MSc in Computer Engineering, Cybersecurity and Artificial - - PowerPoint PPT Presentation

MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 4 Review of matrix algebra and state-space transformations

• Prof. Mauro Franceschelli
• Dept. of Electrical and Electronic Engineering

University of Cagliari, Italy

Wednesday, 25th March 2020

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Outline

Diagonalization Jordan form State transition matrix and modes Transient and steady state behavior

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Diagonalization

Brief review of matrix algebra

• Definition. Given a square matrix ❆ of order n, let λ ∈ R be a scalar and

let ✈ = 0 be a column vector with dimensions n × 1. If it holds ❆ ✈ = λ ✈ then λ is said to be an eigenvalue (”autovalore” in italian) of matrix ❆ associated to the eigenvector (”autovettore”) ✈. In practice: The eigenvalues are the roots of the characteristic polynomial: P(λ) = det(λ■ − ❆) = 0. If λ is an eigenvalue of ❆, then the corresponding eigenvector ✈ is a non-zero solution of the linear system of equations: (λ■ − ❆) ✈ = 0 where 0 is a column vector n × 1 the elements of which are all zeros.

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Diagonalization

Diagonalization procedure

The diagonalization procedure computes a similarity transformation which allows to transform a general matrix ❆ to a diagonal matrix ❆′ = P−1❆P. If matrix ❆ has eigenvalues λ1, λ2, . . . , λn, then matrix ❆′ is: ❆′ =      λ1 · · · λ2 · · · . . . . . . ... . . . · · · λn      . A sufficient condition for a matrix ❆ of order n to be diagonalizable is that it has distinct eigenvalues: λi = λj if i = j.

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Diagonalization

Advantages of a diagonal representation

1 - The state transition matrix can be immediately computed: ❆′ =    λ1 · · · . . . ... . . . · · · λn    = ⇒ e❆

′t =

   eλ1t · · · . . . ... . . . · · · eλnt    2 - The analysis is easier because all state variables are decoupled. For instance, for a system with single input:      ˙ x1(t) = λ1x1(t) + b1u(t) . . . . . . ˙ xn(t) = λnxn(t) + bnu(t)

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Diagonalization

Advantages of a diagonal representation

1 - The state transition matrix can be immediately computed: ❆′ =    λ1 · · · . . . ... . . . · · · λn    = ⇒ e❆

′t =

   eλ1t · · · . . . ... . . . · · · eλnt    2 - The analysis is easier because all state variables are decoupled. For instance, for a system with single input:      ˙ x1(t) = λ1x1(t) + b1u(t) . . . . . . ˙ xn(t) = λnxn(t) + bnu(t)

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Diagonalization

The modal matrix

• Definition. Given a matrix ❆ of dimension n × n, let ✈ 1, ✈ 2, . . . , ✈ n, be a

set of linearly independent eigenvectors corresponding to the eigenvalues λ1, λ2, . . . , λn. We denote modal matrix ❆ the matrix n × n ❱ =

• ✈ 1

✈ 2 · · · ✈ n

• .
• Theorem. If a matrix ❆ has distinct eigenvalues λ1, . . . , λn, then

the corresponding eigenvectors ✈ 1, . . . , ✈ n are linearly indepen- dent. If ❆ does not have n distinct eigenvalues, then the modal matrix exists if and only if to each eigenvalue with algebraic multiplicity ν correspond ν linearly independent eigenvectors.

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Diagonalization

The modal matrix

• Definition. Given a matrix ❆ of dimension n × n, let ✈ 1, ✈ 2, . . . , ✈ n, be a

set of linearly independent eigenvectors corresponding to the eigenvalues λ1, λ2, . . . , λn. We denote modal matrix ❆ the matrix n × n ❱ =

• ✈ 1

✈ 2 · · · ✈ n

• .
• Theorem. If a matrix ❆ has distinct eigenvalues λ1, . . . , λn, then

the corresponding eigenvectors ✈ 1, . . . , ✈ n are linearly indepen- dent. If ❆ does not have n distinct eigenvalues, then the modal matrix exists if and only if to each eigenvalue with algebraic multiplicity ν correspond ν linearly independent eigenvectors.

