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C. Melchiorri (DEI) Automatic Control & System Theory 1

AUTOMATIC CONTROL AND SYSTEM THEORY

MODAL ANALYSIS

Claudio Melchiorri

Dipartimento di Ingegneria dell’Energia Elettrica e dell’Informazione (DEI) Università di Bologna

Email: [email protected]

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Modal Analysis

We wish to study the time response of a dynamics system due to certain input. Recall that the response, of a 1st-order differential equation to a unit step input is an exponential function as follows: Continuous-time system

Discrete-time system

The time response of a continuous/discrete system depends on exponential functions eA t or Ad

k, and their structural properties can analyzed by means of a proper similarity transformation

where T is a constant matrix and A, Ad are Jordan (diagonal) matrix, with a simpler expression with respect to

(1)

(2)

A, Ad

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Modal Analysis

Definition: The time functions that appear in form of eA t or Adk, in (1) or (2) are

known as system linear modes.

System free response can be characterized on the basis of the modal analysis, i.e. of the values of the eigenvalues / eigenvectors. With this respect, two different cases must be considered: 1.  Distinct eigenvalues 2.  Multiple eigenvalues Notice that matrices A and Ad refer to the transformed state variables

T z = x (or equivalently z = T-1 x)

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Modal Analysis – Distinct eigenvalues

Distinct eigenvalues

In this case, matrix A (Ad) is in the diagonal form:

Continous-time case Discrete-time case

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•  The modes of a system are linearly independent since the eigenvalues λ1, …, λn are distinct.

eλ1 t, …, eλn t λ1

k, …, λnk

•  If the initial state x0 belongs to an eigenspace of a particular eigenvalue, then the free evolution of the system will remain in the same eigenspace.

•  Each mode may be ”excited" independently of the others; in general, complex modes are excited in pair, except when the initial state is complex as well

-1.5 -1 -0.5 0 0.5 1 1.5

-1 0

1 -1.5

-1 -0.5

0 0.5

1 1.5

V

x0

•  A matrix can be written in diagonal form iff the minimum polynomial has simple roots only

•  A matrix can be written in diagonal form iff there are n linearly independent eigenvectors

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0 2 4 6 8 0 10 20 30 40 50 60

-1 -0.5 0 0.5 1 -1

-0.5

0

0.5

1

0 2 4 6 8 0

0.5

1

1.5

2

-1 -0.5 0 0.5 1 -1

-0.5

0

0.5

1

0 2 4 6 8 0 0.2 0.4 0.6 0.8

1

-1 -0.5 0 0.5 1 -1

-0.5

0

0.5

1

Eigenvalue on the complex plane

λ = 0.5

λ = 0

λ = -0.5

Continuous-time systems

Initial Condition (IC): x0 = 1 Simple real eigenvalues

eλ t

System response

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C. Melchiorri (DEI) Automatic Control & System Theory 7 Claudio Melchiorri

Modal Analysis – Distinct eigenvalues Eigenvalue on the complex plane System response

λ = 1.5

λ = 1

λ = 0.5

Discrete-time systems

Initial Condition (IC): x0 = 1 Simple real Eigenvalues

λk

0 5 10 0 10 20 30 40 50 60

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 0

0.5

1

1.5

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 0

0.5

1

1.5

-2 -1 0 1 2 -2

-1

0

1

2

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0 5 10 0 0.2 0.4 0.6 0.8

1

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 -0.5

0

0.5

1

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 -2

-1

0

1

2

-2 -1 0 1 2 -2

-1

0

1

2

System response

λ = 0

λ = -0.5

λ = -1

Eigenvalue on the complex plane

Discrete-time systems

Initial Condition (IC): x0 = 1 Simple real Eigenvalues

λk

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0 5 10 -60 -40 -20

0 20 40 60

-2 0 2 -3 -2 -1 0 1 2 3

0 5 10 -2

-1

0

1

2

-2 0 2 -3 -2 -1 0 1 2 3

0 5 10 -2

-1

0

1

2

-2 0 2 -3 -2 -1 0 1 2 3

λ = 0.5 ± 2j

λ = ± 2j

λ = -0.5 ± 2j

Continuous-time systems

Initial Condition: x0 = 1 Simple complex eigenvalues

eσ t cos(ω t)

