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Register a new userParsimonious Volterra System Identification
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In this short paper, we aim at developing algorithms for sparse Volterra system identification when the system to be identified has infinite impulse response. Assuming that the impulse response is represented as a sum of exponentials and given input-output data, the problem of interest is to find the simplest nonlinear Volterra model which is compatible with the a priori information and the collected data. By simplest, we mean the model whose impulse response has the least number of exponentials. The algorithms provided are able to handle both fragmented data and measurement noise. Academic examples at the end of paper show the efficacy of proposed approach.
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A simple nonlinear system modeling algorithm designed to work with limited emph{a priori }knowledge and short data records, is examined. It creates an empirical Volterra series-based model of a system using an $l_{q}$-constrained least squares algorithm with $qgeq 1$. If the system $mleft( cdot right) $ is a continuous and bounded map with a finite memory no longer than some known $tau$, then (for a $D$ parameter model and for a number of measurements $N$) the difference between the resulting model of the system and the best possible theoretical one is guaranteed to be of order $sqrt{N^{-1}ln D}$, even for $Dgeq N$. The performance of models obtained for $q=1,1.5$ and $2$ is tested on the Wiener-Hammerstein benchmark system. The results suggest that the models obtained for $q>1$ are better suited to characterize the nature of the system, while the sparse solutions obtained for $q=1$ yield smaller error values in terms of input-output behavior.
The paper introduces a novel methodology for the identification of coefficients of switched autoregressive linear models. We consider the case when the systems outputs are contaminated by possibly large values of measurement noise. It is assumed that only partial information on the probability distribution of the noise is available. Given input-output data, we aim at identifying switched system coefficients and parameters of the distribution of the noise which are compatible with the collected data. System dynamics are estimated through expected values computation and by exploiting the strong law of large numbers. We demonstrate the efficiency of the proposed approach with several academic examples. The method is shown to be extremely effective in the situations where a large number of measurements is available; cases in which previous approaches based on polynomial or mixed-integer optimization cannot be applied due to very large computational burden.
In this paper, we study the system identification problem for sparse linear time-invariant systems. We propose a sparsity promoting block-regularized estimator to identify the dynamics of the system with only a limited number of input-state data samples. We characterize the properties of this estimator under high-dimensional scaling, where the growth rate of the system dimension is comparable to or even faster than that of the number of available sample trajectories. In particular, using contemporary results on high-dimensional statistics, we show that the proposed estimator results in a small element-wise error, provided that the number of sample trajectories is above a threshold. This threshold depends polynomially on the size of each block and the number of nonzero elements at different rows of input and state matrices, but only logarithmically on the system dimension. A by-product of this result is that the number of sample trajectories required for sparse system identification is significantly smaller than the dimension of the system. Furthermore, we show that, unlike the recently celebrated least-squares estimators for system identification problems, the method developed in this work is capable of textit{exact recovery} of the underlying sparsity structure of the system with the aforementioned number of data samples. Extensive case studies on synthetically generated systems, physical mass-spring networks, and multi-agent systems are offered to demonstrate the effectiveness of the proposed method.
This paper describes the design and implementation of linear controllers with a switching condition for a nonlinear hot air blower system (HABS) process trainer PT326. The system is interfaced with a computer through a USB based data acquisition module and interfacing circuitry. A calibration equation is implemented through computer in order to convert the amplified output of the sensor to temperature. Overall plant is nonlinear; therefore, system identification is performed in three different regions depending upon the plant input. For these three regions, three linear controllers are designed with closed-loop system having small rise time, settling time and overshoot. Switching of controllers is based on regions defined by plant input. In order to avoid the effect of discontinuity, due to switching of linear controllers, parameters of all linear controllers are taken closer to each other. Finally, discretized controllers along with switching condition are implemented for the plant through computer and practical results are demonstrated.
Input design is an important issue for classical system identification methods but has not been investigated for the kernel-based regularization method (KRM) until very recently. In this paper, we consider in the time domain the input design problem of KRMs for LTI system identification. Different from the recent result, we adopt a Bayesian perspective and in particular make use of scalar measures (e.g., the $A$-optimality, $D$-optimality, and $E$-optimality) of the Bayesian mean square error matrix as the design criteria subject to power-constraint on the input. Instead to solve the optimization problem directly, we propose a two-step procedure. In the first step, by making suitable assumptions on the unknown input, we construct a quadratic map (transformation) of the input such that the transformed input design problems are convex, the number of optimization variables is independent of the number of input data, and their global minima can be found efficiently by applying well-developed convex optimization software packages. In the second step, we derive the expression of the optimal input based on the global minima found in the first step by solving the inverse image of the quadratic map. In addition, we derive analytic results for some special types of fixed kernels, which provide insights on the input design and also its dependence on the kernel structure.
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