Amir massoud Farahmand SoloGen@SoloGen.net Chaos Control Part II Amir massoud Farahmand SoloGen@SoloGen.net

Review Why Chaos control?! THE BEGINNING WAS CHAOS! Chaos is Fascinating! Chaos is Everywhere! Chaos is Important! Chaos is a new paradigm shift in science!

Review II What is it?! Nonlinear dynamics Deterministic but looks stochastic Sensitive to initial conditions (positive Bol (Lyapunov) exponents) Strange attractors Dense set of unstable periodic orbits (UPO) Continuous spectrum

Review III Chaos Control: Goals Stabilizing Fixed points Stabilizing Unstable Periodic Orbits Synchronizing of two chaotic dynamics Anti-control of chaos Bifurcation control

Review IV Chaos Control: Methods Linearization of Poincare Map OGY (Ott-Grebogi-York) Time Delayed Feedback Control Impulsive Control OPF (Occasional Proportional Feedback) Open-loop Control Conventional control methods

Chaos Control Conventional control Back-stepping A. Harb, A. Zaher, and M. Zohdy, “Nonlinear recursive chaos control,” ACC2002. Frequency domain methods Circle-like criterion to ensure L2 stability of a T-periodic solution subject to the family of T-periodic forcing inputs. M. Basso, R. Genesio, and L. Giovanardi, A. Tesi, “Frequency domain methods for chaos control,” 2000.

Chaos Control Conventional + Chaotic Taking advantage of inherit properties of chaotic systems Periodic Chaotic systems are dense (according to Devaney definition) Waiting for the sufficient time, every point of the attractor will be visited. If we are sufficiently close to the goal, turn-on the conventional controller, else do nothing! T. Vincent, “Utilizing chaos in control system design,” 2000.

Chaos Control Conventional + Chaotic Henon map Stabilizing to the unstable fixed point Locally optimal LQR design Farahmand, Jabehdar, “Stabilizing Chaotic Systems with Small Control Signal”, unpublished Figure 1 Henon map

Chaos Control Conventional + Chaotic Figure 2. Sample response for two different attraction threshold Figure 5. Settling time for different theta

Chaos Control Conventional + Chaotic Figure 3. Peak of control signal for different theta Figure 4. Controlling energy for different theta

Chaos Control Impulsive control of periodically forced chaotic system Z. Guan, G. Chen, T. Ueta, “On impulsive control of periodically forced pendulum system,” IEEE T-AC, 2000.

Anti-Control of Chaos Definitions and Applications (I) Anti-control of chaos (Chaotification) is Making a non-chaotic system, chaotic. Enhancing chaotic properties of a chaotic system.

Anti-Control of Chaos Definition and Applications (II) Stability is the main focus of traditional control theory. There are some situations that chaotic behavior is desirable Brain and heart regulation Liquid mixing Secure communication Small control (Chaotification of non-chaotic system  chaos control method (small control)  conventional methods )

Anti-Control of Chaos Discrete case (I) Suppose we have a LTI system. If we change its dynamic with a proper feedback such that it is bounded has positive Lyapunov exponent then we may have made it chaotic. We may use Marotto theorem to prove the existence of chaos in the sense of Li and Yorke. X. Wang and G. Chen, “Chaotification via arbitrarily small feedback controls: theory, methods, and applications,” 2000.

Anti-Control of Chaos Discrete case (II)

Anti-Control of Chaos Discrete case (III)

Anti-Control of Chaos Continuous case (I) Approximating a continuous system by its time-delayed version (Discrete map). Making a discrete dynamics chaotic is easy. It has not been proved yet! X. Wang, G. Chen, X. Yu, “Anticontrol of chaos in continuous-time systems via time-delayed feedback,” 2000.

Anti-Control of Chaos Continuous case (II)

Synchronization (I) Carrier Clock, Secure communication, Power systems and … Formulation: Synchronization Unidirectional (Model Reference Control) Mutual

Synchronization (II) Linear coupling

Synchronization (III) Drive-Response concept of Pecora-Carroll L.M. Pecora and T.L. Carol, “Synchronization in chaotic systems,” 1990.

Synchronization of Semipassive systems (I) A. Pogromsky, “Synchronization and adaptive synchronization in semipassive systems,” 1997. Semipassive Systems , Isidori normal form Control Signal

Synchronization of Semipassive systems (II) Lemma: Suppose that previous systems are semipassive with radially unbounded continuous storage function. Then all solutions of the coupled system with following control exist on infinite time interval and are bounded.

Synchronization of Semipassive systems (III) Theorem I: Assume that A1. The functions q, a, b are continuous and locally Lipschitz A2.The system is semipassive A3.There exist C2-smooth PD function V0 and … that A4.The matrix b1+b2 is PD: A5. then there exist …  that goal of synchronization is achieved.

Synchronization of Semipassive systems (IV) Lorenz system (Turbulent dynamics of the thermally induced fluid convection in the atmosphere) Figure 1. error and control signal for linearly coupled system