Showing papers in "International Journal of Control in 1970"

Discrete calculus of variations

Abstract: This paper develops the concept of discrete calculus of variations and then demonstrates its application to optimisation problems for discrete systems. The parallels between this concept and classical calculus of variations are stressed throughout.

On the discrete time matrix Riccati equation of optimal control

Abstract: This paper is concerned with the discrete time matrix Riccati equation. The properties established are those of minimality, convergence, uniqueness and stability. Further the convergence of the policy space approximation technique is proved. These results are analogous to those known for the continuous-time Riccati equation, but the techniques used are simpler.

Diagonalization and inverses for non-linear systems†

Abstract: The problem of decoupling and inverting dynamic systems is considered. Attention is focused on the problems that arise as available linear techniques ore applied to nonlinear systems. The decoupling of non-linear multi-variate differential equations is illustrated in a series of three examples.

An algebraic method for control systems design

Abstract: A computer-oriented algebraic method for the design of duplicator class control systems is established. To involves four steps: (1) Equation fitting, (2) parameter assuming, (3) system simplifying and (4) quotient matching. The power and simplicity of the method are demonstrated by several examples.

A mathematical model of a 200 MW boiler

Abstract: A mathematical model is developed for a 200 MW coal-fired, drum-type natural circulation boiler with reheat. The complete unit is divided, for the convenience of the analysis, into several sections, viz. the economizer, the drum, the downcomer, etc. The dynamic behaviour of each section is described in mathematical terms. The equations are linearized and manipulated into state space form for solution in a digital computer. Results for the more important system variables are presented end discussed. Difficulties found in the modelling are emphasized.

On the linear servomechanism problem

Abstract: This paper considers the problem of the optimal control of a linear servomechanism using output feedback. It is shown that a generalization of the standard optimal control problem can be obtained in such a manner that optimal control is possible by feeding system outputs into a dynamic controller which operates only on past and present values of these outputs. This represents a new approach to the design of controllers for linear systems which may play an important role in applications of optimal control theory. It allows the linear servomechanism problem to be treated, for the first time, in a realistic manner.

Stability, transient behaviour and trajectory bounds of interconnected systems†

Abstract: In practice, many systems are complex and of high dimension. Many such systems may be viewed as being composed of several simpler sub-systems which when connected in an appropriate fashion yield the original composite system. The stability, the transient behaviour and estimates for the trajectory bounds of certain composite systems are analysed in terms of their sub-systems. This is accomplished by defining the stability of sub-systems and of composite systems in terms of certain time-varying sub-sets of the state space which are pre-specified in a given problem. After stating definitions of stability for sub-systems which are under the influence of perturbing forces and for composite systems, theorems are stated and proved which yield sufficient conditions for stability. These theorems involve the existence of Lyapunov-like functions which do not possess any particular definiteness requirements on V and [Vdot]. The time-varying sub-sets of the state space which are utilized in the stability definitions a...

Modal control of linear time invariant systems

Abstract: A procedure is presented for the determination of a linear control law (for a system with one input) such that the resultant closed–loop System has specified eigenvalues. The design procedure is simpler than existing techniques.

Comparison of four numerical algorithms for solving the Liapunov matrix equation

Abstract: This paper summarizes four alternative methods for computing the solution to the Liapunov matrix equation. A comparison of the methods is presented, along with sample computer runs. The necessity for solving this equation arises not only in certain phases of nonlinear stability analysis, but also in the design of optimal systems with fixed control structures. A more general form of this equation, which often appears in structural analysis, can be handled directly by one of the methods, and by the others with minor modifications.

An approximate method of state estimation for non-linear dynamical systems with state-dependent noise†

Abstract: In this paper a method of stochastic linearization is demonstrated for the purpose of establishing an approximate approach to solve filtering problems of non-linear stochastic systems with state-dependent noise in a Markovian framework. The models of both the dynamical system and the observation process are described by non-linear stochastic differential equations of Ito type. The principal line of attack is to expand the non-linear drift term into a certain linear function with coefficients which are determined under the minimal squared error criterion. Two methods of linearization are developed for the non-linear diffusion term. The linearized models are thus characterized by expansion coefficients dependent on both the state estimate and the error covariance. A method is given for the simultaneous treatment of the approximate structure of state estimator dynamics and of the running evaluation of the error covariance, including quantitative aspects of sample path behaviours obtained by digital simulatio...

