The time-optimal design of linear dynamic systems

The following general time-optimal design problem is studied: determine a real constant square matrix,A, subject to specified constraints, to minimize the transit time between specified endpoints while satisfying the vector differential equationdot{x}(t) = Ax(t). Two specific kinds of constraints onAare considered: 1) where the individual elements ofAare free but the matrix as a whole must satisfyQ(A) leq hat{Q}whereQ(A)is a specified homogeneous function of the elements ofAandhat{Q}is a given upper bound and 2) where the elements ofAare individually bounded. Theoretical results show that both problems can be solved by first solving a related minimum cost fixed time problem. The latter problem is solved iteratively by using the generalized Newton-Raphson method for two point boundary value problems to provide a set of linear equations at each iteration. The cost functionQis then minimized subject to these equations using appropriate optimization techniques.     

Parameter identification of linear structural dynamic systems

The parameter estimations of linear multi-degree-of-freedom structural dynamic systems are carried out in time domain. Methods for estimating the system parameters and the modal parameters are presented. The equation of motion is transformed into the state space equation of the observable canonical form, and then into the auto-regressive and moving average model with auxiliary stochastic input (ARMAX) model to process the measurement data contaminated by the system noise as well as the output noise. The parameters of the ARMAX model are estimated by using the sequential prediction error method. Then, the parameters of equation of motion are recovered thereafter. In order to verify the accuracy of the estimation method, analytical simulation studies are performed for a model with two degrees of freedom on the basis of simulated data under various noise conditions. It is shown that the presented methods yield good estimates even under large noise conditions. The method is also applied to the identification of the modal parameters of a building model based on the experimental data.     

Adaptive dynamic programming for linear impulse systems

We investigate the optimization of linear impulse systems with the reinforcement learning based adaptive dynamic programming （ADP） method. For linear impulse systems, the optimal objective function is shown to be a quadric form of the pre-impulse states. The ADP method provides solutions that iteratively converge to the optimal objective function. If an initial guess of the pre-impulse objective function is selected as a quadratic form of the pre-impulse states, the objective function iteratively converges to the optimal one through ADP. Though direct use of the quadratic objective function of the states within the ADP method is theoretically possible, the numerical singularity problem may occur due to the matrix inversion therein when the system dimensionality increases. A neural network based ADP method can circumvent this problem. A neural network with polynomial activation functions is selected to approximate the pr~impulse objective function and trained iteratively using the ADP method to achieve optimal control. After a successful training, optimal impulse control can be derived. Simulations are presented for illustrative purposes.     

Diagonalization algorithms for linear time-varying dynamic systems

On the basis of a mode-vector representation, we show that its time-varying amplitudes and frequencies can be obtained by diagonalizing the system matrix. Next, we reformulate an explicit diagonalizing algorithm that was earlier proposed by Wu. Then, the missing convergence proof is given. Moreover, we present a new and implicit iteration scheme that is closely related to that given by Wu. In both algorithms, the time-varying system matrix is gradually diagonalized by successive algebraic similarity transformations. It is proved that the convergence conditions are essentially the same. Although the class of systems for which the algorithms are applicable is still not fully known, the results of this paper may be of theoretical and practical interest.     

An algorithm for inverting linear dynamic systems

An algorithm is presented for inverting transformations associated with linear dynamic systems. The induced inverse system, when it exists, operates on outputs and generates the corresponding inputs of the system to which it is inverse. The applications for inverse systems are found in such diverse areas as control, coding, filtering, and sensitivity analysis.     

Parity relations for linear uncertain dynamic systems

A new approach for the design of parity relations for linear dynamic systems with additive and multiplicative uncertainties is presented. Instead of cancelling uncertainties following the example of the so-called robust approaches, uncertain parity relations take uncertainties into account as bounded variables. The method is based on the analysis of zonotopes representing the uncertainties. It leads both to Boolean detection results and to an indicator representing the distance to the opposite decision.     

Decoupling of linear systems by dynamic output feedback

Necessary and sufficient conditions for decoupling of linear systems by dynamic output feedback are derived, under the requirement that the decoupled system be internally stable. The conditions are stated in terms of quantities which are directly related to the transfer matrix of the given system. The main issue is resolved through the introduction of a new concept—the strict adjoint. The strict adjoint is a minimal polynomial matrix that diagonalizes a given matrix.This research was supported in part by US Army Research Grant DAAG29-80-C0050 and US Air Force Grant AFOSR76-3034D through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida 32611, U.S.A.     

On the robustness of linear continuous-time dynamic systems

In this note, for a given linear continuous-time dynamic systemdot{x} = Ax + Bu, we determine sufficient conditions for the existence of a constant feedback matrixksuch that the closed-loop system matrixA - Bkadmits a diagonal positive solution of the Lyapunov equation. An algorithm for the determination of suchkmatrix is given and some examples are solved.     

Detecting scalar intermittent faults in linear stochastic dynamic systems

Intermittent faults (IFs) have properties such as intermittency, random magnitude and random duration time. Hence the detection of IFs means: (i) to detect not only all the appearing time but also all the disappearing time of IFs and (ii) to detect the appearing time of an IF before this IF disappears, and the disappearing time of an IF before the subsequent IF appears. Within a statistical framework, the detection of scalar IFs in continuous linear stochastic dynamic systems has been mainly studied. Based on the sliding window, an analytical residual is generated, and two hypothesis tests are implemented to detect the appearing and disappearing times of IFs. In addition, a necessary and sufficient condition for the detectability of IFs is obtained, and the detection speed can be fast enough. Theoretical analysis and numerical simulations fully verify that IFs can be successfully detected.     

