Trajectories Of Closed-loop System Research Articles

Control of a Driftless Bilinear Vector Field on $n$-Sphere

In this paper, we consider a multi-input driftless bilinear system evolving on the $n$ -dimensional sphere $S^{n}$ . We first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on $S^{n}$ . For the general framework, we establish the global controllability on $S^{n}$ and propose two linear control laws on $S^{n}$ that achieve asymptotic stabilization of an equilibrium point with an almost global domain-of-attraction. Further, the asymptotically stable closed-loop system trajectories are shown to be arcs on the geodesics of $S^{n}$ for a particular choice of the equilibrium point. Next, we propose two linear time-varying control laws to achieve trajectory tracking on $S^{n}$ and show the asymptotic stability of the tracking error. A distributed control is designed for the consensus of multiagent bilinear systems on $S^{n}$ with an undirected tree as the communication graph. The consensus manifold is shown to have an almost global domain of attraction.

Noncertainty-equivalence adaptive attitude control of satellite orbiting around an asteroid

The development of an adaptive control system for the attitude control of spacecraft orbiting around uniformly rotating asteroids is the subject of this paper. It is assumed that inertia parameters of spacecraft, and zonal and sectorial harmonic coefficients associated with the gravitational field of asteroid are not known. The objective is to control the Euler (yaw, pitch, roll) angles of the spacecraft along prescribed reference attitude trajectories. Based on the immersion and invariance methodology, a noncertainty-equivalence adaptive (NCEA) control law for the trajectory control of the Euler angles is derived. The design is completed in two steps of a backstepping design process, and filtered signals are used for synthesis. Unlike traditional certainty-equivalence adaptive systems, the estimated parameters of this NCEA law consist of certain nonlinear algebraic functions as well as partial estimates, obtained by integral adaptation law. By the Lyapunov analysis, it is shown that in the closed-loop system, the yaw, pitch, and roll angles trajectory tracking errors asymptotically converge to zero. For the purpose of illustration, attitude control of a spacecraft orbiting around 433 Eros asteroid is considered. Simulation results confirm that this NCEA law can accomplish precise nadir pointing attitude of the spacecraft on elliptical prograde and retrograde orbits despite uncertainties in the spacecraft dynamics.

A neuroadaptive architecture for model reference control of uncertain dynamical systems with performance guarantees

Neuroadaptive control systems have the capability to approximate unstructured system uncertainties on a given compact set using neural networks. Yet, a challenge in their design is to guarantee the closed-loop system trajectories stay in this set such that the universal function approximation property is satisfied and the overall system stability is achieved. To address this challenge, we present and analyze a new neuroadaptive architecture in this paper for model reference control of uncertain dynamical systems with strict performance guarantees. Specifically, the proposed architecture is predicated on a novel set-theoretic framework and has the capability to keep the closed-loop system trajectories within an a-priori, user-defined compact set without violating the universal function approximation property. A transport aircraft example is also given to complement the presented theoretical results.

Robust Output-Feedback Formation Control Design for Nonholonomic Mobile Robot (NMRs)

SummaryThis paper addresses the systematic approach to design formation control for kinematic model of unicycle-type nonholonomic mobile robots. These robots are difficult to stabilize and control due to their nonintegrable constraints. The difficulty of control increases when there is a requirement to control a cluster of nonholonomic mobile robots in specific formation. In this paper, the design of the control scheme is presented in a three-step process. First, a robust state-feedback point-to-point stabilization control is designed using sliding mode control. In the second step, the controller is modified so as to address the tracking problem for time-varying reference trajectories. The proposed control scheme is shown to provide the desired robustness properties in the presence of the parameter variation, in the region of interest. Finally, in third step, tracking problem of a single nonholonomic mobile robot extends to formation control for a group of mobile robots in the leader–follower scenario using integral terminal- based sliding mode control augmented with stabilizing control. Starting with the transformation of the mathematical model of robots, the proposed controller ensures that the robots maintain a constant distance between each other to avoid collision. The main problem with the proposed controller is that it requires all states specially velocities. Therefore, the state-feedback control scheme is then extended to output feedback by incorporating a highgain observer. With the help of Lyapunov analysis and appropriate simulations, it is shown that the proposed output-feedback control scheme achieves the required control objectives. Furthermore, the closed loop system trajectories reach to desired equilibrium point in finite time while maintaining the special pattern.

