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Model-Reference Adaptive Systems (MRAS)

The model-reference adaptive system (MRAS) is an important adaptive controller. It is 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. The block dia. of MRAS system is

shown in Fig. 1. The system has an ordinary feedback loop composed of the process and the

controller. Another feedback loop changes the controller parameters on the basis of feedback from

the error which is the difference between the output of the system and 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 parameters in a MRAS system can be adjusted by using the following

methods.

• Gradient method

• Lyapunov stability theorem

Fig. 1. Block dia. of MRAS system

The 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 lab. at MIT.

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. Consider the loss function

𝐽(𝜃) =1

2𝑒2

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Here the objective is to adjust parameters in such a way that the loss function will be minimum.

To make J small, it is possible to change the parameters in the direction of the negative gradient

of (J). i. e.,

𝒅𝜽

𝒅𝒕= −𝜸

𝝏𝑱

𝝏𝜽= −𝜸𝒆

𝝏𝒆

𝝏𝜽= −𝜸𝛁𝑱 (1)

This is known as MIT rule.

The partial derivative 𝜕𝑒

𝜕𝜃 is called the sensitive derivative of the system and it tells how the error

is influenced by the adjustable parameter. If it is assumed that the parameter changes are slower

than the other variables in the system, then the derivatives 𝜕𝑒

𝜕𝜃 can be evaluated under the

assumption that 𝜃 is constant.

The loss function can also be chosen as

𝐽(𝜃) = |𝑒|

So, the gradient method gives 𝒅𝜽

𝒅𝒕= −𝜸

𝝏𝒆

𝝏𝜽 𝒔𝒊𝒈𝒏 𝒆

The first MRAS was implemented was based on this formula.

Equation (1) is also applicable for many adjustable parameters. In that case, the variable 𝜃 is

interpreted as a vector and 𝜕𝑒

𝜕𝜃 is as the gradient of the error with respect to the parameters.

𝜃𝑖 = [𝜃1 𝜃2 𝜃3 ⋯ 𝜃𝑛] and 𝜕𝑒

𝜕𝜃𝑖= [

𝜕𝑒

𝜕𝜃1

𝜕𝑒

𝜕𝜃2

𝜕𝑒

𝜕𝜃3⋯

𝜕𝑒

𝜕𝜃𝑛].

Note:

• This method can also be applied to nonlinear systems. It can also be used to handle partially

known systems.

• The MIT rule will perform well if the adaptation gain (𝛾) is small. Consequently, it is not

possible to give fixed limits that guarantee stability.

Example: MRAS for a first-order system using MIT rule

Consider a process described by 𝑑𝑦

𝑑𝑡= −𝑎𝑦 + 𝑏𝑢

where u and y are the control variable and the measured output respectively.

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The desired response is given by 𝑑𝑦𝑚

𝑑𝑡= −𝑎𝑚𝑦𝑚 + 𝑏𝑚𝑢𝑐 (model)

where 𝑎𝑚 > 0 and the reference signal is bounded.

Let the controller is given by 𝑢 = 𝜃1𝑢𝑐 − 𝜃2𝑦

Here the controller has two parameters.

So, 𝑑𝑦

𝑑𝑡= −𝑎𝑦 + 𝑏𝑢 = −𝑎𝑦 + 𝑏(𝜃1𝑢𝑐 − 𝜃2𝑦) = −(𝑎 + 𝑏𝜃2)𝑦 + 𝑏𝜃1𝑢𝑐

The input-output relations of the system and the model will be same if

𝑑𝑦

𝑑𝑡=

𝑑𝑦𝑚

𝑑𝑡

∴ −(𝑎 + 𝑏𝜃2)𝑦 + 𝑏𝜃1𝑢𝑐 = −𝑎𝑚𝑦𝑚 + 𝑏𝑚𝑢𝑐

𝜃1 = 𝜃10 =

𝑏𝑚

𝑏; 𝜃2 = 𝜃2

0 =𝑎𝑚−𝑎

𝑏

This is called the perfect model-following.

The process is 𝑑𝑦

𝑑𝑡= −(𝑎 + 𝑏𝜃2)𝑦 + 𝑏𝜃1𝑢𝑐

∴ 𝑦 =𝑏𝜃1

𝑝+𝑎+𝑏𝜃2𝑢𝑐 (where 𝑝 =

𝑑𝑦

𝑑𝑡 is the differential operator)

Here error is 𝑒 = 𝑦 − 𝑦𝑚 . So, the sensitivity derivatives are

𝜕𝑒

𝜕𝜃1=

𝑏

𝑝+𝑎+𝑏𝜃2𝑢𝑐

𝜕𝑒

𝜕𝜃2= −

𝑏2𝜃1

(𝑝+𝑎+𝑏𝜃2)2 𝑢𝑐 = −𝑏

𝑝+𝑎+𝑏𝜃2𝑦

The above expressions depend on the process parameters a and b which are unknown.

