Bounded Input Disturbances Research Articles

Global tracking for robot manipulators using a simple causal PD controller plus feedforward

SUMMARYThis paper shows that a well-known causal PD controller plus feedforward solves the global output feedback tracking control problem of robot manipulators, by requiring only the existence of the robot natural damping, no matter how small. To this end, we first demonstrate that a robot controlled by a causal PD is globally input-to-state stable (ISS) with respect to a bounded input disturbance. Then, we prove that the addition of a feedforward compensation renders the error system uniformly globally asymptotically stable. Furthermore, we present a possible extension to more general nonlinear systems and also to uncertain systems.

PI-Funnel Control for Two Mass Systems

We control a two mass system with uncertain parameter values, encompassing friction and hysteretic effects, modelled as a functional differential equation. As opposed to control strategies invoking identification mechanisms or neural networks, we propose controllers relying on structural system properties only. Output behaviour with pre-specified accuracy is guaranteed by the funnel controller, active damping of the shaft oscillation by the high-pass filter, zero tracking output error in the steady state by the PI-controller and bounded input disturbances are rejected. Finally, the controllers are implemented on a real plant, an electrical drive.

Affine and predictive control policies for a class of nonlinear systems

The input-state linear horizon (ISLH) for a nonlinear discrete-time system is defined as the smallest number of time steps it takes the system input to appear nonlinearly in the state variable. In this paper, we employ the latter concept and show that the class of constraint admissible N -step affine state-feedback policies is equivalent to the associated class of constraint admissible disturbance-feedback policies, provided that N is less than the system’s ISLH. The result generalizes a recent result in [Goulart, P. J., Kerrigan, E. C., Maciejowski, J. M. (2006). Optimization over state feedback policies for robust control with constraints. Automatica, 42(4), 523–533] and is significant because it enables one: (i) to determine a constraint admissible state-feedback policy by employing well-known convex optimization techniques; and (ii) to guarantee robust recursive feasibility of a class of model predictive control (MPC) policies by imposing a suitable terminal constraint. In particular, we propose an input-to-state stabilizing MPC policy for a class of nonlinear systems with bounded disturbance inputs and mixed polytopic constraints on the state and the control input. At each time step, the proposed MPC policy requires the solution of a single convex quadratic programme parameterized by the current system state.

Fault Reconstruction for Lipschitz Nonlinear Descriptor Systems via Linear Matrix Inequality Approach

By using the linear matrix inequality for a nonlinear descriptor system with a Lipschitz constraint, a state-space observer is designed to simultaneously reconstruct the descriptor system states and the fault signals. For a nonlinear descriptor system with bounded input disturbances, a robust state-space observer is proposed to simultaneously reconstruct the descriptor system states and the faults and attenuate the input bounded disturbances. The fault considered can be unbounded (provided that the qth derivative of the fault is zero piecewise or norm bounded); thus, the present fault reconstruction approach can handle a large class of fault signals including time-invariant and time-varying signals. A numerical example is given to illustrate the proposed design approach.

A methodology for the design of robust rollover prevention controllers for automotive vehicles with active steering

In this paper we present a robust controller design methodology for vehicle rollover prevention utilizing active steering. Control design is based on keeping the magnitude of the vehicle load transfer ratio (LTR) below a certain level in the presence of driver steering inputs; we also develop an exact expression for LTR. The proposed controllers have a proportional-integral structure whose gain matrices are obtained using the results of Pancake et al. (2000) (T. Pancake, M. Corless and M. Brockman, “Analysis and control of polytopic uncertain/nonlinear systems in the presence of bounded disturbance inputs”, Proceedings of the American Control Conference, Chicago, IL, June 2000; T. Pancake, M. Corless and M. Brockman, “Analysis and control for a class of uncertain/nonlinear systems in the presence of bounded disturbance inputs”, 2007). These controllers reduce the transient magnitude of the LTR while maintaining the steady state steering response of the vehicle. The controllers can be designed to be robust with respect to vehicle parameters such as speed and centre of gravity height. We also provide a modification to the controllers so that they only activate when the potential for rollover is significant. Numerical simulations demonstrate the efficacy of our approach and the resulting controllers.

