Chapter 10: Perturbation and Averaging Chapter 11

Nonlinear Systems and ControlLecture 5 (Meetings 17-19)

Chapter 10: Perturbation and AveragingChapter 11: Singular Perturbation

Eugenio Schuster

[email protected] Engineering and Mechanics

Lehigh University

Prof. Eugenio Schuster ME 450 - Nonlinear Systems and Control Spring 2022 1 / 40

Asymptotic Methods

Exact closed-form analytic solutions of nonlinear differential equations are possibleonly for special cases. Alternatives:

Numerical Methods

Asymptotic Methods

− Perturbation Method− Averaging Theory− Singular Perturbation

Both Averaging Theory and Singular Perturbation Theory exploit the multipletime-scale (“slow” dynamics vs. “fast” dynamics), revealed by asymptoticmethods, and inherent to many practical problems.


Perturbation Theory

Consider the systemx = f(t, x, ε) (1)

where f : [t0, t1]×D × [−ε0, ε0]→ Rn is “sufficiently smooth” in thesearguments over a domain D ⊂ Rn, with initial condition

x(t0, ε) = η(ε), (2)

where, for generality, the initial state is allowed to depend “smoothly” on ε.

The solutions of (1)-(2) depends on ε, a point that is emphasized by writingthe solution as x(t, ε).

The goal of the perturbation method is to exploit the “smallness” of ε toconstruct approximate solutions that will be valid for sufficiently small |ε|.Simplest approximation: “nominal” or “unperturbed” system, i.e., ε = 0,

x = f(t, x, 0), x(t0) = η(0) , η0 (3)

Suppose this “nominal” or “unperturbed” problem has a unique solutionx0(t) defined on [t0, t1] and x0(t) ∈ D for all t ∈ [t0, t1].


Perturbation Theory

Suppose further that f is continuous in (t, x, ε) and locally Lipschitz in (x, ε),uniformly in t and η is locally Lipschitz in ε for(t, x, ε) ∈ [t0, t1]×D × [−ε0, ε0].

From continuity of solutions with respect to initial states and parameters(Theorem 3.5) whe know that there is a positive constant ε1 < ε0 such thatfor all |ε| < ε1 (for sufficiently small |ε|), the problem (1)-(2) has a uniquesolution x(t, ε) defined on [t0, t1].

Approximating x(t, ε) by x0(t) can be justified by Theorem 3.4, which showsthat there is a positive constant k such that

‖x(t, ε)− x(t0)‖ ≤ k|ε|, ∀|ε| < ε1, ∀t ∈ [t0, t1]

When the approximation error satisfies this bound, we say that the error is oforder O(ε) and write x(t, ε)− x(t0) = O(ε).


Perturbation Theory

We assume that f and η have continuous partial derivatives up to order N + 1and N respectively, with respect to (x, ε) for (t, x, ε) ∈ [t0, t1]×D × [−ε0, ε0]. Toobtain a higher-order approximation of x(t, ε) we construct Taylor series

x(t, ε) =

x(t,0)︷︸︸︷x0(t) +

N−1∑k=1

xk(t)εk + εNxR(t, ε), (4)

xk(t) =1

2k−1∂kx(t, ε)

∂εk

∣∣∣∣ε=0

with initial condition

η(ε) = η(0) +

N−1∑k=1

ηkεk + εNηR(ε), (5)

ηk =1

2k−1∂kη(ε)

∂εk

∣∣∣∣ε=0


Perturbation Theory

Therefore (remember that x(t0, ε) = η(ε)),

x0(t0) = η(0), xk(t0) = ηk, k = 1, 2, . . . , N − 1 (6)

Substituting the Taylor series for x(t, ε) in (1) we obtain

x(t, ε) =

N−1∑k=0

xk(t)εk + εN xR(t, ε) = f(t, x(t, ε), ε) ≡ h(t, ε) (7)

h(t, ε) =

h(t,0)︷︸︸︷h0(t) +

N−1∑k=1

hk(t)εk + εNhR(t, ε) (8)

hk(t) =1

2k−1∂kh(t, ε)

∂εk

∣∣∣∣ε=0

Since the equation holds for all sufficiently small ε, it must hold as an identity inε, i.e., coefficients of like powers of ε must be equal.


