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MSc in Computer Engineering, Cybersecurity and Artificial Intelligence, Fault Diagnosis and Estimation in Dynamical Systems (FDE), a.a. 2019/2020, Lecture 2 A brief review of analysis of continuous-time linear dynamical systems

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

University of Cagliari, Italy

Monday, 23rd March 2020

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Outline

Introductions to systems and models Analysis of IO dynamical systems Modes and their classification

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What is a dynamical system? an example (water tank)

• Input flow rate: q1(t) = a
• Output flow rate: q2(t) = b
• Water volume inside the tank: V (t)

a b h V c d

d dt V (t) = q1(t) − q2(t).

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Difference between systems and models

• Systems are a collection of interacting parts, either natural or artificial, which

evolve in time.

• A model of a system is a mathematical abstraction intended to approximate

and predict the behavior of the system under proper assumptions.

• The term ”dynamical system” refers to a mathematical model. In this course

we often call a ”dynamical system” simply a ”system” for brevity’s sake.

• The properties of dynamical systems can be used for their classification

(different models imply different methodologies for their analysis). Example: A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions.

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Difference between systems and models

• Systems are a collection of interacting parts, either natural or artificial, which

evolve in time.

• A model of a system is a mathematical abstraction intended to approximate

and predict the behavior of the system under proper assumptions.

• The term ”dynamical system” refers to a mathematical model. In this course

we often call a ”dynamical system” simply a ”system” for brevity’s sake.

• The properties of dynamical systems can be used for their classification

(different models imply different methodologies for their analysis). Example: A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions.

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Difference between systems and models

• Systems are a collection of interacting parts, either natural or artificial, which

evolve in time.

• A model of a system is a mathematical abstraction intended to approximate

and predict the behavior of the system under proper assumptions.

• The term ”dynamical system” refers to a mathematical model. In this course

we often call a ”dynamical system” simply a ”system” for brevity’s sake.

• The properties of dynamical systems can be used for their classification

(different models imply different methodologies for their analysis). Example: A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions.

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Difference between systems and models

• Systems are a collection of interacting parts, either natural or artificial, which

evolve in time.

• A model of a system is a mathematical abstraction intended to approximate

and predict the behavior of the system under proper assumptions.

• The term ”dynamical system” refers to a mathematical model. In this course

we often call a ”dynamical system” simply a ”system” for brevity’s sake.

• The properties of dynamical systems can be used for their classification

(different models imply different methodologies for their analysis). Example: A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions.

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Difference between systems and models

• Systems are a collection of interacting parts, either natural or artificial, which

evolve in time.

• A model of a system is a mathematical abstraction intended to approximate

and predict the behavior of the system under proper assumptions.

• The term ”dynamical system” refers to a mathematical model. In this course

we often call a ”dynamical system” simply a ”system” for brevity’s sake.

• The properties of dynamical systems can be used for their classification

(different models imply different methodologies for their analysis). Example: A dynamical system is linear or non-linear depending on whether its model has the property of linearity. On the other hand, a linear dynamical system can approximate the evolution in time of a nonlinear system under certain conditions.

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Linear and non-linear systems

Input/output (IO) models

A model is linear if the principle of superposition principle holds: , cause c1 effect e1 cause c2 effect e2

• =

⇒ cause (ac1 + bc2) effect (ae1 + be2) non-linear: if the superposition principle does not hold. Single Input-Single Output (SISO) Linear IO model: it is represented by a linear differential equation of order n with possibly time-varying coefficients: a0(t)y(t) + · · · + an(t)y (n)(t) = b0(t)u(t) + · · · + bm(t)u(m)(t).

• y (n)(t) represents the n-th derivative of function y(t)

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Linear and non-linear systems

Input/output (IO) models

A model is linear if the principle of superposition principle holds: , cause c1 effect e1 cause c2 effect e2

• =

⇒ cause (ac1 + bc2) effect (ae1 + be2) non-linear: if the superposition principle does not hold. Single Input-Single Output (SISO) Linear IO model: it is represented by a linear differential equation of order n with possibly time-varying coefficients: a0(t)y(t) + · · · + an(t)y (n)(t) = b0(t)u(t) + · · · + bm(t)u(m)(t).

