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Adaptive Control of PDEs
Andrey Smyshlyaev and Miroslav KrsticUniversity of California, San Diego
Backstepping Control Design
Unstable heat equation
ut = uxx +λuu(0) = 0u(1) = control
Desired behavior (target system)
wt = wxx
w(0) = 0w(1) = 0
Backstepping transformation
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller is obtained by setting x = 1 in the transformation
u(1) =Z 1
0k(1,y)u(y)dy
Backstepping Control Design
Unstable heat equation
ut = uxx +λuu(0) = 0u(1) = control
Desired behavior (target system)
wt = wxx
w(0) = 0w(1) = 0
Backstepping transformation
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller is obtained by setting x = 1
u(1) =Z 1
0k(1,y)u(y)dy
Backstepping Control Design
Unstable heat equation
ut = uxx +λuu(0) = 0u(1) = control
Desired behavior (target system)
wt = wxx
w(0) = 0w(1) = 0
Backstepping transformation
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller is obtained by setting x = 1
u(1) =Z 1
0k(1,y)u(y)dy
Backstepping Control Design
Unstable heat equation
ut = uxx +λuu(0) = 0u(1) = control
Desired behavior (target system)
wt = wxx
w(0) = 0w(1) = 0
Backstepping transformation
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller is obtained by setting x = 1
u(1) =Z 1
0k(1,y)u(y)dy
Backstepping Control Design
Unstable heat equation
ut = uxx +λuu(0) = 0u(1) = control
Desired behavior (target system)
wt = wxx
w(0) = 0w(1) = 0
Backstepping transformation
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller is obtained by setting x = 1
u(1) =Z 1
0k(1,y)u(y)dy
wt −wxx = u(x)
[
λ+2ddx
k(x,x)
]
+ k(x,0)ux(0)
+Z x
0u(y)[kxx(x,y)− kyy(x,y)−λk(x,y)]dy
For right hand side to be zero, 3 conditions should be satisfied:
kxx(x,y)− kyy(x,y) = λk(x,y)
k(x,0) = 0
λ+2ddx
k(x,x) = 0
Are these 3 conditions compatible? In other words, is this PDE well posed?
wt −wxx = u(x)
[
λ+2ddx
k(x,x)
]
+ k(x,0)ux(0)
+Z x
0u(y)[kxx(x,y)− kyy(x,y)−λk(x,y)]dy
For right hand side to be zero, 3 conditions should be satisfied:
kxx(x,y)− kyy(x,y) = λk(x,y)
k(x,0) = 0
λ+2ddx
k(x,x) = 0
Are these 3 conditions compatible? In other words, is this PDE well posed?
Control kernel PDE
kxx(x,y)− kyy(x,y) = λk(x,y)
k(x,0) = 0
k(x,x) = −λx2
Exact solution
k(x,y) = −λyI1
(
√
λ(x2− y2)
)
√
λ(x2− y2)
where I1 is a 1st order modified Bessel function of first kind.
Boundary controller
u(1) =
Z 1
0k(1,y)u(y)dy = −
Z 1
0λy
I1
(
√
λ(1− y2)
)
√
λ(1− y2)u(y)dy
Control kernel PDE
kxx(x,y)− kyy(x,y) = λk(x,y)
k(x,0) = 0
k(x,x) = −λx2
Exact solution
k(x,y) = −λyI1
(
√
λ(x2− y2)
)
√
λ(x2− y2)
where I1 is a 1st order modified Bessel function of first kind.
Control Gain
k1(y)
y
λ = 10
λ = 15
λ = 20
λ = 25
0 0.2 0.4 0.6 0.8 1-40
-30
-20
-10
0
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Literature on Adaptive Control of DPS:
• High-gain adaptive feedback (non-identifier based) under a relative degree oneassumption—Logemann and coauthors (Martensson, Ryan, Townley).
• MRAC (with identifiability proofs but with actuation throughout the PDE domain)—Bentsman, Orlov, Hong; Demetriou, Rosen, and coworkers; Solo and Bamieh.
• Methods employing positive realness/passivity—Demetriou, Ito, Curtain; Wen andBalas.
• LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in-put operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan.
• Nonlinear PDEs (Burgers, Kuramoto-Sivashinsky)—Liu and Krstic; Kobayashi.
• Systems on lattices—Jovanovic and Bamieh.
• Experimental boundary controller for a flexible beam—de Queiroz, Dawson, andcoworkers.
