Automation Systems Lecture 3 - Frequency Response Methods, Basic Dynamical Elements Jakub Mozaryn

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Lecture 3 - Frequency Response Methods, Basic Dynamical Elements

Jakub Mozaryn

Institute of Automatic Control and Robotics, Department of Mechatronics, WUT

Warszawa, 2016

Jakub Mozaryn Automation Systems

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Frequency response

The frequency response of a system is defined as the steady-state response of the system to a sinusoidal input signal.

The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state; it differs from the input waveform only in amplitude and phase angle.

In analysis of linear systems frequency response methods are used to examine stability, as well as other properties eg. robustness.

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Define as a function of frequency:

amplitude ratio of the output (response) to the input (cause), phase angle shift between the output and input

A distinction is made between the following frequency characteristics:

amplitude-phase polar plot- Nyquist plot,

amplitude and phase logarithmic plots - Bode plots (diagrams),

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Frequency response

Figure 1 : Determination of frequency response

u(t) = A1sin[ωt] (1)

y (t) = A2sin[ω(t − tϕ)] (2) gdzie:

A_i - amplitude,

ω - angular frequency (constant for input and output), t_ϕ - delay of the output signal phase to input signal.

tϕ< 0 - negative phase shift, tϕ> 0 - positive phase shift.

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Figure 2 : Input signal

Figure 3 : Output signal, negative phase shift

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Frequency response

The phase shift of the output signal relative to the input signal can be expressed as a shift in time by tϕ and then the output signal is described by the function

y (t) = A2sin[ω(t − tϕ)]

or as a shift angle ϕ(ω) = ωt_ϕ, when

y (t) = A₂sin[ωt − ϕ]

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To describe the systems where the signals of sinusoidal variables rhere is used transform in frequency domain G (j ω).

Transform in frequency domain is connected with Fourier transform which designated the transform F (j ω) to function in time domain f (t) according to following equation (also called Fourier Integral)

F (j ω) =

∞

Z

−∞

f (t)e^−jωtdt

Spectral transfer function

Spectral transfer function is the ratio of the Fourier transform of the output signal to the Fourier transform of the input signal.

Gj ω = y (j ω) x (j ω)

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Spectral transfer function

Between spectral transfer function, and transfer function there is formal relation

G (j ω) = G (s)|s=j ω

resulting from the relation between transforms of Laplace and Fourier.

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Using property of Laplace transform - theorem of the shift in the real variable domain

L{f (t + τ )} = L{f (t)}e^{τ s}

spectral transmittance of the object in the case of a sinusoidal signal at an input has the form

G (s) = L {A2(ω)sin[ω(t + tϕ)]}

L {A1sin[ω(t)]} =A2(ω) A1

L {sin[ω(t)]} e^t^ϕ^s

L {sin[ω(t)]} =A2(ω) A1

e^t^ϕ^s

Because

G (j ω) = Y (j ω)

U(j ω), G (j ω) = G (s)|_{s=j ω}, t_ϕ=ϕ(ω) ω therefore

G (j ω) = A2(ω) A1

e^t^ϕ^s|s=j ω= A2(ω) A1

e^t^ϕ^{j ω}= A2(ω) A1

e^{j ϕ(ω)}

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Spectral transmittance

Spectral transmittance can be presented as follows

G (j ω) = A2(ω) A1

e^{j ϕ(ω)}= M(ω)e^{j ϕ(ω)} where:

M(ω) = ^A²_A^(ω)

1 - magnitude of spectral transmittance ϕ(ω) - argument of spectral transmittance

The transmittance can be divided into 2 components G (j ω) = M(ω)e^{j ϕ(ω)}= P(ω) + jQ(ω) where:

P(ω) - real part of spectral transmittance Q(ω) - imaginary part of spectral transmittance

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Polar plot

A plot of a function expressed in polar coordinates, which is a locus of the ending of vector of spectral transmittance G (j ω) with changes ω = 0 → ∞

Figure 4 : Example of polar plot

M(ω) =p

[P(ω)]²+ [Q(ω)]²

ϕ(ω) = arctg Q(ω) P(ω)

P(ω) = M(ω) cos[ϕ(ω)]

Q(ω) = M(ω) sin[ϕ(ω)]

M(ω) = P(ω) cos[ϕ(ω)] + Q(ω) sin[ϕ(ω)]

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Amplitude and phase logarythmic plots - Bode diagrams

Figure 5 : Logarythmic plots

Frequency characteristics

Frequency characteristics of phase and aplitude are presented on two separate plots:

amplitude plot L(ω) = |G (j ω)|

w according to frequency ω , phase plot ϕ = arg G (ω) w according to frequency ω.

