Construction of Control Lyapunov Functions for a Class of Nonlinear Systems

The construction of control Lyapunov functions for a class of nonlinear systems is considered. We develop a method by which a control Lyapunov function for the feedback linearizable part can be constructed systematically via Lyapunov equation. Moreover, by a control Lyapunov function of the feedback linearizable part and a Lyapunov function of the zero dynamics, a control Lyapunov function for the overall nonlinear system is established.

Universal construction of control Lyapunov functions for a class of nonlinear systems

A method is developed by which control Lyapunov functions of a class of nonlinear systems can be constructed systematically. Based on the control Lyapunov function, a feedback control is obtained to stabilize the closed-loop system. In addition, this method is applied to stabilize the Benchmark system. A simulation shows the effectiveness of the method.

Stabilization of discrete nonlinear systems based on control Lyapunov functions

The stabilization of discrete nonlinear systems is studied. Based on control Lyapunov functions, a sufficient and necessary condition for a quadratic function to be a control Lyapunov function is given. From this condition, a continuous state feedback law is constructed explicitly. It can globally asymptotically stabilize the equilibrium of the closed-loop system. A simulation example shows the effectiveness of the proposed method.

A Survey on the Control Lyapunov Function and Control Barrier Function for Nonlinear-Affine Control Systems

This survey provides a brief overview on the control Lyapunov function(CLF)and control barrier function(CBF)for general nonlinear-affine control systems.The problem of control is formulated as an optimization problem where the optimal control policy is derived by solving a constrained quadratic programming(QP)problem.The CLF and CBF respectively characterize the stability objective and the safety objective for the nonlinear control systems.These objectives imply important properties including controllability,convergence,and robustness of control problems.Under this framework,optimal control corresponds to the minimal solution to a constrained QP problem.When uncertainties are explicitly considered,the setting of the CLF and CBF is proposed to study the input-to-state stability and input-to-state safety and to analyze the effect of disturbances.The recent theoretic progress and novel applications of CLF and CBF are systematically reviewed and discussed in this paper.Finally,we provide research directions that are significant for the advance of knowledge in this area.

Direct adaptive control for nonlinear uncertain system based on control Lyapunov function method

The problem of adaptive stabilization of a class of multi-input nonlinear systems with unknown parameters both in the state vector-field and the input vector-field has been considered. By employing the control Lyapunov function method, a direct adaptive controller is designed to complete the global adaptive stability of the uncertain system. At the same time, the controller is also verified to possess the optimality. Example and simulations are provided to illustrate the effectiveness of the proposed method.

Asymptotic controllability of a class of discrete-time systems with disturbances

This article deals with the uniformly globally asymptotic controllability of discrete nonlinear systems with disturbances.It is shown that the system is uniformly globally asymptotic controllability with respect to a closed set if and only if there exists a smooth control Lyapunov function.Further, it is obtained that the control Lyapunov function may be used to construct a feedback law to stabilize the closed-loop system.In addition, it is proved that for periodic discrete systems, the resulted control Lyapunov functions are also time periodic.

The feedback control of fractional order unified chaotic system

This paper studies the stability of the fractional order unified chaotic system. On the unstable equilibrium points, the equivalent passivity＇＇ method is used to design the nonlinear controller. With the definition of fractional derivatives and integrals, the Lyapunov function is constructed by which it is proved that the controlled fractional order system is stable. With Laplace transform theory, the equivalent integer order state equation from the fractional order nonlinear system is obtained, and the system output can be solved. The simulation results validate the effectiveness of the theory.

Improved forwarding control design method and its application

Integrator forwarding is a recursive nonlinear design technique for the stabilization of feed-forward systems. However, this method still has some limitation. An improved design method is proposed to extend the field of application of this technique. This method is used to design a stabilizer for the inertia wheel pendulum system. Moreover, it is shown that the control Lyapunov function which is obtained from this method can also be used to design a globally asymptotically stabilizing controller with optimality.