Approaches to identifier design
• Lyapunov
• Estimation based/Certainty equivalence
– with passive identifier (often called “observer-based” method)
– with swapping identifier (often called the “gradient” method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
Approaches to identifier design
• Lyapunov
• Estimation based/Certainty equivalence
– with passive identifier (often called “observer-based” method)
– with swapping identifier (often called the “gradient” method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
Approaches to identifier design
• Lyapunov
• Estimation based/Certainty equivalence
– with passive identifier (often called “observer-based” method)
– with swapping identifier (often called the “gradient” method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
Approaches to identifier design
• Lyapunov
• Estimation based/Certainty equivalence
– with passive identifier (often called “observer-based” method)
– with swapping identifier (often called the “gradient” method)
This talk:
Part I: State-feedback with passive identifier
Part II: Output feedback with swapping identifier
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
Backstepping (functional parameters)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
Backstepping (functional parameters)
Backstepping (output feedback)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
[CE scheme with passive identifier]
Backstepping (functional parameters)
[Lyapunov-based scheme]
Backstepping (output feedback)
[CE scheme with swapping identifier]
Certainty Equivalence Approach with a Passive Identifier
Introductory Example (Unstable)
ut(x, t) = uxx(x, t)+λu(x, t)
u(0, t) = 0
u(1, t) = boundary control (to be designed)
λ – unknown parameter
Certainty Equivalence Approach with a Passive Identifier
Introductory Example (Unstable)
ut(x, t) = uxx(x, t)+λu(x, t)
u(0, t) = 0
u(1, t) = boundary control (to be designed)
λ – unknown parameter
Boundary controller:
u(1) = −Z 1
0λξ
I1
(
√
λ(1−ξ2)
)
√
λ(1−ξ2)u(ξ)dξ
λ – estimate of λ
Identifier design
Plant:ut = uxx +λu
u(0) = 0
“Observer:”
ut = uxx + λu+ γ2(u− u)R 10 u(x)2dx
u(0) = 0u(1) = u(1)
Update law:˙λ = γ
R 10 (u(x)− u(x))u(x)dx
γ > 0: adaptation gain
Denote estimation error signals:
λ = λ− λ (parameter error)
e = u− u (output error)
Properties guaranteed by the identifier (independent from the controller):
λ(t), ‖e(t)‖ bounded over [0,∞)˙λ(t), ‖ex(t)‖, ‖e(t)‖‖u(t)‖ square integrable over [0,∞)
MAIN RESULT:
Proof that the entire u, u, λ PDE/ODE system is stable even though the controller and
the identifier are separately designed (as stabilizing).
Nontrivial because the adaptive closed loop is nonlinear even when the plant is linear.
Never been done before for PDEs.
Denote estimation error signals:
λ = λ− λ (parameter error)
e = u− u (output error)
Properties guaranteed by the identifier (independent from the controller):
λ(t), ‖e(t)‖ bounded over [0,∞)˙λ(t), ‖ex(t)‖, ‖e(t)‖‖u(t)‖ square integrable over [0,∞)
MAIN RESULT:
Proof that the entire u, u, λ PDE/ODE system is stable even though the controller and
the identifier are separately designed (as stabilizing).
Nontrivial because the adaptive closed loop is nonlinear even when the plant is linear.
Never been done before for PDEs.
The main idea of our proof:
Change of variables (bounded operators because λ is bounded)
w(x) = u(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)u(ξ)dξ
u(x) = w(x)−Z x
0λξ
J1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)w(ξ)dξ
z(x) = e(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)e(ξ)dξ
“Target system”
wt = wxx +˙λ
Z x
0
ξ2
w(ξ)dξ+(
λ+ γ2‖u+ e‖‖u‖)
z
w(0) = 0w(1) = 0
The main idea of our proof:
Change of variables (bounded operators because λ is bounded)
w(x) = u(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)u(ξ)dξ
u(x) = w(x)−Z x
0λξ
J1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)w(ξ)dξ
z(x) = e(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)e(ξ)dξ
“Target system”
wt = wxx +˙λ
Z x
0
ξ2
w(ξ)dξ+(
λ+ γ2‖u+ e‖‖u‖)
z
w(0) = 0w(1) = 0
The main idea of our proof:
Change of variables (bounded operators because λ is bounded)
w(x) = u(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)u(ξ)dξ
u(x) = w(x)−Z x
0λξ
J1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)w(ξ)dξ
z(x) = e(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)e(ξ)dξ
“Target system”
wt = wxx +˙λ
Z x
0
ξ2
w(ξ)dξ+(
λ+ γ2‖u+ e‖‖u‖)
z
w(0) = 0w(1) = 0
The main idea of our proof:
Change of variables (bounded operators because λ is bounded)
w(x) = u(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)u(ξ)dξ
u(x) = w(x)−Z x
0λξ
J1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)w(ξ)dξ
z(x) = e(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)e(ξ)dξ
“Target system”
wt = wxx +˙λ
Z x
0
ξ2
w(ξ)dξ+(
λ+ γ2‖u+ e‖‖u‖)
z
w(0) = 0w(1) = 0
The main idea of our proof:
Change of variables (bounded operators because λ is bounded)
w(x) = u(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)u(ξ)dξ
u(x) = w(x)−Z x
0λξ
J1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)w(ξ)dξ
z(x) = e(x)+Z x
0λξ
I1
(
√
λ(x2−ξ2)
)
√
λ(x2−ξ2)e(ξ)dξ
“Target system”
wt = wxx +˙λ
Z x
0
ξ2
w(ξ)dξ+(
λ+ γ2‖u+ e‖‖u‖)
z
w(0) = 0w(1) = 0
Properties of the closed loop system:
• globally Lipschitz time-varying coupled PDEs
• additive and multiplicative perturbations which are square integrable in time (though
not necessarily bounded)
• the unperturbed system is exp stable
We use the Gronwall lemma to prove that the state is bounded.