Robust limit cycle control in a class of nonlinear discrete-time systems

ABSTRACTThis paper studies generation of robust periodic solutions in a class of nonlinear discrete-time system. The sustained oscillations, with the desired frequency and amplitude, are achieved through the creation of the appropriate elliptic limit cycle in the phase plane of the uncertain closed-loop discrete-time system. In the first step, the nominal control law is designed to enforce the trajectories of the nominal closed-loop system to converge to the desired limit cycle. Next, considering uncertain terms, an additional robustifying term is designed. This term is added to the nominal controller to sustain the desirable stable oscillations in the presence of uncertain terms. The resulted robust controller brings the trajectories of the uncertain closed-loop discrete-time system to a boundary layer (with adjustable width) around the desired limit cycle. Moreover, the domain of attraction of the limit cycle and also the ultimate boundary layer around it are calculated via the Lyapunov analysis. Additionally, in order to verify the applicability of the proposed method, it is implemented on the discretised model of a spring–damper system. Computer simulations confirm the theoretical results in generating robust stable oscillations.

Nonlinear PID-Type Controller for Quadrotor Trajectory Tracking

A novel proportional-integral-derivative (PID)-type motion controller for a quadrotor is introduced in this paper. A rigorous analysis of the closed-loop system trajectories is provided, and gain tuning guidelines are discussed. Real-time experimental results consisting of the implementation of a PID-based scheme, a sliding-mode controller, and the new scheme are given. Gains are selected so that the three tested controllers present the same energy consumption. In order to assess the robustness of the controllers tested, experiments are carried out in the presence of disturbances in one of the actuators. Specifically, the disturbance consists in attenuating the force delivered. Better tracking accuracy is obtained with the introduced nonlinear PID-type algorithm.

An adaptive sliding mode approach for Markovian jump systems with uncertain mode-dependent time-varying delays and partly unknown transition probabilities

This paper deals with the problems of stochastic stability and sliding mode control for a class of continuous-time Markovian jump systems with mode-dependent time-varying delays and partly unknown transition probabilities. The design method is general enough to cover a wide spectrum of systems from those with completely known transition probability rates to those with completely unknown transition probability rates. Based on some mode-dependent Lyapunov-Krasovski functionals and making use of the free-connection weighting matrices, new delay-dependent conditions guaranteeing the existence of linear switching surfaces and the stochastic stability of sliding mode dynamics are derived in terms of linear matrix inequalities (LMIs). Then, a sliding mode controller is designed such that the resulted closed-loop system's trajectories converge to predefined sliding surfaces in a finite time and remain there for all subsequent times. This paper also proposes an adaptive sliding mode controller design method which applies to cases in which mode-dependent time-varying delays are unknown. All the conditions obtained in this paper are in terms of LMI feasibility problems. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Full‐order sliding mode control for finite‐time attitude tracking of rigid spacecraft

This study addresses an attitude tracking problem for a rigid spacecraft with bounded external disturbances. Attitude dynamics of the rigid spacecraft is modelled by a unit quaternion for global representation. Based on this representation, a novel continuous controller is proposed based on the full-order terminal sliding mode method. The finite-time stability of the closed-loop system is assured by Lyapunov and homogeneous finite-time stability theorems. Moreover, the singularity term in the terminal sliding surface is compensated by an identical item in the feedback control law, and this makes the closed-loop system trajectories converge to the equilibrium in finite-time, and hence a high precision is obtained. Finally, simulation results are presented to illustrate the effectiveness of the proposed controller.

Sliding mode control of hybrid switched systems via an event-triggered mechanism

This paper is devoted to solving the problem of sliding mode control for discrete-time switched systems via an event-triggered strategy. First, a new linear switching function combined with corresponding networked sliding mode dynamics is constructed using a time-delay system design method and event-triggering scheme. Then, on the basis of the Lyapunov functional technique and the average dwell time approach, sufficient conditions for the existence of the concerned networked sliding mode control are established in terms of linear matrix inequalities. Furthermore, an event-triggered sliding mode control law is developed to drive the resultant closed-loop system trajectories into a bounded switched region and maintain them therein for subsequent periods. Finally, a verification example is given to show the effectiveness of the proposed new design techniques.

