Nonparametric Procedures for Identification and Control of Linear Dynamic Systems

The linear dynamic systems theory appears to be the most developed part of the control theory. Comparative simplicity of this theory due to the superposition principle dominating there, makes it widely accessible when developing control systems for different kind of processes. Monographs [1, 2, 3] are devoted to numerous analysis and synthesis problems concerning these systems. In the following an approach is developed to linear dynamic plants control algorithms aimed at the estimation of the plant reverse operator for the case when the order of the equation describing the process is unknown.

In the present paper finite-dimensional dynamical control systems described by linear higherorder ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient condition for relative controllability in a given time interval for linear dynamical system with multiple point delays in control is formulated and proved. This condition is generalization to relative controllability case some previous results concerning controllability of linear dynamical systems without multiple point delays in the control. Simple numerical example, which illustrates theoretical result is also given. Finally, some remarks and comments on the existing results for controllability of dynamical systems with delays in control are also presented.

Traditional methods for simulating stochastic differential equations fail when applied to processes evolving in manifolds as these schemes leave the manifold, often after a single time step. In the deterministic case the geometric numerical integration literature is well established, but for stochastic differential equations in manifolds the literature is sparse and underdeveloped. In this paper, we construct a simple geometric Euler numerical scheme for simulating Brownian motion on SO(3) and study its uniform mean square convergence to the underlying stochastic differential equation. Simulations are provided which illustrate the preservation of the manifold structure.

Necessary and sufficient conditions for impulsive controllability of linear dynamical systems are obtained, which provide a novel approach to problems that are basically defined by continuous dynamical systems, but on which only discrete‐time actions are exercised. As an application, impulsive maneuvering of a spacecraft is discussed.

Complete controllability of linear dynamic systems

In this article we present a tutorial introduction to Generalized Proportional-Integral-Derivative Control (GPID). We particularly emphasize the links between flatness and GPID control, in the context of linear dynamic systems. We present several illustrative design examples, some of them of physical flavor. The performance of the controlled systems is evaluated by means of digital computer simulations.

Linear quadratic regulator (LQR) and proportional-integral-derivative (PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

The basic idea behind infinite horizon state feedback k-cost-cumulant (kCC) control is to seek a constant state feedback law which minimizes the value of a finite linear combination of the first k cost cumulants of a traditional integral quadratic cost associated with a linear stochastic system, when there is no specification of a large terminal time. The paper begins with the development of matrix algebraic equations for the cost cumulants. These equations permit the incorporation of classes of linear feedback controllers for linear dynamical systems defined on infinite horizon. The infinite horizon control problem with the optimization goal of a finite linear combination of the first k cost cumulants is then stated. Because the control optimization problem at hand involves matrix equality constraints, the optimal feedback solution is investigated by utilizing a Lagrange multipliers technique. The efficacy and applicability of this control paradigm, based upon the first three cost cumulants, are demonstrated on the second earthquake generation benchmark for response control of cable-stayed bridges (Caicedo, J.M., et al., 2003). The simulation results indicate that the cost cumulants in the infinite horizon case offer significant improvements in robust stability margin while keeping comparable levels of structural performance when comparing to those of the baseline LQG in the benchmark.

Metacognitive Control of Linear Dynamic Systems with Self-confidence Adaptation

This paper addresses to the constrained optimization problem for continuous-time dynamical systems. We extend and modify the existing results in constrained optimization to multi-input open-loop unstable systems, and the generalized nonquadratic cost is introduced. The paper elaborates a new procedure that exploits the beneficial advantages of an innovative design framework. Necessary and sufficient conditions for optimality are embedded and a computationally tractable technique is developed for synthesizing the nonlinear control laws. The key theorem, two propositions and three lemmas are formulated and proved. The procedure is illustrated through an example. This example illustrates the application of the explored optimization methodology to design a bounded smooth controller for the AFTI/F-16 fighter. This control law ensures the optimality in an entire flight envelope and belongs to a specified admissible set that satisfies the bounds imposed on deflections of the control surfaces.

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The methodology of estimation and control theory is considered in terms of response, stability, estimation, and control of linear dynamic systems. Within the context of discrete-time systems, multi-input, multi-output, nth-order linear systems are discussed, and general results for optimal estimation, optimal control, and other topics are presented. The application of these results to modeling human behavior is considered with special emphasis on man-machine system models.

In recent years, multilayer neural networks and recurrent networks have emerged as important components for representing nonlinear transformations and have proved particularly successful in pattern recognition and optimization problems. The authors explore methods for incorporating such networks in adaptive systems for the identification and control of complex nonlinear dynamical systems. >

The problem of recursive robust identification of linear discrete-time single-input single-output dynamic systems with correlated disturbances is considered. Problems related to the construction of optimal robust stochastic approximation algorithms in the min-max sense are demonstrated. Since the optimal solution cannot be achieved in practice, several robustified stochastic approximation algorithms are derived on the basis of a suitable non-linear transformation of normalized residuals, as well as step-by-step optimization with respect to the weighting matrix of the algorithm. The convergence of the developed algorithms is established theoretically using the ordinary differential equation approach. Monte Carlo simulation results are presented for the quantitative performance evaluation of the proposed algorithms. The results indicate the most suitable algorithms for applications in engineering practice.

On-Line Eigenvector Algorithms for the Identification of Dynamic Systems