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(∞)