Showing papers in "IEEE Transactions on Automatic Control in 1983"

Global transformations of nonlinear systems

Abstract: Recent results have established necessary and sufficient conditions for a nonlinear system of the form \dot{x}(t) = f(x(t))-u(t)g(x(t)) . with f(0) = 0 , to be locally equivalent in a neighborhood of the origin in Rnto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that {g\ldot[f\dotg],...,(ad^{n-1}f\ldotg)} span an n -dimensional space and that {g\ldot[f\ldot g],...,(ad^{n-2}f\ldotg)} is an involutive set.

Feedback, minimax sensitivity, and optimal robustness

Abstract: In this paper, we look for feedbacks that minimize the sensitivity function of a linear single-variable feedback system represented by its frequency responses. Sensitivity to disturbances and robustness under plant perturbations are measured in a weighted H^{\infty} norm. In an earlier paper, Zames proposed an approach to feedback design involving the measurement of sensitivity by "multiplicative seminorms," which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty, or where signal power-spectra are not fixed, but belong to sets. The problem was studied in a general setting, and some H^{\infty} examples were solved. Here, a detailed study of the single-variable case is undertaken. The results are extended to unstable plants, and explicit formulas for the general situation of a finite number of right half-plane (RHP) plant zeros or poles are provided. The Q or "approximate-inverse" parametrization of feedbacks that maintain closed-loop stability is extended to the ease of unstable plants. The H^{\infty} and Wiener-Hopf approaches are compared.

Utilization of modified polar coordinates for bearings-only tracking

Abstract: Previous studies have shown that the Cartesian coordinate extended Kalman filter exhibits unstable behavior characteristics when utilized for bearings-only target motion analysis (TMA). In contrast, formulating the TMA estimation problem in modified polar (MP) coordinates leads to an extended Kalman filter which is both stable and asymptotically unbiased. Exact state equations for the MP filter are derived without imposing any restrictions on own-ship motion; thus, prediction accuracy inherent in the traditional Cartesian formulation is completely preserved. In addition, these equations reveal that MP coordinates are well-suited for bearings-only TMA because they automatically decouple observable and unobservable components of the estimated state vector. Such decoupling is shown to prevent covariance matrix ill-conditioning, which is the primary cause of filter instability. Further investigation also confirms that the MP state estimates are asymptotically unbiased. Realistic simulation data are presented to support these findings and to compare algorithm performance with respect to the Cramer-Rao lower bound (ideal) as well as the Cartesian and pseudolinear filters.

Formulation and optimization of cubic polynomial joint trajectories for industrial robots

Abstract: Because of physical constraints, the optimum control of industrial robots is a difficult problem. An alternative approach is to divide the problem into two parts: optimum path planning for off-line processing followed by on-line path tracking. The path tracking can be achieved by adopting the existing approach. The path planning is done at the joint level. Cubic spline functions are used for constructing joint trajectories for industrial robots. The motion of the robot is specified by a sequence of Cartesian knots, i.e., positions and orientations of the hand. For an N -joint robot, these Cartesian knots are transformed into N sets of joint displacements, with one set for each joint. Piecewise cubic polynomials are used to fit the sequence of joint displacements for each of the N joints. Because of the use of the cubic spline function idea, there are only n - 2 equations to be solved for each joint, where n is the number of selected knots. The problem is proved to be uniquely solvable. An algorithm is developed to schedule the time intervals between each pair of adjacent knots such that the total traveling time is minimized subject to the physical constraints on joint velocities, accelerations, and jerks. Fortran programs have been written to implement: 1) the procedure for constructing the cubic polynomial joint trajectories; and 2) the algorithm for minimizing the traveling time. Results are illustrated by means of a numerical example.

Adaptive linear controller for robotic manipulators

Abstract: This paper presents a new approach to the position and velocity control of a manipulator by using an adaptive controller of the self-tuning type for each joint. The complicated manipulator system is modeled by a set of time series difference equations. The parameters of the models are determined by on-line recursive algorithms, which result from minimizing the sum of the squared equation errors. The adaptive controller of each joint is designed on the basis of the difference equation model and a chosen performance criterion. The controller gains are calculated on-line using the model with the estimated values of system parameters. Simulation results are presented to demonstrate the applicability of the approach. Some aspects of the implementation are also discussed.

A new set of invariants for linear systems--Application to reduced order compensator design

Abstract: A new set of invariants for linear systems, weighting the contribution of each state component to the inherent closed-loop LQG behavior of the system is presented, together with applications to model order reduction and reduced order compensator design.

