# WUSTL ESE 543 - Lecture 5 and 6 stab Lyap theory - Copy (56 pages)

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## Lecture 5 and 6 stab Lyap theory - Copy

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- School:
- Washington University in St. Louis
- Course:
- Ese 543 - Control Systems Design by State Space Methods

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Lecture 5 Nonlinear Systems Reference Texts J J Slotine W Li Applied Nonlinear Control Prentice Hall 1995 Khalil H K Nonlinear Systems Prentice Hall 2002 Linear Systems x t Ax t x R Unique equilibrium point if Stable equilibrium if A nx t R is nonsingular Re A 0 System response is a composition of natural modes If x Ax Bu x R nx u R nu t R Principle of superposition If Re A 0 BIBO stability in presence of u Sinusoidal input produces sinusoidal output of same frequency These same properties do not hold for nonlinear systems K A Wise 2 Nonlinear Systems General representation of nonlinear dynamics nx x f t x x R The solution t R x x t x t0 x0 is called state trajectory Definition The nonlinear system is called autonomous if the dynamics is not explicitly depending upon time Definition A state once x t x f x x is called an equilibrium state point of the system if is equal to x it remains equal to x for all future time It can be found by solving the nonlinear algebraic equations 0 f x 3 K A Wise Nonlinear Systems cont General representation of nonlinear dynamics nx x f t x x R t R x f x g x u u R nu x f x u Multiple equilibrium points are possible Stability depends upon initial conditions Can exhibit limit cycles self excited oscillations Exhibit bifurcations Changes in parameters can cause stability and the number of equilibrium points to change 4 K A Wise Equilibrium Points for Linear Systems A linear time invariant system x A x has a single equilibrium point at the origin if A is non singular If A is singular it has an infinity of equilibrium points which are contained in the null space of the matrix A i e by the subspace defined by Ax 0 This implies that the equilibrium points are not isolated Example x x 0 All the points on the x 1 x2 x 2 x2 x2 0 i e x axis are equilibrium points 5 K A Wise Nonlinear Example Duffing Equation 3 x x x 0 Undamped Solve for equilibriums 3 x x 0 0 x 0 i i Is a critical bifurcation value x Called a pitchfork bifurcation 6 K A Wise

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