Examination of Bandwidth Enhancement and Circulant Filter Frequency Cutoff Robustification in Iterative Learning Control
Creators: Zhang, Tianyi
Publisher: name: Columbia University
Year: 2021
Subjects: subject: Mechanical engineering
subject: FOS: Mechanical engineering
subject: Machine learning--Mathematical models
subject: Intelligent control systems
subject: Iterative methods (Mathematics)
subject: Gibbs phenomenon

description: The iterative learning control (ILC) problem considers control tasks that perform a specific tracking command, and the command is to be performed is many times. The system returns to the same initial conditions on the desired trajectory for each repetition, also called run, or iteration. The learning law adjusts the command to a feedback system based on the error observed in the previous run, and aims to converge to zero-tracking error at sampled times as the iterations progress. The ILC problem is an inverse problem: it seeks to converge to that command that produces the desired output. Mathematically that command is given by the inverse of the transfer function of the feedback system, times the desired output. However, in many applications that unique command is often an unstable function of time. A discrete-time system, converted from a continuous-time system fed by a zero-order hold, often has non-minimum phase zeros which become unstable poles in the inverse problem. An inverse discrete-time system will have at least one unstable pole, if the pole-zero excess of the original continuous-time counterpart is equal to or larger than three, and the sample rate is fast enough. The corresponding difference equation has roots larger than one, and the homogeneous solution has components that are the values of these poles to the power of k, with k being the time step. This creates an unstable command growing in magnitude with time step. If the ILC law aims at zero-tracking error for such systems, the command produced by the ILC iterations will ask for a command input that grows exponentially in magnitude with each time step. This thesis examines several ways to circumvent this difficulty, designing filters that prevent the growth in ILC. The sister field of ILC, repetitive control (RC), aims at zero-error at sample times when tracking a periodic command or eliminating a periodic disturbance of known period, or both. Instead of learning from a previous run always starting from the same initial condition, RC learns from the error in the previous period of the periodic command or disturbance. Unlike ILC, the system in RC eventually enters into steady state as time progresses. As a result, one can use frequency response thinking. In ILC, the frequency thinking is not applicable since the output of the system has transients for every run. RC is also an inverse problem and the periodic command to the system converges to the inverse of the system times the desired output. Because what RC needs is zero error after reaching steady state, one can aim to invert the steady state frequency response of the system instead of the system transfer function in order to have a stable solution to the inverse problem. This can be accomplished by designing a Finite Impulse Response (FIR) filter that mimics the steady state frequency response, and which can be used in real time. This dissertation discusses how the digital feedback control system configuration affects the locations of sampling zeros and discusses the effectiveness of RC design methods for these possible sampling zeros. The sampling zeros are zeros introduced by the discretization process from continuous-time system to the discrete-time system. In the RC problem, the feedback control system can have sampling zeros outside the unit circle, and they are challenges for the RC law design. Previous research concentrated on the situation where the sampling zeros of the feedback control system come from a zero-order hold on the input of a continuous-time feedback system, and studied the influence of these zeros including the influence of these sampling zeros as the sampling rate is changed from the asymptotic value of sample time interval approaching zero. Effective RC design methods are developed and tested based for this configuration. In the real world, the feedback control system may not be the continuous-time system. Here we investigate the possible sampling zero locations that can be encountered in digital control systems where the zero-order hold can be in various possible places in the control loop. We show that various new situations can occur. We discuss the sampling zeros location with different feedback system structures, and show that the RC design methods still work. Moreover, we compare the learning rates of different RC design methods and show that the RC design method based on a quadratic fit of the reciprocal of the steady state frequency response will have the desired learning rate features that balance the robustness with efficiency. This dissertation discusses the steady-state response filter of the finite-time signal used in ILC. The ILC problem is sensitive to model errors and unmodelled high frequency dynamics, thus it needs a zero-phase low-pass filter to cutoff learning for frequencies where there is too much model inaccuracy for convergence. But typical zero-phase low-pass filters, like Filtfilt used by MATLAB, gives the filtered results with transients that can destabilize ILC. The associated issues are examined from several points of view. First, the dissertation discusses use of a partial inverse of the feedback system as both learning gain matrix and a low-pass filter to address this problem The approach is used to make a partial system inverse for frequencies where the model is accurate, eliminating the robustness issue. The concept is used as a way to improve a feedback control system performance whose bandwidth is not as high as desired. When the feedback control system design is unable to achieve the desired bandwidth, the partial system inverse for frequency in a range above the bandwidth can boost the bandwidth. If needed ILC can be used to further correct response up to the new bandwidth. The dissertation then discusses Discrete Fourier Transform (DFT) based filters to cut off the learning at high frequencies where model uncertainty is too large for convergence. The concept of a low pass filter is based on steady state frequency response, but ILC is always a finite time problem. This forms a mismatch in the design process, and we seek to address this. A math proof is given showing the DFT based filters directly give the steady-state response of the filter for the finite-time signal which can eliminate the possibility of instability of ILC. However, such filters have problems of frequency leakage and Gibbs phenomenon in applications, produced by the difference between the signal being filtered at the start time and at the final time, This difference applies to the signal filtered for nearly all iterations in ILC. This dissertation discusses the use of single reflection that produced a signal that has the start time and end times matching and then using the original signal portion of the result. In addition, a double reflection of the signal is studied that aims not only to eliminate the discontinuity that produces Gibbs, but also aims to have continuity of the first derivative. It applies a specific kind of double reflection. It is shown mathematically that the two reflection methods reduce the Gibbs phenomenon. A criterion is given to determine when one should consider using such reflection methods on any signal. The numerical simulations demonstrate the benefits of these reflection methods in reducing the tracking error of the system.