On the design of optimal time-invariant compensators for linear stochastic time-invariant systems

Design equations are developed for an optimal limited state feedback controller problem. These equations are developed for the stochastic case, in which both plant noise and measurement noise may be present, and for the case of a dynamic compensator. Four possible approaches to the solution of the nonlinear design equations are described. A fourth-order example illustrates some of the difficulties associated with the solution of these equations and suggests additional areas for study.     

On a general optimal algorithm for multirate output feedbackcontrollers for linear stochastic periodic systems

A modified optimal algorithm for multirate output feedback controllers of linear stochastic periodic systems is developed. By combining the discrete-time linear quadratic regulation (LQR) control problem and the discrete-time stochastic linear quadratic regulation (SLQR) control problem to obtain an extended linear quadratic regulation (ELQR) control problem, one derives a general optimal algorithm to balance the advantages of the optimal transient response of the LQR control problem and the optimal steady-state regulation of the SLQR control problem. In general, the solution of this algorithm is obtained by solving a set of coupled matrix equations. Special cases for which the coupled matrix equations can be reduced to a discrete-time algebraic Riccati equation are discussed. A reducable case is the optimal algorithm derived by H.M. Al-Rahmani and G.F. Franklin (1990), where the system has complete state information and the discrete-time quadratic performance index is transformed from a continuous-time one     

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.     

Observers for optimal estimation of the state of linear stochastic discrete systems

For an n-dimensional linear non-stationary stochastic discrete system with multiplicative and additive noise, the problem of estimation of the plant phase coordinate vector from partially noisy observation is considered. An algorithm is developed which is of lower dimension than the Kalman-Bucy filtering algorithm, namely (n-l)     

On overtaking optimal tracking for linear systems

Optimal tracking problems involving a time-varying linear plant and infinite-horizon quadratic performance index are considered. In general, such problems yield an unbounded performance index for every control and thus must be interpreted as so-called overtaking optimal control problems. A completing-the-square argument is used to derive the overtaking optimal control for a general problem formulation, and various properties are discussed.     

Design of optimal constrained dynamic compensators for non-stationary linear stochastic systems

This paper deals with the design of fixed-order dynamic compensators for non-stationary linear stochastic systems with noisy observations, where the observation noise need not necessarily be white. An integral quadratic performance index defined over a finite time interval is employed and this yields a matrix variational problem in the compensator parameters. It is shown how the optimal, possibly time-varying, compensator parameters rnay be determined by direct solution of this variational problem using a conjugate-gradient technique. Consideration is also given to finding suboptimal compensators that are simpler to implement. In particular, an algorithm is proposed for designing compensators having gains that are constrained to be piecewise-constant functions of time, with provision for optimally choosing the instants at which gains changes occur. An illustrative numerical example is included.     

On controllability of linear stochastic systems

We discuss several concepts of controllability for partially observable stochastic systems: complete controllability, approximate controllability, controllability and S-controllability, and show that complete and approximate controllability notions are equivalent, and in turn are equivalent to the controllability for linear stochastic systems controlled by gaussian processes. We derive necessary and sufficient conditions for these concepts of controllability. These criteria reduce to the well-known rank condition.     

On the periodic coordination of linear stochastic systems

The decentralized stochastic control of a linear dynamic system consisting of several subsystems is considered. A two-level approach is used by the introduction of a coordinator who collects measurements from the local controllers periodically and in return transmits coordinating parameters. Two types of coordination are considered: open-loop feedback and closed loop. The resulting control laws are found to be intuitively attractive.     

On the Hankel-norm approximation of linear stochastic systems

We derive ordering and interlocking properties for the singular values of the block-Hankel matrices corresponding to different spectral factors of a given spectral density matrix. The results are then applied to Hankel-norm approximation of SISO stochastic systems. In particular we show that the minimum phase model may be approximated in Hankel norm by systems of a certain prescribed dimension with better accuracy than any other model with the same output process. We also provide upper bounds on the gain in accuracy obtained by choosing the minimum phase model over some other model.     

On the open-loop solution of linear stochastic optimal control problems

We consider open-loop solutions of linear stochastic optimal control problems with constraints on control variables and probabilistic constraints on state variables. It is shown that this problem reduces to an equivalent linear deterministic optimal control problem with similar constraints and with a new criterion to minimize. Concavity or convexity is preserved. Hence, the machinery available for solving deterministic optimal control problems can be used to get an open-loop solution of the stochastic problem. The convex case is investigated and a bound on the difference between closed-loop and open-loop optimal costs is given.     

