A dynamic observer for a synchronous machine†

The closed-loop optimal control law obtained for a linear time-invariant system always requires the entire state vector to be available for direct measurement. It is seldom that all the state variables of a physical system are available for measurement. A compatible dynamic observer of the Luenberger type is designed to reconstruct the entire state vector for a synchronous machine-infinite bus system. The performance of the system with the observer cascaded to the optimal controller is compared with the performance of the system when all the state variables are available for feedback. The transfer matrix relating the output and input is derived.     

On the determination of the optimal constant output feedback gains for linear multivariable systems

The optimal control of linear time-invariant systems with respect to a quadratic performance criterion is discussed. The problem is posed with the additional constraint that the control vectoru(t)is a linear time-invariant function of the output vectory(t) (u(t) = -Fy(t))rather than of the state vectorx(t). The performance criterion is then averaged, and algebraic necessary conditions for a minimizingFastare found. In addition, an algorithm for computingFastis presented.     

Linear dynamic filtering with noisy input and output

State estimation problems for linear time-invariant systems with noisy inputs and outputs are considered. An efficient recursive algorithm for the smoothing problem is presented. The equivalence between the optimal filter and an appropriately modified Kalman filter is established. The optimal estimate of the input signal is derived from the optimal state estimate. The result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem.     

Discrete-time optimal feedback controllers with time-multiplied performance indexes

A class of optimal state and output feedback control laws for discrete-time time-invariant linear systems which minimizes a class of discrete-time time-multiplied performance indexes is presented. A necessary condition, an existence condition, and a sufficient condition for the control laws are derived. A simple example is given to illustrate the effectiveness of the proposed control laws.     

Lyapunov stability robustness measures for multivariable,continuous, time-invariant,linear systems

In this paper general robustness measure bounds are introduced for any multivariable, continuous, time-invariant, linear systems. Bounds are obtained for allowable non-linear time-varying perturbations such that the resulting system remains stable. Bounds are also derived for linear perturbations. The robustness measures and the related theorems are applied to optimal LQ state feedback, direct output feedback, and to generalized dynamic output feedback designs.     

The design of optimal compensators for linear constant systems with inaccessible states

This paper considers the problem of designing an optimal linear time-invariant dynamic compensator for the regulation of annth-order linear time-invariant plant. The usual quadratic cost on the state and control is averaged over initial plant state values. The globally optimal compensator gains and dynamic order are determined by showing that this problem is mathematically identical to a steady-state stochastic control problem whose optimal solution is known.     

Design of reduced-order optimal estimators directly in the frequency domain

The frequency-domain design of optimal reduced-order estimators ((Caiman filters) for continuous-time linear time-invariant systems containing 0 ≤ k ≤ m noise-free output measurements is considered. In a direct frequency-domain approach to state feedback and estimation the ‘states’ of a system do not appear explicitly, and therefore we here focus our attention mainly on the frequency-domain characteristics of the optimal filter. By suitably modifying known time-domain results, a simple polynomial matrix equation can be derived from which the optimal filter parameters are obtained by spectral factorization. The equivalence between the time- and the frequency-domain results is established with the aid of a non-minimal representation of reduced-order observers.     

Internal behaviour provided by nonlinear l1-optimal controllers

An l₁-optimal linear time-invariant (LTI) compensator may have an order significantly higher than that of the plant, even when the state is measurable. Recently there has been work exploring the use of nonlinear static feedback controllers which provide near optimal performance. Here we consider a class of nonlinear state feedback controllers and derive superposition-like bounds on both the plant state and the controlled output in the event that the plant initial condition, the disturbance, and the noise are all non-zero.     

Data-driven output-feedback fault-tolerant control for unknown dynamic systems with faults changing system dynamics

This paper studies the data-driven output-feedback fault-tolerant control (FTC) problem for unknown dynamic systems with faults changing system dynamics. In a framework of active FTC, two basic issues are addressed: the fault detection employing only the measured input–output information; the controller reconfiguration to achieve optimal output-feedback control in the presence of multiple faults. To detect faults and write the system state via the input–output data, an approach to data-driven design of a residual generator with a full-rank transformation matrix is presented. An output-feedback approximate dynamic programming method is developed to solve the optimal control problem under the condition that the unknown linear time-invariant discrete-time plant has multiple outputs. According to the above results and the proposed input–output data-based value function approximation structure of time-varying plants, a model-free output-feedback FTC scheme considering optimal performance is given. Finally, two numerical examples and a practical example of a DC motor control system are used to demonstrate the effectiveness of the proposed methods.     

A convergent algorithm for computing stabilizing static output feedback gains

We revisit the approach by Cao et al. that uses a fixed-structure control law to find stabilizing static output feedback gains for linear time-invariant systems. By performing singular value decomposition on the output matrix, together with similarity transformations, we present a new stabilization method. Unlike their results that involve a difficult modified Riccati equation whose solution is coupled with other two intermediate matrices that are difficult to find, we obtain Lyapunov equations. We present a convergent algorithm to solve the new design equations for the gains. We will show that our new approach, like theirs, is a dual optimal output feedback linear quadratic regulator theory. Numerical examples are given to illustrate the effectiveness of the algorithm and validate the new method.     

