Multivariable Stochastic Systems Research Articles

Active fault tolerant control using state-space self-tuning control approach

This paper presents a new fault tolerant control scheme for unknown multivariable stochastic systems by modifying the conventional state-space self-tuning control approach. For the detection of faults, a quantitative criterion is developed by comparing the innovation process errors occurring in the Kalman filter estimation algorithm, which, for faulty system recovery, a weighting matrix resetting technique is developed by adjusting and resetting the covariance matrices of the parameter estimate obtained in the Kalman filter estimation algorithm to improve the parameter estimation of the faulty systems. The proposed method can effectively cope with partially abrupt and/or gradual system faults and/or input failures with fault detection. The modified state-space self-tuning control scheme can be applied to the multivariable stochastic faulty system without requiring prior knowledge of system parameters and noise properties.

Minimum entropy filtering for multivariate stochastic systems with non-Gaussian noises

In this note, a minimum entropy filtering algorithm is presented for a class of multivariate dynamic stochastic systems, which are represented by a set of time-varying difference equations and are subjected to the multivariate non-Gaussian stochastic inputs. Several new concepts including the hybrid random vector, hybrid probability and hybrid entropy are firstly established to describe the probabilistic property of the estimation errors. New relationships are provided between the probability density functions (PDFs) of the multivariate stochastic input and output for different mapping cases. Recursive algorithms are then proposed to design the real-time sub-optimal filter so that the hybrid entropy of the estimation error can be minimized. Finally, an improved algorithm is provided through the on-line tuning of the weighting matrices so as to guarantee the local stability of the error system.

Risk sensitive identification of linear stochastic systems

Risk-sensitive identification of AR-processes was first considered in [12]. The purpose of this paper is to extend this original approach to ARMA-processes and even to multi-variable linear stochastic systems. We provide a new definition of a risk-sensitive identification criterion. For this we first consider a recursive identification procedure which is parameterized by a weight-matrix K acting on the stochastic gradient. Using the asymptotic theory of recursive estimation a suitably scaled version of the error process will be approximated by a stationary Gaussian process, see Chapter 4.5, Part II of [1]. The new risk sensitive criterion will be defined in terms of this associated stationary Gaussian process in a familiar manner via an exponential-quadratic cost. The main result of the paper is the minimization of the proposed new criterion with respect to the weight-matrix K over a feasible set EK, see (22), where the cost function is known to be finite, Theorem 6.1. This results will then be extended to the case when minimization over a feasible set E°K is considered, see (26), on the complement of which the cost function is known to be infinite, Theorem 6.1. The starting point of our analysis is an expression of the cost function given in LEQG-theory, in particular a result of [10]. A new expression for the cost function will be also given, using stochastic realization theory, as the mutual information rate between two stochastic processes.

Linear filtering for bilinear stochastic differential systems with unknown inputs

Investigates the problem of state estimation for bilinear stochastic multivariable differential systems in the presence of an additional disturbance, whose statistics are completely unknown.. A linear filter is proposed, based on a suitable decomposition of the state of the bilinear system into two components. The first one is a computable function of the observations while the second component is estimated via a suitable linear filtering algorithm. No a priori information on the disturbance is required for the filter implementation. The proposed filter is robust with respect to the unknown input, in that the covariance of the estimation error is not affected by such input. Numerical simulations show the effectiveness of the proposed filter.

STATE-SPACE SELF-TUNING CONTROL FOR NONLINEAR STOCHASTIC AND CHAOTIC HYBRID SYSTEMS

This paper presents a new state-space self-tuning control scheme for adaptive digital control of continuous-time multivariable nonlinear stochastic and chaotic systems, which have unknown system parameters, system and measurement noises, and inaccessible system states. Instead of using the moving average (MA)-based noise model commonly used for adaptive digital control of linear discrete-time stochastic systems in the literature, an adjustable auto-regressive moving average (ARMA)-based noise model with estimated states is constructed for state-space self-tuning control of nonlinear continuous-time stochastic systems. By taking advantage of a digital redesign methodology, which converts a predesigned high-gain analog tracker/observer into a practically implementable low-gain digital tracker/observer, and by taking the non-negligible computation time delay and a relatively longer sampling period into consideration, a digitally redesigned predictive tracker/observer has been newly developed in this paper for adaptive chaotic orbit tracking. The proposed method enables the development of a digitally implementable advanced control algorithm for nonlinear stochastic and chaotic hybrid systems.

Vector ARMA estimation: a reliable subspace approach

A parameter estimation method for finite-dimensional multivariate linear stochastic systems, which is guaranteed to produce valid models approximating the true underlying system in a computational time of a polynomial order in the system dimension, is presented. This is achieved by combining the main features of certain stochastic subspace identification techniques with sound matrix Schur restabilizing procedures and multivariate covariance fitting, both of which are formulated as linear matrix inequality problems. All aspects of the identification method are discussed, with an emphasis on the two issues mentioned above, and examples of the overall performance are provided for two different systems.

