Input Saturation Nonlinearity Research Articles

Controller design for nonlinear time delay distributed control systems subjected to input saturation nonlinearity and disturbances

This article proposes a state feedback controller synthesis for nonlinear time-delay distributed control systems subjected to input saturation nonlinearity and disturbances. Nonlinear time-delay distributed control system individual states are without time-delay and the states coming from other subsystems have the communication link time-delay. The coupling states time-delay is presumed to be time-varying within a predefined bound. First, we suggest global state feedback controller design and then, we extend the proposed global design technique to more general local state feedback controller scheme by using an auxiliary region of attraction. Linear matrix inequality (LMI)-based solution is anticipated to synthesis global and local state feedback controller by using global and local sector bounded condition, Lyapunov–Krasovskii function, and Lipschitz condition. Which guarantee global and local asymptotic stability of the complete closed-loop system and $$ L_{2} $$ gain reduction of the mapping from $$ d_{p} (t) $$ to $$ z_{p} (t) $$ . Application results are presented to validate the benefits and effectiveness of the anticipated controller design schemes.

Simple adaptive robust output tracking control schemes of uncertain parametric strict‐feedback non‐linear systems with unknown input saturations

In this study, the problem of robust output tracking control is considered for a class of uncertain strict-feedback non-linear systems with any input saturation. A new design approach, called an adaptive design approach, is proposed to deal with saturated input non-linearities. The main merits of the proposed adaptive design approach can be summarised as follows. (i) It is completely unnecessary to know any information on input saturation non-linearities. (ii) By utilising the proposed adaptive design approach, the resulting control schemes will have a specially simple structure. (iii) The adaptive design approach proposed in this study is also easy to understand for the system designers, and by combining the proposed adaptive design approach with other control ones, a number of interesting results can be expected for a rather large class of dynamical systems with saturation in the actuators. Finally, a numerical example is given to illustrate the design procedure, and the corresponding simulations are also implemented to show the validity of the theoretical results.

Takagi–Sugeno Fuzzy-Based CNF Control Approach Considering a Class of Constrained Nonlinear Systems

ABSTRACTThis paper proposes a parallel distributed compensation-based composite nonlinear feedback (CNF) control approach for a class of constrained nonlinear systems, which is described based on Takagi–Sugeno (T-S) fuzzy models with input saturation. By producing invariant sets, the sufficient conditions guarantee the asymptotic stability of the closed-loop system. The presence of input saturation nonlinearity is derived in terms of linear matrix inequality. To the best knowledge of the authors, the research presented here is taken into real consideration as the first attempt to propose a new T-S fuzzy-based CNF control approach. Contrary to the existing CNF outcomes for the linear systems, the proposed one is now applicable to cope with a class of nonlinear systems. In a word, the present CNF control approach comprises two nonlinear parts. It is to note that the first term assures the stability of the closed-loop nonlinear system and provides a fast convergence response, as long as the second term reduces the overshoot of the response, correspondingly. In one such case, the overall CNF control approach satisfies the predefined input saturation, as well. In the sequel, in order to illustrate the merits of the proposed control approach, the continuous stirred tank reactors and the electromagnetic suspension system are chosen to verify the investigated results in comparison with those obtained from the existing results.

Adaptive backstepping control for air-breathing hypersonic vehicles with input nonlinearities

This paper addresses the control problem of air-breathing hypersonic vehicles subject to input nonlinearities, aerodynamic uncertainties and flexible modes. An adaptive backstepping controller and a dynamic inverse controller are developed for the altitude subsystem and the velocity subsystem, respectively, where the former eliminates the problem of “explosion of terms” inherent in backstepping control. Moreover, a modified smooth inverse of the dead-zone is proposed to compensate for the dead-zone effects and reduce the computational burden. Based on this smooth inverse, an input nonlinear pre-compensator is designed to handle input saturation and dead-zone nonlinearities, which leads to a simpler control design for the altitude subsystem subject to these two input nonlinearities. It is proved that the proposed controllers can guarantee that all closed-loop signals are bounded and the tracking errors converge to an arbitrarily small residual set. Simulation results are carried out to demonstrate the effectiveness of the proposed control scheme.

