Disturbance estimation based tracking control for periodic piecewise time-varying delay systems

This work is mainly focused on the design of uncertainty and disturbance estimator-based tracking control scheme for a class of periodic piecewise time-varying delay systems with uncertainties and disturbances. Precisely, the considered system consists of several subsystems, where the ﬁxed dwell time and switching sequence are utilised to achieve the coor-dination between each subsystem. By constructing Lyapunov–Krasovskii functional with time-varying periodic piecewise matrices and employing the matrix polynomial approach, an uncertainty and disturbance estimator-based periodic tracking control design is developed such that the trajectories of the considered system are driven to track the reference model. Moreover, the lumped disturbances so called the summation of uncertainties and exogenous disturbances are estimated with the aid of strictly proper low-pass ﬁlter. Finally, a numerical example is presented to verify the effectiveness and advantages of the developed theoretical results.


INTRODUCTION
Periodicity is a common factor encountered in practice, wherein the certain characteristics are repeated in regular period of time. More specifically, the periodic behaviour occurs in many engineering problems such as helicopter rotors, DC-DC converters, transmission lines with distinct loads, vehicle suspension, rotor bearing systems, and power electronic equipment's with multi-function converters. Due to its significance and widespread applications, the stability analysis of periodic systems has received much attention among many research communities [1][2][3]. Periodic systems with finite number of subsystems in a unit period of time are referred to as periodic piecewise systems (PPSs) [4,5]. Generally, the PPSs are composed of linear time-invariant subsystems and hence it may lose dynamic characteristics of the original periodic systems. Thus, the study of PPSs with time-varying subsystems attains considerable attention among researchers [6][7][8][9][10]. Notably in [6], the authors developed a state feedback control law with time-varying gain matrices to ensure the exponential stability of PPSs with time-varying subsystems. Furthermore in [7], a novel matrix polynomial lemma is proposed to linearize the nonlinearities raised in between the time-varying matrices, wherein This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Control Theory & Applications published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology the system and controller is constituted with distinct width of time-varying coefficients which bring more flexibility to convex optimisation tools. In the work of Zhu et al. [9], the exponential stability of the periodic piecewise time-varying systems (PPTVSs) is ensured by Lyapunov stability theory. Li et al. [10] proposed an H ∞ control with time-varying polynomial controller gain matrix that ensures the exponential stability of PPTVSs with polynomial subsystems.
In practice, the occurrence of delay is inexorable and it is the source of instability or performance degradation of various kinds of dynamical control systems [11][12][13]. Therefore, it is necessary and important to deal with the complications resulting from the combination of time-varying delay and periodicity in various practical systems. Considerable efforts devoted to the study of stability and control problems of delayed PPSs are reported in the literature [14][15][16]. The authors in [14] proposed an H ∞ control scheme for a class of PPSs with time-invariant subsystems, disturbances and time-varying delay, which ensures the exponential stability of the system. Xie and Lam [15] developed a guaranteed cost periodic control scheme for delayed PPSs with time-invariant subsystems. In [16], the non-fragile controller with additive and multiplicative gain perturbations is designed to stabilise the closed-loop delayed PPSs with time-varying subsystems, but it is limited to constant delays. However, to the best of the present author's knowledge, no work on the combination of PPSs with time-varying subsystems and time-varying delays has been reported in the literature.
