On the Use of a Frequency Method for Classifying the Oscillatory Behavior in Nonlinear Control Problems Jorge L. Moiola in collaboration with G. Revel, D. Alonso, G. Itovich and F. I. Robbio Instituto de Investigaciones en Ingeniería Eléctrica (IIIE) Departamento de Ing. Eléctrica y de Computadoras, Universidad Nacional del Sur, Avda. Alem 1253, (8000) Bahía Blanca, Argentina First Iberoamerican Meeting on Geometry, Mechanics and Control Santiago de Compostela, June 23 – 27, 2008

Outline Background and motivationBackground and motivation Harmonic balance approximationsHarmonic balance approximations Hopf bifurcationHopf bifurcation Double Hopf and fold-flip bifurcationsDouble Hopf and fold-flip bifurcations ConclusionsConclusions

Background Approximations play an important role in the analysis and control of nonlinear dynamical systems Several methods are available today Dynamical systems theory (perturbation and averaging) (Buonomo & Di Bello, 1996; Phillipson & Schuster, 2000) Not Discussed Today Control systems theory (frequency-domain approach) (Mees & Chua, 1979; Mees, 1981) Topic for today

Motivation: Frequency Domain Approach Detecting Hopf bifurcation (Moiola & Chen, 1993) Detecting multiple (degenerate) Hopf bifurcations (Moiola & Chen, 1994) Detecting the first period-doubling bifurcation (Rand, 1989; Belhaq & Houssni, 1995; Tesi, Abed, Genesio, Wang, 1996) Detecting cascade and global behaviors of oscillations (Belhaq, Houssni, Freire, Rodrígues-Luis, 2000; Bonani & Gilli, 1999) Reducing computational errors in Floquet multipliers (Choe & Guckenheimer, 1999; Guckenheimer & Meloon, 2000; Lust, 2001) … Detecting nonlinear distortions (Maggio, de Feo, Kennedy, 2004; Robbio, Moiola, Paolini & Chen, 2007)

Hopf Bifurcation →An equilibrium point → a periodic solution If the bifurcation parameter is far away from the critical value, higher-order bifurcation formulas are needed.

Poincaré Map Poincaré map (return map) P : U   is a discrete map defined by the points x  U that return to the cross section  after a time  (x) : Limit cycle L 0 is an orbit that starts in  and returns to the same point at  x 0 is a fixed point of the Poincaré map: P(x 0 ) = x 0 and  (x 0 ) = T

Stability Analysis (I) The stability of the periodic solution can be determined analyzing the state transition matrix Φ(t,0) with J var (µ,t) is periodic with fundamental period monodromy matrix The stability of the periodic solution L 0 is determined by the eigenvalues of the monodromy matrix M(µ)

Stability Analysis (II) The matrix M(µ) has n eigenvalues One of them is always +1, say 1 (  ), since its eigenvector is tangent to the periodic cycle The remaining (n-1) ones… for local stability of the periodic solution Eigenvalues with modulus 1 (| j (  )| =1), are known as critical multipliers. Depending on the way they cross the unit circle, different types of periodic branches emerge. It is stable if | j (  )| < 1, for all j = 2, 3,..., n It is unstable if | j (  )| >1, for some j = 2, 3,..., n

Types of Crossing One eigenvalue at  1 One pair of complex-conjugate eigenvalues One eigenvalue at +1

Types of Cyclic Bifurcations (I) One Eigenvalues at +1 Fold bifurcation: saddle-node, pitchfork or transcritical

Types of Cyclic Bifurcations (II) One Eigenvalue at -1 Period doubling, flip, or sub-harmonic bifurcation

Types of Cyclic Bifurcations (III) A pair of complex-conjugated eigenvalues Neimark - Sacker or Torus bifurcation

Time Domain Analysis vs Frequency Domain Analysis Stability analysis Time domain: eigenvalues of Jacobian matrix Frequency domain: eigenloci using Nyquist criterion Oscillation analysis Time domain: time series Frequency domain: frequency spectrum Bifurcation analysis ? --- Topic for today Control system analysis? --- Not discussed today

Frequency Domain Approach (I) Reformulate a general ODE system There are many representations Following Mees & Chua (1979) A linear system with transfer matrix G(s;  ) A memoryless nonlinear feedback f(  ;  ) Dimension may be reduced !

