Integral action in robust adaptive control

Conditions are studied under which integral action in robust adaptive control is soundly based. The method of analysis amounts to verifying that the key conditions needed for convergence are satisfied. A simple strategy is described for choosing the integral constant and global convergence is established for the resultant algorithm. This illustrates a proof paradigm which could be similarly applied to other problems. The principal assumptions used are that the plant, when augmented by an integrator, has a known degree of controllability and that an overbounding function is known for the unmodeled system response. The continuous-time case is treated, but the corresponding discrete-time results follow mutatis mutandis by simply replacing p =d/dt by δ=q-1/Δ, L ₂ by l₂, ∫ by Σ, and so on     

Order-determination theorems and system identification

Theorems for order-determination without a priori knowledge of upper bounds on the order in MIMO dynamic systems are developed. Also, deterministic procedures are introduced to determine orders and estimate parameters simultaneously by recursively computing the order-determining quantity S_n(a,b,k), which plays a crucial role in order-determination procedures, and the least-squares estimate &thetas;_n(a,b) of &thetas;(p,q), with p and q denoting the true orders     

Adaptive variable structure controller of redundant robots with mobile/fixed obstacles avoidance

In this paper, a variable structure adaptive controller is proposed for redundant robot manipulators constrained by moving obstacles. The main objective of the controller is to force the model states of the robot to track those of a chosen reference model. In addition, the controller is designed directly in Cartesian space and no knowledge on the dynamic model is needed, except its structure. The parameters of the controller are adapted using adaptive laws obtained via Lyapunov stability analysis of the closed loop. The performances of the proposed controller are evaluated using a 3 DOF robot manipulator evolving in a vertical plane constrained by a mobile obstacle. The obtained results show its effectiveness compared to other tested variable structure controllers.     

Mapping nested loop algorithms into multidimensional systolicarrays

Consideration is given to transforming depth p-nested for loop algorithms into q-dimensional systolic VLSI arrays where 1⩽q⩽p-1. Previously, there existed complete characterizations of correct transformation only for the cases where q=p-1 or q=1. This gap is filled by giving formal necessary and sufficient conditions for correct transformation of a p-nested loop algorithm into a q-dimensional systolic array for any q, 1⩽q⩽p-1. Practical methods are presented. The techniques developed are applied to the automatic design of special purpose and programmable systolic arrays. The results also contribute toward automatic compilation onto more general purpose programmable arrays. Synthesis of linear and planar systolic array implementations for a three-dimensional cube-graph algorithm and a reindexed Warshall-Floyd path-finding algorithm are used to illustrate the method     

Mixed H2/H∞ control:a convex optimization approach

The problem of finding an internally stabilizing controller that minimizes a mixed H₂/H_∞ performance measure subject to an inequality constraint on the H_∞ 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 H₂/H_∞ 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_∞ 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     

An efficient algorithm for calculating the likelihood andlikelihood gradient of ARMA models

Exact analytical expressions are obtained for the likelihood and likelihood gradient stationary autoregressive moving average (ARMA) models. Denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p+2q+1)N +o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p+6q+2)N+o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2 p+6q+2)N+o(N) multiply-add operations per iteration     

Discrete system reduction via impulse-response Gramians and itsrelation to q-Markov covers

A method for model reduction of linear discrete systems is proposed. It is based on the impulse-response Gramian proposed by the authors (1989) for discrete systems. This Gramian is an extension of the one proposed for linear continuous systems and contains information on the input-output behavior of the system. The rth-order reduced-order models are made to retain the first r Markov parameters and the first r×r elements of the impulse-response Gramian of the original system. The relation between this method and the q-Markov cover method is also discussed. The method is illustrated by a numerical example     

Adaptive Control of a Two-Link Robot Using Batch Least-Square Identifier

We design a regulation-triggered adaptive controller for robot manipulators to efficiently estimate unknown parameters and to achieve asymptotic stability in the presence of coupled uncertainties.Robot manipulators are widely used in telemanipulation systems where they are subject to model and environmental uncertainties.Using conventional control algorithms on such systems can cause not only poor control performance,but also expensive computational costs and catastrophic instabilities.Therefore,system uncertainties need to be estimated through designing a computationally efficient adaptive control law.We focus on robot manipulators as an example of a highly nonlinear system.As a case study,a 2-DOF manipulator subject to four parametric uncertainties is investigated.First,the dynamic equations of the manipulator are derived,and the corresponding regressor matrix is constructed for the unknown parameters.For a general nonlinear system,a theorem is presented to guarantee the asymptotic stability of the system and the convergence of parameters'estimations.Finally,simulation results are discussed for a two-link manipulator,and the performance of the proposed scheme is thoroughly evaluated.     

