The Frequency Response of Mathieu's Equation
Karthik Chikmagalur
Table of Contents
Motivation
Time-periodic dynamical systems
Three related ideas
Parametric Excitation
Excitation of oscillations through parametric forcing
Parametric Resonance
Destabilization of system response through parametric forcing
Vibrational Stabilization
Parametric stabilization of nominally unstable system behavior
Microcantilever based mass sensing
I¨θ+c˙θ+kθ=M(t,θ)
MEMS devices used as capacitive sensors can operate at high frequencies, with high sensitivities and low power requirements.
Can utilize parametric excitation to decouple sensitivity from bandwidth (Q factor)
Capacitive sensing is also sensitive to parasitic signals from the actuating drive signal.
Parametric Excitation
Can achieve very high Q factors not currently attainable with present resonance mode AFM techniques.
Control of Stirling engines
The simplest dynamical model of a Stirling engine is the Schmidt model:
¨x+c˙x+x−y1+x−y=0
Which exhibits parametric resonance for certain periodic displacer motions y(t).
When linearized about x=0, with y(t)=ϵcos(ωt),
¨x+c˙x+(1+2ϵcos(ωt))x=0
Parametric Resonance
Instability in the system as a result of parametric variation
Vibrational stabilization
A rigid pendulum where the pivot point oscillates in the vertical direction.
¨θ+(−Al¨Y(t)−gl)sin(θ)=0
Vibrational Stabilization
The Kapitza pendulum can be vibrationally stabilized with the right pivot motion AY(t).
Parametric Resonance in Mechanics
Linear stability of Faraday waves when a container is oscillated vertically [benjamin1954]
Kelvin-Helmholtz instability with time-periodic shear [kelly1965]
Vibrations of columns subjected to oscillating axial loads [iwatsubo1974]
Parametric Resonance in Fluid Mechanics: More examples
Acoustic instabilities in flames [sammarco1997] and in Rayleigh-Bénard convection [vittori1998]
Stability of Barotropic and Baroclinic shear flows [poulin2003] and [pedlosky2003]
Modelling time-periodic systems
Mathieu's Equation
The canonical example of a periodic linear system is Mathieu's Equation,
¨x(t)+(±ω2+ϵcost)x(t)=0
+ω2: Is a harmonic oscillator with a periodically time-varying spring constant
−ω2: Exponentially unstable in the absence of parametric forcing, corresponds to a system like the Kapitza pendulum.
Mathieu's Equation with viscous damping
¨x(t)+2νω˙x(t)+(±ω2+ϵcost)x(t)=0
Can be transformed through a change of coordinates ˉx=x(t)eνωt to Mathieu's Equation
The Hill ODE
Mathieu's Equation is a special case of the more general Hill ODE
¨x(t)+f(t)x(t)=0, f(t+T)=f(t)
where f has a single harmonic.
x(t)=Φ(t,t0)x(t0)
The Hill ODE is formally solved using Floquet theory: Its stability is determined by the eigenvalues of the Monodromy map λ[Φ(T,0)], where Φ(t,t0) is the state transition matrix.
The Hill ODE is measure preserving, ddtdetΦ(t,s)=0
Measure preserving
Mathieu's Equation: Stability
¨x(t)+(±ω2+ϵcost)x(t)=0
Parametric Amplification or Resonance
+ω2: A nominally stable (ϵ=0) harmonic oscillator can be destabilized with parametric forcing of very low amplitude if the forcing frequency is chosen carefully.
Vibrational Stabilization
−ω2: The right combination of ϵ and forcing frequency can stabilize the system.
Vibrational stabilization - Averaging
¨x(t)+(−ω2+ϵcost)x(t)=0
Averaging methods relate the stability of a time varying system ˙x=f(t/α,x) to that of a time-invariant averaged system ˙x=F(x) as α→0.
For Mathieu's Equation:
For ϵ2>2ω2, ∃ ω⋆ such that the origin is (Lyapunov) stable for ω<ω⋆
Questions of Interest
Given a viable range of parameters (ω,ϵ), how do they differ in performance?
How sensitive are different possible operating points of a parametric amplifier to noise?
How robust is the vibrationally stabilized region of parameter space to disturbances?