Projective synchronization for two nonidentical time-delayed fractional-order T–S fuzzy neural networks based on mixed $${H_\infty }$$ H ∞ /passive adaptive sliding mode control

This paper deals with the problem of mixed $${H_\infty }$$ /passive projective synchronization for two different fractional-order (FO) T–S fuzzy neural networks with uncertain parameters and time delays. Firstly, a fractional integral sliding surface which is suitable for the considered FO error system is proposed. Second, in terms of the established sliding surface, combining a novel reaching law, a new adaptive sliding mode control law is introduced, which can force the closed-loop dynamic error system trajectories to reach the sliding surface. Then, by giving a continuous frequency distributed model of the FO dynamic networks and the application of FO system stability theory, the projective synchronization conditions are addressed in terms of linear matrix inequality techniques. Based on the conditions, a desired controller which can guarantee the robust stability of the closed-loop system and also ensure a mixed $${H_\infty }$$ /passive performance level is designed. Finally, synchronization of two nonidentical time-delayed FO T–S fuzzy neural networks with uncertain parameters as a simulation example is given to illustrate the effectiveness of the proposed method.

Sliding Mode Control of Discrete-Time Switched Systems with Repeated Scalar Nonlinearities

This note studies the design of sliding mode control (SMC) for discrete-time hybrid stochastic switched systems with repeated scalar nonlinearities. The weighed $\mathcal{H}_{\infty}$ gain performance is considered for the system dynamics to optimize its transient state performance. First, sufficient conditions are given to guarantee the corresponding system is exponentially stable while achieving a desired weighed $\mathcal{H}_{\infty}$ performance. A new switching surface function is constructed by the average dwell time technique and the positive diagonally dominant Lyapunov functional method to further reduce the conservativeness induced by the repeated scalar nonlinearity. Then, the corresponding sliding mode dynamics are obtained and the solvability condition for the desired switching surface function is derived. Furthermore, the synthesis of the proposed SMC law is proposed to force the resulting closed-loop system trajectories onto the pre-specified sliding mode region with a desired level of accuracy. Finally, the feasibility and the effectiveness of the presented new design techniques are illustrated by examples and simulations.

Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties

ABSTRACTIn this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.

Mixed $$H_\infty $$ H ∞ /Passive Projective Synchronization for Nonidentical Uncertain Fractional-Order Neural Networks Based on Adaptive Sliding Mode Control

This paper deals with the problem of mixed $$H_\infty $$ /passive projective synchronization for two different fractional-order (FO) neural networks with uncertain parameters. Firstly, a fractional integral sliding surface which is suitable for the considered FO error system is proposed. Secondly, in terms of the established sliding surface, combining a novel reaching law, a new adaptive sliding mode control law is introduced, which can force the closed-loop dynamic error system trajectories to reach the sliding surface. Then, a continuous frequency distributed model of the FO dynamic networks is given, via the application of FO system stability theory and robust control, the projective synchronization conditions are addressed in terms of linear matrix inequality techniques. Based on the conditions, a desired controller which can guarantee the robust stability of the closed-loop system and also ensure a mixed $$H_\infty $$ /passive performance level is designed. Finally, synchronization of two nonidentical FO neural networks with uncertain parameters as a simulation example is given to illustrate the effectiveness and advantages of the proposed method.

Observer-Based Fuzzy Adaptive Output-Feedback Control of Stochastic Nonlinear Multiple Time-Delay Systems.

This paper is concerned with the observer-based fuzzy output-feedback control for stochastic nonlinear multiple time-delay systems. On the basis of the consistent form of virtual input signals and increasing characteristics of the system upper bound functions, a variable splitting technique is employed to surmount the difficulty occurred in the nonlower-triangular form. In the controller design procedure, a state observer is first designed, and then an adaptive fuzzy output-feedback control method is presented by combining backstepping design together with fuzzy systems' universal approximation capability. The proposed adaptive controller guarantees the semi-global boundedness of closed-loop system trajectories in terms of fourth-moment. Two simulation examples are displayed to demonstrate the feasibility of the suggested controller.

Robust Generation of Limit Cycles in Nonlinear Systems: Application on Two Mechanical Systems

This paper studies inducing robust stable oscillations in nonlinear systems of any order. This goal is achieved through creating stable limit cycles in the closed-loop system. For this purpose, the Lyapunov stability theorem which is suitable for stability analysis of the limit cycles is used. In this approach, the Lyapunov function candidate should have zero value for all the points of the limit cycle and be positive in the other points in the vicinity of it. The proposed robust controller consists of a nominal control law with an additional term that guarantees the robust performance. It is proved that the designed controller results in creating the desirable stable limit cycle in the phase trajectories of the uncertain closed-loop system and leads to induce stable oscillations in the system's output. Additionally, in order to show the applicability of the proposed method, it is applied on two practical systems: a time-periodic microelectromechanical system (MEMS) with parametric errors and a single-link flexible joint robot in the presence of external disturbances. Computer simulations show the effective robust performance of the proposed controllers in generating the robust output oscillations.

