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

High-Gain Observersin

Nonlinear Feedback Control

Lecture # 4Adaptive Control

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

Problem Formulation

Consider a SISO nonlinear system represented globally bythe 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 functionsfi are known smooth nonlinearities which could depend ony, 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 theconstant parameters θ1 to θp are unknown


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

(g0 + θT g) 6= 0 ∀ θ ∈ Ω

The objective is to design an adaptive output feedbackcontroller which guarantees boundedness of all variables ofthe closed-loop system, and tracking of a given referencesignal 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


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 controllersuch that for all Y(0) ∈ Y , for all YR(t) ∈ YR, and for allθ ∈ Ω, all variables of the closed-loop system are boundedfor all t ≥ 0, and

limt→∞

|y(t) − yr(t)| = 0

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


Augment a series of m integrators at the input side of thesystem and treat v = u(m) as the input of the augmentedsystem

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:

xi = xi+1, 1 ≤ i ≤ n− 1

xn = f0(x, z) + θT f(x, z) + (g0 + θT g)v

zi = zi+1, 1 ≤ i ≤ m− 1

zm = v

y = x1


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

The augmented state model has relative degree n and canbe transformed into a globally-defined normal form by thechange of variables

ζi = zi −xn−m+i

g0 + θT g, 1 ≤ i ≤ m

which transforms the z-equations into

ζi = ζi+1, 1 ≤ i ≤ m− 1

ζm = − f0(x,z)+θT f(x,z)

g0+θT g

∣∣∣zi=ζi+xn−m+i/(g0+θT g)


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

Remarks:

The restriction of the coefficient (g0 + gT θ) to beconstant is made for convenience. The result can beextended 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 iscrucial for the derivation of the adaptive controller andmay require redefinition of the physical parameters ofthe system


Example: A single link manipulator with flexible joints andnegligible damping can be represented by

Iq1 +MgL sin q1 + k(q1 − q2) = 0

Jq2 − k(q1 − q2) = u

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

y(4) = gLM

I(y2 sin y−y cos y)−

(k

I+ k

J

)y−gkLM

IJsin y+ k

IJu

θ1 = gLM

I, θ2 =

(k

I+ k

J

), θ3 = gkLM

IJ, θ4 = k

IJ


State Feedback Controller

e1 = y − yr = x1 − yr

e2 = y − yr = x2 − yr...

en = y(n−1) − y(n−1)r = xn − y

(n−1)r

e = Ae+ bf0(e+ Yr, z) + θT f(e+ Yr, z)+ (g0 + θT g)v − y

(n)r

z = A2z + b2v

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

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


e = Ame+ bKe+ f0(e+ Yr, z) + θT f(e+ Yr, z)+ (g0 + θT g)v − y

(n)r

PAm +ATmP = −Q, Q = QT > 0

V = eTPe+ 12θTΓ−1θ, Γ = ΓT > 0, θ = θ − θ

V = −eTQe+ 2eTPb[f0 + θT f + (g0 + θT g)v

+Ke− y(n)r ] + θTΓ−1 ˙

θ

v =−Ke+ y

(n)r − f0(e+ Yr, z) − θT f(e+ Yr, z)

g0 + θT g︸︷︷︸

ψ(e,z,YR ,θ)


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

V = −eTQe+ θTΓ−1[˙θ − Γφ]

˙θ = Γφ ⇒ V = −eTQe

With parameter projection, we can achieve

θTΓ−1[ ˙θ − Γφ] ≤ 0 ⇒ V ≤ −eTQe

while keeping θ(t) ∈ Ω for all t ≥ 0


Example of Parameter Projection:

Ω = θ | ai ≤ θi ≤ bi, 1 ≤ i ≤ p

Ωδ = θ | ai − δ ≤ θi ≤ bi + δ, 1 ≤ i ≤ p ⊂ Ω

Let Γ be a positive diagonal matrix. The adaptive law

˙θ = Proj(θ, φ)

where Proj(θ, φ) is defined on the next page, is locallyLipschitz in (θ, φ) and ensures

