David H. Owens

Multivariable norm optimal iterative learning control with auxiliary optimisation

The paper describes a substantial extension of norm optimal iterative learning control (NOILC) that permits tracking of a class of finite dimensional reference signals whilst simultaneously converging to the solution of a constrained quadratic optimisation problem. The theory is presented in a general functional analytical framework using operators between chosen real Hilbert spaces. This is applied to solve problems in continuous time where tracking is only required at selected intermediate points of the time interval but, simultaneously, the solution is required to minimise a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. Applications to the discrete time case, including the case of multi-rate sampling, are also summarised. The algorithms are motivated by practical need and provide a methodology for reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear whilst maintaining the desired tracking accuracy necessary for task completion. Solutions in terms of NOILC methodologies involving both feedforward and feedback components offer the possibilities of greater robustness than purely feedforward actions. Results describing the inherent robustness of the feedforward implementation are presented and the work is illustrated by experimental results from a robotic manipulator.

An inverse-model approach to multivariable norm optimal iterative learning control with auxiliary optimisation

Motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval, a general optimisation-based iterative learning control (ILC) algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. In practice, the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear. The theory is developed using the well-known norm optimal ILC (NOILC) framework, using general linear, functional operators between real Hilbert spaces. Solutions are derived using feedforward action, convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions. Algorithms are specified for both continuous and discrete-time state-space representations, with the latter including application to multi-rate sampled systems. Experimental results using a robotic manipulator confirm the practical utility of the algorithms and the closeness with which observed results match theoretical predictions.

A Novel Design Framework for Point-to-Point ILC Using Successive Projection

A novel design approach is proposed for point-to-point iterative learning control (ILC), enabling system constraints to be satisfied while simultaneously addressing the requirement for high-performance tracking. It is shown that point-to-point ILC design can be formulated and solved using a successive projection first proposed by J. von Neumann, allowing a number of new point-to-point ILC algorithms to be developed and analyzed. To illustrate this framework, two new algorithms are derived with different convergence and computational properties for the constrained point-to-point ILC design problem. The proposed algorithms are validated on a robotic arm with experimental results demonstrating their effectiveness.

Parameter-optimal iterative learning control using polynomial representations of the inverse plant

Based on the observation that iterative learning control (ILC) can be based on the inverse plant but that the approach can be degraded by modelling errors, particularly at high frequencies, this article investigates the construction and properties of a multi-parameter parameter-optimal ILC algorithm that uses an approximate polynomial representation of the inverse with natural high-frequency attenuation. In its simplest form, the algorithm replicates the original work of Owens and Feng but, more generally, it is capable of producing significant improvements to the observed convergence rate. As the number of parameters increases, convergence rates approach that of the ideal plant inverse algorithm. Introducing compensation into the algorithm provides a formal link to previously published gradient and norm-optimal ILC algorithms and indicates that the polynomial approach can be regarded as approximations to those control laws. Simulation examples verify the theoretical performance predictions.

Influence of Nonminimum Phase Zeros on the Performance of Optimal Continuous-Time Iterative Learning Control

Iterative learning control can be applied to systems that execute the same tracking task over a finite time duration. An execution is known as a trial, and once each is complete, the system resets to the starting location and the next trial begins. All previous trial information is available for use in constructing the control input for the next trial, and the basic idea is to improve tracking performance from trial-to-trial. This brief analyzes the effects of nonminimum phase zeros on the trial-to-trial error norm convergence of norm optimal iterative learning control, a commonly used algorithm, for differential linear systems with supporting experimental results from a test facility.

Singular value distribution of non-minimum phase systems with application to iterative learning control

This paper provides a rigorous mathematical analysis on the singular value distributions of input-output matrices for discrete time non-minimum phase (NMP) systems. It is shown that when the time scale considered is sufficiently long, the input-output matrix of a NMP system has m infinitesimally small singular values, the rest of which are significantly large with a non-zero lower bound, where m is the number of NMP zeros in the NMP systems. It is the existence of these m nearly zero singular values that causes various difficulties in analysis and design for NMP systems. The corresponding singular vector spaces can also be characterised. The analysis results are further applied to a gradient-based iterative learning control algorithm to analyse a well-known problematic slow convergence phenomenon and numerical simulations are presented to verify the theoretical predictions.

Experimental verification of constrained iterative learning control using successive projection

In many practical applications, constraints are often present on, for example, the magnitudes of the control inputs. Recently, based on a novel successive projection framework, two constrained iterative learning control (ILC) algorithms were developed with different convergence properties and computational requirements. This paper investigates the effectiveness of these two methods experimentally on a gantry robot facility, which has been extensively used to test a wide range of linear model based ILC algorithms. The results obtained demonstrate the effectiveness of the algorithms in solving one form of the general constrained ILC problem.

Newton method based iterative learning control of the upper limb

A non-linear iterative learning control algorithm is used for the application of functional electrical stimulation to the human arm. The task is to track trajectories in the horizontal plane and stimulation is applied to the triceps muscle. A model of the system is first produced, and then the equations required to implement the control law are derived. Practical considerations are high-lighted and the issue of parameter selection is discussed. Experimental results are subsequently presented, and are used to confirm that the algorithm is capable of exhibiting robustness together with achieving a high level of performance when practically applied to a control problem.

Point-to-point ILC with accelerated convergence

This paper proposes a novel point-to-point iterative learning control (ILC) algorithm for high performance trajectory tracking applications. Based on a successive project formulation of the point-to-point ILC design problem, two point-to-point ILC design algorithms are derived: one algorithm reCovers the norm optimal point to point ILC algorithm with a desirable physical property of converging to the minimum norm (energy) solution, and the other one (interestingly) accelerates convergence speed which could lead to significant reduction in system configuration time/cost. Numerical results are provided to demonstrate the proposed algorithms effectiveness.