Jari J. Hätönen

An algebraic approach to iterative learning control

In this paper discrete-time iterative learning control (ILC) systems are analysed from an algebraic point of view. The algebraic analysis shows that a linear-time invariant single-input–single-output model can always represented equivalently as a static multivariable plant due to the finiteness of the time-axis. Furthermore, in this framework the ILC synthesis problem becomes a tracking problem of a multi-channel step-function. The internal model principle states that for asymptotic tracking (i.e. convergent learning) it is required that an ILC algorithm has to contain an integrator along the iteration axis, but at the same time the resulting closed-loop system should be stable. The question of stability can then be answered by analysing the closed-loop poles along the iteration axis using standard results from multivariable polynomial systems theory. This convergence theory suggests that time-varying ILC control laws should be typically used instead of time-invariant control laws in order to guarantee good transient tracking behaviour. Based on this suggestion a new adaptive ILC algorithm is derived, which results in monotonic convergence for an arbitrary linear discrete-time plant. This adaptive algorithm also has important implications in terms of future research work—as a concrete example it demonstrates that ILC algorithms containing adaptive and time-varying components can result in enhanced convergence properties when compared to fixed parameter ILC algorithms. Hence it can be expected that further research on adaptive learning mechanisms will provide a new useful source of high-performance ILC algorithms.

Norm-Optimal Iterative Learning Control Applied to Gantry Robots for Automation Applications

This paper is concerned with the practical implementation of the norm-optimal iterative learning control (NOILC) algorithm. Here, the complexity of this algorithm is first considered with respect to real-time control applications, and a new modified version, fast norm-optimal ILC (F-NOILC), is derived for this application, which potentially allows implementation with a sampling rate three times faster that the original algorithm. A performance index is used to assess the experimental results obtained from applying F-NOILC to an industrial gantry robot system and, in particular, the effects of varying the parameters in the cost function, which is at the heart of the norm-optimal approach

Active vibration isolation in a “smart spring” mount using a repetitive control approach

In a variety of different engineering systems there is a requirement to isolate sensitive equipment from foundation vibration or alternatively, isolate the foundation from machinery vibration. Passive solutions to this problem provide some isolation but performance is significantly degraded in the presence of structural compliance. A recently proposed hybrid active/passive solution known as the “Smart Spring” mounting system specifically addresses this problem of compliance. In earlier work on this system the required local controller was based on LQG design on the assumption that the vibration sources are random. The work reported here investigates the application of a repetitive control approach to deal with periodic vibration sources. The industrial potential of the approach has been shown using an experimental facility where isolation results in the region of 50dB have been achieved. Copyright

Discrete-time inverse model-based iterative learning control: stability, monotonicity and robustness

This paper considers the robustness of an inverse iterative learning control algorithm. A simple learning gain results in robust convergence that is monotonic in the least squares sense provided that the multiplicative plant uncertainty satisfies a matrix positivity requirement. The results are extended to the frequency domain using a simple graphical Nyquist test. The analysis extends naturally to a parameter-optimal control setting. Non-monotone convergence is considered by using a simple weighted norm based on exponential weighting of time series.

Newton method based iterative learning control for discrete non-linear systems

Significant progress has been achieved in terms of both theory and industrial applications of iterative learning control (ILC) in the past decade. However, the techniques of solving non-linear ILC problems are still under development. The main result of this paper is a novel non-linear ILC algorithm that utilizes the capability of the Newton method. By setting up links between non-linear ILC problems and non-linear multivariable equations, the Newton method is introduced into the ILC framework. The implementation of the new algorithm allows one to decompose a nonlinear ILC problem into a sequence of linear time-varying ILC problems. Simulations on a discrete non-linear system and a manipulator model display its advantages. Conditions for its semi-local convergence are analysed. Links of ILC with existing non-linear topics are pointed out as ways to construct new non-linear ILC schemes. Potential improvements are discussed for future work.

An Optimality-Based Repetitive Control Algorithm for Discrete-Time Systems

In this paper, we first demonstrate that the zeroth-order hold discretization of a class of well-known continuous-time repetitive control algorithms can result in instability. Having highlighted the potential problem that discretization can pose to controller structures of this form, a new optimality-based repetitive control algorithm is developed for discrete-time systems. Under mild technical conditions on the plant, the algorithm will result in asymptotic convergence for an arbitrary -periodic reference signal and an arbitrary discrete-time linear time-invariant plant. The algorithm has been practically applied to a nonminimum phase spring-mass-damper system. It is shown that it is capable of producing near perfect tracking within a very small number of trials. The robustness of the controller is established for this plant in terms of the bounds on the uncertainty that will allow the system to satisfy the Nyquist stability criterion. A zero-phase filter is designed and implemented. Experimental results illustrate how long-term performance can be maintained using the proposed method.

A new robust iterative learning control algorithm for application on a gantry robot

In this paper a new robust steepest-descent algorithm for discrete-time iterative learning control is introduced for plant models with multiplicative uncertainty. A theoretical analysis of the algorithm shows that if a tuning parameter in the algorithm is selected to be sufficiently large, the algorithm will result in monotonic convergence if the plant uncertainty satisfies a positivity condition. This is a major improvement when compared to the standard steepest-descent algorithm, which lacks a mechanism for finding a balance between convergence speed and robustness. Experimental work on a gantry robot is performed to demonstrate that the algorithm results in near perfect tracking in the limit.

Fast norm-optimal iterative learning control for industrial applications

Norm-optimal iterative learning control has potential to significantly increase the accuracy of many trajectory tracking tasks which can be found in industry. The algorithm can achieve very low levels of tracking error and the number of iterations required to reach minimal error is small compared to many other iterative learning control algorithms. However, in the current format, the algorithm is not attractive to industry because it requires a large number of calculations to be performed at each sample instant. This implies that control hardware must be very fast which is expensive, or that the sample frequency must be reduced which can result in reduced performance. To remedy these problems, a revised version, fast norm-optimal iterative learning control is proposed which is significantly simpler and faster to implement. The new version is tested both in simulation and in practice on a three axis industrial gantry robot.

Convex modifications to an iterative learning control law

In this paper, a predictive norm-optimal iterative learning control algorithm from Amann, Owens, and Rogers (Int. J. Control 69 (2) (1998) 203-226) is analyzed. The main new result of this is that any of the predictive inputs from the predictive algorithm can be used in the control of the plant. This results in a faster convergence rate than that obtained with the approach proposed by Amann, Owens, and Rogers. Furthermore, the nature of the convergence of this new scheme is analysed in detail in terms of the free parameters of the algorithm.

Discrete Fourier Transform Based Iterative Learning Control Design for Linear Plants With Experimental Verification

A general class of iterative learning control law is examined using the Discrete Fourier Transform and it is shown that if the nominal plant satisfies a given uncertainty condition, there exist algorithms that are capable of driving the tracking error monotonically to zero. The effect of the filters appearing in the algorithm on the convergence rate is then examined using a bi-linear mapping into the space of plant uncertainty. A weighting function for the convergence rate is subsequently specified as a design parameter, and it is shown that the filters can be chosen to maximise the weighted convergence rate over a given region of uncertainty space. This permits the designer to manipulate the convergence and robustness properties of the algorithm in a straightforward manner. It is then demonstrated how the change of input over successive trials and the residual error may also be incorporated into the cost function. Experimental results are presented using a non-minimum phase test facility to show the effectiveness of the design method.