Antiresonance in Switched Systems with Only Unstable Modes
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Antiresonance in switched systems with only unstable modes Maurizio Porri∗ Department of Mechanical and Aerospace Engineering and Department of Biomedical Engineering, New York University, Tandon School of Engineering, Brooklyn, New York 11201, USA Russell Jeter and Igor Belykhy Department of Mathematics and Statistics, Georgia State University, P.O. Box 4110, Atlanta, Georgia, 30302-410, USA (Dated: October 10, 2020) Antiresonance is a key property of dynamical systems that leads to the suppression of oscillations at select frequencies. We present the surprising example of a switched system that alternates between unstable modes, but exhibits antiresonance response for a wide range of switching frequencies. Through mathematical analysis, we elucidate the stabilization mechanism and characterize the range of antiresonant frequencies for periodic and stochastic switching. The demonstration of this new physical phenomenon opens the door for a new paradigm in the study and design of switched systems. PACS numbers: 05.45.-a, 46.40.Ff, 02.50.Ey, 45.30.+s Introduction. Switched dynamics are pervasive in the- this question by oering the rst example of a switched oretical physics, neuroscience, and engineering [1]. For system composed of unstable modes that displays a stable example, the temporal patterning of interactions within response in a nite range of switching frequencies. Our active matter systems discontinuously evolves in time system has an unstable average that hinders stability in as their comprising units change their spatial organiza- the fast-switching limit, and the instability of both of the tion [25]. Likewise, synchronization in brain networks modes hampers stability for slow-switching frequencies. emerges from sporadic, on-o synaptic interactions be- Governing equations. We consider a two-dimensional tween spiking and bursting neurons [6]. Just as the func- nonlinear switched system with state vector z(t) = tion of several natural systems is based on switched dy- (x(t); y(t)), alternating in time between two possible namics, so too switch logic is ubiquitous in the design of modes, namely, engineering systems, such as converters and communica- tion networks [7, 8]. z_(t) = Fs(t)(z) (1) Stability is arguably the most fundamental property where the binary signal s(t) switches between 0 and 1 to of these systems [9, 10]. Over the last two decades, a select either of the nonlinear functions F0 or F1. Initially, number of analytical studies have documented a strong we focus on a periodic signal of period T with a duty cycle sensitivity of stability to variations in the switching fre- δ, such that the switch is on for δT units of time and is quency [1115], which has also been registered in applica- o for (1 − δ)T units of time; later, we explore stochastic tions to complex systems [1621]. Switched systems that switching. In order to achieve the desired response, we alternate between stable modes can display an unstable design the modes as behavior at selected switching frequencies [22, 23]. This phenomenon can be viewed as an internal resonance [24], 8 x2y > x_ = x − 2c wherein the switching frequency does not allow the dy- <> 2 2 s = 0 : x + y ; namics in each of the stable modes to decay before the x3 − xy2 > y_ = y + c :> 2 2 onset of a new switch. x + y (2) 8 3 2 Heraclitus' theory of the unity of opposites has sel- y − x y > x_ = x − c <> 2 2 dom defeated our intuition of physical processes, from the s = 1 : x + y ; xy2 particle-wave duality in quantum mechanics [25] to the > y_ = y + 2c matter/anti-matter asymmetry problem [26]. Antireso- : x2 + y2 nance continues to be extensively investigated in acous- where c is a positive constant that determines the cou- tics [27], optics [28], biology [29], and quantum physics pling between the two state components; when possible, [30], with a focus on smooth, stable systems. On the we omit the dependence on time. other hand, antiresonance of switched systems with un- For c = 0, each of the modes in (2) reduces to an un- stable modes has never been documented. stable node with decoupled state components. As c in- Is it possible to induce antiresonance in the form of sta- creases, the two components become nonlinearly coupled, ble response of a system switching between two unstable forming a degenerate xed point, shown in Fig. 1. The modes? In this Letter, we provide a positive answer to lack of dierentiability of the vector eld at the origin 2 2 1.0 1.0 1.0 0.5 0.5 1 0.5 y 0.0 y 0.0 y y 0 0.0 -0.5 -0.5 -0.5 -1 -1.0 -1.0 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -2 -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 x x x x FIG. 1. Streamlines of the two unstable modes of the switched FIG. 2. Trajectories