Approximately consider that 𝑝 + 𝑎 + 𝑏𝜃20 ≈ 𝑝 + 𝑎𝑚

With this approximation we get the following equations for updating the controller parameters:

𝑑𝜃1

𝑑𝑡= −𝛾′ (

𝑏

𝑝 + 𝑎𝑚𝑢𝑐) 𝑒 = −

𝛾′𝑏

𝑎𝑚(

𝑎𝑚

𝑝 + 𝑎𝑚𝑢𝑐) 𝑒 = −𝛾 (

𝑎𝑚

𝑝 + 𝑎𝑚𝑢𝑐) 𝑒

𝑑𝜃2

𝑑𝑡=

𝛾′𝑏

𝑎𝑚(

𝑎𝑚

𝑝 + 𝑎𝑚𝑦) 𝑒 = 𝛾 (

𝑎𝑚

𝑝 + 𝑎𝑚𝑦) 𝑒

In above equations, 𝛾 =𝛾′𝑏

𝑎𝑚.

The sign of parameter b must be known to have the correct sign of 𝛾. The block dia. of MRAS

system is shown in Fig. below.

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Fig. Block dia. of a model-reference controller for a first-order process

Determination of the Adaptation Gain

In case adaptive control law, there is one parameter i.e. adaptation gain (𝛾) that has to be chosen

by the user using the MIT rule. Consider an MRAS for adaptation of a feedforward gain as shown

in Fig. 2. Here 𝑘𝐺(𝑠) is the process with G(s) known and k is an unknown constant. It is assumed

that G(s) is stable. Here the objective is to find a feedforward controller that gives the transfer

function 𝑘0𝐺(𝑠). Here 𝑢𝑐 is the command signal, 𝑦𝑚 is the model output, y is the process output

and 𝜃 is the adjustable parameter. The process transfer function 𝑘𝐺(𝑠) will be equal to model

transfer function 𝑘0𝐺(𝑠) if the parameter 𝜃 is chosen as 𝜃 =𝑘0

𝑘.

Fig. 2. Block dia. of an MRAS for adjustment of a feedforward gain

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The above system is described by the following equations.

𝑦 = 𝑘𝐺(𝑝)𝑢 (where 𝑝 = 𝑑𝑑𝑡⁄ is the differential operator.)

𝑦𝑚 = 𝑘𝑜𝐺(𝑝)𝑢𝑐

𝑢 = 𝜃𝑢𝑐

𝑒 = 𝑦 − 𝑦𝑚 = 𝑘𝐺(𝑝)𝜃𝑢𝑐 − 𝑘𝑜𝐺(𝑝)𝑢𝑐

The sensitivity derivative is given as

𝜕𝑒

𝜕𝜃= 𝑘𝐺(𝑝)𝑢𝑐 =

𝑘

𝑘0

(𝑘0𝐺(𝑝)𝜃𝑢𝑐) =𝑘

𝑘0𝑦𝑚

The MIT rule gives

𝑑𝜃

𝑑𝑡= −𝛾′𝑒

𝑘

𝑘0𝑦𝑚 = −𝛾𝑦𝑚𝑒 (2)

where 𝛾 = 𝛾′ 𝑘

𝑘0 has been introduced.

Eq. (2) gives the law for adjusting the parameter.

𝑑𝜃

𝑑𝑡= −𝛾𝑦𝑚𝑒 = −𝛾𝑦𝑚(𝑦 − 𝑦𝑚)

Putting the expression of y and 𝑦𝑚, we get

𝑑𝜃

𝑑𝑡+ 𝛾𝑦𝑚(𝑘𝐺(𝑝)𝜃𝑢𝑐) = 𝛾𝑦𝑚

2 (3)

This above equation is known as parameter equation.

If G(s) is a rational transfer function, then Eq. (3) will represent a linear time-varying ordinary

differential equation. Such equations may exhibit very complicated behavior. It is not possible to

give a simple analytical characterization of the properties of the system, particularly how they are

influenced by the parameter 𝛾.

Lyapunov Theory

There is no guarantee that an adaptive controller based on the MIT rule will give a stable closed-

loop system. But designed adaptive controller based on Lyapunov theory can guarantee the

stability of the system.

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Stability in the sense of Lyapunov

Consider an autonomous system described by

�̇� = 𝑓(𝑋)

The above system is said to be stable in the sense of Lyapunov about an equilibrium point 𝑋𝑒 if

for every 휀 > 0,there exist 𝛿(휀) > 0 such that ‖𝑥0 − 𝑥𝑒‖ < 𝛿 and ‖𝑥(𝑡) − 𝑥𝑒‖ < 휀 for ∀𝑡 ≥ 0.