Adaptive force and position control of manipulators

The article presents simple methods for the design of adaptive force and position controllers for robot manipulators within the hybrid control architecture. The force controller is composed of an adaptive PID feedback controller, an auxiliary signal, and a force feedforward term, and achieves tracking of desired force setpoints in the constraint directions. The position controller consists of adaptive feedback and feedforward controllers as well as an auxiliary signal, and accomplishes tracking of desired position trajectories in the free directions. The controllers are capable of compensating for dynamic cross-couplings that exist between the position and force control loops in the hybrid control architecture. The adaptive controllers do not require knowledge of the complex dynamic model or parameter values of the manipulator or the environment. The proposed control schemes are computationally fast and suitable for implementation in online control with high sampling rates. The methods are applied to a two-link manipulator for simultaneous force and position control. Simulation results confirm that the adaptive controllers perform remarkably well under different conditions.     

Simultaneous stabilization of single-input single-output discretetime systems

The problem of finding one compensator which simultaneously stabilizes a family of single-input-single-output (SISO) discrete-time plants is considered. The family of plants is described by the transfer functions {Pz, q): q∈Q}, which are generated via a z-transformation. A number of assumptions describing the set of allowable plants are then given. These assumptions include some regularity conditions on the plants and a minimum-phase requirement. The satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator C(z) for simultaneous stabilization. An iterative computation method is provided for control design     

Convolution on mesh connected multicomputers

An efficient parallel algorithm is presented for convolution on a mesh-connected computer with wraparound. The algorithm does not require a broadcast feature for data values, as assumed by previously proposed algorithms. As a result, the algorithm is applicable to both SIMD and MIMD meshes. For an N×N image and a M×M template, the previous algorithms take O (M²q) time on an N×N mesh-connected multicomputer (q is the number of bits in each entry of the convolution matrix). The algorithms have complexity O(M²r), where r=max {number of bits in an image entry, number of bits in a template entry}. In addition to not requiring a broadcast capability, these algorithms are faster for binary images     

The stability of a family of polynomials can be deduced from afinite number 0(k3) of frequency checks

Let φ(s,a)=φ₀(s,a)+ a₁φ₁(s)+a₂ φ₂(s)+ . . .+a_kφ _k(s)=φ₀(s)-q(s, a) be a family of real polynomials in s, with coefficients that depend linearly on parameters a_i which are confined in a k-dimensional hypercube Ω_a . Let φ₀(s) be stable of degree n and the φ_i(s) polynomials (i⩾1) of degree less than n. A Nyquist argument shows that the family φ(s) is stable if and only if the complex number φ₀(jω) lies outside the set of complex points -q(jω,Ω_a) for every real ω. In a previous paper (Automat. Contr. Conf., Atlanta, GA, 1988) the authors have shown that -q(jω,Ω_a ), the so-called `-q locus', is a 2k convex parpolygon. The regularity of this figure simplifies the stability test. In the present paper they again exploit this shape and show that to test for stability only a finite number of frequency checks need to be done; this number is polynomial in k, 0(k³), and these critical frequencies correspond to the real nonnegative roots of some polynomials     

Compound-Poisson software reliability model

The probability density estimation of the number of software failures in the event of clustering or clumping of the software failures is considered. A discrete compound Poisson (CP) prediction model is proposed for the random variable X_rem, which is the remaining number of software failures. The compounding distributions, which are assumed to govern the failure sizes at Poisson arrivals, are respectively taken to be geometric when failures are forgetful and logarithmic-series when failures are contagious. The expected value (μ) of X_rem is calculated as a function of the time-dependent Poisson and compounding distribution based on the failures experienced. Also, the variance/mean parameter for the remaining number of failures, q_rem, is best estimated by q_past from the failures already experienced. Then, one obtains the PDF of the remaining number of failures estimated by CP(μ,q). CP is found to be superior to Poisson where clumping of failures exists. Its predictive validity is comparable to the Musa-Okumoto log-Poisson model in certain cases     

Adaptive Regulation of Amplitude Limited Robot Manipulators With Uncertain Kinematics and Dynamics

Common assumptions in most of the previous robot controllers are that the robot kinematics and manipulator Jacobian are perfectly known and that the robot actuators are able to generate the necessary level of torque inputs. In this note, an amplitude-limited torque input controller is developed for revolute robot manipulators with uncertainty in the kinematic and dynamic models. The adaptive controller yields semiglobal asymptotic regulation of the task-space setpoint error. The advantages of the proposed controller include the ability to actively compensate for unknown parametric effects in the dynamic and kinematic model and the ability to ensure actuator constraints are not breached by calculating the maximum required torque a priori     

Variational calculus for descriptor problems

The first-order, necessary condition for optimality is derived from a variational argument that involves an ad hoc modification of the Bliss method, resulting in a Hamiltonian characterization in terms of Edx,dt, rather than dx/dt, the former being smoother than the latter. This approach sidesteps the regularity conditions of the Lagrange multiplier theory. Under some mild assumptions, the necessary condition for optimality is also sufficient and the optimal control exists. The numerically relevant result is a generalized eigenvector, inverse-free characterization of optimality     