• y (n)(t) represents the n-th derivative of function y(t)

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Linear and non-linear systems

State variable (VS) models

Linear State Variable (SV) model: It is represented by a system of n first order differential equations and output which is linear combination of state variables and inputs:                      ˙ x1(t) = a1,1(t)x1(t) + · · · + a1,n(t)xn(t) + b1,1u1(t) + · · · + b1,rur(t) . . . . . . ˙ xn(t) = an,1(t)x1(t) + · · · + an,n(t)xn(t) + bn,1u1(t) + · · · + bn,rur(t) y1(t) = c1,1(t)x1(t) + · · · + c1,n(t)xn(t) + d1,1u1(t) + · · · + d1,rur(t) . . . . . . yp(t) = cp,1(t)x1(t) + · · · + cp,n(t)xn(t) + dp,1u1(t) + · · · + dp,rur(t) more compactly:

• ˙

①(t) = ❆(t)①(t) + ❇(t)✉(t) ②(t) = ❈(t)①(t) + ❉(t)✉(t) where ❆(t) = {ai,j(t)} matrix n × n; ❇(t) = {bi,j(t)} matrix n × r; ❈(t) = {ci,j(t)} matrix p × n; ❉(t) = {di,j(t)} matrix p × r

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Linear and non-linear systems

State variable (VS) models

Linear State Variable (SV) model: It is represented by a system of n first order differential equations and output which is linear combination of state variables and inputs:                      ˙ x1(t) = a1,1(t)x1(t) + · · · + a1,n(t)xn(t) + b1,1u1(t) + · · · + b1,rur(t) . . . . . . ˙ xn(t) = an,1(t)x1(t) + · · · + an,n(t)xn(t) + bn,1u1(t) + · · · + bn,rur(t) y1(t) = c1,1(t)x1(t) + · · · + c1,n(t)xn(t) + d1,1u1(t) + · · · + d1,rur(t) . . . . . . yp(t) = cp,1(t)x1(t) + · · · + cp,n(t)xn(t) + dp,1u1(t) + · · · + dp,rur(t) more compactly:

• ˙

①(t) = ❆(t)①(t) + ❇(t)✉(t) ②(t) = ❈(t)①(t) + ❉(t)✉(t) where ❆(t) = {ai,j(t)} matrix n × n; ❇(t) = {bi,j(t)} matrix n × r; ❈(t) = {ci,j(t)} matrix p × n; ❉(t) = {di,j(t)} matrix p × r

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Linear and non-linear systems

Importance of linearity

Why linearity is important? 1 Under certain conditions non-linear systems can be linearized around particular states of interest (equilibrium/operating points). 2 The qualitative behavior of linear systems approximates the behavior of many real systems. 3 Under certain conditions (small inputs) the superposition of effects holds for many non-linear systems (critical property in analogue electronics). 4 Powerful methodologies exist for the analysis, control and estimation of linear dynamical systems Example: a spring can be well approximated by a linear dynamical system for small deformations, i.e., the force F exerted by the string is proportional to its stretching x, F = −k · x. For large deformations the spring either fully compresses or stretches out.

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Stationary and non-stationary systems

Stationary (time-invariant) and non-stationary (time-variant) systems

A system is Stationary (time-invariant): if cause and effect can be shifted in time cause c(t) effect e(t) = ⇒ cause c(t − T) effect e(t − T) (applied at time t = 0) (applied after T)

t c(t) t e(t) t c(t-T) T t e(t-T) T

non-stationary (time-variant): otherwise.

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Stationary and non-stationary systems

Stationary IO and SV models

In a stationary system in a given state an input causes the same output independently from when it is applied. Real system maybe be well approximated by stationary system for some period of time (for instance an RC circuit) even if they are not stationary when long periods of time are considered (RC circuit with degraded components due to heavy use). IO stationary model: The IO relationship does not depend explicitly from time: h

• y(t), ˙

y(t), . . . , y(n)(t), u(t), ˙ u(t), . . . , u(m)(t)

• = 0

for linear systems it reduces to a linear differential equation with constant coefficients: a0y(t) + a1 ˙ y(t) + · · · + any(n)(t) = b0u(t) + b1 ˙ u(t) + · · · + bmu(m)(t). SV stationary model: The state and output equation do not depend explicitly from time: ˙ ①(t) = ❢ (①(t), ✉(t)) ②(t) = ❣ (①(t), ✉(t)) for linear systems:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) where A, B, C and D are matrices of constants.

