Iterative learning control methods are represented as powerful tools to control dynamics nowadays. Our new controller based on particular case of iterative learning control is radically different from the presented conventional method, which attempts to stabilize a class of nonlinear systems by satisfying the conditions of Lyapunov Stability Theorem. Since our algorithm is model based, its robu...

Formations that contain a small number of robots are modeled as controlled Lagrangian systems on Jacobi shape space. This allows a block-structured control of position, orientation and shape of the formation. Feedback control laws are derived using control Lyapunov functions. The controlled dynamics converges to the invariant set where desired shape is achieved. Controllers are implemented in a...

Lyapunov exponents of chaotic attractors are hard to estimate, especially for non-smooth systems. One method to estimate the maximal Lyapunov exponent is by using its relationship with the synchronization properties of coupled systems. The maximal Lyapunov exponent is equal to the minimal proportional feedback gain necessary to achieve full state synchronization with a replica system. In this p...

Closed physical systems eventually come to rest, the reason being that due to friction of some kind they continuously lose energy. The mathematical extension of this principle is the concept of a Lyapunov function. A Lyapunov function for a dynamical system, of which the dynamics are modelled by an ordinary differential equation (ODE), is a function that is decreasing along any trajectory of th...

This chapter is about numerical methods for a particular type of equation expressed as a matrix equality. The Lyapunov equation is the most common problem in the class of problems called matrix equations. Other examples of matrix equations: Sylvester equation, Stein equation, Riccati equation. Definition 5.0.1 Consider two square matrices A, W ∈ Rn×n. The problem to find a square matrix X ∈ Rn×...

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration xn+1 = g(xn) can be determined through sublevel sets of a Lyapunov function. In [3] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no neg...

This paper proposes new criteria for stability and performance of nonlinear interconnected systems by smoothly generalizing a recently-developed method of the state-dependent scaling. The criteria are directly related to Lyapunov functions so that they show how to construct Lyapunov functions which establish desired stability and performance explicitly. A set of popular Lyapunov stability crite...

This paper addresses the stability of the switched systems with discrete and distributed time delays. By applying Lyapunov functional and function method, we show that, if the norm of system matrices Bi is small enough, the asymptotic stability is always achieved. Finally, a example is provided to verify technically feasibility and operability of the developed results. Keywords—Switched system,...

We consider nonlinear systems dx=dt = f(x(t)) where Df(x(t)) is known to lie in the convex hull of L matrices A1, : : : , AL 2 R n . For such systems, quadratic Lyapunov functions can be determined using convex programming techniques [1]. This paper describes an algorithm that either nds a quadratic Lyapunov function or terminates with a proof that no quadratic Lyapunov function exists. The alg...

Input-to-state stability (ISS) of a feedback interconnection of two discrete-time ISS systems satisfying an appropriate small gain condition is investigated via the Lyapunov method. In particular, an ISS Lyapunov function for the overall system is constructed from the ISS Lyapunov functions of the two subsystems. We consider parameterized families of discrete-time systems that naturally arise w...