Uncertain Underactuated Systems Research Articles

Prescribed performance control for uncertain underactuated systems via constraint‐following with application to a PVTOL aircraft

SummaryGuaranteeing prescribed transient and steady state performance (TSSP) for mechanical systems is of great significance to their motion control, motivating the prescribed performance control (PPC). This study investigates the PPC for uncertain underactuated systems. We consider time‐varying (possibly fast), bounded uncertainty including large mismatched portions, however the bound is unknown. The control design is implemented via constraint‐following, that is, rendering the system to follow a series of servo constraints. We hence impose the prescribed TSSP on the constraint‐following error. An approximation‐free adaptive robust PPC based on a state transformation technique is designed, achieving approximate constraint‐following and guaranteeing the prescribed TSSP to be satisfied. It employs an adaptive law to online estimate the unknown bounds for uncertainty compensation. It does not assume sufficiently small mismatched uncertainties, hence admitting large mismatched uncertainties. The stability of the controlled underactuated systems is sustained by rigorous proofs. By the proposed approach, the PPC of a planar vertical take‐off and landing aircraft is explored, which also illustrates the effectiveness of the approach.

Adaptive Neural Network Output Feedback Control of Uncertain Underactuated Systems With Actuated and Unactuated State Constraints

Underactuated systems are widely applied in industry, construction, manufacturing, etc., and the complex working environment puts forward higher demands for safety and transient performance. Hence, it is necessary to consider how to simultaneously ensure actuated and unactuated motion constraints by fewer control inputs, especially when systems suffer from model uncertainties, unavailable velocities, etc. Unfortunately, it is still a significant challenge to overcome in real applications and theoretical analysis. Additionally, most existing studies merely consider the specific control objects and few general methods are applicable to a class of underactuated systems. To this end, we design a new adaptive output-feedback controller for a class of uncertain underactuated systems. Compared with existing methods only handling actuated constraints, an important merit of this article is that by introducing the elaborately designed coupling term composed of actuated and unactuated constraints together, all state variables are kept within the preset time-variant ranges and converge to their desired values. Furthermore, a new Lyapunov function candidate is utilized to provide a theoretical guarantee. As far as we know, without the need of exact model knowledge and velocity feedback, this article provides the first solution to achieve accurate motion control and state constraints for both actuated and unactuated variables, which is meaningful both theoretically and practically. Meanwhile, the asymptotic stability of the equilibrium point for the closed-loop system is proven by utilizing Lyapunov techniques and Barbalat’s lemma. For verification, the presented controller is applied to underactuated overhead and rotary cranes, respectively, together with detailed theoretical analysis and experimental validations.

Sliding Mode Control with Gaussian Process Regression for Underactuated Mechanical Systems

This work introduces a new control scheme for uncertain underactuated mechanical systems. The proposed approach is mainly based on sliding mode control, but a Gaussian process regressor is also embedded in the control law for uncertainty estimation and compensation. The convergence properties of the closed-loop signals are analytically proved by means of the Lyapunov stability theory. Numerical simulations with an inverted pendulum on a cart are presented to confirm the improved performance of the proposed control scheme.

Neuro-adaptive fixed-time non-singular fast terminal sliding mode control design for a class of under-actuated nonlinear systems

This paper presents a fixed-time neuro-adaptive control design for a class of uncertain under-actuated nonlinear systems (UNS) using a non-singular fast terminal sliding mode control (TSMC) with a radial basis function (RBF)-based estimator to achieve the convergence and robustness against the uncertainties. The mathematical model of the considered class is reduced into an equivalent regular form. A fast TSMC is designed for the transformed form to improve the control performance and annihilate the associated singularity problem of the conventional TSMC. Lyapunov stability theory ensures the steering of the sliding manifold and system states in fixed time. RBF neural networks are adaptively estimate the nonlinear drift functions. The theoretical design, analysis, and simulations of cart-pendulum and quadcopter demonstrate the feasibility and benefits of the regular form transformation and the designed control design. Comparing the proposed synthesis with the standard literature presents the attractive nature of proposed method for such a class.