θ(0) ∈ Ω ⇒ θ(t) ∈ Ωδ, ∀ t ≥ 0

θTΓ−1[˙θ − Γφ] ≤ 0


[Proj(θ, φ)]i is defined by

γiiφi, if ai ≤ θi ≤ bi or

if θi > bi and φi ≤ 0 or

if θi < ai and φi ≥ 0

γii

[

1 + bi−θi

δ

]

φi, if θi > bi and φi > 0

γii

[

1 + θi−ai

δ

]

φi, if θi < ai and φi < 0


V ≤ −eTQe ⇒ V is bounded ⇒ (e, θ) are bounded

⇒ (x, z, θ) are bounded

Boundedness of θ follows also from θ ∈ Ωδ. With allsignals bounded, we conclude from V ≤ −eTQe that

e(t) → 0 as t → ∞

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

c1 = maxe∈E0

eTPe, and c2 = maxθ∈Ω,θ∈Ωδ

12(θ− θ)TΓ−1(θ− θ)


c3 > c1 + c2 ⇒ e(t) ∈ Edef= 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, θ)|, Si ≥ max |φi(e, z,YR, θ)|where the maximization is taken over all

e ∈ E1def= eTPe ≤ c4, c4 > c3

z ∈ Z, YR ∈ YR, θ ∈ Ωδ


ψs(e, z,YR, θ) = S sat

(

ψ(e, z,YR, θ)S

)

φsi (e, z,YR, θ) = Si sat

(

φi(e, z,YR, θ)Si

)

, 1 ≤ i ≤ p

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


Parameter Convergence Under State Feedback

Recall

ζi = zi − xn−m+i

g0 + θT g, 1 ≤ i ≤ m

ζi = ζi+1, 1 ≤ i ≤ m− 1

ζm = − f0(x,z)+θT 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 ζ(t) be thesolution of the differential equation when z(0) = 0. Definez by

zi(t) = ζi(t) +y

(n−m+i−1)r (t)

g0 + θT g, 1 ≤ i ≤ m


Assumption: For any bounded x(t) such that x(t) → Yr(t)as t → ∞ and any z(0) ∈ Z0, ζ(t) → ζ(t) as t → ∞(consequently, z(t) → z(t) as t → ∞)

It is satisfied in the special case when

ζ = A0ζ + F0(x)

with a Hurwitz matrix A0 and a continuous function F0

Define wr(t) by

wr(t) = f(Yr(t), z(t)) + 1g0+θT g

g[

y(n)r (t) − f0(Yr(t), z(t)) − θT f(Yr(t), z(t))

]


Assumption: y(n+1)r (t) is bounded and wr(t) is persistently

exciting (PE)

The closed-loop system can be written as

e = Ame− bθTw(t)˙θ = 2Γ(t)w(t)bTPe

z = A2z + b2ψ

where w = f + gψ and Γ(t) is a bounded

bounded y(n+1)r (t) ⇒ bounded e ⇒ lim

t→∞e(t) = 0

⇒ w(t) → wr(t) as t → ∞ ⇒ w(t) is PE


Using the PE property of w and the fact that the derivativeof V = eTPe+ 1

2 θTΓθ satisfies V ≤ 0, it can be shown

(by a standard argument in adaptive control) that the originof the system

e

˙θ

=

Am −bwT (t)

2Γ(t)w(t)bTP 0

e

θ

is exponentially stable. Hence

θ(t) → θ as t → ∞


Output Feedback Control

Replace e by e, provided by the high-gain observer:

˙ei = ei+1 + (αi/εi)(e1 − e1), 1 ≤ i ≤ n− 1

˙en = (αn/εn)(e1 − e1)

The roots of

sn + α1sn−1 + · · · + αn−1s+ αn = 0

have negative real parts

ηi =ei − ei

εn−i , 1 ≤ i ≤ n


After finite timeV ≤ −eTQe+ kε

V ≤ −c0eTPe+ kε = −c0V + 12c0θ

TΓ−1θ + εk

θ ∈ Ωδ ⇒ 1

2θTΓ−1θ ≤ c2 ⇒ V ≤ −c0V + c0c2 + εk

On the boundary V = c3, V < 0 wheneverc3 > c2 + εk/c0. Since c3 > c1 + c2, we conclude that, forsufficiently small ε, the set V ≤ c3 ∩ θ ∈ Ωδ ispositively invariant. For sufficiently small ε,

