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### Transcript of High-Gain Observers in Nonlinear Feedback Control Lecture ... Lecture # 4 Adaptive Control High-Gain

• High-Gain Observers in

Nonlinear Feedback Control

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 1/61

• Problem Formulation

Consider a SISO nonlinear system represented globally by the nth-order differential equation

y(n) = f0(·) + p ∑

i=1

fi(·)θi + (

g0 +

p ∑

i=1

giθi

)

u(m)

where u is the control input, y is the measured output, y(i)

denotes the ith derivative of y, and m < n. The functions fi are known smooth nonlinearities which could depend on y, y(1), · · ·, y(n−1), u, u(1), · · ·, u(m−1), e.g.,

f0(·) = f0(y, y(1), · · · , y(n−1), u, u(1), · · · , u(m−1))

The constant parameters g0 to gp are known, while the constant parameters θ1 to θp are unknown

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 2/61

• The vector θ = [θ1, · · · , θp]T belongs to Ω, a known compact convex subset of Rp. Let Ω̂ be a convex subset of Rp which contains Ω in its interior. We assume that

(g0 + θ T g) 6= 0 ∀ θ ∈ Ω̂

The objective is to design an adaptive output feedback controller which guarantees boundedness of all variables of the closed-loop system, and tracking of a given reference signal yr(t), where

YR(t) = [yr(t), y(1)r (t), · · · , y(n−1)r (t), y(n)r (t)]T

is bounded and piecewise continuous

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 3/61

• Y(t) = [y(t), y(1)(t), · · · , y(n−1)(t)]T

Yr(t) = [yr(t), y(1)r (t), · · · , y(n−1)r (t)]T

Y ⊂ Rn and YR ⊂ Rn+1 are given compact sets

Objective: Design an adaptive output feedback controller such that for all Y(0) ∈ Y , for all YR(t) ∈ YR, and for all θ ∈ Ω, all variables of the closed-loop system are bounded for all t ≥ 0, and

lim t→∞

|y(t) − yr(t)| = 0

The compact sets Y , YR, and Ω are arbitrary but known

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 4/61

• Augment a series of m integrators at the input side of the system and treat v = u(m) as the input of the augmented system

z1 = u, z2 = u (1), . . . zm = u

(m−1)

x1 = y, x2 = y (1), . . . xn = y

(n−1)

State-space model of the augmented system:

ẋi = xi+1, 1 ≤ i ≤ n− 1 ẋn = f0(x, z) + θ

T f(x, z) + (g0 + θ T g)v

żi = zi+1, 1 ≤ i ≤ m− 1 żm = v

y = x1

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 5/61

• The initial states of the integrators are chosen such that z(0) ∈ Z0, a compact subset of Rm

The augmented state model has relative degree n and can be transformed into a globally-defined normal form by the change of variables

ζi = zi − xn−m+i g0 + θT g

, 1 ≤ i ≤ m

which transforms the ż-equations into

ζ̇i = ζi+1, 1 ≤ i ≤ m− 1

ζ̇m = − f0(x,z)+θ T f(x,z)

g0+θT g

∣ ∣ ∣ zi=ζi+xn−m+i/(g0+θT g)

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 6/61

• Minimum Phase Assumption: For every θ ∈ Ω, the system has the the property that for any z(0) ∈ Z0 and any bounded x(t), the state ζ(t) is bounded

Remarks:

The restriction of the coefficient (g0 + gT θ) to be constant is made for convenience. The result can be extended to the case when g0 and g are functions of y, y(1), · · ·, y(n−1), u, u(1), · · ·, u(m−1), provided |g0 + gT θ| is globally bounded from below

The linear dependence on the unknown parameters θ is crucial for the derivation of the adaptive controller and may require redefinition of the physical parameters of the system

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 7/61

• Example: A single link manipulator with flexible joints and negligible damping can be represented by

Iq̈1 +MgL sin q1 + k(q1 − q2) = 0 Jq̈2 − k(q1 − q2) = u

q1, q2 are angular positions, u is the torque input, and the physical parameters g, I, J , k, L, and M are all positive y = q1 satisfies the 4th-order differential equation

y(4) = gLM I

(ẏ2 sin y−ÿ cos y)− (

k

I + k

J

) ÿ−gkLM

IJ sin y+ k

IJ u

θ1 = gLM

I , θ2 =

( k

I + k

J

) , θ3 =

gkLM

IJ , θ4 =

k

IJ

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 8/61

• State Feedback Controller

e1 = y − yr = x1 − yr e2 = ẏ − ẏr = x2 − ẏr ...

en = y (n−1) − y(n−1)r = xn − y(n−1)r

ė = Ae+ b{f0(e+ Yr, z) + θT f(e+ Yr, z) + (g0 + θ

T g)v − y(n)r } ż = A2z + b2v

(A, b) and (A2, b2) are controllable canonical pairs that represent chains of n and m integrators, respectively

