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Sliding mode output-feedback stabilization of uncertain nonlinear nonaffine systems
Abstract
The paper considers the variable-structure output-feedback stabilization of nonlinear uncertain nonaffine systems when the state vector is not completely available and the use of observers is required. The strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems. A full-order observer is designed and the control problem is solved by forcing a sliding regime for the observer. Depending on the type of uncertainties, conditions are found under which convergence to the unique ideal solution of both the system and the observer, either exponentially or with a bounded error, is proven.
... (1) A nonlinear observer-based output feedback which is applied to estimate the state of the nonlinear system in the presence of parameter uncertainties and unknown inputs (Bartolini and Punta, 2012). Such an observer does not require any transformation to be designed. ...
... Then, considering the uncertain nonlinear control system (1)-(2), the aim of this paper is to detect and isolate possible faults acting on the system. For fulfilling such a goal, a nonlinear observerbased output feedback (Bartolini and Punta, 2012) is applied to estimate the state of the system April 1, 2016 International Journal of Control IJC2014˙R2 ...
... Consider the following nonlinear Luenberger-like observer (Bartolini and Punta, 2012) ...
This paper considers the fault detection and isolation problem for nonlinear uncertain non-affine systems. The proposed approach is based on an output-feedback stabilization strategy, which exploits a Luenberger-like nonlinear observer. As a main result a residual-based fault detection and isolation approach is developed, which relies on the measurable output, some of its derivatives, which are provided exactly by a uniform high-order sliding-mode differentiator, and the observer’s state. The proposed methodology is able to detect some possible components and actuators faults, and to isolate, under some mild conditions, the actuator faults from those in components. Simulation results illustrate the feasibility of the proposed approach.
... The strategy to achieve the control objective is based on the following observer [11] ˙ ...
... Definition 1: [11] The mapping v * , solution to (10), is defined as the observer's equivalent control relevant to the output y. The solutions (in the Filippov sense) are considered in [0, +∞) for (4) and (9), corresponding to the observer's equivalent control, i.e. ...
... Suppose the fault-free case, i.e. d(t, x) = 0. Then, the following results, given in [11], are established. Consider the case when the uncertainties are bounded by a linear growth and vanishing at the origin, i.e. the case when γ 2 = 0 in (7), the following theorem can be stated. ...
This paper deals with the fault detection prob-lem for nonlinear uncertain non-affine systems. The variable-structure output-feedback stabilization based on observer is required. Depending on the type of uncertainties, conditions are given under which convergence to the unique solution of both the system and the observer, and fault detection are achieved. Simulation results illustrate the feasibility of the proposed approach.
... Therefore, some hypotheses have been proposed to guarantee the controllability of the system and transform non-affine functions. Under the assumption that ∂ f (x, u)/∂u is bounded, mean value theorem or Taylor series expansion is utilized to construct pseudo-affine model, and then traditional control schemes such as adaptive fuzzy control [33][34][35][36], adaptive neural control [37][38][39][40][41][42], sliding mode control [43,44], finite-time containment control [45,46] and PPC [20,32] can be designed based on transformational models. Moreover, indirect or direct adaptive controllers are constructed for a class of nonaffine systems with smooth unknown nonlinear functions [47][48][49][50]. ...
... In what follows, an adaptive neural controller (ANC) is compared with the novel prescribed performance controller (NPPC) designed in this paper. Mean value theorem or Taylor series expansion [33][34][35][36][37][38][39][40][41][42][43][44][45][46] can't be directly utilized to construct pseudo-affine model due to in-differentiable points of non-affine system in this paper. Thus, both controllers are designed based on Assumption 4 and model transformation method in Sect. ...
An error constraint control problem is considered for pure-feedback systems with non-affine functions being possibly in-differentiable. A new constraint variable is used to construct virtual control that guarantees the tracking error within the transient and steady-state performance envelopment. The new error transformation avoids non-differentiable problems and complex deductions caused by traditional error constraint approaches. A locally semi-bounded and continuous condition for non-affine functions is employed to ensure the controllability and transform the closed-loop system into a pseudo-affine form. An auxiliary system with bounded compensation term is proposed for nonlinear systems with input saturation. On the basis of backstepping technique, an adaptive neural controller is designed to handle unknown terms and circumvent repeated differentiations of virtual controls. The boundedness and convergence of the closed-loop system are proved by Lyapunov theory. Asymptotic tracking is achieved without violating control input constraint and error constraint. Two examples are performed to verify the theoretical findings.
