Showing papers in "IEEE Transactions on Automatic Control in 1968"

An innovations approach to least-squares estimation--Part II: Linear smoothing in additive white noise

Abstract: The innovations approach to linear least-squares approximation problems is first to "whiten" the observed data by a causal and invertible operation, and then to treat the resulting simpler white-noise observations problem This technique was successfully used by Bode and Shannon to obtain a simple derivation of the classical Wiener filtering problem for stationary processes over a semi-infinite interval Here we shall extend the technique to handle nonstationary continuous-time processes over finite intervals In Part I we shall apply this method to obtain a simple derivation of the Kalman-Bucy recursive filtering formulas (for both continuous-time and discrete-time processes) and also some minor generalizations thereof

Control of large-scale dynamic systems by aggregation

Abstract: A method is proposed to obtain a model of a dynamic system with a state vector of high dimension. The model is derived by "aggregating" the original system state vector into a lower-dimensional vector. Some properties of the aggregation method are investigated in the paper. The concept of aggregation, a generalization of that of projection, is related to that of state vector partition and is useful not only in building a model of reduced dimension, but also in unifying several topics in the control theory such as regulators with incomplete state feedback, characteristic value computations, model controls, and bounds on the solution of the matrix Riccati equations, etc. Using the quantitative definition of weak coupling proposed by Milne, a suboptimal control policy for the weakly coupled system is derived. Questions of performance degradation and of stability of such suboptimally controlled systems are also answered in the paper.

On state estimation in switching environments

Abstract: Work concerned with the state estimation in linear discrete-time systems operating in Markov dependent switching environments is discussed. The disturbances influencing the system equations and the measurement equations are assumed to come from one of several Gaussian distributions with different means or variances. By defining the noise in this manner, periodic step changes in the inputs which cannot be feasibly measured for economic or technical reasons can be detected and corrected. These changes can be in the amplitudes of the inputs or in the variances of stochastic inputs. A Bayes estimation procedure is applied to the problem of filtering the observations of the system so that an estimate of the system state is obtained. Computer simulations for the optimal and suboptimal estimators are also presented.

Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements

Abstract: This paper presents the derivation of the dynamical equations of a second-order filter which estimates the states of a non-linear plant on the basis of discrete noisy measurements. The filter equations contain terms involving the second-order partial derivatives of the plant and output equations. Simulation results are presented which yield a comparison of the performance of the first-versus the second-order filter when applied to a nonlinear third-order system. The results indicate that the inclusion of second-order terms can markedly improve the filter performance.

Sets of possible states of linear systems given perturbed observations

Abstract: The state equation x_{n} = A_{n}x_{n-1}+B_{n}W_{n} and the output equation y_{n} = C_{n}x_{n}+D_{n}W_{n} relate the finite-dimensional vectors x_{n}, y_{n} and w n . The initial state x 0 is known to belong to a set X 0 and each w n . to a set W n , where X_{0}, W_{1}, . . . , W_{N} are compact and convex. Define S_{n}(y_{1}, ... , y_{n}) as the set of all possible values of x n compatible with observation of outputs y_{1}, ... , y_{n} ; it is compact and convex. S n plays the same role as the a posteriori distribution in the stochastic case and is determined recursively by S_{n-1}(y_{1}, ... , y_{n-1}) and y n . The sets involved are completely characterized by their support functions. The law of evolution of the support function of S n is established. Some special cases and applications are pointed out.

Adaptive step size random search

Abstract: Fixed step size random search for minimization of functions of several parameters is described and compared with the fixed step size gradient method for a particular surface. A theoretical technique, using the optimum step size at each step, is analyzed. A practical adaptive step size random search algorithm is then proposed, and experimental experience is reported that shows the superiority of random search over other methods for sufficiently high dimension.

The numerical solution of A'Q+QA =-C

Abstract: The eigenvalues of the system matrix are used to construct a matrix which is a transformation in state space between two most frequently used canonical forms.

Optimal control of the linear regulator with constant disturbances

Abstract: An optimal control problem for a linear regulator with constant external disturbance is formulated. It is shown that, for a suitably selected quadratic-type performance index, the optimal control is not an explicit function of the external disturbance. Moreover, the optimal control can be synthesized as a time-invariant linear function of the state plus the first time integral of a certain other time-invariant linear function of the state.