ℒ︁2‐Gain analysis and control of uncertain nonlinear systems with bounded disturbance inputs

AbstractThis paper proposes a convex approach to regional stability and ℒ︁2‐gain analysis and control synthesis for a class of nonlinear systems subject to bounded disturbance signals, where the system matrices are allowed to be rational functions of the state and uncertain parameters. To derive sufficient conditions for analysing input‐to‐output properties, we consider polynomial Lyapunov functions of the state and uncertain parameters (assumed to be bounded) and a differential‐algebraic representation of the nonlinear system. The analysis conditions are written in terms of linear matrix inequalities determining a bound on the ℒ︁2‐gain of the input‐to‐output operator for a class of (bounded) admissible disturbance signals. Through a suitable parametrization involving the Lyapunov and control matrices, we also propose a linear (full‐order) output feedback controller with a guaranteed bound on the ℒ︁2‐gain. Numerical examples are used to illustrate the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.

Simple adaptive control for aircraft control surface failures

This work extends the so-called simple adaptive control approach to direct model reference adaptive control of multi-input multi-output systems to include loss of control effectiveness failures. It is proven that all signals are bounded for loss of control effectiveness failures during a bounded input disturbance. A state space approach is introduced for computing the feedforward compensator that is required by the stability result. The adaptive algorithm is applied to a three input model of the linearized lateral dynamics of the F/A-18 aircraft. Simulation results are obtained with single, double, and triple control effectiveness failures of 88% during the occurrence of a lateral gust. These results show that the adaptive controller exhibits improved model following as compared with a fixed gain eigenstructure assignment controller.

Noise Tolerant Iterative Learning Control and Identification for Continuous-Time Systems With Unknown Bounded Input Disturbances

This paper considers the problems of both noise tolerant iterative learning control (ILC) and iterative identification for a class of continuous-time systems with unknown bounded input disturbance and measurement noise. To this aim, we first propose a formulation of an extended ILC scheme using sampled input∕output (I∕O) data. The proposed ILC method has distinctive features as follows. Its learning law works in a prescribed finite-dimensional parameter space and employs I∕O data of all past trials efficiently. Also, the time derivative of tracking error is not required. Then, it is presented how the uncertain parameters can be identified by using the proposed ILC algorithm and how robust it is against measurement noise through a numerical example. Furthermore, its experimental evaluation is performed to demonstrate the effectiveness of the proposed identification scheme.

Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems

For Lipschitz nonlinear descriptor systems with bounded input disturbances, by solving a Lyapunov equation, a robust state-space observer is proposed to simultaneously estimate descriptor system states, actuator faults, their finite times derivatives, and attenuate input disturbances in any desired accuracy. The considered faults can be unbounded (provided that their q th derivatives are bounded), the present fault estimation approaches can handle a large class of faults. By using the estimates of descriptor states and faults, and the linear matrix inequality (LMI) technique, a fault-tolerant control scheme is worked out. The nonlinear fault-tolerant control system can be made solvable, causal, asymptotically stable, and attenuate input uncertainties in terms of the prescribed performance index. Only original coefficient matrices are used in the proposed state-space observers and fault-tolerant controllers; therefore, the present design approaches are preferable in applications. Finally, a numerical example is given to illustrate the design procedure and simulations demonstrate the efficiency of the proposed design.

A New Adaptive Backpropagation Algorithm Based on Lyapunov Stability Theory for Neural Networks

A new adaptive backpropagation (BP) algorithm based on Lyapunov stability theory for neural networks is developed in this paper. It is shown that the candidate of a Lyapunov function V(k) of the tracking error between the output of a neural network and the desired reference signal is chosen first, and the weights of the neural network are then updated, from the output layer to the input layer, in the sense that deltaV(k) = V(k) - V(k - 1) < 0. The output tracking error can then asymptotically converge to zero according to Lyapunov stability theory. Unlike gradient-based BP training algorithms, the new Lyapunov adaptive BP algorithm in this paper is not used for searching the global minimum point along the cost-function surface in the weight space, but it is aimed at constructing an energy surface with a single global minimum point through the adaptive adjustment of the weights as the time goes to infinity. Although a neural network may have bounded input disturbances, the effects of the disturbances can be eliminated, and asymptotic error convergence can be obtained. The new Lyapunov adaptive BP algorithm is then applied to the design of an adaptive filter in the simulation example to show the fast error convergence and strong robustness with respect to large bounded input disturbances.