... Non-affine systems are used in two major areas. In the first category, nonlinear systems are considered with uncertainties in states or control inputs (Lee et al., 2009;Bartolini & Punta, 2012;Huang & Wu, 2011;Huang et al., 2012;Gang et al., 2012). Different methods have been applied in this category, including (Lee et al., 2009) introduced direct adaptive backstepping and a recurrent wavelet neural network method to control a class of non-affine systems. ...
... They simulated a double pendulum as a non-affine system and the results expressed the good performance of the algorithm. (Bartolini & Punta, 2012) presented a variable structure method for controlling a non-affine system with uncertainties. The state vector was not completely available, so an observer was combined in the structure of the controller. ...
The present study uses a combination of a state-dependent Riccati equation (SDRE) controller and SDRE observer to control a class of nonlinear non-affine control systems. This type of nonlinear system features control nonlinearities that change the formulation of the SDRE and its solution techniques. Control law is obtained using the exact solution method for low-order and uncomplicated systems; numerical approaches are available for complex and high order systems. Both solution methods must use integral control formulation or solve the system equation using control law simultaneously to treat control nonlinearities. The closed loop feedback of the system is partially available, providing an observer for estimation of full-state feedback. The efficacy, advantages and compatibility of the controller with the observer guides the choices to the SDRE observer. The stability proof of the SDRE controller in the non-affine structure is presented and four case studies are simulated and discussed. Two were selected from previous studies to verify the proposed algorithm. The third, more complicated model, shows the ability of the controller to solve complex and high order systems. A brief discussion of the suboptimal cheap control problem and an example are also presented.
... The alternative does not resolve; however, the nonlinearities raise the tilting of the rotors and the combination of the states. An unconventional potential key is to realize a rolling-mode output-feedback linearization as offered on (Bartolini, 2012). However, that publication does not cover some essential parts such as strategies to specify a sufficient tilting manifold. ...
This paper presents a hierarchical flight control system for smart tilt-rotor drones. The proposed approach performs high-level mission goals by gradually confirming them into machine-level instructions. The learned data from numerous sensors is spread backside to the greater levels for sensitive decision making. Each vertical takeoff and landing drone is linked through regular wireless communication rules for an accessible multi-agent facility. The proposed flight control system has been effectively employed on several types of smart tilt-rotor drones and has been validated in some applications. Solutions from waypoint navigation, a probabilistic chase-evasion competition, and vision-based object chasing show the capability of the recommended method for intelligent drones.
... This topic is involved with the stabilization of SCSMS. And as for the stabilization issues, scholars have proposed different control schemes [21][22][23][24] to stabilize different systems, in which feedback control is a powerful and universal strategy. Mao et al 25 applied stochastic feedback control to stabilize a given nonlinear hybrid system. ...
... However, the sliding mode controllers mentioned above are all designed for affine systems and are difficult to directly be applied to nonaffine systems. According to the authors' best knowledge, all the sliding mode control designed for nonaffine systems [27][28][29][30][31][32] can only guarantee the infinitetime convergence of the states. The finite-time controllers have faster convergence rate than that of infinite-time controllers. ...
A class of unknown nonaffine pure-feedback nonlinear systems is investigated and a novel output feedback control scheme with low complexity is proposed, based on the sliding mode control theory. The scheme is capable of guaranteeing output tracking error with finite-time convergence and bounded closed loop signals. In this scheme, a novel transformation method is included, which can easily transform the state-feedback control of nonaffine systems into output feedback control of strict-feedback affine systems. Based on the transformed affine systems, a novel finite-time sliding mode control is designed, which is continuous and nonsingular. The control scheme proposed in this work is simple and easy to implement, in which the ''explosion of complexity'' caused by backstepping-like scheme is completely avoided. And the finite-time convergence is successfully achieved. In addition, the scheme is designed based on output feedback control. And the dynamics of the nonaffine nonlinear systems is unknown in the design process. Thus, the system knowledge needed is reduced.
... However, many stabilization and tracking methods such as H∞ control, linear matrix inequality (LMI), state-feedback and output-feedback control are sensitive to uncertainties and disturbances (Pang and Chen, 2008). Several control methods have been proposed to stabilize the uncertain time-delay systems using discontinuous controllers based on sliding mode control (SMC), hybrid control and time-varying strategies (Bartolini and Punta, 2012;Lin and Chen, 2011). ...