The application of Kolmogorov–Rényi statistics to problems of parameter uncertainty in systems design†

Abstract: This paper presents a method for dealing with parameter uncertainty in system design which is based on the study of the statistical properties of an ensemble of systems defined by a given structure and by a priori parameter distributions rather than point parameter estimates. It is assumed that the model of the actual system is a random member of the ensemble. The object of the analysis is to design or modify the properties of the ensemble to ensure a high probability of adequate performance of the actual system. The primary statistical function employed is the sample distribution function. This function is used to estimate the true population distribution of a scalar variable chosen to measure the system property of interest. The sample distribution function is constructed from random samples of this figure of merit generated by a suitable digital computer programme. The accuracy of the estimation of the population distribution by the sample distribution is determined by application of statistical result...

Optimal distributed-parameter control using classical variational theory†

Abstract: The classical calculus of variations approach to the design of finite-dimensional optimal control systems is extended to a general class of fully non-linear distributed-parameter systems with distributed and/or boundary control inputs. As in lumped theory the result is a distributed-parameter two-time-point boundary-value problem in the canonical form of Hamilton, which in the linear case reduces to a partial differential equation of the Riccati type. A computational approximation technique for integrating the canonical equations is given which is based on the Riccati equation and does not require any hill-climbing procedure. The results are finally extended to an important class of mixed distributed-parameter and lumped-parameter systems.

Parameter identification for a class of distributed systems

Abstract: A new practical method, hereby called the ‘moment functional method’, is presented for the identification of the parameters of distributed parameter systems characterized by either the one-dimensional wave or diffusion equation. The method is extended to include systems characterized by a one-dimensional diffusion equation with a coefficient which is a polynomial in time. In this case the method determines the coefficients in the polynomial. The feasibility of the method lies in the on-line generation of linear time-invariant algebraic equations in the unknown system parameters by means of two Poisson filter chains which are fed from three points along the distributed system. The results of simulation studies are presented to illustrate the applicability of the method.

Optimization of linear systems of constrained configuration

Abstract: For the sake of simplicity it is often desirable to restrict the number of feedbacks in a controller. In this case the optimal feedbacks depend on the disturbance to which the system is subjected. Using a quadratic error integral as a measure of the response, three criteria of optimization are considered : The response to a given initial disturbance. The worst response to an initial disturbance of given magnitude. The worst comparison with the unconstrained optimal system. It is shown that for each of these criteria the gradient with respect to the feedbacks can be calculated by a uniform method. The solution may then be found either directly or by a descent procedure. The method is illustrated by an example.

Specific optimal control of the linear regulator using a dynamical controller based on the minimal-order Luenberger observer†

Abstract: The problem is considered of designing the optimal, fixed configuration, dynamical feedback controller of order (n—m) required to control an nth-order linear system with output of order m(m

A bootstrap method for the statistical estimation of model parameters

Abstract: This paper presents the derivation and evaluation of an asymptotically unbiased statistical estimator for the parameters in the vector difference equation canonical description of a linear multi variable system disturbed by correlated plant and measurement noise processes having rational spectral densities. The sequential estimation algorithm utilizes instrumental variables generated by a recursive filter incorporating the current parameter estimate in a novel manner. A spectral factorization technique for the identification of the noise process is discussed. Numerical results comparing the estimation algorithm with other methods are reported for scalar-output and multivariable-output cases.

Identification of linear systems through a Grammian technique

Abstract: A new technique of identification of linear stationary multi-variable systems is presented in this paper. The natural modes are determined accurately by a Gram matrix formulation of the output functions and the weightings of these modes are calculated by the schemes described. The state equations developed from the decomposed outputs provide, up to a linear equivalence upon the state space, a true description of the system. Among the advantages of the method are its relative insensitivity to noise in the data, and the explicit determination of the system order.

Approximate solutions of non-linear non-autonomous second-order differential equations†

Abstract: A method is presented for determining approximate solutions to a class of differential equations characterized by: where resonance phenomena (arising, for example, when ƒ(x, [xdot], t) = x cosω0 t) may be neglected. The approximation is developed from an asymptotic expansion in terms of the amplitude and phase of the solution. Three examples are considered in illustration of the application of the approximation technique, and using an integral error function, solution error is shown graphically for these examples in terms of equation parameters. An expression for the approximate solution is derived which makes it possible to determine solution accuracy for any function f(x, x˙, t) once the approximate amplitude envelope and phase relationships have been derived. Graphical solutions demonstrate the accuracy which can be maintained even up to relatively large values of the parameter µ.

On equivalence of quadratic loss functions

Abstract: When a linear, time.invariant plant is optimized with respect to the performance index where x is the state vector and u the control, the optimal control can be expressed as a feedback law u = –Kx Two pairs of matrices [Q, R] and [Q e, R e], yielding the same control law are equivalent. A necessary and sufficient condition is derived, in the single–input case, for a symmetric non–negative definite Q to be equivalent to a diagonal matrix Q*. This condition is satisfied by a plant described by equations in phase–variable canonical form, and a formula for Q* in terms of Q is given. It is shown that an equivalent Q e can be parameterized by exactly n non–negative parameters. For the multi–input case, Q e and R e must contain at least nr parameters, where n and r are the dimensions of x and u, respectively.

