model reference adaptation systems (mras)

Contents

1.1 Model Reference Adaptation Systems (MRAS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 MIT Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Determination of Adaptation Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1.3 Normalized MIT Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.1.4 Design of MRAS Using Lyapunov Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.2 MATLAB Codes and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.4 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1

1.1. MODEL REFERENCE ADAPTATION SYSTEMS (MRAS) Adaptive Control

1.1 Model Reference Adaptation Systems (MRAS)

MRAS is an important adaptive controller. It may be regarded as an adaptive servo system in which the desired performance is

expressed in terms of a reference model, which gives the desired response to a command signal. This is a convenient way to give

specifications for a servo problem. A block diagram of the system is shown in Figure 1.1. The system has an ordinary feedback

loop composed of the process and the controller in addition to another feedback loop that changes the controller parameters.

The parameters are changed on the basis of feedback from the error, which is the difference between the output of the system

and the output of the reference model. The ordinary feedback loop is called the inner loop, and the parameter adjustment loop

is called the outer loop. The mechanism for adjusting the parameters in a model-reference adaptive system can be obtained in

two ways: by using a gradient method or by applying stability theory.

Figure 1.1: Block diagram of a model-reference adaptive system

1.1.1 MIT Rule

The MIT rule is the original approach to model-reference adaptive control. The name is derived from the fact that it was

developed at the Instrumentation Laboratory (now the Draper Laboratory) at MIT.

To present the MIT rule, we will consider a closed-loop system in which the controller has one adjustable parameter θ.

The desired closed-loop response is specified by a model whose output is ym. Let e be the error between the output y of

the closed-loop system and the output ym of the model. One possibility is to adjust parameters in such a way that the loss

function J(θ) = 12e

2is minimized.

Procedure

Process : G(s) =y

u(1.1)

Model : Gm(s) =ymuc

(1.2)

Control law : u(t) = f(uc, y) (1.3)

Get closed loop from [1.1] & [1.3] :y

uc(1.4)

Error : e = y − ym (1.5)∂e

∂θ=

∂y

∂θ(1.6)

MIT Rule :dθ

dt= −γe∂e

∂θ(1.7)

Mohamed Mohamed El-Sayed Atyya of 40


Examples

1. Gain Adjustment

Gp(s) = θG(s) = θ2

s2 + 2s+ 4

Gm(s) = θoG(s) = θo2

s2 + 2s+ 4; θo = 2

e = y − ym = θG(s)uc − θoG(s)ucdθ

dt= −γe∂e

∂θ= −γG(s)uce = −γym

θoe

Figure 1.2: Gain adjustment block diagram

At γ = 0.5



At γ = 0.7

At γ = 1



At γ = 1.2

At γ = 1.5



2. First Order System Adjustment

dy

dt= −ay + bu

G(s) =Y

U=

b

s+ adymdt

= −amy + bmu

G(s) =YmU

=bm

s+ amUse the control law : u(t) = touc(t)− soy(t)

U = toUc − soY = Ys+ a

b

⇒ Y

Uc=

tos+ab + so

=boto

s+ a+ bsobm = btoam = a+ bso

e = Y − Ym =bto

s+ a+ bsoUc =

bms+ am

Uc

∂e

∂to=

b

s+ a+ bsoUc =

b

s+ amUc

∂e

∂so=

−b2to(s+ a+ bso)2

Uc =−b

s+ a+ bsoY =

−bs+ am

Y

dθ

dt= −γ∂e

∂θe

dtodt

= −γ1

[b

s+ amUc

]e

dsodt

= γ2

[b

s+ amY

]e

Let : a = 1, b = 2, am = 8, bm = 8

Figure 1.3: First order system adjustment block diagram



At γ1 = 2, γ2 = 0.1

At γ1 = 2, γ2 = 1



At γ10 = 2, γ2 = 0.5

At γ1 = 1, γ2 = 0.05



3. Second Order System Adjustment

G(s) =y

u=

s+ b

s2 + a1s+ a0

Gm(s) =ymuc

=s+ bm

s2 + a1ms+ a0m

Let the control law : u(t) = t0uc(t)− s0y(t)

ys2 + a1s+ a0

s+ b= t0uc − s0y

⇒ y

uc=

t0s+ bt0s2 + (a1 + s0)s+ a0 + bs0

e = y − ym∂e

∂t0=

s+ b

s2 + (a1 + s0)s+ a0 + bs0uc ≈ Gm(s)uc

∂e

∂s0=

−t0(s+ b)(s+ b)