A minimax control problem for sampled linear systems

Abstract: A linear differential system is subject to a bounded control and a bounded disturbance. The controller receives the value of the state at a finite number of fixed sampling times. The cost is a convex or concave function of the state at a fixed final time. Given any control law, there is a maximum cost over all perturbations, the guaranteed performance for this control law. It is desired to find the minimum of this number over all control laws. Fenchel's theory of conjugate convex functions is used to set up an algorithm dual to the dynamic programming approach. This algorithm deals with the support functions of reachable sets, more easily determined than other descriptions of these sets. A discrete minimax principle is generally incorrect for problems of this class.

Comments "On pole assignment in multi-input controllable linear systems"

Abstract: A short and direct new proof is given to Wonham's theorem that a time invariant multi-input linear dynamical system is controllable only if its poles can arbitrarily be reassigned in a closed-loop system by means of a constant (state variable) feedback law. The construction provided in the proof is directly applicable as an effective algorithm for this pole assignment.

Limited memory optimal filtering

Abstract: Linear and nonlinear optimal filters with limited memory length are developed. The filter output is the conditional probability density function and, in the linear Gaussian case, is the conditional mean and covariance matrix where the conditioning is only on a fixed amount of most recent data. This is related to maximum-likelihood least-squares estimation. These filters have application in problems where standard filters diverge due to dynamical model errors. This is demonstrated via numerical simulations.

Conditions for asymptotic stability of the discrete minimum-variance linear estimator

Abstract: Stability of the discrete homogeneous linear minimum-variance estimation formulas is investigated. Sufficient conditions for uniform asymptotic stability in the large are derived. The conditions, if satisfied, also imply stochastic controllability and observability of the plant.

Optimization of natural-gas pipeline systems via dynamic programming

Abstract: The complexity and expense of operating natural-gas pipeline systems have made optimum operation and planning of increased interest to the natural-gas pipeline industries. Since the operations of natural-gas pipeline sytems are characterized by inherent nonlinearities and numerous constraints, dynamic programming provides an extremely powerful method for optimizing such systems. This paper summarizes the application of dynamic programming techniques to solve optimization problems that occur in the short-term (transient) terminal distribution and long-term (steady-state) transmission of natural gas.

The design of suboptimal linear time-varying systems

Abstract: This paper deals with the determination of suboptimal feedback laws for the control of linear time-varying systems with quadratic performance criteria. Easily implementable time functions are used to generate the required time-varying gains; free constant parameters in the control law description are chosen so as to minimize an "averaged" control cost. A simple example is included to illustrate the theory.

Singular perturbation method for reducing the model order in optimal control design

Abstract: The optimal control for a high-order model of the plant is approximated by some functions obtained from two low-order models, the second being the sensitivity model of the first. The resuiting system is optimally sensitive with respect to the change of model order. The method is aimed at improving the "low-order design" in critical cases when the plant order is too high for the iterative optimization procedures to be practical.

Analysis and design of model following control systems by state space techniques

Abstract: Analysis and design of model follower control system by state space techniques, giving testing procedure

A simplified viewpoint of hyperstability

Abstract: The definition of a hyperstable system according to Popov is given a network theoretic interpretation, and proofs are presented outlining the connection between passive and hyperstable systems.

An off-axis circle criterion for stability of feedback systems with a monotonic nonlinearity

Abstract: An off-axis circle criterion is stated in this paper as a sufficient condition for the stability of a nonlinear feedback system with a single monotonic nonlinearity. The criterion provides a geometric method for determining a sector (K_{1} with in which the nonlinearity may lie for asymptotic stability. The relation between the criterion stated and the well-known Popov criterion and the circle criterion for time-varying systems is also discussed. Two examples are included to clarify various aspects of the criterion and its relation to existing results.

On the stabilization of controllable systems

Abstract: Conditions are obtained for the asymptotic stabilization of linear dynamical systems through the use of linear state-variable feedback The procedures developed rely on uniform controllability of the given systems, and yield explicit expressions for the transition matrices of the stabilized systems As a consequence of the treatment, explicit expressions are obtained for Liapunov transformations which reduce the stabilized systems to canonical (phase-variable) form Results are given for time-varying multivariable (multi-input) systems

Minimax controller design

Abstract: Given a system with uncertain parameters, a performance index, and a controller structure, it is sometimes appropriate to determine the controller parameters by minimaximizing the performance index. In many of the cases where such a controller is too pessimistic it is appropriate to minimaximize a performance sensitivity. Some properties of these two types of controllers are presented. Since the minimax problems that arise do not in general have a saddle point, they must be solved by an iterative or search procedure. A new algorithm for solving algebraic minimax problems with or without a saddle point is presented and proved to converge. The rate of convergence is discussed. The algorithm is applied to the design of a controller for a string of three vehicles.

