Showing papers on "State variable published in 1971"

Modern Control Theory

Abstract: 1. Background and Preview. 2. Highlights of Classical Control Theory. 3. State Variables and the State Space Description of Dynamic Systems. 4. Fundamentals of Matrix Algebra. 5. Vectors and Linear Vector Spaces. 6. Simultaneous Linear Equations. 7. Eigenvalues and Eigenvectors. 8. Functions of Square Matrices and the Cayley-Hamilton Theorem. 9. Analysis of Continuous and Discrete Time State Equations. 10. Stability. 11. Controllability and Observability for Linear Systems. 12. The Relationship between State Variable and Transfer Function Descriptions of Systems. 13. Design of Linear Feedback Control Systems. 14. An Introduction to Optimal Control Theory. 15. An Introduction to Nonlinear Control Systems.

Thermodynamic Theory of Viscoplasticity

Abstract: Publisher Summary This chapter reviews the thermodynamic foundations of the theory of viscoplasticity. The essential feature of viscoplasticity is the simultaneous description of rheologic and plastic effects of a material. The necessity for simultaneous consideration of viscoelastic and plastic properties of a material is indicated by the experimental investigations of dynamic loads. The thermodynamic theory of elastic-viscoplastic materials presents for finite strains two basic difficulties. The first of these is connected with the kinematic description of plastic deformation. The second difficulty concerns the problem of choice of thermodynamic variables of state. The chapter discusses development of the thermodynamic theory of plasticity for finite strains using the rate-type theory, and generalized the inviscid theory of plasticity to nonisothermal finite deformations. The principle of material frame indifference called “invariance requirements under superposed rigid body motions,” was used and explored the thermodynamical restrictions. The chapter presents the formulation of the thermodynamic theory of a rate-sensitive plastic material within the framework of thermodynamics of a material with internal state variables. In this thermodynamic theory of an inelastic material, the deformation tensor and temperature are considered as thermodynamic state variables, while the components of the inelastic deformation tensor appear as internal state parameters.

Control of linear dynamic systems with set constrained disturbances

Abstract: A linear dynamic system with input and observation uncertainties is studied. The uncertainties are constrained to be contained in specified sets. No probabilistic structure is assumed. The problem of keeping the state of the system in a specified region is investigated. Necessary and sufficient conditions for a solution to the problem and an algorithm that constructs the control are derived for open-loop and closed-loop control laws. The algorithm is also approximated by a bounding ellipsoid algorithm. Two special control laws, linear and "linear-plus-dead-band," are studied, and the regions that contain the state and control are characterized. Ellipsoids that bound the state and control are also derived, and a simple example of linear and linear-plus-dead-band control laws is presented.

Optimal limited state variable feedback controllers for linear systems

Abstract: This paper considers the problem of designing compensators, the dimensions of which are fixed a priori, for linear systems. Two types of compensators are considered: first, static (gain only) compensators which operate directly upon the output signals to generate the controls, and second, dynamic compensators of fixed dimension. The equations that define the parameters of such compensators are developed.

Optimal Stationary Control with State Control Dependent Noise

Abstract: The steady state optimal linear regulator with state and control dependent noise is analyzed in a manner similar to that developed by Wonham [1]. By state dependent noise we mean Gaussian white noise with coefficient linear in the state variable, and similarly for control dependent noise. Using a Lyapunov criterion for the existence of stationary probability distributions due to Zakai, it is possible to treat equations leading to diffusion processes with degenerate differential generators. It is found that if the noise is sufficiently small, then an optimal control exists. Further analysis, again using Lyapunov methods, yields conditions under which an optimal control exists no matter how large the noise is.

Combustion of a premixed system in stagnation flow. I - Theoretical.

