Showing papers in "International Journal of Control in 1995"

Flatness and defect of non-linear systems: introductory theory and examples

Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

Sliding mode stabilization of uncertain systems using only output information

Abstract: A practical procedure is presented for synthesizing output feedback controllers for uncertain systems based on sliding mode concepts. The class of systems to which the results apply is identified and includes the requirement that the nominal linear system is minimum phase. The procedure proposed for the design of the sliding surface uses established output feedback eigenvalue assignment results. It will be shown that all the assumptions imposed on the system pertain to the design of the sliding surface. The proposed controller, which guarantees attainment of a sliding mode despite the presence of uncertainty requires no additional assumptions.

Overtraining, regularization and searching for a minimum, with application to neural networks

Abstract: In this paper we discuss the role of criterion minimization as a means for parameter estimation. Most traditional methods, such as maximum likelihood and prediction error identification are based on these principles. However, somewhat surprisingly, it turns out that it is not always ‘optimal’ to try to find the absolute minimum point of the criterion. The reason is that ‘stopped minimization’ (where the iterations have been terminated before the absolute minimum has been reached) has more or less identical properties as using regularization (adding a parametric penalty term). Regularization is known to have beneficial effects on the variance of the parameter estimates and it reduces the ‘variance contribution’ of the misfit. This also explains the concept of ‘overtraining’ in neural nets. How does one know when to terminate the iterations then? A useful criterion would be to stop iterations when the criterion function applied to a validation data set no longer decreases. However, in this paper, we show th...

Closed-loop control of systems over a communications network with queues

Abstract: Due to remote sensor, actuator and processor locations, many systems need to implement closed loop control through a communication network. Thus, they may thus face the problem of induced random delays caused by the network. These delays may deteriorate the system performance and may even cause instability. The problems get more complicated when the possibility of queue formation at the transmitting side is considered for closed loop data transmission. In this paper, we propose a state prediction/control scheme to control this type of system. The scheme utiliźes knowledge of the amount of data in the queue to enhance prediction. Two automotive examples will be used to demonstrate the performance of the proposed scheme.

Receding horizon control and discontinuous state feedback stabilization

Abstract: This paper addresses three aspects of receding horizon control in discrete-time: (1) feedback stabilization of general nonlinear systems with receding horizon control; (2) the generation of stabili...

Optimal residual decoupling for robust fault diagnosis

Abstract: his paper deals with residual generation for the diagnosis of faults in the presence of disturbances. The emphasis is on modelling errors, represented as multiplicative disturbances, and on parametric faults. These are both characterized as discrepancies in a set of underlying parameters. The residuals are obtained using parity equations. To address the situation when the number of uncertain parameters is too high to allow perfect decoupling, two approximate decoupling methods are introduced. One utilizes rank reduction of the model-error/fault entry matrix via singular value decomposition. The other minimizes a least squares performance index, formulated on the residuals, under a set of equality constraints. It is shown that, by the appropriate construction of the entry matrix or of the performance index and the constraints, a broad variety of structured and directional residual strategies can be implemented.

Adaptive control of a class of nonlinear discrete-time systems

Abstract: We consider adaptive control via state feedback for a class of feedback linearizable discrete-time systems. To parallel the development in adaptive nonlinear continuous-time control, we employ a systematic procedure to design the controllers and the update laws for the so-called parametric-strict-feedback form and the parametric-pure-feedback form. The geometric conditions that characterize this class of systems are given. Depending on the growth conditions of the nonlinearities, global boundedness and convergence are achieved with various update laws. We also develop adaptive indirect schemes with an observer-based identifier for the parametric-strict-feedback form. As a result, the drawback of overparametrization in direct schemes can be removed.

The XY-centring algorithm for the dual LMI problem: a new approach to fixed-order control design

Abstract: Many fixed-order suboptimal control problems with stability, performance and robustness specifications can be reduced to a search for a matrix X > 0 satisfying a linear matrix inequality (LMI) while X −1 satisfies another LMI. This paper defines a certain class of these problems we shall call the ‘dual LMI problem’, and a computational algorithm to solve our dual LMI problem is given. Properties and limitations of the algorithm are discussed in comparison with the existing algorithm (the min/max algorithm). An extension to optimal control problems is provided. Numerical examples for the fixed-order stabilization problem and the static output feedback linear quadratic optimal control problem demonstrate the applicability of the proposed algorithm.

