adaptive control of pdes - miroslav krsticflyingv.ucsd.edu/krstic/teaching/282/slides_adappde.pdfput...
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![Page 1: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/1.jpg)
Adaptive Control of PDEs
Andrey Smyshlyaev and Miroslav KrsticUniversity of California, San Diego
![Page 2: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/2.jpg)
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
![Page 3: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/3.jpg)
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
![Page 4: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/4.jpg)
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
![Page 5: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/5.jpg)
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
![Page 6: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/6.jpg)
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
![Page 7: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/7.jpg)
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?
![Page 8: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/8.jpg)
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?
![Page 9: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/9.jpg)
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
![Page 10: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/10.jpg)
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.
![Page 11: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/11.jpg)
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
![Page 12: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/12.jpg)
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.
![Page 13: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/13.jpg)
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.
![Page 14: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/14.jpg)
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.
![Page 15: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/15.jpg)
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.
![Page 16: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/16.jpg)
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.
![Page 17: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/17.jpg)
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.
![Page 18: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/18.jpg)
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.
![Page 19: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/19.jpg)
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
![Page 20: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/20.jpg)
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
![Page 21: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/21.jpg)
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
![Page 22: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/22.jpg)
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
![Page 23: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/23.jpg)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
![Page 24: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/24.jpg)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
![Page 25: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/25.jpg)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
Backstepping (functional parameters)
![Page 26: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/26.jpg)
Control Architecture
parameters
states
inputs
MRAC Schemes(Demetriou, Rosen;
Bentsman, Orlov, Hong)
Backstepping (constant parameters)
Backstepping (functional parameters)
Backstepping (output feedback)
![Page 27: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/27.jpg)
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]
![Page 28: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/28.jpg)
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
![Page 29: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/29.jpg)
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
![Page 30: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/30.jpg)
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
![Page 31: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/31.jpg)
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.
![Page 32: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/32.jpg)
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.
![Page 33: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/33.jpg)
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
![Page 34: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/34.jpg)
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
![Page 35: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/35.jpg)
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
![Page 36: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/36.jpg)
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
![Page 37: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/37.jpg)
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
![Page 38: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/38.jpg)
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.
![Page 39: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/39.jpg)
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.
![Page 40: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/40.jpg)
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.
![Page 41: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/41.jpg)
Extensions
• 2D and 3D domains with boundary actuation
• reaction-advection-diffusion PDEs with all coefficients constant and unknown.
![Page 42: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/42.jpg)
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.
![Page 43: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/43.jpg)
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
![Page 44: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/44.jpg)
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
![Page 45: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/45.jpg)
Lyapunov Approach
Reaction-advection-diffusion PDEs in 1D:
ut = εuxx+bux+λu
u(0) = 0,
where ε,b,λ are unknown constants.
![Page 46: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/46.jpg)
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
![Page 47: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/47.jpg)
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)
)
![Page 48: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/48.jpg)
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.
![Page 49: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/49.jpg)
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
![Page 50: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/50.jpg)
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
![Page 51: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/51.jpg)
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.
![Page 52: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/52.jpg)
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).
![Page 53: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/53.jpg)
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).
![Page 54: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/54.jpg)
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).
![Page 55: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/55.jpg)
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).
![Page 56: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/56.jpg)
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).
![Page 57: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/57.jpg)
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).
![Page 58: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/58.jpg)
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).
![Page 59: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/59.jpg)
Lyapunov Adaptive Control for Plantswith Spatially Varying Coefficients
![Page 60: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/60.jpg)
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
![Page 61: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/61.jpg)
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
![Page 62: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/62.jpg)
PDE with unknown functional parameter
ut = uxx +λ(x)u
u(0) = 0
u(1) = control
![Page 63: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/63.jpg)
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
![Page 64: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/64.jpg)
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
![Page 65: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/65.jpg)
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
![Page 66: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/66.jpg)
Comparison with Finite Dimension
Plant (λ)
Estimator
λ
Pole PlacementController
Bezout Equation
![Page 67: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/67.jpg)
Comparison with Finite Dimension
Plant (λ)
Estimator
λ
Pole PlacementController
Bezout Equation
BacksteppingController
k - PDE
![Page 68: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/68.jpg)
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
![Page 69: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/69.jpg)
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
![Page 70: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/70.jpg)
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
![Page 71: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/71.jpg)
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
![Page 72: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/72.jpg)
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]
![Page 73: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/73.jpg)
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
![Page 74: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/74.jpg)
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
![Page 75: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/75.jpg)
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
![Page 76: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/76.jpg)
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
![Page 77: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/77.jpg)
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
![Page 78: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/78.jpg)
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
![Page 79: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/79.jpg)
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
![Page 80: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/80.jpg)
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
![Page 81: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/81.jpg)
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
![Page 82: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/82.jpg)
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
![Page 83: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/83.jpg)
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)
![Page 84: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/84.jpg)
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)
![Page 85: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/85.jpg)
Summary
# parameters
# states# controls
![Page 86: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/86.jpg)
Summary
Finite dimensional
adaptive control
70’s-80’s
# parameters
# states# controls
![Page 87: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/87.jpg)
Summary
Finite dimensional
adaptive control
70’s-80’s
MRAC DPS
90’s
# parameters
# states# controls
![Page 88: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/88.jpg)
Summary
Finite dimensional
adaptive control
70’s-80’s
MRAC DPS
90’s
Backsteppingfor const. par.