All variables are bounded

The saturation components of ψs and φs will not beeffective for t ≥ T1


V ≤ −eTQe+ kε ⇒ limT →∞

1

T

∫ T

0eT (t)Qe(t) dt ≤ kε

The mean-square tracking error is of order O(ε)

Sharper Result Under PE Condition:

Singular Perturbation Result:

x = F (t, x, z) +H1(t, x, z, y, ε)

z = G(t, x, z) +H2(t, x, z, y, ε)

εy = Ay + εH3(t, x, z, y, ε)

where A is Hurwitz and H1,H2 vanish at y = 0.


Suppose the solutions of

x = F (t, x, z)

z = G(t, x, z)

are bounded and satisfy

‖x(t)‖ ≤ ke−γ(t−t0)‖x0‖, ∀ t ≥ t0

Then, for sufficiently small ε,∥∥∥∥∥

[

x(t)

y(t)

]∥∥∥∥∥

≤ k1e−γ1(t−t0)

∥∥∥∥∥

[

x(t0)

y(t0)

]∥∥∥∥∥, ∀ t ≥ t0


Under the PE Condition we have

∥∥∥∥∥∥∥

e(t)

θ(t)

η(t)

∥∥∥∥∥∥∥

≤ k1e−γ1(t−t1)

∥∥∥∥∥∥∥

e(t1)

θ(t1)

η(t1)

∥∥∥∥∥∥∥

, ∀ t ≥ t1

for some k1 > 0, γ1 > 0, and any t1 > T1

limt→∞[z(t) − z(t)] = 0

The trajectories (e(t), z(t), θ(t)) under output feedbackapproach the corresponding trajectories under statefeedback as ε → 0


Example

G(s) =c

s(s+ a)

where a and c are unknown, but −1.9 ≤ a ≤ 1.9 and1.1 ≤ c ≤ 2.9. The vector θ = [a c]T belongs to acompact convex hypercube Ω, and we take δ = 0.1.Design an adaptive output controller such that the output yasymptotically tracks a reference signal yr, which is theoutput of the reference model

Gm(s) =ω2

s2 + 2ζωs+ ω2

driven by a command signal r


The system and reference model can be represented by thedifferential equations

y = −ay + cu, (v = u)

yr = −2ζωyr − ω2yr + ω2r

We assume that r and r are bounded, so that yr, yr, yr,

and y(3)r are bounded. The PE condition requires

wr =

[

−yr(yr + ayr)/c

]

to be persistently exciting for all −1.9 ≤ a ≤ 1.9 and1.1 ≤ c ≤ 2.9


This is the case if and only if[

yr

yr

]

is PE , which is true if r(t) is a stationary signal that issufficiently rich of order 2

K = [ω2 2ζω] ⇒ Am =

[

0 1

−ω2 −2ζω

]

P = α

[

2ω(1 + ζ) 1

1 2/ω

]

, α > 0


Γ = γI, γ > 0

u = ψ =1

c[−Ke+ yr + a(e2 + yr)]

φ =

[

φ1

φ2

]

=

[

−2α(e1 + 2e2/ω)(e2 + yr)

2α(e1 + 2e2/ω)ψ

]

where a and c are the parameter estimates. To proceedwith the design we need to calculate the saturation levelsfor ψ, φ1, and φ2

ζ = 0.7, ω = 1, α = 0.5, γ = 10

e(0) ∈ E0 = e | |e1| ≤ 1 and |e2| ≤ 1


c1 = maxe∈E0

eTPe ≤ 7.4α

c2 = maxθ∈Ω,θ∈Ω

‖θ − θ‖2/2γ ≤ 10/γ

c4 = 4.8, e ∈ E1 = eTPe ≤ c4By calculating upper bounds on |ψ|, |φ1|, |φ2| for e ∈ E1,θ ∈ Ωδ, |yr(t)| ≤ 1.5, and |yr(t)| ≤ 1.5, we choose thesaturation levels of ψ, φ1, and φ2 as S = 10.1, S1 = 10,and S2 = 32.2, respectively