[s2 + (a1 + s0)s+ a0 + bs0]2uc

=−(s+ b)

s2 + (a1 + s0)s+ a0 + bs0y =−y2

t0ucdθ

dt= −γ∂e

∂θe

dt0dt

= −γ1Gmuce

ds0

dt= γ2

y2

t0uce = γGmye

Figure 1.4: Second order system adjustment block diagram



Let : G(s) = s+5s2+4s+3 (stable), Gm(s) = s+10

s2+8s+10At γ1 = γ2 = 10

At γ1 = 10, γ2 = 5



At γ1 = 1.2, γ2 = 0.5

At γ1 = 1.2, γ2 = 0.5



Let : G(s) = s+5s2+2s−3 (unstable), Gm(s) = s+10

s2+8s+10

At γ1 = γ2 = 20

At γ1 = 20, γ2 = 10



At γ1 = 20, γ2 = 40

4. Second Order System Adjustment with First Order Controller

G(s) =y

u=

s+ b

s2 + a1s+ a0=N

D

Gm(s) =ymuc

=s+ bm

s3 + a2ms2 + a1ms+ a0m

Let the control law : u(t) =t0

1 + r1suc(t)−

s0

1 + r1sy(t)

y =t0G(s)

1 + r1suc −

s0G(s)

1 + r1sy

y

uc=

t0N

D(1 + r1s) + s0Ne = y − ym

∂e

∂t0=

N

D(1 + r1s) + s0Nuc ≈ Gmuc

∂e

∂s0=

−N2

[D(1 + r1s) + s0N ]2uc ≈ −Gm

y

ucuc = −Gmy

∂e

∂r1=

−NDs[D(1 + r1s) + s0N ]2

uc ≈ −GmDs

D(1 + r1s) + s0Nuc

≈ −Gmuc∂θ

∂t= −γ∂e

∂θe

∂t0∂t

= −γ1[Gmuc]e ⇒ t0 = −γ1

s[Gmuc]e

∂s0

∂t= γ2[Gmy]e ⇒ s0 =

γ2

s[Gmy]e



∂r1

∂t= γ3[Gmuc]e ⇒ r1 =

γ3

s[Gmuc]e

Let:

G(s) =s+ 2

s2 + s+ 6Gm(s) =

s+ 24

s3 + 9s2 + 26s+ 24

Figure 1.5: Second order system adjustment with first order controller block diagram

For Step Input:

γ1 = 50, γ2 = 25, γ3 = −100



For square Input:

γ1 = 7.5, γ2 = 3.75, γ3 = −15



For Sinusoidal Input:

γ1 = 7.5, γ2 = 3.75, γ3 = −15



1.1.2 Determination of Adaptation Gain

• Consider the plant transfer function G(s).

• Multiply the denominator of G(s) by s and add the term µ to get the characteristic equation

sG(s) + µ = 0

where, µ = γymuck.

• Find µ that places all the roots in left half of S − plane.

• If ymuck = constant ⇒ γ =µ

ymuck


Examples

1. First order systemLet :

B = 1, A = s+ 1, µ = 1



2. Second order systemLet :

B = 1, A = s2 + s+ 1, µ = 0.4



1.1.3 Normalized MIT Rule

Procedure

Process : G(s) =y

u(1.8)

Model : Gm(s) =ymuc

(1.9)

Control law : u(t) = f(uc, y) (1.10)

Get closed loop from [1.8] & [1.10] :y

uc(1.11)

Error : e = y − ym (1.12)∂e

∂θ=

∂y

∂θ= −ϕ (1.13)

Normalized MIT Rule :dθ

dt= γ

ϕe

α + ϕTϕ(1.14)

α > 0 (1.15)

Examples

1. Gain Adjustment

Gp(s) = θG(s) = θ2

s2 + 2s+ 4

Gm(s) = θoG(s) = θo2

s2 + 2s+ 4; θo = 2

e = y − ym = θG(s)uc − θoG(s)uc

ϕ = −∂e∂θ

= −G(s)uc = −ymθ0

dθ

dt= γ

ϕe

α + ϕTϕ= −γ yme/θ0

α + (ym/θ0)2= −γ yme

α + y2m




At

γ = 1.2, α = 0.1




dy

dt= −ay + bu

G(s) =Y

U=

b

s+ adymdt

= −amy + bmu

G(s) =YmU

=bm

s+ amUse the control law : u(t) = touc(t)− soy(t)