Perturbation Theory

Matching the like powers of ε we obtain

ε0 : x0(t) = f(t, x0, 0), x0(t0) = η0 (9)

which, not surprisingly, is the unperturbed problem (3).

ε1 : x1(t) = A(t)x1 + g1(t, x0(t)), x1(t0) = η1

ε2 : x2(t) = A(t)x2 + g2(t, x0(t), x1(t)), x2(t0) = η2...

εk : xk(t) = A(t)xk + gk(t, x0(t), . . . , xk−1(t)), xk(t0) = ηk (10)

for k = 1, 2, . . . , N − 1, where A(t) is the Jacobian [∂f/∂x] evaluated atx = x0(t) and ε = 0, and the term gk(t, x0(t), . . . , xk−1(t)) is a polynomial inx1(t), . . . , xk−1(t) with coefficients depending continuously on t and x0(t).


Perturbation Theory

The zeroth-order term h0(t) is given by

h0(t) = f(t, x(t, 0), 0) = f(t, x0(t), 0)

Therefore, by matching coefficients of ε0, we obtain

x0 = f(t, x0, 0), x0(t0) = η0


Perturbation Theory

The first-order term h1(t) is given by

h1(t) =∂

∂εf(t, x(t, ε), ε)

∣∣∣∣ε=0

=

∂

∂xf(t, x(t, ε), ε)

∂x

∂ε(t, ε) +

∂

∂εf(t, x(t, ε), ε)

∣∣∣∣ε=0

=∂f

∂x(t, x0(t), 0)x1(t) +

∂f

∂ε(t, x0(t), 0)

Therefore, by matching coefficients of ε1, we obtain

x1 =∂f

∂x(t, x0, 0)x1 +

∂f

∂ε(t, x0, 0) , A(t)x1 + g1(t, x0), x1(t0) = η1

where A(t) , ∂f∂x (t, x0, 0) and g1(t, x0) , ∂f

∂ε (t, x0, 0).


Perturbation Theory

The process can be continued to derive equations for x2, x3, ...

This involves higher-order derivatives of f w.r.t. x, which makes itcumbersome.

There is no need for writing the equations in general form (equation for x2can be found in the book).

Once the idea is understood, equations can be obtained for specific problemof interest (see example).


Perturbation Theory

Example 10.2: Consider the Van der Pol state equation

x+ ε(x2 − 1)x+ x = 0


Perturbation Theory

Theorem 10.1: Suppose

f and its partial derivatives with respect to (x, ε) up to order N arecontinuous in (t, x, ε) for (t, x, ε) ∈ [t0, t1]×D × [−ε0, ε0].

η and its derivatives up to order N are continuous for ε ∈ [−ε0, ε0]

the nominal problem (3) has a unique solution x0(t) defined on [t0, t1] andx0(t) ∈ D for all t ∈ [t0, t1].

Then, there exists ε∗ > 0 such that for all |ε| < ε∗, the problem (1)-(2) has aunique solution x(t, ε) defined on [t0, t1], which satisfies

x(t, ε)−N−1∑k=0

xk(t)εk = O(εN ).


Perturbation Theory

Proof: We want to show that∑N−1k=0 xk(t)εk is indeed an O(εN ) approximation

of x(t, ε). Consider the approximation error

e = x−N−1∑k=0

xk(t)εk

Differentiating both sides of this equation with respect to t and substitutingfor the derivatives of x and xk from (1), (9) and (10), it can be shown that esatisfies the equation

e = A(t)e+ ρ1(t, e, ε) + ρ2(t, ε), e(t0) = εNηR(ε) (11)

