Control Of Nonlinear Mechanical Systems Research Articles

Cooperative Passivity-Based Control of Nonlinear Mechanical Systems

In this work, we propose two cooperative passivity-based control methods for networks of mechanical systems. By cooperatively synchronizing the end-effector coordinates of the individual agents, we achieve cooperation between systems of different types. The underlying passivity property of our control approaches ensures that cooperation is stable and robust. Neither of the two approaches rely on the modeling information of neighbors, locally, which simplifies the interconnection of applicable systems and makes the approaches modular in their use. Our first approach is a generalized cooperative Interconnection-and-Damping Assignment passivity-based control (IDA-PBC) scheme for networks of fully actuated and underactuated systems. Our approach leverages the definition of end-effector coordinates in existing single-agent IDA-PBC solutions for underactuated systems to satisfy the matching conditions, independently of the cooperative control input. Accordingly, our approach integrates a large set of existing single-agent solutions and facilitates cooperative control between these and fully actuated systems. Our second approach proposes agent outputs composed of their end-effector coordinates and velocities to guarantee cooperative stability for networks of fully actuated systems in the presence of communication delays. We validate both approaches in simulation and experiments.

Adaptive Robust Control for Nonlinear Mechanical Systems With Inequality Constraints and Uncertainties

The inequality constraints, system nonlinearities, parameter uncertainties, and external disturbances are always unavoidable in practical mechanical systems. This article proposes an adaptive robust control (ARC) algorithm from the view of servo constraint following to tackle the control problem of mechanical systems subject to the above factors. For the inequality constraints, a creative diffeomorphism which could convert the two-sided bounded state variables to the unbounded ones is explored, which could render the transformed nonlinear system free from inequality constraints. For the system uncertainties, a leakage-type ARC algorithm is developed, which could render the system the practical stability. The permanent magnet linear motor (PMLM) system is utilized as a typical application to verify the proposed state transformation and ARC approach. Numerical simulations show that the displacement of the PMLM system could well track the desired trajectory without violating the given bound line.

Symplectic discrete-time energy-based control for nonlinear mechanical systems

We present a novel approach for discrete-time state feedback control implementation which reduces the deteriorating effects of sampling on stability and performance in digitally controlled nonlinear mechanical systems. We translate the argument of energy shaping to discrete time by using the symplectic implicit midpoint rule. The method is motivated by recent results for linear systems, where feedback imposes closed-loop behavior that exactly represents the symplectic discretization of a desired target system. For the nonlinear case, the sampled system and the target dynamics are approximated with second order accuracy using the implicit midpoint rule. The implicit nature of the resulting state feedback requires the numerical solution of an in general nonlinear system of algebraic equations in every sampling interval. For an implementation with pure position feedback, the velocities/momenta have to be approximated in the sampling instants, which gives a clear interpretation of our approach in terms of the Störmer–Verlet integration scheme on a staggered grid. Both the Hamiltonian and the Lagrangian perspective are adopted. We present discrete-time versions of impedance or energy shaping plus damping injection control as well as computed torque tracking control in the simulation examples to illustrate the performance and stability gain compared to the quasi-continuous implementation. We discuss computational aspects and show the structural advantages of the implicit midpoint rule compared to other integration schemes in the appendix.

Predictor-Based Control for Nonlinear Mechanical Systems with Measurement Delay.

In this paper, a controller for nonlinear systems with delay in the measurement using a predictor-based strategy is proposed. The nonlinear systems considered in this work are the class of mechanical systems of triangular form and which mathematical model can be obtained by means of the Euler-Lagrange formulation. A predictor for the mechanical nonlinear system is applied in order to get a delay-free system. Then, a tracking PD controller is designed using the predictor to compensate the delay effect. In order to demonstrate the effectiveness of the control scheme proposed, two examples are presented, one to show the performance of the predictor and the other one to show the use of such predictor in order to control the system. Then, a tracking controller for time-varying references is designed for mechanical systems with measurement delay, and the performance is shown during simulation. Additionally, it is considered the case when external disturbances are present in the system and to deal with them a method of estimation of the disturbance is proposed. The results are shown and discussed and finally the conclusions for this work are given and ideas for future work are proposed.

Quasi fixed-time fault-tolerant control for nonlinear mechanical systems with enhanced performance

A novel quasi fixed-time prescribed performance control approach is investigated for mechanical systems with consideration of unknown dynamics and actuator faults. First, a fixed-time stable performance function is developed to quantitatively characterize the transient and steady-state responses. Then, a quasi fixed-time adaptive controller with a pretty simple computational structure is devised based on the developed performance function. Different from the existing finite-time/fixed-time control schemes, the prominent advantages are twofold: (1) no fractional-order state or output information is required, wherein, the order of the states involved in the developed controller just equals to one. (2) the fixed convergence time can be prespecified by the uses instead of relying on several parameters. Finally, applications to a 2-link robotic manipulator are employed to validate the effectiveness of the proposed control approach.