Abstract: The theoretical steady-state and transient characteristics of a premixed combustible system in laminar axisymmetric stagnation flow are studied by way of numerical solution of the governing boundary-layer equations. Computations, which are based on the oxidation reaction of carbon monoxide in humid air, invoke a minimum of simplifying assumptions. Steady-state results include prediction of (1) profiles of the state variables, (2) extinction of blow-off conditions, (3) ignition conditions, and (4) the heat flux to the stagnation point. Computer simulations of the unsteady state provide an indication of the stability of a given steady state to perturbations. Comparisons between some of the predictions of the mathematical model and the results of laboratory experiments will be reported in Part II.

Design of time-variable multivariable systems by decoupling and by the inverse

Abstract: System decoupling by state variable feedback is demonstrated for the time-variable case. Conditions under which decoupling is possible are formulated and the matrices effecting decoupling are given. The inverse of a general time-variable multivariable system is also determined. By applying this inverse a new synthesis method is developed.

A second-order method for state estimation of non-linear dynamical systems†

Abstract: The paper is concerned with further elaboration of the concept of statistical approximation of a given nonlinear function and its application to the problem of state estimation of a non-linear nois...

On the identification of nonlinear dynamical systems

Abstract: Two classes of nonlinear dynamical systems driven by independent noise on which linear discrete-time scalar measurements are performed also in the presence of additive independent noise are considered. The evolution operators for these classes are described, respectively, by algebraic and trigonometric polynomials in the state variables. Such polynomials frequently appear in equations describing physical systems. On the assumption that certain moments of the noise sources are known, a framework is developed which leads to convergent stochastic approximation algorithms for the identification of system parameters.

Lower order generalized aggregated model and suboptimal control

Abstract: 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.

Controllability and pole assignment for discrete time linear systems defined over arbitrary fields

Abstract: Discrete time finite dimensional autonomous linear systems, investigating controllability and pole assignment to closed loop transfer matrix by choice of state variable feedback gain

Models and performance functionals for load frequency control in interconnected power systems

Abstract: Particular state variable representations are developed for the load frequency control of a linearized two-area interconnected power system. Associated cost functionals are developed emphasizing differences between the various state variable configurations. Comparison of the resulting modern optimal feedback controllers with the conventional are made.

Optimal control of systems with state-dependent time delay†

Abstract: Necessary conditions for optimal control of systems containing a time delay that is a function of the state of the system and of time are derived by utilizing calculus of variations. The time delay may be in the state vector and in the control vector. The state vector and the control vector can be constrained by inequality constraints. A transformation to eliminate state variable inequality constraints by increasing the dimensions of state space, developed by Jacobson for an undelayed system, is extended to a system with time delays. Necessary conditions to obtain an optimal delay are shown, and an example of finding an optimal delay is included. A gradient algorithm for systems with state dependent time delays has been developed.

Memory effects and internal changes of a material

Abstract: A thermodynamic theory of a material with memory and internal changes is proposed. A conception to treat such material as a generalized simple body is shown. Fading memory phenomenon is used to describe the rheological properties and internal state variables are introduced to describe the internal structural changes during plastic deformation. The application of this theory to the description of an elastic-viscoplastic material is given. The discussion of the behavior of an elastic-viscoplastic material during particular processes is presented.

Inventory Models with Forecasting and Dependent Demand

Abstract: In general, the single-product inventory model in which demands in successive periods are not independent is difficult to treat. This paper defines a large class of such problems, when there is a positive lead time for delivery, which can be treated by the classical formulation with a single state variable. All results which hold for inventory models with a constant delivery lag can be shown to hold also for this model. An application is made to a system in which demands are generated by part failure and in which a portion of these failures are repaired after a given constant or probabilistic time.

An application of the linear-quadratic tracking problem to economic stabilization policy

Abstract: Short-term economic stabilization policy is approached as a problem in optimal control. The optimal control problem is defined as a dual discrete time tracking problem (nominal state and nominal policy trajectories are tracked) for a linear time-invariant system with a quadratic cost functional. This problem is solved analytically, and the solution is applied to a 28 state variable quarterly econometric model. The experimental results demonstrate that this approach is valuable both as a tool for policy planning and as a method of analyzing the dynamic properties of econometric models.