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Stationary and non-stationary systems

Stationary IO and SV models

In a stationary system in a given state an input causes the same output independently from when it is applied. Real system maybe be well approximated by stationary system for some period of time (for instance an RC circuit) even if they are not stationary when long periods of time are considered (RC circuit with degraded components due to heavy use). IO stationary model: The IO relationship does not depend explicitly from time: h

• y(t), ˙

y(t), . . . , y(n)(t), u(t), ˙ u(t), . . . , u(m)(t)

• = 0

for linear systems it reduces to a linear differential equation with constant coefficients: a0y(t) + a1 ˙ y(t) + · · · + any(n)(t) = b0u(t) + b1 ˙ u(t) + · · · + bmu(m)(t). SV stationary model: The state and output equation do not depend explicitly from time: ˙ ①(t) = ❢ (①(t), ✉(t)) ②(t) = ❣ (①(t), ✉(t)) for linear systems:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) where A, B, C and D are matrices of constants.

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Stationary and non-stationary systems

Stationary IO and SV models

In a stationary system in a given state an input causes the same output independently from when it is applied. Real system maybe be well approximated by stationary system for some period of time (for instance an RC circuit) even if they are not stationary when long periods of time are considered (RC circuit with degraded components due to heavy use). IO stationary model: The IO relationship does not depend explicitly from time: h

• y(t), ˙

y(t), . . . , y(n)(t), u(t), ˙ u(t), . . . , u(m)(t)

• = 0

for linear systems it reduces to a linear differential equation with constant coefficients: a0y(t) + a1 ˙ y(t) + · · · + any(n)(t) = b0u(t) + b1 ˙ u(t) + · · · + bmu(m)(t). SV stationary model: The state and output equation do not depend explicitly from time: ˙ ①(t) = ❢ (①(t), ✉(t)) ②(t) = ❣ (①(t), ✉(t)) for linear systems:

• ˙

①(t) = ❆①(t) + ❇✉(t) ②(t) = ❈①(t) + ❉✉(t) where A, B, C and D are matrices of constants.

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Outline

Introductions to systems and models Analysis of IO dynamical systems Modes and their classification

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Analysis of linear dynamical systems

IO model with standard notation

A model of a linear, stationary, SISO system is a differential equation an dny(t) dtn + · · · + a0y(t) = bm dmu(t) dtm + · · · + b0u(t) where with standard notation we denote t ∈ R: independent time variable; u : R → R: input variable. u(t) is a scalar; y : R → R: output variable. y(t) is a scalar n: order of the system. ai ∈ R per i = 0, . . . , n, e bi ∈ R per i = 0, . . . , m, n ≥ m: real coefficients

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Analysis of linear dynamical systems

Time-evolution of a dynamical system

The time-evolution of the dynamical system is the solution of the differential equation obtained by computing y(t) for t ≥ t0. It is needed: the initial conditions y(t)|t=t0 = y0, dy(t) dt

• t=t0

= y′

0,

· · · dn−1y(t) dtn−1

• t=t0

= y(n−1) , the input signal: u(t) for t ≥ t0 The information on the past evolution of the system for t ∈ (−∞, t0] is contained in the initial state ①(t0). The initial conditions are related to the initial state and yield the same information Initial state Initial conditions ①(t0) = 0 y0 = · · · = y(n−1) = 0 ①(t0) = 0 (∃i) y(i) = 0

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Analysis of linear dynamical systems

Time-evolution of a dynamical system

The time-evolution of the dynamical system is the solution of the differential equation obtained by computing y(t) for t ≥ t0. It is needed: the initial conditions y(t)|t=t0 = y0, dy(t) dt

• t=t0

= y′

0,

· · · dn−1y(t) dtn−1

• t=t0

= y(n−1) , the input signal: u(t) for t ≥ t0 The information on the past evolution of the system for t ∈ (−∞, t0] is contained in the initial state ①(t0). The initial conditions are related to the initial state and yield the same information Initial state Initial conditions ①(t0) = 0 y0 = · · · = y(n−1) = 0 ①(t0) = 0 (∃i) y(i) = 0

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Natural and forced evolution

Natural and forced evolution

For t ≥ t0 it holds: y(t) = yn(t) + yf (t) yℓ(t): natural evolution (or natural response) The time-evolution (or response) of the system starting from the given initial conditions with u(t) = 0 for t ≥ t0. yf (t): forced evolution (or forced response) The time-evolution (or response) of the system with an input u(t) = 0 and zero initial conditions yt0 = · · · = y(n−1)

t0

= 0.