An improved adaptive fuzzy backstepping control for nonlinear mechanical systems with mismatched uncertainties

ABSTRACTWhen the nonlinear mechanical system has mismatched uncertainties, it is difficult to design control algorithm to achieve high precision trajectory tracking control. Traditional backstepping control is an effective control method for uncertain, mismatched nonlinear systems, but there is inherent problem of “complexity due to the explosion of terms”. In this paper, based on the backstepping method, only one fuzzy system is used to approximate the unknown nonlinear functions, the unknown control gain and the differential of virtual control law of each subsystem. In order to reduce the influence of fuzzy approximation error and external interference, the above control scheme is further improved by designing special adjustable control parameters and first order low pass filter. The improved control scheme not only improves the control precision of the system obviously, but also solves the problem of “explosion of terms”, and greatly reduces the initial control input, and provides the conditions for the practical application. The simulation results show the effectiveness of the proposed methods.

Adjoint-Based Optimization Procedure for Active Vibration Control of Nonlinear Mechanical Systems

In this paper, a new computational algorithm for the numerical solution of the adjoint equations for the nonlinear optimal control problem is introduced. To this end, the main features of the optimal control theory are briefly reviewed and effectively employed to derive the adjoint equations for the active control of a mechanical system forced by external excitations. A general nonlinear formulation of the cost functional is assumed, and a feedforward (open-loop) control scheme is considered in the analytical structure of the control architecture. By doing so, the adjoint equations resulting from the optimal control theory enter into the formulation of a nonlinear differential-algebraic two-point boundary value problem, which mathematically describes the solution of the motion control problem under consideration. For the numerical solution of the problem at hand, an adjoint-based control optimization computational procedure is developed in this work to effectively and efficiently compute a nonlinear optimal control policy. A numerical example is provided in the paper to show the principal analytical aspects of the adjoint method. In particular, the feasibility and the effectiveness of the proposed adjoint-based numerical procedure are demonstrated for the reduction of the mechanical vibrations of a nonlinear two degrees-of-freedom dynamical system.

Control of the spatial motion of a multilink inverted pendulum using a torque applied to the first link

An approach to constructing a control for non-linear mechanical systems in which the number of degrees of freedom exceeds the dimension of the generalized control forces is developed. An n-link pendulum with two-degree-of-freedom joints, controlled by a torque applied to the first link, is considered as an example. Such a pendulum has 2n different equilibrium positions. A feedback control with an absolute value constraint, which transfers the pendulum from the neighbourhood of an arbitrary equilibrium position to this equilibrium position in a finite time, is constructed. For this purpose, the equations of motion of the pendulum are linearized in the neighbourhood of the equilibrium position under consideration, complete controllability of the linear model is established, and a control is constructed for it using the linear matrix inequality technique. The applicability of the control law obtained to the solution of the problem of controlling a non-linear multilink pendulum is verified. ©2014

Optimal Linear Quadratic Regulators for Control of Nonlinear Mechanical Systems with Redundant Degrees-of-Freedom

Optimal Linear Quadratic Regulators for Control of Nonlinear Mechanical Systems with Redundant Degrees-of-Freedom

Design and Control of Nonlinear Mechanical Systems for Minimum Time

This paper presents an integrated methodology for optimal design and control of nonlinear flexible mechanical systems, including minimum time problems. This formulation is implemented in an optimum design code and it is applied to the nonlinear behavior dynamic response. Damping and stiffness characteristics plus control driven forces are considered as decision variables. A conceptual separation between time variant and time invariant design parameters is presented, this way including the design space into the control space and considering the design variables as control variables not depending on time. By using time integrals through all the derivations, design and control problems are unified. In the optimization process we can use both types of variables simultaneously or by interdependent levels. For treating minimum time problems, a unit time interval is mapped onto the original time interval, then treating equally time variant and time invariant problems. The dynamic response and its sensitivity are discretized via space-time finite elements, and may be integrated either by at-once integration or step-by-step. Adjoint system approach is used to calculate the sensitivities.

A response spectral decomposition approach for the control of nonlinear mechanical systems

This study investigates a control methodology for dynamic systems based on representing all possible system responses under all possible values of the control variables. The underlying idea is to extend the system along the 'control dimension' and explicitly account for the dependence of the system state on control variables. A spectral discretisation along the 'control dimension' is employed. The optimal control values are chosen to obtain the desired parameterised system response. This approach has the advantage of allowing an inexpensive search for the optimal control values at each time increment without having to run the full nonlinear dynamic model. Numerical studies for the control of linear and nonlinear quarter-car models riding on various terrain profiles show promising results.

