1.□ Remark 1.For the transformed systems (27) and (28), note that the matrix pair is observable and all the time-delay terms in the system description are associated with the output and input only. Therefore, a Luenberger-type state observer (or a functional state observer) can be readily designed. Now, in order to make full use of Theorem 1, it is necessary for us to be able to systematically determine the unknowns , , , , (j = 3, 4, …, n + 1) and , (k = 5, 6, …, n + 1) such that conditions (21)–(26) of Theorem 1 are satisfied. Accordingly, in the following, we derive a systematic procedure for solving these unknowns. This will result in an effective algorithm for computing a generalised state transformation (20) which transforms the original system (1)–(3) into the new observable form (27) and (28). Let us denote the following recursive matrices: (38) (39) (40) (41)Let us now consider (21)–(24). By substituting (15)–(19) into (21)–(24) and using (38)–(41), we obtain the following equations: (42) Let us now express (42) in the following compact form: (43)where are as defined below (44) (45) (46) (47) (48) (49) (50) (51)where , , , , , , and are as defined below (52) (53) (54) (55)From (43), a solution for χn +1 exists if and only if (56)Next, to determine the remaining unknowns , and (j = 3, 4, …, n + 1), let us look at the solvability of (25) and (26). Substituting (15)–(19) into (25) and (26), using (38)–(41) and after some rearranging, we obtain the following equation expressed in a compact vector–matrix form: (57)where (58) (59) (60)In (59) and (60), , , and (k = 5, 6, …, n + 1) are defined as follows: (61) (62) (63) (64) Remark 2.It is clear from (59) and (60), Zn +1 and Tn +1 are two known constant matrices since , (k = 4, 5, …, n + 1), , (ℓ=4, 5, …, n) have already been derived from the solution to (43). From (57), a solution for ζn +1 always exists if and only if (65) Accordingly, we present an effective algorithm to transform a general n -order time-delay system (n ≥ 3) with two outputs into the observable form (27) and (28). Algorithm 1: Step 1: Obtain matrices Xn +1 and Yn +1 according to (48)–(55). Check if condition (56) is satisfied or not. If so, obtain χn +1 where , where denotes the Moore–Penrose inverse of Xn +1. Step 2: Substitute , (k = 4, 5, …, n + 1) and , (ℓ=4, 5, …, n) into (59) and (60) and obtain Zn +1 and Tn +1. Check if condition (65) is satisfied or not. If so, obtain , where denotes the Moore–Penrose inverse of Zn +1. Step 3: From (15)–(19), obtain matrices Mi and Ni(i = 1, 2, …, n + 1) and hence the state transformation (20). Finally, obtain a transformed system according to (27) and (28). 2.2 Backward transformation As we have explained in the previous section, transformed systems (27) and (28) are observable and a satisfactory state observer, , can be easily designed to estimate the state vector z (t). After a satisfactory state observer has been designed, we can use state transformation (20) to obtain an observer for x (t). In the following, we discuss how to solve for given that is known. In this paper, we refer to this problem as the backward transformation problem. Since x1 (t) = y1 (t) and x2 (t) = y2 (t) are known, let us express Mi and Ni(i = 3, 4, …, n + 1) as Mi = [mi 1 mi 2 … min] and Ni = [ni 1 ni 2 … nin]. Also, let us define , , , , , and as (66) (67) (68)where in (67) and (68), x denotes any arbitrary scalar, f (i, 1), i = 3, 4, …, n + 1, is defined as From (20), for i = 3, …, n + 1, we obtain the following: (69)where is obtained from (68) with zi(t) being replaced by . Case 1: or . When , (69) is reduced to (70)Thus, if has a left inverse , i.e. , then is obtained as follows: (71)When , (69) is reduced to (72)The above equation implies that instead of an instantaneous state observer, a delayed state observer, i.e. , is now obtained, where (73)Case 2: and has full column rank. If there exists a matrix such that where P1 and Q1 are as defined in (67), and are arbitrary matrices. Then, by pre-multiplying both sides of (69) by P, we obtain (74)From (74), we obtain (75)where contains the first (n − 2) rows of . Clearly, from (75), we can first obtain . Due to the particular structure of the matrices P1 and Q1, can next be derived from using and . Therefore, in the same manner, we can obtain the rest of , i = 5, 6, …, n. Case 3: and does not have full column rank. There are cases where some of the columns of matrix are zeroes and therefore some states of that correspond to those columns cannot be obtained. In such a situation, we first eliminate those state variables from . Accordingly, let , where 1 ≤ q < (n − 2). This means that there are (n − 2 − q) states of that cannot be obtained. Let be those states that have been eliminated from . Also, let be the remaining state variables of . Then, (69) can be rearranged and