where

ρ1(t, e, ε) = f(t, e+

N−1∑k=0

xk(t)εk, ε)− f(t,

N−1∑k=0

xk(t)εk, ε)−A(t)e

ρ2(t, ε) = f(t,

N−1∑k=0

xk(t)εk, ε)− f(t, x0(t), 0)−N−1∑k=1

[A(t)xk(t) + gk(·)]εk


Perturbation Theory

e = x− x0 −N−1∑k=1

xkεk

= f(t, x, ε)− f(t, x0(t), 0)−N−1∑k=1

[A(t)xk + gk(t, x0(t), . . . , xk−1(t))]εk

= f(t, e+

N−1∑k=0

xk(t)εk, ε)− f(t, x0(t), 0)−N−1∑k=1

[A(t)xk + gk(·)]εk

= A(t)e

−A(t)e+ f(t, e+

N−1∑k=0

xk(t)εk, ε)− f(t,

N−1∑k=0

xk(t)εk, ε)︸︷︷︸ρ1(t,e,ε)

+f(t,

N−1∑k=0

xk(t)εk, ε)− f(t, x0(t), 0)−N−1∑k=1

[A(t)xk + gk(·)]εk︸︷︷︸ρ2(t,ε)


Perturbation Theory

By assumption, x0(t) ∈ D for all t ∈ [t0, t1] and bounded. Hence, there existλ > 0 and ε1 > 0 such that for all ≤ l ≤ λ and |ε| ≤ ε1, the functions x0(t),∑N−1k=0 xk(t)εk , and e+

∑N−1k=0 xk(t)εk belong to a compact subset of D.

It can easily verified that

ρ1(t, 0, ε) = 0

‖ρ1(t, e2, ε)− ρ1(t, e1, ε)‖ ≤ k1‖e2 − e1‖‖ρ2(t, ε)‖ ≤ k2|ε|N

for all t ∈ [t0, t1], e1, e2 ∈ Bλ, ε ∈ [−ε1, ε1], for some constants k1, k2 > 0.

Equation (11) can be viewed as a perturbation of

e0 = A(t)e0 + ρ1(t, e0, ε), e0(t0) = 0, (12)

which has a unique solution e0(t, ε) ≡ 0 for t ∈ [t0, t1].

Application of Theorem 3.5 shows that (11) has a unique solution on [t0, t1]for sufficiently small |ε|.Application of Theorem 3.4 shows that

‖e(t, ε)‖ = ‖e(t, ε)− e0(t, ε)‖ = O(εN ).


Perturbation Theory

Theorem 10.1 can be extended to the infinite time interval [t0,∞) under someadditional stability conditions (exponential stability of the unperturbed system).

Theorem 10.2 Let D ⊂ Rn be a domain that contains the origin and suppose

f and its partial derivatives with respect to (x, ε) up to order N arecontinuous in (t, x, ε) and bounded for (t, x, ε) ∈ [t0,∞)×D0 × [−ε0, ε0] forevery compact set D0 ⊂ D. If N = 1, [∂f/∂x](t, x, ε) is Lipschitz in (x, ε),uniformly in t

η and its derivatives up to order N are continuous for ε ∈ [−ε0, ε0]

origin is an exponentially stable equilibrium point of the nominal system (3)

there is a Lyapunov function V (t, x) that satisfies the conditions of Theorem4.9 for the nominal system (3) for (t, x) ∈ [t0,∞)×D and W1(x) ≤ c is acompact subset of D.


Perturbation Theory

Then, for each compact set Ω ⊂ W2(x) ≤ ρc, 0 < ρ < 1, there exists ε∗ > 0such that for all t0 ≥ 0, η0 ⊂ Ω, |ε| < ε∗, the problem (1)-(2) has a uniquesolution x(t, ε), uniformly bounded on [t0,∞), which satisfies

x(t, ε)−N−1∑k=0

xk(t)εk = O(εN ),

where O(εN ) holds uniformly in t for all t ≥ t0.


Averaging Theory

The averaging method applies to a system of the form

x = εf(t, x, ε) (13)

where ε is a small positive parameter and f(t, x, ε) is T -periodic in t; that is

f(t+ T, x, ε) = f(t, x, ε) ∀(t, x, ε) ∈ [0,∞)×D × [0, ε0] (14)

for some domain D ∈ Rn.

The method approximates the solution of this system by the solution of an“averaged system,” obtained by averaging f(t, x, ε) at ε = 0.