Here ‖. ‖ represents the Euclidean norm. The ‖𝑋‖ = √𝑥12 + 𝑥2

2 + ⋯ + 𝑥𝑛2, which represents a

hyper-spherical region.

A dynamic system is Lyapunov stable about an equilibrium point if state trajectories are confined

to a bounded region whenever the initial condition is chosen sufficiently close to equilibrium point.

Asymptotically Stability Theorem

An equilibrium state 𝑋𝑒 of an autonomous system is asymptotically stable if

• it is stable and 𝛿𝑎 < 𝛿

• exist if ‖𝑥0 − 𝑥𝑒‖ < 𝛿𝑎 and ‖𝑥(𝑡) − 𝑥𝑒‖ → 0 as 𝑡 → ∞.

For asymptotically stable, state trajectories converge to the equilibrium point.

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Instability Theorem

A dynamic system is unstable about an equilibrium point if state trajectories leave a bounded

region whenever the initial condition is chosen sufficiently close to equilibrium point.

Definiteness of scalar function

Positive definite function:

The function 𝑓(𝑥) is positive definite if 𝑓(0) = 0 and 𝑓(𝑥) > 0 for all 𝑥 ∈ 𝑅𝑛.

Example: 𝑓(𝑥1, 𝑥2) = 𝑥12 + 𝑥2

2.

Positive semi-definite Function:

The function 𝑓(𝑥) is positive semidefinite if 𝑓(0) = 0 and 𝑓(𝑥) ≥ 0 for all 𝑥 ∈ 𝑅𝑛.

Example: 𝑓(𝑥1, 𝑥2) = (𝑥1 + 𝑥2)2.

Negative definite function:

If the function −𝑓(𝑥) is positive definite then function is called negative definite function.

Example: 𝑓(𝑥1, 𝑥2) = −(𝑥12 + 𝑥2

2).

Negative semi-definite function:

If the function −𝑓(𝑥) is positive semidefinite then function is called negative semidefinite

function.

Example: 𝑓(𝑥1, 𝑥2) = −(𝑥1 + 𝑥2)2

Indefinite function:

A function is not definite or semidefinite in either sense is defined to be indefinite function. For

some values of x, f(x) is positive and for some values x, it is negative.

Example: 𝑓(𝑥1, 𝑥2) = 𝑥1𝑥2 + 𝑥22.

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Sign definiteness of quadratic function

Consider a continuous function V (x) that can be expressed as

𝑉(𝑥) = 𝑥𝑇𝑃𝑥

where 𝑃 ∈ 𝑅𝑛×𝑛 is a symmetric matrix.

If the matrix P is positive definite, then the corresponding quadratic form 𝑉(𝑥)is also will be

positive definite.

The sign of matrix P can be determined using Sylvester’s criterion and also form the eigenvalues

of the matrix P.

Sylvester’s criterion:

A symmetric matrix P is positive definite if and only if all its principle minors are positive.

Consider P is a 3 × 3 symmetric matrix and 𝑃 = [𝑃11 𝑃12 𝑃13

𝑃12 𝑃22 𝑃23

𝑃13 𝑃23 𝑃33

]

If 𝑃11 > 0; |𝑃11 𝑃12

𝑃12 𝑃22| > 0 and |

𝑃11 𝑃12 𝑃13

𝑃12 𝑃22 𝑃23

𝑃13 𝑃23 𝑃33

| > 0, then P will be positive definite.

Alternatively, the sign definiteness of P can be determined from the eigen values of P also. The

matrix P will be positive definite, if and only if all the eigenvalues of P are positive. If

eigenvalues are positive or zero, then P will be positive semidefinite.

Lyapunov stability criterion (Direct method)

In this approach, stability of the system can be determined without solving the differential

equation. So, this approach is known as direct method. The direct method of Lyapunov stability

criterion is based upon the concept of energy and the relation of stored energy with system stability.

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Consider a spring-mass-damper system as shown in Fig. above. Here the dynamics of the system

is given by-

�̈� + 𝐷�̇� + 𝐾𝑥 = 0

Where x is the linear displacement of the body, M is the mass of the body and here M=1, D is the

damping co-efficient and K is the spring constant.

Here state variables are-

𝑥 = 𝑥1 ; �̇� = 𝑥2

So the equations are-

�̇�1 = 𝑥2

�̇�2 = −𝐷𝑥2 − 𝐾𝑥1

For this system, the energy storage elements are mass and the spring. So the internal energy stored

by the system-

𝑉(𝑋) =1

2𝑥2

2 +1

2𝐾𝑥1

2

The rate of change of energy is given by-

�̇�(𝑋) = 𝑥2�̇�2 + 𝐾𝑥1�̇�1

= 𝑥2(−𝐷𝑥2 − 𝐾𝑥1) + 𝐾𝑥1𝑥2

= −𝐷𝑥22

From the above equation, it can be observed that the rate of change of internal stored energy is

negative. So here internal stored energy decreases. So, the system is here a stable system.