This paper presents sliding mode control to achieve finite-time robust-tracking and model-following for uncertain dynamical systems with time-varying state-delays and disturbances. The selection of sliding surface and the existence of the sliding mode are two significant issues, which have been addressed. The control law is designed to guarantee the existence of the sliding mode around the linear sliding surface in finite time. Simulation results are presented to show the efficiency of the proposed scheme as a promising way for controlling similar nonlinear systems.
This paper proposes a novel output constrained non-affine control structure by combing backstepping adaptive control with coordinate transformation technique for Generic Hypersonic Vehicles (GHVs) attitude tracking problem suffering from stochastic uncertainties. Firstly, by introducing several specific nonlinear functions, the output constrained control problem is transformed into a stabilization problem of several new variables. Then a novel adaptive control scheme that employee the boundary estimation technique and Nussbaum-type functions to counteract the effect of non-affine aerodynamic characteristics is integrated to address the attitude tracking problem. Meanwhile, two Neural Networks (NNs) are introduced to approximate and compensate the unknown nonlinearities. With the aid of a four-order Lyapunov function, the closed-loop attitude control system is proved to be stochastically stable. As a result, the output tracking errors can be guaranteed to stay in the predefined constraints during the whole control process and finally converge to an acceptable small value. Comparative simulation results are presented to demonstrate the effectiveness and advantages of the proposed control strategy.
This article studies an adaptive control problem for nonaffine nonlinear systems with linear parametric uncertainty. Inspired by the affine system case, the studied nonaffine nonlinear systems are specified such that the ideal control input is implicitly defined by a matching equation stemming from a Lyapunov‐based analysis. The core idea of control design is to solve the matching equation in a dynamic manner. Toward this end, a dynamic state‐feedback law, together with an adaptation law for the unknown parameters, is developed such that global asymptotic stabilization is theoretically guaranteed. The proposed method is validated on a Duffing‐Holmes oscillator with a nonaffine input and a multi‐input numerical example.
This work considers an input and output constraint control problem for pure‐feedback systems with nonaffine functions possibly being in‐differentiable. A locally semibounded and continuous condition for nonaffine functions is presented to guarantee the controllability, and the nonaffine system is transformed to an equivalent pseudoaffine one based on the mild condition. Combined with backstepping technique, a novel prescribed performance controller with new performance functions is constructed to circumvent high frequency chattering in control input. An auxiliary system with bounded compensation term is utilized in this paper, successfully avoiding the overrun of control input. The methodology achieves the desired transient and steady‐state performance and presents excellent robustness against the system uncertainty. Finally, two numerical simulations are performed to demonstrate the effectiveness of the proposed approach.
The position and attitude control of a six‐degree‐of‐freedom vehicle is dealt with in this paper. The actuation system is assumed to consist of a set of mono‐directional devices suitably located and directed. The model of the system is characterized by a large amount of uncertainties, disturbances and measurement errors. The state is assumed fully available and the navigation problems generating smooth references for the state trajectories are supposed already solved. A new attitude guidance algorithm has been developed to enhance robustness with respect to a class of nonsmooth measurement errors. The use of the simplex‐based sliding mode methodology reveals to be simultaneously suitable for the design of the actuation system (position and orientation of the actuators) and the implementation of the control strategy. The chattering phenomenon is strongly attenuated by the introduction of integrators in the input channel and, consequently to this choice, a suitable mechanism to avoid the unbounded growth of the individual thrust of the actuators is designed, while at the same time, achieving a direct control of power losses. The performances of the proposed control scheme are demonstrated by simulation by using mathematical models available in the literature.
This paper presents a disturbance rejection method for an affine nonlinear system. The control system is constructed based on the equivalent-input-disturbance (EID) approach. An affine nonlinear state observer is used to reconstruct the state of the affine nonlinear system and to estimate an EID. The well-known differential mean value theorem enables us to describe the closed-loop system in the state space as a linear-parameter-varying system. This makes it easy to derive sufficient conditions of global uniform ultimate boundedness in term of linear matrix inequalities (LMIs) by using a Lyapunov function and convexity theory. Controllers are designed based on the LMIs. A numerical example is used to illustrate the design of the control system. And a comparison between the EID-based control and the sliding-mode control demonstrates the effectiveness and advantages of the EID-based control method. © 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd.