Frequency Domain Approach (II) Equilibrium points are solutions of The transfer function of the linearized system is G(s;  )J(  ), with Eigenvalues are zeros of If one eigenvalue crosses over the imaginary axis, function h(,s;  ) has one root at =  1 Bifurcation Condition: from 0 =

Graphical Hopf Bifurcation Condition There is a unique eigenvalue which passes the critical point (-1 + i0) Determinant Transversality condition Stability index for the Hopf bifurcation Precise formulas are omitted here

Graphical Hopf Bifurcation Analysis 2 nd order harmonic balance approximation (Mees & Chua, IEEE Trans. CAS, 1979) Eigenlocus 4 th order harmonic balance (Mees, 1981)

Approximations of Periodic Solutions (I) 2 nd order SOLUTION:

Approximations of Periodic Solutions (II) From of 2 nd order approximation With q-order approximation of the periodic solution with obtained iteratively from

In the step q : N iterations Computational Algorithm

Computation of the Monodromy Matrix (I) It is necessary to integrate The original nonlinear system The variational equation This calculus is only possible if we have an analytical expression of the periodic solution Approximate the monodromy matrix M q The eigenvalues of M q are the approximated values of the characteristic multipliers

Computation of the Monodromy Matrix (II) Since one multiplier must be +1, this property allows to determine the precision of the approximation The multiplier can be used for If we are trying to find a cyclic bifurcation the difference in the value +1, gives the error in the determination of the bifurcation. If we are approximating the cycle, the variation of this eigenvalue to the theoretical value +1, gives an indication of the validity of the approximation.

Example 1: Genesio-Tesi System (1996)

Example 1: Genesio-Tesi System (I)

Example 1: Genesio-Tesi System (II) SISO realization Hopf bifurcation NO YES

Example 1: Genesio-Tesi System (III) Calculating the 2 nd order harmonic balance Since f(e 1 ) is quadratic Obtaining an analytic expression of the approximated orbit Iterative procedure

Example 1: Genesio-Tesi System (IV) 6 th order harmonic balance approximation  PD M3M M3M3 AUTO

Example 1: Genesio-Tesi System (V)

Double Hopf Bifurcation (I) Two pairs of complex conjugated eigenvalues of the linearized system cross the imaginary axis at two different frequencies Recent studies on voltage collapse in electrical power systems (Dobson et al., IEEE Trans. CAS, 2001) Nonlinear dynamics in high performance compressors for jets (Coller, Automatica, 2003)

Double Hopf Bifurcation (II) Recall: characteristic polynomial Using: generalized Nyquist stability criterion (MacFarlane & Postlethwaite, Int. Journal of Control, 1977) Obtained: Hopf bifurcation condition

Non-resonant Double Hopf Existence of non-resonant double Hopf bifurcation 1 : 1 resonance occurs when

Resonant Double Hopf Existence of resonant double Hopf bifurcation Whitney umbrella Using quasi-analytical approximations of limit cycles in the vicinity of Hopf bifurcation curves

Double Hopf Bifurcation: An Application Double LC resonant circuit (Yu, Nonlinear Dynamics, 2002) Nonlinear element

Double Hopf Bifurcation Diagram of the Hopf and Neimark-Sacker curves obtained from the frequency domain approach

Resonant double Hopf bifurcation: An application Modified Yu’s circuit by using a controlled current source (Itovich & Moiola, Nonlinear Dynamics, 2005) Nonlinear element

Whitney umbrella for the controlled system Hopf bifurcation curves of the modified Yu’s circuit

Other Bifurcation Curves “Cusp” point of limit cycles Neimark-Sacker bifurcation Hopf, Neimark-Sacker and folds of limit cycles

Neimark-Sacker curves in the modified Yu’s circuit * LOCBIF— frequency domain method

Fold curves in the modified Yu’s circuit * LOCBIF— frequency domain method

Continuation of Limit Cycles (I) * LOCBIF— frequency domain methodo XPP-AUTO

Continuation of Limit Cycles (II) * LOCBIF— frequency domain methodo XPP-AUTO

State variables Parameters Bifurcation parameters A Modified Circuit

Model Features The origin is the only equilibrium point Two pair of complex eigenvalues on the imaginary axis with frequencies Both frequencies are equal for

