Robustification and Optimization in Repetitive Control For Minimum Phase and Non-Minimum Phase Systems

Repetitive control (RC) is a control method that specifically aims to converge to zero tracking error of a control systems that execute a periodic command or have periodic disturbances of known period. It uses the error of one period back to adjust the command in the present period. In theory, RC can completely eliminate periodic disturbance effects. RC has applications in many fields such as high-precision manufacturing in robotics, computer disk drives, and active vibration isolation in spacecraft. The first topic treated in this dissertation develops several simple RC design methods that are somewhat analogous to PID controller design in classical control. From the early days of digital control, emulation methods were developed based on a Forward Rule, a Backward Rule, Tustin’s Formula, a modification using prewarping, and a pole-zero mapping method. These allowed one to convert a candidate controller design to discrete time in a simple way. We investigate to what extent they can be used to simplify RC design. A particular design is developed from modification of the pole-zero mapping rules, which is simple and sheds light on the robustness of repetitive control designs. RC convergence requires less than 90 degree model phase error at all frequencies up to Nyquist. A zero-phase cutoff filter is normally used to robustify to high frequency model error when this limit is exceeded. The result is stabilization at the expense of failure to cancel errors above the cutoff. The second topic investigates a series of methods to use data to make real time updates of the frequency response model, allowing one to increase or eliminate the frequency cutoff. These include the use of a moving window employing a recursive discrete Fourier transform (DFT), and use of a real time projection algorithm from adaptive control for each frequency. The results can be used directly to make repetitive control corrections that cancel each error frequency, or they can be used to update a repetitive control FIR compensator. The aim is to reduce the final error level by using real time frequency response model updates to successively increase the cutoff frequency, each time creating the improved model needed to produce convergence zero error up to the higher cutoff. Non-minimum phase systems present a difficult design challenge to the sister field of Iterative Learning Control. The third topic investigates to what extent the same challenges appear in RC. One challenge is that the intrinsic non-minimum phase zero mapped from continuous time is close to the pole of repetitive controller at +1 creating behavior similar to pole-zero cancellation. The near pole-zero cancellation causes slow learning at DC and low frequencies. The Min-Max cost function over the learning rate is presented. The Min-Max can be reformulated as a Quadratically Constrained Linear Programming problem. This approach is shown to be an RC design approach that addresses the main challenge of non-minimum phase systems to have a reasonable learning rate at DC. Although it was illustrated that using the Min-Max objective improves learning at DC and low frequencies compared to other designs, the method requires model accuracy at high frequencies. In the real world, models usually have error at high frequencies. The fourth topic addresses how one can merge the quadratic penalty to the Min-Max cost function to increase robustness at high frequencies. The topic also considers limiting the Min-Max optimization to some frequencies interval and applying an FIR zero-phase low-pass filter to cutoff the learning for frequencies above that interval

Phase correction for Learning Feedforward Control

Intelligent mechatronics makes it possible to compensate for effects that are difficult to compensate for by construction or by linear control, by including some intelligence into the system. The compensation of state dependent effects, e.g. friction, cogging and mass deviation, can be realised by learning feedforward control. This method identifies these disturbing effects as function of their states and compensates for these, before they introduce an error. Because the effects are learnt as function of their states, this method can be used for non-repetitive motions. The learning of state dependent effects relies on the update signal that is used. In previous work, the feedback control signal was used as an error measure between the approximation and the true state dependent effect. If the effects introduce a signal that contains frequencies near the bandwidth, the phase shift between this signal and the feedback signal might seriously degenerate the performance of the approximation. The use of phase correction overcomes this problem. This is validated by a set of simulations and experiments that show the necessity of the phase corrected scheme

Synthesis and Analysis of Design Methods in Linear Repetitive, Iterative Learning and Model Predictive Control