It should be noted that the exact mathematical model of the system is primarily required to achieve the desired system performance, but in practice, the exact modelling of physical systems is not always guaranteed because of the differences or errors between the mathematical model and the real physical systems, which are commonly referred to as model uncertainties. Moreover, the existence of uncertainties is categorised by several factors such as modelling errors, variation in real parameters, aging of system components, temperature fluctuations, and operating point shifting. Furthermore, unexpected errors in modelling the real-world systems happen frequently. Due to the limited knowledge of physical phenomena, uncertainties are regarded as unknown dynamics which are frequently a source of instability and very difficult to design a robust control law. Therefore, it is necessary and important to formulate the system with model uncertainties. A great number of fruitful works on the stability and stabilisation analysis of uncertain dynamical systems have been discussed in [17][18][19][20].
On the other hand, the existence of abrupt disturbances are unpredictable and may severely affect the dynamics of the system [21,22]. Because of the combination of uncertainties and disturbances, much effort on the control design is needed for the stabilisation of system trajectories. In this regard, it is more significant to estimate the uncertainties, non-linearities and exogenous disturbances in control systems. At the early stage, several control approaches were proposed for various dynamical systems to compensate the disturbance effects. Most of the control approaches require some assumptions about the unpredictable aspects like bound values of unknown uncertainties and disturbances, but in practice they are not always measurable. In order to conquer the above complexity, an effective technique called UDE-based strategy was proposed in [23] and it is widely used for many dynamical systems [23][24][25][26][27][28][29]. The UDE-based control approach proposed by Zhong and Rees [23] ensures better tracking performance of the linear time-invariant systems and also estimates the lumped disturbances with respect to the appropriate bandwidth selection of the low-pass filter.
In unpredictable environments, the desired behaviour of the system trajectories is one of the key objectives to maintain better performance of the control systems. The tracking control problem is one of the most eminent and essential research topics which has enormous applications in science and engineering [30][31][32][33][34][35]. Owing to these perspectives, the authors in [31] developed an H ∞ tracking control scheme with periodic control gain parameters for a class of PPSs with time-invariant subsystems, wherein the desired tracking performance of the system states to the reference state is attained within a finite-period of time. Unfortunately, only one published paper addressed the problem in designing the tracking control for PPSs with timeinvariant subsystems. To the best of the present author's knowledge, the tracking control problem of PPSs with time-varying subsystems has not yet been investigated. In order to fulfill such research gap and also inspired by the significance of the works in [6-10, 23 31, 36, 37], a disturbance and uncertainty estimation based tracking control is designed for a class of uncertain time-varying delayed PPTVSs with disturbances. To be precise, the PPTV control gain matrix is obtained by solving a set of delay-dependent stability conditions, which is derived by considering periodically time-varying Lyapunov-Krasovskii functional. Especially, the matrix polynomial approach proposed in [7] is utilised to linearize the non-linearities raised in between the time-varying matrices. Further, the main contributions of this work are listed as follows: 1. Tracking control design with periodic behaviour is developed for a class of uncertain PPTVSs with time-varying delay and disturbances for the first time. 2. Inspired by the work in [31], a new periodic piecewise time-varying reference model with time-varying delay and bounded input is considered for the addressed system. 3. To tackle the impact of lumped disturbances, that is the total sum of uncertainties and unexpected disturbances, the lowpass filter is implemented with appropriate bandwidth value and to facilitate the complexities in derivations. Moreover, here the general assumptions on lumped disturbances are relaxed together with less tuning parameters. 4. The proposed periodic control design is more effective for state tracking, which is endorsed by presenting simulation results.

