Abstract
This paper concerns a Padé-approximation-based preview repetitive control (PRC) scheme for continuous-time linear systems. Firstly, the state space representation of the repetitive controller is transformed into a nondelayed one by Padé approximation. Then, an augmented dynamic system is constructed by using the nominal state equation with the error system and the state equation of a repetitive controller. Next, by using optimal control theory, a Padé-approximation-based PRC law is obtained. It consists of state feedback, error integral compensation, output integral of repetitive controller, and preview compensation. Finally, the effectiveness of the method is verified by a numerical simulation.
1. Introduction
Preview control (PC) is an extended feedforward control method. By making full use of the known future reference signal or disturbance signal, the performance of the closed-loop system can be improved. PC has received considerable attention since it was introduced by Tomizuka in 1970s. For more than five decades, the most extensive research into preview control has focused on the linear quadratic optimal control problem with preview compensation [1–5], especially for discrete-time systems. Subsequently, linear matrix inequality (LMI) technique has been extensively used to handle the PC problems for uncertain discrete-time systems [6]. Combining LMI-based PC with other control schemes, some new concepts are proposed, such as adaptive PC [7], fault tolerant PC [8], PC [9], observer-based PC [10], and distributed PC [11]. However, in continuous-time systems, the methods that can be used to deal with PC issue are relatively limited, and differential operation is widely used instead of differential operation [12–16].
On the contrary, repetitive control (RC) is an effective technique to improve the performance of tracking periodic reference or suppressing periodic disturbance. It was originated in 1980s by Inoue et al. [17] and then developed by Hara et al. [18] and Doh et al. [19]. Over the last a couple of decades, a great deal of research has been devoted to the theory and applications of the RC. A summary of RC works can be referred to in [20], and recent literatures of RC can be found in [21–24].
The preview repetitive control strategy(PRC), which combines preview control and repetitive control, can significantly improve the control performance of the closed-loop system, especially the tracking of periodic signals. Through the difference operator, the relationship between the system and the future signal and the relationship between the repetitive controller and the tracking error are established. Then, preview information can be fused with the plant object to form an augmented autonomous discrete system [25–27]. Recently, benefiting from LMI technology, PRC law can combine with other control schemes, such as robust sliding-mode PRC law [28], generalized-discrete-time optimal PRC method [29], and robust guaranteed-cost PRC scheme [30]. In addition, Li [31] proposed an observer-based PRC strategy for uncertain discrete-time systems using two-dimensional model approach. In [32], the design method of adaptive fuzzy finite time control for switched pure feedback nonlinear systems with given performance based on the observer is discussed. The problem of output feedback control for discrete-time systems with two quantized signals in measurement output and control input was discussed in [33].
It is worth pointing out that the above studies about PRC only focus on discrete-time system and less on continuous-time systems. In [34], the optimal PRC for continuous-time linear system was obtained and the result was applied to the tracking problem of permanent magnet synchronous motor drive system. However, due to the time-delay part of RC, Padé-approximation technique can be used to reduce the complexity of constructing augmented error system. This observation drives our current research.
This paper focuses on the Padé-approximation-based PRC problem of linear continuous-time systems. Main contributions are summarized as follows. (1) The state space representation of the repetitive controller is transformed into a delay-free state space representation by Padé approximation. (2) An augmented dynamic system is obtained by processing the repetitive controller and tracking error signal. (3) Based on the optimal control theory, the regulator problem of the augmented system is solved, from which the optimal PRC law for the original system can be derived.
This paper is organized as follows. Section 2 presents the problem formulation and the basic assumptions. The Padé-approximation-based PRC law is derived in Section 3. Section 4 provides a numerical simulation. Finally, some conclusions are drawn in Section 5.
2. Problem Statement
Consider the following linear system:where is the state vector, is the control input, and is the output of the plant. are constant matrices with appropriate dimensions. For simplicity, this paper only considers the single-input single-output plant .
Figure 1 shows the basic configuration of the repetitive control system, where is the controlled plant, is a periodic reference input with period , and is a repetitive controller. The output of the repetitive controller iswhereis the tracking error.
The following assumptions will be needed throughout the paper.
Assumption 1. The reference input to be tracked by is a periodic reference signal and the period is . Furthermore, is piecewise differentiable. At some nondifferential points , we take the left derivative or the right derivative instead of .
Assumption 2. The periodic reference input is previewable in the sense that the future value of is available at each instant of time . The value of is presented as the preview length of the reference input .
Assumption 3. System (1) is finite spectrum controllability. Namely, the pair is stabilizable, and the following condition holds:
Assumption 4. is detectable.
Remark 1. The overall block diagram in Figure 2 consists of four blocks: (i) repetitive controller block, (ii) preview control compensator block, (iii) state feedback controller block, and (iv) error integral compensator block. The repetitive controller block generates the periodic signal of period and is used to improve the learning performance between repetition periods. The preview control compensator block is used to improve the tracking performance by using the future value of the desired reference input signal. The state feedback controller block is used to improve the stability of the system in each period. The error integral compensator block is used to reduce static error.
The objective of this brief is to develop a Padé-approximation-based PRC law as described in Figure 2 such that, in the steady state, the output vector of system (1) tracks the desired output without static error, namely,
Note that, in Figure 1,where and are Laplace transforms of and , respectively.