Nash game‐based adaptive robust control design optimization for the underactuated mechanical system with fuzzy evidence theory

AbstractThis article proposes a novel Nash game‐theoretical optimal adaptive robust control design approach to address the constraint‐following control problem for the uncertain underactuated mechanical systems with fuzzy evidence theory. First, the uncertainty is considered bounded and the bound is unknown but lies in a specified fuzzy evidence number. Second, a deterministic adaptive robust control scheme is proposed based on the servo constraint following control method, which renders the uncertain underactuated mechanical system to follow the specified constraints accurately with deterministic performance (guarantee uniform boundedness and uniform ultimate boundedness). It is shown that the designed self‐adjusting leakage‐type adaptive law can compensate for the uncertainty and avoid overcompensation. Third, based on the performance analysis and the fuzzy evidence description of uncertainty, the Nash game theory is introduced into the multi‐parameter optimization design for the two tunable control gains selected as two players. The cost functions for two players are relevant to the system constraint‐following performance and control cost. Then we can obtain the optimal control gains by seeking the Nash equilibrium which is always proved to exist. Ultimately, the simulation results on the two‐wheeled self‐balancing robot demonstrate the availability of the proposed control scheme and the optimal design approach for the underactuated mechanical systems with uncertainties.

Robust Backstepping Sliding Mode Control Design for a Class of Underactuated Electro–Mechanical Nonlinear Systems

This paper presents a backstepping sliding mode control (BSMC) scheme for uncertain underactuated nonlinear systems. Since, underactuated electro-mechanical nonlinear systems (UEMNS) operate under the fewer number of inputs than the degree of freedom, therefore, the control of such systems remains a complex task. In this work, a class of UEMNS is considered which can be transformed into the so-called regular form. In the design process, these UEMNS are first transformed into regular form. In this form, the systems are properly subdivided into series cascaded blocks in which the one of the blocks is indirectly driven by the applied control input whereas the second block is directly controlled by the control input. This structure is apparently suitable for backstepping design. Hence, all the controller steps are designed via the proposed backstepping sliding mode technique. The step by step stability is proved rigorously by considering the Lyapunov approach. In term of benefits, this designed control law provides low-frequency vibration as compared to conventional SMC along with robust performance in the presence of matched uncertainties. This claim is verified via the numerical simulation results of the cart-pendulum system. For further confirmation, these results are also compared with the standard literature results to demonstrate the benefits and effectiveness of the aforesaid control scheme for such class.

Robust Tracking in Underactuated Systems Using Flatness-Based ADRC With Cascade Observers

Abstract In this work, the problem of trajectory tracking in uncertain underactuated systems is considered. To solve it, a combination of differential flatness and active disturbance rejection control (ADRC) is proposed. The controller design is synthesized in the absence of detailed knowledge of the system model and focuses on dealing with over-amplification of measurement noise, typically seen in conventional single high-gain observer-centered control approaches. The introduced solution is based on fully utilizing the information already available about the governed system, without the necessity for additional measurement devices. To be easily implementable, it is expressed in an industry familiar error-based form with a straightforward tuning method. Through experimental verification, the proposed approach is shown to enhance the disturbance-rejection capabilities of the standard ADRC structure and reduce its sensitivity to measurement noise, thus increasing its practical appeal.

Towards structure-independent stabilization for uncertain underactuated Euler–Lagrange systems

Available control methods for underactuated Euler–Lagrange (EL) systems rely on structure-specific constraints that may be appropriate for some systems, but restrictive for others. A generalized (structure-independent) control framework is to a large extent missing, especially in the presence of uncertainty. This paper introduces an adaptive-robust control framework for a quite general class of uncertain underactuated EL systems. Compared to existing literature, the important attributes of the proposed solution are: (i) avoiding structure-specific restrictions, namely, symmetry condition property of the mass matrix, and a priori bounds on non-actuated states or state derivatives; (ii) considering Coriolis, centripetal, friction and gravity terms to be unknown, while only requiring the knowledge of maximum perturbation around a nominal value of the mass matrix; (iii) handling state-dependent uncertainties irrespective of their linear or nonlinear in parameters structure. These features significantly widen the range of underactuated EL systems the proposed solution can handle in comparison to the available methods. Stability is studied analytically and the performance is verified in simulation using offshore boom crane dynamics.

