Showing papers in "IEEE Transactions on Automatic Control in 1991"

Systematic design of adaptive controllers for feedback linearizable systems

Abstract: A systematic procedure for the design of adaptive regulation and tracking schemes for a class of feedback linearizable nonlinear systems is developed. The coordinate-free geometric conditions, which characterize this class of systems, do not constrain the growth of the nonlinearities. Instead, they require that the nonlinear system be transformable into the so-called parametric-pure feedback form. When this form is strict, the proposed scheme guarantees global regulation and tracking properties, and substantially enlarges the class of nonlinear systems with unknown parameters for which global stabilization can be achieved. The main results use simple analytical tools, familiar to most control engineers. >

Linear systems with state and control constraints: the theory and application of maximal output admissible sets

Abstract: The initial state of an unforced linear system is output admissible with respect to a constraint set Y if the resulting output function satisfies the pointwise-in-time condition y(t) in Y, t>or=0. The set of all possible such initial conditions is the maximal output admissible set O/sub infinity /. The properties of O/sub infinity / and its characterization are investigated. In the discrete-time case, it is generally possible to represent O/sub infinity / or a close approximation of it, by a finite number of functional inequalities. Practical algorithms for generating the functions are described. In the continuous-time case simple representations of the maximal output admissible set are not available, however, it is shown that the discrete-time results may be used to obtain approximate representations. >

Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems

Abstract: Conditions under which a nonlinear system can be rendered passive via smooth state feedback are derived. It is shown that, as in the case of linear systems, this is possible if and only if the system in question has relative degree one and is weakly minimum phase. It is proven that weakly minimum phase nonlinear systems with relative degree one can be globally asymptotically stabilized by smooth state feedback, provided that suitable controllability-like rank conditions are satisfied. This result incorporates and extends a number of stabilization schemes recently proposed for global asymptotic stabilization of certain classes of nonlinear systems. >

Paradigms and puzzles in the theory of dynamical systems

Abstract: A self-contained exposition is given of an approach to mathematical models, in particular, to the theory of dynamical systems. The basic ingredients form a triptych, with the behavior of a system in the center, and behavioral equations with latent variables as side panels. The author discusses a variety of representation and parametrization problems, in particular, questions related to input/output and state models. The proposed concept of a dynamical system leads to a new view of the notions of controllability and observability, and of the interconnection of systems, in particular, to what constitutes a feedback control law. The final issue addressed is that of system identification. It is argued that exact system identification leads to the question of computing the most powerful unfalsified model. >

The attitude control problem

Abstract: A general framework for the analysis of the attitude tracking control problem for a rigid body is presented. A large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a Lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body. The controllers share the common structure of a proportional-derivative feedback plus some feedforward which can be zero (the model-independent case), the Coriolis torque compensation, or an adaptive compensation. These controller structures are compared in terms of the requirement on the a priori model information, guaranteed transient performance, and robustness. The global stability of the Luh-Walker-Paul robot end-effector controller is also analyzed in this framework. >

Filtering and smoothing in an H/sup infinity / setting

Abstract: The problems of filtering and smoothing are considered for linear systems in an H/sup infinity / setting, i.e. the plant and measurement noises have bounded energies (are in L/sub 2/), but are otherwise arbitrary. Two distinct situations for the initial condition of the system are considered; the initial condition is assumed known in one case, while in the other the initial condition is not known but the initial condition, the plant, and measurement noise are in some weighted ball of R/sup n/XL/sub 2/. Finite-horizon and infinite-horizon cases are considered. Necessary and sufficient conditions are presented for the existence of estimators (both filters and smoothers) that achieve a prescribed performance bound, and algorithms that result in performance within the bounds are developed. In case of smoothers, the optimal smoother is also presented. The approach uses basic quadratic optimization theory in a time-domain setting, as a consequence of which both linear time-varying and time-invariant systems can be considered with equal ease. (In the smoothing problem, for linear time-varying systems, one considers only the finite-horizon case). >

Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics

Abstract: Continuing the development of the structured singular value approach to robust control design, the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty. The problem is shown to be equivalent to a smooth constrained finite-dimensional optimization problem. In view of the fact that the functional to be maximized may have several local extrema, an upper bound on mu whose computation is numerically tractable is established; this leads to a sufficient condition of robust stability and performance. A historical perspective on the development of the mu theory is included. >

System identification using Laguerre models

Abstract: The traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear-in-the-parameters predictor models leads to approximations of very high order in cases of rapid sampling and/or dispersion in time constants. By using prior information about the time constants of the system more appropriate expansions, related to Laguerre networks, are introduced and analyzed. It is shown that the model order can be reduced, compared to ARX (FIR, AR) modeling, by using Laguerre models. Furthermore, the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the Laguerre parameter. Consistency (error bounds), persistence of excitation conditions. and asymptotic statistical properties are investigated. This analysis is based on the result that the covariance matrix of the regression vector of a Laguerre model has a Toeplitz structure. >