Logarithmic gain (unit - decibel) L(ω) = 10log10M²(ω)

= 20 log M(ω)[dB]

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In complex automation systems it’s often possible to extract a number of the simple, indivisible, functional elements, connected together. Their properties can be assigned with certain accuracy to few basic

mathematical models. Abstract elements with properties corresponding to those models are basic (or elementary) linear dynamical elements.

Methods of description of basic elements:

equations of motion, transfer fuction, static characteristic, transient response,

transfer function in the frequency domain, polar plot of a transfer function (Nyquist plot)

logarithmic plots in the frequency domain (Bode plots)

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Basic dynamical elements

y (t) = ku(t) Proportional element

(non-inertial)

Tdy (t)

dt + y (t) = ku(t) First-Order Lag

Tdy (t)

dt = u(t), lub dy (t)

dt = ku(t) Integrator y (t) = Tdu(t)

dt Differentiator (ideal)

Tdy (t)

dt + y (t) = T_ddu(t)

dt Differentiator (real)

T²d²y (t)

dt + 2ξTdy (t)

dt + y (t) = ku(t)

Second Order Lag, if 0 < ξ < 1

y (t) = u(t − T0) Delay

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Figure 6 : Examples of proportional components: a) two-port network, b) lever, c) hydraulic lever

a)

U2(t) = R2

R₁+ R₂U1(t) b)

y (t) = b ax (t) c)

F2(t) = d₂² d₁²F1(t)

Equation of motion

y (t) = ku(t) where: k - gain

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Proportional (Non-inertial) element

Dynamic equation y (t) = ku(t) Static characteristic

y = ku Transfer fuction

G (s) = Y (s) U(s) = k

Figure 7 : Static characteristic of proportional element

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Transient response

y (t) = L⁻¹[ust

1

sk] = kust

Figure 8 : Transient response of proportional element

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Proportional (Non-inertial) element

Spectral transmittance Gj ω= G (s)|s=j ω = k P(ω) = k, Q(ω) = 0

M(ω) = k L(ω) = 20 log k[dB]

ϕ(ω) = 0 _{Figure 9 :} Nyquist plot

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Amplitude diagram L(ω) = 20 log k[dB]

Phase diagram ϕ(ω) = 0

Figure 10 : Bode plots

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First Order Lag Element

Figure 11 : First Order Lag Element - spinning disc on a shaft

Jd ω(t)

d t + Rω(t) = M(t) J

R d ω(t)

d t + ω(t) = 1 RM(t)

where: R - coefficient of viscosity in bearings, J - moment of inertia, M - torque, ω - angular velocity.

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Figure 12 : First Order Lag Element - RL network

U₁(t) = LdI (t)

dt + U₂(t) I (t) = U₂(t)

R L

R dU2(t)

dt + U₂(t) = U₁(t)

where: L - inductance, R - ressistance, U1(t) - input voltage, U2(t) - output voltage.

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First Order Lag Element

Equations of motion a) disc rotating on the shaft

J R

d ω(t)

d t + ω(t) = 1 RM(t) b) RL network

L R

dU2(t)

dt + U2(t) = U1(t)

Equation of a motion Tdy (t)

dt + y (t) = ku(t) where: T - time constant.

