2.153 Adaptive Control Lecture 9 Closed-loop Reference...
Transcript of 2.153 Adaptive Control Lecture 9 Closed-loop Reference...
2.153 Adaptive ControlLecture 9
Closed-loop Reference Models and Transients
Anuradha Annaswamy
( [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
θ(t) = −sign(kp)exp k(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
θ(t) = −sign(kp)exp k(t) = −sign(kp)er
( [email protected] ) 2 / 11
Stability and ConvergenceLeads to Error Model 3: e = ame+ θ
Tω
V =1
2
(e2 + |kp|θ
Tθ
)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
( [email protected] ) 3 / 11
Stability and ConvergenceLeads to Error Model 3: e = ame+ θ
Tω
V =1
2
(e2 + |kp|θ
Tθ
)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
( [email protected] ) 3 / 11
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
( [email protected] ) 11 / 11
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
( [email protected] ) 11 / 11
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
( [email protected] ) 11 / 11