Regulation is proved using Barbalat’s lemma.
Properties of the closed loop system:
• globally Lipschitz time-varying coupled PDEs
• additive and multiplicative perturbations which are square integrable in time (though
not necessarily bounded)
• the unperturbed system is exp stable
We use the Gronwall lemma to prove that the state is bounded.
Regulation is proved using Barbalat’s lemma.
Properties of the closed loop system:
• globally Lipschitz time-varying coupled PDEs
• additive and multiplicative perturbations which are square integrable in time (though
not necessarily bounded)
• the unperturbed system is exp stable
We use the Gronwall lemma to prove that the state is bounded.
Regulation is proved using Barbalat’s lemma.
Extensions
• 2D and 3D domains with boundary actuation
• reaction-advection-diffusion PDEs with all coefficients constant and unknown.
Simulation Example ( 2D with 1D actuation)
ut = ε(uxx +uyy)+b1ux +b2uy +λu
Domain: rectangle 0≤ x ≤ 1, 0≤ y ≤ 2 with actuation applied on the side with x = 1 and
Dirichlet boundary conditions on the other three sides.
Unknown parameters: ε = 1, b1 = 1, b2 = 2, λ = 22→ unstable eigenvalues at 8.4 and 1.
t = 0
t = 0.5 t = 2
t = 0.25
u
u
u
u
21
0 01
2
01
2
01
2
xy
x x
xy
yy
00.5
1
00.5
1
00.5
1
00.5
1
-15
-10
-5
0
5
10
15
-15
-10
-5
0
5
10
15
-15
-10
-5
0
5
10
15
-15
-10
-5
0
5
10
15
The closed loop state u(x, t) at different times
ε
b2
b1
λ
t t
tt0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5
0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5
0
5
10
15
20
25
-1
0
1
2
3
0
1
2
3
4
0
1
2
3
The parameter estimates versus time
Lyapunov Approach
Reaction-advection-diffusion PDEs in 1D:
ut = εuxx+bux+λu
u(0) = 0,
where ε,b,λ are unknown constants.
Lyapunov Approach
Reaction-advection-diffusion PDEs in 1D:
ut = εuxx+bux+λu
u(0) = 0,
where ε,b,λ are unknown constants.
Controller:
u(1) = −Z 1
0αξe−
β2(1−ξ)
I1
(
√
α(1−ξ2)
)
√
α(1−ξ2)u(ξ)dξ ,
Update laws:
˙α = γ‖w‖2
1+‖w‖2
˙β = γR 10 w(x)
R x0 ϕ(x,ξ)w(ξ)dξdx
1+‖w‖2
w(x) = u(x)−Z x
0k(x,ξ)u(ξ)dξ
ϕ(x,ξ) = divk(x,ξ)+Z x
ξ(divk(x,σ))l(σ,ξ)dσ
k(x,ξ) = −αξe−β2(x−ξ)
I1
(
√
α(x2−ξ2)
)
√
α(x2−ξ2)
l(x,ξ) = −αξe−β2(x−ξ)
J1
(
√
α(x2−ξ2)
)
√
α(x2−ξ2)
divk(x,ξ) =1ξk(x,ξ)+ αe−
β2(x−ξ) ξ
x+ξI2
(√
α(x2−ξ2)
)
Lyapunov function:
V =12
(
log(1+‖w‖2)+εγ
(
α2+ β2)
)
Nonstandard but positive definite. The log is inspired by Praly’s work on nonlinear adaptive
control in 1992.
Lyapunov function:
V =12
(
log(1+‖w‖2)+εγ
(
α2+ β2)
)
Nonstandard but positive definite. The log is inspired by Praly’s work on nonlinear adaptive
control in 1992.