PROBLEM FORMULATION AND PRELIMINARIES
Consider a class of uncertain linear PPTVSs with time-varying state delay and disturbances in the following compact form: where (t ) ∈ ℜ n , u(t ) ∈ ℜ m and (t ) ∈ L 2 [0, ∞), respectively, represent the state, control and disturbance vectors; , B(t ) and B (t ) are known time-varying appropriate dimensioned periodic matrices with the fundamental period T p , that is, A(t ) and A (t ) represent the unknown time-varying uncertainties of the considered system (1); (t ) denotes the delay in system dynamics which satisfies the conditions 0 ≤ (t ) ≤̄anḋ(t ) ≤̂, wherein the parameters̄> 0 and < 1 are known scalars. Now, the each periodic time interval represent the uncertainties of system dynamics over each sub-interval with the following structure: where  i ,  Ai and  Bi are appropriate dimensioned matrices, and Γ T i (t )Γ i (t ) ≤ I . The system (1) is then reformulated as follows: (3) of control input vector is assumed to be full column rank matrix.

Remark 1.
Inspired by the work in [4], the system uncertainties are described in periodic piecewise form . Further, the structure defined in (2) consists of partially known matrices  i (t ),  Ai and  Bi . Unlike the conventional description of uncertainties, in this work, the coupling of time-varying term i (t ) and unknown matrix Γ i (t ) exist in the uncertainties. It ends up as a non-convex variable that is difficult to decouple in stability analysis, which brings more technical challenges. Moreover, this non-convex variable affects the fixed dwell time switching.
Remark 2. The addressed PPTVS is formulated by the inherent dynamic characteristics of practical time-varying systems with periodically varying elements. In particular, the system (1) is partitioned into finite number of sub-intervals with fixed dwell time T i . The selection of T i , i = 1, 2, … ,  is based on the variation of the system dynamics. Subsequently, the corresponding subsystem over each sub-interval is approximated by an averaged model [2]. Based on this concept, the continuous-time periodic system (1) is remodelled as PPTVS in (3) with PPTV matrix coefficients.
In order to study the state tracking control problem, for all t ∈ [ T p + t i−1 , T p + t i ), let us consider the following PPTV reference model: where r (t ) and r (t ) are the reference state and uniformly bounded input vector, respectively; A ri (t ) = A ri + are the given appropriate dimensioned PPTV matrices with the fundamental period T p .
The main purpose of this work is to develop a robust tracking control u(t ), ensuring the asymptotic convergence of the error vector (t ) = r (t ) − (t ), that is, the states of the system (3) asymptotically track the states of the reference model (4). In other words, the error dynamics model should be asymptotically stable, where K i (t ) is the appropriate dimensioned error feedback PPTV gain matrix to be determined.
Combining the equations (3)-(5), we obtain where B i is the collection of both undesirable uncertainties and unknown disturbances in the addressed PPTVS (3). It should be noted that the control law defined in the equation (6) is not readily applicable because of the presence of unknown terms of the system (3). Thus, the unknown lumped disturbance u (t ) is observed with the aid of the dynamics of system (3): However, the state derivativė(t ) in the relation (7) will cause algebraic loops. To overcome this issue, the UDE-based control strategy in [23] is adopted to estimate the lumped disturbance (7). Precisely, u (t ) is estimated by using the low-pass filter with appropriate bandwidth. In this connection, a low-pass filter  g (s) = 1 1+ fs is utilised to estimate the unknown lumped disturbance with respect to the appropriate bandwidth f of the filter. Now, the lumped disturbance u (t ) can be accurately estimated by passing it into the developed low-pass filter  g (s). Then, the estimated lumped disturbance is given by where , }} * ′′ denotes the convolution operator and q g (t ) represents the impulse response of the filter  g (s). Then, by substituting the estimated lumped disturbance u e (t ) in (6), we get the UDE-based control law as follows: By employing the linear property of the convolution operator and inverse Laplace transform, the UDE-based control law in the above equation is reformulated as where  −1 {.} denotes the inverse Laplace operator. Now, by substituting the low-pass filter  g (s) in the above equation and by simple calculation, we obtain Remark 3. It should be noted that the lumped disturbance may degrade the system performance. The proposed UDEbased controller effectively estimates the lumped disturbance and ensures better tracking performance of the considered PPTVS. More specifically, the significant application of the PPTVSs in many real-world problems and the necessity of disturbance rejection motivates this work to design the UDEbased tracking control scheme for PPTVSs. The proposed UDE-based control scheme consists of three distinct components, namely, reference model, filter design and error feedback gain. In particular, the reference model is used to determine the set-point response. The filter approach is utilised to estimate the unpredictable factors of the addressed system and the disturbance response is determined by the error feedback gain. Owing to this perspective, the developed control law (11) paves a way to obtain a better tracking performance between the states of the actual system and those of the reference model.