U = toUc − soY = Ys+ a

b

⇒ Y

Uc=

tos+ab + so

=boto

s+ a+ bsobm = btoam = a+ bso

e = Y − Ym =bto

s+ a+ bsoUc =

bms+ am

Uc

∂e

∂to=

b

s+ a+ bsoUc =

b

s+ amUc

≈ Gm(s)Uc = ym∂e

∂so=

−b2to(s+ a+ bso)2

Uc =−b

s+ a+ bsoY =

−bs+ am

Y

≈ −Gm(s)Ydθ

dt= γ

ϕe

α + ϕTϕdtodt

= −γ1yme

α1 + y2m

dsodt

= γ2Gm(s)ye

α2 + (Gm(s)y)2

Let : a = 1, b = 2, am = 8, bm = 8




At

γ1 = 5, γ2 = 10, α1 = 0.1, α2 = 5




G(s) =y

u=

s+ b

s2 + a1s+ a0

Gm(s) =ymuc

=s+ bm

s2 + a1ms+ a0m

Let the control law : u(t) = t0uc(t)− s0y(t)

ys2 + a1s+ a0

s+ b= t0uc − s0y

⇒ y

uc=

t0s+ bt0s2 + (a1 + s0)s+ a0 + bs0

e = y − ym∂e

∂t0=

s+ b

s2 + (a1 + s0)s+ a0 + bs0uc ≈ Gm(s)uc = ym

∂e

∂s0=

−t0(s+ b)(s+ b)

[s2 + (a1 + s0)s+ a0 + bs0]2uc

=−(s+ b)

s2 + (a1 + s0)s+ a0 + bs0y =−y2

t0ucdθ

dt= γ

ϕe

α + ϕTϕdt0dt

= −γ1yme

α1 + y2m

ds0

dt= γ2

y2ucα2u2

c + y4

Figure 1.9: Second order system adjustment block diagram



Let :

G(s) =s+ 5

s2 + 4s+ 3(stable), Gm(s) =

s+ 10

s2 + 8s+ 10, γ1 = 15, γ2 = 0.1, α1 = 15,

α2 = 20



Let :

G(s) =s+ 5

s2 + 2s− 3(unstable), Gm(s) =

s+ 10

s2 + 8s+ 10, γ1 = 400, γ2 = 1,

α1 = α2 = 0.001



4. Second Order System Adjustment with First Order Controller

G(s) =y

u=

s+ b

s2 + a1s+ a0=N

D

Gm(s) =ymuc

=s+ bm

s3 + a2ms2 + a1ms+ a0m

Let the control law : u(t) =t0

1 + r1suc(t)−

s0

1 + r1sy(t)

y =t0G(s)

1 + r1suc −

s0G(s)

1 + r1sy

y

uc=

t0N

D(1 + r1s) + s0Ne = y − ym

∂e

∂t0=

N

D(1 + r1s) + s0Nuc ≈ Gmuc = ym

∂e

∂s0=

−N2

[D(1 + r1s) + s0N ]2uc ≈ −Gm

y

ucuc = −Gmy

∂e

∂r1=

−NDs[D(1 + r1s) + s0N ]2

uc ≈ −GmDs

D(1 + r1s) + s0Nuc

≈ −Gmuc = −ymdθ

dt= γ

ϕe

α + ϕTϕ∂t0∂t

= −γ1yme

α1 + y2m

∂s0

∂t= γ2

Gmye

α2 + (Gmy)2

∂r1

∂t= γ3

yme

α3 + y2m

Let:

G(s) =s+ 2

s2 + s+ 6, Gm(s) =

s+ 24

s3 + 9s2 + 26s+ 24,

γ1 = 400, γ2 = 33.3333, γ3 = −50, α1 = 100, α2 = 100, α3 = 400



1.1.4 Design of MRAS Using Lyapunov Theory

We will now show how Lyapunovs stability theory can be used to construct algorithms for adjusting parameters in adaptive

systems. To do this, we first derive a differential equation for the error, e = y− ym. This differential equation contains the

adjustable parameters. We then attempt to find a Lyapunov function and an adaptation mechanism such that the error will

go to zero.When using the Lyaponov theory for adaptive systems, we find that dV/dt is usually only negative semi-definite.

The procedure is to determine the error equation and a Lyapunov function with a bounded second derivative.

