Abstract
In this paper, a new method for the design of the preview controller for a class of discrete-time systems is proposed based on the virtual system. Firstly, by taking the known future reference signal as the output, the virtual system with similar structures to the controlled system is constructed. Then, the augmented error system is received by translating the controlled system to it and by integrating the error equation. Thus, the tracking problem of the controlled system is transformed into the regulation problem of the augmented error system. Secondly, in view of the minimum principle, the optimal controller of the augmented error system is acquired, and the preview controller of the controlled system is also gained. Further, by discussing the stabilizability and detectability of the augmented error system, the conditions for the existence of the unique positive semidefinite solution to an algebraic Riccati equation are obtained. By using the method in this paper, making difference and dimension expansion for the state equation in designing the augmented error system is avoided and the output can track the reference signals better. Finally, the numerical simulation shows the effectiveness of the proposed controller.
Highlights
When a part of future reference signals or exogenous disturbance signals are known, they can be utilized by preview control and the tracking performance of the closed-loop system can be improved [1]
In [19], the augmented error system is constructed by applying difference to both sides of the state equation and to the error equation, which makes the preview tracking problem of the discrete-time system transformed into a standard linear quadratic regulation (LQR) problem. en, the optimal controller of the augmented error system can be obtained by directly using the existing results
− KxΓr k + Mr + cr k + Mr, where P is the unique positive semidefinite solution, satisfying the algebraic Riccati equation (37), K, Kr(i)(i 0, 1, 2, . . . , Mr) meet (47)–(49), and S1, S2 meet (39) and (40). It can be seen from (52) that the optimal preview controller of system (1) consists of four parts. e first part is the sum of tracking errors term Kqki −01e(i), which is originated from the employment of the integrator. e second part is the state feedback Kxx(k). e third part is the preview compensation of reference signals Mi 0r Kr(i)[r(k + i) − r(k+ Mr + i)]. e last one −KxΓr(k + Mr) + cr(k + Mr) is the preview complement from the virtual system
Summary
When a part of future reference signals or exogenous disturbance signals are known, they can be utilized by preview control and the tracking performance of the closed-loop system can be improved [1]. E augmented error system is a common approach to solve the preview control problem It is proposed by Tomizuka and Rosenthal [19] in 1979 and has been applied to both discrete-time systems and continuous-time systems. Due to a group of identities about future disturbance signals being added to the augmented error system, the obtained controller for the controlled system contains preview compensation Based on this technique, Katayama et al solved the tracking problem of the discrete-time system with preview reference signals in [20], and the conclusion obtained is extended to the condition of continuous-time systems in [21]. In [22], the tracking problem of the continuous-time linear system with both reference signals and disturbance signals previewable is further solved.
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