Traditional and Robust Predictor-Based Control for Mixing Systems in Cement Plants

This paper presents a rational matrix represented model of homogenizing processes and their disturbances in a cement plant and also general features as a multivariable stochastic system. A control method and actually applied results are presented, which is composed of PI-controllers with Smith predictors with an optimum mixing calculation. From robustness performances necessary for the process, this paper also reports performance comparison results by simulation experiments when applying a robust Smith predictor based two-degree of freedom controller with the traditional control and also with the extended horizon adaptive control.

A New Suboptimal Approach to the Filtering Problem for Bilinear Stochastic Differential Systems

The aim of this paper is to present a new approach to the filtering problem for the class of bilinear stochastic multivariable systems, consisting in searching for suboptimal state-estimates instead of the conditional statistics. As a first result, a finite-dimensional optimal linear filter for the considered class of systems is defined. Then, the more general problem of designing polynomial finite-dimensional filters is considered. The equations of a finite-dimensional filter are given, producing a state-estimate which is optimal in a class of polynomial transformations of the measurements with arbitrarily fixed degree. Numerical simulations show the effectiveness of the proposed filter.

Robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability

Presents two robust solutions to the control of the output probability density function for general multi-input and multi-output stochastic systems. The control inputs of the system appear as a set of variables in the probability density functions of the system output, and the signal available to the controller is the measured probability density function of the system output. A type of dynamic probability density model is formulated by using a B-spline neural network with all its weights dynamically related to the control input. It has been shown that the so-formed robust control algorithms can control the shape of the output probability density function and can guaranteed the closed-loop stability when the system is subjected to a bounded unknown input. An illustrative example is included to demonstrate the use of the developed control algorithms, and desired results have been obtained.

Robust covariance control for perturbed stochastic multivariable system via variable structure control

Based on the concept of variable structure control, this paper investigates the steady state covariance assignment problem for perturbed stochastic multivariable systems. By using the invariance property of variable structure systems, the matched perturbation of the system disappears on the sliding mode. With the aid of Ito-formula, the controller u( t) is proposed. Combining the sliding phase and hitting phase of the system design, the control feedback gain matrix G is derived to satisfy the steady state covariance assignment.

Stochastic Process Representation and Analysis of the Homogenizing Process in a Cement Plant

This paper presents a predicting method for standard deviations of chemical composition fluctuations of raw meal at a kiln inlet using chemical analysis results of boring samples for raw materials in a quarry. A stochastic process representation for their fluctuations at the inlet of the plant is presented to specify their time domain fluctuations. The authors have also presented dedicated transfer functions which represent homogenizing effect of each constituent process and the raw material mixing control system. Using these representations, the Bode-diagrams to obtain decreasing performances of the fluctuations are finally presented. The theoretical calculation result of the standard deviation is compared with the result measured in an actual plant. As a result of a succession of these analyses, moreover, the authors could also present the structure of the mixing control system as a multivariable stochastic control system.

A New Formula for the Multivariable Predictor

Model Based Predictive Control (MBPC) is a control strategy based on the explicit use of a model to predict the process output over a long-range time period. To compute the step ahead predictor, one would solve polynomial equations in recursive way. This paper deals with a new formula for computing the predictor of stochastic multivariable linear systems. This formula delimits the role of each model parameter and avoids the recursive resolution of diophantine equations. The process and prediction model which are used unite the most commonly used models in predictive controllers.

Covariance control for stochastic multivariable systems with hysteresis nonlinearity

By using the describing function approximation method, the covariance control of a multivariable stochastic system with hysteresis nonlinearities is studied. Since the describing function for a hysteresis nonlinearity is always a complex form, this complex component in the system makes the problem more complicated. This paper proposes a condition under which the designed state feedback gain is guaranteed to be real and the specified state covariance X of the closed loop system is also achieved.

Multivariable Adaptive Control of Systems with Unknown Dead-Zone Nonlinearities

This paper presents an indirect adaptive control scheme of stochastic multivariable systems with unknown dead-zone nonlinearities. The scheme is applicable even to systems which may possess an arbitrary interactor matrix and be nonminimum phase. Since identification is separated from the calculation of controller design and a single adaptation algorithm is used, the indirect scheme provides more flexibility and has the advantage that much fewer estimated parameters and memory space are required. The proposed scheme ensures the global stability and convergence.

