Patricio E. Valenzuela

Optimal input design for non-linear dynamic systems: A graph theory approach

In this article a new algorithm for the design of stationary input sequences for system identification is presented. The stationary input signal is generated by optimizing an approximation of a scalar function of the information matrix, based on stationary input sequences generated from prime cycles, which describe the set of finite Markov chains of a given order. This method can be used for solving input design problems for nonlinear systems. In particular it can handle amplitude constraints on the input. Numerical examples show that the new algorithm is computationally attractive and that is consistent with previously reported results.

A graph theoretical approach to input design for identification of nonlinear dynamical models

In this paper the problem of optimal input design for model identification is studied. The optimal input signal is designed by maximizing a scalar cost function of the information matrix, where the input signal is a realization of a stationary process with finite memory, with its range being a finite set of values. It is shown that the feasible set for this problem can be associated with the prime cycles in the graph of possible values and transitions for the input signal. A realization of the optimal input signal is generated by running a Markov chain associated with the feasible set, where the transition matrix is built using a novel algorithm developed for de Bruijn graphs. The proposed method can be used to design inputs for nonlinear output-error systems, which are not covered in previous results. In particular, since the input is restricted to a finite alphabet, it can naturally handle amplitude constraints. Finally, our approach relies on convex optimization even for systems having a nonlinear structure. A numerical example shows that the algorithm can be successfully used to perform input design for nonlinear output-error models.

A graph/particle-based method for experiment design in nonlinear systems

Abstract We propose an extended method for experiment design in nonlinear state space models. The proposed input design technique optimizes a scalar cost function of the information matrix, by computing the optimal stationary probability mass function (pmf) from which an input sequence is sampled. The feasible set of the stationary pmf is a polytope, allowing it to be expressed as a convex combination of its extreme points. The extreme points in the feasible set of pmfs can be computed using graph theory. Therefore, the final information matrix can be approximated as a convex combination of the information matrices associated with each extreme point. For nonlinear systems, the information matrices for each extreme point can be computed by using particle methods. Numerical examples show that the proposed technique can be successfully employed for experiment design in nonlinear systems.

A novel input design approach for systems with quantized output data

In this paper, we explore the problem of input design for systems with quantized measurements. For the input design problem, we calculate and optimize a function of the Fisher Information Matrix (FIM). The calculation of the FIM is greatly simplified by using known relationships of the derivative of the likelihood function, and the auxiliary function arising from the Expectation Maximization (EM) algorithm. To optimize the FIM, we design an experiment using a recently published method based on graph theory. A numerical example shows that the proposed experiment can be successfully used in quantized systems.

Optimal tracking performance for unstable tall plant models

This article focuses on the best achievable tracking performance of unstable tall plant models. The work is presented for discrete time, LTI systems, when an exponentially decaying signal is considered as reference. Closed form expressions for the best tracking performance for one and two degree of freedom control architectures are presented. As an application of those results, they are used in an example to compute the performance gains in the control of originally tall systems which have been squared-up by adding control input channels.

Performance bounds for SIMO and squared-up plant models

This paper presents performance bounds in the control of SIMO plant models, and the variation of those bounds when input channels are added to the plant so as to make it square. Our work focuses on discrete-time, linear time invariant (LTI) MIMO systems, minimizing the squared regulation error when an impulsive input disturbance is injected into the plant. Closed form expressions for the optimal regulation performance are developed, and the sensitivity of the performance to channel addition is also investigated. Finally, a numerical example is presented to illustrate the main results of the paper.

Applications oriented input design for closed-loop system identification: a graph-theory approach

A new approach to experimental design for identification of closed-loop models is presented. The method considers the design of an experiment by minimizing experimental cost, subject to probabilistic bounds on the input and output signals, and quality constraints on the identified model. The input and output bounds are common in many industrial processes due to physical limitations of actuators. The aforementioned constraints make the problem non-convex. By assuming that the experiment is a realization of a stationary process with finite memory and finite alphabet, we use results from graph-theory to relax the problem. The key feature of this approach is that the problem becomes convex even for non-linear feedback systems. A numerical example shows that the proposed technique is an attractive alternative for closed-loop system identification.

Optimal Enforcement of Causality in Non-Parametric Transfer Function Estimation

Traditionally, non-parametric impulse and frequency response functions are estimated by taking the ratio of power spectral density estimates. However, this approach may often lead to non-causal estimates. In this letter, we derive a closed form expression for the impulse response estimator by smoothed empirical transfer function estimate, which allows optimal enforcement of causality on non-parametric estimators based on spectral analysis. The new method is shown to be asymptotically unbiased and of minimum covariance in a positive semidefinite sense among a broad class of linear estimators. Numerical simulations illustrate the performance of the new estimator.

On robust input design for nonlinear dynamical models

We present a method for robust input design for nonlinear state-space models. The method optimizes a scalar cost function of the Fisher information matrix over a set of marginal distributions of st ...

On Adaptive Boosting for System Identification

In the field of machine learning, the algorithm Adaptive Boosting has been successfully applied to a wide range of regression and classification problems. However, to the best of the authors’ knowledge, the use of this algorithm to estimate dynamical systems has not been exploited. In this brief, we explore the connection between Adaptive Boosting and system identification, and give examples of an identification method that makes use of this connection. We prove that the resulting estimate converges to the true underlying system for an output-error model structure under reasonable assumptions in the large sample limit and derive a bound of the model mismatch for the noise-free case.