Bifurcation Analysis The analysis is centered on the double Hopf Organizing center Generates periodic and quasi-periodic orbits Useful to interpret interactions between oscillatory modes The analysis is performed varying η 1 (C 1 ) and η 2 (R) for fixed values of η 3 (current source) Four cases with dissimilar behavior (unfolding): η 3 = 0, η 3 = , η 3 = and η 3 =

Case a) η 3 = 0 Region 1 Cycle from H 1 Region 2 Stable equilibrium Unstable equilibrium Stable cycle Cycle from H 2 Region 3 Unstable cycle Unstable equilibrium Stable cycle 2D torus from H 2 cycle Region 4 Unstable 2D torus Stable cycle Unstable equilibrium Stable cycle 2D torus collapses with H 1 cycle Region 5 Stable cycle Unstable equilibrium Unstable cycle Collapse of H 1 cycle Región 6 Stable cycle Unstable equilibrium Collapse of H 2 cycle

Case b) η 3 = Region 1 Region 2 Region 3 Region 5 Region 6 Same as before 2D torus from H 1 cycle Region 7 Stable 2D torus Unstable cycle Unstable equilibrium Unstable cycle 2D torus collapses with H 2 cycle

Case c) η 3 = Same as before Region 1 Region 2 Region 3 Region 6 Region 7 Collapse of H 1 cycle Region 8 Stable 2D torus Unstable cycle Unstable equilibrium 2D torus collapses with H 2 cycle

Case d) η 3 = Region 9 Unstable equilibrium Cycle from H 2 Región 10 Unstable cycle Unstable equilibrium Cycle from H 1 Región 11 Unstable cycle Unstable equilibrium Unstable cycle 2D torus from H 1 cycle Region 12 Unstable 2D torus Unstable cycle Unstable equilibrium Unstable cycle Region 13 Stable 3D tours & Unstable 2D torus Unstable cycle Unstable equilibrium Unstable cycle Region 14 Stable 2D torus Unstable cycle Unstable equilibrium Unstable cycle 2D torus collapses with H 2 cycle Region 15 Collapse of H 2 cycle Region 16 Stable cycle Unstable equilibrium Unstable cycle Stable equilibrium Unstable cycle Collapse of H 1 cycle

Simulations: Case d) regions 12, 13 & 14 (η 2 = 1.88) Region 12: Unstable 2D torus Region 13: Stable 3D torus Region 14: Stable 2D torus

Simulations: Case d) region 15 (η 2 = 1.88) Region 15: Stable limit Cycle

Fold-flip Bifurcation Also, there are two fold-flip bifurcations for η 3 = Both are located over a tiny period doubling (or flip) bifurcation “bubble” close to the double Hopf point This is a co-dimension 2 bifurcation of periodic orbits Floquet multipliers at +1 (fold) and -1 (filp) Tangent intersection between a cyclic fold and a flip curve Rare in continuous systems Studied via numerical continuation

Cyclic Fold Bifurcation

Flip Bifurcation

Schematic Representation Two different unfoldings Continuations above and below the FF points Global phenomena make possible the coexistence of both FF points Double Hopf Flip “bubble” Cyclic Fold

Above FF 1 : η 2 = aa’

Below FF 1 : η 2 = bb’

Region 1 Stable cycle Unstable cycle Unstable PD cycle Normal Form of FF 1 Region 2 Stable cycle Unstable cycle Region 3 No cycles Region 4 Unstable cycle

Above FF 2 : η 2 = cc’

Below FF 2 : η 2 = dd’

Normal Form of FF 2 Region 5 Unstable PD cycle Region 6 Unstable cycle Unstable PD cycle Region 7 Unstable cycle Region 8 Unstable PD cycle Unstable cycle

Applications in Power Systems

Conclusions Frequency domain approach to bifurcation analysis Reducing dimension, so reducing the complexity of analysis Graphical and numerical, therefore bypassing some very sophisticated mathematical analysis Applicable to almost all kinds of bifurcation analysis Relatively heavy computations are generally needed May be combined with LOCBIF and XPP-AUTO Other applications Nonlinear signal distortion analysis in electric oscillators Bifurcation and chaos control

Thank you!