So, an energy function is associated with the direct method of the Lyapunov stability criterion and

that energy function should be positive definite. This energy function can be determined easily if

the physical system is known to us. But it is very difficult to determine energy function from the

mathematical equations of the system if the system components are not known to us. In that case

fictious energy function is considered. This fictious energy function is known as Lyapunov

function and the Lyapunov function should be positive definite function.

Lyapunov’s stability theorem (direct method)

Consider an autonomous system described by

�̇� = 𝑓(𝑋) (4)

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Suppose that there exists a scalar function 𝑉(𝑥) which for some real number 휀 > 0, satisfies the

following properties for all x in the region ‖𝑥‖ ≤ 휀:

• V(x) is positive definite function i.e. 𝑉(𝑥) > 0; 𝑥 ≠ 0 and 𝑉(0) = 0.

• V(x) has continuous partial derivatives with respect to all components of x.

The equilibrium state 𝑥𝑒 = 0 of the system given by equation (4) is

• asymptotically stable if �̇�(𝑥) < 0, 𝑥 ≠ 0, i.e., �̇�(𝑥) is a negative definite function.

• asymptotically stable in-the-large if �̇�(𝑥) < 0, 𝑥 ≠ 0 and in addition 𝑉(𝑥) → ∞ as

‖𝑥‖ → ∞.

Lyapunov’s instability theorem (direct method)

Consider an autonomous system described by

�̇� = 𝑓(𝑋) (5)

Suppose that there exists a scalar function 𝑉(𝑥) which for some real number 휀 > 0, satisfies the

following properties for all x in the region ‖𝑥‖ ≤ 휀:

• V(x) is positive definite function i.e. 𝑉(𝑥) > 0; 𝑥 ≠ 0 and 𝑉(0) = 0.

• V(x) has continuous partial derivatives with respect to all components of x.

The equilibrium state 𝑥𝑒 = 0 of the system given by equation (5) is unstable if �̇�(𝑥) > 0, 𝑥 ≠ 0,

i.e., �̇�(𝑥) is a positive definite function.

Lyapunov function for linear system

Using Lyapunov’s direct method, stability of both linear and non-linear system can be determined.

Consider a linear autonomous system described by

�̇� = 𝐴𝑥 (6)

where A is 𝑛 × 𝑛 real constant matrix.

The linear system described by equ. (6) is globally asymptotically stable at the origin if and only

if, for any given symmetric positive definite matrix Q, there exists a symmetric positive definite

matrix P that satisfies the matrix equation

𝐴𝑇𝑃 + 𝑃𝐴 = −𝑄

Proof: Consider Lyapunov function 𝑉(𝑥) = 𝑥𝑇𝑃𝑥

where 𝑉(𝑥) is positive definite function.

Then �̇�(𝑥) = �̇�𝑇𝑃𝑥 + 𝑥𝑇𝑃�̇�

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= 𝑥𝑇𝐴𝑇𝑃𝑥 + 𝑥𝑇𝑃𝐴𝑥

= 𝑥𝑇(𝐴𝑇𝑃 + 𝑃𝐴)𝑥

= −𝑥𝑇𝑄𝑥

∴ 𝐴𝑇𝑃 + 𝑃𝐴 = −𝑄

The above equation is known as Lyapunov equation for linear system.

As Q is positive definite, �̇�(𝑥) will be negative definite. So, the system will be stable.

Norm of x is defined as ‖𝑥‖ = (𝑥𝑇𝑃𝑥)1

2⁄

So 𝑉(𝑥) = ‖𝑥‖2

𝑉(𝑥) → ∞ as ‖𝑥‖ → ∞

So, the system is here globally asymptotically stable at the origin.

Note:

• The system will be asymptotically stable if for positive definite matrix Q, the solution of

Lyapunov equation P is also becomes positive definite.

• The matrix P is here symmetric matrix.

• For convenience the matrix Q is often chosen as identity matrix.

Lyapunov Theory for Time-variable Systems

Consider a time-varying system described by the equation

𝑑𝑥

𝑑𝑡= 𝑓(𝑥, 𝑡) (7)

The origin is an equilibrium point of equ.(7) if 𝑓(0, 𝑡) = 0, ∀𝑡 ≥ 0. It is assumed that f is such

that solutions exist for all 𝑡 ≥ 𝑡0. It is also assumed that f is piecewise continuous in t and locally

Lipschitz in x in a neighborhood of 𝑥(𝑡) = 0.

In time-varying case, the solution depends on t as well as on the starting time 𝑡0. This implies that

in the definition of stability in the sense of Lyapunov 𝛿 depends on 휀 and 𝑡0. The definition on

stability can be refined to give uniform stability properties with respect to the initial time.