Adaptive neural control for stochastic pure‐feedback non‐linear time‐delay systems with output constraint and asymmetric input saturation

In this study, the adaptive tracking control is investigated for a class of stochastic pure-feedback non-linear time-delay systems with output constraint and asymmetric input saturation non-linearity. First, the Gaussian error function is employed to represent a continuous differentiable asymmetric saturation model, and the barrier Lyapunov function is designed to cope with the output constraints. Then, the appropriate Lyapunov–Krasovskii functional and the property of hyperbolic tangent functions are used to address the effects of the unknown time-delay terms, and the neural network is employed to approximate the unknown non-linearities. At last, based on Lyapunov stability theory, a robust adaptive neural controller is proposed, which decreases the number of learning parameters and thus avoids the over-estimation problem. Under the designed neural controller, all the closed-loop signals are guaranteed to be 4-moment (or 2 moment) semi-globally uniformly ultimately bounded and the tracking error converges to a small neighbourhood of the origin for bounded initial conditions. Two simulation examples are presented to further illustrate the effectiveness of the designed method.

Stabilisation of locally Lipschitz non‐linear systems under input saturation and quantisation

This study addresses the control of locally Lipschitz non-linear systems under quantisation and input saturation non-linearities. The non-linear dynamics of the systems are taken to be locally Lipschitz, rather than the conventional globally Lipschitz counterpart, to consider a generalised form of the Lipschitz non-linear systems. An ellipsoidal region containing the origin is constructed, for which the non-linear dynamics satisfy the Lipschitz condition in contrast to the conventional approaches. Saturation and quantisation non-linearities are dealt using a generalised local sector condition and a bound on the quantisation noise. A regional control strategy for the stabilisation of non-linear systems using state feedback is devised by employing these conditions, which is further extended to attain robustness against external perturbations. The proposed control strategies guarantee convergence of the states of a non-linear system inside a bounded reducible region in the neighbourhood of the origin. In contrast to the conventional approaches, the present study considers ellipsoidally Lipschitz non-linear systems, supports various types of quantisers, ensures attenuation of the disturbances, and provides a clear picture of the region of stability. An example of the application of the proposed control strategies for a modified Chua's circuit is demonstrated.

Adaptive neural control for nonstrict-feedback time-delay systems with input and output constraints

An adaptive tracking control is investigated for a class of nonstrict-feedback nonlinear systems with time delays subject to input saturation nonlinearity and output constraint. First, the Gaussian error function is used to express the continuous differentiable asymmetric saturation model, and a barrier Lyapunov function is designed to ensure that the output parameters are restricted. Then, an appropriate Lyapunov–Krasovskii functional is chosen to deal with the unknown time-delay terms, and the neural network is used to model the unknown nonlinearities. Finally, based on Lyapunov stability theory, an adaptive neural controller is designed to establish the closed-loop system stability. The example is provided to further illustrate the effectiveness and applicability of the proposed approach.

Adaptive neural synchronization control of chaotic systems with unknown control directions under input saturation

In this paper, we investigate the synchronization problem of chaotic systems with unknown control direction under input saturation nonlinearity. By utilizing backstepping technique, an adaptive neural synchronization control scheme is developed via neural approximation technique and Nussbaum-type function method. To deal with the problem of sharp corner of saturation, a tangent function based smooth function is used to approximate input saturation. To facilitate controller design, an auxiliary signal is constructed to augment the plant. Two Nussbaum functions are respectively used to deal with unknown control direction and compensate for the time-varying gain arising from the partial differentiation of the smooth function of input saturation. Tracking-differentiator is introduced to handle the problem of “explosion of computation” in the backstepping design. Adaptive robust effects are designed to deal with the “disturbance-like” term and tracking-differentiator estimation error. Lyapunov based analysis gives the transient performance and convergence of synchronization errors. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

Robust model predictive control synthesis for state-delayed systems with randomly occurring input saturation nonlinearities

In this paper, a model predictive control algorithm is presented for linear parameter varying systems with both state delays and randomly occurring input saturation. The input saturation is assumed to be occurred randomly with Bernoulli-distributed white sequences. A constant sate feedback law is designed at each time instant to ensure the robust stability of the closed-loop system with respect to polytopic uncertainties. The optimization of model predictive controller is cast into solving a linear matrix inequalities optimization problem. Then, the results are extended to gain-scheduled approach in which a set of state feedback laws are designed for each vertex of the system model. The state feedback law is scheduled by the time varying model parameters to achieve less conservatism in controller design. Finally, two examples are employed to show the effectiveness of the proposed algorithms.

Vibration control of an axially moving accelerated/decelerated belt system with input saturation

This article describes an investigation of a boundary control for vibration suppression of an axially moving accelerated or decelerated belt system with input saturation. Firstly, after considering the effects of the high acceleration or deceleration and unknown distributed disturbance, an infinite-dimensional model of the belt system is described by a nonhomogeneous partial differential equation and a set of ordinary differential equations. Secondly, by synthesizing boundary control techniques and Lyapunov’s direct method, a boundary control is developed to suppress the belt’s vibration and to stabilize the belt system at its equilibrium position globally; an auxiliary system is proposed to compensate for the nonlinear input saturation characteristic; a disturbance adaptation law is employed to mitigate the effects of unknown boundary disturbance; and the S-curve acceleration/deceleration method is adopted to plan the belt’s axial speed. Thirdly, with the proposed boundary control, the wellposedness of the closed-loop belt system is mathematically demonstrated and uniformly bounded stability of the closed-loop system is achieved without any discretization of the system dynamic model. Finally, simulation results are presented to verify the validity and effectiveness of the proposed control scheme.

