2.153 adaptive control fall 2019 lecture 2: simple adaptive...
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2.153 Adaptive ControlFall 2019
Lecture 2: Simple Adaptive Systems: Identification
Anuradha Annaswamy
aanna@mit.edu
September 9, 2019
( aanna@mit.edu) September 9, 2019 1 / 14
Parameter Adaptation - Recursive Schemes
Adaptive Control: The control of Uncertain Systems
Adaptive Control (in this Course):The control of Linear Time-invariant Plants with Unknown Parameters
( aanna@mit.edu) September 9, 2019 2 / 14
Parameter Adaptation - Recursive Schemes
Adaptive Control: The control of Uncertain Systems
Adaptive Control (in this Course):The control of Linear Time-invariant Plants with Unknown Parameters
( aanna@mit.edu) September 9, 2019 2 / 14
Adaptive Control: A Parametric Framework
Nonlinear, time-varying, with unknown parameter θ
x = f(x, u, θ, t) y = h(x, u, θ, t)
Linear Time-Varying (LTV) with unknown parameter θ
x = A(θ, t)x+B(θ, t)u y = C(θ, t)x+D(θ, t)u
Linear Time-Invariant (LTI) with unknown parameter θ
x = A(θ)x+B(θ)u y = C(θ)x+D(θ)u
System to be controlled (open-loop): PlantControlled System (closed-loop): System
( aanna@mit.edu) September 9, 2019 3 / 14
Adaptive Control: A Parametric Framework
Nonlinear, time-varying, with unknown parameter θ
x = f(x, u, θ, t) y = h(x, u, θ, t)
Linear Time-Varying (LTV) with unknown parameter θ
x = A(θ, t)x+B(θ, t)u y = C(θ, t)x+D(θ, t)u
Linear Time-Invariant (LTI) with unknown parameter θ
x = A(θ)x+B(θ)u y = C(θ)x+D(θ)u
System to be controlled (open-loop): Plant
Controlled System (closed-loop): System
( aanna@mit.edu) September 9, 2019 3 / 14
Adaptive Control: A Parametric Framework
Nonlinear, time-varying, with unknown parameter θ
x = f(x, u, θ, t) y = h(x, u, θ, t)
Linear Time-Varying (LTV) with unknown parameter θ
x = A(θ, t)x+B(θ, t)u y = C(θ, t)x+D(θ, t)u
Linear Time-Invariant (LTI) with unknown parameter θ
x = A(θ)x+B(θ)u y = C(θ)x+D(θ)u
System to be controlled (open-loop): PlantControlled System (closed-loop): System
( aanna@mit.edu) September 9, 2019 3 / 14
Direct and Indirect Adaptive Control
θp: Plant parameter - unknown; θc: Control parameter
Indirect Adaptive Control: Estimate θp as θp. Compute θc using θp.
θp → θp → θc
Direct Adaptive Control: Directly estimate θc as θc. Compute the plantestimate θp using θc
θp → θc → θc
( aanna@mit.edu) September 9, 2019 4 / 14
Direct and Indirect Adaptive Control
θp: Plant parameter - unknown; θc: Control parameter
Indirect Adaptive Control: Estimate θp as θp. Compute θc using θp.
θp → θp → θc
Direct Adaptive Control: Directly estimate θc as θc. Compute the plantestimate θp using θc
θp → θc → θc
( aanna@mit.edu) September 9, 2019 4 / 14
Direct and Indirect Adaptive Control
θp: Plant parameter - unknown; θc: Control parameter
Indirect Adaptive Control: Estimate θp as θp. Compute θc using θp.
θp → θp → θc
Direct Adaptive Control: Directly estimate θc as θc. Compute the plantestimate θp using θc
θp → θc → θc
( aanna@mit.edu) September 9, 2019 4 / 14
Identification of a Single Parameter
θ: Unknown, scalar
y(t) = θu(t)
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 5 / 14
Identification of a Single Parameter
θ: Unknown, scalar
y(t) = θu(t)
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 5 / 14
Identification of a Vector Parameter
y(t) = θTu(t)
y ∈ R, θ ∈ Rn, u : R+ → Rn
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 6 / 14
Identification of a Vector Parameter
y(t) = θTu(t)
y ∈ R,
θ ∈ Rn, u : R+ → Rn
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 6 / 14
Identification of a Vector Parameter
y(t) = θTu(t)
y ∈ R, θ ∈ Rn,
u : R+ → Rn
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 6 / 14
Identification of a Vector Parameter
y(t) = θTu(t)
y ∈ R, θ ∈ Rn, u : R+ → Rn
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 6 / 14
Identification of a Vector Parameter
y(t) = θTu(t)
y ∈ R, θ ∈ Rn, u : R+ → Rn
Identify θ using measurements {u(t), y(t)}.
( aanna@mit.edu) September 9, 2019 6 / 14
Identification of a Single Parameter - Recursive Scheme
y(t) = θu(t)
θ: Unknown, scalar
Identify θ as θ(t) at every instant
( aanna@mit.edu) September 9, 2019 7 / 14
Identification of a Single Parameter - Recursive Scheme
y(t) = θu(t)
θ: Unknown, scalar Identify θ as θ(t) at every instant
( aanna@mit.edu) September 9, 2019 7 / 14
Identification of a Vector Parameter - Recursive Scheme
y(t) = θTu(t)
y ∈ R, θ ∈ Rn, u : R+ → Rn
Identify θ as θ(t) at every instant
( aanna@mit.edu) September 9, 2019 8 / 14
Error Model 1
θ: Unknown, u(t) and e(t) can be measured at each instant t.
( aanna@mit.edu) September 9, 2019 9 / 14
Identification of a Parameter in a Dynamic System
Simplest Transfer Function of a Motor:
V : Voltage input ω: Angular Velocity output
K,J,B: Physical parameters
Plant:K
Js+B=
a1s+ θ1
K,J,B unknown ⇒ a1, θ1 unknown
( aanna@mit.edu) September 9, 2019 10 / 14
Identification of a Parameter in a Dynamic System
Simplest Transfer Function of a Motor:
V : Voltage input ω: Angular Velocity output
K,J,B: Physical parameters
Plant:K
Js+B=
a1s+ θ1
K,J,B unknown ⇒ a1, θ1 unknown
( aanna@mit.edu) September 9, 2019 10 / 14
One way of identifying parameters a1 and θ1
Assume that a1 is known.
Identify θ1 as θ. θ = θ − θ1
Plant: ω = −θ1ω + u u = a1V
( aanna@mit.edu) September 9, 2019 11 / 14
One way of identifying parameters a1 and θ1
Assume that a1 is known. Identify θ1 as θ.
θ = θ − θ1
Plant: ω = −θ1ω + u u = a1V
( aanna@mit.edu) September 9, 2019 11 / 14
One way of identifying parameters a1 and θ1
Assume that a1 is known. Identify θ1 as θ. θ = θ − θ1
Plant: ω = −θ1ω + u u = a1V
( aanna@mit.edu) September 9, 2019 11 / 14
Error Model 3
e = −θ1e+ θu
( aanna@mit.edu) September 9, 2019 12 / 14
An alternate procedure for identifying θ1:
a1s+ θ1
=
a1s+ θm
1− θm − θ1s+ θm
θ ≡ θm − θ1
( aanna@mit.edu) September 9, 2019 13 / 14
Reading for Today’s Lecture
Chapter 3: Sections 3.1, 3.2, 3.3.1Additional Reading: Hassan Khalil, Nonlinear Systems; Slotine and Li,Nonlinear Control
( aanna@mit.edu) September 9, 2019 14 / 14
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