Optimal predictive control of continuous nonlinear systems

Abstract: A new approach for the design of nonlinear feedback tracking controllers is presented in this paper. The response of a nonlinear, continuous-time system is first predicted by appropriate functional expansions. Then, a control law is developed by minimizing the local difference between the predicted and desired responses. Closed-loop stability and robustness of the controller are discussed. Some relations and distinctions between the current predictive controller and geometric control approach are explored. The uniqueness of the current control law is demonstrated by successfully solving a class of tracked vehicle motion control problems for which some major existing nonlinear control methods are not applicable.

Non-holonomic control systems: from steering to stabilization with sinusoids

Abstract: We present a control law for globally asymptotically stabilizing a class of controllable nonlinear systems without drift. The control law converts into closed loop feedback earlier strategies for open loop steering of non-holonomic systems using sinusoids at integrally related frequencies. The global result is obtained by introducing saturation functions. Simulation results for stabilizing a simple kinematic model of an automobile are included.

Discrete-time adaptive control of systems with unknown deadzones

Abstract: An adaptive deadzone inverse approach is developed for control of discrete-time systems with unknown deadzones at the input of linear dynamics. The proposed controller consists of an adaptive deadzone inverse, whose parameters are updated to cancel the effect of the deadzone, and a linear part, which is designed to achieve tracking of a reference signal by the system output. When the linear part of the system is known, its parameters are used to specify the parameters of the linear part of the controller. When the linear part of the plant is unknown, the linear control parameters are adaptively estimated. The control error caused by the adaptive deadzone inverse is expressed as a linearly parametrizable part plus a bounded disturbance. Adaptive laws which employ parameter projection are used to update the controller parameters. They are robust with respect to the bounded disturbance, and they result in parameter estimates needed to implement an adaptive deadzone inverse. We prove the closed-loop signal bo...

Observer design in receding-horizon predictive control

Abstract: This paper considers the robust implementation of a class of predictive control methods represented by GPC. Such controllers are in two-degrees-of-freedom form where there are dynamics in both the forward and the feedback paths. The ‘tuning knobs’ of predictive controllers determine the characteristic polynomial PC , and for a given PC the observer or prefiltering polynomial T in the feedback path determines the robustness of the closed loop. Previous intuitive guidelines on the selection of T are shown to be limited in their effectiveness. For an open-loop stable plant, a simple criterion is provided which allows the feedback dynamics to be specified so as to enhance robustness. The T polynomial is then chosen to satisfy this criterion. In addition, robust design through T is related to an H∞ -optimal control scheme using the so-called Q-parametrization. Despite its simplicity, the new proposed approach to the design of T is seen to result in robustness comparable with that obtained from the H∞ method.

Improved structure selection for nonlinear models based on term clustering

Abstract: In this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification. It is argued that if a certain type of term in a nonlinear model is spurious, the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model. Once the spurious clusters have been detected, the corresponding terms can be deleted from the set of candidate terms. The consequences of doing this are (i) a drastic reduction in the size of the set of candidate terms and, consequently, a substantial gain in computation time is achieved; (ii) the final estimated model is more likely to reproduce the dynamics of the original system; and (iii) the final model is more robust to overparametrization. Numerical examples are included to illustrate the new procedure.

Design of a global sliding-mode controller for a motor drive with bounded control

Abstract: A global sliding-mode controller can ensure sliding behaviour throughout an entire response for applications of a motor drive. The desired specifications, input disturbances and uncertainties of parameters are considered in the design, and the range of allowable reference input (RARI) is then estimated under the constraint on control activities and the sliding mode requirement; conversely, the designers can specify the demands on the capability of a servo driver to achieve a desired performance. To verify the proposed scheme, we performed experiments on its implementation in a DC brushless servo system. The advantages of the proposed scheme were indicated by comparison with the conventional sliding-mode scheme; the coherence of the theoretical estimates and the experimental results were tabulated for further validation of the proposed scheme.

