2.153 Adaptive Control Lecture 9 Closed-loop Reference...

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2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha Annaswamy [email protected] ( [email protected] ) 1 / 11

Transcript of 2.153 Adaptive Control Lecture 9 Closed-loop Reference...

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2.153 Adaptive ControlLecture 9

Closed-loop Reference Models and Transients

Anuradha Annaswamy

[email protected]

( [email protected] ) 1 / 11

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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

θ(t) = −sign(kp)exp k(t) = −sign(kp)er

( [email protected] ) 2 / 11

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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

θ(t) = −sign(kp)exp k(t) = −sign(kp)er

( [email protected] ) 2 / 11

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Stability and ConvergenceLeads to Error Model 3: e = ame+ θ

V =1

2

(e2 + |kp|θ

)V = ee+ θ

T ˙θ

= ame2 + kpeθ

Tω + |kp|θT ˙θ

= ame2 + θT (kpeω + |kp|

˙θ) = ame

2 ≤ 0

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

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Stability and ConvergenceLeads to Error Model 3: e = ame+ θ

V =1

2

(e2 + |kp|θ

)V = ee+ θ

T ˙θ

= ame2 + kpeθ

Tω + |kp|θT ˙θ

= ame2 + θT (kpeω + |kp|

˙θ) = ame

2 ≤ 0

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

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Adaptive Gain ExampleSimulation 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]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =1

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =10

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100

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Adaptive Gain ExampleSimulation 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]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =1

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =10

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100

( [email protected] ) 4 / 11

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Adaptive Gain ExampleSimulation 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]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =1

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =10

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

0 10 20 30−3

−2

−1

0

1

time [s]Parameter

θk

γ =100

( [email protected] ) 4 / 11

Page 9: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Closed-Loop Reference Model

Plant: xp = apxp + kpu

Closed-loop Reference Model: xcm = amxcm + kmr − `ec

Controller: u = θ(t)xp + k(t)r

Adaptive law:˙θ = −γsgn(bp)e

cφ˜θ> =

[θ k

]and φ> =

[xp r

]1 Stability is guaranteed

2 limt→∞ ec(t) = 0

( [email protected] ) 5 / 11

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Closed-Loop Reference Model

Plant: xp = apxp + kpu

Closed-loop Reference Model: xcm = amxcm + kmr − `ec

Controller: u = θ(t)xp + k(t)r

Adaptive law:˙θ = −γsgn(bp)e

cφ˜θ> =

[θ k

]and φ> =

[xp r

]1 Stability is guaranteed2 limt→∞ e

c(t) = 0

( [email protected] ) 5 / 11

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Closed-Loop Reference Model

Plant: xp = apxp + kpu

Closed-loop Reference Model: xcm = amxcm + kmr − `ec

Controller: u = θ(t)xp + k(t)r

Adaptive law:˙θ = −γsgn(bp)e

cφ˜θ> =

[θ k

]and φ> =

[xp r

]1 Stability is guaranteed2 limt→∞ e

c(t) = 0

( [email protected] ) 5 / 11

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Closed-Loop Reference Model

Plant: xp = apxp + kpu

Closed-loop Reference Model: xcm = amxcm + kmr − `ec

Controller: u = θ(t)xp + k(t)r

Adaptive law:˙θ = −γsgn(bp)e

cφ˜θ> =

[θ k

]and φ> =

[xp r

]

1 Stability is guaranteed2 limt→∞ e

c(t) = 0

( [email protected] ) 5 / 11

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Closed-Loop Reference Model

Plant: xp = apxp + kpu

Closed-loop Reference Model: xcm = amxcm + kmr − `ec

Controller: u = θ(t)xp + k(t)r

Adaptive law:˙θ = −γsgn(bp)e

cφ˜θ> =

[θ k

]and φ> =

[xp r

]1 Stability is guaranteed2 limt→∞ e

c(t) = 0( [email protected] ) 5 / 11

Page 14: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Transient Performance With CRM

CRM gain ` affects:

L2 norm of ec(t)

L∞ norm of xcm(t)