Averaging Theory

x = εf(t, x, ε) (15)

The right hand side of (15) is multiplied by ε > 0. When ε is small, the solution xwill vary “slowly” with t relative to the periodic fluctuation of f(t, x, ε).

It is intuitively clear that if the response of the system is much slower than theexcitation, then such response will be determined predominantly by the average ofthe excitation, i.e.,

x = εfav(x) (16)

where

fav(x) =1

T

∫ T

0

f(τ, x, 0)dτ


Averaging Theory

Theorem 10.4: Let f(t, x, ε) and its partial derivatives with respect to (x, ε) upto second order be continuous and bounded for (t, x, ε) ∈ [t0,∞)×D0 × [0, ε0]for every compact set D0 ⊂ D, where D ⊂ Rn is a domain. Suppose f isT -periodic in t for some T > 0 and ε is a positive parameter. Let x(t, ε) andxav(εt) denotes the solutions of

x = εf(t, x, ε) (17)

x = εfav(x), fav(x) =1

T

∫ T

0

f(τ, x, 0)dτ (18)

If xav(εt) ∈ D ∀t ∈ [0, b/ε] and x(0, ε)− xav(0) = O(ε), then there existsε∗ > 0 such that for all 0 < ε < ε∗, x(t, ε) is defined and

x(t, ε)− xav(εt) = O(ε) for all t ∈ [0, b/ε]


Averaging Theory

Theorem 10.4 (cont’):

If the origin x = 0 ∈ D is an exponentially stable equilibrium point of theaveraged system (18), Ω ⊂ D is a compact subset of its region of attraction,xav(0) ∈ Ω, and x(0, ε)− xav(0) = O(ε), then there exists ε∗ > 0 such thatfor all 0 < ε < ε∗, x(t, ε) is defined and

x(t, ε)− xav(εt) = O(ε) for all t ∈ [0,∞]

If the origin x = 0 ∈ D is an exponentially stable equilibrium point of theaveraged system (18), then there exists ε∗ > 0 and k > 0 such that, for all0 < ε < ε∗, (17) has a unique, exponentially stable, T -periodic solutionx(t, ε) with the property

‖x(t, ε)‖ ≤ kε

NOTE: If f(t, 0, ε) = 0 for all (t, ε) ∈ [0,∞)× [0, ε0], the origin will be anequilibrium point of (17). By the uniqueness of the T -periodic solution x(t, ε), itfollows that x(t, ε) is the trivial solution x = 0. In this case, the theorem ensuresthat the origin is an exponentially stable equilibrium of (17).


Averaging Theory

Example: Weakly Nonlinear Second-order Oscillators

y + y = εy(1− y2)


Averaging Theory

The averaging method applies to system (17) in cases more general than the casewhen f(t, x, ε) is periodic in t. In particular, it applies when the function f(t, x, 0)has a well-defined average

fav(x) = limT→∞

1

T

∫ t+T

t

f(τ, x, 0)dτ (19)

and ∥∥∥∥∥ 1

T

∫ t+T

t

f(τ, x, 0)dτ − fav(x)

∥∥∥∥∥ ≤ kσ(T ) (20)

where k is a positive constant and σ, called the convergence function, is a strictlydecreasing, continuous, bounded function such that σ(T )→ 0 as T →∞.


Averaging Theory

Theorem 10.5: Similar to Theorem 10.4 except that the estimates O(ε) arereplaced by estimates O(α(ε)), where α is a class K.


Singular Perturbation Theory

Consider the “standard” singular perturbation model

x = f(t, x, z, ε) (21)

εz = g(t, x, z, ε) (22)

ε→ 0 implies a drastic change in the dynamics. Differential equation for zdegenerates into algebraic equation

0 = g(t, x, z, 0) (23)

Suppose this equation has at least one root

z = h(t, x) (24)

The reduced model (slow model) is then given by

x = f(t, x, h(t, x)) (25)



The discontinuity of solutions caused by singular perturbations can beavoided if analyzed in separate time scales.

The multi-scale approach is a fundamental characteristic of singularperturbation methods.