Intelligent robotic control

Abstract: A theoretical approach is developed to deal with man-machine interactive systems requiring advanced decision making in unpredictable environments. The hierarchical method consists of a three-layer control of "increasing intelligence and decreasing precision." The lowest level consists of several controllers designed for effective control with existing hardware using an approximation theory of optimal control. The next level is that of a coordinator which utilizes new computer architectures to effectively control the overall hardware system. The highest level is the organizer which supervises the performance of the overall system. Both highest levels are computer implemented and the research involved is in developing the appropriate architecture and software to accommodate others. The lowest level, aimed for end-point control tasks, is dominated by typical hardware control methods. The coexistence of the two approaches makes the method novel. Application of intelligent control techniques to robotics and manipulative systems is considered.

Stabilization of uncertain systems via linear control

Abstract: This note considers the problem of stabilizing a linear dynamical system (Σ) whose state equation includes a time-varying uncertain parameter vector q(\cdot) . Given the dynamics \dot{x}(t)=A(q(t))x(t)+ B(q(t))u(t) and a bounding set Q for the values q(t) , the objective is to choose a control law u(t)=p(x(t)) guaranteeing uniform asymptotic stability for all admissible variations of q(\cdot) . Our results differ from previous work in one fundamental way; that is, we show that when working with linear controllers, it is possible to dispense with all assumptions on B(\cdot) which have been made by previous authors (e.g., see [1]-[9]). This elimination of hypotheses on B(\cdot) is accomplished roughly as follows: the system (\Sigma) {\underline {\underline \Delta}} (A(q), B(q)) is shown to be equivalent to another system (\Sigma^{+}) {\underline {\underline \Delta}} (A^{+}(q), B^{+}) as far as stabilization is concerned. Since B^{+} is a constant matrix (independent of q ), the desired result is readily obtained.

An LQ-solution to a control problem associated with a solar thermal central receiver

Abstract: The linearized process dynamics of the steam boiler in a solar-powered central receiver change abruptly when clouds interfere with the sun's rays The steam temperature regulator used to maintain proper exit steam conditions must control a system with variable structure and discontinuous state trajectories This paper investigates the quadratic-optimal control of such a system, and gives the design equations for the optimal regulator

Descriptor variable systems and optimal state regulation

Abstract: Linear systems of the form E\dot{x} = Ax + Bu with E singular are treated. It is desired to find a control which drives the system asymptotically to the origin, minimizing a quadratic cost functional. No restrictions are placed on initial conditions. The cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control. Through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order Riccati equation.

Nonlinear Kalman filtering techniques for terrain-aided navigation

Abstract: The application of nonlinear Kalman filtering techniques to the continuous updating of an inertial navigation system using individual radar terrain-clearance measurements has been investigated. During this investigation, three different approaches for handling the highly nonlinear terrain measurement function were developed and their performance was established. These were 1) a simple first-order extended Kalman filter using local derivatives of the terrain surface, 2) a modified stochastic linearization technique which adaptively fits a least squares plane to the terrain surface and treats the associated fit error as an additional noise source, 3) a parallel Kalman filter technique utilizing a bank of reduced-order filters that was especially important in applications with large initial position uncertainties. Theoretical and simulation results are presented.

Large deviations and rare events in the study of stochastic algorithms

Abstract: New asymptotics formulas for the mean exit time from an almost stable domain of a discrete-time Markov process are obtained. An original fast simulation method is also proposed. The mathematical background involves the large deviation theorems and approximations by a diffusion process. We are chiefly concerned with the classical Robbins-Monroe algorithm. The validity of the results are tested on examples from the ALOHA system (a satellite type communication algorithm).

Stability analysis of single loop control systems with saturation and antireset-windup circuits

Abstract: "Reset-windup" appears when regulators with integral action are used with saturating actuators. Antireset-windup (ARW) circuits are in practice used in most standard control equipment today. Their design is based on intuition and simulation. This paper presents analysis of systems with ARW circuits based on nonlinear stability theory. The analysis gives possibilities to compare the performance of different ARW circuits. It predicts the system response adequately without need for extensive simulations.

