Worst-case system identification in l/sub 1/: error bounds, optimal models and model reduction

A time domain identification procedure is developed which yields upper bounds for the l1- and H∞-norm of the model error for a given nominal model. In the procedure use is made of measurement data and a priori information, consisting of a time domain bound on the noise and information about the decay-rate of the pulse response of the model error. The upper bounds are determined by parametrizing the model error and calculating the worst-case model error by solving a set of linear programming problems.

Historically, one of the basic issues in control systems design has been robustness: the ability of a controlled plant to withstand variations in or lack of knowledge of its dynamics. Even if the dynamics of a system are accurately known for purposes of implementation, it is often desirable to design a control system based on a simplified model. Consequently it is essential to be able to guarantee a reasonable performance not only for the nominal plant, but also for its neighbouring perturbations: this is the issue of robustness. Since the beginning of this decade major advances have been made in this area, notably using the H -approach; this term is meant to cover the solution of sensitivity reduction, approximation and model reduction, robustness and related control design problems using the mathematics of Hardy spaces and related areas in Harmonic Analysis. This book contains the proceedings of the NATO Advanced Research Workshop on Modelling, Robustness and Sensitivity Reduction in Control Systems, which was held at the University of Groningen, December 1986. Its aim was to explore the development of H -design techniques and its ramifications in Systems Theory in a unified and systematic way with the emphasis on recent advances and future directions in this fast developing area. In particular the following inter-related aspects were addressed: H -mathematical foundations, model approximation and robustness in control design, optimal sensitivity reduction, modelling and system identification and signal processing.

Identification of linear models in view of robust control design requires the identification of a control‐relevant nominal model, and a quantification of model uncertainty. In this paper a procedure is presented to quantify the model uncertainty of any prespecified nominal model, from a sequence of measurement data of input and output signals from a plant. By employing a nonparametric empirical transfer function estimate (ETFE), we are able to split the model uncertainty into three parts: the inherent uncertainty in the data due to data imperfections, the unmodelled dynamics in the nominal model, and the uncertainty due to interpolation. A frequency‐dependent hard error bound is constructed, and results are given for tightening the bound through appropriate input design.

This paper deals with uncertain model structures, model validation and robust performance analysis of active magnetic bearings. The dynamics of active magnetic bearing systems are characterized by their instability and complex dynamics of rotor and electromagnets. The feedback control is indispensable to stabilize the system, further the closed-loop systems of magnetic bearings should have robustness for stability and performance against model uncertainties. First we derive a nominal mathematical model of AMBs as a linear state-space model under some assumption and idealization, then we consider the discrepancy between the real physical systems and the obtained nominal design model. This discrepancy can be expressed as the structured uncertainties by Linear Fractional Transformation. These uncertainties include linearization error, parametric uncertainties, unmodeled dynamics, and gyroscopic effect. Then we set the interconnection structure which contains the above structurally represented uncertainties. Next we design a robust controller which achieves robust performance condition. Finally, we validate the interconnection structure with the nominal model and uncertainties, and analyze the robustness of stability and performance of the closed-loop system via the mixed structured singular value.

We propose a numerical method of frequency-weighted model reduction. The model to be reduced (an original model) is a stable SISO discrete-time model described by high-order state-space equations. We design the reduced-order model so that it can interpolate 1st- and 2nd-order information of the original model at complex frequency points (interpolation points) in the unit circle. The characteristics of the reduced-order model greatly depend on the choice of the interpolation points. The proposed model reduction method is a numerical one that chooses the interpolation points by searching in the unit circle to find the reduced-order model such that L∞-norm of the reduction error is less than a prescribed value. This method has the following features that show that it is an effective numerical method of the frequency-weighted model reduction. i) The reduced-order model is guaranteed to be stable. ii) The procedure for finding the reduced-order model is simple and requires a relatively small amount of computation. iii) The order of the reduced-order model can be controlled by choosing the number of interpolation points. © 1998 Scripta Technica, Electr Eng Jpn, 126(2): 31–39, 1999

The three-tank (3-tank) liquid-level control system commonly exists in industries under the proportional-integral-derivative (PID) control loop. However, the proportional-integral-derivative-accelerated (PIDA) can yield better responses than the PID for higher order plant. In this paper, the robust PIDA controller design by the cuckoo search (CS) for the 3-tank liquid-level control system is proposed. Results obtained will be compared with the PID controller designed by the CS. As results, the responses of the 3-tank liquid-level controlled system with the robust PIDA controller optimally designed by the CS are better than the PID controller designed by the CS. Due to model uncertainty within ±10% of each parameter of the nominal plant model, the robust PIDA controller optimized by the CS can provide the very satisfactory responses in both time and frequency domains with robust performance and robust stability.