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Diagonalization

The modal matrix

Given a matrix ❆ of dimension n × n let ❱ be its modal matrix. The matrix ❆′ is

• btained by the similarity transformation ❆′ = ❱ −1❆❱ is a diagonal matrix.
• Proof. By the definition of eigenvalue and eigenvector it holds

λi✈ i = ❆✈ i (i = 1, . . . , n) = ⇒

• λ1✈ 1

λ2✈ 2 · · · λn✈ n

• =
• ❆✈ 1

❆✈ 2 · · · ❆✈ n

• thus
• ✈ 1

✈ 2 · · · ✈ n

• 

    λ1 · · · λ2 · · · . . . . . . ... . . . · · · λn      = ❆

• ✈ 1

✈ 2 · · · ✈ n

• therefore

❱ ❆′ = ❆❱ = ⇒ ❆′ = ❱ −1❆❱

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Diagonalization

The modal matrix

Given a matrix ❆ of dimension n × n let ❱ be its modal matrix. The matrix ❆′ is

• btained by the similarity transformation ❆′ = ❱ −1❆❱ is a diagonal matrix.
• Proof. By the definition of eigenvalue and eigenvector it holds

λi✈ i = ❆✈ i (i = 1, . . . , n) = ⇒

• λ1✈ 1

λ2✈ 2 · · · λn✈ n

• =
• ❆✈ 1

❆✈ 2 · · · ❆✈ n

• thus
• ✈ 1

✈ 2 · · · ✈ n

• 

    λ1 · · · λ2 · · · . . . . . . ... . . . · · · λn      = ❆

• ✈ 1

✈ 2 · · · ✈ n

• therefore

❱ ❆′ = ❆❱ = ⇒ ❆′ = ❱ −1❆❱

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Diagonalization

Example (homework)

Given the representation:           

• ˙

x1(t) ˙ x2(t)

• =
• −1

1 −2

• x1(t)

x2(t)

• +
• 1
• u(t)
• y1(t)

y2(t)

• =
• 2

1 2

• x1(t)

x2(t)

• +
• 1.5
• u(t)

show that the a change of variables x(t) = ❱ z(t) based on the modal matrix ❱ = 1 1 −1

• transforms the system into a diagonal form

           ˙ z1(t) ˙ z2(t)

• =

−1 −2

• z1(t)

z2(t)

• +
• 1

−1

• u(t)
• y1(t)

y2(t)

• =
• 2

1 −2

• z1(t)

z2(t)

• +
• 1.5
• u(t)

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Diagonalization

System analysis via diagonalization

Given a representation in which matrix ❆ has modal matrix ❱ , consider the diagonal system obtained by the diagonalization procedure:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) = ⇒

• ˙

③(t) = ❆′③(t) + ❇′✉(t) ②(t) = ❈ ′①(t) + ❉✉(t) By the properties of the state transition matrix and the Lagrange formula, it holds: e❆t = ❱ e❆

′t❱ −1

The natural state evolution ①n(t) starting from the initial state ①(0) is ①n(t) = ❱ ③n(t) = ❱ e❆

′t③(0) = ❱ e❆ ′t❱ −1①(0)

The forced evolution ①f (t) for a given input signal ✉(t) is ①f (t) = ❱ ③f (t) = ❱ t e❆

′(t−τ)❇′✉(τ)dτ 9 / 35

Diagonalization

System analysis via diagonalization

Given a representation in which matrix ❆ has modal matrix ❱ , consider the diagonal system obtained by the diagonalization procedure:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) = ⇒

• ˙

③(t) = ❆′③(t) + ❇′✉(t) ②(t) = ❈ ′①(t) + ❉✉(t) By the properties of the state transition matrix and the Lagrange formula, it holds: e❆t = ❱ e❆

′t❱ −1

The natural state evolution ①n(t) starting from the initial state ①(0) is ①n(t) = ❱ ③n(t) = ❱ e❆

′t③(0) = ❱ e❆ ′t❱ −1①(0)

The forced evolution ①f (t) for a given input signal ✉(t) is ①f (t) = ❱ ③f (t) = ❱ t e❆

′(t−τ)❇′✉(τ)dτ 9 / 35

Diagonalization

System analysis via diagonalization

Given a representation in which matrix ❆ has modal matrix ❱ , consider the diagonal system obtained by the diagonalization procedure:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) = ⇒

