Showing papers in "IEEE Transactions on Automatic Control in 1979"

Finite spectrum assignment problem for systems with delays

Abstract: In this paper linear systems with delays in state and/or control variables are considered. The objective is to design a feedback law which yields a finite spectrum of the closed-loop system, located at an arbitrarily preassigned set of n points in the complex plane. It is shown that in case of systems with delays in control only the problem is solvable if and only if some function space controllability criterion is met. The solution is then easily obtainable by standard spectrum assignment methods, while the resulting feedback law involves integrals over the past control. In case of delays in state variables it is shown that a technique based on the finite Laplace transform, related to a recent work on function space controllability, leads to a constructive design procedure. The resulting feedback consists of proportional and (finite interval) integral terms over present and past values of state variables. Some indications on how to combine these results in case of systems including both state and control delays are given. Sensitivity of the design to parameter variations is briefly analyzed.

Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems

Abstract: The extended Kalman filter is an approximate filter for nonlinear systems, based on first-order linearization. Its use for the joint parameter and state estimation problem for linear systems with unknown parameters is well known and widely spread. Here a convergence analysis of this method is given. It is shown that in general, the estimates may be biased or divergent and the causes for this are displayed. Some common special cases where convergence is guaranteed are also given. The analysis gives insight into the convergence mechanisms and it is shown that with a modification of the algorithm, global convergence results can be obtained for a general case. The scheme can then be interpreted as maximization of the likelihood function for the estimation problem, or as a recursive prediction error algorithm.

A Hessenberg-Schur method for the problem AX + XB= C

Abstract: One of the most effective methods for solving the matrix equation AX+XB=C is the Bartels-Stewart algorithm. Key to this technique is the orthogonal reduction of A and B to triangular form using the QR algorithm for eigenvalues. A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form. The resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B . The stability of the new method is demonstrated through a roundoff error analysis and supported by numerical tests. Finally, it is shown how the techniques described can be applied and generalized to other matrix equation problems.

Uncertain dynamical systems--A Lyapunov min-max approach

Abstract: A class of dynamical systems in the presence of uncertainty is formulated by contingent differential equations. Asymptotic stability (in the sense of Lyapunov) is then developed via generalized dynamical systems (GDS's). The uncertainty is deterministic; the only assumption is that its value belongs to a known compact set. Application to variable structure and model reference control systems are discussed.

Invertibility of multivariable nonlinear control systems

Abstract: This paper gives sufficient conditions for the invertibility of nonlinear control systems of the form \dot{x}= A(x)+ u_{1}B_{1}(x) +... + u_{m} B_{m}(x);y=c(x) , where the state space is a real analytic manifold. These conditions are also necessary in the case of single-input nonlinear systems and multivariable time-invariant linear systems. For invertible systems we construct nonlinear inverse systems.

Modeling and control of constrained dynamic systems with application to biped locomotion in the frontal plane

Abstract: The general problem of control of state-space constrained dynamic systems is considered. Based on the derivation of the constraint forces as explicit functions of the state and the input, a general model is presented that encompasses both the constrained case and the corresponding unconstrained case. Stability and point-to-point motion of these systems in the vicinity of an operating point are considered under operating conditions which either maintain or deliberately violate the constraints. Algorithms for computation of the necessary feedback gains in the vicinity of the operating point are discussed. For a three-link biped model, several motions in the vicinity of the vertical stance are considered, and the necessary feedback gains are derived. Digital computer simulation of some biped motions are carried out to serve as examples and to demonstrate use of the theory. This work is to be regarded as an elementary step in better understanding of human motor control and in the design of robots and prosthetic devices.

The convergence of AML

Abstract: In this work it is shown that provided a certain positive real condition is satisfied, the AML recursion for the parameters of a scalar ARMAX time series model converges with probability one without the need of monitoring. Previous proofs of convergence had effectively required that the recursion be monitored.

The optimal linear-quadratic time-invariant regulator with cheap control

Abstract: The infinite-time linear-quadratic regulator is considered as the weighting on the control energy tends to zero (cheap control). First, a study is made of the qualitative behavior of the limiting optimal state and control trajectories. In particular, the orders of initial singularity are found and related to the excess of poles over zeros in the plant. Secondly, it is found for which initial conditions the limiting minimum cost is zero (perfect regulation). This generalizes an earlier result of Kwakernaak and Sivan. Finally, a simple extension is made to the steady-state LQG problem with cheap control and accurate observations.