Example:

Gearbox (1:n)

Motor

Link

Torsional Springk: Elasticity Constant

1

3ml2θl +Blθl +

mgl

2sin θl + k

(θl +

1

nθm

)= 0

Jmθm +Bmθm +k

n

(θl +

1

nθm

)= τm



Denote z = k(θl + 1nθm), µ = 1/k. Note that µz = θl + 1

nθm.

Set Bl = Bm = 0 for simplicity.

1

3ml2θl +

mgl

2sin θl + z = 0

µz = −3g

2lsin θl −

3z

ml2− z

n2Jm+

τmnJm

Therefore,

1

3ml2θl +

mgl

2sin θl + z = 0

µz +3g

2lsin θl +

(3

ml2+

1

n2Jm

)z =

τmnJm



Quasi-steady state: µ→ 0

z =1

3ml2 + 1

n2Jm

(τmnJm

− 3g

2lsin θl

)Reduced model: Replace into θl equation. This is the “rigid” model.

1

3ml2θl +

mgl

2sin θl +

13ml2 + 1

n2Jm

(τmnJm

− 3g

2lsin θl

)= 0

1

3ml2θl +

(mgl

2−

3g2l

3ml2 + 1

n2Jm

)sin θl = − 1

3nJmml2 + 1

n

τm

1

3ml2θl +

(mgl

2n2Jm3ml2 + 1

n2Jm

)sin θl = − 1

3nJmml2 + 1

n

τm

1

3ml2θl +

(mgl2

3n2Jmml2 + 1

)sin θl = − 1

3nJmml2 + 1

n

τm



Consider the “standard” singular perturbation model

x = f(t, x, z, ε), x(t0) = ξ(ε), x ∈ Rnεz = g(t, x, z, ε), z(t0) = η(ε), z ∈ Rm (26)

Let x(t, ε) and z(t, ε) denote the solution of the full model.

Reduced Model (approximate slow response) + Quasi-Steady State:

˙x = f(t, x, h(t, x), 0), x(t0) = ξ(0) = ξ0 (27)

z = h(t, x(t)), z(t0) = h(t0, ξ0) (28)

Discrepancy (fast response):

z(t, ε)− h(t, x(t)) (29)



Lety = z − h(t, x) (30)

Then, we have

x = f(t, x, y + h(t, x), ε), x(t0) = ξ(ε)εy = g(t, x, y + h(t, x), ε)− ε∂h∂t− ε∂h∂xf(t, x, y + h(t, x), ε), y(t0) = η(ε)− h(t0, ξ(ε))

(31)

We rescale time:

τ =t− t0ε→ ε

dt=

1

dτ(32)

Then, we have (like a regular perturbation)

dydτ = g(t, x, y + h(t, x), ε)− ε∂h∂t− ε∂h∂xf(t, x, y + h(t, x), ε), y(0) = η(ε)− h(t0, ξ(ε))

(33)

Letting ε = 0 freezes t = t0 + ετ , and x(t0 + ετ, ε).



Boundary Layer Model (unperturbed model):

dy

dτ= g(t, x, y + h(t, x), 0) (34)

at each (t, x).

Q: Does y(τ)→ 0 for any t0 and x(t0, 0)?A: Need to study stability of equilibrium at y = 0 of Boundary Layer Model.



Theorem 11.1 (Tikhonov’s theorem): If

The “reduced” system x = f(t, x, h(t, x), 0) has a unique solution x(t)on [t0, t1],

The equilibrium point y = 0 of the “boundary-layer” systemdydτ = g(t, x, y + h(t, x), 0) is e.s. uniformly in (t, x),

then, ∃ε∗, µ > 0 such that for all ε ∈ (0, ε∗) and for all|z(t0, ε = 0)− h(t0, x(t0, ε = 0))| < µ, the full singular perturbation problem hasa solution x(t, ε) and z(t, ε) on [t0, t1], and

x(t, ε)− x(t) = O(ε)

z(t, ε)− h(t, x(t))− y(t/ε) = O(ε)

hold uniformly for t ∈ [t0, t1], where y(τ) is the solution of the boundary layermodel. Moreover, for all tb > t0, ∃ε∗∗ < ε∗ s.t.

|z(t, ε)− h(t, x(t))| = O(ε), ∀t ∈ [tb, t1],∀ε < ε∗∗



Theorem 11.2: If the origin of the “reduced” system x = f(t, x, h(t, x), 0) is e.s.and the origin of the “boundary-layer” system dy

dτ = g(t, x, y + h(t, x), 0) is e.s.uniformly in (t, x), then the conclusions of Theorem 11.1 hold for t1 =∞whenever |x(t0, ε = 0)| is sufficiently small.



