Adaptive Rejection of Unknown Sinusoidal Disturbances in Linear SISO Uncertain Systems

1 I n t r o d u c t i o n Many mechanical systems are subjected to periodic disturbances which may adversely influence the mission performance, effectiveness, accuracy of operation, and life cycle. Examples of such systems include: torque pulsations in DC and stepper motors at the frequency of rotation and its integer harmonics [1], rotating pump in a cryocooler expander [2], rotating magnetic bearing [3], disk drive systems with repeatable runout error [4], cutter radical eccentricity in milling operation [5], and steel casting processes [6]. Traditionally, the design of the controller is influenced by a-prior knowledge of the disturbance signals to a large extent; for example, the type of signal and its bandwidth [7]. When the disturbance is partially known or directly measurable, it is often possible to directly cancel the disturbance based on the measurements, for instance, using adaptive feedforward control (AFC) algorithms [8], and linear quadratic c o n trol (LQG) methods [9]. When disturbance signals are completely unknown or not directly measurable, adap1Author of the correspondence, Phone: (864)656-5642, Fax: (864)656-4435. rive techniques such as internal model techniques [10], external model principle [7], predict model techniques [11], and disturbance observers [12] can be used. These methods utilize the fact that the disturbance signals can be regarded as the output of an autonomous systern, and adaptive regulation is achieved through estimation of the disturbance model on-line and then using the estimated model in the control design [13]. In the adaptive internal model principle, the problem of adaptation to unknown frequency of the disturbance is solved together with the problem of adapting to uncertain plant parameters [14]. Adaptive implementation of the internal model principle for such arrangement usually increases the order of compensator. A controller for the resulting over parameterized model is then used to cancel the disturbance effects. Drawbacks of the approach based on the adaptive internal model principle are discussed in [15]. Other alternatives are to model a state space system which generates the disturbance as an output. If the disturbance can be directly measurable, a feedforward canceling technique can be applied. For unknown plants, it is difficult to develop a controller if the plant is nonminimum phase. By utilizing a recursive parameter identification algorithm to estimate external disturbances, the problem of disturbance rejection of nonminimum phase plants has been studied [7]. This paper considers the problem of adaptively rejecting disturbance inputs in which the frequency and amplitude are unknown. In contrast to disturbance model estimation techniques, the technique proposed here is based on constructing a set of stabilizing controllers using a state estimate observer, which does not require disturbance model estimation. The adaptation scheme is based on the use of tuning functions [16] whose performance bounds are computable and can be utilized for systematic improvement of transient performance. Utilizing only the system output, a virtual control input is used in place of unmeasurable and unknown signals. Although the technique proposed here is applicable to a more general class of disturbance (i.e.