# High-Gain Observers in Nonlinear Feedback Control Lecture ... the HGO (8) can be linear or nonlinear

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High-Gain Observers in

Nonlinear Feedback Control

Lecture # 5 Sampled-Data Control

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 1/??

Problem Setup

Consider a nonlinear system represented by

ẋ = Ax+Bφ(x, z, u)

ż = ψ(x, z, u)

y = Cx, ζ = Θ(x, z)

u ∈ R is the control input, y ∈ R and ζ ∈ Rm are measured outputs, and x ∈ Rr and z ∈ Rl are the states

A =

0 1 . . . . . . 0

0 0 1 . . . 0

... ...

0 . . . . . . 0 1

0 . . . . . . . . . 0

, B =

0

0

...

0

1

, C = [

1 0 . . . . . . 0

]

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 2/??

Assumption: φ, ψ, Θ are locally Lipschitz; φ(0, 0, 0) = 0, ψ(0, 0, 0) = 0, Θ(0, 0) = 0

Partial State Feedback:

u = γ(x, ζ)

stabilizes the origin of the closed-loop system

χ̇ = F (χ, γ(x, ζ))

χ =

[

x

z

]

, F (χ, u) =

[

Ax+Bφ(x, z, u)

ψ(x, z, u)

]

γ(0, 0) = 0; γ is locally Lipschitz and globally bounded in x

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 3/??

Output Feedback Control:

u = γ(x̂, ζ)

˙̂x = Ax̂+Bφo(x̂, ζ, u) +H(y − Cx̂)

HT =

[ α1

ε , α2

ε2 , · · ·

αr

εr

]

ε is a small positive parameter and αi are chosen such that the roots of

sr + α1s r−1 + · · · + αr−1s+ αr = 0

have negative real parts.

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 4/??

We consider a zero-order-hold system with uniform sampling period T . The continuous-time observer is discretized and implemented in discrete time as a difference equation To study the asymptotic behavior as ε and T tend to zero, we restrict their ratio

0 < r1 ≤ ε/T ≤ r2 < ∞

r1 and r2, independent of ε The limit T/ε → 0 would destroy the high-gain property of the observer The limit T/ε → ∞ would destroy the fast sampling property and may lead to instability of the discretized observer

T = αε, r1 ≤ α ≤ r2

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 5/27

To avoid inherent ill-conditioning of the observer when ε is very small, apply the scaling

ϕ = Dx̂, D = diag [1, ε, . . . , εr−1]

ϕ̇ = 1 ε [Aoϕ+Hoy + ε

rBφo(x̂, ζ, u)]

x̂ = D−1ϕ

Ao = A−HoC, H T o =

[

α1 α2 . . . . . . αr

]

Ao is Hurwitz

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 6/??

Linear Observer (φ0 = 0)

The observer can be descretized by an one of several available methods to arrive at the discrete-time model

q(k + 1) = Adq(k) +Bdy(k)

x̂(k) = D−1[Cdq(k) +Ddy(k)]

where {Ad, Bd, Cd, Dd} depend on the discretization method. The matrices for the Forward Difference (FD), Backward Difference (BD) and Bilinear Transformation (BT) methods are shown in the following table. The state q is ϕ in the FD method, but different in the other two cases.

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 7/??

FD BD BT

Ad (I + αAo) (I − αAo) −1

︸ ︷︷ ︸

M1

(I + α

2 Ao)(I −

α

2 Ao)

−1

︸ ︷︷ ︸

N2M2

Bd αHo αM1Ho αM2Ho

Cd I M1 M2

Dd 0 αM1Ho α 2M2Ho

In the case of the FD method, we assume that α is chosen such that the eigenvalues of Ad are in the interior of the unit circle. For the other two methods, this condition holds for any choice of α

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 8/??

Nonlinear Observer (φ0 6= 0)

The observer is discretized using the Forward Difference method, to obtain

q(k + 1) = Adq(k) +Bdy(k) + αε rBφo(x̂(k), ζ(k), u(k))

x̂(k) = D−1Cdq(k)

where Ad, Bd, and Cd are given in the FD column of Table

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 9/??

Closed-loop Analysis

The plant dynamics are given by

χ̇ = F (χ, u)

The solution of over [kT, (k + 1)T ] is given by

χ(t) = χ(k) + (t− kT )F (χ(k), u(k))

+ ∫ t kT [F (χ(τ ), u(k)) − F (χ(k), u(k))] dτ

By the Lipschitz property of F and the Gronwall-Bellman inequality

‖χ(t) − χ(k)‖ ≤ 1L1

[

e(t−kT )L1 − 1 ]

‖F (χ(k), u(k))‖

∀ t ∈ [kT, kT + T ]

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 10/??