Adaptive neural dynamic surface sliding mode control for uncertain nonlinear systems with unknown input saturation

In this article, an adaptive neural dynamic surface sliding mode control scheme is proposed for uncertain nonlinear systems with unknown input saturation. The non-smooth input saturation nonlinearity is firstly approximated by a smooth non-affine function, which can be further transformed into an affine form according to the mean value theorem. Then, one simple sigmoid neural network is employed to approximate the uncertain nonlinearity including the input saturation, and the approximation error is estimated using an adaptive learning law. Virtual controls are designed in each step by combing the dynamic surface control and integral sliding mode technique, and thus the problem of complexity explosion inherent in the conventional backstepping method is avoided. With the proposed control scheme, no prior knowledge is required on the bound of input saturation, and comparative simulations are given to illustrate the effectiveness and superior performance.

Sector-condition-based results for adaptive control and synchronization of chaotic systems under input saturation

This paper addresses the design of adaptive feedback controllers for two problems (namely, stabilization and synchronization) of chaotic systems with unknown parameters by considering input saturation constraints. A novel generalized sector condition is developed to deal with the saturation nonlinearities for synthesizing the nonlinear and the adaptive controllers for the stabilization and synchronization control objectives. By application of the proposed sector condition and rigorous regional stability analysis, control and adaptation laws are formulated to guarantee local stabilization of a nonlinear system under actuator saturation. Further, simple control and adaptation laws are developed to synchronize two chaotic systems under uncertain parameters and input saturation nonlinearity. Numerical simulation results for Rössler and FitzHugh–Nagumo models are provided to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization control methodologies.

Adaptive neural data‐based compensation control of non‐linear systems with dynamic uncertainties and input saturation

In this study, an adaptive neural backstepping control scheme is proposed for a class of strict-feedback non-linear systems with unmodelled dynamics, dynamic disturbances and input saturation. To solve the difficulties from the unmodelled dynamics and input saturation, a dynamic signal and smooth function in non-affine structure subject to the control input signal are introduced, respectively. Radial basis function (RBF) neural networks are used to approximate the packaged unknown non-linearities, and an adaptive neural control approach is developed via backstepping, which guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded in mean square. The main contributions of this note lie in that a control strategy is provided for a class of strict-feedback non-linear systems with unmodelled dynamics uncertainties and input saturation, and the proposed control scheme does not require any information of the bound of input saturation non-linearity. Simulation results are used to show the effectiveness of the proposed control scheme.

H∞Loop Shaping Control of Input Saturated Systems with Norm-Bounded Parametric Uncertainty

This paper proposes a gain-scheduling control design strategy for a class of linear systems with the presence of both input saturation constraints and norm-bounded parametric uncertainty. LMI conditions are derived in order to obtain a gain-scheduled controller that ensures the robust stability and performance of the closed loop system. The main steps to obtain such a controller are given. Differently from other gain-scheduled approaches in the literature, this one focuses on the problem ofH∞loop shaping control design with input saturation nonlinearity and norm-bounded uncertainty to reduce the effect of the disturbance input on the controlled outputs. Here, the design problem has been formulated in the four-blockH∞synthesis framework, in which it is possible to describe the parametric uncertainty and the input saturation nonlinearity as perturbations to normalized coprime factors of the shaped plant. As a result, the shaped plant is represented as a linear parameter-varying (LPV) system while the norm-bounded uncertainty and input saturation are incorporated. This procedure yields a linear parameter-varying structure for the controller that ensures the stability of the polytopic LPV shaped plant from the vertex property. Finally, the effectiveness of the method is illustrated through application to a physical system: a VTOL “vertical taking-off landing” helicopter.