Eigenvalue on the complex plane System Response

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0 5 10 -1.5 -1

-0.5 0

0.5 1

1.5

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 -1.5 -1

-0.5 0

0.5 1

1.5

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 -1.5 -1

-0.5 0

0.5 1

1.5

-2 -1 0 1 2 -2

-1

0

1

2

λ = e § j π/4

λ = e § j π/2

λ = 0.5 e § j π/4

Eigenvalue on the complex plane System response

Discrete-Time systems

Initial condition: x0 = 1 Simple complex eigenvalues

eσ t cos(k ω)

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Modal Analysis – Multiple eigenvalues Continous-time systems - In this case, we have

where each block has an upper-triangular form

The modes of the system are

eAt =

eJ1t 0 0 . . . 00 eJ2t 0 . . . 0

...0 eJnt

λi eigenvalue µi multiplicity

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Modal Analysis – Multiple eigenvalues Discrete-time systems - In this case, we have

where each block has an upper-triangular form

Akd =

Jk1 0 0 . . . 00 Jk

2 0 . . . 0...

0 Jkn

The modes of the system are

λi eigenvalue µi multiplicity

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Modal Analysis – Multiple eigenvalues

•  Invariance: if the initial state xo belongs to the subspace generated by a chain (miniblock) relative to an eigenvalue λ, then the trajectory is completely contained in the same subspace.

•  Interdependance: it is not possible to excite (independently) the individual modes belonging to a miniblock of the Jordan matrix form.

•  x0 is initial state

Continuous-Time System

Discrete-Time system

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Modal Analysis - Example

Modes m1 = eλ t and m2 = t e λ t with λ = 0

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 0

0.5

1

1.5

2

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 0 2 4 6 8

10

Modes m1 = eλ t and m2 = t e λ t with λ = -0.5

•  Given a system in Jordan form with an eigenvalues λ with multiplicity 2:

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Modal Analysis - Example

Modes m1 = λk and m2 = k λk-1 with λ = 1

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 0

5

10

15

-2 -1 0 1 2 -2

-1

0

1

2

0 5 10 0

0.5

1

1.5

•  Given a system in Jordan form with an eigenvalues λ with multiplicity 2:

Modes m1 = λk and m2 = k λk-1 with λ = 0.5

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Modal Analysis – complex conjugate eigenvalues

•  By using the real form of the Jordan matrix, it is possible to consider the case of pairs of complex conjiugate eigenvalues with multeplicity greater than 1 (modes are real in any case)

•  Discrete-time case

•  Continous-time case

…

…

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Modal Analysis – Modes convergence characteristic

•  Given a stationary LS (continuous- or discrete-time), the mode m(t) (real or complex) defined for t ≥ 0 is: •  Convergent if:

•  Limited but not convergent, if a positive real number M > 0 exists such that

•  Non limited if for all M > 0, a time instant t exists such that

•  Proposition 1: The modes of system are: •  Convergent iff all λ(A) have negative real part ( Re{λi} < 0) •  Limited iff all λ(A) have non positive real part ( Re{λi} ≤ 0) and those with null

real part part ( Re{λi} = 0) are associated with Jordan mini-blocks with unitary dimension (simple roots of the minimum polynomial)

•  Proposition 2: The modes of the system are: •  Convergent iff all λ(A) have modulus less than 1 ( |λi| < 1) •  Limited iff all λ(A) have modulus } ≤ 1 1 ( |λi| ≤ 1) and those with unit

modulus are associated with Jordan mini-blocks with unitary dimension (simple roots of the minimum polynomial)

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Modal Analysis – Dominant modes

•  The time evolution of a linear dynamic system expressed by

with initial condition x(0) = x0 is given by

as t (or k) goes to infinity, the modes tend to go to zero, remain constant, or diverge depending on the value of the eigenvalues.

0 1 2 3 4 5 0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

Tempo (sec)

•  Some of them, however, tend to zero more quickly than others, so their influence on the asymptotic behaviour of the system becomes negligible with increasing time Dominante mode

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• Let us consider for simplicity the case of distinctreal eigenvalues

• Definition: Let λI be the eigenvalues of matrix A. The eigenvalue λi is the dominant eigenvalue of the matrix A if the following relation holds:

•  Continous time case

•  Discrete-time case

• Property: as t (or k) goes to infinity, the free evolution x(t) of a linear time-invariant system (x(k) in discrete-time) tends to flatten along the subspace corresponding to the dominant eigenvalue, i.e. the eigenvalue λ1 that has the larger real part (modulus).


