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

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

  • 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.