Regional controllability of distributed parameter systems

Abstract: The purpose of this paper is to show how one can achieve regional controllability for distributed systems. First, we give a definition and some properties of this new concept, then we concentrate on the determination of a control achieving regional controllability with minimum energy. A direct method is developed and is illustrated by a hyperbolic system with point control and a parabolic system with zone control. We also give an important remark that converts the problem into a constrained optimization one.

Model validation tests for multivariable nonlinear models including neural networks

Abstract: A fast and concise MTMO nonlinear model validity test procedure is derived, based on higher order correlation functions, to form a global-to-local hierarchical validation diagnosis of identified MEMO linear and nonlinear models. The new procedure is applied to four MIMO nonlinear system models including a neural network training example, to demonstrate the effectiveness of the tests.

Nonlinear system identification using neural state space models, applicable to robust control design

Abstract: Prediction error learning algorithms for neural state space models are developed, both for the deterministic case and the stochastic case with measurement and process noise. For the stochastic case, a predictor with direct parametrization of the Kalman gain by a neural net architecture is proposed. Expressions for the gradients are derived by applying Narendra's sensitivity model approach. Finally a linear fractional transformation representation is given for neural state space models, which makes it possible to use these models, obtained from input/output measurements on a plant, in a standard robust performance control scheme.

Robust time-delay control of multimode systems

Abstract: This paper presents a procedure for the design of open loop controllers for flexible structures using multiple step inputs delayed in time The controller attenuates the residual vibration by cancelling the complex poles of the system Robustness is achieved by locating additional zeros at the cancelled poles of the system The paper begins by addressing the control of a single mode and examines the effect of user selected time-delays on robustness and the reference input Next, a procedure for the design of robust time-delay controllers for multiple modes with user selected time-delays is considered This is followed by a design of a minimum time-delay controller, such that the step input magnitudes are constrained to values between 0 and 1 Two examples, a spring-mass system and a single-link flexible-arm robot are used to illustrate the effectiveness of the proposed controller

Inherent design limitations for linear sampled-data feedback systems

Abstract: There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

Output feedback stabilization using variable structure control

Abstract: We consider the stabilization of a class of multivariable nonlinear systems, about an equilibrium point at the origin, using variable structure output feedback control In particular, the system can be transformed into a normal form with no zero dynamics A robust high-grain observer is used to estimate the state variables while rejecting the effect of disturbance A globally bounded discontinuous variable structure controller is designed to compensate for modelling error We show that the controller can stabilize the closed-loop system and does not suffer from the peaking phenomenon that exists in previous designs

Optimal loop-shaping for systems with large parameter uncertainty via linear programming

Abstract: This paper describes an optimization method for designing feedback systems subject to large parameter uncertainty. Following the design philosophy of the Quantitative Feedback Theory (Horowitz and Sidi 1972, 1978) the objective is to minimize the magnitude of the open loop L(jω) at high frequencies subject to: (a) low and intermediate-frequency bounds capturing the closed-loop robust-performance objectives; (b) a universal-frequency bound on L(jω) which limits the effects of disturbances; and (c) realization constraints on L(s) in the form of Bode's integral. This last relation is discretized at a number of frequencies and defines, together with (a) and (b), the overall set of linear constraints in a resulting linear programming optimization problem.

Identification of a class of nonlinear continuous-time systems using Hartley modulating functions

Abstract: Most of the existing approaches to identification of nonlinear dynamic systems involve matching a given input-output behaviour with empirical discrete-time approximations such as artificial neural networks, Kolmogorov-Gabor polynomials, radial basis function networks, etc. Techniques for dealing with physically-based continuous-time models are either applicable to only a restricted class of systems or are computationally very demanding. In this paper a new methodology is presented that is applicable to a large class of nonlinear continuous-time systems, by defining a set of Hartley modulating functions for characterizing the continuous process signals. The advantages of this new class of modulating functions are that a set of algebraic equations with real coefficients results, the formulations are free from boundary conditions, and the computations can be made using fast algorithms for the discrete Hartley transformation. The resulting estimation scheme is applied to different categories of nonlinear syst...

Robust tracking and model following for uncertain time-delay systems

Abstract: In this paper, a new controller design method for uncertain time-delay systems is proposed. We develop a nonlinear switching controller to force the outputs of the uncertain time-delay system to track, the outputs of a non-delay reference model that represents the ideal response of the controlled uncertain time-delay system. We also demonstrate the attractiveness of the switching hypersurface and show that, along the hypersurface, the trajectories are insensitive to system parameter variations.