Indirect adaptive pole-placement control of MIMO stochastic systems: self-tuning results

In this paper, we consider indirect adaptive pole-placement control (APPC) of linear multivariable stochastic systems. Instead of the canonical representation often used in the literature, we propose using a non-minimal but otherwise uniquely identifiable pseudo-canonical parameterization that is more suitable for multivariable ARMAX model identification. To identify the plant, we use the weighted extended least-squares (WELS) algorithm, a least-squares method with slowly decreasing weights which was introduced in Bercu (1995). The pole-placement controller parameters are then calculated by using a certain perturbation of the parameter estimates such that the linear models corresponding to the perturbed estimates are uniformly controllable and observable. We prove that with a reasonable amount of prior information, the resulting APPC scheme is globally stabilizing and asymptotically self-tuning regardless of the degree of persistency of external excitation. These results represent the most complete study of stochastic multivariable APPC systems to this date.

Multivariable Adaptive Control of Backlash Nonlinear Systems with Arbitrary Interactor Matrix

This paper presents an indirect adaptive control scheme of stochastic multivariable systems with backlash inputs. The scheme is applicable even to systems which may possess an arbitrary interactor matrix and be nonminimum phase. Since identification is separated from the calculation of controller design and a single adaptation algorithm is used, the indirect scheme provides more flexibility and has the advantage that much fewer estimated parameters and memory space are required. The proposed scheme ensures the global stability and convergence. Simulation results verify the effectiveness of the proposed algorithm.

Optimal controller design for discrete saturating systems

A design procedure is proposed for synthesizing a linear-quadratic-Gaussian (LQG) optimal controller to obtain excess robustness optimization against noise spectral uncertainties, non-linear time-varying (NLTV) unmodelled dynamics in the discrete saturating systems. A robust stability criterion is derived for multivariable stochastic discrete-time systems with NLTV unmodelled dynamics and constrained actuators. An algorithm based on the robust stabilization criterion is presented for synthesizing a robust controller not only to minimize the least favourable cost functional J but also to maximize the excess robustness with respect to H∞-norm criterion by selecting two adequate weighting matrices Q(z) and R(z). We shall adopt the transfer function approach and two weighting matrices in the cost functional are shaped via the inverse LQG method to achieve excess robustness optimization.

Robust LQG optimal controller design for multivariable discrete saturating systems with noise spectral uncertainties and nonlinear time-varying unmodeled dynamics

A design procedure is proposed for robust LQG (linear-quadratic-Gaussian) optimal controller synthesis against noise spectral uncertainties, nonlinear time-varying (NLTV) unmodeled dynamics in discrete saturating systems. A robust stability criterion is derived for multivariable stochastic discrete-time systems with NLTV unmodeled dynamics and constrained actuators. An algorithm based on the robust stabilization criterion is presented for synthesizing a robust controller not only to minimize the least favorable cost functional J but also to satisfy the robust stabilization criterion by specifying an appropriate weighting scalar in the cost functional. A necessary and sufficient condition for the solvability of such a robust stabilization problem is derived by means of Nevanlinna-Pick interpolation theory. The Weiner Z-domain solution for controller synthesis, the saddle point theory, and the properties of the Schur operator (Class S) are employed to treat this problem. Finally, a numerical example is given to illustrate the results.

Robust LQG optimal controller design for multivariable discrete saturating systems with noise spectral uncertainties and non-linear time-varying unmodelled dynamics

A design procedure is proposed for robust linear-quadratic-gaussian (LQG) optimal controller synthesis against noise spectral uncertainties, non-linear time-varying (NLTV) unmodelled dynamics in discrete saturating systems. A robust stability criterion is derived for multivariable stochastic discrete-time systems with NLTV unmodelled dynamics and constrained actuators. An algorithm based on the robust stabilization creterion is presented for synthesing a robust controller not only to minimize the least favourable cost functional but also to satisfy the robust stabilization criterion by specifying an appropriate weighting scalar in the cost functional. A necessary and sufficient condition for the solvability of such a robust stabilization problem is derived by means of the Nevanlinna-Pick interpolation theory. The Wiener Z-domain solution for controller synthesis, the saddle point theory, and the properties of Schur operator (Class S) are employed to treat this problem. Finally, a numerical example is given to illustrate the results.

Multi-step ahead predictions for multi variable linear stochastic systems via model recursion

The multi-step ahead prediction method for the output of a multi-variable linear discrete-time stochastic system, directly based on the recursion of system model, is developed. The basic procedures are: (1) rewrite the original system model into a one-step ahead recursive version; (2) repeatedly use this recursion step by step and thus successively gain all the required multi-step ahead predictions. It is proved that this model recursion method is strictly equivalent to the existing method using the recursion of multi-variable Diophantine equation, under the condition of white noise. The proposed method possesses advantages both in computational reduction and programming simplicity, and has successfully been used in a multi-variable modified generalized predictive controller.