Addressing these questions requires understanding the input-output characteristics of the system
Impulse response at different operating points
¨x(t)+(±ω2+ϵcost)x(t)=δ(t)
Input-Output characteristics
"Transfer function" analysis: Let ˉω=2πT. Then
Lifting
Lifting allows us to represent a time-periodic system as a time invariant one in a space with higher dimensional input and output spaces
Representing periodic systems as LTI
CONTINUOUS TIME, PERIODIC
where x(t)∈Rn, u(t)∈Rp, y(t)∈Rm.
DISCRETE TIME, LTI
where ˆxk∈Rn, ˆuk∈Lp2[0,T], ˆvk∈Lm2[0,T].
Lifting of Signals
f∈LpN,e[0,∞), 1≤p<∞: Extended space of continuous time N -vector signals
ˆf∈lpLp[0,T]: Space of sequences which take values in Lp[0,T].
The lifting operator WT:Lp[0,∞)→lpLp[0,T], such that
ˆf=WTf, ˆfi(t)=f(Ti+t), 0≤t≤T
WT breaks up the signal f into an infinite number of pieces, each of which is a shifted copy of f restricted to an interval [0,T].
Lifting of Systems
Let DT and S be delay and shift operators:
Then WTDT=SWT
For any linear operator G:Lp[0,∞)→Lp[0,∞), define its lifting as ˆG:=WTGW−1T, ˆG:lpLp[0,T]→lpLp[0,T]
If G is T -periodic, then it commutes with the delay operator DT, GDT=DTG
Then ˆGS=SˆG, so ˆG is shift-invariant (LTI)
Representing periodic systems as LTI
CONTINUOUS TIME, PERIODIC
where x(t)∈Rn, u(t)∈Rp, y(t)∈Rm.
DISCRETE TIME, LTI
where ˆxk∈Rn, ˆuk∈Lp2[0,T], ˆvk∈Lm2[0,T].
Solution in terms of ˆG
ˆG is shift-invariant (in k) and so has a simple solution: ˆyk=k−1∑j=0ˆCˆAk−j−1ˆBˆwj+ˆDˆwk
ˆG has the semi-infinite Toeplitz representation ˆG≡[ˆD000…ˆCˆBˆD00…ˆCˆAˆBˆCˆBˆD0…ˆCˆA2ˆBˆCˆAˆBˆCˆBˆD…⋮⋱⋱⋱⋱]
Relating the Lifting operator to the Harmonic transfer operator
Numerical Solution through Discretization and Lifting
Discretizing the Hill ODE
Discretized symplectically with N steps over period T, then lifted
Method details
First order Euler backward/forward for the two components:
xn=x(nΔ),fn=f(nΔ)
Symplectic:
ϕTk(01−10)ϕk=(01−10)
Poles of the lifted system
Found as the eigenvalues of the Monodromy map ˆA=Φ(T,0)
The Eigenvalues are constrained to the unit circle (when the system is stable) or the real axis (when unstable).
Since Φ(0,0)=I, detΦ(t,0)=1∀t>0
λ1=1λ2 and
λ⋆1=λ2 or λ1,λ2∈R
Eigenvalues of the Monodromy map
The argument of the eigenvalues is always 0 or π at the stability boundaries. In the stable regions the poles are e±jˉω, 0≤ˉω≤π.
Singular values of the lifted system
The lifted system has a single mode that grows at resonance. This suggests that its frequency response close to resonance is similar to that of a second order system.
Characterizing the free response
Transfer function of ˆG:
ˆG(z)=ˆC(zI−ˆA)−1ˆB⏟rank 2+ˆD⏟∞ rank but finite and bounded
Near resonance, (zI−ˆA)−1 is near-singular and outstrips ˆD in magnitude, so it depends on z through factors (z−e±jˉω)−1:
ˆY(z)≈cˆu1(ejˉω)z−ejˉω+c⋆ˆu1(e−jˉω)z−e−jˉω
ˆu1(ejˉω): First left singular vector at resonance
Characterizing the free response
y(t)=p(t)exp(jˉω⌊tT⌋), p(t+T)=p(t)
If ˉω is rational: periodic function, otherwise "almost periodic"
y(t)WT→ˆyk=pejˉωk
ˆyl=pejˉω(l−1)⇔ˆY(z)=pz−ejˉω
ˆY(z)≈cˆu1(ejˉω)z−ejˉω+c⋆ˆu1(e−jˉω)z−e−jˉω⇔y(t)≈ℜ[c ˆu1,t(ejˉω) exp{jˉω(⌊tT⌋−1)}]
The free response of Mathieu's Equation can be characterized as the product of copies of a "constant" function ˆu1(ejˉω) and an "almost periodic" complex exponential.