expressed as follows: (76)where and . If matrix has full column rank, then by following Case 2, we can obtain 2.3 Numerical example and simulation results To illustrate the effectiveness of our results and also to show how easy it is to apply Algorithm 1, let us consider the motivated fifth-order example that we considered earlier. According to Step 1, condition (56) is found to be satisfied and hence we obtain , , , , , , , , , , , , , , , , and . Next, in Step 2, by substituting , , , , , , , , and into (59) and (60), condition (65) is also found to be satisfied. Hence, we obtain , , , , and . Then, according to Step 3, we obtain state transformations (4)–(9). Finally, a transformed system of the forms (27) and (28) is obtained, which is the same as in (10) and (11). Now, we consider the backward transformation problem for this example. We have where This corresponds to Case 1 and since has a left inverse, where from (73) we obtain the delayed state observer (12)–(14). 2.4 Simulation results In order to obtain simulation results, let us now design a Luenberger-type state observer for transformed systems (10) and (11), where and , , , Γ, Γ1, Γ2, are as given in (10) and (11). Since the pair is observable, matrix can be easily obtained through the well-known pole-assignment technique. By choosing the eigenvalues of at, say, (− 4, − 4.1, − 4.2, − 4.5, − 4.6, − 5), matrix L is obtained as below Figs. 1, 2–3 show the responses of x3 (t − τ), x4 (t − τ), x5 (t − τ) and their estimations, i.e. , , , for the case where τ = 1 s with arbitrary initial conditions and input It is clear from Figs. 1, 2–3 that the designed observer able to track the delayed version of the state vector, as expected. Remark 3.Our method offers several advantages over the method reported in [15]. Firstly, the method [15] requires that the observability matrix, Qk(s), be column unimodular, where Qk(s) is defined as follows: with k ≤ n being the smallest integer such that rank(Qk(s)) = rank(Qn(s)) for all s in . Note that, there are many time-delay systems where the matrix pair (A, C) is not observable. In such cases, it is not hard to prove that Qk(s) is not column unimodular and therefore the method [15] fails to work. In contrast, our computational method does not require the pair (A, C) be observable, as we have shown this in the motivated example. Secondly, to apply the method [15], one has to implement several steps with symbolics computation, for instance, to test the column unimodular of Qk(s) or to find a left inverse of Qk(d), wherein all the elements of Qk(d) are polynomials of the time-delay operator d (where dx (t) = x (t − τ) [15]). In the general case, to obtain a left inverse, , of Qk(d), one has to solve a system of equations where the coefficients and unknowns are polynomials of d. Due to the nature of symbolics computation, those steps are not easy to carry out and they take considerable times, especially, when the order of the system is high. Whereas, our method offers a systematic procedure and it is computationally simple and easy to implement. Fig. 1Open in figure viewerPowerPoint Responses of and x3 (t − 1) Fig. 2Open in figure viewerPowerPoint Responses of and x4 (t − 1) Fig. 3Open in figure viewerPowerPoint Responses of and x5 (t − 1) In the following remark, we discuss the case where the delay is time varying and unknown. Remark 4.Let us now consider systems (1)–(3) when the time delay τ is replaced by a time-varying delay, τ (t), that is assumed to be unknown but to be bounded within the interval [τ1, τ2], where τ1 > 0 and τ2 > 0. Using the idea about performing the approximation for x (t − τ (t)) in [19, 23], in this case, we approximate the term x (t − τ (t)) by x (t − τa), where τa = (τ1 + τ2)/2. The approximation error can be expressed as (77)It is not difficult to prove that , for all v (t) ∈ L2, where L2 is the space of square integrable functions with and || · || denotes the Euclidean norm.Then, systems (1)–(3) can be written as (78) (79) (80)If conditions (21)–(26) of Theorem 1 are satisfied then the change of coordinate (20), where in (20) the time delay τ is replaced by τa, transforms systems (78)–(80) into the following form: (81) (82)where , , , , , Γ, Γ1, Γ2 and Γ3 are defined as in Theorem 1 and 3 Conclusion In this paper, we have presented a systematic method for deriving state transformations of a class of time-delay systems with two outputs. The reported result is significant because the proposed state transformations can be used to transform time-delay systems into new coordinates where all the time-delay terms in the system description are associated with the output and input only. 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Publication Year: 2015
Publication Date: 2015-10-01
Language: en
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