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Natural and forced evolution

Initial instant of time

• Usually, we consider the initial instant of time as t0 = 0.
• If t0 = 0, we can simply make a variable change τ = t − t0 and solve the

differential equation for τ = 0

• Once the analytic expression of the system response is computed as

function of τ, we can substitute for τ = t − t0 and obtain y(t).

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The characteristic polynomial

Homogeneous equation and characteristic polynomial

For a given IO system given by: an dny(t) dtn + · · · + a1 dy(t) dt + a0y(t) = bm dmu(t) dtm + · · · + b1 du(t) dt + b0u(t) we call the associated homogeneous equation: an dny(t) dtn + · · · + a1 dy(t) dt + a0y(t) = 0 Thus, to compute the natural evolution we compute the solution of the associated homogeneous equation by letting u(t) = 0 for t ≥ 0. The characteristic polynomial of the homogeneous equation is the polynomial of order n on the variable λ P(λ) = anλn + an−1λn−1 + · · · + a1λ + a0 =

n

• i=0

aiλi.

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The characteristic polynomial

Homogeneous equation and characteristic polynomial

For a given IO system given by: an dny(t) dtn + · · · + a1 dy(t) dt + a0y(t) = bm dmu(t) dtm + · · · + b1 du(t) dt + b0u(t) we call the associated homogeneous equation: an dny(t) dtn + · · · + a1 dy(t) dt + a0y(t) = 0 Thus, to compute the natural evolution we compute the solution of the associated homogeneous equation by letting u(t) = 0 for t ≥ 0. The characteristic polynomial of the homogeneous equation is the polynomial of order n on the variable λ P(λ) = anλn + an−1λn−1 + · · · + a1λ + a0 =

n

• i=0

aiλi.

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The characteristic polynomial

The fundamental theorem of algebra

A polynomial with real coefficients of order n has n roots (real or complex conjugate numbers), solution of P(λ) = 0. In general there are r ≤ n distinct roots pi each with multiplicity νi (with n = r

i=1 νi):

n

• λ1

· · · λ1

• ν1

λ2 · · · λ2

• ν2

· · · λr · · · λr

• νr

If all roots have unitary algebraic multiplicity, then: n

• λ1

λ2 · · · λn−1 λn

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The characteristic polynomial

Solution of the homogeneous equation

The solution of the homogeneous equation, i.e., the natural evolution of the system, is yℓ(t) =

n

• i=1

νi−1

• k=0

Riktkeλit where λi, i = 1, . . . , n are the roots of the characteristic polynomial P(λ) = 0, νi is the multiplicity of the i-th roots and Rij are coefficients which depend upon the initial conditions. In this course we sometimes denote by α the real part of λ and by ω its imaginary

• part. Thus λ = α + jω.

We call modes of evolution associated to a root λi of multiplicity ν the ν functions of time eλit, teλit, · · · , tν−1eλit. Thus, a system with characteristic polynomial of order n has exactly n modes.

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The characteristic polynomial

Solution of the homogeneous equation

The solution of the homogeneous equation, i.e., the natural evolution of the system, is yℓ(t) =

n

• i=1

νi−1

• k=0

Riktkeλit where λi, i = 1, . . . , n are the roots of the characteristic polynomial P(λ) = 0, νi is the multiplicity of the i-th roots and Rij are coefficients which depend upon the initial conditions. In this course we sometimes denote by α the real part of λ and by ω its imaginary

• part. Thus λ = α + jω.

We call modes of evolution associated to a root λi of multiplicity ν the ν functions of time eλit, teλit, · · · , tν−1eλit. Thus, a system with characteristic polynomial of order n has exactly n modes.

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The characteristic polynomial

Solution of the homogeneous equation

The solution of the homogeneous equation, i.e., the natural evolution of the system, is yℓ(t) =

n

• i=1

νi−1

• k=0

Riktkeλit where λi, i = 1, . . . , n are the roots of the characteristic polynomial P(λ) = 0, νi is the multiplicity of the i-th roots and Rij are coefficients which depend upon the initial conditions. In this course we sometimes denote by α the real part of λ and by ω its imaginary

• part. Thus λ = α + jω.

