Uncertain Multivariable Nonlinear Systems Research Articles

Local input‐to‐state stability analysis on compact set for disturbance observer‐based uncertain multivariable nonlinear control systems

AbstractThis article proposes a local input‐to‐state stability (LISS) analysis as a practical stability analysis approach. In existing LISS analysis approaches, a region including a state trajectory is not specified, and the decrement of a Lyapunov function must be guaranteed in the entire domain of the system state. This reduces the advantage of LISS by limiting the state region, and limits the class of systems for which stability can be guaranteed. To expand the class of systems for which stability can be guaranteed, the LISS Lyapunov function on a compact set (LISS LF CS) is proposed to guarantee LISS. By using the LISS LF CS, the region containing the state trajectory was revealed and the LISS condition was relaxed. In addition, the convergence rate and steady state error bound were determined for systems with LISS LF CS. The input‐to‐state stability (ISS) and LISS were analyzed for a multivariable disturbance observer‐based control system using the ISS and LISS LFs, respectively. The advantages of the proposed LISS analysis are demonstrated by comparing the ISS and LISS conditions. The analytical results were verified by conducting numerical simulations.

Recursive filtering adaptive neural fault-tolerant control for uncertain multivariable nonlinear systems

This paper synthesizes a recursive filtering adaptive neural fault-tolerant controller for uncertain multivariable nonlinear systems. The proposed control scheme adopts an estimation/cancellation strategy to deal with uncertainties and/or disturbances. The nonlinear uncertainties are approximated by a Gaussian radial basis function (GRBF)-based neural network incorporated with a piecewise constant adaptive law, where the adaptive law will generate adaptive parameters by solving the error dynamics between the real system and the state predictor with the neglection of unknowns, and recursive least squares (RLS) is applied to distribute the total uncertainty estimates into matched and mismatched components. The cooperation of GRBF learning method and piecewise constant adaptive law relaxes the stringent constraint on the hardware CPU speed and achieves fast adaptation. The filtering control law delivers a satisfactory performance with guaranteed robustness. Two numerical examples are provided to illustrate the effectiveness of the proposed control architecture via comparisons.

Filtered adaptive constrained sampled‐data control for uncertain multivariable nonlinear systems

AbstractA filtered adaptive constrained sampled‐data controller for uncertain multivariable nonlinear systems in the presence of various constraints is synthesized in this paper. A piecewise constant adaptive law drives that estimation error dynamics to zero at each sampling time instant yields adaptive parameters. The filtered control scheme consists of two components. Based on an estimation/cancellation strategy, a disturbance rejection control law is designed to compensate the nonlinear uncertainties within the bandwidth of low‐pass filters, whereas a constraint violation avoidance control law is designed to solve an online constrained optimization problem. Although a reduced sampling time helps to minimize the estimation error caused by the neglect of unknowns, the resulting aggressive signals put more restrictions on the control law. Greater sacrifice of tracking performance is required to satisfy the constraints. The constraints violation avoidance control law is in favor of a larger sampling time. Sufficient conditions are given to guarantee the stability of the closed‐loop system with the sampled‐data controller, where the input/output signals are held constant over the sampling period. Numerical examples are provided to validate the theoretical results, comparisons between the constrained sampled‐data controller and unconstrained adaptive controller with the implementation of different sampling times are carried out.

Filtered adaptive constrained sampled-data output feedback control for uncertain multivariable nonlinear systems

This paper synthesises a constrained sampled-data output feedback controller for uncertain multivariable nonlinear systems in the presence of various constraints, using the filtered adaptive control architecture. Nonlinear uncertainties are estimated online by solving the error dynamics between the output predictor and the real system with the neglection of unknowns, the yielded adaptive parameters are piecewise constant. In the proposed control scheme, the nonlinear uncertainties are compensated for within the bandwidth of low-pass filters, while the trade-off between tracking and constraints violation avoidance is formulated as a numerical constrained optimisation problem which is solved periodically. Priority is given to constraints violation avoidance at the cost of deteriorated tracking performance. System transformations are performed to find the sufficient conditions that can guarantee the stability of the closed-loop system with the sampled-data controller, where both the states and control inputs are held constant over the sampling period.

Observer-based parametric decoupling controller design for a class of multi-variable non-linear uncertain systems

ABSTRACTThis paper presents a novel decoupling control strategy for Lipschitz multi-variable non-linear uncertain systems. Using the explicit parametric design, an observer-based output feedback controller has been developed with free parameters while the closed-loop system can be further described by transfer function matrix with these free parameters. The coupling effects of the systems would be attenuated if the free parameters are optimised where the performance criterion is given based on the norm of the transfer functions. Moreover, the sufficient conditions of stabilization have been obtained for observer, controller and closed-loop system, respectively. Following the procedure of the presented control strategy, an illustrative numerical example is given to demonstrate the effectiveness of the presented control strategy. In addition, the similar design approach has been discussed for filtering problem which is a potential extension of the presented control strategy.

