This paper presents a means to approximate the dynamic and static equations of stochastic nonlinear systems and to estimate state variables based on radial basis function neural network (RBFNN). After a nonparametric approximate model of the system is constructed from a priori experiments or simulations, a suboptimal filter is designed based on the upper bound error in approximating the original unknown plant with nonlinear state and output equations. The procedures for both training and state estimation are described along with discussions on approximation error. Nonlinear systems with linear output equations are considered as a special case of the general formulation. Finally, applications of the proposed RBFNN to the state estimation of highly nonlinear systems are presented to demonstrate the performance and effectiveness of the method. >

Radial basis function neural network for approximation and estimation of nonlinear stochastic dynamic systems

First-order necessary conditions for quadratically optimal, steady-state,fixed-order dynamic compensation of a linear, time-invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form. In contrast to the pair of matrix Riccati equations for the full-order LQG case, the optimal steady-state fixed-order dynamic compensator is characterized by four matrix equations (two modified Riccati equations and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains. The coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced-order controller case.

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The optimal projection equations for fixed-order dynamic compensation

A sequential numerical algorithm is described which obtains gains minimizing a broad class of performance indexes, including the standard LQ case. The primary contribution is a proof that the algorithm converges to a local minimum under nonrestrictive assumptions. Numerical examples illustrate the theory. The second example demonstrates an important LQ design technique which permits the designer to prespecify the feedback structure, subject to the requirement of output feedback stabilizability.

Convergence of a numerical algorithm for calculating optimal output feedback gains

Iterative methods for finding the optimal constant feedback gains for parametric LQ problems, notably for optimal constant output feedback problems, are surveyed. The connections of several methods to loss function expansions are discussed with important implications to the understanding of their convergence properties. Especially, the descent Anderson-Moore method, Levine-Athans like methods, and the Newton method are considered. Convergence results are also included. The initialization problem and the output feedback stabilization problem are also discussed. Furthermore, it is shown that the concepts and methods surveyed in this paper are useful in solving many realistic generalized parametric LQ problems as well, notably so-called robust parametric LQ problems.

Computational methods for parametric LQ problems--A survey

We consider the design of optimal state feedback gains subject to structural constraints on the distributed controllers. These constraints are in the form of sparsity requirements for the feedback matrix, implying that each controller has access to information from only a limited number of subsystems. The minimizer of this constrained optimal control problem is sought using the augmented Lagrangian method. Notably, this approach does not require a stabilizing structured gain to initialize the optimization algorithm. Motivated by the structure of the necessary conditions for optimality of the augmented Lagrangian, we develop an alternating descent method to determine the structured optimal gain. We also utilize the sensitivity interpretation of the Lagrange multiplier to identify favorable communication architectures for structured optimal design. Examples are provided to illustrate the effectiveness of the developed method.

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Augmented Lagrangian Approach to Design of Structured Optimal State Feedback Gains

An algorithm is given for optimization of constant parameters in the controller for a time invariant linear system. The algorithm may be applied to linear controllers of arbitrary structure with any degree of decentralization in the distribution of information. Most important, it is not necessary to provide an initial controller of the specified structure which stabilizes the system. Rather, the structural constraints are initially relaxed to whatever degree necessary to obtain a stabilizing controller. The algorithm then gradually restores the constraints, converges to a local optimum solution, when one exists, and stabilizes any open loop system possessing a controllable unstable subspace. The necessary conditions while the optimum parameters must satisfy are developed for the arbitrarily constrained controller. The method of satisfying these necessary conditions involves use of a standard conjugate direction search together with a combined multiplier and penalty method for invoking the structural constraints. While attention is focused on the deterministic control problem, it is noted that systems with driving disturbances and measurement uncertainty may be handled with no significant modification of the algorithm. The result is a very practical tool for system design via parameter optimization.

Parameter optimization in linear systems with arbitrarily constrained controller structure

Estimation of traffic variables using a linear model of traffic flow

Two approaches are proposed for on line identification of parameters in a class of nonlinear discrete time systems. The system is modelled by state equations in which state and input variables enter nonlinearly in general polynomial form while unknown parameters and random disturbances enter linearly. States and inputs are observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite but unknown covariance. One method of parameter estimation is based upon a least squares approach; the second is a stochastic approximation algorithm (SSA). In each case the algorithm is motivated by minimization of an appropriate error criterion and modified to eliminate bias errors. Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability one. Examples are given illustrating convergence of a recursive form of the LSA. This recursive form still requires inversion of a matrix at each step. The SAA requires no matrix inversions but experience to date with the algorithm indicates that convergence is slow relative to that of the LSA.

Parameter identification in a class of nonlinear systems

Measurement selection for linear multivariable control systems

Algorithms for the robust detection and estimation of soft failures which have very modest computational requirements are presented. Soft failures are characterized by deviations in the statistical parameters of system inputs and measurement errors. The on-line detection scheme utilizes a single Kalman filter and requires the implementation of Wald Sequential Detectors (WSD). The algorithm consists of repeated application of the Sequential Probability Ratio Test (SPRT) to the filter innovations and use of the sequence of Sample Numbers (SN) for each SPRT termination for the detection of a failure. The proposed scheme for the estimation of a failed parameter consists of a bank of detectors and utilizes the properties of the Average Sample Number (ASN). The capabilities of the scheme, including its robustness for the isolation of single parameter failures, are illustrated via simulation results for a maneuvering target problem.

Charles H. Knapp

Papers

Parameter optimization in linear systems with arbitrarily constrained controller structure

Estimation of traffic variables using a linear model of traffic flow

Parameter identification in a class of nonlinear systems

Measurement selection for linear multivariable control systems

Robust detection and estimation of soft failures in linear systems