Stable:
V ≤−µ‖wx‖2
1+‖w‖2
Target system
wt = εwxx+bwx−cw+ εβZ x
0ϕ(x,ξ)w(ξ)dξ+ εαw
+˙β
Z x
0ψ(x,ξ)w(ξ)dξ+ ˙α
Z x
0
ξ2
e−β2(x−ξ)w(ξ)dξ
w(0) = 0
w(1) = 0
where
ψ(x,ξ) =x−ξ
2k(x,ξ)+
12
Z x
ξ(x−σ)k(x,σ)l(σ,ξ)dσ
Target system
wt = εwxx+bwx−cw+ εβZ x
0ϕ(x,ξ)w(ξ)dξ+ εαw
+˙β
Z x
0ψ(x,ξ)w(ξ)dξ+ ˙α
Z x
0
ξ2
e−β2(x−ξ)w(ξ)dξ
w(0) = 0
w(1) = 0
where
ψ(x,ξ) =x−ξ
2k(x,ξ)+
12
Z x
ξ(x−σ)k(x,σ)l(σ,ξ)dσ
Must use parameter projection in the update laws (with a priori bounds on ε,b,λ) to prevent
the destabilizing effect of ˙α and ˙β.
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Comparison between the Lyapunov and the Certainty-Equivalence/Passive-Identifier Approaches
• Lyapunov does not require an “observer” PDE.
• CE/Passive update law much simpler.
• Lyapunov provides a complete Lyapunov function → tighter control on transient per-
formance
• CE/Passive does not require projection.
• Lyapunov does not require a separate estimate of ε.
• Lyapunov does not require the measurement of ux (CE/Passive does when ε or b are
unknown).
Lyapunov Adaptive Control for Plantswith Spatially Varying Coefficients
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
PDE with unknown functional parameter
ut = ε(x)uxx +b(x)ux +λ(x)u+g(x)u(0)+Z x
0f (x,y)u(y)dy
ux(0) = −qu(0)
u(1) = control
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
Control gain PDE:
kxx(x,y)− kyy(x,y) = λ(y)k(x,y)
k(x,0) = 0
k(x,x) = −12
Z x
0λ(s)ds
Approximate solution:
k0(x,y) = −14
Z
x+y2
x−y2
λ(ξ)dξ
ki+1(x,y) = ki(x,y)+14
Z
x+y2
x−y2
Z
x−y2
0λ(ξ−η)ki(ξ+η,ξ−η)dηdξ
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
Control gain PDE:
kxx(x,y)− kyy(x,y) = λ(y)k(x,y)
k(x,0) = 0
k(x,x) = −12
Z x
0λ(s)ds
Approximate solution:
k0(x,y) = −14
Z
x+y2
x−y2
λ(ξ)dξ
ki+1(x,y) = ki(x,y)+14
Z
x+y2
x−y2
Z
x−y2
0λ(ξ−η)ki(ξ+η,ξ−η)dηdξ
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
Control gain PDE:
kxx(x,y)− kyy(x,y) = λ(y)k(x,y)
k(x,0) = 0
k(x,x) = −12
Z x
0λ(s)ds
Approximate solution:
k0(x,y) = −14
Z
x+y2
x−y2
λ(ξ)dξ
ki+1(x,y) = ki(x,y)+14
Z
x+y2
x−y2
Z
x−y2
0λ(ξ−η)ki(ξ+η,ξ−η)dηdξ
Comparison with Finite Dimension
Plant (λ)
Estimator
λ
Pole PlacementController
Bezout Equation
Comparison with Finite Dimension
Plant (λ)
Estimator
λ
Pole PlacementController
Bezout Equation
BacksteppingController
k - PDE
Adaptive Scheme
Update law
λt(x, t) = γu(x)
(
w(x)− R 1x kn(ξ,x)w(ξ)dξ
)
1+‖w‖2
where
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller
u(1) =
Z 1
0kn(1,y)u(y)dy
Adaptive Scheme
Update law
λt(x, t) = γu(x)
(
w(x)− R 1x kn(ξ,x)w(ξ)dξ
)
1+‖w‖2
where
w(x) = u(x)−Z x
0k(x,y)u(y)dy
Controller
u(1) =
Z 1
0kn(1,y)u(y)dy
Adaptive Scheme
Update law
λt(x, t) = γu(x)
(
w(x)− R 1x kn(ξ,x)w(ξ)dξ
)
1+‖w‖2
where
w(x) = u(x)−Z x
0kn(x,y)u(y)dy
Controller
u(1) =
Z 1
0kn(1,y)u(y)dy
Adaptive Scheme
Update law
λt(x, t) = γu(x)
(
w(x)− R 1x kn(ξ,x)w(ξ)dξ
)
1+‖w‖2
where
w(x) = u(x)−Z x
0kn(x,y)u(y)dy
Controller
u(1) =
Z 1
0kn(1,y)u(y)dy
Main Result
There exist γ∗ and n∗ such that for any γ ∈ (0,γ∗) and for any n ≥ n∗ the solutions u(x, t),
λ(x, t) of the closed loop system are uniformly bounded and
limt→∞
u(x, t) = 0 for all x ∈ [0,1]
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w(t)‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤− (1− γ/γ∗)1+‖w(t)‖2‖wx(t)‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Update law
λt(x, t) = γu(x)
(
w(x)−R 1
x kn(ξ,x)w(ξ)dξ)
1+‖w‖2
Proof Idea
Target system
wt = wxx−Z x
0kn
t (x,y)u(y)dy+
Z x
0λ(y)(kn(x,y)− kn+1(x,y))u(y)dy
+ λ(x)u(x)−Z x
0λ(y)kn(x,y)u(y)dy
w(0) = 0
w(1) = 0
Lyapunov function
V =12
log(1+‖w‖2)+12γ‖λ‖2
Nonstandard but positive definite. Inspired by Praly’s work on nonlinear adaptive control in
1992.