Simulation: a = 1, c = 2 (known), ǫ = 0.01, zero initialconditions

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2(a

) y

y_r

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

Time

(b)

u


Simulation: a = 1, c = 2 (known), ǫ = 0.01 zero initialconditions

0 2 4 6 8 10 12 14 16 18 20−0.15

−0.1

−0.05

0

0.05(a

)

e=y−y_r

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

Time

(b)

Estimate of a


Simulation: a = −1, c = 2 (unknown), ǫ = 0.01, zero initialconditions except y(0) = 1 and θ2(0) = 1.1

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2(a

)

y

y_r

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

Time

(b)

u



0 5 10 15 20 25 30 35 40 45 50−0.5

0

0.5

1(a

)

e=y−y_r

0 5 10 15 20 25 30 35 40 45 50−2

−1

0

1

2

3

Time

(b)

Estimate of a

Estimate of c



0 0.05 0.1 0.15 0.2 0.25 0.3−20

−10

0

10

20(a

)

u

0 0.05 0.1 0.15 0.2 0.25 0.3−20

0

20

40

60

Time

(b)

e_2

Estimate of e_2


Example:

ξ1 = ξ2 + θξ21ξ2 = u+ ξ3

ξ3 = −ξ3 + y

y = ξ1

y(3) = (u+ y − y) + 2θ(yy + y2 + yy) + u

The system satisfies all the assumptions. Add an integratorat the input, take z = u, and set v = u as the control input.The PE condition requires y(4)(t) to be bounded and

wr(t) = 2[yr(t)yr(t) + y2r(t) + yr(t)yr(t)]

to be persistently exciting.


e1 = y − yr, e2 = y − yr, e3 = y − yr

K = [2 4 3] assigns the eigenvalues of Am at −1 and−1 ± j. The matrix P satisfies PAm +ATmP = −I

ψ = −Ke− u− y + y − 2θ(yy + y2 + yy) + y(3)r

φ = 4(eTPb)(yy + y2 + yy)

θ ∈ Ω = θ | 0 ≤ θ ≤ 2Take δ = 0.1, γ = 10

maxθ∈Ω,θ∈Ωδ

1

2γ(θ − θ)2 = 0.2205


Take c4 = 2.4 which allows for c1 = 2

e(0) ∈ E0 = eTPe ≤ 2

e ∈ E1 = eTPe ≤ 2.4Using the zero dynamics equation

ζ = −ζ − [y + 2θ(yy + y2 + yy)]

where ζ = z − y. Taking z(0) = 0 and assuming that themagnitudes of yr and its first two derivatives are boundedby 0.1, it can be shown that for all e ∈ E1, |z| is bounded by33


By calculating upper bounds on ψ and φ for all e ∈ E1,|z| ≤ 33 and θ ∈ Ωδ, the saturation levels S = 75 andS1 = 25 are chosen for ψ and φ, respectively. Theestimates ei are provided by the high-gain observer

ǫ ˙e1 = e2 + (2/ε)(e1 − e1)

ǫ ˙e2 = e3 + (4/ε2)(e1 − e1)

ǫ ˙e3 = (3/ε3)(e1 − e1)

The output feedback controller is obtained by replacing ei inψ and φ by ei and taking v = ψs, u =

∫v

Simulation: θ = 1, yr = 0.1 sin t, ǫ = 0.01, zero initialconditions for all variables except y(0) = 1. In one graphresults are shown for y(0) = 1 and y(0) = 0.45


0 2 4 6 8 10 12 14−0.5

0

0.5

1

1.5

2

(a)

Tracking error

0 2 4 6 8 10 12 14−8

−6

−4

−2

0

2

Time

(b) u


0 0.05 0.1 0.15 0.2 0.25 0.3−100

−50

0

50

100

(a)

v

0 2 4 6 8 10 12 14−20

−10

0

10

20

Time

(b)

v


0 2 4 6 8 10 12 14−0.5

0

0.5

1

1.5

2

(a)