MAIN RESULT
This section will focus on deriving sufficient conditions in the form of linear matrix inequalities ensuring the tracking performance between the states of the addressed system (1) and the desired reference model (4). In other words, the asymptotic stabilisation of the PPTV error system (5) with the aid of the developed UDE-based periodic tracking control (11) is ensured. More specifically, Lemma 1 (matrix polynomial approach) is incorporated for linearising the non-linearities raised in between the periodic time-varying matrices. Before establishing the stability conditions, we consider the continuous time-varying positive-definite matrix P (t ) with the periodic nature of period T p , that is, . Based on the above consideration, the positive-definite matrix P i (t ) is defined as follows: Moreover, the schematic representation of PPTV positive-definite matrix P (t ) is given in Figure 1.
Theorem 1. The PPTV error system (5) with Assumption 1 is considered along with the known scalars̄> 0 and̂. If there exist a symmetric positive-definite matrices P i, j , R k and appropriate dimensioned matrices Y i, j , i = 1, 2, … , , j = 1, 2, … , M i − 1, k = 1, 2, 3, such that the following LMI constraints hold: where Π i, j,0 (5) is asymptotically stable. Moreover, to obtain the error feedback PPTV gain matrix, the following relation together with the time-varying matrix functions P i (t ) in (12) is utilised: Proof. In order to establish the desired delay-dependent stability constraint that ensures the required result, the time-varying Lyapunov-Krasovskii functional candidate for the error system (5) with respect to the PPTV matrix P (t ) defined in (12) is formulated in the following form: where P i, j , R 1 , R 2 and R 3 are the positive-definite matrices. The time derivative of the above considered time-varying Lyapunov-Krasovskii functional along the solution of (5) is computed aṡ Further, by employing Lemma 2 to the single integral term in (20), we get . By calculating the time derivative of the matrix function P (t ) in (12), we obtainṖ Now by combining the inequalities (20)-(22), we geṫ By simple manipulation, the time-varying piecewise periodic matrices in Π with the fundamental period T p can be expanded in the form of the following matrix polynomial: where i (t ), i, j (t ) ∈ [0, 1) and the elements of the matrices Π i, j,0 , Π i, j,1 , Π i, j,2 and Π i, j,3 are the same as in the statement of the theorem. By using Lemma 1 and the conditions (13)- (16) in the equation (24), we obtain the matrix Π < 0. Based on the above discussion and Lyapunov stability theory, the asymptotic stability of the error system (5) is ensured. This completes the proof. □ It is assumed that the time-varying delay is a positive constant in the addressed system (1), that is (t ) = , then the error system (5) is deduced as follows: Correspondingly, in the follow up, the control law is reformulated as: Subsequently, the stability conditions in respect of error system (25) is provided in the sequel theorem. For brevity purposes, the notations considered in Theorem 1 are utilised and also the similar steps of Theorem 1 are omitted in the proof of the theorem.
whereΠ i, j,0 (25) is asymptotically stable. Moreover, the error feedback PPTV gain matrix is calculated by using the relation given in (17).
Proof. In order to prove this theorem, the Lyapunov-Krasovskii functional for system (25) is constructed as similar to (19), which is described in the following line: Then, by calculating the time derivative of (31) together with the simple manipulation as same as in Theorem 1, we obtain the matrix polynomial as follows: where i (t ), i, j (t ) ∈ [0, 1) and the elements of the matri-cesΠ i, j,0 ,Π i, j,1 ,Π i, j,2 andΠ i, j,3 are same as in (27)- (30). This directly implies that the closed-loop error system (25) is asymptotically stable, which completes the proof of this theorem. □ In addition, the following corollary gives a set of sufficient conditions to ensuring the asymptotic stability of the addressed system (1) in the absence of time-varying delay. In particular, the proof of the corollary is similar to the one of Theorem 1 with respect to the first term of the Lyapunov-Krasovskii functional defined in Equation (19). In this connection, the asymptotic stability of the delay-free piecewise periodic error system (5) is ensured.
then the system (5) without delay is asymptotically stable. Moreover, the relations (12), (17) and (18) are utilised to obtain the error feedback PPTV gain matrix K i (t ).

NUMERICAL SIMULATION
The inherent potential and applicability of the control law developed in this paper are exemplified by providing a numerical examples together with its simulation results. More specifically, in this section, the robustness of the control scheme is verified by showing a better tracking performance of the considered examples with the desired reference system.

