Using control authority to transform nonlinear models into linear ones is one of the most commonly used ideas of practical nonlinear control design. Generally, the trick helps one to recognize \simple\nonlinear feedback design tasks

Lyapunov analysis, which uses monotonicity of a given function of system state along trajectories of a given dynamical system, is a major tool of nonlinear system analysis It is possible, however, to use monotonicity of volumes of subsets of the state space to predict certain properties of system behavior. This lecture gives an introduction to suc methods

12.1 Systems with controllable linearizations A relatively straightforward case of local controllability analysis is defined by systems with controllable linearizations 12.1.1 Controllability of linearized system Let To: 0, THR, uo: 0, T]H Rm be a

where f:R\×Rn×R→ R\ and g:R\×R\×R→ R are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory co(t), yo(t), and that the derivative

Proof Existence and uniqueness of r(t, u)and A(t)follow from Theorem 3. 1. Hence, in order to prove differentiability and the formula for the derivative, it is sufficient to show that there exist a function C: R++R+ such that C(r)/r-0 as r-0 and E>0 such

This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems 6.1 Stability of an equilibria

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS

In particular, when o=0, this yields the definition of a Lyapunov function Finding, for a given supply rate, a valid storage function(or at least proving that one exists)is a major challenge in constructive analysis of nonlinear systems. The most com-

with x(0)=I exist and are unique on the time interval t E [ 0, 1] for allTER\.Then discrete time system(4. 1)with f(5)=r(, i)describes the evolution of continuous time system(4.)at discrete time samples. In particular, if a is continuous then so is f Let us call a point in the closure of X locally attractive for system(4. 1)if there exists

Definition A real-valued function V: X H R defined on state space X of a system with behavior set B and state r:B×[0,∞)→ X is called a Lyapunov function if tHv(t)=v(a(t))=v(a(z(), t)) is a non-increasing function of time for every z E B according to this definition, Lyapunov