Synchronization of Two Fractional-Order Chaotic Systems via Nonsingular Terminal Fuzzy Sliding Mode Control

The synchronization of two fractional-order complex chaotic systems is discussed in this paper. The parameter uncertainty and external disturbance are included in the system model, and the synchronization of the considered chaotic systems is implemented based on the finite-time concept. First, a novel fractional-order nonsingular terminal sliding surface which is suitable for the considered fractional-order systems is proposed. It is proven that once the state trajectories of the system reach the proposed sliding surface they will converge to the origin within a given finite time. Second, in terms of the established nonsingular terminal sliding surface, combining the fuzzy control and the sliding mode control schemes, a novel robust single fuzzy sliding mode control law is introduced, which can force the closed-loop dynamic error system trajectories to reach the sliding surface over a finite time. Finally, using the fractional Lyapunov stability theorem, the stability of the proposed method is proven. The proposed method is implemented for synchronization of two fractional-order Genesio-Tesi chaotic systems with uncertain parameters and external disturbances to verify the effectiveness of the proposed fractional-order nonsingular terminal fuzzy sliding mode controller.

A robust approach for trajectory tracking control of a quadrotor with experimental validation

In this paper, a new control scheme for motion control of quadrotors is presented. The proposed model based controller makes use of simple estimations of vehicle parameters. Besides, the controller is developed under properties of the vehicle model. A rigorous analysis is developed to prove that the closed-loop system trajectories are uniformly ultimately bounded. The validity of the proposed control scheme is tested by numerical simulations and experimental results. This validation shows that the proposed controller guarantees that the quadrotor performs tracking of a desired pose (position and orientation) trajectory with good accuracy. The new scheme has been compared with respect to other robust controllers. The results indicate better tracking accuracy for the new scheme.

Adaptive sliding mode controller based on super-twist observer for tethered satellite system

ABSTRACTIn this work, the sliding mode control based on the super-twist observer is presented. The parameters of the controller as well as the observer are admitted to be time-varying and depending on available current measurements. In view of that, the considered controller is referred to as an adaptive one. It is shown that the deviations of the generated state estimates from real state values together with a distance of the closed-loop system trajectories to a desired sliding surface reach a μ-zone around the origin in finite time. The application of the suggested controller is illustrated for the orientation of a tethered satellite system in a required position.

Nonlinear adaptive trajectory control of multi-input multi-output submarines with input constraints

The paper presents a new nonlinear adaptive control system for the dive-plane control of multi-input multi-output submarine models with input constraints, in the presence of external disturbances. It is assumed that all the system parameters, except the signs of the principal minors of the input matrix, are unknown. The objective is to design an adaptive control law for the tracking of reference depth and pitch angle trajectories, despite constraints on the rotation angles of the bow and stern hydroplanes. For the derivation of a singularity-free adaptation law, the decomposition of the input matrix as a product of a positive-definite symmetric matrix, a diagonal matrix, and an upper triangular matrix (termed SDU decomposition) is obtained. Then an adaptive control system is designed which accomplishes uniform ultimate boundedness of the closed-loop system trajectories. The control system includes an auxiliary dynamic system to counter the effect of control input saturation. Simulation results show that the designed adaptive law accomplishes precise depth and pitch angle trajectory control, despite symmetric and non-symmetric input constraints, parametric uncertainties and external disturbance inputs. These results also show that the designed saturating adaptive controller can accomplish certain dive-plane maneuvers using either of the hydroplanes; but the bow hydroplane can execute larger maneuvers than the stern hydroplane. The robust performance of the controller, despite non-symmetric input constraints, is useful in situations in which partial failure of actuators causes rotation of the hydroplanes in unequal ranges.

State and actuator fault estimation observer design integrated in a riderless bicycle stabilization system

This paper deals with an observer design for Linear Parameter Varying (LPV) systems with high-order time-varying parameter dependency. The proposed design, considered as the main contribution of this paper, corresponds to an observer for the estimation of the actuator fault and the system state, considering measurement noise at the system outputs. The observer gains are computed by considering the extension of linear systems theory to polynomial LPV systems, in such a way that the observer reaches the characteristics of LPV systems. As a result, the actuator fault estimation is ready to be used in a Fault Tolerant Control scheme, where the estimated state with reduced noise should be used to generate the control law. The effectiveness of the proposed methodology has been tested using a riderless bicycle model with dependency on the translational velocity v, where the control objective corresponds to the system stabilization towards the upright position despite the variation of v along the closed-loop system trajectories.