Design and analysis of a dynamic positioning system based on Kalman filtering and optimal control

Abstract: -A novel adaptive filtering technique is described for a class of systems with unknown disturbances. The estimator includes both a self-tuning filter and a Kalman filter. The state estimates are employed in a closed-loop feedback control scheme wbich is designed via the usual linear quadratic approach. The approach was developed for application to the dynamic ship positioning control problem and has the advantage that existing nonadaptive Kalman filtering systems may be easily modified to include the self-tuning feature. work was supported b\\l GEC Electrical Projects Ltd., and the United Manuscript received April 21, 1982: revised September 27, 1982. This Kingdom Science and Engineering Research Council. P. T.-K. Fung \\vas with the Department of Electrical Engineering, University of Strathclyde, Glasgow, Scotland. He is now with the Space and Electronic Group, Spar Aerospace Ltd.. Weston. Ont.. Canada. vrrsity of Strathclyde. Glasgow, Scotland. M. J. Grimble is with the Department of Electrical Engineering, Uni

Continuation methods: Theory and applications

Abstract: This paper surveys in a tutorial fashion theoretical and applications aspects of the confirmation method for the solution of large scale system engineering problems. The continuation method is motivated and defined. The existence of the method is formulated in terms of degree theory a la Algebraic Topology. Examples are given throughout the paper and applications to engineering and economic problems are cited and described.

On the structure of balanced and other principal representations of SISO systems

Abstract: A new matrix W co which can be considered as a cross-Gramian matrix which contains information about both controllability and observability is defined for single-input, single-output, linear systems. Using this matrix, the structural properties of linear systems are studied in the context of principal component analysis. The matrix W co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability Gramians. The importance of this matrix in model-order reduction is highlighted.

Dynamic ship positioning using a self-tuning Kalman filter

Abstract: A novel adaptive filtering technique is described for a class of systems with unknown disturbances. The estimator includes both a self-tuning filter and a Kalman filter. The state estimates are employed in a closed-loop feedback control scheme which is designed via the usual linear quadratic approach. The approach was developed for application to the dynamic ship positioning control problem and has the advantage that existing nonadaptive Kalman filtering systems may be easily modified to include the self-tuning feature.

Decoupled Kalman filters for phased array radar tracking

Abstract: Kalman filters have been used in numerous phased array radars to track satellites, reentry vehicles, and missiles. This paper considers the design of these filters to reduce computational requirements, ill-conditioning, and the effects of nonlinearities. Several special coordinate systems used to represent the Kalman filter error covariance matrix are described. These covariance coordinates facilitate the approximate decoupling required for practical filter design. A tutorial discussion and analysis of ill-conditioning in Kalman filters is used to motivate these design considerations. This analysis also explains several well-known phenomena reported in the literature. In addition, a discussion of nonlinearities and methods to mitigate their ill effects is included.

Second-order convergence analysis of stochastic adaptive linear filtering

Abstract: The convergence of an adaptive filtering vector is studied, when it is governed by the mean-square-error gradient algorithm with constant step size. We consider the mean-square deviation between the optimal filter and the actual one during the steady state. This quantity is known to be essentially proportional to the step size of the algorithm. However, previous analyses were either heuristic, or based upon the assumption that successive observations were independent, which is far from being realistic. Actually, in most applications, two successive observation vectors share a large number of components and thus they are strongly correlated. In this work, we deal with the case of correlated observations and prove that the mean-square deviation is actually of the same order (or less) than the step size of the algorithm. This result is proved without any boundedness or barrier assumption for the algorithm, as it has been done previously in the literature to ensure the nondivergence. Our assumptions are reduced to the finite strong-memory assumption and the finite-moments assumption for the observation. They are satisfied in a very wide class of practical applications.

An anatomy of industrial robots and their controls

Abstract: In the United States, commercially available industrial robots perform very well in limited areas of industrial tasks such as arc welding, paint spraying, etc. These tasks mainly involve synchronization but no task interaction. A close examination of the basic structure and controls of the robots reveals their resulting limitations which lead to unnatural specifications and inefficient performance of task interactions. It is our opinion that, to expand the range of robot tasks to include labor intensive jobs such as product assembly, sensors of multiple purposes must be added onto the robots and integrated into their control systems. Computer command language must be developed to enable nonexpert users to operate the robots, and a work-method must be available for analyzing robot time-motion so that the robots can be programmed to achieve best efficiency with least production cost.

Optimal flow control of a class of queueing networks in equilibrium

Abstract: The problem of optimum flow control of a class of queueing systems which appears as a model of datagram and virtual circuit computer communication networks is investigated. This class "intuitively" has the property that by increasing the load on the network, both the average throughput and the average time delay also increase. It is shown that the control that achieves the maximum throughput under a bounded average time delay criterion can be specified by a "window" flow control mechanism (bang-bang control). The window size L , the maximum number of unacknowledged packets in the system, can be easily derived from the preassigned upper bound on the time delay T , the Norton equivalent of the queueing system μ, and the maximum admissible total load on the network c .

