4_eq_finding - 4 `Equilibrium Finding Key points Nonlinear systems may have a number of equilibrium points(from zero to infinity These are obtained from

# 4_eq_finding - 4 `Equilibrium Finding Key points Nonlinear...

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4 ` Equilibrium Finding We consider systems that can be written in the following general form, where x is the state of the system, u is the control input, and f is a nonlinear function. ) , ( u x f x = & m n u x , Let u = u e = constant. At an equilibrium point, 0 ) , ( = e e u x f . Key points Nonlinear systems may have a number of equilibrium points (from zero to infinity). These are obtained from the solution of n algebraic equations in n unknowns. The global implicit function theorem states condition for uniqueness of an equilibrium point. Numeral solutions to obtain the equilibrium points can be obtained using several methods, including (but not limited to) the method of Newton-Raphson and steepest descent techniques.
To obtain the equilibrium points, one has to solve n algebraic equations in n unknowns. How can we find out if an equilibrium point is unique? See next section. Global Implicit Function Theorem ) , ( e u x f x = & Define x x f x J = ) ( the Jacobian of f.

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