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

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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

    Lecture # 4 Adaptive 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