A comparison of two linear methods of estimating the parameters ofARMA models

A finite-order stationary and minimum-phase ARMA (autoregressive moving-average) (p,q) model is equivalent to an infinite-order AR (autoregressive) model. Two methods of estimating the parameters of the ARMA (p,q) model by solving only linear equations are based on or closely related to this equivalence relation. One method was derived directly from the equivalence relation by D. Graupe et al. (ibid., vol.AC-20, p.104-107, Feb. 1975). The other was derived by S. Li and B.W. Dickinson (ibid., vol.AC-31, p.275-278, Mar. 1986 and IEEE Trans. Acoust. Speech Signal Process., vol.ASSP-36, p.502-512, Apr. 1988) based on an iterated least-squares regression approach. The end results bear close resemblance to those of Graupe et al. The two methods are compared, and ways to improve the parameter estimates are suggested     

Adaptive tracking control of manipulators: Theory and experiments

This paper considers the trajectory tracking problem for uncertain robot manipulators and proposes two adaptive controllers as solutions to this problem. The first controller is derived under the assumption that the manipulator state is measurable, while the second strategy is developed for those applications in which only position measurements are available. The adaptive schemes are very general and computationally efficient since they do not require knowledge of either the mathematical model or the parameter values of the manipulator dynamics, and are implemented without calculation of the robot inverse dynamics or inverse kinematic transformation. It is shown that the control strategies ensure uniform boundedness of all signals in the presence of bounded disturbances, and that the ultimate size of the tracking errors can be made arbitrarily small. Experimental results are presented for a PUMA 560 manipulator and demonstrate that accurate and robust trajectory tracking can be achieved by using the proposed controllers.     

Adaptive control for robot manipulators using multiple parameter models

In this paper, we propose multiple parameter models based adaptive switching control system for robot manipulators. We first uniformly distribute the parameter set into a finite number of smaller compact subsets. Then, distributed candidate controllers are designed for each of these smaller compact subsets. Using Lyapunov inequality, a candidate controller is identified from the finite set of distributed candidate controllers that best estimates the plant at each instant of time. The design reduced the observer-controller gains by reducing modeling errors and uncertainties via identifying an appropriate control/model via choosing largest guaranteed decrease in the value of the Lyapunov function energy function. Compared with CE based CAC design, the proposed design requires smaller observer-controller gains to ensure stability and tracking performance in the presence of large-scale modeling errors and disturbance uncertainties. In contrast with existing adaptive method, multiple model based distributed hybrid design can be used to reduce the energy consumption of the industrial robotic manipulator for large scale industrial automation by reducing actuator input energy. Finally, the proposed hybrid adaptive control design is experimentally tested on a 3-DOF Phantom^TM robot manipulator to demonstrate the theoretical development for real-time applications.

     

Robust control of robot manipulators with fast load adaptation

Many adaptive robot controllers have been proposed in the literature to solve manipulator trajectory tracking problems for high-speed operations in the presence of parameter uncertainties. However, most of these controllers stem from the applications of the existing adaptive control theory, which is traditionally focused on tracking slowly time-varying parameters. In fact, manipulator dynamics have fast transient processes for high-speed operations and load changes are abrupt. These observations motivate the present research to incorporate change detection techniques into self-tuning schemes for tracking abrupt load variations and achieving fast load adaptation. To this end, a robustly global stabilizing controller for a robot model with parametric and non-parametric uncertainies is developed based on the Lyapunov second method, and it is then made adaptive via the self-tuning regulator concept. The two-model approach to online change detection in load is used and the estimation algorithm is reinitialized once load changes are detected. This allows a much faster adaptive identification of load parameters than the ordinary forgetting factor approach. Simulation results demonstrate that the proposed controller achieves better tracking accuracy than the existing adaptive and non-adaptive controllers.     

Motion control of mechanical manipulators

The theory and implementation results of a recently developed class of adaptive and repetitive controllers used for motion control of robot manipulators are presented. The repetitive controller, which learns the input torque corresponding to a repetitive desired trajectory, requires no explicit knowledge of the manipulator equations of motion. The adaptive controller, on the other hand, which estimates the robot dynamic parameters on-line, may be used for more general trajectories but requires more detailed modeling information. Both schemes are computationally efficient and require no acceleration feedback of any kind; only standard position and velocity feedback information is utilized.The performance of both the adaptive and repetitive controllers was experimentally evaluated on an IBM 7545 robot. The experimental results of these controllers confirm a significant improvement in tracking accuracy over conventional Computed Torque and PD controllers.This work was supported by the National Science Foundation under grant MSS-8910427.