Stable:
V ≤−14
(1− γ/γ∗)1+‖w‖2 ‖wx‖2
Simulation Example
ut = uxx +b(x)ux +λ(x)u
00.1
0.20.3
0.40.5
0
0.5
1−10
−5
0
5
10
tx
The closed-loop state u(x, t)
00.1
0.20.3
0.40.5
0
0.5
15
10
15
20
25
tx
00.1
0.20.3
0.40.5
0
0.5
1−1
0
1
2
3
tx
The parameter estimates λ(x, t) and b(x, t)
0 0.2 0.4 0.6 0.8 10
5
10
15
20
25 estimate true parameter
λ(x, t = ∞) versus true λ(x)
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
estimate
true parameter
b(x, t = ∞) versus true b(x)
Summary
# parameters
# states# controls
Summary
Finite dimensional
adaptive control
70’s-80’s
# parameters
# states# controls
Summary
Finite dimensional
adaptive control
70’s-80’s
MRAC DPS
90’s
# parameters
# states# controls
Summary
Finite dimensional
adaptive control
70’s-80’s
MRAC DPS
90’s
Backsteppingfor const. par.
# parameters
# states# controls
Summary
Finite dimensional
adaptive control
70’s-80’s
MRAC DPS
90’s
Backstepping
for func. par.
Backsteppingfor const. par.
# parameters
# states# controls
Output Feedback Adaptive Control
Benchmark Plant with Unknown Parameter in the Domain
Parabolic PDE with unknown parameter multiplying the measured output u(0, t)
ut(x, t) = uxx(x, t)+gu(0, t) ,
ux(0, t) = 0,
u(1, t) = boundary control (to be designed)
Transfer function:
u(0,s) =s
g+(s−g)cosh√
su(1,s)
Infinite relative degree and unstable for g > 2.
Virtually all adaptive control results for PDEs are for relative degree one problems.
Basic fact from finite dimensional adaptive control: relative degree 3 much harder than
relative degrees 1 or 2.
Benchmark Plant with Unknown Parameter in the Domain
Parabolic PDE with unknown parameter multiplying the measured output u(0, t)
ut(x, t) = uxx(x, t)+gu(0, t) ,
ux(0, t) = 0,
u(1, t) = boundary control (to be designed)
Transfer function:
u(0,s) =s
g+(s−g)cosh√
su(1,s)
Infinite relative degree and unstable for g > 2.
Virtually all adaptive control results for PDEs are for relative degree one problems.
Basic fact from finite dimensional adaptive control: relative degree 3 much harder than
relative degrees 1 or 2.
Benchmark Plant with Unknown Parameter in the Domain
Parabolic PDE with unknown parameter multiplying the measured output u(0, t)
ut(x, t) = uxx(x, t)+gu(0, t) ,
ux(0, t) = 0,
u(1, t) = boundary control (to be designed)
Transfer function:
u(0,s) =s
g+(s−g)cosh√
su(1,s)
Infinite relative degree and unstable for g > 2.
Virtually all adaptive control results for PDEs are for relative degree one problems.
Basic fact from finite dimensional adaptive control: relative degree 3 much harder than
relative degrees 1 or 2.
Benchmark Plant with Unknown Parameter in the Domain
Parabolic PDE with unknown parameter multiplying the measured output u(0, t)
ut(x, t) = uxx(x, t)+gu(0, t) ,
ux(0, t) = 0,
u(1, t) = boundary control (to be designed)
Transfer function:
u(0,s) =s
g+(s−g)cosh√
su(1,s)
Infinite relative degree and unstable for g > 2.
Virtually all adaptive control results for PDEs are for relative degree one problems.
Basic fact from finite dimensional adaptive control: relative degree 3 much harder than
relative degrees 1 or 2.
Controller
u(1) = −Z 1
0
√
gsinh√
g(1− x)(gv(x)+η(x)) dx
g – estimate of g.
Input filter
ηt = ηxx
ηx(0) = 0
η(1) = u(1)
Output filter
vt = vxx +u(0)
vx(0) = 0
v(1) = 0
Controller
u(1) = −Z 1
0
√
gsinh√
g(1− x)(gv(x)+η(x)) dx
g – estimate of g.
