This paper concerns the design problem of optimal preview repetitive control (OPRC) for impulse-free continuous-time descriptor systems. First, by using a linear transformation, the descriptor system is transformed into a normal system with relatively low dimensionality and an algebraic system. Based on this, an augmented system which contains the error vector and the derivative of the normal system is constructed. Then, applying the lift technique and introducing a new performance index, the OPRC designing problem is converted into a regulation problem. Based on the optimal control theory, the regulator problem of the augmented system is solved. Furthermore, the explicit OPRC law for the original system is deduced. Different from the existing results, the preview feed-forward and error integral compensations are added into the OPRC system, which can significantly improve the tracking performance. Finally, a numerical example simulation result verifies the validity of the proposed method.

#### Keywords

- repetitive control
- preview control
- optimal control
- continuous-time descriptor systems

Descriptor systems (also known as singular systems), which are first proposed by Rosenbrock in the 1970s [1] are a class of more general dynamic systems than general ones. They have been widely used in many fields in recent years [2–4], and increasing attention has been paid to the study of descriptor systems [5–7]. Moreover, many characteristics of analysis and synthesis in general systems theory also have been extended to descriptor systems [8, 9].

Preview control (PC) is a kind of feed-forward control method. By fully utiliing the known future knowledge of references or disturbances, it can improve the tracking performance of the controlled system [10, 11]. For discrete-time systems, by using a difference operator, the preview compensation can be constructed as part of the augmented discrete-time systems. Therefore, the PC problem of discrete-time systems can be converted into a stabilisation problem. So far, the most extensive researches of PC for discrete-time systems are based on linear quadratic regulator (LQR) or linear matrix inequality (LMI) [12–14]. Moreover, the problems of H PC [15], adaptive PC [16] and sliding mode-based PC [17] are also investigated.

For continuous-time systems, the preview compensation cannot be used directly as a part of the augmented dynamic systems. An alternative method is to use the derivatives of the state equation and the tracking error. However, the obtained augmented dynamic systems are non-autonomous ones. As a result, the difficulty and complexity are greatly increased. By differentiating both the output error signal and the two sides of the state equation, the optimal PC problem of continuous-time systems is solved in [18–20]. By taking the derivative of the control input, a new optimal PC was deduced [21]. The optimal PC problem of impulse-free descriptor systems was studied [23]. More recently, for continuous-time descriptor multi-agent systems, a cooperative optimal preview tracking problem was considered in [24] by using the LQR technique.

On the other hand, repetitive control (RC) is an effective technique for improving the performance of systems that track periodic reference or reject periodic disturbance [25]. It was originated by Inoue et al. [25] and subsequently developed by many researchers, such as Hara [26] and Doh [27]. Nowadays, a great deal of research has been focused on the theory and applications of RC, and various structures and algorithms have been devised (see, for example, [28–32]). The combination of preview and RC, which is called preview RC, can significantly improve the control performance of closed-loop systems. Since the relationships between the system and the future signal as well as the output of the RC were established by the difference operator, preview RC for discrete-time systems has become very popular in many research fields.

A discrete-time preview RC method [34] was first presented based on LQ optimal control. Using ARMAX models, a new design method for discrete-time preview RC was proposed [35]. The design method of discrete-time sliding mode preview repetitive servo systems was introduced [36, 37]. Very recently, an LMI-based robust guaranteed-cost preview RC for uncertain discrete-time systems was proposed [38].

It is worth pointing out that although there have been fruitful results about discrete-time preview RC, there have been only a few studies on designing preview RC for continuous-time systems. This paper focuses on the preview RC problem of linear continuous-time descriptor systems. The main contributions are as follows:

The descriptor system is transformed into a restricted equivalent one with an algebraic system by taking non-singular linear transformation.

Based on the lift technique, an augmented system is constructed, which contains the output of the basic RC;

Different from the existing PC methods, the augmented state vector and the derivative of the tracking error, as well as the derivative of the control input, are introduced into the quadratic performance index; (3) Based on the optimal control, the regulator problem of the augmented system is solved, from which the explicit optimal preview repetitive controller (OPRC) for the original system is deduced.