A direct approach to the design of asymptotically optimal control lers

Abstract: Many optimal control solutions require a complete set of measurements of current state variables, which may not be fully available It is reasonable to ask whether compensators cannot be designed in such a way that the desirable qualities of the optimal control are reproduced One method of constructing a compensator that generates an asymptotically optimal control is to generate an estimate of the complete set of state variables by an auxiliary dynamic system, such as an observer or a Kalman filter It can be shown, however, that a simpler design is often possible by employing the fact that usually a linear combination of the set of state variables is all that is required to reconstruct the optimal control A simple direct method of determining such a controller is presented in this paper,

Sub-optimal nonlinear controllers for relay and saturating control systemst

Abstract: The optimal control signal for saturating or relay control systems is a complicated function of several state variables. A sub-optimal technique is to use instead a function which is simply a linear combination of functions each of which has only one state variable as argument. When the plant consists of a set of pure integrators, previous work has shown how to reduce the resulting sub-optimal controller design problem to the determination of a few parameters. In the present paper the choice of these parameters is considered. For a third-order system it is shown that the motion can be depicted in a ‘reduced state space’ having only two coordinates. The portrait in this state plane enables pathological behaviour (instability and excessively slow sliding motion caused by incorrect adjustment of controller parameters) to be recognized and avoided. The sub-optimal controller can then be adjusted so that settling times about 30° greater than the optimal values can be obtained. The reduced state space is also u...

Instability of Optimal Stochastic Control Systems under Parameter Variations

Abstract: The deterioration of a linear optimal stochastic control scheme, designed under the assumptions of the certainty-equivalence principle (the optimal filter and controller, determined independently, combine to give a totally optimal system), is investigated when the parameters of the actual system do not coincide with the design values. This linear suboptimal stochastic system is described by a covariance matrix composed of covariances of the estimates of the state variables, the errors in the estimates of the state variables, and the correlation between these errors and the estimates. In particular, this paper is concerned with the covariance matrix resulting from a single state dynamical system and a scalar linear measurement function of both the state variable and the control variable (e.g., accelerometer measurements). A modeling error in the control variable coefficient of the measurement function may induce instability in the stochastic system with either unstable or stable dynamics. Furthermore, the absolute magnitude of the error in the control variable coefficient directly influences system stability, not the relative error. Thus, relatively small errors compared to the design value of this coefficient may be quite important. 4 LTHOUGH there are many studies on divergence of op-£^- tirnal filters, little attention has been given to the effect of modeling inaccuracies on optimal linear stochastic control systems. Here we extend Fitzgerald's1 investigation of Kalman filter divergence to optimal linear stochastic control systems. These systems are designed under the certainty equivalence principle2 which states that if the expected value of a quadratic function of the state and the control variables is to be minimized subject to linear dynamics, the optimal system is composed of an optimal filter in cascade with an optimal controller. This separation is possible because the estimate in the state is uncorrelated with the error in this estimate. If the parameters in the assumed model of the dynamics or the measurement device deviate from the parameters of the actual system, the estimate and the error in the estimate become correlated. The behavior of the system because of the gains based on an inaccurate model is studied by considering the coupled matrix covariance equation composed of the Covariances of the error in the estimate, the estimate, and the estimate with its error. Some of the characteristics of this linear matrix equation are studied through a scalar linear dynamic equation. The errors in system parameters enter into the 2X2 covariance equation in a dimensionless form allowing the following general results to be obtained: 1) The stochastic control system may be unstable when the nonoptimal filter and deterministic control systems individually are stable. 2) Instability occurs only when the error in the parameter exceeds a finite threshold value. 3) If the measurement is a linear function of the control variable as well as the state (e.g., accelerometer measurements) and there are errors in the coefficient of the control, then instability of the total system may occur for both stable and unstable dynamical systems. 4) The filter or control gains are not functions of the coefficient of the control variable in the measurement function. Consequently, Presented as Paper 70-36 at the AIAA 8th Aerospace Sciences