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•  Let vi be the eigenvectors associated with the eigenvalues λi. Since the eigenvectors define a basis of the state space, any initial condition (IC) x0 can be expressed as the linear combination of its components in the basis defined by the eigenvectors vi

where the component x0,i can be expressed as

x0,i = vi*T x0

i.e. the scalar product of x0 and the row product of v*I of T-1

•  The free evolution of x(t) and x(k) corresponding to the IC x0 are

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•  The free evolution x(t) and x(k) corresponding to the IC x0 are:

•  Regardless of IC, if x0,1 ≠ 0 the time evolution of the system tends to the eigenspace of the dominant mode

0 0.2 0.4 0.6 0.8 1

0 0.5

1 0 0.2 0.4 0.6 0.8

1

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•  In the case of complex conjugate eigenvalues, the pair of eigenvalues λ1,2= σ ± j ω is called dominant if (continuous time):

•  If v1 is the eigenvector corresponding to λ1 and v1R, v1I are its real and imaginary parts (vectors as well), it can be shown that

the free motion of the system tends to the subspace defined by v1R, v1I •  Similar considerations also apply in the case of discrete-time systems with

dominant complex conjugate eigenvalues.

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Modal Analysis – Numerical Example

•  Determine the subspace corresponding to the dominant mode of the linear time-invariant continuous system

•  Characteristic polynomial of A:

•  So, there are three coincident eigenvalues in λ = 2. The corresponding eigenvector v1 is obtained by solving the system of equations:

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The generalized eigenvectors v2 and v3 are obtained from

By using the transformation matrix

One obtains

Modal – Numerical Example

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Modal Analysis – Numerical Example

•  if x3 ≠ 0 we can write,

the direction along which the path x(t) “flatens” is then

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Invariant systems

In mathematical terms, an invariant is a property or characteristic that does not change when subjected to a transformation. Examples:

•  Real part and absolute value of a complex number with the operation of conjugation; •  The degree of a polynomial with a linear transformation of the variables; •  The eigenvalues or singular values of a matrix with a similarity transformation; •  The Euclidean norm (length of vectors) with an orthonormal transformation (rotation); •  The angle between two vectors with an orthonormal transformation (rotation); •  …

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Invariant systems

Given: •  R3: Vector space •  {e1, e2, e3}: Main basis E (column of matrix I) •  V: Subspace •  { v1, v2}: Basis of the subspace V •  V = [v1. v2]: Base matrix of the subspace V

Given a matrix An x n, the space V is said invariant in A if

A V ⊆ V Properties: •  The sum of two invariants is an invariant •  The intersection of two invariants is an invariant

•  V is invariant in A if

e1

e3

e2

v1 v2

V

E

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Invariant systems

e1

e3

e2

v1 v2

V

Example: let V is invariant in A if The proof is immediate

Example: let V is invariant in A if In this case, the property is not verified

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Some geometric properties Change of basis •  Instead of basis E ={e1, e2, e3}, we consider a new

basis H ={h1, h2, h3 } with matrix T = [h1, h2, h3] non-singular.

•  Let x be the components of point p in E, and z the components in H. Then:

x = T z, z = T-1 x

•  Given an n x n matrix A in E, in the new basis H one obtains a new (similar) matrix A1

A1 = T-1 A T In fact

e1

e3

e2

v1 v2

V

e1

e3

e2

h1

h3 h2

p

E

H

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Some geometric properties

•  Given matrix A, let consider an invariant subspace V of A with dimensions k < n, and with basis matrix V, n x k. Assuming

T = [V, V’]

be non singular, the matrix A1 = T-1 A T has the following structure

T

T-1 A = T A1 T-1 A1 = T-1 A T

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Some geometric properties •  Example: The vector is expressed in E as:

p = [3, 2, 5]T; Let

be the matrix defining the new basis H. Then

p’ = T-1 p = [-0.6548, 3.7892, 4.8180]T

with

e1

e3

e2

h1

h3 h2

p

e1

e3

e2

h1

h3

h2

p

2 3

1

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• Example: Given matrix A and the invariant described by V

• Then

k = 2

n - k = 1

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• Example: Given matrix A and the invariant described by V

• Then


k = 1

n - k = 2

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AUTOMATIC CONTROL AND SYSTEM THEORY

MODAL ANALYSIS

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