12 Synchronization and control Henk Nijmeijer, Henri Huijberts, - PowerPoint PPT Presentation

12 Synchronization and control Henk Nijmeijer, Henri Huijberts, Sasha Pogromsky Eindhoven University of Technology, University of London Cooperation with: Ilya Blekhman, Sasha Fradkov Alejandro Rodriguez-Angeles, Torsten Lilge, Iven Mareels, Giovanni Santoboni Rob Willems, 13

12 12 Synchronization and control Introduction Henk Nijmeijer, Henri Huijberts, Sasha Pogromsky History of synchronization Eindhoven University of Technology, University of London • Christiaan Huijgens (1670): pendulum clocks synchronize... • Rayleigh (1877): organ tubes sound in unison..., tuning forks,... • 1900: electrical and electromechanical systems (Van der Pol) Cooperation with: Ilya Blekhman, Sasha Fradkov Alejandro Rodriguez-Angeles, Torsten Lilge, • Rotating bodies, e.g. moon vs earth Iven Mareels, Giovanni Santoboni • Chaos synchronization Rob Willems • (Secure) communication: Pecora and Carroll (1990) 13 • Controlled synchronization: communication, ship mooring, robot coordination, sound and light show,... 12 12 This presentation Contents • Control view on synchronization • Introduction (Henk Nijmeijer) • Dynamics • An observer view on synchronization (Henri Huijberts) • Observer theory vs synchronization • Communication and Synchronization (Henri Huijberts) • Controlled synchronization • Synchronization in diffusive networks (Henk Nijmeijer) • Examples of simple/chaotic systems • Controlled synchronization (Henri Huijberts) • Theory: literature • Coordination of mechanical systems (Henk Nijmeijer) • No complete review

12 12 Huijgens’ notebook, “Horloges Marines (Et Sympathie Des Horloges)”, Christiaan Huijgens, 1 March, 1665. born April 14, 1629 , The Hague, died July 8, 1695 , The Hague 12 12 Women living together have synchronous menstrual cycles. A figure from Huijgens’ notebook, 22 Febr. 1665

12 12 Synchronization in networks: different synchronization modes, partial synchronization. An application of master slave synchronization 12 12 This workshop: • Control theory: observers, controlled synchronization • Dynamics: synchronization in networks • Applications: communication and coordination of mechanical systems

12 12 z Contents • Introduction • An observer view on synchronization • Communication and synchronization • Synchronization in diffusive networks • Controlled synchronization y x • Coordination of mechanical systems The Lorenz attractor. 12 12 Synchronization and observers Two points of view on synchronization Two points of view on synchronization Peccora and Carroll (1990), Lorenz system Peccora and Carroll (1990), Lorenz system Transmitter (master) system : Receiver (slave) system Transmitter (master) system : (”copy” of master) x 1 = σ ( x 2 − x 1 ) ˙ x 1 = σ ( x 2 − x 1 ) ˙ ˙ x 2 = rx 1 − x 2 − x 1 x 3 ˙ x 2 = rx 1 − � � x 2 − x 1 � x 3 x 2 = rx 1 − x 2 − x 1 x 3 ˙ ˙ x 3 = − bx 3 + x 1 x 2 ˙ � x 3 = − b � x 3 + x 1 � x 2 x 3 = − bx 3 + x 1 x 2 ˙ No � x 1 -dynamics, because x 1 already known.

