Roland Longchamp

Modeling and set point control of closed-chain mechanisms: theory and experiment

We derive a reduced model, that is, a model in terms of independent generalized coordinates, for the equations of motion of closed-chain mechanisms. We highlight the fact that the model has two special characteristics which make it different from models of open-chain mechanisms. First, it is defined locally in the generalized coordinates. We therefore characterize the domain of validity of the model in which the mechanism satisfies the constraints and is not in a singular configuration. Second, it is an implicit model, that is, parts of the equations of motion are not expressed explicitly. Despite the implicit nature of the equations of motion, we show that closed-chain mechanisms still satisfy a skew symmetry property, and that proportional derivative (PD)-based control with so-called simple gravity compensation guarantees (local) asymptotic stability. We discuss the computational issues involved in the implementation of the proposed controller. The proposed modeling and PD control approach is illustrated experimentally using the Rice planar delta robot which was built to experiment with closed-chain mechanisms.

Robust control of polytopic systems by convex optimization

Robust control synthesis of linear time-invariant SISO polytopic systems is investigated using the polynomial approach. A convex set of all stabilizing controllers for a polytopic system is given over an infinite-dimensional space. A finite-dimensional approximation of this set is obtained using the orthonormal basis functions and represented by a set of LMIs thanks to the KYP lemma. Then, an LMI based convex optimization problem for robust pole placement with sensitivity function shaping in two- and infinity-norm is proposed. The simulation results show the effectiveness of the proposed method.

PID controller tuning using Bode's integrals

A new method for PID controller tuning based on Bodes integrals is proposed. It is shown that the derivatives of amplitude and phase of a plant model with respect to frequency can be approximated by Bodes integrals without any model of the plant. This information can be used to design a PID controller for slope adjustment of the Nyquist diagram and improve the closed-loop performance. Besides, the derivatives can be also employed to estimate the gradient and the Hessian of a frequency criterion in an iterative PID controller tuning method. The frequency criterion is defined as the sum of squared errors between the desired and measured gain margin, phase margin and crossover frequency. The method benefits from specific feedback relay tests to determine the gain margin, the phase margin and the crossover frequency of the closed-loop system. Simulation examples and experimental results illustrate the effectiveness and the simplicity of the proposed method to design and tune the PID controllers.

Influence of zero locations on the number of step-response extrema

Abstract A new bounding theorem for the number of extrema that may occur in the step-response of a stable linear system is presented. The derivation of an easily-computed upper bound is given to complement literature results which have previously established the existence of a lower bound. The theorem requires knowledge of the pole-zero configuration of the transfer-function and is applicable to stable systems with real zeros and real poles.

On the Consistency of Certain Identification Methods for Linear Parameter Varying Systems

The consistency of certain identification methods for Linear Parameter Varying systems is considered. More precisely, methods for the identification of SISO input-output models are analysed. In order to perform a consistency analysis the application of ergodicity is required, which is not obviously applicable with these types of time-varying systems. It is therefore shown that, when the scheduling parameter satisfies certain conditions, ergodicity type results can be applied to the methods considered. An analysis is then carried out for two cases: that of noise-free measurements of the scheduling parameter, and then the more realistic case of noisy scheduling parameter measurements. The latter is shown to be an errors-in-variables type problem. In both cases the least squares technique is shown to typically give biased estimates and the instrumental variables method is proposed as a way of resolving this. The analysis carried out is reinforced by results in simulation.

Fixed-Order Controller Design for Polytopic Systems Using LMIs

Convex parameterization of fixed-order robust stabilizing controllers for systems with polytopic uncertainty is represented as a linear matrix inequality (LMI) using the Kalman-Yakubovich-Popov (KYP) lemma. This parameterization is a convex inner approximation of the whole nonconvex set of stabilizing controllers, and depends on the choice of a central polynomial. It is shown that, with an appropriate choice of the central polynomial, the set of all stabilizing fixed-order controllers that place the closed-loop poles of a polytopic system in a disk centered on the real axis can be outbounded with some LMIs. These LMIs can be used for robust pole placement of polytopic systems.

A statistical analysis of certain iterative learning control algorithms

Iterative learning control (ILC) is a technique used to improve the tracking performance of systems carrying out repetitive tasks, which are affected by deterministic disturbances. The achievable performance is greatly degraded, however, when non-repeating, stochastic disturbances are present. This paper aims to compare a number of different ILC algorithms, proposed to be more robust to the presence of these disturbances, first by a statistical analysis and then by simulation results and their application to a linear motor. New expressions for the expected value and variance of the controlled error are developed for each algorithm. The different algorithms are then tested in simulation and finally applied to the linear motor system to test their performance in practice. A filtered ILC algorithm is proposed when the noise and desired output spectra are separated. Otherwise an algorithm with a decreasing gain gives good robustness to noise and achievable precision but at a slower convergence rate.

Influence of Zero Locations on the Number of Extrema in the Step Response

Despite the power of modern control techniques, the problem of determining the number of extrema expected in the step response of a transfer function remains open. This note gives an easily computed upper bound for the number of extrema. In several cases this bound is sufficient to determine whether the step response of a closed-loop system will display overshoot. The analysis also permits determining the existence of inverse-response behavior. Criteria helpful in a pole-placement design context are given.

A closed form inverse dynamics model of the delta parallel robot

Abstract A complete kinematics and inverse dynamics model in closed form of the Delta Parallel Robot is devel- opped. The modelling approach uses only inverse kinematics and Newton laws to obtain an inverse dynamics model called “in the two spaces” since it is parametrized by the robot’s state in both task space and joint space.The proposed modelling process can be applied to all fully parallel robot structures and it is shown that the number of computations required to evaluate the model is not larger than the standard case of a serial robot.

Iterative Learning Control based on Stochastic Approximation

In this paper stochastic approximation theory is used to produce Iterative Learning Control (ILC) algorithms which are less sensitive to stochastic disturbances, a typical problem for the learning process of standard ILC algorithms. Two algorithms are developed, one to obtain zero mean controlled error and one to minimise the mean 2-norm of the controlled error. The former requires a certain knowledge of the system but in the presence of noise can give reasonably rapid convergence. The latter can either use a model or be model free by employing a second experiment.