• ˙

③(t) = ❆′③(t) + ❇′✉(t) ②(t) = ❈ ′①(t) + ❉✉(t) By the properties of the state transition matrix and the Lagrange formula, it holds: e❆t = ❱ e❆

′t❱ −1

The natural state evolution ①n(t) starting from the initial state ①(0) is ①n(t) = ❱ ③n(t) = ❱ e❆

′t③(0) = ❱ e❆ ′t❱ −1①(0)

The forced evolution ①f (t) for a given input signal ✉(t) is ①f (t) = ❱ ③f (t) = ❱ t e❆

′(t−τ)❇′✉(τ)dτ 9 / 35

Diagonalization

System analysis via diagonalization

Given a representation in which matrix ❆ has modal matrix ❱ , consider the diagonal system obtained by the diagonalization procedure:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) = ⇒

• ˙

③(t) = ❆′③(t) + ❇′✉(t) ②(t) = ❈ ′①(t) + ❉✉(t) By the properties of the state transition matrix and the Lagrange formula, it holds: e❆t = ❱ e❆

′t❱ −1

The natural state evolution ①n(t) starting from the initial state ①(0) is ①n(t) = ❱ ③n(t) = ❱ e❆

′t③(0) = ❱ e❆ ′t❱ −1①(0)

The forced evolution ①f (t) for a given input signal ✉(t) is ①f (t) = ❱ ③f (t) = ❱ t e❆

′(t−τ)❇′✉(τ)dτ 9 / 35

Outline

Diagonalization Jordan form State transition matrix and modes Transient and steady state behavior

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The Jordan form

Introduction

In the case in which matrix ❆ has eigenvalues with non-unitary algebraic multiplicity, there may not exist n linearly independent eigenvectors to build a modal matrix: therefore, it does not always exist a similarity transformation which transforms the system in a diagonal form with all real coefficients. Nevertheless, it is always possible by extending the concept of eigenvector, to determine a set n of linearly independent generalized eigenvectors. Such vectors can be used to build a generalized modal matrix ❱ which allows, by similarity, to transform matrix ❆ to the so-called Jordan form ❏ = ❱ −1❆❱ , a canonical form which is ”block diagonal” which generalizes the diagonal form.

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The Jordan form

Introduction

In the case in which matrix ❆ has eigenvalues with non-unitary algebraic multiplicity, there may not exist n linearly independent eigenvectors to build a modal matrix: therefore, it does not always exist a similarity transformation which transforms the system in a diagonal form with all real coefficients. Nevertheless, it is always possible by extending the concept of eigenvector, to determine a set n of linearly independent generalized eigenvectors. Such vectors can be used to build a generalized modal matrix ❱ which allows, by similarity, to transform matrix ❆ to the so-called Jordan form ❏ = ❱ −1❆❱ , a canonical form which is ”block diagonal” which generalizes the diagonal form.

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The Jordan form

Introduction

In the case in which matrix ❆ has eigenvalues with non-unitary algebraic multiplicity, there may not exist n linearly independent eigenvectors to build a modal matrix: therefore, it does not always exist a similarity transformation which transforms the system in a diagonal form with all real coefficients. Nevertheless, it is always possible by extending the concept of eigenvector, to determine a set n of linearly independent generalized eigenvectors. Such vectors can be used to build a generalized modal matrix ❱ which allows, by similarity, to transform matrix ❆ to the so-called Jordan form ❏ = ❱ −1❆❱ , a canonical form which is ”block diagonal” which generalizes the diagonal form.

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The Jordan form

The Jordan block

Given a complex number λ ∈ C and an integer p ≥ 1 we define the Jordan block of order p associated to λ the p × p square matrix:          λ 1 · · · λ 1 · · · λ · · · . . . . . . . . . ... . . . . . . · · · λ 1 · · · λ         

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The Jordan form

The Jordan form

A matrix ❏ is said to be in Jordan form if it is a block diagonal matrix ❏ =      ❏1 · · · ❏2 · · · . . . . . . ... . . . · · · ❏q      where each block along its diagonal is a Jordan block ❏i. More then one Jordan blocks can be associated to the same eigenvalue. The Jordan form is a generalization of the diagonal form.