of the periodically-switched system in (1) system given in (2) for , with color indicating the inten- c = 4 for c = 4 and δ = 0:5 and dierent values of T (left): T = 0:1 sity of the vector eld for (left) and (right). s = 0 s = 1 (solid red), T = 2 (green), and T = 5 (dashed red). Initial condition is (1; 0) for all cases. Streamlines of the average system in (1) for c = 4 and δ = 0:5, with color indicating the does not allow one to characterize the xed point follow- intensity of the vector eld (right). ing standard practice. Although unstable, the xed point has features of an unstable node and a stable focus. A the nonlinear interplay between the intrinsic dynamics of typical trajectory will rst rotate about the origin, reduc- each mode at intermediate switching frequencies. ing its distance from it like a stable focus, and then it will Antiresonance at intermediate switching. Surprisingly, escape to innity along the principal unstable direction there could be a wide range of intermediate switching like an unstable node. The principal unstable directions periods that induce the stability of the switched system are the - and -axis for the rst ( ) and second y x s = 0 so that the origin becomes a stable focus, as shown in ( ) modes, respectively. The interplay between these s = 1 Fig. 2. The stabilization mechanism relies on switching contrasting features is the key for the antiresonance of slow enough to be attracted toward the origin during the the switched system. rotation, but fast enough to avoid reaching the unstable Slow- and fast-switching limits. In the limit case T 1 principal directions. For and , numerical the time-scale for each mode the dynamics of the c = 4 δ = 0:5 simulations suggest that the range of that guarantees switched system is unstable, as it would spend a large T convergence toward the origin is The fraction of time in one of the unstable modes before it T 2 (1:18; 4:26): emergence of this antiresonant response is completely could re-switch to the other [12]. After spiraling toward open-loop, whereby switching takes place without any the origin for almost an entire quadrant, each trajectory knowledge of the present state of the system. In the fol- will approach the principal unstable direction of the mode lowing, we provide a mathematical proof for the existence and travel away from the origin. Switching will cause the of this antiresonance window. trajectory to experience a sudden turn and after each pe- Mathematical analysis of stability. System (1) with riod, the distance from the origin will increase, as demon- modes (2) can be conveniently written in polar coordi- strated in Fig. 2. nates, by dening x = r cos θ and y = r sin θ, so that An unstable response is also registered for the fast- switching case, in which T 1, as shown in Fig. 2. This r_ = r + cr (2s − 1) sin θ cos θ; (4a) result can be explained by examining the average system [13], which is obtained by averaging (2) θ_ = c (1 − s) cos2 θ + cs sin2 θ: (4b) (3δ − 2)x2y − δy3 This representation indicates that the nonlinearity is x_ = x + c ; limited to the angular coordinate and that the system x2 + y2 (3) (3δ − 1)xy2 + (1 − δ)x3 has a triangular structure, with the angular coordinate y_ = y + c : evolving independent of the radial one. This structure x2 + y2 is amenable to a closed-form solution, which we present This system has an unstable focus, without any principal after examining stability under periodic switching to gain unstable direction. Except for the origin, all the trajec- insight into more complex switching processes. tories spiral out to innity. To study the stability of the periodically-switched sys- Although often used in the analytical treatment of tem, we perform the change of variable ρ = 1=r, so that switched systems [12, 13], the slow- and fast-switching (4a) becomes limits oer a close representation of the switched system ρ_ = −ρ − cρ (2s − 1) sin θ cos θ: (5) when the switching period is much larger and smaller than the time-scale of the individual modes. Predict- This change of variables maps the origin to innity and ing the onset of an antiresonance requires the study of vice-versa, thereby turning the study of antiresonance 3 into the study of resonance. Most importantly, a Carte- 1 sian representation of the transformed system, through 0 ) λ , yields the following linear switched ( ζ = (ρ cos θ; ρ sin θ) -1 system: Re -2 −1 −s(t)c -3 ζ_(t) = A(t)ζ(t); with A(t) = ; (6) 0 1 2 3 4 5 6 7 (1 − s(t))c −1 T System (6) periodically switches between two linear sys- 1 tems determined by two Hurwitz matrices, A when the 0 0 ) λ switch is o (s = 0) and A1 when the switch is on (s = 1), ( -1 such that Re -2 −1 0 −1 −c -3 A = ; and A = : (7) 0 1 2 3 4 5 6 7 0 c −1 1 0 −1 T These matrices have repeated eigenvalues λ = −1, which 1 yield a stable degenerate node in both modes.