Input filter
ηt = ηxx
ηx(0) = 0
η(1) = u(1)
Output filter
vt = vxx +u(0)
vx(0) = 0
v(1) = 0
“Swapping” identifier
Gradient update law
˙g = γv(0, t)
1+ v2(0, t)(u(0)− gv(0)−η(0))
and the same input and output filters as in the previous slide.
Denote estimation error signals:
g = g− g (parameter error)
e(0) = u(0)− gv(0)−η (output error)
Properties guaranteed by the identifier (independent from the controller):
g(t) bounded over [0,∞)
˙g(t),e(0, t)
√
1+ v2(0)bounded and square integrable over [0,∞)
“Swapping” identifier
Gradient update law
˙g = γv(0, t)
1+ v2(0, t)(u(0)− gv(0)−η(0))
and the same input and output filters as in the previous slide.
Denote estimation error signals:
g = g− g (parameter error)
e(0) = u(0)− gv(0)−η (output error)
Properties guaranteed by the identifier (independent from the controller):
g(t) bounded over [0,∞)
˙g(t),e(0, t)
√
1+ v2(0)bounded and square integrable over [0,∞)
MAIN RESULT:
Proof that the entire u,v,η, g PDE/ODE system is stable
The main idea of our proof:
Change of variables (bounded operator because g is bounded)
w(x) = gv(x)+η(x)+Z x
0
√
gsinh√
g(x− y)(gv(y)+η(y))dy
“Target system”
wt = wxx + gcosh√
gx e(0, t)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
Rewrite the output filter as
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
The main idea of our proof:
Change of variables (bounded operator because g is bounded)
w(x) = gv(x)+η(x)+Z x
0
√
gsinh√
g(x− y)(gv(y)+η(y))dy
“Target system”
wt = wxx + gcosh√
gx e(0, t)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
Rewrite the output filter as
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
The main idea of our proof:
Change of variables (bounded operator because g is bounded)
w(x) = gv(x)+η(x)+Z x
0
√
gsinh√
g(x− y)(gv(y)+η(y))dy
“Target system”
wt = wxx + gcosh√
gx e(0, t)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
Rewrite the output filter as
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
The main idea of our proof:
Change of variables (bounded operator because g is bounded)
w(x) = gv(x)+η(x)+Z x
0
√
gsinh√
g(x− y)(gv(y)+η(y))dy
“Target system”
wt = wxx + gcosh√
gx e(0, t)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
Rewrite the output filter as
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
The main idea of our proof:
Change of variables (bounded operator because g is bounded)
w(x) = gv(x)+η(x)+Z x
0
√
gsinh√
g(x− y)(gv(y)+η(y))dy
“Target system”
wt = wxx + gcosh√
gx e(0, t)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
Rewrite the output filter as
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
The main idea of our proof:
wt = wxx + gcosh√
gx e(0)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
— two interconnected time varying PDEs
— additive and multiplicative perturbations which are bounded and square integrable in
time
We use the Gronwall lemma to prove that ‖wx(t)‖ and ‖vx(t)‖ are bounded.
Regulation is proved using Barbalat’s lemma.
The main idea of our proof:
wt = wxx + gcosh√
gx e(0)+ ˙g
{
v+Z x
0
sinh√
g(x− y)√g
(gv(y)+w(y)) dy
}
wx(0) = 0
w(1) = 0
vt = vxx +w(0)+ e(0)
vx(0) = 0
v(1) = 0
— two interconnected time varying PDEs
— additive and multiplicative perturbations which are bounded and square integrable in
time
We use the Gronwall lemma to prove that ‖wx(t)‖ and ‖vx(t)‖ are bounded.
Regulation is proved using Barbalat’s lemma.