Closed-loop sensitivity reduction of linear optimal control systems

Abstract: Inequalities are derived which demonstrate the reduction of sensitivity to plant-parameter variations of closed-loop linear optimal systems (as compared to nominally equivalent open-loop systems) according to a particular measure of sensitivity. The shortcomings of some other measures of sensitivity are discussed and illustrated in numerical examples. Numerical results for the sensitivity comparison of a first-order system and a third-order pitch-axis control system are given.

On the design of linear systems with piecewise-constant feedback gains

Abstract: This paper presents the theory and the computational algorithm required to determine the piecewise-constant feedback gains for the optimal control of a linear system with quadratic performance index over a finite control interval [t_{0} T] . A third order example illustrates the relation of the piecewise-constant gains to the truly optimal time-varying gains.

New guidance

Abstract: Since the day of the first Administrative Committe meeting in 1953, over 2000 papers have been processed for possible publication in this Transactions, not including the numerous technical correspondence items, and there have been 45 issues of the Transactions. An enormous amount of much appreciated support and effort has been given by a vast number of reviewers and the committee members. The work involved to create each issue has been considerable, but it has been rewarded with the formation of acquaintances throughout the world and with a sense of accomplishment that does not always emerge from professional society activities. Fourteen years have also passed and at last, for the first time, starting with the next issue, there will be new guidance for this Transactions: a new editor, new associate editors, and a Group Administrative Committee without one of the original committee members. The group is indeed fortunate in being able to obtain the services of Prof. John B. Lewis as the new editor.

An analysis of the divergence problem in the Kalman filter

Abstract: The effect of modeling errors in a linear discrete stochastic system upon the Kalman filter state estimates is investigated. Errors in both plant dynamics and noise covariances are permitted. The errors are characterized in such a manner that a linear recursion relation for the actual estimation error covariances can be derived. Conditions which guarantee that the covariance matrix remains bounded are described in terms of the asymptotic stability of the homogeneous part of the covariance equation and the boundedness of the forcing terms in the inhomogeneous equation.

Eigenvalue sensitivity and state-variable selection

Abstract: The first part of this paper presents a new measure of sensitivity specifically applicable to the realization of a linear discrete system on a digital computer. It is also shown that the sensitivity of the eigenvalues to parameter inaccuracies in the realization depends strongly on the choice of state variables. From these considerations, a realization is obtained which is "best" for a large class of systems of interest with regard to minimizing storage requirements, arithmetic operations, parameter accuracy, and eigenvalue sensitivity. The second half of the paper considers the very practical problem of determining the number of bits accuracy required in the computer-stored parameters of the system to achieve satisfactory performance. For the realization found to be a best compromise, equations are obtained for determining these bit requirements. Examples are given showing the application of this realization to the computer implementation of a discrete filter, and a comparison is given to other possible realizations.

Best least-squares representation of signals by exponentials

Abstract: In this paper the approximation of a given real time function over (0, \infty) by a linear combination of a given number n of exponentials is considered, such that the integrated squared error is minimized over both the n coefficients of the linear combination and the n exponents used. The usual necessary condition for stationarity of the integrated squared error leads to a set of 2n simultaneous equations, nonlinear in the exponents. This condition is interpreted in the geometric language of abstract vector spaces, and an equivalent condition involving only the exponents, with the coefficients suppressed, is developed. It is next indicated how this latter condition can be applied to signals which are not known analytically, but only, for example, as voltages recorded on magnetic tape, or as a table of sampled values. The condition still in effect requires solution of nonlinear algebraic equations, and a linear iterative method is proposed for this purpose. Finally, the procedure is illustrated with a simple example.

On a class of linear stochastic differential games

Abstract: The solution for a class of stochastic pursuit-evasion differential games between two linear dynamic systems is given. This class includes the classical interception game in Euclidean space. The performance index which is optimized is quadratic, and one of the two players has imperfect (noisy) knowledge of the states of the two systems. The "certainty-equivalence principle' or, equivalently, the technique of separating the estimator and the controller which characterizes the standard stochastic control problem is shown to be applicable to this class of differential games.