Precision tracking control of non-minimum phase systems with zero phase error

Abstract: In this paper, a method is proposed for designing the command feedforward controller for zero phase error tracking control of systems having unacceptable zeros. The unacceptable zeros include both non-minimum phase zeros and lightly damped zeros. In the proposed approach, a digital preview filter along with the acceptable part of the system's inverse model is designed to form the command feedforward controller. Using preview information of the input trajectory, the designed tracking controller guarantees zero phase error and high precision tracking performance. In the proposed design, the best tracking performance can be achieved by minimizing a weighted penalty function, which controls the gain error between the desired output and the actual output. Two design cases were examined. The first is the design based on a uniform weighting function. In this study, the optimal solution of the design can be analytically obtained and three prior approaches were shown to be the special cases of the proposed solutio...

A robust PI-controller for infinite-dimensional systems

Abstract: In this paper, we deal with single-input single-output systems of the form on a separable Hilbert space H, where the operator A is the generator of an exponentially stable C0-semigroup on H, b ϵ H, C is a A -admissible linear operator and w is an arbitrary constant disturbance vector in H. We propose a low-gain PI-controller which stabilizes and regulates the system such that, for a given reference constant yr, y(t) tends to yr independently of w as t → + ∞ . Our result generalizes the previous one of Pohjolainen (1982) in that the semigroup is not necessarily holomorphic. A numerical example will be given to illustrate the application of the theory.

Optimal nonlinear, but continuous, feedback control of systems with saturating actuators

Abstract: This paper presents a modification of the optimal saturating feedback control laws given by Frankena and Sivan (1979) and Ryan (1982 a) for asymptotically stable systems. Unlike their results, which involve bang-bang action and singular extremals, the modified control law is continuous. Specifically, the new control law is linear inside a cylinder set and saturated elsewhere. Using steady-state Hamilton-Jacobi-Bellman theory, this control law is shown to be optimal for a modified performance functional with a discontinuous integrand.

Recursive identification using feedforward neural networks

Abstract: The paper is concerned with the identification of an unknown nonlinear dynamical system when only the inputs and outputs are accessible for measurement. Under certain assumptions it is shown that, generically, the system can be realized by a recursive input-output model. Furthermore, relying on the approximation properties of neural networks and the existence of effective training algorithms, it is demonstrated how an effective identification model can be constructed. Simulation results are presented to complement the theoretical discussions.

Decentralized control and overlapping decomposition of mechanical systems–Part 1. System decomposition

Abstract: This paper improves a new inclusion concept, called ‘txtension’ to mechanical systems described by matrix second-order equations. Recent work on specialization of the inclusion principle has been generalized to the expansion and contraction of input, state and output spaces, but without the possibility of constructing completely general contractible controllers. In this paper, necessary and sufficient conditions for extensions and contractible feedback controllers are proved. They enable the construction of general contractible controllers without any constraint on the structure of controllers for mechanical systems.

Decentralized control design: uncertain systems with strong interconnections

Abstract: We design decentralized controls for interconnected non-linear dynamical systems with uncertainties. This uncertainty is (possibly fast) time-varying. It may appear in each system as uncertain parameter and input disturbance. It also may appear in the interconnections. No statistical information about the uncertainty is imposed; only its possible bound is assumed to be known. The design takes the interconnections into explicit account. That is, the interconnections are to be compensated rather than to be tolerated. Under a mild assumption, interconnections with arbitrary non-linear bound can be addressed. This renders all previous works on this theme as special cases.

Improving computational efficiency of model predictive control algorithm using wavelet transformation

Abstract: We propose an MPC algorithm with desirable stability and constraint-handling properties, yet which retains computational simplicity. The closed-loop stability and constraint-handling are achieved by formulating an infinite-horizon objective and then solving an equivalent finite-horizon problem. On-line computational requirements for multi-time-scale systems, are reduced substantially by techniques called ‘blocking’ and ‘condensation’. We show that the blocking and condensation techniques are conveniently applied after expressing the MPC objective and constraint equations in terms of wavelet bases.