Output Feedback with Swapping Identifier
Parabolic PDE with unknown functional parameter
ut(x, t) = uxx(x, t)+λ(x)u(x, t) ,
ux(0, t) = 0,
u(1, t) = boundary control
Measurement : u(0, t)
Infinite relative degree plant with arbitrarily many (finite) number of unstable poles
First, convert the plant into observer canonical form (generalization for PDEs)
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
Output Feedback with Swapping Identifier
Parabolic PDE with unknown functional parameter
ut(x, t) = uxx(x, t)+b(x)ux(x, t)+λ(x)u(x, t)+g(x)u(0, t)+
Z x
0f (x,y)u(y)dy ,
ux(0, t) = 0,
u(1, t) = boundary control
Measurement : u(0, t)
Infinite relative degree plant with arbitrarily many (finite) number of unstable poles
First, convert the plant into observer canonical form (generalization for PDEs)
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
Output Feedback with Swapping Identifier
Parabolic PDE with unknown functional parameter
ut(x, t) = uxx(x, t)+λ(x)u(x, t) ,
ux(0, t) = 0,
u(1, t) = boundary control
Measurement : u(0, t)
Infinite relative degree plant with arbitrarily many (finite) number of unstable poles
First, convert the plant into observer canonical form (generalization for PDEs)
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
Output Feedback with Swapping Identifier
Parabolic PDE with unknown functional parameter
ut(x, t) = uxx(x, t)+λ(x)u(x, t) ,
ux(0, t) = 0,
u(1, t) = boundary control
Measurement : u(0, t)
Infinite relative degree plant with arbitrarily many (finite) number of unstable poles
First, convert the plant into observer canonical form (generalization for PDEs)
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
Output Feedback with Swapping Identifier
Parabolic PDE with unknown functional parameter
ut(x, t) = uxx(x, t)+λ(x)u(x, t) ,
ux(0, t) = 0,
u(1, t) = boundary control
Measurement : u(0, t)
Infinite relative degree plant with arbitrarily many (finite) number of unstable poles
First, convert the plant into observer canonical form (generalization for PDEs)
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
Observer canonical form
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
To put the plant into this form we use the transformation
v(x) = u(x)−Z x
0p(x,y)u(y)dy
pxx(x,y) = pyy(x,y)+λ(y)p(x,y)
p(1,y) = 0
p(x,x) =12
Z 1
xλ(s)ds
θ(x) and θ1 are the new unknown parameters:
θ(x) = −py(x,0) θ1 = −p(0,0)
Observer canonical form
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
To put the plant into this form we use the transformation
v(x) = u(x)−Z x
0p(x,y)u(y)dy
pxx(x,y) = pyy(x,y)+λ(y)p(x,y)
p(1,y) = 0
p(x,x) =12
Z 1
xλ(s)ds
θ(x) and θ1 are the new unknown parameters:
θ(x) = −py(x,0) θ1 = −p(0,0)
Observer canonical form
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
We are going to directly estimate θ(x) and θ1 without identifying
the original plant parameter λ(x).
Note:
— v(0) = u(0) and therefore v(0) is measured.
— unknown parameters multiply the measured output v(0)
Observer canonical form
vt(x, t) = vxx(x, t)+θ(x)v(0, t)
vx(0) = θ1v(0, t)
v(1) = u(1)
We are going to directly estimate θ(x) and θ1 without identifying
the original plant parameter λ(x).
Note:
— v(0) = u(0) and therefore v(0) is measured.
— unknown parameters multiply the measured output v(0)
Input filter
ψt = ψxx
ψx(0) = 0
ψ(1) = u(1)
Only one filter (the plant has no zeros)
Output filters
Ft = Fxx +δ(x−ξ)u(0) ξ ∈ [0,1]
Fx(0) = 0
F(1) = 0
φt = φxx
φx(0) = u(0)
φ(1) = 0
Algebraic way to represent F(x,ξ):