Estimation and prediction for maneuvering target trajectories

Abstract: The Kalman filter is well suited for application to the problem of anti-aircraft gun fire control. In this paper we make use of the Kalman filter theory, to develop an accurate, numerically efficient scheme for estimating and predicting the present and future position of maneuvering fixed-wing aircraft. This scheme was implemented in a radar tracker gun fire control system and tested against a variety of fixed-wing aircraft targets. Actual field test results are presented to demonstrate the high accuracy pointing which can be achieved by this approach.

Projected Newton methods and optimization of multicommodity flows

Abstract: A superlinearly convergent Newton like method for linearly constrained optimization problems is adapted for solution of multicommodity network flow problems of the type arising in communication and transportation networks. We show that the method can be implemented approximately by making use of conjugate gradient iterations without the need to compute explicitly the Hessian matrix. Preliminary computational results suggest that this type of method is capable of yielding highly accurate solutions of nonlinear multicommodity flow problems far more efficiently than any of the methods available at present.

Linear feedback decoupling--Transfer function analysis

Abstract: The problem of linear system decoupling is examined based on recent results on linear feedback. New insight is obtained, through which resolution of the decoupliug problem is accomplished by calculations, performed directly on the given transfer matrix. Computation of the decoupling compensators follows by easy constructions. The problem of feedback block decoupling with internal stability, is also formulated and resolved.

Linear time-variable systems: Balancing and model reduction

Abstract: A "uniformly balanced" realization for linear time-variable systems is defined. This representation is characterized by the fact that its controllability, and observability Gramians are equal and diagonal. Existence and uniqueness of the uniformly balanced realization is studied. Such a framework has many remarkable properties and leads to a novel method for approximating time-variable systems, where the subsystems of the balanced realization can be taken as a reduced model. The reduced model is examined from the point of view of stability, controllability, and observability.

Simultaneous stabilization and simultaneous pole-placement by nonswitching dynamic compensation

Abstract: In this paper, motivated by questions in fault tolerance, we investigate the existence of a compensator which simultaneously renders a given r -tuple of plants internally stable. Sufficient conditions are derived for simultaneous pole-assignability of the generic r -tuple by dynamic output feedback, which are also shown to be necessary (and equivalent to generic stabilizability) in the case where the number of either input or output channels is one. We also derive an upper bound on the order of a simultaneous pole-assigning compensator. If r = 1 , this reduces to the condition derived by Brasch and Pearson, while if r = 2 , this contains the recent theorem by Vidyasagar and Viswanadham. The cases r \geq 3 are new.

Design of linear systems with saturating linear control and bounded states

Abstract: A theorem is developed, based on the circle criterion, which gives sufficient conditions for the stability of a saturating linear control for a linear system with bounded states. This theorem can be applied to determine a range of gains for the linear control which ensures stability. It is shown by example that this approach results in an increased speed of response over that obtained by previous methods which do not account for state bounds.

Optimal instrumental variable estimation and approximate implementations

Abstract: The accuracy properties of instrumental variables (IV) methods are investigated. Extensions such as prefiltering of data and use of additional instruments are included in the analysis. The parameter estimates are shown to be asymptotically Gaussian distributed. An explicit expression is given for the covariance matrix of their distribution. The covariance matrix is then taken as a (multivariable) measure of accuracy. It is shown how it can be optimized by an appropriate selection of instruments and prefilter. The so obtained optimal instrumental variable estimates cannot be used directly since the true system and the statistical properties of the disturbance must be known in order to compute the optimal instruments and prefilters. A multistep procedure consisting of three or four simple steps is then proposed as a way to overcome this difficulty. This procedure includes modeling of the disturbance as an ARMA process using a statistically efficient method such as a prediction error method. The statistical properties of the estimates obtained with the multistep procedure are also analyzed. Those estimates are shown to be asymptotically Gaussian distributed as well. The covariance matrix of the estimation errors is compared to that corresponding to a prediction error method. For some model structures these two approaches give the same asymptotic accuracy. The conclusion is that the multistep procedure, which is quite easy to implement and also has nice uniqueness properties, can be viewed as an interesting alternative to prediction error methods.

The state vector reconstruction for linear systems with unknown inputs

Abstract: Necessary and sufficient conditions for the existence of an observer for linear multivariable systems with unknown inputs are presented in Theorem 1. Proof of the theorem gives an algorithm of full order observer matrices calculation.