12 12 e 1 = x 1 − � x 1 , e 2 = x 2 − � x 2 , e 3 = x 3 − � x 3 e 2 = x 2 − � x 2 , e 3 = x 3 − � x 3 e 1 = σ ( e 2 − e 1 ) ˙ e 2 = − e 2 − x 1 e 3 ˙ e 2 = − e 2 − x 1 e 3 ˙ e 3 = − be 3 + x 1 e 2 ˙ e 3 = − be 3 + x 1 e 2 ˙ V ( e 2 , e 3 ) = e 2 2 + e 2 Lyapunov function: 3 V ( e 1 , e 2 , e 3 ) = 1 /σe 2 1 + e 2 2 + e 2 Lyapunov function: 3 V = − 2 e 2 ˙ 2 − 2 be 2 3 V = − 2( e 1 − 1 / 2 e 2 ) 2 − 3 / 2 e 2 ˙ 2 − 2 be 2 3 ( e 2 , e 3 ) → (0 , 0) , as t → ∞ ( e 1 , e 2 , e 3 ) → (0 , 0 , 0) , as t → ∞ Control viewpoint: Slave is partial observer for master. Control viewpoint: slave is full observer for master. 12 12 ˙ x 1 = σ ( x 2 − x 1 ) ˙ x 1 = σ ( � � x 2 − � x 1 ) ˙ x 2 = rx 1 − x 2 − x 1 x 3 ˙ x 2 = rx 1 − � � x 2 − x 1 � x 3 Two points of view on synchronization ˙ x 3 = − bx 3 + x 1 x 2 ˙ � x 3 = − b � x 3 + x 1 � x 2 Peccora and Carroll (1990), Lorenz system Two related problems: Transmitter (master) system : Receiver (slave) system (”copy” of master) System � ˙ x 1 = σ ( x 2 − x 1 ) ˙ � x 1 = σ ( � x 2 − � x 1 ) y(t) System � ˙ x 2 = rx 1 − x 2 − x 1 x 3 ˙ � x 2 = rx 1 − � x 2 − x 1 � x 3 System System � � y(t) ˙ x 3 = − bx 3 + x 1 x 2 ˙ x 3 = − b � � x 3 + x 1 � x 2 e(t) x(t) x(t) x(t) Observer � e(t) Synchronization/observer problem Convergent systems, Demidovich

12 12 Example Synchronization/Observer problem statement x 1 = x 2 ˙ x 2 = ax 1 + bx 2 , ˙ y = x 1 x = f ( x ) ˙ observer: estimate for ( x 1 , x 2 ) y = h ( x ) How to find observer? Try ”Pecora & Carroll copy”: x ( t ) ∈ R n , state; y ( t ) ∈ R , output, or measurement. ˙ x 1 = � � x 2 Observer: given y ( t ), t ≥ 0, reconstruct asymptotically x ( t ) , t ≥ 0. ˙ x 2 = ay + b � � x 2 = ax 1 + b � x 2 Reduced observer: given y ( t ), t ≥ 0, reconstruct asymptotically x ( t ) modulo y ( t ). e 1 = x 1 − � x 1 , e 2 = x 2 − � x 2 So if slave can be chosen freely, synchronization problem is equivalent to e 1 = e 2 , ˙ e 2 = be 2 ˙ observer problem. Does give reconstruction of x 2 iff b < 0, but not reconstruction of x 1 ! 12 12 Example x 1 = x 2 ˙ x 2 = ax 1 + bx 2 , ˙ y = x 1 Alternative: reduced observer: estimate for x 2 . ˙ x 1 = � � x 2 + k 1 e 1 How to find reduced observer? Try ”Pecora & Carroll copy”: ˙ x 2 = a � � x 1 + b � x 2 + k 2 e 1 ˙ x 2 = ay + b � � x 2 = ax 1 + b � x 2 Suitable k 1 and k 2 yield ( e 1 , e 2 ) → (0 , 0) , as t → ∞ e 2 = x 2 − � x 2 (Reduced observer: similar) e 2 = be 2 ˙ Asymptotic reconstruction if and only if b < 0.