L2 norm of θ(t), k(t)

L2 norm of u(t) (under investigation)

( [email protected] ) 6 / 11

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

Lyapunov Function: V (ec, ˜θ) = 12ec2 + 1

2γ−1|kp| ˜θ> ˜θ

Derivative of V : V (ec, ˜θ) = (am + `)ec2 ≤ 0

Integrate V :∫∞0 V (ec(τ), ˜θ(τ))dτ = V (∞)− V (0)

⇒ −(am + `)∫∞0 (ec(τ))2dτ = V (0)− V (∞)

⇒∫∞0 ec(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

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

Lyapunov Function: V (ec, ˜θ) = 12ec2 + 1

2γ−1|kp| ˜θ> ˜θ

Derivative of V : V (ec, ˜θ) = (am + `)ec2 ≤ 0

Integrate V :∫∞0 V (ec(τ), ˜θ(τ))dτ = V (∞)− V (0)

⇒ −(am + `)∫∞0 (ec(τ))2dτ = V (0)− V (∞)

⇒∫∞0 ec(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

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

Lyapunov Function: V (ec, ˜θ) = 12ec2 + 1

2γ−1|kp| ˜θ> ˜θ

Derivative of V : V (ec, ˜θ) = (am + `)ec2 ≤ 0

Integrate V :∫∞0 V (ec(τ), ˜θ(τ))dτ = V (∞)− V (0)

⇒ −(am + `)∫∞0 (ec(τ))2dτ = V (0)− V (∞)

⇒∫∞0 ec(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

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

Lyapunov Function: V (ec, ˜θ) = 12ec2 + 1

2γ−1|kp| ˜θ> ˜θ

Derivative of V : V (ec, ˜θ) = (am + `)ec2 ≤ 0

Integrate V :∫∞0 V (ec(τ), ˜θ(τ))dτ = V (∞)− V (0)

⇒ −(am + `)∫∞0 (ec(τ))2dτ = V (0)− V (∞)

⇒∫∞0 ec(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

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

Lyapunov Function: V (ec, ˜θ) = 12ec2 + 1

2γ−1|kp| ˜θ> ˜θ

Derivative of V : V (ec, ˜θ) = (am + `)ec2 ≤ 0

Integrate V :∫∞0 V (ec(τ), ˜θ(τ))dτ = V (∞)− V (0)

⇒ −(am + `)∫∞0 (ec(τ))2dτ = V (0)− V (∞)

⇒∫∞0 ec(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

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

Lyapunov Function: V (ec, ˜θ) = 12ec2 + 1

2γ−1|kp| ˜θ> ˜θ

Derivative of V : V (ec, ˜θ) = (am + `)ec2 ≤ 0

Integrate V :∫∞0 V (ec(τ), ˜θ(τ))dτ = V (∞)− V (0)

⇒ −(am + `)∫∞0 (ec(τ))2dτ = V (0)− V (∞)

⇒∫∞0 ec(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

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Transient Performance With CRM: L2 norm of k(t)

Adaptive Law: k = −γsgn(kp)ecr

Square and Integrate:∫∞0 |k|

2dτ =∫∞0 γ2ec(t)2r2dτ

⇒ ‖k(t)‖2L2≤ γ2‖r(t)‖2L∞

‖ec(t)‖2L2

⇒ ‖k(t)‖2L2≤γ2‖r(t)‖2L∞

V (0)

|am + `|

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Transient Performance With CRM: L2 norm of k(t)

Adaptive Law: k = −γsgn(kp)ecr

Square and Integrate:∫∞0 |k|

2dτ =∫∞0 γ2ec(t)2r2dτ

⇒ ‖k(t)‖2L2≤ γ2‖r(t)‖2L∞

‖ec(t)‖2L2

⇒ ‖k(t)‖2L2≤γ2‖r(t)‖2L∞

V (0)

|am + `|

( [email protected] ) 8 / 11

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Transient Performance With CRM: L2 norm of k(t)