Uniform Lyapunov Stability

The solution 𝑥(𝑡) = 0 of equ.(7) is uniformly stable if for 휀 > 0 there exists a number 𝛿(휀) > 0,

independent of 𝑡0, such that

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‖𝑥(𝑡0)‖ < 𝛿 ⇒ ‖𝑥(𝑡)‖ < 휀 ∀𝑡 ≥ 𝑡0 ≥ 0

The solution is uniformly asymptotically stable if it is uniformly stable and there is 𝑐 > 0,

independent of 𝑡0, such that 𝑥(𝑡) ⟶ 0 as 𝑡 ⟶ ∞, uniformly in 𝑡0 for all ‖𝑥(𝑡0)‖ < 𝑐.

Definition of Class K function

A continuous function 𝛼: [0, 𝛼) → [0, ∞) is said to belong to class K if it is strictly increasing and

𝛼(0) = 0. It is said to belong to class 𝐾∞ if 𝛼 = ∞ and 𝛼(𝑟) → ∞ as 𝑟 → ∞.

Lyapunov’s stability theorem: Time-varying systems

Let x=0 be an equilibrium point of equ. (7) and 𝐷 = {𝑥 ∈ 𝑅𝑛|‖𝑥‖ < 𝑟}. Let V be a continuously

differentiable function such that

𝛼1(‖𝑥‖) ≤ 𝑉(𝑥, 𝑡) ≤ 𝛼2(‖𝑥‖) (8)

𝑑𝑉

𝑑𝑡=

𝜕𝑉

𝜕𝑡+

𝜕𝑉

𝜕𝑥𝑓(𝑥, 𝑡) ≤ −𝛼3(‖𝑥‖)

for ∀𝑡 ≥ 0 where 𝛼1, 𝛼2 and 𝛼3 are class K functions. Then x=0 is uniformly asymptotically stable.

Note:

• A function 𝑉(𝑥, 𝑡) satisfying the left inequality of (8) is said to be positive definite. A

function satisfying the right inequality of (8) is said to be decrescent.

• To show stability for time-variable systems, it is necessary to bound the function 𝑉(𝑥, 𝑡)

by a function that doesn’t depend on t.

• When using Lyapunov theory on adaptive control problem, it can be often found that 𝑑𝑉𝑑𝑡⁄

only is negative semidefinite. So additional conditions must be imposed on the system.

Barbalat’s lemma

If g is a real function of a real variable t, defined and uniformly continuous for 𝑡 ≥ 0 and if the

limit of the integral ∫ 𝑔(𝑠)𝑑𝑠𝑡

0 as t tends to infinity exists and is a finite number then lim

𝑡→∞𝑔(𝑡) = 0.

Boundedness and convergence set

Let 𝐷 = {𝑥 ∈ 𝑅𝑛|‖𝑥‖ < 𝑟} and suppose that 𝑓(𝑥, 𝑡) is locally Lipschitz on 𝐷 × [0, ∞). Let V be

a continuously differentiable function such that

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𝛼1(‖𝑥‖) ≤ 𝑉(𝑥, 𝑡) ≤ 𝛼2(‖𝑥‖)

𝑑𝑉

𝑑𝑡=

𝜕𝑉

𝜕𝑡+

𝜕𝑉

𝜕𝑥𝑓(𝑥, 𝑡) ≤ −𝑊(𝑥) ≤ 0

for ∀𝑡 ≥ 0, ∀𝑥 ∈ 𝐷 where 𝛼1and 𝛼2 are class K functions defined on [0, r) and W(x) is continuous

on D.

It is assumed that 𝑑𝑉𝑑𝑡⁄ is uniformly continuous in t. Then all solutions of equ.(7) with ‖𝑥(𝑡0)‖ <

𝛼2−1(𝛼1(𝑟)) are bounded and satisfy

𝑊(𝑥(𝑡)) → 0 as 𝑡 → ∞.

If all the assumptions hold globally and 𝛼1 belongs to class 𝐾∞, the statement is true for all 𝑥(𝑡0) ∈

𝑅𝑛.

This theorem states that the states of the system are bounded and that they approach the set

{𝑥 ∈ 𝐷|𝑊(𝑥) = 0}.

Design of MRAS using Lyapunov Theory

Lyapunov’s stability theory can also be used to construct algorithms for adjusting parameters in

adaptive system. In this case, a differential equation for the error 𝑒 = 𝑦 − 𝑦𝑚 has to be derived.

This equation contains the adjustable parameters. Then a Lyapunov function and an adaptation

mechanism have to be found such that the error will go to zero.

Example: First order MRAS based on stability theory

Consider a process described by 𝑑𝑦

𝑑𝑡= −𝑎𝑦 + 𝑏𝑢

The desired response is given by 𝑑𝑦𝑚

𝑑𝑡= −𝑎𝑚𝑦𝑚 + 𝑏𝑚𝑢𝑐

where 𝑎𝑚 > 0 and the reference signal is bounded.