This paper deals with the multi-degree cyclic single-hoist scheduling problem with time window constraints, in which multiple identical parts enter and leave the system during each cycle. We propose an analytical mathematical model and a branch-and-bound algorithm so as to find a cyclic sequence of hoist moves that maximises the throughput. The branch-and-bound algorithm implicitly enumerates the sequence of hoist moves and requires the solution of a specific set of linear programming problems (LPPs). Computational results on benchmark instances and randomly generated test instances are presented.

A time Petri net is said to be critically observable at a given time instant if the markings consistent with any observation at the time instant are included either in the set of critical markings or non-critical markings. This work studies the verification problem of critical observability for timed discrete event systems modeled by bounded labeled time Petri nets. The proposed method is a two-fold process: a preliminary verification of critical observability for the underlying logic labeled Petri net and a further verification considering the time constraint associated with each transition. The first step is based on the concurrent composition of a reachability graph of the logic net. If the logic net is critically observable, then the time net is also critically observable at any given time instant. Otherwise, the second step is to design an algorithm to compute all pairs of transition-class sequences that violate critical observability at the given time instant, and then a set of linear programming problems is exploited to check critical observability for the corresponding timed system. <i>Note to Practitioners</i>&#x2014;Timed discrete event systems provide a theoretical model for safety-critical real applications such as air traffic management, smart grid, and industrial control, which are vulnerable to malicious attack and destruction at some particular time instants such that a system may be misled to dangerous states. Critical observability of a timed discrete event system is a property with which the predefined dangerous states can be determined and detected from the observation by an observer at a given time instant. This research aims to offer a systematic approach to check critical observability for timed discrete event systems modeled by bounded labeled time Petri nets, which can also present some new ideas and insights for practitioners in the field of safety-critical systems.

A projection and division method for problems of linear dynamic programming

A water distribution system often consists of a large number of components. However, funds are, in general, not available to completely update/upgrade the deteriorating water distribution system's components. Limited funds have to be optimally distributed among the system's components on the basis of the contribution of each component to the overall system's availability. The large dimension and high degree of nonlinearity of models of deteriorating large‐scale water distribution networks make the maintenance‐related problem difficult to handle as a whole. In this paper a multilevel approach is proposed for water distribution systems that maximizes the availability of the overall system under a given cost constraint. This decomposition method tackles a large‐scale nonlinear programming problem by solving a set of linear programming problems with a smaller dimension iteratively, thus greatly reducing a large‐scale system's complexity.

Gradient-based non-linear microstructure design

Optimal adjustment algorithm for p coordinates is a generalization of the optimal pair adjustment algorithm for linear programming, which in turn is based on von Neumann’s algorithm. Its main advantages are simplicity and quick progress in the early iterations. In this work, to accelerate the convergence of the interior point method, few iterations of this generalized algorithm are applied to the Mehrotra’s heuristic, which determines the starting point for the interior point method in the PCx software. Computational experiments in a set of linear programming problems have shown that this approach reduces the total number of iterations and the running time for many of them, including large-scale ones.

In the present paper, it will be reported original results concerning the application of Neural Networks (NN) in mobile robot in trajectory tracking control. This work combines a feedback linearization based on a nominal model and an NN adaptive dynamic compensation. In mobile robot with uncertain dynamic parameters, two controllers are implemented separately: a kinematic controller and an inverse dynamic controller. The uncertainty in the nominal dynamic model is compensated by a neural adaptive feedback controller. The resulting adaptive controller is efficient and robust in the sense that it succeeds to achieve a good tracking performance with a small computational effort. The learning laws were deduced by Lyapunov's stability analysis. Finally, the performance of the control system is verified through experiments.

Given a nominal statistical model, we consider the minimax estimation problem consisting of finding the best least-squares estimator for the least favorable statistical model within a neighborhood of the nominal model. The neighborhood is formed by placing a bound on the Kullback-Leibler (KL) divergence between the actual and nominal models. For a Gaussian nominal model and a finite observations interval, or for a stationary Gaussian process over an infinite interval, the usual noncausal Wiener filter remains optimal. However, the worst case performance of the filter is affected by the size of the neighborhood representing the model uncertainty. On the other hand, standard causal least-squares estimators are not optimal, and a characterization is provided for the causal estimator and the corresponding least favorable model. The causal estimator takes the form of a risk-sensitive estimator with an appropriately selected risk sensitivity coefficient.

Estimation of Frequency Domain Model Uncertainties with Application to Robust Controller Design