Iterative Learning Control and Adaptive Control for Systems with Unstable Discrete-Time Inverse

de Bowen Wang

Iterative Learning Control (ILC) considers systems which perform the given desired trajectory repetitively. The command for the upcoming iteration is updated after every iteration based on the previous recorded error, aiming to converge to zero error in the real-world. Iterative Learning Control can be considered as an inverse problem, solving for the needed input that produces the desired output. However, digital control systems need to convert differential equations to digital form. For a majority of real world systems this introduces one or more zeros of the system z-transfer function outside the unit circle making the inverse system unstable. The resulting control input that produces zero error at the sample times following the desired trajectory is unstable, growing exponentially in magnitude each time step. The tracking error between time steps is also growing exponentially defeating the intended objective of zero tracking error. One way to address the instability in the inverse of non-minimum phase systems is to use basis functions. Besides addressing the unstable inverse issue, using basis functions also has several other advantages. First, it significantly reduces the computation burden in solving for the input command, as the number of basis functions chosen is usually much smaller than the number of time steps in one iteration. Second, it allows the designer to choose the frequency to cut off the learning process, which provides stability robustness to unmodelled high frequency dynamics eliminating the need to otherwise include a low-pass filter. In addition, choosing basis functions intelligently can lead to fast convergence of the learning process. All these benefits come at the expense of no longer asking for zero tracking error, but only aiming to correct the tracking error in the span of the chosen basis functions. Two kinds of matched basis functions are presented in this dissertation, frequency-response based basis functions and singular vector basis functions, respectively. In addition, basis functions are developed to directly capture the system transients that result from initial conditions and hence are not associated with forcing functions. The newly developed transient basis functions are particularly helpful in reducing the level of tracking error and constraining the magnitude of input control when the desired trajectory does not have a smooth start-up, corresponding to a smooth transition from the system state before the initial time, and the system state immediately after time zero on the desired trajectory. Another topic that has been investigated is the error accumulation in the unaddressed part of the output space, the part not covered by the span of the output basis functions, under different model conditions. It has been both proved mathematically and validated by numerical experiments that the error in the unaddressed space will remain constant when using an error-free model, and the unaddressed error will demonstrate a process of accumulation and finally converge to a constant level in the presence of model error. The same phenomenon is shown to apply when using unmatched basis functions. There will be unaddressed error accumulation even in the absence of model error, suggesting that matched basis functions should be used whenever possible. Another way to address the often unstable nature of the inverse of non-minimum phase systems is to use the in-house developed stable inverse theory Longman JiLLL, which can also be incorporated into other control algorithms including One-Step Ahead Control and Indirect Adaptive Control in addition to Iterative Learning Control. Using this stable inverse theory, One-Step Ahead Control has been generalized to apply to systems whose discrete-time inverses are unstable. The generalized one-step ahead control can be viewed as a Model Predictive Control that achieves zero tracking error with a control input bounded by the actuator constraints. In situations where one feels not confident about the system model, adaptive control can be applied to update the model parameters while achieving zero tracking error.