Example:

x = −x+ z

εz = tan−1(1− z − x)



Stability Analysis (autonomous singularly perturbed system):

x = f(x, z), f(0, 0) = 0 (35)

εz = g(x, z), g(0, 0) = 0 (36)

Therefore, the origin is an isolated equilibrium. Let z = h(x) be an isolated rootof 0 = g(x, z). Let us shift equilibrium of boundary layer model to the origin byusing y = z − h(x), h(0) = 0 (because g(0, 0) = 0),

x = f(x, y + h(x)) (37)

εy = g(x, y + h(x))− ε∂h∂xf(x, y + h(x)) (38)

Assuming that ‖h(x)‖ ≤ ζ(‖x‖), where ζ is class K, the map y = z − h(x) isstability preserving, i.e. the origin of (35)-(36) is a.s. if and only if the origin of(37)-(38) is a.s.



The reduced system

x = f(x, h(x)) (39)

has an equilibrium at x = 0 and the boundary-layer system

dy

dτ= g(x, y + h(x)) (40)

with τ = t/ε (x ∼ fixed parameter) has equilibrium at y = 0.

We assume that for each subsystem, the origin is a.s. and we have Lyapunovfunctions that satisfy the conditions of Lyapunov’s theorem. In the case of theboundary layer system we require a.s. of the origin to hold uniformly in the frozenparameter x. Study stability of (x, y) = 0 via a composite Lyapunov form basedon Lyapunov functions for x, y guaranteed from converse Lyapunov theorems.



Theorem 11.3: Consider the singular perturbed system (37)-(38). Assume thereare Lyapunov functions V (x) and W (x, y) (W1(y) ≤W (x, y) ≤W2(y)) thatsatisfy

∂V

∂xf(x, h(x)) ≤ −α1ψ

21(x) (41)

∂W

∂yg(x, y + h(x)) ≤ −α2ψ

22(y) (42)

∂V

∂x[f(x, y + h(x))− f(x, h(x))] ≤ β1ψ1(x)ψ2(y) (43)[

∂W

∂x− ∂W

∂y

∂h

∂x

]f(x, y + h(x)) ≤ β2ψ1(x)ψ2(y) + γψ2

2(y) (44)

where ψ1 and ψ2 are p.d.f. and α1, α2, β1, β2, γ ≥ 0.



Theorem 11.3 (cont’d): Let

εd ,α1α2

α1γ + 14d(1−d) [(1− d)β1 + dβ2]2

(45)

andε∗ ,

α1α2

α1γ + β1β2(46)

The, the origin of the singular perturbed system (37)-(38) is a.s. for all0 < ε < ε∗. Moreover

ν(x, y) = (1− d)V (x) + dW (x, y), 0 < d < 1 (47)

is a Lyapunov function for ε ∈ (0, εd).



Example:

x = −x+ z

εz = − tan−1(x+ z)



Theorem 11.4:

x = f(t, x, z, ε) f(t, 0, 0, ε) = 0

εz = g(t, x, z, ε) g(t, 0, 0, ε) = 0, g(t, x, h(t, x), 0) = 0, h(t, 0) = 0

If the origin of the “reduced” system x = f(t, x, h(t, x), 0) is e.s. and the origin ofthe “boundary-layer” system dy

dτ = g(t, x, y + h(t, x), 0) is e.s. uniformly in (t, x),then ∃ε∗ such that for all ε < ε∗ the origin of the full system is e.s.


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Chapter 10: Perturbation and Averaging Chapter 11