Linear recursive state estimators under uncertain observations

Abstract: For linear systems with uncertain observations, we investigate the existence of recursive least-squares state estimators. The uncertainty in the observations is caused by a binary switching sequence γ k , which is specified by a conditional probability distribution and which enters the observation equation as z_{k} = \gamma_{k} H_{k} x_{k}+\upsilon_{k} . Conditions are established which lead to a recursive filter for x k , and a procedure for constructing a mixture sequence {\gamma_{k}} that satisfies these conditions is given. Such mixture sequences model the transmission of data in multichannels as in remote sensing situations as well as data links with random interruptions. They can also serve as models for communication in the presence of multiplicative noise.

Closed-loop Stackelberg strategies with applications in the optimal control of multilevel systems

Abstract: This paper develops a new approach to obtain the closed-loop Stackelberg (CLS) solution of an important class of two-person nonzero-sum dynamic games characterized by linear state dynamics and quadratic cost functionals. The new technique makes use of an important property of nonunique representations of a closed-loop strategy, and it relates the CLS solution to a particular representation of the optimal solution of a team problem. It is shown that, under certain conditions, the CLS strategies for the leader are linear and of the one-step memory type, while those of the follower can be realized in linear feedback form. Exact expressions are given for the optimal coefficient matrices involved, which can be determined recursively. These results are then extended to multilevel discrete-time control of linear-quadratic systems which are characterized by one central controller and K second-level controllers. Conditions are obtained under which a one-step memory strategy of the central controller forces the other controllers to a team-optimal solution, while each one of the K second-level controllers is in fact minimizing its own cost function.

On the metric complexity of casual linear systems: ε -Entropy and ε -Dimension for continuous time

Abstract: Estimates of e-entropy and e-dimension in the Kolmogorov sense are obtained for a class of causal, linear, time-invariant, continuous-time systems under the assumptions that impulse responses, satisfy an exponential order condition |f(t)| \leq Ce ^{-at} , and frequency responses satisfy an attenuation condition |F(j\omega)|\leg K\omega^{-1} . The dependence of e-entropy and e-dimension on the accuracy e is characterized by order, type, and power indexes. Similar results for the discrete-time case are reviewed and compared.

Upper and lower bounds on the solution of the algebraic Riccati equation

Abstract: Given an algebraic matrix Riccati equation A'K+ KA - KBB'K + Q =0 , the fundamental inequalities which are satisfied by the extremal eigenvalues of the positive definite solution K , are established. It Is illustrated that these resultant estimations appear to be considerably tighter than previously available results in many cases. Similar results are obtained for the discrete algebraic matrix Riccati equation.

Stabilization of linear systems with time-varying delay

Abstract: The problem of stabilizing a linear system with time-varying delay by means of linear feedback without delay is considered. A sufficient condition on the open loop system is presented for the possiblity of actualizlng the arbitrarily prescribed degree of stability in the closed loop system. Under the condition, the feedback gain can be constructed from the knowledge of the upper bounds of the delay and its derivative; no additional information on the delay is necessary.

Proportional minus delay controller

Abstract: A new type of controller, which utilizes the time-delay effect, is proposed. It is shown that the conventional P-controller equipped with an appropriate time-delay performs an averaged derivative action and thus can replace the PD-controller, showing quick responses to input changes but being insensitive to high-frequency noise.

Nonclassical control problems and Stackelberg games

Abstract: A nonclassical control problem, where the control depends on state and time, and its partial derivative with respect to the state appears in the state equation and in the cost function is analyzed. Stackelberg dynamic games which lead to such nonclassical control problems are considered and studied.

System identification via membership set constraints with energy constrained noise

Abstract: In this paper the identification of constant unknown parameters of a linear system is studied. Rather than finding an estimate of the parameters vector, a membership set for this vector is constructed such that any vector in this set is consistent with the measurements and noise specifications. The usual statistical specification of the noise is replaced here by energy constraints. Convergence of the membership set to a single point (the "true" vector) is studied. The convergence results are related to the convergence of the identification algorithm in the probability sense.