Repetitive Control (RC) seeks to converge to zero tracking error of a feedback control system performing periodic command as time progresses, or to cancel the influence of a periodic disturbance as time progresses, by observing the error in the previous period. Iterative Learning Control (ILC) is similar, it aims to converge to zero tracking error of system repeatedly performing the same task, and also adjusting the command to the feedback controller each repetition based on the error in the previous repetition. Compared to the conventional feedback control design methods, RC and ILC improve the performance over repetitions, and both aiming at zero tracking error in the real world instead of in a mathematical model. Linear Model Predictive Control (LMPC) normally does not aim for zero tracking error following a desired trajectory, but aims to minimize a quadratic cost function to the prediction horizon, and then apply the first control action. Then repeat the process each time step. The usual quadratic cost is a trade-off function between tracking accuracy and control effort and hence is not asking for zero error. It is also not specialized to periodic command or periodic disturbance as RC is, but does require that one knows the future desired command up to the prediction horizon. The objective of this dissertation is to present various design schemes of improving the tracking performance in a control system based on ILC, RC and LMPC. The dissertation contains four major chapters. The first chapter studies the optimization of the design parameters, in particular as related to measurement noise, and the need of a cutoff filter when dealing with actuator limitations, robustness to model error. The results aim to guide the user in tuning the design parameters available when creating a repetitive control system. In the second chapter, we investigate how ILC laws can be converted for use in RC to improve performance. And robustification by adding control penalty in cost function is compared to use a frequency cutoff filter. The third chapter develops a method to create desired trajectories with a zero tracking interval without involving an unstable inverse solution. An easily implementable feedback version is created to optimize the same cost every time step from the current measured position. An ILC algorithm is also created to iteratively learn to give local zero error in the real world while using an imperfect model. This approach also gives a method to apply ILC to endpoint problem without specifying an arbitrary trajectory to follow to reach the endpoint. This creates a method for ILC to apply to such problems without asking for accurate tracking of a somewhat arbitrary trajectory to accomplish learning to reach the desired endpoint. The last chapter outlines a set of uses for a stable inverse in control applications, including Linear Model Predictive Control (LMPC), and LMPC applied to Repetitive Control (RC-LMPC), and a generalized form of a one-step ahead control. An important characteristic is that this approach has the property of converging to zero tracking error in a small number of time steps, which is finite time convergence instead of asymptotic convergence as time tends to infinity

Sparse Time-Frequency decomposition for multiple signals with same frequencies

In this paper, we consider multiple signals sharing same instantaneous frequencies. This kind of data is very common in scientific and engineering problems. To take advantage of this special structure, we modify our data-driven time-frequency analysis by updating the instantaneous frequencies simultaneously. Moreover, based on the simultaneously sparsity approximation and fast Fourier transform, some efficient algorithms is developed. Since the information of multiple signals is used, this method is very robust to the perturbation of noise. And it is applicable to the general nonperiodic signals even with missing samples or outliers. Several synthetic and real signals are used to test this method. The performances of this method are very promising

From Model-Based to Data-Driven Discrete-Time Iterative Learning Control

This dissertation presents a series of new results of iterative learning control (ILC) that progresses from model-based ILC algorithms to data-driven ILC algorithms. ILC is a type of trial-and-error algorithm to learn by repetitions in practice to follow a pre-defined finite-time maneuver with high tracking accuracy. Mathematically ILC constructs a contraction mapping between the tracking errors of successive iterations, and aims to converge to a tracking accuracy approaching the reproducibility level of the hardware. It produces feedforward commands based on measurements from previous iterations to eliminates tracking errors from the bandwidth limitation of these feedback controllers, transient responses, model inaccuracies, unknown repeating disturbance, etc. Generally, ILC uses an a priori model to form the contraction mapping that guarantees monotonic decay of the tracking error. However, un-modeled high frequency dynamics may destabilize the control system. The existing infinite impulse response filtering techniques to stop the learning at such frequencies, have initial condition issues that can cause an otherwise stable ILC law to become unstable. A circulant form of zero-phase filtering for finite-time trajectories is proposed here to avoid such issues. This work addresses the problem of possible lack of stability robustness when ILC uses an imperfect a prior model. Besides the computation of feedforward commands, measurements from previous iterations can also be used to update the dynamic model. In other words, as the learning progresses, an iterative data-driven model development is made. This leads to adaptive ILC methods. An indirect adaptive linear ILC method to speed up the desired maneuver is presented here. The updates of the system model are realized by embedding an observer in ILC to estimate the system Markov parameters. This method can be used to increase the productivity or to produce high tracking accuracy when the desired trajectory is too fast for feedback control to be effective. When it comes to nonlinear ILC, data is used to update a progression of models along a homotopy, i.e., the ILC method presented in this thesis uses data to repeatedly create bilinear models in a homotopy approaching the desired trajectory. The improvement here makes use of Carleman bilinearized models to capture more nonlinear dynamics, with the potential for faster convergence when compared to existing methods based on linearized models. The last work presented here finally uses model-free reinforcement learning (RL) to eliminate the need for an a priori model. It is analogous to direct adaptive control using data to directly produce the gains in the ILC law without use of a model. An off-policy RL method is first developed by extending a model-free model predictive control method and then applied in the trial domain for ILC. Adjustments of the ILC learning law and the RL recursion equation for state-value function updates allow the collection of enough data while improving the tracking accuracy without much safety concerns. This algorithm can be seen as the first step to bridge ILC and RL aiming to address nonlinear systems