Adaptive Control of PDEs - Miroslav put operators and exp stable dynamics)—Duncan, Maslowski,...

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Transcript of Adaptive Control of PDEs - Miroslav put operators and exp stable dynamics)—Duncan, Maslowski,...

  • Adaptive Control of PDEs

    Andrey Smyshlyaev and Miroslav Krstic University of California, San Diego

  • Backstepping Control Design

    Unstable heat equation

    ut = uxx +λu u(0) = 0 u(1) = control

    Desired behavior (target system)

    wt = wxx w(0) = 0 w(1) = 0

    Backstepping transformation

    w(x) = u(x)− Z x

    0 k(x,y)u(y)dy

    Controller is obtained by setting x = 1 in the transformation

    u(1) = Z 1

    0 k(1,y)u(y)dy

  • Backstepping Control Design

    Unstable heat equation

    ut = uxx +λu u(0) = 0 u(1) = control

    Desired behavior (target system)

    wt = wxx w(0) = 0 w(1) = 0

    Backstepping transformation

    w(x) = u(x)− Z x

    0 k(x,y)u(y)dy

    Controller is obtained by setting x = 1

    u(1) = Z 1

    0 k(1,y)u(y)dy

  • Backstepping Control Design

    Unstable heat equation

    ut = uxx +λu u(0) = 0 u(1) = control

    Desired behavior (target system)

    wt = wxx w(0) = 0 w(1) = 0

    Backstepping transformation

    w(x) = u(x)− Z x

    0 k(x,y)u(y)dy

    Controller is obtained by setting x = 1

    u(1) = Z 1

    0 k(1,y)u(y)dy

  • Backstepping Control Design

    Unstable heat equation

    ut = uxx +λu u(0) = 0 u(1) = control

    Desired behavior (target system)

    wt = wxx w(0) = 0 w(1) = 0

    Backstepping transformation

    w(x) = u(x)− Z x

    0 k(x,y)u(y)dy

    Controller is obtained by setting x = 1

    u(1) = Z 1

    0 k(1,y)u(y)dy

  • Backstepping Control Design

    Unstable heat equation

    ut = uxx +λu u(0) = 0 u(1) = control

    Desired behavior (target system)

    wt = wxx w(0) = 0 w(1) = 0

    Backstepping transformation

    w(x) = u(x)− Z x

    0 k(x,y)u(y)dy

    Controller is obtained by setting x = 1

    u(1) = Z 1

    0 k(1,y)u(y)dy

  • wt −wxx = u(x) [

    λ+2 d dx

    k(x,x)

    ]

    + k(x,0)ux(0)

    + Z x

    0 u(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+2 d dx

    k(x,x) = 0

    Are these 3 conditions compatible? In other words, is this PDE well posed?

  • wt −wxx = u(x) [

    λ+2 d dx

    k(x,x)

    ]

    + k(x,0)ux(0)

    + Z x

    0 u(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+2 d dx

    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) = −λx 2

    Exact solution

    k(x,y) = −λy I1

    (

    λ(x2− y2) )

    λ(x2− y2)

    where I1 is a 1st order modified Bessel function of first kind.

    Boundary controller

    u(1) = Z 1

    0 k(1,y)u(y)dy = −

    Z 1

    0 λy

    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) = −λx 2

    Exact solution

    k(x,y) = −λy I1

    (

    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 one assumption—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 and Balas.

    • 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, and coworkers.

  • Literature on Adaptive Control of DPS:

    • High-gain adaptive feedback (non-identifier based) under a relative degree one assumption—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 and Balas.

    • 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, and coworkers.

  • Literature on Adaptive Control of DPS:

    • High-gain adaptive feedback (non-identifier based) under a relative degree one assumption—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 and Balas.

    • 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, and coworkers.

  • Literature on Adaptive Control of DPS:

    • High-gain adaptive feedback (non-identifier based) under a relative degree one assumption—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 and Balas.

    • 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, and coworkers.

  • Literature on Adaptive Control of DPS:

    • High-gain adaptive feedback (non-identifier based) under a relative degree one assumption—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 and Balas.

    • 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, and coworkers.

  • Literature on Adaptive Control of DPS:

    • High-gain adaptive feedback (non-identifier based) under a relative degree one assumption—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 and Balas.

    • 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, and coworkers.

  • Literature on Adaptive Control of DPS:

    • High-gain adaptive feedback (non-identifier based) under a relative degree one assumption—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 and Balas.

    • LQR with least-squares adaptation (for stochastic evolution eqns with unbounded in- put operators and exp stable dynamics)—Dunca