# parameters
# states# controls
![Page 89: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/89.jpg)
Summary
Finite dimensional
adaptive control
70’s-80’s
MRAC DPS
90’s
Backstepping
for func. par.
Backsteppingfor const. par.
# parameters
# states# controls
![Page 90: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/90.jpg)
Output Feedback Adaptive Control
![Page 91: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/91.jpg)
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.
![Page 92: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/92.jpg)
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.
![Page 93: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/93.jpg)
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.
![Page 94: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/94.jpg)
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.
![Page 95: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/95.jpg)
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
![Page 96: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/96.jpg)
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
![Page 97: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/97.jpg)
“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,∞)
![Page 98: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/98.jpg)
“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,∞)
![Page 99: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/99.jpg)
MAIN RESULT:
Proof that the entire u,v,η, g PDE/ODE system is stable
![Page 100: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/100.jpg)
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
![Page 101: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/101.jpg)
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
![Page 102: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/102.jpg)
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
![Page 103: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/103.jpg)
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
![Page 104: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/104.jpg)
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
![Page 105: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/105.jpg)
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.
![Page 106: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/106.jpg)
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.
![Page 107: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/107.jpg)
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)
![Page 108: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/108.jpg)
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)
![Page 109: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/109.jpg)
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)
![Page 110: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/110.jpg)
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)
![Page 111: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/111.jpg)
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)
![Page 112: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/112.jpg)
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)
![Page 113: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/113.jpg)
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)
![Page 114: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/114.jpg)
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)
![Page 115: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/115.jpg)
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)
![Page 116: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/116.jpg)
Input filter
ψt = ψxx
ψx(0) = 0
ψ(1) = u(1)
Only one filter (the plant has no zeros)
![Page 117: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/117.jpg)
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
![Page 118: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/118.jpg)
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
![Page 119: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/119.jpg)
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
![Page 120: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/120.jpg)
Adaptive scheme
ut = uxx +λ(x)u
ux(0) = 0u(1) u(0)
ψ φ
θ(x), θ1
output filterinput filter
identifier
controller
![Page 121: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/121.jpg)
Adaptive scheme
vt = vxx +θv(0)
vx(0) = θ1v(0)
u(1) u(0)
ψ φ
θ(x), θ1
output filterinput filter
identifier
controller
![Page 122: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/122.jpg)
Adaptive scheme
vt = vxx +θv(0)
vx(0) = θ1v(0)
u(1) u(0)
output filterinput filter ψ φ
θ(x), θ1 identifier
controller
![Page 123: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/123.jpg)
Adaptive scheme
vt = vxx +θv(0)
vx(0) = θ1v(0)
u(1) u(0)
ψ φ
θ(x), θ1
output filterinput filter
identifier
controller
![Page 124: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/124.jpg)
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
![Page 125: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/125.jpg)
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
]
![Page 126: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/126.jpg)
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
)
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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)
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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.
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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.
![Page 130: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/130.jpg)
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.
![Page 131: Adaptive Control of PDEs - Miroslav Krsticflyingv.ucsd.edu/krstic/teaching/282/Slides_adapPDE.pdfput operators and exp stable dynamics)—Duncan, Maslowski, Pasik-Duncan. • Nonlinear](https://reader035.vdocuments.pub/reader035/viewer/2022070904/5f6eac7cfd079c74fe344906/html5/thumbnails/131.jpg)
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)
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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
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Simulation Results
u(0)
tt
u(1)
Control effort Output evolution
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Simulation Results (parameter estimates)
θ1
θ2
t
tx
θ
0 0.5 1−20
−15
−10
−5
0
5
x
θ
θ(∞)