FAULT ESTIMATION FOR NONLINEAR DESCRIPTOR SYSTEMS WITH LIPSCHITZ CONSTRAINTS VIA LMI APPROACH

By using the linear matrix inequality (LMI), for a nonlinear descriptor system with Lipschitz constraint, a state-space observer is designed to simultaneously estimate the descriptor system states, the fault signals and the finite times derivatives of the faults. For a nonlinear descriptor system with bounded input disturbances, a robust state-space observer is proposed to simultaneously estimate the descriptor system states, the faults, the finite times derivatives of the faults, and attenuate the input disturbances. The considered fault can be unbounded (provided that the qth derivative of the fault is zero piecewise or norm bounded), the present fault estimation approach can thus handle a large class of fault signals. A numerical example is included to illustrate the present design approach.

PD observer parametrization design for descriptor systems

Based on the design of proportional and derivative (PD) observers, a new parametrization of all observers for descriptor systems is developed. Without using any additional transformations, the present observer parametrization is convenient and efficient to be performed. The connections between the proposed parametrization and the known results are established. For descriptor systems with bounded input disturbances, the estimation error dynamics are parametrized with respect to the disturbance vectors. An example is included to illustrate the parametrization procedures.

Output-feedback model-reference sliding mode control of uncertain multivariable systems

This note considers the robust output tracking problem using a model-reference sliding mode controller for linear multivariable systems of relative degree one. It is shown that the closed loop system is globally exponentially stable and the performance is insensitive to bounded input disturbances and parameter uncertainties. The strategy is based on output-feedback unit vector control to generate sliding mode. The only required a priori information about the plant high frequency gain matrix K/sub p/ is the knowledge of a matrix S/sub p/ such that -K/sub p/S/sub p/ is Hurwitz which relaxes the positive definiteness requirement usually needed by other methods.

Decentralized control and disturbance attenuation for large-scale nonlinear systems in generalized output-feedback canonical form

A global decentralized robust adaptive output-feedback dynamic compensator is proposed for stabilization, tracking, and disturbance attenuation of the decentralized generalized output-feedback canonical form. This represents the largest class for which decentralized robust adaptive output-feedback tracking and disturbance attenuation results are currently available. The system is allowed to contain unknown parameters multiplying output-dependent nonlinearities, and, also, unknown nonlinearities satisfying certain bounds. Under the assumption that a constant matrix can be found for each subsystem to achieve a certain property, it is shown that reduced-order observers and backstepping controllers can be designed to achieve practical stabilization of the tracking error in each subsystem in the presence of bounded disturbance inputs. Sufficient conditions under which asymptotic tracking and stabilization can be achieved are also obtained. Signal gains from disturbance inputs to tracking errors are presented in the input-to-output-practical-stability and integral-input-to-output-practical-stability frameworks. A particular case in which the standard L 2 -gain disturbance attenuation is achieved is also provided.

Dual mode predictive tracking of piecewise constant references for constrained linear systems

In this paper, the attention is focused on the tracking of a piecewise constant reference for an LTI system in presence of state-control constraints and bounded disturbance inputs. Exploiting robust controlled invariant sets in the extended state-setpoint space, we propose a novel predictive tracking algorithm which yields robust constraint satisfaction and good tracking performance with a reasonable on-line computational burden. The main idea is to adopt a dual-mode controller that operates as a predictive regulator whenever the desired setpoint is feasible or in feasibility recovery mode otherwise. Simulation results demonstrate the improved tracking performance of the proposed method with respect to available techniques.