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The Jordan form

Examples

The matrices shown next are in Jordan form         2 1 2 1 2 2 3 1 3         ,   2 2 3   ,   2 1 2   .

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The Jordan form

The Jordan form

Proposition Any square matrix ❆ can be transformed by similarity into matrix ❏ In Jordan form. Such a form is unique with the exception of permutations of the diagonal blocks. Let λ be an eigenvalue of algebraic multiplicity ν and geometric multiplicity µ. Then, there are µ Jordan blocks corresponding to λ. If pi is the order of the generic Jordan block i, for i = 1, . . . , µ, then µ

i=1 pi = ν.

Note: The geometric multiplicity µ of an eigenvalue tells the number of linearly independent eigenvectors associated to it and it holds 1 ≤ µ ≤ ν.

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The Jordan form

Examples

❏1 =         2 1 2 1 2 2 3 1 3         , ❏2 =   2 2 3   , ❏3 =   2 1 2   . ❏1: The eigenvalue 2 has algebraic multiplicity 4 and geometric multiplicity 2, the eigenvalue 3 has algebraic multiplicity equal to 2 and geometric multiplicity 2. The algebraic and geometric multiplicity of the eigenvalues of J2 and J3 are .... (homework)

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The Jordan form

Examples

❏1 =         2 1 2 1 2 2 3 1 3         , ❏2 =   2 2 3   , ❏3 =   2 1 2   . ❏1: The eigenvalue 2 has algebraic multiplicity 4 and geometric multiplicity 2, the eigenvalue 3 has algebraic multiplicity equal to 2 and geometric multiplicity 2. The algebraic and geometric multiplicity of the eigenvalues of J2 and J3 are .... (homework)

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The Jordan form

Examples

Let ❆ be a 5 × 5 matrix with an eigenvalue λ1 with algebraic multiplicity ν1 = 4 and an eigenvalue λ2 with algebraic multiplicity equal to ν2 = 1. Its Jordan form has a Jordan block of order 1 corresponding to the eigenvalue λ2 The Jordan blocks associated to λ1 depend on its geometric multiplicity µ1 ≤ ν1 = 4.

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The Jordan form

Example

These are the possible cases: µ1 = 4. ❏1 =       λ1 λ1 λ1 λ1 λ2       and it is diagonalizable. µ1 = 3. ❏2 =       λ1 1 λ1 λ1 λ1 λ2       .

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The Jordan form

Examples

µ1 = 2. ❏3 =       λ1 1 λ1 λ1 1 λ1 λ2       , ❏4 =       λ1 1 λ1 1 λ1 λ1 λ2       . µ1 = 1. ❏5 =       λ1 1 λ1 1 λ1 1 λ1 λ2       .

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The Jordan form

State transition matrix in Jordan form

Given a matrix ❏ in Jordan form ❏ =      ❏1 · · · ❏2 · · · . . . . . . ... . . . · · · ❏q      , its matrix exponential is e❏t =       e❏ 1t · · · e❏ 2t · · · . . . . . . ... . . . · · · e❏ qt       . (1)

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The Jordan form

State transition matrix in Jordan form

Furthermore, if ❏i is a generic block of order p, its matrix exponential is e❏ it =                   eλt teλt

t2 2!eλt

· · ·

tp−3 (p−3)!eλt tp−2 (p−2)!eλt tp−1 (p−1)!eλt

eλt teλt · · ·

tp−4 (p−4)!eλt tp−3 (p−3)!eλt tp−2 (p−2)!eλt

eλt · · ·

tp−5 (p−5)!eλt tp−4 (p−4)!eλt tp−3 (p−3)!eλt

. . . . . . . . . ... . . . . . . . . . · · · eλt teλt

t2 2!eλt

· · · eλt teλt · · · eλt                   .

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The Jordan form

State transition matrix in Jordan form

Given a matrix ❆ of order n with n eigenvalues λ1, λ2, . . . , λn, let ❱ be a modal matrix which allows to obtain the Jordan form ❏ = ❱ −1❆❱ . It holds e❆t = ❱ e❏t❱ −1.

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Outline

Diagonalization Jordan form State transition matrix and modes Transient and steady state behavior

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State transition matrix and modes

State transition matrix

In the past lecture in the study of input-output models, we characterized the so-called modes of evolution, i.e., those functions of time which characterize the evolution of a linear dynamical system The very same concept applies to state variables models.

