Padé-Approximation-Based Preview Repetitive Control for Continuous-Time Linear Systems

.is 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. .en, 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.


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 closedloop 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][2][3][4][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], H ∞ 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][13][14][15][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][22][23][24]. e preview repetitive control strategy(PRC), which combines preview control and repetitive control, can significantly improve the control performance of the closedloop system, especially the tracking of periodic signals. rough 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. en, preview information can be fused with the plant object to form an augmented autonomous discrete system [25][26][27]. Recently, benefiting from LMI technology, PRC law can combine with other control schemes, such as robust sliding-mode PRC law [28], generalized-discretetime optimal PRC method [29], and robust guaranteed-cost PRC scheme [30]. In addition, Li [31] proposed an observerbased 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. e 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 continuoustime 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. is observation drives our current research.
is paper focuses on the Padé-approximation-based PRC problem of linear continuous-time systems. Main contributions are summarized as follows. (1) e 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.
is paper is organized as follows. Section 2 presents the problem formulation and the basic assumptions. e Padéapproximation-based PRC law is derived in Section 3. Section 4 provides a numerical simulation. Finally, some conclusions are drawn in Section 5.

Problem Statement
Consider the following linear system: where x p (t) ∈ R n is the state vector, u(t) ∈ R q is the control input, and y(t) ∈ R m is the output of the plant. A, B, C, and D are constant matrices with appropriate dimensions. For simplicity, this paper only considers the single-input single-output plant (q � m � 1). Figure 1 shows the basic configuration of the repetitive control system, where G(s) is the controlled plant, r(t) is a periodic reference input with period L, and C R (s) is a repetitive controller. e output of the repetitive controller where is the tracking error. e following assumptions will be needed throughout the paper.

Assumption 1.
e reference input r(t) ∈ R m to be tracked by y(t) is a periodic reference signal and the period is L. Furthermore, r(t) is piecewise differentiable. At some nondifferential points t, we take the left derivative _ r(t − 0) or the right derivative _ r(t + 0) instead of _ r(t).
Assumption 2. e periodic reference input r(t) is previewable in the sense that the future value of r(σ)(t ≤ σ ≤ t + l r ) is available at each instant of time t. e value of l r is presented as the preview length of the reference input r(t). (1) is finite spectrum controllability. Namely, the pair (A, B) is stabilizable, and the following condition holds: Remark 1. e 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. e repetitive controller block generates the periodic signal of period L and is used to improve the learning performance between repetition periods. e preview control compensator block is used to improve the tracking performance by using the future value of the desired reference input signal. e state feedback controller block is used to improve the stability of the system in each period. e error integral compensator block is used to reduce static error. e 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 y(t) of system (1) tracks the desired output r(t) without static error, namely, Note that, in Figure 1, where V(s) and E(s) are Laplace transforms of v(t) and e(t), respectively. From Padé-approximation formula [35,36], for simplicity, let Substituting (8) into the formula of (6) yields rough 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.

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 n × 1 augmented vector z(t), where n � n + 2m, from (1), (2), and (10), we get the following augmented dynamic system: where A, B, and D 1 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 u(t) that minimizes subject to the state equation (12), where Q z ∈ R n×n and R ∈ R q×q are both positive definite matrices, erefore, the problem of PRC design based on Padé approximation can be transformed into solving the optimal control input u(t) of system (12) under the performance index (14). To this end, we have to ensure the existence conditions of stabilizability of (A, B) and the detectability of (Q (1/2) z , A). Lemma 1. (A, B) is stabilizable if and only if Assumption 3 holds.
Proof. Let C + be the closed right-half of the complex plane. According to PBH rank test, (A, B) is stabilizable if and only if the matrix (A − sI n , B) is of full row rank for any s ∈ C + . en, according to the structure of the matrices A and B, we obtain When s � 0, we have Because ( When s > 0, by elementary transformation, we obtain e following theorems give the main results of this paper. Under the performance index (14) and Assumptions 1-4, the Padé-approximation-based optimal preview repetitive controller u(t) of system (1) is given by where F x , F e , and F v are the feedback gains given by K � K x K e K v ∈ R n×n is the unique semidefinite solution of the algebraic Riccati equation (ARE): In addition, A c 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 is where λ(t) is the adjoint vector; by Lemma 3, we can obtain en, Based on the optimal control theory [34], the following regular equations are obtained: Combining the first equation of (29) into (12) yields where K(t) ∈ R n×n and g(t) ∈ R n×1 satisfy the boundary conditions: e above conditions can be rewritten as Some simple calculations yield where Φ(t, τ) is the state-transition matrix of (35) which satisfies the initial condition , the solution of (35) is given by where Under the boundary condition g(t a ) � 0, it follows from (37) that where t a ∧(t + l r ): � min t a , t + l r . Now, we let t a ⟶ ∞. en, it is well known that, under Lemmas 1 and 2 that the matrix pair (A, B) is stabilizable and (Q (1/2) z , A) is detectable, the solution K(t) of (34) converges to one constant matrix K, which is the unique nonnegative definite solution of the ARE (23). Moreover, A c (τ) converges to A c , which is stable, and Φ(t, τ) reduces to e − A c (t− τ) as t a ⟶ ∞. erefore, Furthermore, substituting (38) into (40), we obtain Combining (29), (31), and (41), the optimal control input _ u(t) is given by Integrating both sides of the above formula over [− L, t](L > l r ) yields Mathematical Problems in Engineering 5 Note that x(t) � 0, u(t) � 0, and r(t) � 0, for t ≤ 0. e above formula can be rewritten as For the last part of equation (44), exchange the integration sequence and integrate in [− L, t]. We can obtain eorem 1. e proof is completed. e structure of the closed-loop system is shown in Figure 3. In fact, taking the two sides' Laplace transforms of (20), we have where F 1 (s) is computed as Here, N L (s) is given by en, U(s) can be rewritten as erefore, from (48), we see that the configuration of the closed-loop system is as shown in Figure 3.

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. e numerical simulation example is We employ the Padé-approximation-based PRC law (20) to track the following reference input: Let the initial condition be x(0) � 0 0 0 T , x(t) � 0, u(t) � 0, and r(t) � 0(− 10 ≤ t ≤ 0). e preview lengths are given by l r � 0, l r � 0.05, and l r � 0.1. 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 l r � 0.1, the control system achieves the best tracking performance.
In Figures 6 and 7, with l r � 0.1, 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.

Conclusions
In this study, the state space representation of the repetitive controller was transformed into a nondelayed one by Padé approximation. en, 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. e 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
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 7