P. Sekhavat

Calculation of Lyapunov Exponents using Nonstandard Finite Difference Discretization Scheme: A Case Study

Stability analysis of nonsmooth systems, using Lyapunov direct method, is a difficult task. An alternative way of studying the stability in such systems is to employ the concept of Lyapunov exponents. However, two difficulties in the calculation of Lyapunov exponents, i.e. numerical artifacts and low computational efficiency, often prohibit the application of this otherwise powerful method. In this paper, a case study of the stability analysis of a switching contact task control system using the concept of Lyapunov exponents is presented. The goal is to carry out a numerical investigation of the method of nonstandard finite difference discretization, for constructing discrete models of differential equations that describe the Lyapunov exponents for such a nonsmooth system. It is shown that, as compared with the standard fourth-order Runge-Kutta method, the nonstandard finite difference discretization scheme provides numerically stable results with a larger critical integration step-size and less computation time. Therefore, from numerical stability and computational efficiency viewpoints, the nonstandard finite difference discretization method is meritorious and should be given consideration when stability analysis of systems using Lyapunov exponents is of concern.

On design of continuous Lyapunov's feedback control

A common approach to Lyapunovs stability control is to design a controller such that a Lyapunov function can be derived for the control system to ensure stability. This procedure often leads to a discontinuous controller. When the controller is implemented, the discontinuous terms are replaced with continuous functions to avoid chattering of the control signal. Two associated problems have been overlooked during this procedure. One is that discontinuous control systems are non-smooth, which violates the fundamental assumptions of solution theories and the applicability of Lyapunovs stability theory is questionable. Another problem is that the replacement of discontinuous terms may weaken stability, which can be critical. In this paper, we discuss proper stability analysis of discontinuous control systems using the extended Lyapunovs second method based on Filippovs solution concept for non-smooth systems. We further propose to utilize the concept of Lyapunov exponents to quantitatively analyze the stability of continuous control systems obtained by replacing the discontinuous terms in the discontinuous controllers. An example involving the stabilization of a two-link non-fixed-base robotic manipulator is presented for demonstration. This research fills the gap in designing continuous Lyapunovs stability controllers regarding limited available Lyapunov functions.

Impact Control in Hydraulic Actuators

Every manipulator contact task that begins with a transition from free motion to constraint motion may exhibit impacts that could drive the system unstable. Stabilization of manipulators during this transition is, therefore, an important issue in contact task control design. This paper presents a discontinuous controller to regulate the transition mode in hydraulic actuators. The controller, upon sensing a nonzero force, positions the actuator at the location where the force was sensed, thus, exerting minimal force on a non-moving environment. The scheme does not require force or velocity feedback as they are difficult to measure throughout the short transition phase. Also, no knowledge about the environment or hydraulic parameters is required for control action. Due to the discontinuity of the control law, the control system is nonsmooth. First, the existence, continuation and uniqueness of Filippovs solution to the system are proven. Next, the extension of Lyapunov stability theory to nonsmooth systems is employed to guarantee the global asymptotic convergence of the entire systems state towards the equilibrium point. Complete dynamic characteristics of hydraulic functions and Hertz-type contact model are included in the stability analysis. Experiments are conducted to verify the practicality and effectiveness of the proposed controller. They include actuator collisions with hard and soft environments and with various approach velocities.

Asymptotic Force Control of Hydraulic Actuators With Friction: Theory and Experiments

This paper documents the development, theoretical analysis and experimental evaluation of a Lyapunov-based nonlinear control scheme for asymptotic force regulation of hydraulic actuators with friction. The complete discontinuous model of actuator friction, servo-valve dynamics, and nonlinear hydraulic functions are all included in the theoretical solution and stability analyses of the resulting nonsmooth system. The frictionless contact force is modeled as a linear stiffness. Filippov’s solution theory and the extension of LaSalle’s invariance principle to nonsmooth systems are employed to prove the asymptotic convergence of the system trajectories towards equilibria. Experiments complement the theoretical analysis in providing a solid foundation for implementation of the proposed control scheme for asymptotic force regulation of the hydraulic actuators despite friction effects.© 2004 ASME

Asymptotic Impact Control of Hydraulic Actuators With Friction

The focus of this work is stabilization of hydraulic actuators during the transition from free motion to constraint motion and regulating the intermediate impacts that could drive the system unstable. In our past research, we introduced Lyapunov-based nonlinear control schemes capable of fulfilling the above goal by resting the implement on the surface of the environment before starting the sustained-contact motion. The hydraulic actuator’s stick-slip friction effect was, however, either not included in the analysis or not compensated by the control action. In this paper, the application of our previously introduced friction compensating position control scheme is extended to impact regulation of a hydraulic actuator. Theoretical solution and stability analyses as well as actual experiments prove that such control scheme is also effective for asymptotic impact control (with no position steady-state error) of hydraulic actuators in the presence of actuator’s dry friction.Copyright

Contact task stability analysis via Lyapunov exponents

In this paper, a new application of Lyapunov exponents is introduced for stability analysis of contact task control in robotics. The dynamic model is derived including a nonlinear model of the contact which allows bouncing. The system is controlled by a discontinuous controller composed of laws for free and constrained motions that are switched based on detection of the contact force. The stability analysis of such a nonsmooth system using Lyapunovs direct method is extremely difficult. A model based algorithm for the calculation of Lyapunov exponents in nonsmooth systems is employed to prove the overall stability of the system and the required equations and transition conditions are obtained. Numerical results demonstrate the stability and provide valuable insight into the effect of approach velocity and controller gains on it.

An improved design procedure of Lyapunov feedback control

An improved procedure of Lyapunov feedback control is developed. The procedure includes two parts. First, a discontinuous controller is designed. The extended Lyapunov stability theory is used for the stability analysis of the resulting non-smooth control system. Next, the discontinuous terms in the controller are replaced with continuous functions. This weakens the stability but avoids the chattering of the controller. The trade-off in the stability is analyzed using the concept of Lyapunov exponents. We demonstrate the effectiveness of the design procedure by stabilizing a two-link base-excited inverted pendulum system around the upright position. We believe that our improved design procedure makes the Lyapunov feedback control more applicable.

Overall Stability Analysis of Hydraulic Actuator’s Switching Contact Control Using the Concept of Lyapunov Exponents

Control law switching in a complete contact task control of hydraulic actuators results in a highly nonlinear nonsmooth system. Stability analysis of such a system using Lyapunov direct method involves restrictive modeling assumptions. This paper presents an alternative approach for stability analysis using the concept of Lyapunov exponents. Advantages and challenges of applying the method on a hydraulically actuated switching contact task control system are discussed in detail. Issues considered in the analysis include solution analysis of nonlinear and linearized systems, linearization of nonlinear equations at the instants of discontinuity, existence of Lyapunov exponents, and stability of numerical computations. Theoretical results are complemented with test experiments.

Cascade control of hydraulic actuators during contact tasks

Due to the complex interactions between hydraulic components, the contact task control of hydraulic manipulators is a challenging problem. In this paper, a controller is proposed which incorporates the nonlinear hydraulic effects and actuator-environment dynamics into the control procedure. The approach allows partitioning the hydraulics control from the actuator-environment interaction control so that, a different control algorithm could be applied to each part. The communication between the two parts is performed through an appropriate input-output arrangement. In this paper, a proportional control with feedforward compensation is applied to the hydraulic portion of the system and a bounce-less contact controller is applied to the actuator-environment interaction. Simulation and experimental results demonstrate that the proposed controller remains stable following contact with the environment.