Fxx(x,ξ) = Fξξ(x,ξ)
F(0,ξ) = −φ(ξ)
Fx(0,ξ) = 0
Fξ(x,0) = F(x,1) = 0
F(x,ξ) = −2∞∑
n=0cos
π(2n+1)x2
cosπ(2n+1)ξ
2
Z 1
0cos
π(2n+1)s2
φ(s)ds
Output filters
Ft = Fxx +δ(x−ξ)u(0) ξ ∈ [0,1]
Fx(0) = 0
F(1) = 0
φt = φxx
φx(0) = u(0)
φ(1) = 0
Algebraic way to represent F(x,ξ) through φ(x):
Fxx(x,ξ) = Fξξ(x,ξ)
F(0,ξ) = −φ(ξ)
Fx(0,ξ) = 0
Fξ(x,0) = F(x,1) = 0
F(x,ξ) = −2∞∑
n=0cos
π(2n+1)x2
cosπ(2n+1)ξ
2
Z 1
0cos
π(2n+1)s2
φ(s)ds
Output filters
Ft = Fxx +δ(x−ξ)u(0) ξ ∈ [0,1]
Fx(0) = 0
F(1) = 0
φt = φxx
φx(0) = u(0)
φ(1) = 0
Algebraic way to represent F(x,ξ) through φ(x):
Fxx(x,ξ) = Fξξ(x,ξ)
F(0,ξ) = −φ(ξ)
Fx(0,ξ) = 0
Fξ(x,0) = F(x,1) = 0
F(x,ξ) = −2∞∑
n=0cos
π(2n+1)x2
cosπ(2n+1)ξ
2
Z 1
0cos
π(2n+1)s2
φ(s)ds
Adaptive scheme
ut = uxx +λ(x)u
ux(0) = 0u(1) u(0)
ψ φ
θ(x), θ1
output filterinput filter
identifier
controller
Adaptive scheme
vt = vxx +θv(0)
vx(0) = θ1v(0)
u(1) u(0)
ψ φ
θ(x), θ1
output filterinput filter
identifier
controller
Adaptive scheme
vt = vxx +θv(0)
vx(0) = θ1v(0)
u(1) u(0)
output filterinput filter ψ φ
θ(x), θ1 identifier
controller
Adaptive scheme
vt = vxx +θv(0)
vx(0) = θ1v(0)
u(1) u(0)
ψ φ
θ(x), θ1
output filterinput filter
identifier
controller
Observer error
e(x) = v(x)−[
ψ(x)+θ1φ(x)+
Z 1
0θ(ξ)F(x,ξ)dξ
]
is exponentially stable:
et = exx
ex(0) = 0
e(1) = 0
Static parametric model
e(0) = v(0)−ψ(0)−θ1φ(0)+Z 1
0θ(ξ)φ(ξ)dξ
Observer error
e(x) = v(x)−[
ψ(x)+θ1φ(x)+
Z 1
0θ(ξ)F(x,ξ)dξ
]
is exponentially stable:
et = exx
ex(0) = 0
e(1) = 0
Static parametric model
e(0) = v(0)−[
ψ(0)+θ1φ(0)−Z 1
0θ(ξ)φ(ξ)dξ
]
Update laws (least squares)
θt(x, t) =
R 10 γ(x,y, t)φ(y)dy+ γ0(x, t)φ(0)
1+‖φ‖2+φ2(0)
(
v(0)−ψ(0)− θ1φ(0)+
Z 1
0θ(ξ)φ(ξ)dξ
)
˙θ1 =
R 10 γ0(y, t)φ(y)dy+ γ1(t)φ(0)
1+‖φ‖2+φ2(0)
(
v(0)−ψ(0)− θ1φ(0)+
Z 1
0θ(ξ)φ(ξ)dξ
)
Riccati adaptation gains
γt(x,y, t) = −R 10 γ(x,s)φ(s)ds
R 10 γ(y,s)φ(s)ds+ γ0(x)γ0(y)φ2(0)
1+‖φ‖2+φ2(0)
−φ(0)γ0(y)R 10 γ(x,s)φ(s)ds+φ(0)γ0(x)
R 10 γ(y,s)φ(s)ds
1+‖φ‖2+φ2(0)
γ0(x) = −
(
R 10 γ(x,s)φ(s)ds+ γ0(x)φ(0)
)(
R 10 γ0(s)φ(s)ds+ γ1φ(0)
)
1+‖φ‖2+φ2(0)
γ1 = −
(
R 10 γ0(s)φ(s)ds+ γ1φ(0)
)2
1+‖φ‖2+φ2(0)
Controller
u(1) =
Z 1
0k(1− y)
(
ψ(y)+ θ1φ(y)+
Z 1
0θ(ξ)F(y,ξ)dξ
)
dy
with k(x) given by the integral equation in one variable
k(x) = θ1−Z x
0θ(y)dy−
Z x
0
[
θ1−Z x−y
0θ(s)ds
]
k(y)dy
This equation is solved at each time step.
MAIN RESULT:
Proof that the entire u, φ, ψ, θ, θ1 PDE/ODE system is asymptotically stable.
Controller
u(1) =
Z 1
0k(1− y)
(
ψ(y)+ θ1φ(y)+
Z 1
0θ(ξ)F(y,ξ)dξ
)
dy
with k(x) given by the integral equation in one variable
k(x) = θ1−Z x
0θ(y)dy−
Z x
0
[
θ1−Z x−y
0θ(s)ds
]
k(y)dy
This equation is solved at each time step.
MAIN RESULT:
Proof that the entire u, φ, ψ, θ, θ1 PDE/ODE system is asymptotically stable.
Controller
u(1) =
Z 1
0k(1− y)
(
ψ(y)+ θ1φ(y)+
Z 1
0θ(ξ)F(y,ξ)dξ
)
dy
with k(x) given by the integral equation in one variable
k(x) = θ1−Z x
0θ(y)dy−
Z x
0
[
θ1−Z x−y
0θ(s)ds
]
k(y)dy
This equation is solved at each time step.
MAIN RESULT:
Proof that the entire u, φ, ψ, θ, θ1 PDE/ODE system is asymptotically stable.
Simulation Example
ut = uxx+b(x)ux+λ(x)u
ux(0) = 0
Reference signal: ur(0, t) = 3sin6t b(x) = 3−2x2 λ(x) = 16+3sin(2πx)
Simulation Example
ut = uxx+b(x)ux+λ(x)u
ux(0) = 0
Reference signal: ur(0, t) = 3sin6t b(x) = 3−2x2 λ(x) = 16+3sin(2πx)
u(x, t)
tx
Simulation Results
u(0)
tt
u(1)
Control effort Output evolution
Simulation Results (parameter estimates)
θ1
θ2
t
tx
θ
0 0.5 1−20
−15
−10
−5
0
5
x
θ
θ(∞)
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