Optimum fuzzy combination of robust decoupled sliding mode and adaptive feedback linearization controllers for uncertain under-actuated nonlinear systems

In this paper, an optimum fuzzy combination of decoupled sliding mode control (DSMC) and adaptive feedback linearization (FBL) is presented for under-actuated nonlinear systems with uncertainties. At first, DSMC and FBL are separately designed and the feedback linearization parameters are adapted by using the gradient descent approach; then, a fuzzy weighted summation of the two controllers is introduced. The multi-objective non-dominated sorting genetic algorithm II (NSGAII) is employed to properly choose the control gains. The objective functions for the optimization process are defined as the integrals of time multiplied by absolute errors, which have been minimized, simultaneously. In addition, the ball and beam system, as a well-known and popular benchmark, is investigated to test and challenge the proposed strategy. Simulation results illustrate that this approach would provide better performance in terms of settling time and control input compared with other methods reported in the literature.

Terminal adaptive control for a class of uncertain underactuated systems

This paper presents a backstepping-based terminal adaptive control solution for a class of uncertain underactuated systems with two degrees of freedom. The control scheme developed in this work is based on the finite time stability approach for the terminal convergence of system outputs. Due to the physical restrictions associated with underactuated systems, classical backstepping approach is not applicable to this class of nonlinear systems and hence an approximate version is developed by designing a suitable error surface. Control term is developed in two steps: first, non-singular pseudo-control terms are developed for the dimensions of the configuration space and thereafter a feasible control term is designed to ensure the finite time convergence of error surface. The error surface is designed such that its finite time convergence in turn implies the terminal convergence of configuration variables. To ensure a well-defined control term over the entire state space, control components are embedded with corrective terms. Corrective terms are designed so as to avoid the singularity of the control term and to preserve the desired controller response up to certain extent. A fuzzy logic system is used for the approximation of unknown systems dynamics. Simulation results demonstrate the effectiveness of the control approach.

An adaptive fuzzy sliding mode controller for uncertain underactuated mechanical systems

Underactuated mechanical systems are frequently encountered in several industrial and real-world applications such as robotic manipulators with elastic components, aerospace vehicles, marine vessels, and overhead container cranes. The design of accurate controllers for this kind of mechanical system can become very challenging, especially if a high level of uncertainty is involved. In this paper, an adaptive fuzzy inference system is combined with a sliding mode controller in order to enhance the control performance of uncertain underactuated mechanical systems. The proposed scheme can deal with a large class of multiple-input–multiple-output underactuated systems subject to parameter uncertainties, unmodeled dynamics, and external disturbances. The convergence properties of the resulting intelligent controller are proved by means of a Lyapunov-like stability analysis. Experimental results obtained with an overhead container crane demonstrate not only the feasibility of the proposed scheme, but also its improved efficacy for both stabilization and trajectory tracking problems.

Uniform ultimate boundedness for underactuated mechanical systems as mismatched uncertainty disappeared

We propose to design control for uncertain underactuated mechanical systems. The underactuated mechanical system is to follow prescribed holonomic or nonholonomic constraints. The uncertainty in the system does not in general fall within the range space of the input matrix, which is a major obstacle for control design. To resolve this difficulty, we decompose the uncertainty into matched uncertainty and mismatched uncertainty in a unique manner using the geometric structural characteristics of the system. A control scheme is designed to guarantee uniform boundedness and uniform ultimate boundedness of a constraint-following performance measure. The control is based solely on the matched uncertainty. The mismatched uncertainty turns out to be disappeared, as far as the performance analysis is concerned, since it is orthogonal to the geometric space of interest. For demonstrations, a vehicle/inverted pendulum platform is selected. We charge the system to follow either holonomic or nonholonomic constraint. The simulation shows the system performance, in following the prescribed constraint, is superior.

Adaptive backstepping control for a class of MIMO uncertain underactuated systems with input constraints

This paper presents a backstepping methodology-based semi generic adaptive controller scheme for a class of multi-input multi-output (MIMO) uncertain underactuated systems in presence of actuator constraints. To develop a feasible controller scheme for multi-input multi-output underactuated systems, (n – p + 1) dimensions of the n dimensional configuration space are stabilised by using one dimension of the input space. This control term is developed by applying hierarchical methodology whereas as remaining p – 1 input dimensions are assigned as dedicated control terms to solve the control problem of remaining dimensions of the configuration space. Backstepping technique is used to develop the classical control terms whereas wavelet networks are used to approximate the uncertain dynamics as well as the nonlinear effects of actuator saturation. The proposed scheme, thus, relaxes the constraint of completely and accurately known system dynamics as well as the requirement of measuring the clipped portion of the control effort. A robust control term is used to attenuate the approximation error to a prescribed level. Uniform ultimate boundedness (UUB) stability of the closed loop system is verified in the Lyapunov sense. Simulation results illustrate the effectiveness of theoretical development.