Mixed H/sub 2//H/sub infinity / control: a convex optimization approach

Abstract: The problem of finding an internally stabilizing controller that minimizes a mixed H/sub 2//H/sub infinity / performance measure subject to an inequality constraint on the H/sub infinity / norm of another closed-loop transfer function is considered. This problem can be interpreted and motivated as a problem of optimal nominal performance subject to a robust stability constraint. Both the state-feedback and output-feedback problems are considered. It is shown that in the state-feedback case one can come arbitrarily close to the optimal (even over full information controllers) mixed H/sub 2//H/sub infinity / performance measure using constant gain state feedback. Moreover, the state-feedback problem can be converted into a convex optimization problem over a bounded subset of (n*n and n*q, where n and q are, respectively, the state and input dimensions) real matrices. Using the central H/sub infinity / estimator, it is shown that the output feedback problem can be reduced to a state-feedback problem. In this case, the dimension of the resulting controller does not exceed the dimension of the generalized plant. >

A simple PD controller for robots with elastic joints

Abstract: The point-to-point control of manipulators having elastic joints is considered. It is shown that a simple PD (proportional plus derivative) controller, similar to that used for rigid robots, suffices to globally stabilize the elastic joint robots about a reference position. A robustness analysis is also given with respect to uncertainties on the robot parameters. The results of numerical simulation tests of a manipulator with three revolute elastic joints are presented. >

Control oriented system identification: a worst-case/deterministic approach in H/sub infinity /

Abstract: The authors formulate and solve two related control-oriented system identification problems for stable linear shift-invariant distributed parameter plants In each of these problems the assumed a priori information is minimal, consisting only of a lower bound on the relative stability of the plant, an upper bound on a certain gain associated with the plant, and an upper bound on the noise level The first of these problems involves identification of a point sample of the plant frequency response from a noisy, finite, output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest This problem leads naturally to the second problem, which involves identification of the plant transfer function in H/sub infinity / from a finite number of noisy point samples of the plant frequency response Concrete plans for identification algorithms are provided for each of these two problems >

The peaking phenomenon and the global stabilization of nonlinear systems

Abstract: The problem of global stabilization is considered for a class of cascade systems. The first part of the cascade is a linear controllable system and the second part is a nonlinear system receiving the inputs from the states of the first part. With zero input, the equilibrium of the nonlinear part is globally asymptotically stable. In linear systems, a peaking phenomenon occurs when high-gain feedback is used to produce eigenvalues with very negative real parts. It is established that the destabilizing effects of peaking can be reduced if the nonlinearities have sufficiently slow growth. A detailed analysis of the peaking phenomenon is provided. The tradeoffs between linear peaking and nonlinear growth conditions are examined. >

Asymptotic stabilization of minimum phase nonlinear systems

Abstract: How a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown. More precisely, conditions under which, for every compact set of initial states, it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described. The theory includes the development of globally defined transformations of the system equations to their global normal form. >

Distributed scheduling based on due dates and buffer priorities

Abstract: Several distributed scheduling policies are analyzed for a large semiconductor manufacturing facility, where jobs of wafers, each with a desired due date, follow essentially the same route through the manufacturing system, returning several times to many of the service centers for the processing of successive layers. It is shown that for a single nonacyclic flow line the first-buffer-first-serve policy, which assigns priorities to buffers in the order that they are visited, is stable, whenever the arrival rate, allowing for some burstiness, is less than the system capacity. The last-buffer-first-serve policy (LBFS), where the priority ordering is reversed, is also stable. The earliest-due-date policy, where priority is based on the due date of a part, as well as another due-date-based policy of interest called the least slack policy (LS), where priority is based on the slack of a part, defined as the due date minus an estimate of the remaining delay, are also proved to be stable. >

A globally convergent angular velocity observer for rigid body motion

Abstract: The problem of obtaining the angular velocity of a rigid body from orientation and torque measurements only, without noisy numerical differentiation, is considered. A novel angular velocity/angular momentum observer for rigid body motion is presented. Using Euler quaternions and a mechanical energy function approach, it is shown that the observer estimates converge globally and that the convergence is eventually exponential. It is hoped that a mechanical energy function approach to rigid body control can be combined with the observer presented to lead to a globally stable, nonlinear, observer-based, rigid-body controller in which the observer and controller errors can be separated, in much the same way as one can separate controller and observer poles in the output feedback controllers of linear system theory. >