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State transition matrix and modes

State transition matrix

Given a matrix ❏ in Jordan form, consider the corresponding state transition matrix e❏t. In a given Jordan block of order p associated to the eigenvalue λ show up the functions of time: eλt, teλt, · · · , tp−1eλt, multiplied by some coefficients. If several Jordan blocks are associated to the same eigenvalue and the larger block has dimension p, then the term of maximum order associated to the eigenvalue is tp−1eλt.

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State transition matrix and modes

State transition matrix

Given a matrix ❏ in Jordan form, consider the corresponding state transition matrix e❏t. In a given Jordan block of order p associated to the eigenvalue λ show up the functions of time: eλt, teλt, · · · , tp−1eλt, multiplied by some coefficients. If several Jordan blocks are associated to the same eigenvalue and the larger block has dimension p, then the term of maximum order associated to the eigenvalue is tp−1eλt.

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State transition matrix and modes

State transition matrix

Given a matrix ❏ in Jordan form, consider the corresponding state transition matrix e❏t. In a given Jordan block of order p associated to the eigenvalue λ show up the functions of time: eλt, teλt, · · · , tp−1eλt, multiplied by some coefficients. If several Jordan blocks are associated to the same eigenvalue and the larger block has dimension p, then the term of maximum order associated to the eigenvalue is tp−1eλt.

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State transition matrix and modes

State transition matrix

Consider a matrix ❆. Since such a matrix can be transformed via similarity to a Jordan form, its state transition matrix can be computed as: e❆t = ❱ e❏t❱ −1. Therefore, each of its elements is a linear combination of the functions we just described.

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State transition matrix and modes

Minimal polynomial and modes

Given a matrix ❆ with r distinct eigenvalues λi each with geometric multiplicity µi and largest Jordan block of dimension pi, we define the minimal polynomial as Pmin(λ) =

r

• i=1

(λ − λi)pi. For each eigenvalue λi, i.e., each root of the minimal polynomial with algebraic multiplicity νi we can associate pi functions eλit, teλit, · · · , tpi−1eλit, which we call modes. Each element of the state transition matrix e❆t is a linear combination of modes.

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State transition matrix and modes

Example 1

Consider a matrix ❆ with two eigenvalues λ1 = −1 and λ2 = −2 with unitary algebraic multiplicity and, therefore, largest Jordan block of order

• ne. The minimal polynomial of ❆ is :

Pmin(λ) = P(λ) = (λ + 1)(λ + 2). The modes are e−t and e−2t. Each element of matrix e❆t is a linear combination of these modes.

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State transition matrix and modes

Example 2

Let ❆ be a matrix with a single eigenvalue λ = 3 of algebraic multiplicity 4 and geometric multiplicity 2. It can be put into the Jordan form ❏ =     3 1 3 1 3 3     . The characteristic polynomial and the minimal polynomial are respectively: P(λ) = (λ − λi)ν = (λ − 3)4 e Pmin(λ) = (λ − λi)p = (λ − 3)3. The corresponding modes are e3t, te3t and t2e3t (even if the eigenvalue λ = 3 has algebraic multiplicity equal to ν = 4 there is no mode corresponding to t3e3t).

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Outline

Diagonalization Jordan form State transition matrix and modes Transient and steady state behavior

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Steady state and transient response

Definition

The steady state response (risposta a regime permanente) yss(t) with respect to a given input, is the function of time which, independently from the initial state, corresponds to the evolution to which the system converges as time grows. In a SV model the output is a linear combination

• f the state and the input: y(t) = ❈x(t) + ❉u(t).

Under the next conditions a linear dynamical system admits a steady state: (A)The eigenvalues have all negative real part; (B)The inputs are a linear combination of exponential ramps (this includes constant inputs). Let the forced response yf (t) = yf .o(t) + yf .p include two terms, one which which vanishes asymptotically (yf .o(t)) and one which persists for all t ≥ 0 (yf .p). In such a case: y(t) = yn(t) + yf (t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) .

