This paper presents a method of designing a state-observer based modified repetitive-control system that provides a given level of disturbance attenuation for a class of strictly proper linear plants. Since the time delay in a repetitive controller can be treated as a kind of disturbance, we convert the system design problem into a standard state-feedback control problem for a linear time-invariant system. The Lyapunov functional and the singular-value decomposition of the output matrix are used to derive a linear-matrix-inequality (LMI) based design algorithm for the parameters of the feedback controller and the state-observer. A numerical example demonstrates the validity of the method. 1. Introduction In control engineering practice, many systems exhibit repetitive behavior, such as a robot manipulator, a hard disk drive, and many other servo systems. Repetitive control [1], or RC for short, has proven to be a useful control strategy for a system with a periodic reference input and/or disturbance signal [2–4]. The distinguishing feature of RC is that it contains a pure-delay positive-feedback loop, which is the internal model of a periodic signal. For a given periodic reference input, a repetitive controller gradually reduces the tracking error through repeated learning actions [5], which involves adding the control input of the previous period to that of the present period to regulate the present control input. This theoretically guarantees gradual improvement and finally eliminates any tracking error and provides very precise control, which is a chief characteristic of the human learning process. From the standpoint of system theory, an RC system (RCS) is a neutral-type delay system. Asymptotic tracking and stabilization of the control system are possible only when the relative degree of the compensated plant is zero [5]. To use RC on a strictly proper plant, that is, the case that most control engineering applications deal with, the repetitive controller has to be modified by the insertion of a low-pass filter into the time-delay feedback line. The resulting system is called a modified RCS (MRCS). Since a modified repetitive controller is just an approximate model of a periodic signal, there exists a steady-state tracking error; that is, in an MRCS, the low-pass filter relaxes the stabilization condition but degrades the tracking precision [6]. RC is similar to iterative learning control (ILC), which is another well-known method that makes use of previous control trials. However, as pointed out by [7–9] and others, there are significant
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