A novel approach to the design of unknown input observers

Abstract: A novel state estimator design scheme for linear dynamical systems driven by partially unknown inputs is presented. It is assumed that there is no information available about the unknown inputs, and thus no prior assumption is made about the nature of these inputs. A simple approach for designing a reduced-order unknown input observer (UIO) with pole-placement capability is proposed. By carefully examining the dynamic system involved and simple algebraic manipulations, it is possible to rewrite equations eliminating the unknown inputs from part of the system and to put them into a form where it could be partitioned into two interconnected subsystems, one of which is directly driven by known inputs only. This makes it possible to use a conventional Luenberger observer with a slight modification for the purpose of estimating the state of the system. As a result, it is also possible to state similar necessary and sufficient conditions to those of a conventional observer for the existence of a stable estimator and also arbitrary placement of the eigenvalues of the observer. The design and computational complexities involved in designing UIOs are greatly reduced in the proposed approach. >

An adaptive controller which provides an arbitrarily good transient and steady-state response

Abstract: A model reference adaptive control problem is posed In the problem, the objective is not the usual one of forcing the error between the plant output and the reference model output asymptotically to zero, but instead, it is that of forcing this error to be less than a (arbitrarily small) prespecified constant after a (arbitrarily short) prespecified period of time, with a (arbitrarily small) prespecified upper bound on the amount of overshoot It is shown that to achieve this goal for a stabilizable and detectable, single-input single-output linear time-invariant (LTI) plant, it is necessary and sufficient that the plant be minimum phase Knowledge of an upper bound on the plant order, of the relative degree, and of the sign of the high-frequency gain is not required The controller proposed consists of an LTI compensator together with a switching mechanism to adjust the compensator parameters If an upper bound on the relative degree is available, the compensator has dynamics of order equal to this upper bound less one; otherwise, the order of the compensator is adjusted as well as its parameters >

Feedback gains for correcting small perturbations to standing posture

Abstract: A dynamical model of the neuro-musculo-skeletal mechanics of a cat hindlimb is developed to investigate the feedback regulation of standing posture under small perturbations. The model is a three-joint limb, moving only in the sagittal plane, driven by 10 musculotendon actuators, each with response dynamics dependent on activation kinetics and muscle kinematics. Under small perturbations, the nonlinear postural regulation mechanism is approximately linear. Sensors exist which could provide state feedback. Thus, the linear quadratic regulator is proposed as a model for the structure of the feedback controller for regulation of small perturbations. System states are chosen to correspond to the known outputs of physiological sensors: muscle forces (sensed by tendon organs), a combination of muscle lengths and velocities (sensed by spindle organs), joint angles and velocities (sensed by joint receptors), and motoneuron activities (sensed by Renshaw cells). Thus, the feedback gain matrices computed can be related to the spinal neural circuits. Several proposals for control strategy have been tested under this formulation. It is shown that a strategy of regulating all the states leads to controllers that best mimic the externally measured behavior of real cats. >

A new adaptive learning rule

Abstract: A method is presented for nonlinear function identification and application to learning control. The control objective is to identify and compensate for a nonlinear disturbance function. The nonlinear disturbance function is represented as an integral of a predefined kernel function multiplied by an unknown influence function. Sufficient conditions for the existence of such a representation are provided. Similarly, the nonlinear function estimate is generated by an integral of the predefined kernel multiplied by an influence function estimate. Using the time history of the plant, the learning rule indirectly estimates the unknown function by updating the influence function estimate. It is shown that the estimate function converges to the actual disturbance asymptotically. Consequently, the controller achieves the disturbance cancellation asymptotically. The method is extended to repetitive control applications. It is applied to the control of robot manipulators. Simulation and actual real-time implementation results using the Berkeley/NSK robot arm show that the proposed learning algorithm is more robust and converges at a faster rate than conventional repetitive controllers. >

Input-output stability of sampled-data systems

Abstract: The authors study the L/sub p/ input-output stability of a continuous-time controller, using the usual arrangement of periodic sampling and zero-order hold. It is noted that even if the hybrid system is exponentially stable, this arrangement does not yield L/sub p/ (1 >

Memoryless stabilization of uncertain dynamic delay systems: Riccati equation approach

Abstract: The authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems. The uncertainties are time varying and within a given compact set. This method is an extension of the Riccati equation approach proposed by I.R. Petersen and C.V. Hollot (1986). The extension is straightforward. Also the uncertainties do not need to satisfy the matching conditions. >

Position control of a PM stepper motor by exact linearization

Abstract: The authors consider the position control of a permanent magnet (PM) stepper motor using the exact linearization method. This nonlinear controller takes into account the full dynamics of the stepper motor. In particular, the phase shift between voltage and current in each phase is automatically taken into account. The feedback linearization controller makes the stepper motor into a fast accurate positioning system. The authors consider the feedback linearization technique for the PM stepper motor and show, when the detent torque is neglected, how it quite naturally leads to the well-known DQ transformation of electric machine theory. The authors indicate how constant load torques may be asymptotically rejected by using a nonlinear observer. >