Observer-Based Adaptive Iterative Learning Control for a Class of Nonlinear Time Delay Systems with Input Saturation

This paper presents an adaptive iterative learning control scheme for the output tracking of a class of nonlinear systems with unknown time-varying delays and input saturation nonlinearity. An observer is presented to estimate the states and linear matrix inequality (LMI) method is employed for observer design. The assumption of identical initial condition for ILC is relaxed by introducing boundary layer function. The possible singularity problem is avoided by introducing hyperbolic tangent function. The uncertainties with time-varying delays are compensated for by the combination of appropriate Lyapunov-Krasovskii functional and Young’s inequality. Both time-varying and time-invariant radial basis function neural networks are employed to deal with system uncertainties. On the basis of a property of hyperbolic tangent function, the system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapunov-like composite energy function in two cases, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

Robust Saturated Finite Time Output Feedback Attitude Stabilization for Rigid Spacecraft

This paper investigates the velocity-free feedback control problem associated with finite time attitude stabilization of a rigid spacecraft subject to external disturbance and input saturation. First of all, to address the lack of angular velocity measurement, a novel, fast, finite time convergent observer is proposed to recover the unknown angular velocity information in a finite time under external disturbance. Then, a finite time output feedback controller is proposed using the designed finite time observer, in which the control input saturation nonlinearity is explicitly considered in the proposed finite time stabilizer. Rigorous proofs show that the proposed observer can achieve finite time stability, the controller rigorously enforces actuator-magnitude constraints, and the attitude of the rigid spacecraft will converge to the equilibrium in a finite time. Numerical simulations illustrate the spacecraft performance obtained using the proposed controllers.

Adaptive neural tracking control for stochastic nonlinear strict-feedback systems with unknown input saturation

In this paper, the problem of adaptive neural tracking control is considered for a class of single-input/single-output (SISO) strict-feedback stochastic nonlinear systems with input saturation. To deal with the non-smooth input saturation nonlinearity, a smooth nonaffine function of the control input signal is used to approximate the input saturation function. Classical adaptive technique and backstepping are used for control synthesis. Based on the mean-value theorem, a novel adaptive neural control scheme is systematically derived without requiring the prior knowledge of bound of input saturation. It is shown that under the action of the proposed adaptive controller all the signals of the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. Two simulation examples are provided to demonstrate the effectiveness of the presented results.

Adaptive steering control for uncertain ship dynamics and stability analysis

Marine transportation offers a cost-effective and viable alternative to cargo aircraft, trains, and trucks in general. Traffic control in congested waterways has recently become a challenging task demanding higher accuracy in ship navigation and advanced methods for collision or grounding avoidance. Modern ship control systems call for new technologies to be applied to ship steering which involves station-keeping and course-changing maneuvers. Uncertain cargo ship dynamics require robust steering control design techniques potentially reducing the effects of external conditions and minimizing path deviations in case particularly strong lateral wind or wave forces are experienced. Among additional issues to be addressed are the possible conflicts between input limitations and controller performance. We present an adaptive steering control design for uncertain ship dynamics subject to input constraints while avoiding performance compromises under changing environmental conditions. An adaptive law is combined with a control design including a Linear Quadratic (LQ) controller and a Riccati based anti-windup compensator using Certainty Equivalence Principle for asymptotically stable plants with saturation limits imposed on the control input. If the desired linear performance can be recovered for nominal systems subject to input saturation by implementing an LQ control augmented with an anti-windup compensator, our analysis on its adaptive counterpart reveals that the perturbation terms due to plant parameter errors in the adaptive scheme do not cause any unbounded signals in the closed-loop, and the system remains stable. Continuous-time lumped Hurwitz systems with input saturation nonlinearities and unknown plant parameters can benefit from the resulting indirect adaptive control design. Several ship maneuvering scenarios are simulated to verify the effectiveness of our approach.

The trajectory tracking problem for an unmanned four-rotor system: flatness-based approach

In this article we present a flatness-based controller that solves the trajectory tracking problem for an unmanned four-rotor aircraft. A simple version of the dynamical system obtained from a suitable coordinate transformation of the Euler–Lagrange model of the system is used. This model, combined with a suitable input saturation nonlinearity, also avoids the inconvenient system singularities. The corresponding tracking error is shown to be globally asymptotically stable but only locally exponentially stable, while the controller maintains the four-rotor system motions inside a feasible region. Numerical simulations are presented to assesses the effectiveness of the proposed control strategy.

Parametric Lyapunov Equation Approach to Stabilization of Discrete-Time Systems With Input Delay and Saturation

This paper studies the problem of stabilization of discrete-time linear systems with input delay and saturation nonlinearity. By exploring some further intricate properties of the recently developed parametric Lyapunov equation-based low-gain feedback design approach, solutions are proposed to solve the problems by both state feedback and output feedback. This new approach is not only simpler than the existing method that is based on the eigenstructure assignment technique, but also provides explicit conditions on the low-gain parameter to guarantee the stability of the closed-loop system. Moreover, it is possible by adjusting the low-gain parameter online to achieve global results when the system is subject to both input saturation and time-delay. Also, the delay in the input is allowed to be time-varying in some cases.