We call modes of evolution associated to a root λi of multiplicity ν the ν functions of time eλit, teλit, · · · , tν−1eλit. Thus, a system with characteristic polynomial of order n has exactly n modes.

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The characteristic polynomial

Example

The homogeneous differential equation 3d4y(t) dt4 + 21d3y(t) dt3 + 45d2y(t) dt2 + 39dy(t) dt + 12y(t) = 0. has characteristic polynomial P(λ) = 3λ4 + 21λ3 + 45λ2 + 39λ + 12 = 3(λ + 1)3(λ + 4) with roots λ1 = −1 with multiplicity ν1 = 3, λ2 = −4 with multiplicity ν2 = 1. The four corresponding modes are: e−t, te−t, t2e−t e e−4t.

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Outline

Introductions to systems and models Analysis of IO dynamical systems Modes and their classification

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Modes and their classification

Definitions

Aperiodic modes: are associated with a real root λ = α ∈ R with multiplicity ν: tkeαt for k = 0, . . . , ν − 1. Pseudoperiodic modes: are associated with a couple of complex conjugate roots λ, λ⋆ = α ± jω ∈ C with multiplicity ν: tkeαt cos(ωt + φk) per k = 0, . . . , ν − 1,

• r equivalently

tkeαt cos(ωt) e tkeαt sin(ωt) per k = 0, . . . , ν − 1.

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Modes and their classification

Definitions

A system of order n has n modes. Nevertheless, when considering a couple of complex conjugate roots, we usually talk about a single “pseudoperiodic” mode associated to the couple of roots”. Therefore the mode Meαt cos(ωt + φ) counts for two. This notation is necessary to avoid denoting the modes with complex

• function. In any case, the evolution y(t) is always a real function of a the

real variable t.

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Modes and their classification

Definitions

A system of order n has n modes. Nevertheless, when considering a couple of complex conjugate roots, we usually talk about a single “pseudoperiodic” mode associated to the couple of roots”. Therefore the mode Meαt cos(ωt + φ) counts for two. This notation is necessary to avoid denoting the modes with complex

• function. In any case, the evolution y(t) is always a real function of a the

real variable t.

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Aperiodic modes

Time constant

Non-zero real root λi = α = 0 The fundamental parameter which characterizes this aperiodic mode is tkeαt with k = 1, . . . , νi is the time constant defined as τ = − 1 α. We can therefore represent such a mode in two equivalent forms tkeαt = tke− t

τ .

The exponent t

τ is an adimensional number and τ has the dimension of

time. Zero real root λi = α = 0: the time constant is not defined.

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Aperiodic modes

Roots with unitary multiplicity

Real root α with multiplicity ν = 1 = ⇒ and a single mode eαt. α < 0: stable (or converging) mode. As t grows it converges asymptotically to 0. α = 0: marginally stable (or constant) mode. For any value of t ≥ 0 it holds e0t = 1. α > 0: unstable (or diverging) mode. As t grows it the mode grows to infinity (∞).

1 eα’ t = e− t/τ’ (α’>0) e0 t = 1 (α=0) eα’’ t = e− t/τ’’ (α’’<0) t [s]

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Aperiodic modes

Physical interpretation of the time constant

The mode eαt with t equal to integer multiple values τ equals: t τ 2τ 3τ 4τ 5τ e− t

τ

1 0.37 0.14 0.05 0.02 0.01 Settling time x%: It is the time necessary for the magnitude of the mode to be reduced to a x% of its initial value. Such quantity is denoted as ta,x%. It holds ta,5% = 3τ, ta,2% = 4τ, ta,1% = 5τ. A stable mode can be considered as vanished after a time equal to about 4 ÷ 5 times its time constant τ: at such a time its value is reduced to 2% ÷ 1% of its initial value.

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Aperiodic modes

Speed of a mode

A stable mode vanishes faster if its time constant is smaller. Let α′ < α′′ < 0: the first mode is said faster than the second because τ ′ < τ ′′ (the second is slower than the first).