Choose a matrix K such that Am = A− bK is Hurwitz

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 9/61

• ė = Ame+ b{Ke+ f0(e+ Yr, z) + θT f(e+ Yr, z) + (g0 + θ

T g)v − y(n)r }

PAm +A T mP = −Q, Q = QT > 0

V = eTPe+ 1 2 θ̃TΓ−1θ̃, Γ = ΓT > 0, θ̃ = θ̂ − θ

V̇ = −eTQe+ 2eTPb[f0 + θT f + (g0 + θT g)v +Ke− y(n)r ] + θ̃TΓ−1 ˙̂θ

v = −Ke+ y(n)r − f0(e+ Yr, z) − θ̂T f(e+ Yr, z)

g0 + θ̂T g ︸ ︷︷ ︸

ψ(e,z,YR ,θ̂)

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 10/61

• 2eTPb[f(e+ Yr, z) + gψ(e, z,YR, θ̂)] = φ(e, z,YR, θ̂)

V̇ = −eTQe+ θ̃TΓ−1[ ˙̂θ − Γφ] ˙̂ θ = Γφ ⇒ V̇ = −eTQe

With parameter projection, we can achieve

θ̃TΓ−1[ ˙̂θ − Γφ] ≤ 0 ⇒ V̇ ≤ −eTQe

while keeping θ̂(t) ∈ Ω̂ for all t ≥ 0

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 11/61

• Example of Parameter Projection:

Ω = {θ | ai ≤ θi ≤ bi}, 1 ≤ i ≤ p}

Ωδ = {θ | ai − δ ≤ θi ≤ bi + δ}, 1 ≤ i ≤ p} ⊂ Ω̂ Let Γ be a positive diagonal matrix. The adaptive law

˙̂ θ = Proj(θ̂, φ)

where Proj(θ̂, φ) is defined on the next page, is locally Lipschitz in (θ̂, φ) and ensures

θ̂(0) ∈ Ω ⇒ θ̂(t) ∈ Ωδ, ∀ t ≥ 0

θ̃TΓ−1[ ˙̂θ − Γφ] ≤ 0

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 12/61

• [Proj(θ̂, φ)]i is defined by 





γiiφi, if ai ≤ θ̂i ≤ bi or if θ̂i > bi and φi ≤ 0 or if θ̂i < ai and φi ≥ 0

γii

[

1 + bi−θ̂i δ

]

φi, if θ̂i > bi and φi > 0

γii

[

1 + θ̂i−aiδ

]

φi, if θ̂i < ai and φi < 0

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 13/61

• V̇ ≤ −eTQe ⇒ V is bounded ⇒ (e, θ̃) are bounded

⇒ (x, z, θ̂) are bounded

Boundedness of θ̂ follows also from θ̂ ∈ Ωδ. With all signals bounded, we conclude from V̇ ≤ −eTQe that

e(t) → 0 as t → ∞

In preparation for output feedback, we saturate ψ and φ outside the compact set of interest. Let E0 and Z0 be compact sets such that e(0) ∈ E0, and z(0) ∈ Z0

c1 = max e∈E0

eTPe, and c2 = max θ∈Ω,θ̂∈Ωδ

1 2(θ̂− θ)

TΓ−1(θ̂− θ)

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 14/61

• c3 > c1 + c2 ⇒ e(t) ∈ E def= {eTPe ≤ c3}, ∀ t ≥ 0

Find a compact set Z such that

z(0) ∈ Z0 and e(t) ∈ E ∀ t ≥ 0 ⇒ z(t) ∈ Z ∀ t ≥ 0

S ≥ max |ψ(e, z,YR, θ̂)|, Si ≥ max |φi(e, z,YR, θ̂)| where the maximization is taken over all

e ∈ E1 def= {eTPe ≤ c4}, c4 > c3

z ∈ Z, YR ∈ YR, θ̂ ∈ Ωδ

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 15/61

• ψs(e, z,YR, θ̂) = S sat (

ψ(e, z,YR, θ̂) S

)

φsi (e, z,YR, θ̂) = Si sat (

φi(e, z,YR, θ̂) Si

)

, 1 ≤ i ≤ p

Replace ψ and ψ is the control and adaptive law by ψs and φs. For all e(0) ∈ E0, z(0) ∈ Z0, and θ̂(0) ∈ Ω, we have |ψ| ≤ S and |φi| ≤ Si for all t ≥ 0. Hence the saturation functions will not be effective and the state feedback adaptive controller with ψ and φ replaced by ψs and φs will result in the same performance

High-Gain ObserversinNonlinear Feedback ControlLecture # 4Adaptive Control – p. 16/61

• Parameter Convergence Under State Feedback

Recall

ζi = zi − xn−m+i g0 + θT g

, 1 ≤ i ≤ m

ζ̇i = ζi+1, 1 ≤ i ≤ m− 1

ζ̇m = − f0(x,z)+θ T f(x,z)

g0+θT g

∣ ∣ ∣ zi=ζi+xn−m+i/(g0+θT g)

Set x(t) = Yr(t) in the above equations and let ζ̄(t) be the solution of the differential equation when z(0) = 0. Define z̄ by

z̄i(t) = ζ̄i(t) + y

(n−m+i