Lower order generalized aggregated model and suboptimal control

A method for reducing the order of a linear time-invariant dynamic system is presented. It is shown that it is possible to retain the predominant eigenvalues (or any other set of eigenvalues) of the exact system in the lower order model that possesses the property that its state is an aggregation of the state variables of the original system. Also it is shown that the output of the reduced order model can be constrained to contain all the modes of the exact output and to be close to the actual output of the original system within a specified tolerance. The performance of the original system is investigated for an optimal output regulator problem, when it is controlled on the assumption that its behavior is governed by that of the lower order model. Relations are obtained for the performance degradation that results with the above suboptimal control policy. Numerical examples show that the suboptimal control can be used in practice to lessen the computational complexity required for the higher order optimal control. The stability of the suboptimal control is not guaranteed; however, it is reasonable to expect it to be asymptotically stable when the order of reduction is not excessively high, because the outputs of the exact and lower order models are tolerably close.     

Output stabilization via pole placement of discrete-time linearperiodic systems

The output stabilization problem for discrete-time linear periodic systems is solved. Both the state-feedback control law and the state-predictor are based on a suitable time-invariant state-sampled reformulation associated with a periodic system. Preliminary concepts of periodic system theory are briefly recalled. In particular, the structural properties of a linear discrete-time periodic system are properly related to those of a time-invariant system associated with it. By resorting to such a time-invariant reformulation, the output stabilization problem via pole placement is solved. The stabilizing controller is constituted by a control law and an asymptotic state predictor, both of which are shown to be periodic     

Solution of the optimal constant output feedback problem by conjugate gradients

The problem of finding the optimal, constant output feedback matrix for linear time-invariant multivariable systems with quadratic cost is reconsidered. Simple formulae for the gradient matrix are developed and used in a Fletcher-Powell-Davidon algorithm. Computational results are presented.     

Synthesis of sub-optimal feedback controls for a class of distributed parameter systems†

Technique are derived for the synthesis of sub-optimal feedback controls for parabolic and first-order hyperbolic systems. An explicit result for the time-invariant gain of a specified controller is obtained by a least square approximation of the closed-loop control to the optimal open-loop control. If a least square approximation of the state trajectories is used, a parameter search is shown to give the time-invariant gain. A time-varying gain can be obtained by a re-definition of the original optimal control problem, again with the controller functionality specified. The only require-mont in the closed-loop synthesis is that an optimal open-loop solution exists and is computable.     

An estimator for uncertain output matrix systems

A linear time-invariant optimal estimator for uncertain output matrix systems is considered. Uncertainty of the output matrix is represented by a set of multiple parameters, and the optimal estimator is introduced by minimizing the expected cost function of the estimation error. When the number of measured variables is small, the order of the optimal estimator does not depend on the number of models, and meeting the computational demand becomes quite feasible     

Robust memoryless H∞ controller design for linear time-delay systems with norm-bounded time-varying uncertainty

We consider the robust H_∞ control problem for linear time-delay systems with norm-bounded time-varying uncertainty. Based on the Riccati-equation approach, we present a method for designing a robust memoryless linear time-invariant state feedback control law, which stabilizes the system and reduces the effect of the disturbance input on the controlled output to a prescribed level for all admissible uncertainties.     

Modified tracking performance limitations of unstable linear SIMO feedback control systems

In this paper, the optimal modified performance of the single-input multiple-output (SIMO) linear time-invariant (LTI) systems is investigated. A new modified tracking performance index is proposed to avoid the invalidity of the variance of the tracking error when the steady state tracking error is not a zero vector. The optimal modified performance is related to the scaled factor, the inner factor and the spectral density of the reference input signal. The output dimension and pole direction also play an important role in the optimal modified performance. The strong restrictions on the reference signal directions and the assumption on the plant with an integrator in the existing conclusions are not necessary. Finally, a numerical example is discussed to demonstrate the efficiency of the derived results.     

A new optimal multirate control of linear periodic andtime-invariant systems

The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates     

Dead-beat control and the Riccati equation

Connections between dead-beat control strategies and optimal control policies for linear, time-invariant, discrete-time systems are established. The performance index of the system is quadratic and only the terminal state of the system is penalized. An explicit solution to the singular Riccati equation, associated with this optimization problem, is given. Properties of the time-variable gain matrices, generating the optimal cantrol policy, are presented. In particular, necessary and sufficient conditions for each of these gain matrices to be a time-invariant deadbeat controller are given.     

Robust fault detection filter design for a class of time-delay systems via equivalent transformation

This paper is concerned with the robust fault detection filter (RFDF) design for a class of linear timeinvariant systems (LTISs) with output state time delays. Although existing results in literatures study the RFDF for timedelay systems, few is concerned with the output state time-delay systems. The basic idea of our study is to eliminate the time delays of system and transform it to a delay-free system (i.e., a linear time-invariant system without time delays) by the bicausal change of coordinates approach. Then, we design the RFDF for the delay-free LTIS, which is equivalent to the original system with time delays. We first introduce a class of systems with output state time delays, whose fault can be detected by using the RFDF design approach for delay-free systems. Then, since the RFDF design problem can be formulated as a standard H-infinity-model matching problem, it is solved by using H-infinity-optimization LMI techniques. In the last, the adaptive threshold of fault detection is chosen and an illustrative design example is used to demonstrate the validity of the design approach.