Let the controller is given by 𝑢 = 𝜃1𝑢𝑐 − 𝜃2𝑦

Here error is 𝑒 = 𝑦 − 𝑦𝑚 and the objective is to make the error small.

𝑑𝑒

𝑑𝑡=

𝑑𝑦

𝑑𝑡−

𝑑𝑦𝑚

𝑑𝑡 = −𝑎𝑦 + 𝑏𝑢 + 𝑎𝑚𝑦𝑚 − 𝑏𝑚𝑢𝑐 = −𝑎𝑦 + 𝑏(𝜃1𝑢𝑐 − 𝜃2𝑦) + 𝑎𝑚𝑦𝑚 − 𝑏𝑚𝑢𝑐

= −𝑎𝑦 + 𝑏(𝜃1𝑢𝑐 − 𝜃2𝑦) + 𝑎𝑚𝑦𝑚 − 𝑏𝑚𝑢𝑐

= −𝑎𝑦 + 𝑏(𝜃1𝑢𝑐 − 𝜃2𝑦) + 𝑎𝑚(𝑦 − 𝑒) − 𝑏𝑚𝑢𝑐

= −𝑎𝑚𝑒 − (𝑏𝜃2 + 𝑎 − 𝑎𝑚)𝑦 + (𝑏𝜃1 − 𝑏𝑚)𝑢𝑐

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Here a parameter adjustment mechanism has to be constructed that will drive the parameters

𝜃1and 𝜃2 to their desired values.

Consider the Lyapunov function as

𝑉(𝑒, 𝜃1, 𝜃2) =1

2[𝑒2 +

1

𝑏𝛾(𝑏𝜃2 + 𝑎 − 𝑎𝑚)2 +

1

𝑏𝛾(𝑏𝜃1 − 𝑏𝑚)2] assuming 𝑏𝛾 > 0

The above function is zero when e is zero and the controller parameters are equal to the correct

values. The derivative of the function must be negative.

𝑑𝑉

𝑑𝑡= 𝑒

𝑑𝑒

𝑑𝑡+

1

𝛾(𝑏𝜃2 + 𝑎 − 𝑎𝑚)

𝑑𝜃2

𝑑𝑡+

1

𝛾(𝑏𝜃1 − 𝑏𝑚)

𝑑𝜃1

𝑑𝑡

= 𝑒(−𝑎𝑚𝑒 − (𝑏𝜃2 + 𝑎 − 𝑎𝑚)𝑦 + (𝑏𝜃1 − 𝑏𝑚)𝑢𝑐) +1

𝛾(𝑏𝜃2 + 𝑎 − 𝑎𝑚)

𝑑𝜃2

𝑑𝑡+

1

𝛾(𝑏𝜃1 − 𝑏𝑚)

𝑑𝜃1

𝑑𝑡

= −𝑎𝑚𝑒2 +1

𝛾(𝑏𝜃2 + 𝑎 − 𝑎𝑚) (

𝑑𝜃2

𝑑𝑡− 𝛾𝑦𝑒) +

1

𝛾(𝑏𝜃1 − 𝑏𝑚) (

𝑑𝜃1

𝑑𝑡+ 𝛾𝑢𝑐𝑒)

If the parameters are updated as

𝑑𝜃2

𝑑𝑡= 𝛾𝑦𝑒;

𝑑𝜃1

𝑑𝑡= −𝛾𝑢𝑐𝑒

then 𝑑𝑉

𝑑𝑡= −𝑎𝑚𝑒2

The derivative of V with respect to time is negative semidefinite but not negative definite. This

implies that 𝑉(𝑡) ≤ 𝑉(0) and thus that e, 𝜃1 and 𝜃2 must be bounded. So 𝑦 = 𝑒 + 𝑦𝑚 also be

bounded.

To apply the theorem of boundedness,

𝑑2𝑉

𝑑𝑡2= −2𝑎𝑚𝑒

𝑑𝑒

𝑑𝑡= −2𝑎𝑚𝑒(−𝑎𝑚𝑒 − (𝑏𝜃2 + 𝑎 − 𝑎𝑚)𝑦 + (𝑏𝜃1 − 𝑏𝑚)𝑢𝑐)

As here 𝑢𝑐, e and y are bounded, so �̈� is also bounded. Hence 𝑑𝑉𝑑𝑡⁄ is uniformly continuous. So,

the error e will go to zero. The block dia. of the system is show in Fig. 3. The adjustment rule

obtained from Lyapunov theory is simpler than the MIT rule because Lyapunov theory does not

require filtering of the signals.

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15 | P a g e

Fig.3. Block dia. of an MRAS based on Lyapunov theory for a first-order system

State Space Systems

Lyapunov’s theory can be used to derive stable MRAS for general linear systems. The following

steps have to be followed.

1. Find a controller structure

2. Derive the error equation

3. Find a Lyapunov function and use it to derive a parameter updating law such that the error

will go to zero.