Free response vs first LSV of ˆG
Impulse response (full numerical solution)
Low rank approximation (using first LSV)
The H2 norm of Mathieu's Equation
The H2 norm is a measure of the system's input-amplification ‖G‖2H2:=Tr(∫∞0G(t)G⋆(t)dt) Square average of the norms of the responses to a set of unit inputs that excite all "parts" of the system.
The H2 norm of Mathieu's Equation
The H2 norm is a measure of the system's input-amplification ‖G‖2H2:=Tr(∫∞0G(t)G⋆(t)dt) Square average of the norms of the responses to a set of unit inputs that excite all "parts" of the system.
When G is T -periodic,
‖G‖2H2:=1T∫T0Tr(∫∞0G(t,s)G⋆(t,s)ds)dt
Collects contributions from impulses applied at all times 0≤t≤T.
Under the lifting WT, G→ˆG,
The H2 norm - Stochastic interpretation
W=∑∞k=1ˆAk−1ˆBˆB⋆ˆA⋆(k−1)⇔ˆAWˆA⋆−W=−ˆBˆB⋆
Alternatively, the H2 norm is computed as the trace of the steady state error covariance when the system is fed zero-mean stationary white noise:
limt→∞1T∫t+TtTr{E[y(τ)y⋆(τ)]}dτ
Stochastic interpretation - details
The H2 norm of Mathieu's Equation
Conclusions
The input-output description of Mathieu's Equation is amenable to a low-rank approximation in the vibrationally stabilized region.
The free response of Mathieu's Equation can be approximated by the product of a periodic and an almost periodic function.
The H2 norm of Mathieu's Equation (external input to state) has a minimum in the vibrationally stabilized region.
In the stable region, the H2 norm achieves a minimum at some ϵ≠0 for each fixed ω2.
Conclusions
Work submitted to The American Control Conference, 2020:
- Analysis of Mathieu's Equation through Discretization and Lifting
- Poles of the system and singular values
- Free response characterization
- The H2 norm of Mathieu's equation
Other approaches:
- Asymptotic analyses of Mathieu's equation, the eigenvalues of Φ(T,0) and the H2 norm
- Estimates of the H2 norm through full numerical simulation of the impulse response
Ongoing and future work
Optimal choice of operating regime
The H2 norm is one measure of noise amplification.
- What are the properties of the minima?
- Can the result be generalized?
How robust is Vibrational control? (H∞ norm analysis)
Part I: Dynamics known to be time-periodic
Noise in parametrically forced systems
Study phase noise in parametric oscillators
w(t)
E[w(t)]=ˉw
E[w(t)w(t+τ)]=Rw(τ)
G⟶
x(t)
E[x(t)]=ˉx(t) where x(t+T)=x(t)
E[x(t)x⋆(t+τ)]=Cx(t,τ) where Cx(t+T,τ)=Cx(t,τ)
Periodic systems produce cyclostationary output when the input is stationary.
Studying cyclostationarity
x(t)WT⟶ˆxkˉx(t):=E[x(t)]WT⟶ˉˆx:=E[ˆxk]Cx(t,τ):=E[x(t)x⋆(t+τ)]WT⟶Rˆx(l):=E[ˆxkˆx⋆k+l]Sx(α,ω):=∫T0∫∞τ=−∞Cx(t,τ)e−jωτe−jαtdτdt⏟CyclostationaryWT⟶Pˆx(θ):=∑lRˆx(l)e−lθ⏟Stationary
Part II: Nature of dynamics unknown
System identification from output
Is the output cyclostationary?
Distinguish between LTI and Time-Periodic dynamics from data
System identification of time-periodic systems
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