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Steady state and transient response

Definition

The steady state response (risposta a regime permanente) yss(t) with respect to a given input, is the function of time which, independently from the initial state, corresponds to the evolution to which the system converges as time grows. In a SV model the output is a linear combination

• f the state and the input: y(t) = ❈x(t) + ❉u(t).

Under the next conditions a linear dynamical system admits a steady state: (A)The eigenvalues have all negative real part; (B)The inputs are a linear combination of exponential ramps (this includes constant inputs). Let the forced response yf (t) = yf .o(t) + yf .p include two terms, one which which vanishes asymptotically (yf .o(t)) and one which persists for all t ≥ 0 (yf .p). In such a case: y(t) = yn(t) + yf (t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) .

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Steady state and transient response

Definition

The steady state response (risposta a regime permanente) yss(t) with respect to a given input, is the function of time which, independently from the initial state, corresponds to the evolution to which the system converges as time grows. In a SV model the output is a linear combination

• f the state and the input: y(t) = ❈x(t) + ❉u(t).

Under the next conditions a linear dynamical system admits a steady state: (A)The eigenvalues have all negative real part; (B)The inputs are a linear combination of exponential ramps (this includes constant inputs). Let the forced response yf (t) = yf .o(t) + yf .p include two terms, one which which vanishes asymptotically (yf .o(t)) and one which persists for all t ≥ 0 (yf .p). In such a case: y(t) = yn(t) + yf (t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) .

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Steady state and transient response

Definition

The steady state response (risposta a regime permanente) yss(t) with respect to a given input, is the function of time which, independently from the initial state, corresponds to the evolution to which the system converges as time grows. In a SV model the output is a linear combination

• f the state and the input: y(t) = ❈x(t) + ❉u(t).

Under the next conditions a linear dynamical system admits a steady state: (A)The eigenvalues have all negative real part; (B)The inputs are a linear combination of exponential ramps (this includes constant inputs). Let the forced response yf (t) = yf .o(t) + yf .p include two terms, one which which vanishes asymptotically (yf .o(t)) and one which persists for all t ≥ 0 (yf .p). In such a case: y(t) = yn(t) + yf (t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) .

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Steady state and transient response

Definition

The steady state response (risposta a regime permanente) yss(t) with respect to a given input, is the function of time which, independently from the initial state, corresponds to the evolution to which the system converges as time grows. In a SV model the output is a linear combination

• f the state and the input: y(t) = ❈x(t) + ❉u(t).

Under the next conditions a linear dynamical system admits a steady state: (A)The eigenvalues have all negative real part; (B)The inputs are a linear combination of exponential ramps (this includes constant inputs). Let the forced response yf (t) = yf .o(t) + yf .p include two terms, one which which vanishes asymptotically (yf .o(t)) and one which persists for all t ≥ 0 (yf .p). In such a case: y(t) = yn(t) + yf (t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) .

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Steady state and transient response

Definition

y(t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) . The term yt(t) = yn(t) + yf .o(t) is said transient response The terms yn(t) and yf .o(t) are linear combinations of the systems’ modes and due to assumption (A): lim

t→∞ yn(t) = 0,

e lim

t→∞ yf .o(t) = 0.

The transient response vanishes asymptotically to zero as t grows; it can be considered extinguished after a time ¯ τ related to the time constants of the modes.

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Steady state and transient response

Definition

y(t) = yn(t) + yf .o(t)

• yt(t)

+ yf .p(t) yss(t) . The term yss(t) = yf .p(t) is said steady state response. It does not depend on the initial state of the system and for t > ¯ τ it holds y(t) ≈ yr(t), which means that the total response is characterized only by modes of evolution introduced by the input.

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Steady state and transient response

Example

Consider a system with an input signal u(t) = e0.1t for t ≥ 0. Suppose its initial conditions are such that its natural evolution is yn(t) =

• 2e−t − 1e−2t + te−2t

, and its forced evolution is yf (t) = yf .o(t) + yf .p(t) with yf .o(t) =

• −2.73e−t + 1.88e−2t + 0.95te−2t

, yf .p(t) = 0.85e0.1t, After a time ¯ τ = 6 the transient term vanishes and the output coincides with the steady state term.

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Steady state and transient response

Example

1 2 3 4 5 6 7 8 9 t[s] −1 −0.5 0.5 1 1.5 2 2.5 yr(t) y(t) yt(t)

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