From Padé-approximation formula [35, 36],for simplicity, let
Substituting (8) into the formula of (6) yields
Through the inverse Laplace transform of formula (9), we obtain
Remark 2. Different from (2), there is no time delay in the state space model of the RC system represented by (10), so an augmented error system without time delay can be obtained.
3. Design Controller of Preview Repetitive Control
In this section, by using the optimal control theory, we will design a Padé-approximation-based preview repetitive controller for plant (1).
Introducing the augmented vector ,where , from (1), (2), and (10), we get the following augmented dynamic system:where , and are constant matrices defined by
System (12) is called an augmented dynamic system of (1). To obtain an optimal controller, we wish to find a controller that minimizessubject to the state equation (12), where and are both positive definite matrices, , , , and .
Therefore, the problem of PRC design based on Padé approximation can be transformed into solving the optimal control input of system (12) under the performance index (14). To this end, we have to ensure the existence conditions of stabilizability of and the detectability of .
Lemma 1. is stabilizable if and only if Assumption 3 holds.
Proof. Let be the closed right-half of the complex plane. According to PBH rank test, is stabilizable if and only if the matrix is of full row rank for any . Then, according to the structure of the matrices and , we obtainWhen , we haveBecause is reversible, the matrix is of full row rank if and only ifWhen , by elementary transformation, we obtainBecause is reversible, the matrix is of full row rank if and only if is of full row rank (namely, the pair is stabilizable). The proof is completed.
Lemma 2. If is a positive definite matrix, then is detectable.
Proof. According to PBH rank test, is detectable if and only if has full column rank for any , where is the closed right-half of the complex plane. Since is a positive definite matrix, obviously, is nonsingular. Thus, it is evident that is of full column rank. The proof is completed.
Lemma 3. For a quadratic form , where and are symmetric matrices which do not depend on , then
The following theorems give the main results of this paper.
Theorem 1. Let , , and , for . Under the performance index (14) and Assumptions 1–4, the Padé-approximation-based optimal preview repetitive controller of system (1) is given bywhere
, , and are the feedback gains given by
is the unique semidefinite solution of the algebraic Riccati equation (ARE):
In addition, in (21) is a stable matrix defined by
Proof. According to the theory of linear quadratic regulation, when system (12) is under performance index (14), the Hamilton function iswhere is the adjoint vector; by Lemma 3, we can obtainThen,Based on the optimal control theory [34], the following regular equations are obtained:That is,Combining the first equation of (29) into (12) yieldsSetwhere and satisfy the boundary conditions:The above conditions can be rewritten asSome simple calculations yieldThen, setwhere is the state-transition matrix of (35) which satisfies the initial condition and . Then, for , the solution of (35) is given bywhereUnder the boundary condition , it follows from (37) thatwhere .
Now, we let . Then, it is well known that, under Lemmas 1 and 2 that the matrix pair is stabilizable and is detectable, the solution of (34) converges to one constant matrix , which is the unique nonnegative definite solution of the ARE (23). Moreover, converges to , which is stable, and reduces to as . Therefore,Furthermore, substituting (38) into (40), we obtainCombining (29), (31), and (41), the optimal control input is given byIntegrating both sides of the above formula over yieldsNote that , , and , for . The above formula can be rewritten asFor the last part of equation (44), exchange the integration sequence and integrate in . We can obtain Theorem 1.
The proof is completed.
The structure of the closed-loop system is shown in Figure 3. In fact, taking the two sides’ Laplace transforms of (20), we havewhere is computed asHere, is given byThen, can be rewritten asTherefore, from (48), we see that the configuration of the closed-loop system is as shown in Figure 3.
4. Simulation Results
In this section, to demonstrate the effectiveness of the proposed Padé-approximation-based PRC law design, the same scenario in [37] was considered to demonstrate the validity of our method. The numerical simulation example is
We employ the Padé-approximation-based PRC law (20) to track the following reference input:
The weight matrices in performance index (14) are chosen as
Let the initial condition be , , , and . The preview lengths are given by , , and .
Figures 4 and 5 show the situation of output responses and tracking errors under different preview lengths. Obviously, the Padé-approximation-based RC with preview compensation can effectively reduce the static error and peak overshoot in the process of tracking reference signal (50). With the increase of preview lengths, the peak overshoot and static error will be reduced. However, when the preview length reaches a certain degree, it has little effect on the output response. When , the control system achieves the best tracking performance.
In Figures 6 and 7, with , compare the results of the proposed Padé-based approximation PRC law with those of the LMI-based RC law designed in [37, 38]. It can be seen that the controller proposed in this paper can obviously improve the response speed and tracking accuracy of the system.
From the above simulation results, it is clear to see that the designed Padé-approximation-based PRC law provides the expected tracking performance, which illustrates the significance of the proposed controller design method.
5. Conclusions
In this study, the state space representation of the repetitive controller was transformed into a nondelayed one by Padé approximation. Then, an augmented dynamic system was constructed by using the nominal state equation with the error system and the state equation of the repetitive controller. Finally, by optimal control theory, a Padé-approximation-based PRC law was obtained. The simulation results showed the superiority of the control method. It is worth mentioning that the proposed Padé-approximation-based PRC can be further designed for linear time-delay systems, which is our future research work.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported in part by the Natural Science Foundation of Hunan Province, under Grant 2020JJ6037, the National Natural Science Foundation of China, under Grant 61573298, and the scientific research project of Hunan Province Department of Education, under Grant 20A116.