1 e−3 t = eα’ t = e− t/τ’ e−t = eα’’ t = e− t/τ’’ t [s] τ’=1/3 τ’’=1 4τ’ 5τ’

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Aperiodic modes

Roots with multiplicity greater than one

To these roots are associated the aperiodic modes: eαt, teαt, . . . , tν−1eαt. For such modes tkeαt with k > 0 we can distinguish two cases: α < 0: stable mode for each value of k ≥ 1. α ≥ 0: unstable mode for each value of k ≥ 1.

t eα’t (α’>0) t e0t = t (α=0) t eα’’t = t e−t/τ’’ (α’’<0) t [s] τ’’ tkeα’t (α’>0) tke0t = tk (α=0) tkeα’’t = tke− t/τ’’ (α’’<0) t [s] kτ’’

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Aperiodic modes

Geometric and physical interpretation of the time-constant

Geometric interpretation A mode tkeαt with α < 0, has a single maximum for t = kτ. Physical interpretation For a stable mode tkeαt with k ≥ 1 we can say that it is faster if its time constant τ = − 1

α, is smaller.

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Pseudoperiodic modes

Definition

Lets consider the generic mode tkeαt cos(ωt) associated to a couple of complex conjugate roots λ, λ⋆ = α ± jω. Fundamental parameters time constant (if α = 0): τ = − 1

α

natural frequency: ωn = √ α2 + ω2 damping coefficient: ζ = − α

ωn = − α √ α2+ω2 ∈ [−1, 1]

Depending on the dumping coefficient we some particular cases: ζ = 1: if α < 0 and ω = 0 (two real identical negative roots ) ζ = 0: if α = 0 (two imaginary conjugate roots) ζ = −1: if α > 0 and ω = 0 (two real identical positive roots)

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Pseudoperiodic modes

Definition

Lets consider the generic mode tkeαt cos(ωt) associated to a couple of complex conjugate roots λ, λ⋆ = α ± jω. Fundamental parameters time constant (if α = 0): τ = − 1

α

natural frequency: ωn = √ α2 + ω2 damping coefficient: ζ = − α

ωn = − α √ α2+ω2 ∈ [−1, 1]

Depending on the dumping coefficient we some particular cases: ζ = 1: if α < 0 and ω = 0 (two real identical negative roots ) ζ = 0: if α = 0 (two imaginary conjugate roots) ζ = −1: if α > 0 and ω = 0 (two real identical positive roots)

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Pseudoperiodic modes

Roots of unitary multiplicity

A couple of roots λ, λ⋆ = α ± jω with multiplicity ν = 1 = ⇒ single mode eαt cos(ωt). The mode is bounded by the curves −eαt e eαt. eαt cos(ωt) =      −eαt se t = (2h + 1)π ω, h ∈ N; eαt se t = 2h π ω, h ∈ N. Three cases: α < 0: stable mode. As t grows the bounding curves converge asymptotically to zero. α = 0: marginally stable mode. It is cos(ωt). α > 0: Unstable mode. As t grows the bounding curves converge to ∞.

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Pseudoperiodic modes

Stable mode (α < 0, ζ > 0, τ > 0)

−1 1 eαt cos(ωt) (α<0) eαt −eαt t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω

The time-constant tells how fast the bounding curves vanish.

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Pseudoperiodic modes

Stable mode (α < 0, ζ > 0, τ > 0) (cont.)

As the damping coefficient ζ is reduced the oscillations are more pronounced. Example: α = α < 0 e ω = 2ω ⇒ ζ =

−α √ α2+ω2 > −α

√

α2+ω 2 = ζ.

−1 1 eαt cos(ωt) (α<0) eαt −eαt t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω −1 1 eαt cos(2ωt) (α<0) eαt −eαt t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω

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Pseudoperiodic modes

Marginally stable mode (α = 0, ζ = 0)

−1 1 e0t cos(ωt) = cos(ωt) t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω

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Pseudoperiodic modes

Unstable mode (α > 0, ζ < 0, τ < 0)

−1 1 eαt cos(ωt) (α>0) eαt −eαt t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω

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Pseudoperiodic modes

Roots with multiplicity greater than one

Corresponding modes: eαt cos(ωt), teαt cos(ωt), . . . , tν−1eαt cos(ωt). For the modes tkeαt cos(ωt) with k > 0 there are two cases: α < 0: stable mode for each value k ≥ 1. α ≥ 0: unstable mode for each value k ≥ 1.

tkeαt cos(ωt) (α<0) tkeαt −tkeαt t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω tkeαt cos(ωt) (α>0) tkeαt −tkeαt t [s] π/ω 2π/ω 3π/ω 4π/ω 5π/ω

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