Adaptive Law: k = −γsgn(kp)ecr

Square and Integrate:∫∞0 |k|

2dτ =∫∞0 γ2ec(t)2r2dτ

⇒ ‖k(t)‖2L2≤ γ2‖r(t)‖2L∞

‖ec(t)‖2L2

⇒ ‖k(t)‖2L2≤γ2‖r(t)‖2L∞

V (0)

|am + `|

( [email protected] ) 8 / 11

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Transient Performance With CRM: L2 norm of k(t)

Adaptive Law: k = −γsgn(kp)ecr

Square and Integrate:∫∞0 |k|

2dτ =∫∞0 γ2ec(t)2r2dτ

⇒ ‖k(t)‖2L2≤ γ2‖r(t)‖2L∞

‖ec(t)‖2L2

⇒ ‖k(t)‖2L2≤γ2‖r(t)‖2L∞

V (0)

|am + `|

( [email protected] ) 8 / 11

Page 25: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Transient Performance With CRM: L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒xcm(t) =exp(amt)x

cm(0) +

∫ t0 kmexp(am(t− τ))r(τ)dτ

+(−`)∫ t0 exp(am(t− τ))ec(τ)dτ

Noted: (−`)∫ t0 exp(am(t− τ))ec(τ)dτ ≤ `√

2am‖ec‖L2

⇒ ‖xcm(t)‖2L∞≤ 2‖xom‖2L∞

+ `2

am‖ec‖2L2

⇒ ‖xcm(t)‖2L∞ ≤ 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

where: ‖xom‖L∞ = supt ‖exp(amt)xcm(0) +

∫ t0 bmexp(am(t− τ))r(τ)dτ‖

( [email protected] ) 9 / 11

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Transient Performance With CRM: L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒xcm(t) =exp(amt)x

cm(0) +

∫ t0 kmexp(am(t− τ))r(τ)dτ

+(−`)∫ t0 exp(am(t− τ))ec(τ)dτ

Noted: (−`)∫ t0 exp(am(t− τ))ec(τ)dτ ≤ `√

2am‖ec‖L2

⇒ ‖xcm(t)‖2L∞≤ 2‖xom‖2L∞

+ `2

am‖ec‖2L2

⇒ ‖xcm(t)‖2L∞ ≤ 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

where: ‖xom‖L∞ = supt ‖exp(amt)xcm(0) +

∫ t0 bmexp(am(t− τ))r(τ)dτ‖

( [email protected] ) 9 / 11

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Transient Performance With CRM: L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒xcm(t) =exp(amt)x

cm(0) +

∫ t0 kmexp(am(t− τ))r(τ)dτ

+(−`)∫ t0 exp(am(t− τ))ec(τ)dτ

Noted: (−`)∫ t0 exp(am(t− τ))ec(τ)dτ ≤ `√

2am‖ec‖L2

⇒ ‖xcm(t)‖2L∞≤ 2‖xom‖2L∞

+ `2

am‖ec‖2L2

⇒ ‖xcm(t)‖2L∞ ≤ 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

where: ‖xom‖L∞ = supt ‖exp(amt)xcm(0) +

∫ t0 bmexp(am(t− τ))r(τ)dτ‖

( [email protected] ) 9 / 11

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Transient Performance With CRM: L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒xcm(t) =exp(amt)x

cm(0) +

∫ t0 kmexp(am(t− τ))r(τ)dτ

+(−`)∫ t0 exp(am(t− τ))ec(τ)dτ

Noted: (−`)∫ t0 exp(am(t− τ))ec(τ)dτ ≤ `√

2am‖ec‖L2

⇒ ‖xcm(t)‖2L∞≤ 2‖xom‖2L∞

+ `2

am‖ec‖2L2

⇒ ‖xcm(t)‖2L∞ ≤ 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

where: ‖xom‖L∞ = supt ‖exp(amt)xcm(0) +

∫ t0 bmexp(am(t− τ))r(τ)dτ‖

( [email protected] ) 9 / 11

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Transient Performance With CRM: L2 norm of θ(t)

L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒ ‖xcm(t)‖2L∞ = 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

L2 norm of θ

Adaptive Law: θ = −γsgn(kp)ec(ec + xcm)