Consider a linear system described by 𝑑𝑥

𝑑𝑡= 𝐴𝑥 + 𝐵𝑢

Assume that it is desired to find a control law so that the response to command signal is given by 𝑑𝑥𝑚

𝑑𝑡= 𝐴𝑚𝑥𝑚 + 𝐵𝑚𝑢𝑐 (8)

A general linear control law for the system is given by 𝑢 = 𝑀𝑢𝑐 − 𝐿𝑥

So the closed-loop system becomes 𝑑𝑥

𝑑𝑡= 𝐴𝑥 + 𝐵𝑢 = 𝐴𝑥 + 𝐵(𝑀𝑢𝑐 − 𝐿𝑥)

𝑑𝑥

𝑑𝑡= (𝐴 − 𝐵𝐿)𝑥 + 𝐵𝑀𝑢𝑐 = 𝐴𝑐(𝜃)𝑥 + 𝐵𝑐(𝜃)𝑢𝑐 (9)

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16 | P a g e

All parameters in the matrices L and M may be chosen freely. There may also be constraints among

the parameters. The general case can be captured by assuming that the closed-loop system is

described by equ.(9) where matrices 𝐴𝑐 and 𝐵𝑐 depend on a parameter 𝜃.

Compatibility conditions:

It is not always possible to find parameters 𝜃 such that equ.(8) is equivalent to equ.(9). A sufficient

condition is that there exists a parameter value 𝜃0 such that

𝐴𝑐(𝜃0) = 𝐴𝑚 and 𝐵𝑐(𝜃0) = 𝐵𝑚

When all parameters in the control law can be chosen freely, it implies that

𝐴 − 𝐴𝑚 = 𝐵𝐿; 𝐵𝑚 = 𝐵𝑀

The error equation

The error is defined as 𝑒 = 𝑥 − 𝑥𝑚

∴𝑑𝑒

𝑑𝑡=

𝑑𝑥

𝑑𝑡−

𝑑𝑥𝑚

𝑑𝑡= 𝐴𝑥 + 𝐵𝑢 − 𝐴𝑚𝑥𝑚 − 𝐵𝑚𝑢𝑐 = 𝐴𝑥 + 𝐵𝑢 − 𝐴𝑚𝑥𝑚 − 𝐵𝑚𝑢𝑐 + 𝐴𝑚𝑥 − 𝐴𝑚𝑥

= (𝐴𝑚𝑥 − 𝐴𝑚𝑥𝑚) + (𝐴𝑥 − 𝐴𝑚𝑥) + 𝐵𝑢 − 𝐵𝑚𝑢𝑐

= 𝐴𝑚𝑒 + (𝐴 − 𝐴𝑚)𝑥 + 𝐵(𝑀𝑢𝑐 − 𝐿𝑥) − 𝐵𝑚𝑢𝑐

= 𝐴𝑚𝑒 + (𝐴 − 𝐴𝑚 − 𝐵𝐿)𝑥 + (𝐵𝑀 − 𝐵𝑚)𝑢𝑐

= 𝐴𝑚𝑒 + (𝐴𝑐(𝜃) − 𝐴𝑚)𝑥 + (𝐵𝑐(𝜃) − 𝐵𝑚)𝑢𝑐

= 𝐴𝑚𝑒 + Ψ(𝜃 − 𝜃0) (when 𝜃0 exist)

Consider the Lyapunov function as

𝑉(𝑒, 𝜃) =1

2(𝛾𝑒𝑇𝑃𝑒 + (𝜃 − 𝜃0)𝑇(𝜃 − 𝜃0))

where P is a positive definite matrix. So, the function V also is here positive definite.

𝑑𝑉

𝑑𝑡= −

𝛾

2𝑒𝑇𝑄𝑒 + 𝛾(𝜃 − 𝜃0)Ψ𝑇𝑃𝑒 + (𝜃 − 𝜃0)𝑇

𝑑𝜃

𝑑𝑡

= −𝛾

2𝑒𝑇𝑄𝑒 + (𝜃 − 𝜃0)𝑇 (

𝑑𝜃

𝑑𝑡+ γΨ𝑇𝑃𝑒)

where Q is positive definite and 𝐴𝑚𝑇 𝑃 + 𝑃𝐴𝑚 = −𝑄

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17 | P a g e

As Q and P are positive definite matrices and the above condition is satisfied, so 𝐴𝑚 is here stable.

If the parameter adjustment law is chosen as 𝑑𝜃

𝑑𝑡= −γΨ𝑇𝑃𝑒 then

𝑑𝑉

𝑑𝑡= −

𝛾

2𝑒𝑇𝑄𝑒

The time derivative of Lyapunov function is negative semidefinite. So the error goes to zero.