アクチュエータ系ダイナミクスを考慮したロボットの軌道追従フィードバック階層制御

In the case of control for nonlinear mechanical systems, which have complex actuator dynamics, the total system becomes high-order and nonlinear. Generally speaking, it is not easy to tune feedback gains for state feedback. In this paper, we propose a feedback control scheme for motion control of nonlinear high-order systems. We prove that the proposed scheme can improve the trajectory tracking performance of a robot manipulator system with actuator dynamics. Moreover, some simulation results demonstrate the effectiveness of the proposed control scheme.

Nonlinear control of mechanical systems with an unactuated cyclic variable

Numerous robotic tasks associated with underactuation have been studied in the literature. For a large number of these in the plane, the mechanical models have a cyclic variable, the cyclic variable is unactuated, and all shape variables are independently actuated. This paper formulates and solves two control problems for this class of models. If the generalized momentum conjugate to the cyclic variable is not conserved, conditions are found for the existence of a set of outputs that yields a system with a one-dimensional exponentially stable zero dynamics-i.e., an exponentially minimum-phase system-along with a dynamic extension that renders the system locally input-output decouplable. If the generalized momentum conjugate to the cyclic variable is conserved, a reduced system is constructed and conditions are found for the existence of a set of outputs that yields an empty zero dynamics, along with a dynamic extension that renders the system feedback linearizable. A common element in these two feedback problems is the construction of a scalar function of the configuration variables that has relative degree three with respect to one of the input components. The function arises by partially integrating the conjugate momentum. The results are illustrated on two balancing tasks and on a ballistic flip motion.

Open Problems in Nonlinear Control of Mechanical Systems

Open Problems in Nonlinear Control of Mechanical Systems

Analysis of Proportional—Derivative and Nonlinear Control of Mechanical Systems with Dynamic Backlash

In this paper, we focus on the analysis and control of a simple rigid-body mechanical system with clearance. Contrary to most of the existing works in the literature concerning control, we explicitly treat all the nonlinear non-smooth characteristics of this system considered as a rigid-body mechanical system with unilateral constraints and impacts (dynamic backlash). The model is therefore a hybrid dynamical system, mixing discrete events as well as continuous states. The regulation and tracking capabilities of the proportional—derivative (PD) scheme are investigated. In particular, a complete proof of the existence of a limit cycle for non-collocated PD control is provided, including viability constraints. It is concluded that tracking requires the development of specific control schemes. Consequently, we propose a hybrid control that may be used to track some desired trajectories in conjunction with a PD input. Throughout the paper, the particular features of unilaterally constrained mechanical systems are taken into account, such as the fundamental viability property of closed-loop solutions and controls. This work is a new approach to be considered for application in several areas including the control of kinematic chains with joint clearance and vibro-impact systems, as well as liquid slosh control. Numerical results are presented to illustrate the possible performance of the proposed control scheme and its robustness properties.

Analysis of Proportional-Derivative and Nonlinear Control of Mechanical Systems with Dynamic Backlash

Analysis of Proportional-Derivative and Nonlinear Control of Mechanical Systems with Dynamic Backlash

BAYES PARAMETER IDENTIFICATION WITH REFERENCE TO NONLINEAR OPTIMAL CONTROL

This paper deals with the task of parameter identification using the Bayes estimation method, which makes it possible to take into account the differing consequences of positive and negative estimation errors. The calculation procedures are based on the kernel estimators technique. The final result constitutes a complete algorithm usable for specifying the value of the Bayes estimator on the basis of an experimentally obtained random sample. An elaborated method is provided for numerical computations. In particular, an exemplary application to a random time-optimal control for nonlinear mechanical systems is described.

Passivity-based Visual Feedback Control of Nonlinear Mechanical Systems

This paper investigates the relative rigid body motion (positions and rotations) control problem with visual information. Firstly the model of the relative rigid body motion and the nonlinear observer are considered in order to derive the visual feedback system. Secondly stability and L2-gain performance analysis are discussed based on passivity. Finally we present simulation results to confirm the effectiveness of the proposed visual feedback control design.

Global tracking control of underactuated ships by Lyapunov's direct method

This paper studies the global tracking problem for an underactuated ship with only two propellers. Under sufficient conditions of persistent excitation, two constructive solutions are proposed by application of Lyapunov's direct method. Both solutions exploit the inherent cascade interconnected structure of the ship dynamics and generate explicit Lyapunov functions. A common feature of the second solution and the recent cascade approach of Lefeber (Tracking control of nonlinear mechanical systems, Ph.D. Thesis, University of Twente, 2000) is that global exponential tracking is achieved, but at the expense of transient performance. Extension to unmeasured thruster dynamics is also considered. Simulation results validate the proposed tracking methodology.

Control of Nonlinear Mechanical Systems

Control of Nonlinear Mechanical Systems