Subband coding of cyclostationary signals

Fig. 4. The control input using the boundary-layer technique. formed LTV system to retain the dynamical features of the original LTV system. The update law of the transformation matrix using a timevarying differential Sylvester equation has been developed to guarantee the uniformly exponential convergence of the closed-loop dynamics on the sliding mode. A robust sliding-mode output regulator using only the output measurements has been designed to drive the system output to converge to zero in a finite time. A simulation example has been carried out with the results that good convergence and robustness properties with respect to the bounded input disturbance are obtained. Further work is to extend the scheme to the control of general SISO and MIMO LTV systems with uncertainties. REFERENCES

Lyapunov-Based Control of a Bridge Using Magneto-Rheological Fluid Dampers

This study focuses on the effect of semi-active magneto-rheological fluid (MRF) dampers in reducing the response of a scaled bridge structure subjected to a random loading. A fluid-mechanics based model that can characterize the nonlinear dynamic behavior of MRF dampers is employed. A state-variable model for an integrated system of a scaled, two-span bridge and two MRF dampers is established. A feedback on-off control law is employed based on Lyapunov approach that guarantees the system stability for uncertain bounded input disturbances. An output feedback Lyapunov controller is implemented in the experimental study to verify the presented method. Both the theoretical and experimental studies show that the Lyapunov based control systems can effectively reduce the relative displacement between the deck and the abutment of the bridge when subjected to various input motions. In addition, when comparing to the experiment results, it is demonstrated that the proposed analytical model can predict the response of the system, accurately.

Robust switching adaptive control of multi-input nonlinear systems

During the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters. New design tools such as adaptive feedback linearization, adaptive back-stepping, control Lyapunov functions (CLFs) and robust control Lyapunov functions (RCLFs), nonlinear damping and switching adaptive control have been introduced. Most of the results developed are applicable to single-input feedback-linearizable systems and parametric-strict-feedback systems. These results, however, cannot be applied to multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. In this paper, we develop a general procedure for designing robust adaptive controllers for a large class of multi-input nonlinear systems. This class of nonlinear systems includes as a special case multi-input feedback-linearizable systems, parametric-pure-feedback systems and systems that admit a linear-in-the-parameters CLF. The proposed approach uses tools from the theory of RCLF and the switching adaptive controllers proposed by the authors for overcoming the problem of computing the feedback control law when the estimation model becomes uncontrollable. The proposed control approach has also been shown to be robust with respect to exogenous bounded input disturbances.

A Switching State Feedback Control of Constrained Systems with Exogenous Inputs

This paper considers switching state feedback control of systems with unknown but bounded disturbance inputs subject to state and control constraints, and resolves convergence problems which arise with the effects of unknown disturbances. This difficulties have not been handled by the switching control laws which were proposed in the literatures. The proposed switching control law assures convergence of the state and successive controller switching even in the presence of unknown disturbances. The main idea is to construct an upper bound of the state reachable set ' such that convergence of the state is assured by restricting this set to some subset of the state space. The resulting optimization problems which provide a sequence of feedback gains are more difficult to solve than those with no disturbances. However, this optimization problems may be solved with some heuristic manners. In addition, required on-line computational burden is the same as that in case of no disturbance inputs.

Adaptive λ-tracking for a class of infinite-dimensional systems

For a class of high-gain stabilizable multivariable linear infinite-dimensional systems we present an adaptive control law which achieves approximate asymptotic tracking in the sense that the tracking error tends asymptotically to a ball centred at 0 and of arbitrary prescribed radius λ>0. This control strategy, called λ-tracking, combines proportional error feedback with a simple nonlinear adaptation of the feedback gain. It does not involve any parameter estimation algorithms, nor is it based on the internal model principle. The class of reference signals is W 1,∞, the Sobolev space of absolutely continuous functions which are bounded and have essentially bounded derivative. The control strategy is robust with respect to output measurement noise in W 1,∞ and bounded input disturbances. We apply our results to retarded systems and integrodifferential systems.