Adaptation of a Feedforward Gain

Fig.4. Block dia. of an MRAS for adjustment of a feedforward gain

Lyapunov theory can also be used to derive parameter adjustment laws for the problem of adjusting

a feedforward gain. Consider the problem as shown in Fig.4 where kG(s) is the plant transfer

function. Here G(s) is known but k is unknown. The desired response is given by here k0G(s). The

error is given by

𝑒 = 𝑦 − 𝑦𝑚 = 𝑘𝐺(𝑝)𝜃𝑢𝑐 − 𝑘𝑜𝐺(𝑝)𝑢𝑐 = 𝑘𝐺(𝑝)(𝜃 − 𝜃0)𝑢𝑐

where 𝜃0 =𝑘0

𝑘.

The state space model of the system is written as

𝑑𝑥

𝑑𝑡= 𝐴𝑥 + 𝐵(𝜃 − 𝜃0)𝑢𝑐

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18 | P a g e

𝑒 = 𝐶𝑥

If the homogeneous system �̇� = 𝐴𝑥 is asymptotically stable, there exist positive definite matrices

P and Q such that

𝐴𝑇𝑃 + 𝑃𝐴 = −𝑄

Consider a Lyapunov function

𝑉 =1

2(𝛾𝑥𝑇𝑃𝑥 + (𝜃 − 𝜃0)2)

𝑑𝑉

𝑑𝑡=

𝛾

2(

𝑑𝑥

𝑑𝑡

𝑇

𝑃𝑥 + 𝑥𝑇𝑃𝑑𝑥

𝑑𝑡) + (𝜃 − 𝜃0)

𝑑𝜃

𝑑𝑡

=𝛾

2((𝐴𝑥 + 𝐵𝑢𝑐(𝜃 − 𝜃0))𝑇𝑃𝑥 + 𝑥𝑇𝑃(𝐴𝑥 + 𝐵𝑢𝑐(𝜃 − 𝜃0))) + (𝜃 − 𝜃0)

𝑑𝜃

𝑑𝑡

= −𝛾

2𝑥𝑇𝑄𝑥 + (𝜃 − 𝜃0) (

𝑑𝜃

𝑑𝑡+ 𝛾𝑢𝑐𝐵𝑇𝑃𝑥)

If the parameter adjustment law is chosen as

𝑑𝜃

𝑑𝑡= −𝛾𝑢𝑐𝐵𝑇𝑃𝑥

then Lyapunov function will be negative for 𝑥 ≠ 0. So the state vector x and the error 𝑒 = 𝐶𝑥 will

go to zero as time t goes to infinity.

Output feedback

The parameter adjustment law that uses output feedback can be obtained if the Lyapunov

function is chosen as

𝐵𝑇𝑃 = 𝐶

where C is the output matrix of the system.

So 𝐵𝑇𝑃𝑥 = 𝐶𝑥 = 𝑒

Here the adjustment rule becomes 𝑑𝜃

𝑑𝑡= −𝛾𝑢𝑐𝑒.

Note:

Positive real transfer function

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19 | P a g e

A rational transfer function G with real coefficients is positive real if

𝑅𝑒 𝐺(𝑠) ≥ 0 for 𝑅𝑒 𝑠 ≥ 0

A transfer function G is strictly positive real if 𝐺(𝑠 − 휀) is positive real for some real 휀 > 0.

Kalman-Yakubovich lemma

Consider a linear time-invariant system as 𝑑𝑥

𝑑𝑡= 𝐴𝑥 + 𝐵𝑢; 𝑦 = 𝐶𝑥 which is completely

controllable and observable.

The transfer function of the system is given by 𝐺(𝑠) = 𝐶(𝑠𝐼 − 𝐴)−1𝐵.

The transfer function is strictly positive real if and only if there exist positive definite matrices P

and Q such that

𝐴𝑇𝑃 + 𝑃𝐴 = −𝑄 and 𝐵𝑇𝑃 = 𝐶

MRAS using the Lyapunov rule

Consider the problem of adapting a feedforward gain that can be represented as

𝑑𝑥

𝑑𝑡= 𝐴𝑥 + 𝐵(𝜃 − 𝜃0)𝑢𝑐

𝑒 = 𝐶𝑥

Assume that the transfer function G is strictly positive real. The parameter adjustment rule is

given by-

𝑑𝜃

𝑑𝑡= −𝛾𝑢𝑐𝑒

where 𝛾 is a positive constant which makes the output error e go to zero.

The block dia. of the adaptive system for feedforward gain compensation obtained by Lyapunov

rule is shown in Fig.5.

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Fig. 5. Block dia. of the adaptive system for feedforward gain compensation obtained by

Lyapunov rule

Note: The adjustment rules obtained using Lyapunov’s theory guarantee that the error goes to zero

but it cannot be asserted that the parameters converge to their correct values. This adjustment rules

have the remarkable property that arbitrarily high adaptation gains can be used.