12 12 Nonlinear systems? Linear systems (no complex dynamics): x ∈ R n x = f ( x ) , ˙ y = h ( x ) x ∈ R n x = Ax, ˙ y = Cx Try: ˙ x = f ( � x ) + k ( � x, y ) � ˙ x = A � � x + K ( y − � y ) � y = C � x with k ( � x, y ) = 0 if h ( � x ) = y Required for synchronization: x → � x , as t → ∞ for any initial x (0) , � x (0). e = ( A − KC ) e ˙ Find k ( · , · )! 12 12 When does an observer exist? 1. Linear error dynamics Example: Chua circuit A − KC has arbitrary pole location iff system is observable, i.e.   x 1 = α ( − x 1 + x 2 − ϕ ( x 1 )) ˙ C x 2 = x 1 − x 2 + x 3 ˙     CA   x 3 = − λx 2 ˙   rank = n  .  .   .   CA n − 1 ϕ ( x 1 ) = m 1 x 1 + m 2 ( | x 1 + 1 | − | x 1 − 1 | ) with m 1 = − 5 / 7, m 2 = − 3 / 7, 23 < λ < 31, α = 15 . 6. Thus synchronization � x → x for suitably chosen K . Double scroll chaotic attractor Output: y = x 1 Thus nonlinearity ϕ ( x 1 ) is measurable!

12 12 2. Linearizable Error Dynamics Example: R¨ ossler system, a, b, c > 0 z x 1 = − x 2 − x 3 ˙ x 2 = x 1 + ax 2 ˙ x 3 = c + x 3 ( x 1 − b ) ˙ y = x 3 NB x 3 (0) > 0 = ⇒ x 3 ( t ) > 0, ∀ t > 0. y New coordinates: ( ξ 1 , ξ 2 , ξ 3 ) = ( x 1 , x 2 , log x 3 ) x The double scroll attractor. New output: y = log y = ξ 3 � 12 12 Observer: ˙ x 1 = α ( − � � x 1 + � x 2 − ϕ ( x 1 )) + k 1 e 1 z ˙ x 2 = � � x 1 − � x 2 + � x 3 + k 2 e 1 ˙ x 3 = − λ � � x 2 + k 3 e 1 e 1 = ( k 1 − α ) e 1 + e 2 ˙ e 2 = ( k 2 + 1) e 1 − e 2 + e 3 ˙ e 3 = k 3 e 1 − λe 2 ˙ y Note: Linear Observable System, ( e 1 , e 2 , e 3 ) → (0 , 0 , 0) for suitable x choice of k 1 , k 2 , k 3 . Arbitrarily fast! Trajectories of the R¨ ossler system. Similar design for Lur’e systems.

12 12 3. High-gain Observer In new coordinates: ˙ x = f ( x ) , ˙ y = h ( x ) ξ 1 = − ξ 2 − e � 3 ˙ ξ 2 = ξ 1 + aξ 2 Assume: ˙ ξ 3 = ξ 1 + ( − b + ce − � 3 ) • f ( x ) satisfies Lipschitz condition on positively invariant compact y = ξ 3 � domain Ω. • The n functions h ( x ) , L f h ( x ) , L 2 f h ( x ) , . . . (Iterated Lie derivatives of • Linear part is observable h in the direction of f ) define new coordinates on domain Ω. • Nonlinear part is measurable There exists an observer of the form Observer: ˙ ˙ � x = f ( � x ) + K ( y − h ( � x )) ξ 1 = − � � ξ 2 − e � 3 + k 1 ( ξ 3 − � ξ 3 ) ˙ ξ 2 = � � ξ 1 + a � ξ 2 + k 2 ( ξ 3 − � ξ 3 ) ˙ ξ 3 = � � ξ 1 + ( − b + ce − � 3 ) + k 3 ( ξ 3 − � with K suitable ( n, 1)-vector. ξ 3 ) Example: Lorenz system on compact domain. 12 12 4. Time rescaling Suppose that for the system x = f ( x ) , ˙ y = h ( x ) Error dynamics: there exist new coordinates ξ such that e 1 = − e 2 + k 1 e 3 ˙ ˙ ξ = s ( y )( Aξ + ϕ ( y )) , y = Cξ e 2 = e 1 + ae 2 + k 2 e 3 ˙ with some s ( y ) > 0. e 3 = e 1 + k 3 e 3 ˙ New time: dτ = s ( y ) dt . Suitable k 1 , k 2 , k 3 = ⇒ synchronization dξ dτ = ( Aξ + ϕ ( y )) In new time – linear error dynamics provided ( A, C ) is observable (detectable).