Diego Eckhard

Brief paper: Virtual Reference Feedback Tuning for non-minimum phase plants

Model Reference control design methods fail when the plant has one or more non minimum phase zeros that are not included in the reference model, leading possibly to an unstable closed loop. This is a very serious problem for data-based control design methods where the plant is typically unknown. For Iterative Feedback Tuning a procedure was proposed in [1] to overcome this difficulty. In this paper we extend this idea to Virtual Reference Feedback Tuning, another data-based control design method. We present a simple two-step procedure that can cope with the situation where the unknown plant may or may not have non minimum phase zeros.

On the Tracking Problem for Linear Systems subject to Control Saturation

Abstract This paper addresses the problem of tracking constant references for linear systems subject to control saturation. Considering an unitary output feedback loop, containing an integral action, conditions in LMI form are proposed to compute a state feedback and an integrator anti-windup gain. These conditions ensure that the trajectories of the closed-loop system are bounded in an invariant ellipsoidal set, provided that the initial conditions are taken in this set and the references and the disturbances belong to a certain admissible set. Based on these conditions, optimization problems aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances are proposed.

Robust convergence of the steepest descent method for data-based control

Iterative data-based controller tuning consists of iterative adjustment of the controller parameters towards the parameter values which minimise an H 2 performance criterion. The convergence to the global minimum of the performance criterion depends on the initial controller parameters and on the step size of each iteration. This article presents convergence properties of iterative algorithms when they are affected by disturbances.

Input design as a tool to improve the convergence of PEM

The Prediction Error Method (PEM) is related to an optimization problem built on input/output data collected from the system to be identified. It is often hard to find the global solution of this optimization problem because the corresponding objective function presents local minima and/or the search space is constrained to a nonconvex set. The shape of the cost function, and hence the difficulty in solving the optimization problem, depends directly on the experimental conditions, more specifically on the spectrum of the input/output data collected from the system. Therefore, it seems plausible to improve the convergence to the global minimum by properly choosing the spectrum of the input; in this paper, we address this problem. We present a condition for convergence to the global minimum of the cost function and propose its inclusion in the input design. We present the application of the proposed approach to case studies where the algorithms tend to get trapped in nonglobal minima.

On the convergence of the Prediction Error Method to its global minimum

The Prediction Error Method (PEM) is related to an optimization problem built on input/output data collected from the system to be identified. It is often hard to find the global solution of this optimization problem because the corresponding objective function presents local minima and/or the search space is constrained to a nonconvex set. The existence of local minima, and hence the difficulty in solving the optimization, depends mainly on the experimental conditions, more specifically on the spectrum of the input/output data collected from the system. It is therefore possible to avoid the existence of local minima by properly choosing the spectrum of the input; in this paper we show how to perform this choice. We present sufficient conditions for the convergence of PEM to the global minimum and from these conditions we derive two approaches to avoid the existence of nonglobal minima. We present the application of one of these two approaches to a case study where standard identification toolboxes tend to get trapped in nonglobal minima.

Data-driven model reference control design by prediction error identification ☆

Abstract This paper deals with Data-Driven (DD) control design in a Model Reference (MR) framework. We present a new DD method for tuning the parameters of a controller with a fixed structure. Because the method originates from embedding the control design problem in the Prediction Error identification of an optimal controller, it is baptized as Optimal Controller Identification (OCI). Incorporating different levels of prior information about the optimal controller leads to different design choices, which allows to shape the bias and variance errors in its estimation. It is shown that the limit case where all available prior information is incorporated is tantamount to model-based design. Thus, this methodology also provides a framework in which model-based design and DD design can be fairly and objectively compared. This comparison reveals that DD design essentially outperforms model-based design by providing better bias shaping, except in the full order controller case, in which there is no bias and model-based design provides smaller variance. The practical effectiveness of the design methodology is illustrated with experimental results.

Model Reference Control Design by Prediction Error Identification

Abstract This paper studies a one-shot (non-iterative) data-based method for Model Reference (MR) control design. It shows that the optimal controller can be obtained as the solution of a Prediction Error (PE) identification problem that directly estimates the controller parameters through a reparametrization of the input-output model. The standard tools of PE Identification can thus be used to analyze the statistical properties (bias and variance) of the estimated controller. It also shows that, for MR control design, direct and indirect data-based methods are essentially equivalent.

On the global convergence of identification of output error models

Abstract The Output Error Method is related to an optimization problem based on a multimodal criterion. Iterative algorithms like the steepest descent are usually used to look for the global minimum of the criterion. These algorithms can get stuck at a local minimum. This paper presents sufficient conditions about the convergence of the steepest descent algorithm to the global minimum of the cost function. Moreover, it presents constraints to the input spectrum which ensure that the convergence conditions are satisfied. These constraints are convex and can easily be included in an experiment design approach to ensure the convergence of the iterative algorithms to the global minimum of the criterion.

Data-based controller tuning: Improving the convergence rate

Data-based control design methods most often consist of iterative adjustment of the controllers parameters towards the parameter values which minimize an H2 performance criterion. Typically, batches of input-output data collected from the system are used to feed directly a gradient descent optimization - no process model is used. The convergence to the global minimum of the performance criterion depends on the initial controller parameters, as well as on the size and direction of the steps taken at each iteration. This paper discusses these issues and provides a method for choosing the search direction and the step size at each optimization step so that convergence to the global minimum is obtained with high convergence rate.

Cost function shaping of the output error criterion

Identification of an output error model using the prediction error method leads to an optimization problem built on input/output data collected from the system to be identified. It is often hard to ...