This paper studies the fault accommodation problem for linear systems with time-varying delay, system uncertainties and external disturbances. A novel intermediate estimator-based fault tolerant control (FTC) scheme is proposed, where an intermediate estimator is constructed to estimate the states and the faults simultaneously, based on the estimation; an FTC compensation scheme is designed to eliminate the effects of faults, system uncertainties and time-varying delay. It is proved that the states of the closed-loop system are uniformly ultimately bounded. A simulation example shows the effectiveness of the proposed method.
In this paper the output tracking problem for a class of systems with unstable zero-dynamics is addressed. The state is assumed not measurable. The output of the dynamical system to be controlled has to track a signal, which is the sum of a known number of sinusoids with unknown frequencies, amplitudes and phases. The non-minimum phase nature of the considered systems prevents the direct tracking by standard sliding mode methods, which are known to generate unstable behaviors of the internal dynamics. The proposed method relies on the availability of a flat output and its time derivatives which are function of the unavailable state, therefore a nonlinear observer is needed. Due to the uncertainty in the frequencies and in the parameters defining the relationship between the output of the system and the flat states, adaptive indirect methods are applied.
The proposition of U-model concept (in terms of ‘providing concise and applicable solutions for complex problems’) and a corresponding basic U-control design algorithm was originated in the first author's PhD thesis. The term of U-model appeared (not rigorously defined) for the first time in the first author's other journal paper, which established a framework for using linear polynomial control system design approaches to design nonlinear polynomial control systems (in brief, linear polynomial approaches → nonlinear polynomial plants). This paper represents the next milestone work – using linear state-space approaches to design nonlinear polynomial control systems (in brief, linear state-space approaches → nonlinear polynomial plants). The overall aim of the study is to establish a framework, defined as the U-block model, which provides a generic prototype for using linear state-space-based approaches to design the control systems with smooth nonlinear plants/processes described by polynomial models. For analysing the feasibility and effectiveness, sliding mode control design approach is selected as an exemplary case study. Numerical simulation studies provide a user-friendly step-by-step procedure for the readers/users with interest in their ad hoc applications. In formality, this is the first paper to present the U-model-oriented control system design in a formal way and to study the associated properties and theorems. The previous publications, in the main, have been algorithm-based studies and simulation demonstrations. In some sense, this paper can be treated as a landmark for the U-model-based research from intuitive/heuristic stage to rigour/formal/comprehensive studies.
This study presents a generalized procedure for designing recurrent neural network enhanced control of time-varying-delayed nonlinear dynamic systems with non-affine triangle structure and pure-feedback prototype. Under the framework, recurrent neural network is developed to accommodate the on-line approximation, which the weights of the neural network are iteratively and adaptively updated through system state vector. Based on the neural network online approximation model, an indirect adaptive neural network controller is designed, by means of dynamic compensation, to deal with some of the challenging issues encountered in such complex nonlinear control systems. Taking consideration of the correctness, rigorousness, and generality of the new development, the Lyapunov stability theory is referred to prove that the closed-loop control system is uniformly ultimately bounded stable and the output of the system is converged to a small neighborhood of the desired trajectory. Two bench mark tests are simulated to demonstrate the effectiveness and efficiency of the procedure. In addition these could be the show cases for potential readers/users to digest and/or apply the procedure to their ad hoc problems.
For a class of systems who suffers from disturbances, an original output feedback sliding mode control method is presented based on a novel tracking error observer with disturbance estimator. The mathematical models of the systems are not required to be with high accuracy, and the disturbances can be vanishing or nonvanishing, while the bounds of disturbances are unknown. By constructing a differential sliding surface and employing reaching law approach, a sliding mode controller is obtained. On the basis of an extended disturbance estimator, a creative tracking error observer is produced. By using the observation of tracking error and the estimation of disturbance, the sliding mode controller is implementable. It is proved that the disturbance estimation error and tracking observation error are bounded, the sliding surface is reachable and the closed-loop system is robustly stable. The simulations on a servomotor positioning system and a five-degree-of-freedom active magnetic bearings system verify the effect of the proposed method.