State and output feedback fuzzy variable structure controllers for multivariable nonlinear systems subject to input nonlinearities

This paper presents three fuzzy adaptive controllers for a class of uncertain multivariable nonlinear systems with both sector nonlinearities and dead zones: two first controllers are state feedbacks and the last controller is an output feedback. The design of the first controller concerns systems with symmetric and positive definite control–gain matrix, while the second control design is extended to the case of nonsymmetric control–gain matrix thanks to an appropriate decomposition, namely the product of a symmetric positive definite matrix, a diagonal matrix with diagonal entries +1 or −1, and a unity upper triangular matrix. The third controller is an output feedback extension of the second controller. In this controller, a high-gain observer is incorporated to estimate the unmeasurable states. An appropriate adaptive fuzzy logic system is used to reasonably approximate the uncertain functions. A Lyapunov approach is adopted to derive the parameter adaptation laws and prove the stability of those control systems as well as the exponential convergence of their underlying tracking errors within an adjustable region. The effectiveness of the proposed fuzzy adaptive controllers is illustrated through simulation results.

Observer-Based Robust Adaptive Fuzzy Control for MIMO Nonlinear Uncertain Systems with Delayed Output

An observer-based robust adaptive fuzzy control scheme is presented to tackle the problem of the robust stability and the tracking control for a class of multiinput multioutput (MIMO) nonlinear uncertain systems with delayed output. Because the nonlinear system functions and the uncertainties of the controlled system including structural uncertainties are supposed to be unknown, fuzzy logic systems are utilized to approximate these nonlinear system functions and the upper bounded functions of the uncertainties. Moreover, the upper bound of uncertainties caused by these fuzzy modeling errors is also estimated. In addition, the state observer based on state variable filters is designed to estimate all states which are not available for measurement in the controlled system. By constructing an appropriate Lyapunov function and using strictly positive-real (SPR) stability theorem, the proposed robust adaptive fuzzy controller not only guarantees the robust stability of a class of multivariable nonlinear uncertain systems with delayed output but also maintains a good tracking performance. Finally, some simulation results are illustrated to verify the effectiveness of the proposed control approach.

A Novel Adaptive Fuzzy Control Approach for Uncertain Multivariable Nonlinear Systems with Time Delays

Abstract A novel adaptive fuzzy controller for a class of uncertain multivariable nonlinear systems with time delays is proposed in this paper. By combining fuzzy Takagi-Sugeno (T-S) models and fuzzy logic systems, the modeling error and the uncertainties are not required to satisfy the constraint conditions. In the meantime, fuzzy T-S models are used to approximate the nonlinear systems and fuzzy logic systems are used to compensate the modeling error and the uncertainties. By using state feedback control law and adaptive law, free parameters can be tuned on-line based on Lyapunov synthesis approach. The adaptive fuzzy control scheme is constructed such that the closed-loop system is uniformly ultimately bounded and satisfies the desired  H performance. Simulation results demonstrate the effectiveness of the developed control scheme.

Adaptive control for nonlinear uncertain systems with actuator amplitude and rate saturation constraints

AbstractA direct adaptive nonlinear tracking control framework for multivariable nonlinear uncertain systems with actuator amplitude and rate saturation constraints is developed. To guarantee asymptotic stability of the closed‐loop tracking error dynamics in the face of amplitude and rate saturation constraints, the control signal to a given reference (governor or supervisor) system is modified to effectively robustify the error dynamics to the saturation constraints. Illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.

UNIT VECTOR CONTROL OF UNCERTAIN MULTIVARIABLE NONLINEAR SYSTEMS

A unit vector based output-feedback model-reference adaptive controller (UV-MRAC) for a class of uncertain multivariable nonlinear systems is proposed. This paper generalizes a previous controller which can be applied to plants with uniform relative degree one. Here, the relative degree can be arbitrary. The resulting sliding mode controller is applicable to plants with nonlinear state dependent disturbances which are possibly unmatched with respect to the plant input. The closed loop system has global or semi-global exponential stability with respect to some small residual set and the controller is free of peaking phenomena which could appear in high gain observer based schemes.

A NEW CHARACTERIZATION OF STABLE NEURAL NETWORK CONTROL FOR DISCRETE-TIME UNCERTAIN SYSTEMS

A novel neuro adaptive control framework for discrete-time multivariable nonlinear uncertain systems is developed. The proposed framework is Lyapunov-based and guarantees, instead of ultimate boundedness, partial asymptotic stability of the closed-loop system; that is, Lyapunov stability of the closed-loop system states and attraction with respect to the plant states. Unlike standard neural network approximation, we assume that the approximation error can be confined in a small gain-type norm-bounded conic sector over a compact set. This helps to couple tools from robust control with adaptive laws in discrete time to prove partial asymptotic stability of the closed-loop system. Finally, an illustrative numerical example is provided to demonstrate the efficacy of the proposed approach.

A Lyapunov-based adaptive control framework for discrete-time non-linear systems with exogenous disturbances

A direct adaptive non-linear control framework for discrete-time multivariable non-linear uncertain systems with exogenous bounded disturbances is developed. The adaptive non-linear controller addresses adaptive stabilization, disturbance rejection and adaptive tracking. The proposed framework is Lyapunov-based and guarantees partial asymptotic stability of the closed-loop system; that is, asymptotic stability with respect to part of the closed-loop system states associated with the plant. In the case of bounded energy ℓ 2 disturbances the proposed approach guarantees a non-expansivity constraint on the closed-loop input–output map. Finally, three illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.

Robust stabilization control design for multivariable nonlinear uncertain systems

Robust stabilization control design for multivariable nonlinear uncertain systems