Gabriel Solari

Asymptotic accuracy of iterative feedback tuning

Iterative feedback tuning (IFT) is a widely used procedure for controller tuning. It is a sequence of iteratively performed special experiments on the plant interlaced with periods of data collection under normal operating conditions. In this note, we derive the asymptotic convergence rate of IFT for disturbance rejection, which is one of the main fields of application.

Unbiased estimation of the Hessian for iterative feedback tuning (IFT)

Iterative feedback tuning (IFT) is a data-based method for the optimal tuning of a low order controller. The tuning of the controller parameters is performed iteratively, using a generalized Robbins-Monro type gradient descent scheme. An update step of the controller parameters is performed at each iteration on the basis of data obtained partly during normal operating conditions and partly from some special experiments. These data come from the closed loop system with the current controller. This paper presents a simple improvement to the IFT scheme: it is shown that one can compute an unbiased estimate of the Hessian on the basis of additional experiments on the closed loop system.

Prefiltering in iterative feedback tuning: optimization of the prefilter for accuracy

Iterative feedback tuning (IFT) is a data-based method for the tuning of restricted complexity controllers. At each iteration, an update for the controller parameters is estimated from data obtained partly from the normal operation of the closed loop system and partly from a special experiment, in which the output signal obtained under normal operation is fed back at the reference input. The choice of a prefilter for the input data to the special experiment is a degree of freedom of the method. In this note, the prefilter is designed in order to enhance the accuracy of the IFT update. The optimal prefilter produces a covariance of the new controller parameter vector that is strictly smaller than the covariance obtained with the standard constant prefilter.

Identification for control: Optimal input intended to identify a minimum variance controller

It is well known that the quality of the parameters identified during an identification experiment depends on the applied excitation signal. Prediction error identification using full order parametric models delivers an ellipsoidal region in which the true parameters lie with some prescribed probability level. This ellipsoidal region is determined by the covariance matrix of the parameters. Input design strategies aim at the minimization of some measure of this covariance matrix. We show that it is possible to optimize the input in an identification experiment with respect to a performance cost function of a closed-loop system involving explicitly the dependence of the designed controller on the identified model. In the present contribution we focus on finding the optimal input for the estimation of the parameters of a minimum variance controller, without the intermediate step of first minimizing some measure of the model parameter accuracy. We do this in conjunction with using covariance formulas which are not asymptotic in the model order, which is rather new in the domain of optimal input design. The identification procedure is performed in closed-loop. Besides optimizing the input power spectrum for the identification experiment, we also address the question of optimality of the controller. It is a wide belief that the minimum variance controller should be the optimal choice, since we perform an experiment for designing a minimum variance controller. However, we show that this may not always be the case, but rather depends on the model structure.

Closed-loop optimal experiment design: The partial correlation approach

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant systems in closed loop. The optimisation is performed jointly over the controller and the external input. We use a partial correlation approach, i.e. we parameterize the set of “admissible controller” - “external input” pairs by a finite set of matrix-valued trigonometric moments. Our main contribution is twofold. First we derive a description of the set of admissible finite-dimensional moments by a linear matrix inequality. Optimal input design problems with semi-definite constraints and criteria which are linear in these moments can then be cast as semi-definite programs and solved by standard semi-definite programming packages. Secondly, we develop algorithms to recover the controller and the power spectrum of the external input from the optimal moment vector. This furnishes the user a complete and very general procedure to solve the input design problems of the considered class. Our results can be applied to multi-input multi-output systems, but for pedagogical reasons we present here the single-input single-output case. We also assume that the true system is in the model set.

Closed-loop Optimal Input Design: the Partial Correlation Approach

Abstract We consider optimal experiment design for parametric prediction error system identification of linear time-invariant multi-input multi-output systems in closed-loop. The optimization is performed jointly over the controller and the external input. We use a partial correlation approach, i.e. parametrize the set of admissible controller - external input pairs by a finite set of matrix-valued trigonometric moments. Our main contribution is to derive a description of the set of admissible finite-dimensional moment vectors by a linear matrix inequality. Optimal input design problems with constraints and criteria which are linear in these moments can then be cast as semi-definite programs and solved by standard semi-definite programming packages. Our results can be applied to most of the usual model structures, but we assume that the true system is in the model set.

Closed-Loop Optimal Experiment Design: Solution via Moment Extension

We consider optimal experiment design for parametric prediction error system identification of linear time-invariant multiple-input multiple-output systems in closed-loop when the true system is in the model set. The optimization is performed jointly over the controller and the spectrum of the external excitation, which can be reparametrized as a joint spectral density matrix. The optimal solution consists of first computing a finite set of generalized moments of this spectrum as the solution of a semi-definite program. A second step then consists of constructing a spectrum that matches this finite set of optimal moments and satisfies some constraints due to the particular closed-loop nature of the optimization problem. This problem can be seen as a moment extension problem under constraints. Here we first show that the so-called central extension always satisfies these constraints, leading to a constructive procedure for the optimal controller and excitation spectrum. We then show that one can construct a broader set of parametrized optimal solutions that also satisfy the constraints; the additional degrees of freedom can then be used to achieve additional objectives.

Optimal prefiltering in iterative feedback tuning 1

Abstract Iterative Feedback Tuning (IFT) is a widely used procedure for controller tuning. It is a sequence of iteratively performed special experiments on the plant interlaced with periods of data collection under normal operating conditions. In this paper we derive the asymptotic convergence rate of IFT for disturbance rejection, which is one of the main fields of application. Further we present a method to improve the convergence of IFT by prefiltering the input data for the special experiment. At each iteration step the optimal prefilter is computed from data collected under normal operating conditions of the plant.

Optimization of the prefilter in iterative feedback tuning for improved accuracy of the controller parameter update

Iterative feedback tuning (IFT) is a data-based method for the tuning of restricted complexity controllers. At each iteration, an update for the parameters of the controller is estimated from data obtained partly from the normal operation of the closed loop system and partly from a special experiment. The choice of a prefilter for the input data to the special experiment is a degree of freedom of the method. In the present contribution, the prefilter is designed in order to enhance the accuracy of the IFT update.