Performance robustness of discrete-time systems with structured uncertainty

Abstract: Given an interconnection of a nominal discrete-time plant and a stabilizing controller together with structured, norm-bounded, nonlinear/time-varying perturbations, necessary and sufficient conditions for robust stability and performance of the system are provided. It is shown that performance robustness is equivalent to stability robustness in the sense that both problems can be dealt with in the framework of a general stability robustness problem. The resulting stability robustness problem is shown to be equivalent to a simple algebraic one, the solution of which provides the desired necessary and sufficient conditions for performance/stability robustness. These conditions provide an effective tool for robustness analysis and can be applied to a large class of problems. In particular, it is shown that some known results can be obtained immediately as special cases of these conditions. >

Stochastic robustness of linear time-invariant control systems

Abstract: A simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described. Monte Carlo evaluation of the system's eigenvalues allows the probability of instability and the related stochastic root locus to be estimated. This analysis approach treats not only Gaussian parameter uncertainties but also nonGaussian cases, including uncertain but bound variations. Confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation. Trivial extensions of the procedure admit consideration of alternate discriminants; thus, the probabilities that stipulated degrees of instability will be exceeded or that closed-loop roots will leave desirable regions can also be estimated. Results are particularly amenable to graphical presentation. >

The AAstrom-Wittenmark self-tuning regulator revisited and ELS-based adaptive trackers

Abstract: Although considerable progress has been made in stochastic adaptive control, the problem concerning the convergence of the original self-tuning regulator proposed by K.J. AAstrom and B. Wittenmark (1973) is still open. Since it is attractive in theory and important in applications, this problem is examined and a rigorous proof of its stability and optimality is given. Related problems such as convergence of the extended-least-squares (ELS)-based adaptive tracker are also considered. >

An optimal one-way multigrid algorithm for discrete-time stochastic control

Abstract: The numerical solution of discrete-time stationary infinite-horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately, within a desired accuracy. After a discussion of problem discretization, the authors introduce a multigrid version of the successive approximation algorithm that proceeds 'one way' from coarse to fine grids, and analyze its computational requirements as a function of the desired accuracy and of the discount factor. They also study the effects of a certain mixing (ergodicity) condition on the algorithm's performance. It is shown that the one-way multigrid algorithm improves upon the complexity of its single-grid variant and is, in a certain sense, optimal. >

Application of Kalman filtering to the calibration and alignment of inertial navigation systems

Abstract: The development of Kalman filters for the calibration and alignment of complex inertial guidance systems is presented. The attitude error equation is augmented by gyro and accelerometer model parameters. The mechanization of the algorithm involves preprocessing the raw measurements to reduce the computational load. Results for simulated data show that preprocessing has very little effect on the performance of the filter. Other topics discussed include gyro and accelerometer models and a technique for generating parameter excitation trajectories. >

H/sub 2/-optimal sampled-data control

Abstract: The H/sub 2/-optimal control of continuous-time linear time-invariant systems by sampled-data controllers is discussed. Two different solutions, state space and operator theoretic, are given. In both cases, the H/sub 2/ sampled-data problem is shown to be equivalent to a certain discrete-time H/sub 2/ problem. Other topics discussed include input-output stability of sampled-data systems, performance recovery in digital implementation of analog controllers, and sampled-data control of systems with the possibility of multiple-time delays. >

Robust tracking control of robots by a linear feedback law

Abstract: For the trajectory following problem of a robot manipulator, a simple linear robust fedback control law with constant gain matrix is proposed that makes the resulting error system uniformly ultimately bounded. This control law is very easy to implement by simply choosing a feedback gain according to the coefficients of a polynomial function of the tracking errors which is a bounding function for the terms in the Lagrange-Euler formulation. In the limit as the gain approaches infinity the error system becomes globally asymptotically stable. >

Large deviations theory and efficient simulation of excessive backlogs in a GI/GI/m queue

Abstract: The problem of using importance sampling to estimate the average time to buffer overflow in a stable GI/GI/m queue is considered. Using the notion of busy cycles, estimation of the expected time to buffer overflow is reduced to the problem of estimating p/sub n/=P (buffer overflow during a cycle) where n is the buffer size. The probability p/sub n/ is a large deviations probability (p/sub n/ vanishes exponentially fast as n to infinity ). A rigorous analysis of the method is presented. It is demonstrated that the exponentially twisted distribution of S. Parekh and J. Walrand (1989) has the following strong asymptotic-optimality property within the nonparametric class of all GI/GI importance sampling simulation distributions. As n to infinity , the computational cost of the optimal twisted distribution of large deviations theory grows less than exponentially fast, and conversely, all other GI/GI simulation distributions incur a computational cost that grows with strictly positive exponential rate. >