Example 1. A class of PPTVS is considered in the form of
Equation (1), which is composed of three subsystems with the fundamental period T p = 3. To be precise, the coefficient PPTV matrices of the addressed system are graphically represented in Figure 2. In addition, the piecewise time-varying matrix parameters of the corresponding system (1) for = 0, 1, 2, … are given as follows: Subsystem 1: and T 1 = 0.4.  and T 3 = 1.5.
Furthermore, the reference input and unpredictable exogenous disturbance signals, respectively, are considered as: and Further, the delay and its derivative bounds are chosen as = 0.7 and̂= 0.8, respectively. Based on the parameters chosen above, the sufficient conditions (13)- (16) presented in Theorem 1 are solved via the MATLAB LMI toolbox, which yields the unknown PPTV matrices. Moreover, the variation of obtained matrices P (t ) and Y (t ) are plotted in Figures 4 and 5, , the state tracking performance of the system (1) along with the intricate reference states in the presence and absence of the developed feedback control law (11) is plotted in Figure 7, wherein the poor tracking performance without the control law reveals the necessity of the control input (11). Moreover, with the aid of the proposed UDE-based control scheme, the lumped disturbances are estimated effectively and their simulation is given in Figure 8. To be precise, the lumped disturbances are estimated by designing the low-pass filter and the estimated term is incorporated into the controller design to reject the inexorable degradation factors and also for guaranteeing a better tracking performance of the addressed system. Further, the input vector w r (t ) of the reference system (4) is graphically represented in Figure 9. Accordingly, the external disturbance signal is plotted in Figure 10.
In particular, the low-pass filter is designed with the bandwidth value f = 0.001 to filter the unpredictable and inexorable signals. Also, it is noted that the accuracy of the filter performance is purely based on the selection of band width values. In order to reveal this fact, the tracking error value for various bandwidth values is plotted in Figure 11. It is observed from the figure that the robustness of the tracking performance is increased with respect to the selection of bandwidth values. From the simulation results, it is easy to concluded that the proposed UDE-based periodic tracking controller not only estimates the lumped disturbances but also provides better tracking performance of the addressed model.

Example 2.
In this second example, the effectiveness of the developed theoretical results in Theorem 2 and Corollary 1 will

Subsystem 3:
,Ā 3 (t ) =   Moreover, the remaining parameters are considered as same in Example 1 along by selecting the delay bound as = 2. Based on the parameters considered above, the gain matrices are calculated by solving the conditions derived in Theorem 2 for constant time delay case and in Corollary 1 for delay free case. Precisely, the variation of the obtained matrices P (t ), Y (t ) and K (t ) are plotted in Figure 12, wherein the subfigure (a) represents the result obtained via Theorem 2 and correspondingly the subfigure (b) stands for delay free systems. From this figure, it is clearly revealed that the effort given to the delay free case was significantly minimised compared with time-delay systems.
Further, for the simulation purposes, the disturbance input is chosen as w(t ) = sin(15t ) and the reference input as w r (t ) = 20 cos(10t ). Subsequently, under the initial conditions (0) =  Figure 13 shows the tracking performance of the system (1) with constant time delay based on the proposed control law in (26). In Figure 14, the lumped disturbances and its estimation are plotted, where the efficiency of the control law is clearly revealed which ensures the effective estimation of unpredictable factors. Further, by solving the constraints given in Theorem 2 and Corollary 1, respectively for distinct delay values 0.5, 1, 2 and 0, the corresponding response of tracking error states are plotted in Figure 15. From this figure, it is seen that the convergence rate of the error system is decreased when the delay values are increasing. Thus, this simulation result affirms the efficiency of Theorem 2 and Corollary 1, wherein the effective tracking performance is ensured even in the presence of time-delays, uncertainties and disturbances.

CONCLUSION
In this paper, the periodic tracking control problem is studied for a class of PPTVSs with uncertainties, time-varying state delay and external disturbances. Precisely, the proposed UDEbased control scheme ensures the robust tracking performance between the system and reference states. It is noted that the low-pass filter effectively estimates the lumped disturbances, whose estimated signal is incorporated into the formulation of control design. To facilitate the stability analysis, a set of LMI-based conditions is derived by constructing an appropriate PPTV Lyapunov-Krasovskii functional with the fundamental period. Also, with the aid of the matrix polynomial approach, the feasible solution is ensured and also the PPTV controller gain matrices are obtained by solving the developed LMI constraints. Finally, a numerical example with simulation results is presented to exhibit the inherent potential and applicability of the proposed control strategy.