⇒ |θ|2 = γ2ec2(ec + xcm)2 ≤ 2γ2ec2ec2 + 2γ2ec2xc2m

⇒ ‖θ‖L2 = 4γ2V (0)2

|am+`| + 2γ2‖xcm‖2L∞‖ec‖2L2

⇒ ‖θ‖L2 =4γ2V (0)2

|am + `|+ 2γ2‖xcm‖2L∞

V (0)

|am + `|

( [email protected] ) 10 / 11

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Transient Performance With CRM: L2 norm of θ(t)

L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒ ‖xcm(t)‖2L∞ = 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

L2 norm of θ

Adaptive Law: θ = −γsgn(kp)ec(ec + xcm)

⇒ |θ|2 = γ2ec2(ec + xcm)2 ≤ 2γ2ec2ec2 + 2γ2ec2xc2m

⇒ ‖θ‖L2 = 4γ2V (0)2

|am+`| + 2γ2‖xcm‖2L∞‖ec‖2L2

⇒ ‖θ‖L2 =4γ2V (0)2

|am + `|+ 2γ2‖xcm‖2L∞

V (0)

|am + `|

( [email protected] ) 10 / 11

Page 31: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Transient Performance With CRM: L2 norm of θ(t)

L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒ ‖xcm(t)‖2L∞ = 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

L2 norm of θ

Adaptive Law: θ = −γsgn(kp)ec(ec + xcm)

⇒ |θ|2 = γ2ec2(ec + xcm)2 ≤ 2γ2ec2ec2 + 2γ2ec2xc2m

⇒ ‖θ‖L2 = 4γ2V (0)2

|am+`| + 2γ2‖xcm‖2L∞‖ec‖2L2

⇒ ‖θ‖L2 =4γ2V (0)2

|am + `|+ 2γ2‖xcm‖2L∞

V (0)

|am + `|

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Page 32: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Transient Performance With CRM: L2 norm of θ(t)

L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒ ‖xcm(t)‖2L∞ = 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

L2 norm of θ

Adaptive Law: θ = −γsgn(kp)ec(ec + xcm)

⇒ |θ|2 = γ2ec2(ec + xcm)2 ≤ 2γ2ec2ec2 + 2γ2ec2xc2m

⇒ ‖θ‖L2 = 4γ2V (0)2

|am+`| + 2γ2‖xcm‖2L∞‖ec‖2L2

⇒ ‖θ‖L2 =4γ2V (0)2

|am + `|+ 2γ2‖xcm‖2L∞

V (0)

|am + `|

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Page 33: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Transient Performance With CRM: L2 norm of θ(t)

L∞ norm of xcm(t)

CRM: xcm = amxcm + kmr − `ec

⇒ ‖xcm(t)‖2L∞ = 2‖xom‖2L∞ +`2

am

V (0)

|am + `|

L2 norm of θ

Adaptive Law: θ = −γsgn(kp)ec(ec + xcm)

⇒ |θ|2 = γ2ec2(ec + xcm)2 ≤ 2γ2ec2ec2 + 2γ2ec2xc2m

⇒ ‖θ‖L2 = 4γ2V (0)2

|am+`| + 2γ2‖xcm‖2L∞‖ec‖2L2

⇒ ‖θ‖L2 =4γ2V (0)2

|am + `|+ 2γ2‖xcm‖2L∞

V (0)

|am + `|

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Page 34: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Example With CRMSimulation 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]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-10

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-100

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-1000

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Page 35: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Example With CRMSimulation 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]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-10

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-100

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-1000

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Page 36: 2.153 Adaptive Control Lecture 9 Closed-loop Reference ...mzqu.mit.edu/material/lect/lecture9.pdf · 2.153 Adaptive Control Lecture 9 Closed-loop Reference Models and Transients Anuradha

Example With CRMSimulation 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]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-10

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-100

0 10 20 30

0

0.5

1

1.5

2

2.5

3

time [s]

State

xmxp

xcm

0 10 20 30−3

−2

−1

0

1

time [s]

Parameter

θk

γ =100, ℓ =-1000

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