On the generalized pid control of linear dynamic systems

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.

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.

A linear dynamic system is considered. Orthogonal series procedures are developed to recover the state and its derivative. The uniform mean square error convergence is investigated. An application to identification and open-loop control is discussed.

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 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.

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.

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.

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

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Complete controllability of linear dynamic systems

This article presents the central finite-dimensional H ∞ controller for linear time-varying systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, this article reduces the original H ∞ controller problem to the corresponding H 2 controller problem, using the technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), ‘State-space Solutions to Standard H 2 and H Infinity Control Problems’, IEEE Transactions Automatic Control, 34, 831–847]. This article yields the central suboptimal H ∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H 2 controller obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008), ‘Optimal LQG Controller for Linear Systems with Unknown Parameters’, Journal of The Franklin Institute, 345, 293–302]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H ∞ controller for linear systems with exactly known parameter values.

By complementarily fusing the robust stabilizability condition, the orthogonal-functions approach (OFA) and the hybrid Taguchi-genetic algorithm (HTGA), an integrative method is proposed in this paper to design the robust-stable and quadratic-optimal static output feedback controller such that i) the linear singular control system with structured parameter uncertainties is regular, impulse-free and asymptotically stable and ii) a quadratic finite-horizon integral performance index for the linear nominal singular control system can be minimized. Based on some essential properties of matrix measures, a new sufficient condition is presented for ensuring that the linear singular system with structured and quadratically-coupled structured parameter uncertainties is regular, impulse free and asymptotically stable. By using the OFA and the robust stabilizability condition, the dynamic-optimization problem for the robust-stable and quadratic-optimal static output feedback control design of the linear uncertain singular system is transformed into a static-constrained-optimization problem represented by the algebraic equations with constraint of robust stabilizability condition; thus greatly simplifying the robust-stable and quadratic-optimal static output feedback control design problem of the linear uncertain singular system. Then, for the static-constrained-optimization problem, the HTGA is employed to find the robust-stable and quadratic-optimal static output feedback controller of the linear uncertain singular control system. One design example of the robust-stable and quadratic-optimal static output feedback controller for a mass-spring-damper mechanical system with structured parameter uncertainties is given to demonstrate the applicability of the proposed integrative approach.

This paper investigates the application of associative reinforcement learning techniques to the optimal control of linear discrete-time dynamic systems. Associative reinforcement learning involves the trial and error interaction with a dynamic system to determine the control actions that optimally achieve some desired performance index. The methodology can be applied either online or off-line and in a model based or model free manner. Associative reinforcement learning techniques are applied to the optimal regulator (LQR) control of discrete-time linear systems. Adaptive critic designs are implemented and the convergence speed compared for the different approaches. These methods can determine the optimal state and state/action value function and the optimal policy without requiring system models. (4 pages)

Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept