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2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/  · PDF file 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

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  • 2.153 Adaptive Control Lecture 9

    Closed-loop Reference Models and Transients

    Anuradha Annaswamy

    [email protected]

    ( [email protected] ) 1 / 11

  • Return to Adaptive Control

    Choose u so that e(t)→ 0 as t→∞. kp, ap are unknown.

    u(t) = θ(t)xp + k(t)r

    θ̇(t) = −sign(kp)exp k̇(t) = −sign(kp)er

    ( [email protected] ) 2 / 11

  • Return to Adaptive Control

    Choose u so that e(t)→ 0 as t→∞. kp, ap are unknown.

    u(t) = θ(t)xp + k(t)r

    θ̇(t) = −sign(kp)exp k̇(t) = −sign(kp)er

    ( [email protected] ) 2 / 11

  • Stability and Convergence Leads to Error Model 3: ė = ame+ θ̃

    T ω

    V = 1

    2

    ( e2 + |kp|θ̃

    T θ̃

    ) V̇ = eė+ θ̃

    T ˙̃ θ

    = ame 2 + kpeθ̃

    Tω + |kp|θ̃ T ˙̃ θ

    = ame 2 + θ̃T (kpeω + |kp|

    ˙̃ θ) = ame

    2 ≤ 0

    ⇒ e(t) and θ̃(t) are bounded for all t ≥ t0; e(t)→ 0

    ( [email protected] ) 3 / 11

  • Stability and Convergence Leads to Error Model 3: ė = ame+ θ̃

    T ω

    V = 1

    2

    ( e2 + |kp|θ̃

    T θ̃

    ) V̇ = eė+ θ̃

    T ˙̃ θ

    = ame 2 + kpeθ̃

    Tω + |kp|θ̃ T ˙̃ θ

    = ame 2 + θ̃T (kpeω + |kp|

    ˙̃ θ) = ame

    2 ≤ 0

    ⇒ e(t) and θ̃(t) are bounded for all t ≥ t0; e(t)→ 0

    ( [email protected] ) 3 / 11

  • Adaptive Gain Example Simulation Parameters: am = −1, km = 1, ap = 1, kp = 2

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =1

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =10

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =100

    ( [email protected] ) 4 / 11

  • Adaptive Gain Example Simulation Parameters: am = −1, km = 1, ap = 1, kp = 2

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =1

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s] S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =10

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =100

    ( [email protected] ) 4 / 11

  • Adaptive Gain Example Simulation Parameters: am = −1, km = 1, ap = 1, kp = 2

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =1

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s] S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s]

    P a ra m et er

    θ k

    γ =10

    0 10 20 30

    0

    0.5

    1

    1.5

    2

    2.5

    3

    time [s]

    S ta te

    xm xp

    0 10 20 30 −3

    −2

    −1

    0

    1

    time [s] P a ra m et er

    θ k

    γ =100

    ( [email protected] ) 4 / 11

  • Closed-Loop Reference Model

    Plant: ẋp = apxp + kpu

    Closed-loop Reference Model: ẋcm = amx c m + kmr − `ec

    Controller: u = θ(t)xp + k(t)r

    Adaptive law: ˙̄̃ θ = −γsgn(bp)ecφ ˜̄θ> =

    [ θ̃ k̃

    ] and φ> =

    [ xp r

    ] 1 Stability is guaranteed

    2 limt→∞ e c(t) = 0

    ( [email protected] ) 5 / 11

  • Closed-Loop Reference Model

    Plant: ẋp = apxp + kpu

    Closed-loop Reference Model: ẋcm = amx c m + kmr − `ec

    Controller: u = θ(t)xp + k(t)r

    Adaptive law: ˙̄̃ θ = −γsgn(bp)ecφ ˜̄θ> =

    [ θ̃ k̃

    ] and φ> =

    [ xp r

    ] 1 Stability is guaranteed 2 limt→∞ e

    c(t) = 0

    ( [email protected] ) 5 / 11

  • Closed-Loop Reference Model

    Plant: ẋp = apxp + kpu

    Closed-loop Reference Model: ẋcm = amx c m + kmr − `ec

    Controller: u = θ(t)xp + k(t)r

    Adaptive law: ˙̄̃ θ = −γsgn(bp)ecφ ˜̄θ> =

    [ θ̃ k̃

    ] and φ> =

    [ xp r

    ] 1 Stability is guaranteed 2 limt→∞ e

    c(t) = 0

    ( [email protected] ) 5 / 11

  • Closed-Loop Reference Model

    Plant: ẋp = apxp + kpu

    Closed-loop Reference Model: ẋcm = amx c m + kmr − `ec

    Controller: u = θ(t)xp + k(t)r

    Adaptive law: ˙̄̃ θ = −γsgn(bp)ecφ ˜̄θ> =

    [ θ̃ k̃

    ] and φ> =

    [ xp r

    ]

    1 Stability is guaranteed 2 limt→∞ e

    c(t) = 0

    ( [email protected] ) 5 / 11

  • Closed-Loop Reference Model

    Plant: ẋp = apxp + kpu

    Closed-loop Reference Model: ẋcm = amx c m + kmr − `ec

    Controller: u = θ(t)xp + k(t)r

    Adaptive law: ˙̄̃ θ = −γsgn(bp)ecφ ˜̄θ> =

    [ θ̃ k̃

    ] and φ> =

    [ xp r

    ] 1 Stability is guaranteed 2 limt→∞ e

    c(t) = 0 ( [email protected] ) 5 / 11

  • Transient Performance With CRM

    CRM gain ` affects:

    L2 norm of ec(t) L∞ norm of xcm(t)

    L2 norm of θ̇(t), k̇(t) L2 norm of u(t) (under investigation)

    ( [email protected] ) 6 / 11

  • Transient Performance With CRM: L2 norm of ec(t)

    Lyapunov Function: V (ec, ˜̄θ) = 12e c2 + 12γ

    −1|kp| ˜̄θ> ˜̄θ

    Derivative of V : V̇ (ec, ˜̄θ) = (am + `)e c2 ≤ 0

    Integrate V̇ : ∫∞ 0 V̇ (e

    c(τ), ˜̄θ(τ))dτ = V (∞)− V (0)

    ⇒ −(am + `) ∫∞ 0 (e

    c(τ))2dτ = V (0)− V (∞)

    ⇒ ∫∞ 0 e

    c(t)2dτ ≤ V (0)|am+`|

    ⇒ ‖ec(t)‖L2 =

    √ V (0)

    |am + `|

    where: V (0) = 12e(0) 2 +

    |kp| 2γ

    ˜̄θ>(0)˜̄θ(0)

    ( [email protected] ) 7 / 11

  • Transient Performance With CRM: L2 norm of ec(t)

    Lyapunov Function: V (ec, ˜̄θ) = 12e c2 + 12γ

    −1|kp| ˜̄θ> ˜̄θ

    Derivative of V : V̇ (ec, ˜̄θ) = (am + `)e c2 ≤ 0

    Integrate V̇ : ∫∞ 0 V̇ (e

    c(τ), ˜̄θ(τ))dτ = V (∞)− V (0)

    ⇒ −(am + `) ∫∞ 0 (e

    c(τ))2dτ = V (0)− V (∞)

    ⇒ ∫∞ 0 e

    c(t)2dτ ≤ V (0)|am+`|

    ⇒ ‖ec(t)‖L2 =

    √ V (0)

    |am + `|

    where: V (0) = 12e(0) 2 +

    |kp| 2γ

    ˜̄θ>(0)˜̄θ(0)

    ( [email protected] ) 7 / 11

  • Transient Performance With CRM: L2 norm of ec(t)

    Lyapunov Function: V (ec, ˜̄θ) = 12e c2 + 12γ

    −1|kp| ˜̄θ> ˜̄θ

    Derivative of V : V̇ (ec, ˜̄θ) = (am + `)e c2 ≤ 0

    Integrate V̇ : ∫∞ 0 V̇ (e

    c(τ), ˜̄θ(τ))dτ = V (∞)− V (0)

    ⇒ −(am + `) ∫∞ 0 (e

    c(τ))2dτ = V (0)− V (∞)

    ⇒ ∫∞ 0 e

    c(t)2dτ ≤ V (0)|am+`|

    ⇒ ‖ec(t)‖L2 =

    √ V (0)

    |am + `|

    where: V (0) = 12e(0) 2 +

    |kp| 2γ

    ˜̄θ>(0)˜̄θ(0)

    ( [email protected] ) 7 / 11

  • Transient Performance With CRM: L2 norm of ec(t)

    Lyapunov Function: V (ec, ˜̄θ) = 12e c2 + 12γ

    −1|kp| ˜̄θ> ˜̄θ

    Derivative of V : V̇ (ec, ˜̄θ) = (am + `)e c2 ≤ 0

    Integrate V̇ : ∫∞ 0 V̇ (e

    c(τ), ˜̄θ(τ))dτ = V (∞)− V (0)

    ⇒ −(am + `) ∫∞ 0 (e

    c(τ))2dτ = V (0)− V (∞)

    ⇒ ∫∞ 0 e

    c(t)2dτ ≤ V (0)|am+`|

    ⇒ ‖ec(t)‖L2 =

    √ V (0)

    |am + `|

    where: V (0) = 12e(0) 2 +

    |kp| 2γ

    ˜̄θ>(0)˜̄θ(0)

    ( [email protected] ) 7 / 11

  • Transient Performance With CRM: L2 norm of ec(t)

    Lyapunov Function: V (ec, ˜̄θ) = 12e c2 + 12γ

    −1|kp| ˜̄θ> ˜̄θ

    Derivative of V : V̇ (ec, ˜̄θ) = (am + `)e c2 ≤ 0

    Integrate V̇ : ∫∞ 0 V̇ (e

    c(τ), ˜̄θ(τ))dτ = V (∞)− V (0)

    ⇒ −(am + `) ∫∞ 0 (e

    c(τ))2dτ = V (0)− V (∞)

    ⇒ ∫∞ 0 e

    c(t)2dτ ≤ V (0)|am+`|

    ⇒ ‖ec(t)‖L2 =

    √ V (0)

    |am + `|

    where: V (0) = 12e(0) 2 +

    |kp| 2γ

    ˜̄θ>(0)˜̄θ(0)

    ( [email protected] ) 7 / 11