Identification of linear dynamic systems operating in a networked environment

This paper studies a networked system identification problem, which aims at identifying mathematical models required in networked control/estimation/filtering systems. Specifically, we consider the off-line identification of open-loop stable linear time-invariant processes working in a networked environment. In the networked environment, how the actuators (D/A conversion) operate plays a key role in the complexity of the related identification problems. In particular, it is reasonable to consider the configuration of event-driven actuators subject to random network-induced delays and packet dropouts; as a result, the networked identification problem is formulated as the one to identify continuous-time linear time-invariant models, based on the general non-uniformly non-synchronized sampled data. A modified version of the simplified refined instrumental variable method is proposed to solve this problem, and is validated in a networked identification experiment based on the Matlab/Simulink simulator TrueTime.     

Enforcing stability constraints in set-membership identification of linear dynamic systems

In this paper, we consider the identification of linear systems, a priori known to be stable, from input–output data corrupted by bounded noise. By taking explicitly into account a priori information on system stability, a formal definition of the feasible parameter set for a stable linear system is provided. On the basis of a detailed analysis of the geometrical structure of the feasible set, convex relaxation techniques are presented to solve nonconvex optimization problems arising in the computation of parameter uncertainty intervals. Properties of the computed relaxed bounds are discussed. A simulated example is presented to show the effectiveness of the proposed technique.     

On dynamic reconstruction of input in the second-order linear systems

Consideration was given to the problem of dynamic reconstruction of input in the second-order linear systems. Procedures for its real-time solution were specified. They feature dynamicity, which means that the current values of the approximated input are generated in real time, and stability, which means that for a sufficiently small measurement error approximation is arbitrarily precise.     

Identifiability of MIMO linear dynamic systems operating in closed loop

Identification of multi-input multi-output (MIMO) linear dynamic systems is considered for the case when the measurements are obtained during closed loop operation. Both noise free and noisy feedback situations are analysed. For the case where the disturbances in the feedback path are a full rank stochastic process it is shown that, under certain mild conditions, physically meaningful models for the forward and reverse paths can be uniquely determined. For the case where the feedback path is noise free it is shown that the forward path model can be uniquely determined provided the regulator satisfies certain minimal complexity requirements.     

On a method for simplifying linear dynamic systems

Davison and Chidambara have exposed methods [1], [5] for approximating a linear dynamic system of high dimension by a model of lower dimension, neglecting the fast decaying modes associated with eigenvalues of high magnitude. In spite of a number of correspondence items from the authors it is not clear which model should be used since the examples given might as well show that either method is correct or incorrect according to the cases. In fact it will appear in the following that the problem lies in the existence of null eigenvalues and that a modification of Davison's model in that case will solve the problem, at least for step inputs; for general inputs it will also be clear that some care must be taken, which was not evident in the previous papers.     

Adaptive dynamic surface control for linear multivariable systems

In this paper, an output-feedback adaptive control is presented for linear time-invariant multivariable plants. By using the dynamic surface control technique, it is shown that the explosion of complexity problem in multivariable backstepping design can be eliminated. The proposed scheme has the following features: (1) The L_∞ performance of the system’s tracking error can be guaranteed, (2) it has least number of updated parameters in comparison with other multivariable adaptive schemes, and (3) the adaptive law is necessary only at the first design step, which significantly reduces the design procedure. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.     

Solutions of linear dynamic systems by generalized orthogonal polynomials

Generalized orthogonal polynomials that represent all types of orthogonal polynomial are introduced in this paper. Using the idea of orthogonal polynomial functions that can be expressed by power series, and vice versa, the operational matrix for integration of a generalized orthogonal polynomial is first derived and then applied to solve the equations of linear dynamic systems. The characteristics of each kind of orthogonal polynomial in relation to solving linear dynamic systems is demonstrated. The computational strategy for finding the expansion coefficients of the state variables is very simple, straightforward and easy. The operational matrix is simpler than those of conventional orthogonal polynomials. Hence the expansion coefficients are more easily calculated from the proposed recursive formula when compared with those obtained from conventional orthogonal polynomial approximations.     

Stability of linear dynamic systems with random time-dependent parameters

We present conditions of asymptotic stability for the solution of a system of linear differential equations that depend on stochastic processes.Translated from Kibernetika, No. 3, pp. 70–72, 75, May–June 1990.     

Diagnosis of linear dynamic systems by the nonparametric method

Consideration was given to the problem of fault diagnosis of the linear dynamic systems by the nonparametric method distinguished for the fact that the system parameters may be unknown. An approach was proposed to fault localization. Methods of making decisions from the results of diagnosis were examined.     

Control of linear dynamic systems with set constrained disturbances

A linear dynamic system with input and observation uncertainties is studied. The uncertainties are constrained to be contained in specified sets. No probabilistic structure is assumed. The problem of keeping the state of the system in a specified region is investigated. Necessary and sufficient conditions for a solution to the problem and an algorithm that constructs the control are derived for open-loop and closed-loop control laws. The algorithm is also approximated by a bounding ellipsoid algorithm. Two special control laws, linear and "linear-plus-dead-band," are studied, and the regions that contain the state and control are characterized. Ellipsoids that bound the state and control are also derived, and a simple example of linear and linear-plus-dead-band control laws is presented.