Output Feedback Self-tuning Wavenet Control for Underactuated Euler-Lagrange Systems

There exist some physical systems whose underactuation and limited sensing represent a challenge in engineering. The former requires a model-based nesting or virtual control schemes to control the underactuated degrees of freedom through the actuated ones, while the latter imposes stringent requirements to design the controller using only partial access to the state, called output feedback. In this paper, we address the output feedback control design of class of uncertain underactuated nonlinear systems that is not model-based neither require state estimation. Output feedback and underactuation are circumvented by synthesizing an efficient RFB-based superwavelets neural network that approximates inverse dynamics of the projected actuated subsystem. Using multiresolution scaling with only position, feedback gains are adapted online based on a self-tuning scheme. An illustrative simulation study is presented for the underactuated inverted pendulum-on-a-cart using only angular position.

Adaptive backstepping control for a class of MIMO uncertain underactuated systems with input constraints

Adaptive backstepping control for a class of MIMO uncertain underactuated systems with input constraints

Gain scheduled dynamic surface control for a class of underactuated mechanical systems using neural network disturbance observer

A gain scheduled dynamic surface control (GSDSC) based on neural network disturbance observer (NNDOB) is developed for a class of uncertain underactuated mechanical systems with multiple equilibria. Equilibrium manifold linearization (EML) transforms the nonlinear system into an equivalent model using scheduling variables. The NNDOB with a filtering adaptive law is constructed to approximate arbitrary disturbances. The dynamic surface controller is designed based on the NNDOB and EML model, which efficiently solves the mismatches and overcomes the problem of “explosion of complexity” resulting from repeated differentiation of virtual control variables. Moreover, the robust stability of the closed-loop system is proved by Lyapunov stability theorem. Finally, we apply the proposed control scheme to a power line inspection robotic system. The simulation results illustrate the approximation capability of NNDOB and the effectiveness of the presented control scheme.

Adaptive Control Solution for a Class of MIMO Uncertain Underactuated Systems with Saturating Inputs

This paper addresses the issue of controller design for a class of multi-input multi-output (MIMO) uncertain underactuated systems with saturating inputs. A systematic controller framework, composed of a hierarchically generated control term, meant to ensure the stabilization of a particular portion of system dynamics and some dedicated control terms designed to solve the tracking problem of the remaining system dynamics is presented. Wavelet neural networks are used as adaptive tuners to approximate the system uncertainties also to reshape the control terms so as to deal with the saturation nonlinearity in an antiwindup paradigm. Gradient based tuning laws are developed for the online tuning of adjustable parameters of the wavelet network. A Lyapunov based stability analysis is carried out to ensure the uniformly ultimately bounded (UUB) stability of the closed loop system. Finally, a simulation is carried out which supports the theoretical development.

Robust stabilization of underactuated nonlinear systems: A fast terminal sliding mode approach

This paper presents a fast terminal sliding mode based control design strategy for a class of uncertain underactuated nonlinear systems. Strategically, this development encompasses those electro-mechanical underactuated systems which can be transformed into the so-called regular form. The novelty of the proposed technique lies in the hierarchical development of a fast terminal sliding attractor design for the considered class. Having established sliding mode along the designed manifold, the close loop dynamics become finite time stable which, consequently, result in high precision. In addition, the adverse effects of the chattering phenomenon are reduced via strong reachability condition and the robustness of the system against uncertainties is confirmed theoretically. A simulation as well as experimental study of an inverted pendulum is presented to demonstrate the applicability of the proposed technique.

Structurally optimized wavelet network based adaptive control for a class of uncertain underactuated systems with actuator saturation

Structurally optimized wavelet network based adaptive control for a class of uncertain underactuated systems with actuator saturation

Backstepping-based adaptive control for underactuated systems

A backstepping-based adaptive control scheme is presented for uncertain underactuated systems. On the basis of backstepping approach, virtual control components are first constructed and final control term is then derived by defining an integrated error surface. Proposed control scheme utilises wavelet neural network for the approximation of system uncertainties and thus relaxes the constraint of fully known system dynamics. Update laws for wavelet network parameters are derived in Lyapunov sense. Proposed control scheme guarantees the uniform ultimate boundedness of the closed loop adaptive system. Simulation included illustrates the effectiveness of the proposed approach.