Hierarchical-control-based output synchronization of coexisting attractor networks

This paper introduces the concept of hierarchical-control-based output synchronization of coexisting attractor networks. Within the new framework, each dynamic node is made passive at first utilizing intra-control around its own arena. Then each dynamic node is viewed as one agent, and on account of that, the solution of output synchronization of coexisting attractor networks is transformed into a multi-agent consensus problem, which is made possible by virtue of local interaction between individual neighbours; this distributed working way of coordination is coined as inter-control, which is only specified by the topological structure of the network. Provided that the network is connected and balanced, the output synchronization would come true naturally via synergy between intra and inter-control actions, where the rightness is proved theoretically via convex composite Lyapunov functions. For completeness, several illustrative examples are presented to further elucidate the novelty and efficacy of the proposed scheme.

Spring-Loaded Inverted Pendulum Hopping via Hybrid Averaging and Control Lyapunov Function

The Spring-Loaded Inverted Pendulum(SLIP)has been regarded as a canonical model for hopping and running dynamics of legged robots.This paper presents a novel control of the actuated-SLIP hopping on unknown terrains.We propose that in the neighborhood of the desired stable hybrid limit cycle,the local dynamical behavior of a hybrid system can be expressed by a set of phase coordinates and transverse coordinates.Under some acceptable assumptions,the hybrid averaging theorem is applied on the SLIP non-integrable dynamics to simplify the controller design.Using the inherent symmetry of SLIP dynamics,a control Lyapunov function-based hybrid averaging controller is developed to ensure the exponential stability of the desired gait orbit.This results in a set of linear constraints on the control signal,which can be readily solved by a quadratic programming optimization.Furthermore,a novel method is introduced to improve the robustness against unknown disturbances through the online constraint adjustment.The proposed controller is evaluated in various simulations,demonstrating the SLIP hopping on diverse terrains,including flat,sin-wave,and unregular terrains.The performance of the approach is also validated on a quadruped robot SCIT Dog for generating dynamic gaits such as pronking.

Adaptive stabilization for cascade nonlinear systems

An adaptive controller of full state feedback for certain cascade nonlinear systems achieving input-to-state stability with respect to unknown bounded disturbance is designed using backstepping and control Lyapunov function (CLF) techniques. We show that unknown bounded disturbance can be estimated by update laws, which requires less information on unknown disturbance, as a part of stabilizing control. The design method achieves the desired property: global robust stability. Our contribution is illustrated with the example of a disturbed pendulum.

Global stabilization of nonlinear systems with uncertain structure

The global stabilization problem of nonlinear systems with uncertain structure is dealt with. Based on control Lyapunov function （CLF）, a sufficient and necessary condition for Lyapunov stabilization is given. From the condition, several simply sufficient conditions for the globally asymptotical stability are deduced. A state feedback control law is designed to globally asymptotically stabilize the equilibrium of the closed system. Last, a simulation shows the effectiveness of the method.

Fuzzy adaptive tracking control within the full envelope for an unmanned aerial vehicle

Motivated by the autopilot of an unmanned aerial vehicle（UAV） with a wide flight envelope span experiencing large parametric variations in the presence of uncertainties, a fuzzy adaptive tracking controller（FATC） is proposed. The controller consists of a fuzzy baseline controller and an adaptive increment, and the main highlight is that the fuzzy baseline controller and adaptation laws are both based on the fuzzy multiple Lyapunov function approach, which helps to reduce the conservatism for the large envelope and guarantees satisfactory tracking performances with strong robustness simultaneously within the whole envelope. The constraint condition of the fuzzy baseline controller is provided in the form of linear matrix inequality（LMI）, and it specifies the satisfactory tracking performances in the absence of uncertainties. The adaptive increment ensures the uniformly ultimately bounded（UUB） predication errors to recover satisfactory responses in the presence of uncertainties. Simulation results show that the proposed controller helps to achieve high-accuracy tracking of airspeed and altitude desirable commands with strong robustness to uncertainties throughout the entire flight envelope.