The design of optimal reduced-order stochastic observers fordiscrete-time linear systems

The minimum-variance-state estimation of linear discrete-time systems with random white-noise input and partially noisy measurements is investigated. An observer of minimal order is found which attains the minimum-variance estimation error. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible, and free of measurement noise. An explicit expression is found for the transfer-function matrix of this observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained     

Adaptive sampled-data based linear quadratic optimal control of stochastic systems

The problem of sampled-data (SD) based adaptive linear quadratic (LQ) optimal control is considered for linear stochastic continuous-time systems with unknown parameters and disturbances. To overcome the difficulties caused by the unknown parameters and incompleteness of the state information, and to probe into the influence of sample size on system performance, a cost-biased parameter estimator and an adaptive control design method are presented. Under the assumption that the unknown parameter belongs to a known finite set, some sufficient conditions ensuring the convergence of the parameter estimate are obtained. It is shown that when the sample step size is small, the SD-based adaptive control is LQ optimal for the corresponding discretized system, and sub-optimal compared with that of the case where the parameter is known and the information is complete.     

On optimal choice of sampling strategies for linear system identification

In this paper we study the effect of the sampling strategy on the achievable accuracy in linear system identification experiments. We consider the experiment as being composed of a number of subexperiments and we impose the constraint that the sampling rate should be constant in each of the subexperiments. We investigate the effect of having different sampling rates in each of the subexperiments and we show that the optimal information return can be achieved by use of a finite number of sub-experiments. We develop the theory in two parts : the first relating to the identification of a stochastic process from output observations ; the second relating to the identification of an input-output transfer function from noisy observations. We present a number of examples which show that the appropriate choice of sampling strategy can be of paramount importance in identification experiments. We also show that the design can be extended in a straightforward manner to safeguard against a single non-informative experiment in the case of diffuse prior distribution for the parameters.     

A method for optimal control and filtering of multitime-scale linear singularly-perturbed stochastic systems

In this paper we introduce a transformation for the exact closed-loop decomposition of the optimal Kalman filter and the linear quadratic optimal controller of multi time scale continuous-time, linear, singularly-perturbed stochastic systems. The solution of the corresponding algebraic regulator and filter Riccati equations are obtained in terms of solutions of reduced-order subsystem, algebraic, Riccati equations corresponding to the system time scales. We have also obtained N completely independent reduced-order subsystem Kalman filters working in parallel in different time scales. This allows parallel processing of information with lower-order, different rates Kalman filters consistent with the system time scales.     

The design of optimal reduced order stochastic observers fordiscrete-time linear systems

The minimum variance state estimation of linear discrete-time systems with random white noise input and partially noisy measurements is investigated. An observer of minimal-order that attains the minimum-variance estimation error is found. The structure of this observer is shown to depend strongly on the geometry of the system. This geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate. The transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise. An explicit expression is found for the transfer function matrix of the observer, from which a simple solution to the linear discrete-time singular optimal filtering problem is obtained     

On efficient parametric identification methods for linear discrete stochastic systems

We construct a numerically stable algorithm (with respect to machine rounding errors) of adaptive Kalman filtering in order to solve the parametric identification problem for linear stationary stochastic discrete systems. We solve the problem in the state space. The proposed algorithm is formulated in terms of an orthogonal square-root covariance filter which lets us avoid a standard implementation of the Kalman filter.     

On linear quadratic optimal control of linear time-varying singular systems

Linear time-varying singular systems are treated in this paper. We focus on systems with constant-rank E matrices. It is shown that the existence of state feedback for impulse elimination is both sufficient and necessary for the existence of linear-quadratic optimal control. Also optimal control exists if and only if the corresponding fast subsystem is impulse-controllable. The results obtained are extensions of the existing time-invariant theory.     

Reduced-order optimal state estimator for linear systems with partially noise corrupted measurement

The problem of reduced-order Optimal state estimation for linear systems with singular noise covariance matrix is studied. It is shown that the optimal estimator is somewhat different from the Kalman filter. The state estimator problem in the singular case can be cast as a constrained optimization problem. Solving this optimization problem yields the truly optimal estimator. The estimator derived here is of the form of the hybrid estimator of Fairman [7]. However, the derivations here are somewhat more direct.