Linear control with incomplete state feedback and known initial-state statistics†

Abstract: A method is proposed for the design of a linear, time-invariant state feedback control for a linear, time-invariant, finite-dimensional system with infinite duration of control (regulator problem), which makes use of only those state variables which are available for measurement, providing that these are sufficient to render the system stable. The expected value vector and the covariance matrix for the initial state are presumed to be known. The cost function is quadratic and is expressed in terms of the initial state statistics and the cost-weighting matrix. The necessary conditions are derived for the minimization of the expected value of the cost. The minimization results in a set of m simultaneous polynomial equations in m unknowns where m is the product of the number of the available state variables and the number of control signals formed out of these. The theory is illustrated by a simple example of a third-order system.

Some circuit theory concepts revisited

Abstract: The concept of causality is introduced in a generalized form. Related system properties are then developed in an axiomatic manner. The unification of the development is illustrated through examples and discussions on related literature

Further results on “Approximate solutions of non-linear, non-autonomous second-order differential equations”†

Abstract: A method is presented for determining more accurate approximate solutions to a class of differential equations characterized by where resonance effects may be neglected. The results are an extension of work presented in a previous paper (Barkham and Soudack 1969). The extension involves the relaxation of the restriction that the fundamental oscillation frequency is amplitude invariant. Since, in general, the frequency is amplitude dependent for the system under consideration, relaxation of the aforementioned restriction leads to markedly improved approximate solutions. The notation used in the previous paper will be used here without redefinition of terms.

Feedback control of distributed parameter systems with spatially concentrated controls

Abstract: The problem of designing an optimal feedback controller is discussed for a distributed parameter system governed by a partial differential equation of parabolic type. A quadratic performance index is evaluated with spatially concentrated and time-discrete control, and controllability of this system is investigated. Also an estimator with multi-pointwise observation is constructed. Computational algorithms of these controller and estimator are derived, and to show their validities, a few numerical examples are given

Performance characteristics of an adaptive hydraulic servo-mechanism†

Abstract: This paper presents the results of an extensive set of experiments which were carried out on an adaptive hydraulic servo-mechanism: this system embodied adaptive controllers designed on the basis of Liapunov's direct method.

Sensitivity analysis of discrete Kalman filters.

Abstract: This paper investigates the sensitivity of discrete Kalman filters to erroneous models. Both parameter and structure (state dimensionality) sensitivity are considered, as well as deterministic and random parameter errors. Iterative algorithms are derived for the calculation of the actual filter error covariance matrix for the case of known (deterministic) modelling errors. For the case of random statistical and dynamical modelling errors, an optimal mean-square error estimate of the actual system performance is derived.

On filtering and smoothing algorithms for non-linear state estimation†

Abstract: This paper discusses a summary derivation of maximum a posteriori estimation for continuous and discrete non-linear systems. It is known that with Gaussian a priori statistics, the maximum a posteriori estimate is equivalent to an appropriate least squares fit. Filtering, fixed interval and fixed point smoothing algorithms for approximate non-linear estimation are obtained for the least squares eurve fit using ‘ running time ’ and ‘ fixed time ’ invariant embedding. Examples illustrating the use of the algorithms are presented.

On the state estimation for non-linear dynamic systems †

Abstract: The object of this paper is to present an approximate technique for state estimation of non-linear dynamical systems under noisy observations. The conditional cumulant is introduced, by which the conditional probability density can be characterized. A set of dynamical equations satisfied by conditional cumulants is derived, and an approximate method is proposed for computing the cumulants. The relation of the cumulant method to the stochastic linearization technique is also discussed. Finally the state estimation problem for linear stochaatic system with state-dependent disturbance is solved to illustrate the use of the Gaussian approximation.

On attempts to reduce the sensitivity of the optimal linear regulator to a parameter change

Abstract: One suggested method is to adjoin a quadratic form in the sensitivity vector to the integrand of the cost functional and to find the feedback, linear in the state and sensitivity vectors, which minimizes the augmented cost functional. Several authors have incorrectly assumed that the augmented system can be treated as another optimal regulator problem. In this paper it is shown that the problem, when correctly formulated, has no solution in the sense that there can be no finite state-independent feedback matrices which will satisfy the necessary conditions for an optimum.

Identification of stochastic distributed-parameter systems†

Abstract: The parameter estimation problem is treated for a general class of distributed-parameter systems disturbed by random inputs. The policy of transforming the problem into an equivalent lumped-parameter form amenable to existing statistical parameter estimation methods is followed. A computed example illustrates the effectiveness of the theory.