The rest of this paper is organised as follows. Section 1 presents the problem formulation and assumptions. The OPRC is derived in Section 2. A numerical example is provided in Section 3. Finally, some conclusions are drawn in Section 4.

Figure 1 shows the basic configuration of a continuous-time RC system, where G(s) is the controlled plant, r(t) is a periodic reference input with period L, and C_{R}(s) is an RC. The output of the RC, v(t), is the tracking error.

This paper focuses on OPRC designing of a plant with a relative degree of zero, which is described by the following regular and impulse-free continuous-time descriptor system.
^{n} is the state, ^{q} is the control input, and ^{p} is the output of the plant. ^{p} be a periodic reference signal to be tracked by _{r}_{r}_{r}

The objective of this paper is to develop an RC with preview compensation such that in the steady-state, the output vector

The following assumptions and lemmas are needed.

Assume that system (3) is regular and there exists a _{0} such that det(_{0}

The matrix pair (

The matrix pair (

The matrix

In this subsection, we transform system (3) into a restricted equivalent form by non-singular linear transformation.

Taking non-singular linear transformation

Therefore, the regularity, impulse-free, stability and detectability of systems (3) and (10) are equivalent. On the other hand, it follows from assumption A2 that system (10) is impulse-free, which implies that _{22} is non-singular [3].

Since _{22} is non-singular, from the second equation of (10), we get

Substituting (11) into the first and the third equation of (10), respectively, we have

For

Eqs (22) and (23) are augmented dynamic systems. Correspondingly, we wish to find the optimal controller that minimises the following performance index [39]
^{q×q}, _{e}^{p×p} are positive-definite matrices, and _{z}_{x} Q_{xe}_{x}^{m×m}_{xe}^{p×p} are nonnegative-definite and positive-definite matrices, respectively.

Now, the design problem is transformed into the design of a control input

The following lemmas related to stabilisability (controllability) and detectability (observability) give the existence conditions of the optimal control law for the augmented dynamic system.

_{p}

_{22} is non-singular, by some calculation, we have

Clearly, the matrices

Based on Lemma 3 to Lemma, we have the following Lemma 6.

_{xe} is positive-definite and the martrix pair

_{xe}

Based on the theory of optimal control, we obtain the main result of this paper as follows.

_{e}, F_{v} and F_{x} are given by_{x} K_{e}_{c} in

^{n̅×n̅}, Δ^{n̅×1} and

Using Eq. (40) in Eq. (42), we can obtain
_{a}

Now, let _{a}_{a}_{c} is stable, with _{r}_{a}, τ

On the other hand, substituting Eq. (42) with _{x} K_{e}_{r}

_{x}x_{1} (_{v}v

_{d} and i_{q} are the d – q axis currents, u_{d} and u_{q} are the d – q axis voltages. R and L denotes the stator resistance and inductance per phase respectively, p is the number of pole pairs, φf is the permanent magnet flux, J is the rotor moment of inertia, B is the viscous friction factor and T_{L} also represents the load torque

To decouple the speed and currents, the vector control strategy of _{d}_{L}

The parameters of PMSM system are set as

Given the structure of

The numerical simulation results are shown in Figure 3 to Figure 6. Figures 3 and 4 show the output response and tracking error with different preview lengths. It can be seen that the tracking quality of the output signal can be improved by adjusting the preview lengths. But when the preview lengths reach a certain degree, there is almost no effect on the output response.

For comparison, the LMI-based RC [28, 29] simulation is also provided. The simulation results between RC and OPRC with preview lengths _{r}

This paper considered the design method of OPRC for impulse-free continuous-time descriptor systems. Based on the characteristics of impulse-free descriptor systems, the descriptor system was transformed into a restricted equivalent form. By using the lift technique and the construction of the augmented system, the OPRC designing problem was converted into a regulator problem. The optimal controller for the augmented system was designed by making use of standard optimal control theory, and the OPRC of the original system was then obtained, in which the integrator compensation was innovatively added. The effectiveness of the proposed method has been shown by a numerical example.

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