χ(k + 1) = χ(k) + εαF (χ(k), u(k)) + ε2Φ(χ(k), u(k), ε)

where Φ is locally Lipschitz in (χ, u) and uniformly bounded in ε, for sufficiently small ε

ẋ = Ax+Bφ(X , u)

x(k + 1) = eATx(k) + ∫ T 0 e

AtBdtφ(X (k), u(k))

+ εr+1D−1R(X (k), u(k), ε)

η1 = x1 − x̂1

εr−1 , . . . , ηr−1 =

xr−1 − x̂r−1

ε , ηr = xr − x̂r

η = 1

εr−1 D(x− x̂), D = diag [1, ε, . . . , εr−1]

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 11/??

η = 1

εr−1 D(x− Cdq −DdCx)

η(k + 1) = Afη(k) + 1

εr−1 MDx(k) + ε[·]

Af = CdAdC −1 d

M = (I −DdC)e Aα −Af (I −DdC) − CdBdC

MD =

{

O(ε2) for the FD and BD methods O(ε3) for the BT method

(1/εr−1)MD will have negative powers of ε for large r

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 12/??

ξ = η − 1

εr−1 LDx

ξ(k + 1) = η(k + 1) − 1

εr−1 LDx(k + 1)

= Af [ξ(k) + 1

εr−1 LDx(k)]

− 1

εr−1 MDx(k) + ε[·]

− 1

εr−1 LD

[

eATx(k) +

∫ T

0 eAtBdtφ

]

+ε2LR

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 13/??

DeAT = eAαD, DeAtB = εr−1eAt/εB

These properties follow from

eAt =

r−1∑

k=1

tk

k! Ak, εDA = AD, DB = εr−1B

ξ(k+1) = Afξ(k)+ 1

εr−1 [AfL+M −Le

Aα]Dx(k)+ · · ·

Choose L to satisfy the equation

AfL+M − Le Aα = 0

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 14/??

ξ = 1

εr−1 [(I − L−DdC)Dx− Cd q]

The closed-loop sampled-data system can be represented at the sampling points by the discrete-time model

χ(k + 1) = χ(k) + εαF (χ(k), u(k)) + ε2Φ(χ(k), u(k), ε)

ξ(k + 1) = Afξ(k) + εΓ(χ(k), u(k), x̂(k), ε)

u(k) = γ(x̂(k), ζ(k))

x̂(k) = [I − εN2(ε)]x(k) +N1(ε)ξ(k)

where F,Φ, γ,Γ are locally Lipschitz; Φ,Γ are uniformly bounded in ε, for sufficiently small ε; γ,Γ are globally bounded in x̂; the matrices N1,N2 are analytic functions of ε

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 15/??

Separation Principle

Theorem: Let R be the region of attraction of χ̇ = F (χ, γ(x, ζ)), S be any compact set in the interior of R, Q be any compact subset of Rr, and suppose χ(0) ∈ S and x̂(0) ∈ Q. Then

there exists ε∗1 > 0 such that for every ε ∈ (0, ε ∗ 1], the

solutions (χ(k), ξ(k)) of the closed-loop systems are bounded for all k ≥ 0, and χ(t) is bounded for all t ≥ 0;

given any τ1 > 0, there exist ε∗2, T1 and an integer k ∗

(all dependent on τ1) such that, for every ε ∈ (0, ε∗2],

‖ξ(k)‖ ≤ τ1, ∀ k ≥ k ∗, ‖χ(t)‖ ≤ τ1, ∀ t ≥ T1

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 16/??

given any τ2 > 0, there exists ε∗3 > 0 such that, for every ε ∈ (0, ε∗3],

‖χ(t) − χs(t)‖ ≤ τ2, ∀ t ≥ 0

where χs(t) is the solution of χ̇ = F (χ, γ(x, ζ)) with χs(0) = χ(0)

if the origin of χ̇ = F (χ, γ(x, ζ)) is exponentially stable and the functions F and γ are continuously differentiable in the neighborhood of the origin, then there exists ε∗4 > 0 such that for every ε ∈ (0, ε

∗ 4], the

origin of the (discrete-time) closed-loop system is exponentially stable and S × Q is a subset of its region of attraction in the (χ, x̂) space.

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 17/??

The Pendubot Experiment

High-Gain ObserversinNonlinear Feedback ControlLecture # 5Sampled-Data Control – p. 18/??

1714 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL.

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