General Case

Given a continuous time system and the target dynamics

x = Ax+Bu, ˙xm = Amxm +Bmuc

Consider the controller and the error signals

u(t) = Muc(t)− Lx(t), e(t) = x(t)− xm(t)

If the model-matching problem is solvable, then the error dynamics is

de

dt= Ax+Bu− Amxm −Bmuc

= Ame+ (A− Am −BL)x+ (BM −Bm)uc

= Ame+ Ψ(x, uc) •(θ − θ0

)Consider the following Lyapunov function candidate

V =1

2

[eTPe+

1

γ

(θ − θ0

)T (θ − θ0

)]The time-derivative of V is

V =1

2eT[PAm + AT

mP]e+

(θ − θ0

)TΨTPe+

1

γ

(θ − θ0

)Tθ

If we solve the Lyapunov equation for P = P T > 0

PAm + ATmP = −Q, Q > 0

and choose the update law as

θ = −γΨTPe = −γΨT (x, uc) • P • (x− xm)

then

V = −1

2eT (t)Qe(t)

and we conclude that e(t)→ 0



Examples

1. Gain Adjustment

Process : x = −ax+ buModel : ˙xm = −amxm + bmuc

a = amControl law : u(t) = m1uc(t)

de

dt= −ame+ (−a+ am)x+ (bm1 − bm)uc = −ame+ (bm1 − bm)uc

= −ame+ Ψ(x, uc) •(θ − θ0

)Ψ =

ucb

θ = m1

θ0 =bmb

θ = −γucbPe = −γuce


Leta = am = 4, b = 2, bm = 4, γ = 3




Process : x = −ax+ buModel : ˙xm = −amxm + bmuc

Control law : u(t) = m1uc(t)− l1x(t)de

dt= −ame+ (−a+ am − bl1)x+ (bm1 − bm)uc

= −ame+ Ψ(x, uc) •(θ − θ0

)Ψ =

[−xb

ucb

]θ = [l1 m1]T

θ0 =

[am − ab

bmb

][l1m1

]=

[γ1xe−γ2uce

]


Leta = 2, am = 8, b = 1, bm = 8, γ1 = 0.1, γ2 = 3




Process : G(s) =s+ 5

s2 + 4s+ 3

Model : Gm(s) =s+ 10

s2 + 8s+ 10

A =

[0 1−3 −4

]B =

[01

]Am =

[0 1−10 −8

]Bm =

[01

]Control law : u(t) = m1uc −

[l1 0

]x(t)

de

dt=

[0 1−10 −8

]e+

[0 0

7− l1 0

]x(t) +

[0

m1 − 1

]uc

Ψ = [−x uc]

θ = [l1 m1]T[l1m1

]=

[γ1xe−γ2uce

]

Figure 1.13: second order system adjustment block diagram

Atγ1 = 0.01, γ2 = 0.4



4. Second Order System Adjustment (another solution)

Process : G(s) =s+ 5

s2 + 4s+ 3

Model : Gm(s) =s+ 10

s2 + 8s+ 10

A =

[0 1−3 −4

]B =

[01

]Am =

[0 1−10 −8

]Bm =

[01

]Control law : u(t) = m1uc −

[l1 l2

]x(t)

de

dt=

[0 1−10 −8

]e+

[0 0

7− l1 4− l2

]x(t) +

[0

m1 − 1

]uc

Ψ = [−x − x uc]

θ = [l1 l2 m1]T l1l2m1

=

γ1xeγ2xe−γ3uce

Figure 1.14: second order system adjustment block diagram

Atγ1 = 0.01, γ2 = 0.8, γ2 = 0.5


1.2. MATLAB CODES AND SIMULATION Adaptive Control

1.2 MATLAB Codes and Simulation

1 http://goo.gl/2nhYkk

2 http://goo.gl/RqIX4D

3 http://goo.gl/rOCcS6

4 http://goo.gl/Bx5Tnn

1 http://goo.gl/trRRzu

2 http://goo.gl/QrSHSW

1 http://goo.gl/hdrI90

2 http://goo.gl/DDXsR9

3 http://goo.gl/G67l2g

4 http://goo.gl/QJWWvw

1 http://goo.gl/YzgkEL

2 http://goo.gl/86ZvGi

3 http://goo.gl/cSxmF3

4 http://goo.gl/MW7zTw

1.3 References

1. Karl Johan Astrom, Adaptive Control, 2nd Edition.

2. Leonid B. Freidovich, lecture 12.

1.4 Contacts

[email protected]


http://goo.gl/2nhYkk

http://goo.gl/RqIX4D

http://goo.gl/rOCcS6

http://goo.gl/Bx5Tnn

http://goo.gl/trRRzu

http://goo.gl/QrSHSW

http://goo.gl/hdrI90

http://goo.gl/DDXsR9

http://goo.gl/G67l2g

http://goo.gl/QJWWvw

http://goo.gl/YzgkEL

http://goo.gl/86ZvGi

http://goo.gl/cSxmF3

http://goo.gl/MW7zTw

model reference adaptation systems (mras)

Engineering