Optimal estimator model for human spatial orientation

Abstract: A model is being developed to predict pilot dynamic spatial orientation in response to multisensory stimuli Motion stimuli are first processed by dynamic models of the visual, vestibular, tactile, and proprioceptive sensors Central nervous system function is then modeled as a steady-state Kalman filter which blends information from the various sensors to form an estimate of spatial orientation Where necessary, this linear central estimator has been augmented with nonlinear elements to reflect more accurately some highly nonlinear human response characteristics Computer implementation of the model has shown agreement with several important qualitative characteristics of human spatial orientation, and it is felt that with further modification and additional experimental data the model can be improved and extended Possible means are described for extending the model to better represent the active pilot with varying skill and work load levels

The propagation of parametric uncertainty via polytopes

Abstract: We consider the discrete dynamical system x(k+1)= A(k)x(k) with n2independently varying uncertainties in the entries of A (\cdot) . Although the set of possible states X(k) at time k is not necessarily convex, we show that the convex hull of this set can be recursively propagated forward in time.

A stochastic realization approach to the smoothing problem

Abstract: The purpose of this paper is to develop a theory of smoothing for finite dimensional linear stochastic systems in the context of stochastic realization theory. The basic idea is to embed the given stochastic system in a class of similar systems all having the same output process and the same Kalman-Bucy filter. This class has a lattice structure with a smallest and a largest element; these two elements completely determine the smoothing estimates. This approach enables us to obtain stochastic interpretations of many important smoothing formulas and to explain the relationship between them.

Near-optimal control of composite systems: The multi time-scale approach

Abstract: This short note formulates an m -level hierarchical system structure, with the lower level subsystems being faster than the one above. In the context of linear time-invariant systems and the regulator problem a near-optimal solution is found using results in singular perturbation theory. An example from power systems is given, and computational and conceptual advantages of the method are indicated.

Stochastic functional fourier series, Volterra series, and nonlinear systems analysis

Abstract: A functional Fourier series is developed with emphasis on applications to the nonlinear systems analysis. In analogy to Fourier coefficients, Fourier kernels are introduced and can be determined through a cross correlation between the output and the orthogonal basis function of the stochastic input. This applies for the class of strict-sense stationary white inputs, except for a singularity problem incurred with inputs distributed at quantized levels. The input may be correlated if it is zero-mean Gaussian. The Wiener expansion is treated as an example corresponding to the white Gaussian input and this modifies the Lee-Schetzen algorithm for Wiener kernel estimation conceptually and computationally. The Poisson-distributed white input is dealt with as another example. Possible links between the Fourier and Volterra series expansions are investigated. A mutual relationship between the Wiener and Volterra kernels is presented for a subclass of analytic nonlinear systems. Connections to the Cameron-Martin expansion are examined as well The analysis suggests precautions in the interpretation of Wiener kernel data from white-noise identification experiments.

Optimal nonuniform sampling interval and test-input design for identification of physiological systems from very limited data

Abstract: Optimal design of test-inputs and sampling intervals in experiments for linear system identification is treated as a nonlinear integer optimization problem. The criterion is a function of the Fisher information matrix, the inverse of which gives a lower bound for the covariance matrix of the parameter estimates. Emphasis is placed on optimum design of nonuniform data sampling intervals when experimental constraints allow only a limited number of discrete-time measurements of the output. A solution algorithm based on a steepest descent strategy is developed and applied to the design of a biologic experiment for estimating the parameters of a model of the dynamics of thyroid hormone metabolism. The effects on parameter accuracy of different model representations are demonstrated numerically, a canonical representation yielding far poorer accuracies than the original process model for nonoptimal sampling schedules, but comparable accuracies when these schedules are optimized. Several objective functions for optimization are compared. The overall results indicate that sampling schedule optimization is a very fruitful approach to maximizing expected parameter estimation accuracies when the sample size is small.

