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1 Introduction to Nonlinear Systems, Objective The main goal of this course is to provide to the students a solid. background in analysis and design of nonlinear control systems. Why analysis and not only simulation, Every day computers are becoming more and more powerful to simulate complex. Simulation combined with good intuition can provide useful insight into system s. Nevertheless, It is not feasible to rely only on simulations when trying to obtain guarantees of. stability and performance of nonlinear systems since crucial cases may be missed. Analysis tools provide the means to obtain formal mathematical proofs. certificates about the system s behavior, results may be surprising i e something we had not thought to simulate. 1 Introduction to Nonlinear Systems, Why study nonlinear systems.

Nonlinear versus linear systems, Huge body of work in analysis and control of linear systems. most models currently available are linear but most real systems are nonlinear. dynamics of linear systems are not rich enough to describe many commonly. observed phenomena, 1 Introduction to Nonlinear Systems. Examples of essentially nonlinear phenomena, Finite escape time i e the state can go to infinity in finite time while this is. impossible to happen for linear systems, Multiple isolated equilibria while linear systems can only have one isolated. equilibrium point that is one steady state operating point. Limit cycles oscillation of fixed amplitude and frequency irrespective of the. initial state, Subharmonic harmonic or almost periodic oscillations.

A stable linear system under a periodic input produces an output of the same frequency. A nonlinear system can oscillate with frequencies which are submultiples or multiples of. the input frequency It may even generate an almost periodic oscillation i e sum of. periodic oscillations with frequencies which are not multiples of each other. Other complex dynamic behavior for example chaos biforcations discontinuous. 1 Introduction to Nonlinear Systems, State space model. State equation, Output equation, x1 u1 f1 t x u, 2 3 2 3 2 3. 6 x2 7 6 u2 7 6 f2 t x u 7, x 6 7 u 6 7 f t x u 6, 6 7 6 7 6 7. xn um fn t x u, where x Rn is the state variable u Rm is the input signal and y Rq the output. signal The symbol x dx, denotes the derivative of x with respect to time t.

1 Introduction to Nonlinear Systems, State space model. State equation, x f t x u 1, Output equation, y h t x u 2. Particular cases, Linear Systems where the state model takes the form. x A t x B t u, y C t x D t u, Unforced state equation. i e it does not depend explicitly on the input u e g consider the case that there. is a state feedback u t x and therefore the closed loop system is given by. x f t x t x f t x, Unforced autonomous or time invariant state equation.

1 Introduction to Nonlinear Systems, Example Pendulum. There is a frictional force assumed to be proportional to the linear speed of the mass. m Using the Newton s second law of motion at the tangential direction. ml mg sin kl, where m is the mass l is the length of the rope and k the frictional constant. 1 Introduction to Nonlinear Systems, Example Pendulum. ml mg sin kl, State model, x 2 gl sin x1 k, What are the equilibrium points. 1 Introduction to Nonlinear Systems, Equilibrium point.

A point x x in the state space is said to be an equilibrium point of. x t0 x x t x t t0, that is if the state starts at x it will remain at x for all future time. For autonomous systems the equilibrium points are the real roots of f x 0. The equilibrium points can be of two kinds, isolated that is there are no other equilibrium points in its vicinity. continuum of equilibrium points, Much of nonlinear analysis is based on studying the behavior of a system around its. equilibrium points, 1 Introduction to Nonlinear Systems. Example Pendulum, State model, x 2 gl sin x1 k, Equilibrium points.

0 gl sin x1 k, which implies that x2 0 and sin x1 0 Thus the equilibrium points are. What is the behavior of the system near the equilibrium points. 1 Introduction to Nonlinear Systems, Qualitative behavior of 2nd order linear time invariant systems. x Ax x R2 A R2 2, Apply a similarity transformation M to A. M 1 AM J M R2 2, where J is the real Jordan form of A which depending on the eigenvalues of A may. take one of the three forms, with k being either 0 or 1.

1 Introduction to Nonlinear Systems, Case 1 Both eigenvalues are real with 1 6 2 6 0. The associated eigenvectors v1 v2 R2 1 must satisfy. A v1 v2 v1 v2 M v1 v2 M 1 AM J, This represents a change of coordinates. and we obtain in the new referential, z M 1 x M 1 Ax M 1 AM z z. 1 Introduction to Nonlinear Systems, Case 1 Both eigenvalues are real with 1 6 2 6 0. For a given initial state z1 z2 0 the solution is given by. z1 t z1 0 e 1 t, z2 t z2 0 e 2 t, Eliminating time t.

z2 t z1 t 2 1, z1 t 1 z1 t, e 1 t t ln, z1 0 1 z1 0. ln z 1 0 ln z 1 0 1, z2 t z2 0 e 1 1 z2 0 e 1, At this point several combinations of the eigenvalues can arise. 1 Introduction to Nonlinear Systems, Case 1 Both eigenvalues are real with 1 6 2 6 0. a 1 2 0 In this case e 1 t e 1 t 0 and the curves are parabolic Consider. without loss of generality that 2 1, Phase portrait. The equilibrium point x 0 is called a stable node, 1 Introduction to Nonlinear Systems.

Case 1 Both eigenvalues are real with 1 6 2 6 0, b 1 2 0 The phase portrait will retain the same character but with the. trajectories directions reversed In this case the equilibrium point x 0 is called. an unstable node, c The eigenvalues have opposite signs Consider for example the case 2 0 1. ln z 1 0 1, z2 t z2 0 e 1, The exponent 1, is negative thus we have hyperbolic curves. In this case the equilibrium point is called a saddle point. Subharmonic harmonic or almost periodic oscillations A stable linear system under a periodic input produces an output of the same frequency A nonlinear system can oscillate with frequencies which are submultiples or multiples of the input frequency It may even generate an almost periodic oscillation i e sum of