Modal theory of state observers

Abstract: The paper presents a modal theory of state observers for the control of certain classes of multivariable time-invariant linear systems whose plant matrices possess distinct eigenvalues and whose state variables are not all available for feedback It is shown that the theory leads to an algorithm which greatly facilitates the design of state observers whose dynamical characteristics may be arbitrarily prescribed

Der Entwurf linearer Regelungssysteme im Zustandsraum (Design of a Linear Control System Involving State-Space Vector Feedback),

Abstract: : A simple method for obtaining a desired characteristic equation for a system x(k+1)=Ax(k) by state variable feedback u= -k' x is derived. It does not require a transformation of the state equations to a canonical form. The result is: Obtain k' by substituting the given dynamic matrix A into the desired characteristic polynomial and premultiplying by the last row of the inverse of the controllability matrix. The dual result for the observers of order n and n-1 is shown. (Author)

On a method of calculation in modal control

Abstract: The problems of modal control for linear multivariable time-invariant systems incorporating multi-loop single-input proportional controllers are formulated and solved in a manner which is thought to be both novel and potentially useful. For such systems, it is shown that all eigenvalues may be specified exactly provided that the number of loops is equal to the number of state variables and that the usual conditions for controllability are satisfied. A mathematical corollary of this is that, in general, all the eigenvalues of a given matrix may be changed to new specified values by the addition of an appropriate unit rank matrix to the original matrix. Methods are discussed which enable approximate values of prescribed eigenvalues to be obtained in cases where the number of loops falls short of the number of state variables or where a number of state variables are non-measurable. Problems associated with eigenvalue/gain sensitivity and loop failure are also discussed and illustrative examples are provided.

Sub-optimal solutions of linear control and eigenvalue pattern problems†

Abstract: The present paper deals with sub-optimal solutions for problems of minimizing integral cost fimetioiuils which consider the state variables alone for stationary linear processes. The control vector and its time derivative are assumed to be bounded. The resulting control policies are linear with respect to the state vector. The performance of the proposed sub-optimal policies is compared with that of classical control systems and a procedure for sub-optimal eigenvalue patterns is presented for systems of any order.

Inverse stability problem and its applications

Abstract: Based on Zubov's procedure for constructing a Liapunoy function and the solution of linear undetermined equations, the paper develops a method to determine a dynamical system, the stability or instability boundary of which is given as a closed form in the state space. It is shown that the results obtained can be applied to the synthesis of an asymptotically stable system or a finite time stable system.

A Dual Mode Excitation Control of Synchronous Machine for Suppressing its Hunting

Abstract: To suppress the hunting of synchronous machine caused by stepwise load change, a dual mode field excitation control was investigated for conventional (one-axis) machine and 2-axis machine. For one-axis machine, a dual mode is applied successively, the first one being a bang-bang suboptimal control mode decided by use of energy function instead of Pontryagin's switching function, this mode is applicable to dynamic equation described in nonlinear form. The second one is a linear feedback control mode composed of a linear combination of state variables, and it will be adopted to the range near the steady state operating point where the equation will be linearized. For 2-axis machine at the instant of load change the original excitation introduced by mixing these two controls is efficiently applied. From the results of the investigation it is recognized that the employment of 2-axis synchronous machine is a potential means for the improvement of system stability.

Comments on "On the inverse optimum control problem for a class of nonlinear autonomous systems"

Abstract: The purpose of this correspondence is to indicate an error in Thau's paper in the expression for the part of the loss function involving the state variables, and to derive a corrected and simplified version for this expression. Furthermore, a condition in the frequency-domain is presented that is necessary as well as sufficient for the loss function involving the state variables to be nonnegative definite.