This paper considers the design problem of a fuzzy output tracking controller via a finite number of actuators for a class of nonlinear distributed parameter systems with unknown matched perturbations described by semi-linear first-order hyperbolic partial differential equations (PDEs). Firstly, the T-S fuzzy hyperbolic PDE model is assumed to be used to accurately represent the semi-linear first-order hyperbolic PDE system. Subsequently, based on the T-S fuzzy PDE model, the classic variable structure control technique is adopted to develop a fuzzy output tracking controller such that the output of the semi-linear PDE system exponentially tracks a desired reference and the state of the system is uniformly bounded in the Filippov sense. By constructing a strict Lyapunov function, the sufficient condition on the existence of the fuzzy output tracking controller is derived and presented in terms of algebraic linear matrix inequalities (LMIs) in space with an equality constraint. It has been shown that when these algebraic LMIs with an equality constraint are feasible, fuzzy tracking control design can be obtained via the existing convex optimization techniques. Finally, the developed design methodology is successfully applied to the control of a semi-linear Lotka–Volterra PDE system and the achieved results illustrate its effectiveness.
The note considers the variable-structure control of nonlinear known nonaffine systems when the state vector is not completely available and the use of observers is required. The strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems. A new reduced-order observer is proposed and conditions are found under which it is proven the convergence to the unique ideal solution of both system and observer. The control problem is solved by forcing a sliding regime for the observer, while satisfying an exponential stability criterion for the observation error state equation.
The super-twisting second-order sliding-mode algorithm is modified in order to design a velocity observer for uncertain mechanical systems. The finite time convergence of the observer is proved. Thus, the observer can be designed independently of the controller. A discrete version of the observer is considered and the corresponding accuracy is estimated.
One of the major obstacles in the use of efficient tools for designing control systems is the high order of equations that describe their behaviour. In many cases they may be reduced to a lower order model by neglecting small time constants or rejecting fast components of the system overall motion. Fast motions may, for instance, be caused by small impedances in equations of electromechanical energy converters [155], time constants of electric motors in systems controlling slow processes [68], nonrigidity of flying vehicles construction [74] and many other reasons. The design of control systems resting upon the use of low-order models may be carried out both by analytical and by various computational techniques. (Application of computational techniques to the design of control systems may be seriously hindered not only by their high dimension, but also by the fact that the computational problems in such systems are generally ill-posed and require ad-hoc methods to be developed).
The main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises. The proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal. Such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for Lipschitz’s constant of the derivative.
A global exponential observer is built explicitly for some classes of nonlinear variable structure observed control systems. By using piecewise-continuous output-feedback control laws, or the (continuous) equivalent control for the observer, the unavailable state vector is asymptotically kept on the sliding manifold with exponential decaying error. Some explicit examples are presented.
The present research work aims at the development of a systematic method to arbitrarily assign the zero dynamics of a nonlinear system by constructing the requisite synthetic output maps. The minimum-phase synthetic output maps constructed can be made statically equivalent to the original output maps, and therefore, they could be directly used for nonminimum-phase compensation purposes. Specifically, the mathematical formulation of the problem is realized via a system of first-order nonlinear singular PDEs and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of singular PDEs can be proven to be locally analytic and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package. The minimum-phase synthetic output maps that induce the prescribed zero dynamics for the original nonlinear system can be computed on the basis of the solution of the aforementioned system of singular PDEs. Moreover, static equivalence to the original output map can be readily established by a simple algebraic construction.
The purpose of the present paper is to study stability of constrained nonlinear control systems. Usually, this is done by reducing the constrained control system to an unconstrained control system with respect to the constrained manifold. Different from these approaches, an alternative is proposed here which allows to establish stability without an explicit knowledge of the constrained manifold and the reduced unconstrained control system. The main result is a simple stability theorem. Despite its simplicity, the theorem can be applied to a broad class of stability problems, for example, the stability analysis of unstructured nonlinear differential–algebraic equations of higher index and to LaSalle's invariance principle. Furthermore, the theorem allows a computationally efficient stability analysis of the class of constrained polynomial control systems using semidefinite programming and the sum of squares decomposition.
States of nonlinear sliding mode control systems are considered, which approximately verify the sliding condition on the unbounded time horizon. If the equivalent control achieves uniform exponential stability, it is shown that such (real) states are ultimately bounded with arbitrarily small bound, provided the sliding error is sufficiently small. The proof requires a new converse Lyapunov theorem for constrained motions.