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This paper considers the suboptimal stochastic control of linear discrete-time dynamical systems with unknown or stochastically varying parameters. The suboptimal scheme is based upon the use of the open-loop-feedback-optimal (O.L.F.O.) method. The state and parameter estimates are generated by an extended Kalman filter algorithm. Numerical results for first order systems are presented.
This paper considers the suboptimal stochastic control of linear discrete-time dynamical systems with uncertain or stochastically varying parameters. The suboptimal scheme is based upon the use of the open-loop-feedback-optimal (OLFO) method. The state and parameter estimates are generated by an extended Kalman filter algorithm. Numerical results for first-order systems are presented.
The following paper is presented in viewgraph format and covers topics including: (1) linear state feedback control system; (2) Kalman filter state estimation; (3) relation between residual and stochastic part of output; (4) obtaining Kalman filter gain; (5) state estimation under unknown system model and unknown noises; and (6) relationship between filter Markov parameters and system Markov parameters.
This paper is a report on new methods for on-line testing of single-input/single-output linear and time-invariant systems. No interruption of system operation is necessary, since the signals used in this test procedure are the normal operating signals. Test results indicate that detectable system errors using finite amounts of data are well within tolerance for normal operation in all but the most stringent applications.
Dynamical system with time evolution described by second order linear differential equation in complex Banach space, discussing controllability
This paper delves into designing feedback control gains for a continuous-time linear quadratic regulator (LQR) problem that is constrained to certain predefined structure with unknown state matrix. We bring forth the ideas from reinforcement learning (RL) in conjunction with sufficient stability and performance guarantees in order to design these structured gains using the trajectory measurements of states and controls. Here we first formulate a model-based framework using dynamic programming (DP) to embed the structural constraint to the LQR gain computation in the continuous-time setting, and then subsequently, formulate a policy iteration RL algorithm that can alleviate the requirement of known state matrix in conjunction with maintaining the feedback gain structure. The design enables a distributed learning control design which is necessary for many large-scale cyber-physical systems. Theoretical guarantees are provided for stability and convergence of the structured reinforcement learning (SRL) algorithm. We validate our theoretical results with numerical simulations on a multi-agent networked linear time-invariant (LTI) dynamic system.
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
We construct and analyze multigrid preconditioners for discretizations of operators of the form ${\mathcal D}_{\lambda}+{\mathcal K}^*{\mathcal K}$, where $D_{\lambda}$ is the multiplication with a relatively smooth function $\lambda>0$ and ${\mathcal K}$ is a compact linear operator. These systems arise when applying interior point methods to the minimization problem $\min_{u} \frac{1}{2}(|\!|{\mathcal K} u-f|\!|^2 +\beta|\!|u|\!|^2)$ with box-constraints $\underline{u}\leqslant u\leqslant\overline{u}$ on the controls. The presented preconditioning technique is closely related to the one developed by Draganescu and Dupont [Math. Comp., 77 (2008), pp. 2001–2038] for the associated unconstrained problem and is intended for large-scale problems. As in that work, the quality of the resulting preconditioners is shown to increase as $h\downarrow 0$, but it decreases as the smoothness of $\lambda$ declines. We test this algorithm on a Tikhonov-regularized backward parabolic equation with box-constraints on the control and on a standard elliptic-constrained optimization problem. In both cases it is shown that the number of linear iterations per optimization step, as well as the total number of finest-scale matrix-vector multiplications, is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.
A number of issues are discussed relating to the analysis of a linear damped oscillator equation with a forcing term where the right side of the equation, g(t), is known only at fixed, equal time intervals. (A particular class of such equations has application to civil earthquake engineering.) It is noted that the equation is not a differential equation because the condition of uniqueness does not obtain. A general computational method is presented for calculating x(t), based on an unconditionally stable finite difference technique. The work generalizes the results given in the recent paper of Ly (1984).