The application of model-referenced adaptive control to robotic manipulators

Abstract: The achievement of quality dynamic performance in manipulator systems is difficult using conventional control methods because of both the inherent geometric nonlinearities of these systems and the dependence of the system dynamics on the characteristics of manipulated objects A model-referenced adaptive control law is developed for maintaining uniformly good performance over a wide range of motions and payloads The effectiveness of the approach is demonstrated in several simulations and the system stability as a function of input is investigated Also developed is a 'learning signal' approach designed to minimize initial transients arising from abrupt changes in the inertial payload

Disturbance localization and output deadbeat control through an observer in discrete-time linear multivariable systems

Abstract: This paper considers the problem of disturbance localization for the system x(i + 1) = Ax(i) + Bu(i) + Dd(i), y(i) = Cx(i), w(i) = Ex(i) , with disturbance d(i) , measurement output y(i) , and controlled output w(i) . It is shown that the problem is solvable by using an observer if and only if V^{\ast} \supset 2^{\ast} where V*is the largest ( A,B )-invariant subspace in \ker E and 2*is the least ( A,\ker C )-conditioned invariant subspace containing Im D . Also, it is shown that there exists a controller using an observer that achieves simultaneous disturbance localization and output deadbeat control if and only if the system is controllable modulo \ker E and, in addition, V^{\ast} \supset O^{\ast} where O*is the unknown input unconstructible subspace. A simple algorithm is proposed to design such a controller. This algorithm comprises those of designing the optimal output deadbeat state feedback controller and an unknown input observer.

On the irreducibility condition in the structural controllability theorem

Abstract: It is known that the structural system (A,B) is structurally controllable if and only if the corresponding matrix [A B] is generically full rank and irreducible. In this paper it is shown that the irreducibility condition alone implies that every nonzero mode of (A,B) is generically controllable. This result provides an easy proof to the structural controllability theorem stated above. In addition, it is shown that the basic structure of the Jordan canonical form of (A,B) remains unaffected, in the generic sense, under the variation of the free parameters of (A,B).

Unification of some continuous-time adaptive control schemes

Abstract: Model reference adaptive regulators based on input-output descriptions are examined. By reinterpreting the concept of "augmented error," it is shown that there are no essential differences between the model reference adaptive algorithms and the self-tuning regulators. Both types of schemes can be thought of as composed of a parameter estimator and a control law, based on the parameter estimates. It is shown that many schemes proposed are special cases of a general algorithm. The positive real condition for model reference adaptive systems is also examined. It is shown that this condition is a consequence of the choice of the estimator and that it is not crucial for stability.

The development of frequency-response methods in automatic control [Perspectives]

Abstract: In 1976 the Control Systems Society named three distinguished control systems specialists as Consulting Editors. One of the charges to these men was to submit an invited paper on a topic of their choice for publication without the usual IDC review procedures. At the same time Professor A.G.J. MacFarlane, Professor of Control Engineering at Cambridge University, was invited by the IDC to prepare an IEEE Press reprint book of important papers on frequency-domain methods in control and systems engineering. The coincidence of these two decisions has led to the following paper. "The Development of Frequency-Response Methods in Automatic Control" is one part of the IEEE Press book, Frequency-Response Methodr in Control Systems, edited by A.G.J. MacFarlane and sponsored by the Control Systems Society. The book will appear in mid-1979. The paper has been selected by Consulting Editor Nathaniel Nichols and should be of substantial interest to TRANSACTIONS readers. It also conveys some of the spirit and content of the book which may be purchased from IEEE Press when available.

Model reduction by Chebyshev polynomial techniques

Abstract: The problem of reduced-order modeling of high-order, linear, time-invariant, single-input, single-output systems is considered. A method is proposed, based on manipulating two Chebyshev polynomial series, one representing the frequency response characteristics of the high-order system and the other representing the approximating low-order model. The method can be viewed as generalizing the classical Pade approximation problem, with the Chebyshev polynomial series expansion being over a desired frequency interval instead of a power series about a single frequency point. Two different approaches to the problem are considered. Firstly, approximation is carried out in the s -plane by a Chebyshev polynomial series. Then, modified Chebyshev polynomials are introduced and a mapping to a new plane is defined. It turns out that in the new plane the advantages of the generalized Chebyshev-Pade approximations are retained while what is actually being solved is the classical Pade problem.