2.153 adaptive control fall 2019 lecture 2: simple adaptive...

2.153 Adaptive ControlFall 2019

Lecture 2: Simple Adaptive Systems: Identification

Anuradha Annaswamy

[email protected]

September 9, 2019

( [email protected]) 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


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




x = f(x, u, θ, t) y = h(x, u, θ, t)





System to be controlled (open-loop): Plant

Controlled System (closed-loop): System




x = f(x, u, θ, t) y = h(x, u, θ, t)





System to be controlled (open-loop): PlantControlled System (closed-loop): System


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


Identification of a Single Parameter

θ: Unknown, scalar

y(t) = θu(t)

Identify θ using measurements {u(t), y(t)}.


Identification of a Vector Parameter

y(t) = θTu(t)

y ∈ R, θ ∈ Rn, u : R+ → Rn




y(t) = θTu(t)

y ∈ R,

θ ∈ Rn, u : R+ → Rn




y(t) = θTu(t)

y ∈ R, θ ∈ Rn,

u : R+ → Rn




y(t) = θTu(t)

y ∈ R, θ ∈ Rn, u : R+ → Rn



Identification of a Single Parameter - Recursive Scheme

y(t) = θu(t)

θ: Unknown, scalar

Identify θ as θ(t) at every instant


Identification of a Single Parameter - Recursive Scheme

y(t) = θu(t)

θ: Unknown, scalar Identify θ as θ(t) at every instant


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


Error Model 1

θ: Unknown, u(t) and e(t) can be measured at each instant t.


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


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



Assume that a1 is known. Identify θ1 as θ.

θ = θ − θ1




Assume that a1 is known. Identify θ1 as θ. θ = θ − θ1



Error Model 3

e = −θ1e+ θu


An alternate procedure for identifying θ1:

a1s+ θ1

=

a1s+ θm

1− θm − θ1s+ θm

θ ≡ θm − θ1


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


2.153 adaptive control fall 2019 lecture 2: simple adaptive...

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