The forced steady response of a single degree of freedom system involving a large nonlinearity, represented by unsymmetric piecewise-linear stiffness, is determined by a harmonic balance Newton-Raphson method with the application of the fast Fourier transformation (FFT) algorithm. All possible subharmonic, harmonic, and superharmonic responses are sought. The responses of two systems involving subharmonic and superharmonic dominant motions are calculated by the newly developed method and compared with previously published findings. The results obtained by using this method reveal the details of the response of the systems more efficiently than previous methods.
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Multiple-input, multiple-output mathematical models combined. Report presents mathematical background for combination of linear multiple-input, multiple-output equations of subsystems into overall matrix equations for whole system. Development of such system mathematical models essential for design of complicated control systems.
Using concepts of (C,A)-invariant and unobservability (complementary observability) subspaces, a geometric formulation of the failure detection and identification filter problem is stated. Using these geometric concepts, it is shown that it is possible to design a causal linear time-invariant processor that can be used to detect and uniquely identify a component failure in a linear time-invariant system, assuming: (1) The components can fail simultaneously, and (2) The components can fail only one at a time. In addition, a geometric formulation of Beard's failure detection filter problem is stated. This new formulation completely clarifies of output separability and mutual detectability introduced by Beard and also exploits the dual relationship between a restricted version of the failure detection and identification problem and the control decoupling problem. Moreover, the frequency domain interpretation of the results is used to relate the concepts of failure sensitive observers with the generalized parity relations introduced by Chow. This interpretation unifies the various failure detection and identification concepts and design procedures.
Control theory design package offers engineer full range of subroutines to manipulate and solve Linear-Quadratic-Gaussian types of problems. ORACLS is rigorous tool, intended for multi-input and multi-output dynamic systems in both continuous and discrete form. Written in FORTRAN.
A hybrid technique, called the block iterative LU solver, is proposed for solving the linear equations resulting from a finite element numerical analysis of certain fluid dynamics problems where the equations are weakly coupled between distinct sets of variables. Either the block Jacobi iterative method or the block Gauss-Seidel iterative solver is combined with LU decomposition.
The application of the minimal polynomial extrapolation (MPE) and the reduced rank extrapolation (RRE) to a vector sequence obtained by the linear iterative technique x(sub j) + 1 = Ax(sub j) = b,j = 1,2,..., is considered. Both methods produce a two dimensional array of approximations s(sub n,k) to the solution of the system (I - A)x = b. Here, s(sub n,k) is obtained from the vectors x(sub j), n is less than or equal to j is less than or equal to n + k + 1. It was observed in an earlier publication by the first author that the sequence s(sub n,k), k = 1,2,..., for n greater than 0, but fixed, possesses better convergence properties than the sequence s(sub 0,k), k = 1,2,.... A detailed theoretical explanation for this phenomenon is provided in the present work. This explanation is heavily based on approximations by incomplete polynomials. It is demonstrated by numerical examples when the matrix A is sparse that cycling with s(sub n,k) for n greater than 0, but fixed, produces better convergence rates and costs less computationally than cycling with s(sub 0,k). It is also illustrated numerically with a convection-diffusion problem that the former may produce excellent results where the latter may fail completely. As has been shown in an earlier publication, the results produced by s(sub 0,k) are identical to the corresponding results obtained by applying the Arnoldi method or generalized minimal residual scheme (GMRES) to the system (I - A)x = b.
An on-line minimum variance (MV) parameter identifier is developed which embodies both accuracy and computational efficiency. The formulation results in a linear estimation problem with both additive and multiplicative noise (AMN). The resulting filter which utilizes both the covariance of the parameter vector itself and the covariance of the error in identification is proven to be mean-square convergent and mean-square consistent. The MV parameter identification scheme is then used to construct a stable state and parameter estimation algorithm.
In this brief paper we examine a Riccati equation decomposition due to Reid and Lainiotis and apply the result to the continuous time-invariant linear filtering problem. Exploitation of the time-invariant structure leads to integration-free covariance recursions which are of use in covariance analyses and in filter implementations. A super-linearly convergent iterative solution to the algebraic Riccati equation (ARE) is developed. The resulting algorithm, arranged in a square-root form, is thought to be numerically stable and competitive with other ARE solution methods. Certain covariance relations that are relevant to the fixed-point and fixed-lag smoothing problems are also discussed.