0 2 4 6 8 10 12 14−8

−6

−4

−2

0

2

Time

(b)

State (solid); Output (dashed); (a) Output; (b) Control u


Extensions (Aloliwi and Khalil (1997))

In traditional adaptive control results, PE is required to showparameter convergence, but not to show tracking errorconvergence, as required in the output feedback controller

Assumption: wr satisfies one of the following:

wr is PE

wr = 0

There is a constant nonsingular matrix L, possiblydependent on θ, such that

Lwr(t) =

[

wr1(t)

0

]


θTL−1 = [θT1 , θT2 ]

e = Ame− bθTw + Λ(·)= Ame− bθTL−1Lwr + bθT (wr − w) + Λ(·)= Ame− bθT1wr1 + +bθT (wr − w) + Λ(·)

[

e˙θ1

]

=

[

Am −bGwTr12Γ1Gwr1bTP 0

] [

e

θ1

]

+

[

Λs

Λe

]

k1 ≤ G ≤ k2, ‖Λe‖ ≤ k3‖e‖

‖Λs‖ ≤ k4‖e‖ + k5‖ζ‖ + k6‖η‖, ζ = ζ − ζ

The linear part is exponentially stable


Assumption: The zero dynamics are exponentially stable

W = αV + βV1 + V2 + Vη, α > 0, β > 0

V = eTPe+ 12θTΓ−1θ, V1 is a Lyapunov function for the

zero dynamics in terms of ζ, V2 is a Lypaunov function forthe exponentially stable dynamics in terms of (e, θ1), andVη is a Lyapunov function for the observer dynamics interms of η

W ≤ −

‖e‖‖θ1‖‖ζ‖‖η‖

T

M

‖e‖‖θ1‖‖ζ‖‖η‖


With a particular choice of α and β, it can be shown that Mis positive definite for sufficiently small ε

‖e(t)‖‖θ1(t)‖‖ζ(t)‖‖η(t)‖

→ 0 as t → ∞


Bounded Disturbance:

V ≤ −eTQe+ k1ε+ k2d1

where d1 is a bound on the disturbance. Themean-square tracking error if of the order O(ε+ d1)

If wr is PEW ≤ −k1W + k2d1

All variables, including θ, converge to a ball centered at theorigin, whose size is of the order of O(

√d1).


Adaptive Approximation-Based (Neural Network) Control

Farrell and Polycarpou (2006)

Control Architecture

Function Approximator

Training Algorithm


Control Architecture

xi = xi+1, 1 ≤ i ≤ n− 1

xn = [f0(x) + f(x)] + [g0(x) + g(x)]u

[g0(x) + g(x)] 6= 0

f0 and g0 are known while f and g are unknown

Feedback Linearization:

u =1

g0(x) + g(x)[−f0(x) − f(x) + v]

f and g are approximations of f and g


Function Approximator

f(x) =N∑

i

θiφi(x, σ) = θTφ(x, σ), ∀ x ∈ D

Approximation Structures:

Polynomials

Splines

Radial Basis Functions

Multilayer Perceptron

Fuzzy Approximation

wavelets


Universal Approximation:

In a given compact subset of the domain D, theapproximation error

∆f (x) = f(x) −N∑

i

θiφi(x, σ)

can be made arbitrarily small by choosing N sufficientlylarge


Learning Algorithm

Fix σ in f(x) = θTφ(x, σ)

f(x) = θTφ(x)

On-line Approximator: f(x) = θTφ(x)

Lyapunov-Based Adaptive Laws


Seshagiri and Khalil (2000)

y(n) = F (·) +G(·)u(m), G(·) ≥ k1 > 0, m < n

F (·) = F (y, y(1), . . . , y(n−1), u, u(1), . . . , u(m−1))

Dynamic Extension:

- - -· · · - - -

∫ ∫ ∫

Plantv = u(m) u(m−1) u(1) u y

State Variables:

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

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


Extended State Model:

xi = xi+1, 1 ≤ i ≤ n− 1

xn = F (x, z) +G(x, z)v

zi = zi+1, 1 ≤ i ≤ m− 1

zm = v

y = x1

Dimension = n+m, Relative Degree = n

Normal Form:xi = xi+1, 1 ≤ i ≤ n− 1

xn = F (x, z) +G(x, z)v

ζ = f0(ζ, x) ISS

y = x1


For all (x, z) ∈ X × Z (compact)

F (x, z) = θTf φf (x, z) + δf (x, z), F (x, z) = θTf φf (x, z)

G(x, z) = θTg φg(x, z) + δg(x, z), G(x, z) = θTg φg(x, z)

θ =

[

θfθg

]

, θ =

[

θfθg

]

, ψ =

[

φfφgv

]

θ = θ − θ, δ = δf + δgv, |δf | ≤ λ, |δg| ≤ λ (small)

θ ∈ Θ (compact), θ ∈ Θ (compact), Θ ⊂ Θ

Assumption: ∀ (x, z, θ) ∈ X × Z × Θ, G ≥ k0 > λ > 0

F +Gv = F + Gv − θTψ + δ


Tracking: limt→∞[y(t) − yr(t)] = 0 (bounded

yr, . . . , y(n)r )

ei = xi − y(i−1)r , 1 ≤ i ≤ n

e = Ae+B[F + Gv − y(n)r − θTψ + δ]

v =−Ke+ y

(n)r − F + v1

G

e = (A−BK)︸︷︷︸

Hurwitz

e+B(−θTψ + v1 + δ)

P (A−BK) + (A−BK)TP = −Q, Q = QT > 0


V = eTPe+1

2γθT θ

V = −eTQe+1

γθ( ˙θ − γψ) + 2eTPB(v1 + δ)

˙θ = γψ with parameter projection s.t. θ ∈ Θ

|δ| ≤ |δf | + |δg| |v| ≤ λρ(·)︸︷︷︸

known

+ (λ/k0)︸︷︷︸

(< 1, known)

|v1|

v1 = − λρ

1 − (λ/k0)sat

(

eTPBρ

2µ

)

, µ > 0

V ≤ −eTQe+λµ

1 − (λ/k0)


Determine the approximation set X × Z. Let e(0) ∈ E0

Choose c0 > 0 such that E0 ⊂ eTPe ≤ c0

Choose c > c0 (all forthcoming sets will depend on c)

V = eTPe+1

2γθT θ ≤ c ⇒ eTPe ≤ c ⇒ x ∈ X (compact)

ISS of ζ = f0(ζ, x) ⇒

∂V1(ζ)

∂ζf0(ζ, x) ≤ −α1(‖ζ‖), ∀ ‖ζ‖ ≥ α2(‖x‖)

x ∈ X ⇒ ζ ∈ V1(ζ) ≤ c1 ⇒ z ∈ Z (compact)


For all θ ∈ Θ,1

2θT θ ≤ c2

V ≤ −keTPe+λµ

1 − (λ/k0)≤ −kV +

kc2

γ+

λµ

1 − (λ/k0)

Choose µ (small enough) and γ (large enough) such that

c >c2

γ+

λµ

k(1 − (λ/k0))

Ωdef= V1(ζ) ≤ c1 × V (e, θ) ≤ c × θ ∈ Θ

is compact and positively invariant


Desired ultimate bound: ‖e(t)‖ ≤ b

eTPe ≤ λmin(P )b2 ⇒ ‖e‖ ≤ b

Choose µ (small enough) and γ (large enough) such that

λmin(P )b2 >c2

γ+

λµ

k(1 − (λ/k0))

A def= V1(ζ) ≤ c1 × V (e, θ) ≤ λmin(P )b2 × θ ∈ Θ

is compact, positively invariant, and all trajectories startingin Ω enter A in finite time


Saturate e outside the compact set eTPe ≤ c

High-Gain Observer:

˙ei = ei+1 + (αi/εi)(e1 − e1), 1 ≤ i ≤ n− 1

˙en = (αn/εn)(e1 − e1)

sn + α1sn−1 + · · · + αn is Hurwitz

Replace e by e

The output feedback controller recovers the performance ofthe state feedback controller as ε tends to zero


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

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