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Dynamic equation

Tdy (t)

dt + y (t) = ku(t) Static characteristic

y = ku Transfer function

G (s) = Y (s) U(s) = k

Ts + 1

Figure 13 : Static characteristic of First Order Lag Element

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First Order Lag Element

Transient response

y (t) = L⁻¹[ust

1 s

k Ts + 1]

= u_stk

1 − e^−t^T

Figure 14 : Transient response of First Order Lag Element

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Spectral transmittance Gj ω = G (s)|s=j ω= k

Ts + 1|s=j ω = k

Tj ω + 1 = P(ω) + jQ(ω)

P(ω) = k

T²ω²+ 1, Q(ω) = −kT ω T²ω²+ 1

Figure 15 : Nyquist plot, ωs- corner frequency

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First Order Lag Element

Amplitude diagram

M(ω) = k

√

T²ω²+ 1 L(ω) = 20 log k − 20 logp

T²ω²+ 1[dB]

for

ω 1 T = ωs

L(ω) = 20 log k[dB]

for

ω 1 T = ωs

L(ω) = (20 log k−20 logp

T²ω²+ 1)[dB]

Phase diagram

ϕ = −arctg(T ω)

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Figure 17 : Integrators a) hydraulic integrator, b) gearbox

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Integrator

a)

Q = (

αb s

2

ρ(pz− ps) )

x (t) = Bx (t)

Q₁= Q₂= Bx (t) = Ady (t) dt A

B dy (t)

dt = x (t) b)

d ϕ(t) dt = ω

rx (t)

Equation of motion Tdy (t)

dt = u(t) or

dy (t)

dt = ku(t)

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Dynamic equation

Tdy (t) dt = u(t) Static characteristic

u = 0 Trasfer function G (s) = Y (s)

U(s) = 1 Ts

Figure 18 : Static characteristic of the integrator

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Integrator

Transient response

y (t) = L⁻¹[ust

1 s

1 Ts] = ust

t T

Figure 19 : Transient response of the integrator

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Spectral transmittance

G_{j ω} = G (s)|_{s=j ω} = 1

Ts|_{s=j ω} = 1

Tj ω = −j 1 T ω P(ω) = 0, Q(ω) = − 1

T ω

Figure 20 : Nyquist plot

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Integrator

M(ω) = 1 T ω Amplitude diagram L(ω) = 20 log 1

T ω

= −20 log T ω[dB]

Phase diagram

ϕ(ω) = arctg−_{T ω}¹ 0

= arctg(−∞) = −π 2

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a) Tachometric generator

Figure 22 : Differentiator - tachometric generator

U_y(t) =d θ(t) dt b) Liquid feeder (syringe)

Figure 23 : Differentiator - liquid feeder

Q(t) = Adx (t) dt

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Differentiator - ideal

Dynamic equation

y (t) = T_ddu(t) dt Static characteristic

y = 0 Transfer function

G (s) = Y (s) U(s) = Tds

Figure 24 : Static characteristic of ideal differentiator

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Transient response

y (t) = L⁻¹[ust

1

sTds] = ustTdδ(t)

Figure 25 : Transient response of ideal differentiator

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Differentiator - ideal

Gj ω= Tds|s=j ω= jTdω P(ω) = 0, Q(ω) = Tdω

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Amplitude diagram M(ω) = Tdω L(ω) = 20 log Tdω[dB]

Phase diagram ϕ(ω) = arctgT_dω

0

= arctg(∞) =π 2

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Differentiator - real

a) absorber

Figure 28 : Differentiator - absorber

A du(t)

dt −dy (t) dt

= Q = k∆p

∆pA = Cy (t), ∆p = C Ay A²

kC dy (t)

dt + y (t) = A² kC

du(t) dt Tdy (t)

dt + y (t) = T_ddu(t) dt

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b) RC network

Figure 29 : Differentiator - RC network

RCdU2(t)

dt + U2(t) = dU1(t) dt Tdy (t)

dt + y (t) = T_ddu(t) dt

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Differentiator- real

Dynamic equation

Tdy (t)

dt + y (t) = Td

du(t) dt , kd =Td

T Static characteristic

y = 0 Transfer function

G (s) = Y (s)

U(s) = T_ds Ts + 1

Figure 30 : Static characteristic of differentiator

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Transient response

y (t) = L⁻¹[ust

1 s

Tds

Ts + 1] = ust

Td

T e⁻^T^t

= ustkde⁻^T^t

Figure 31 : Transient response of differentiator

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Differentiator - real

Gj ω = Tds

Ts + 1|s=j ω = Tdj ω Tj ω + 1

P(ω) = TdT ω²

T²ω²+ 1, Q(ω) = Tdω T²ω²+ 1

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Amplitude diagram M(ω) = Tdω

√T²ω²+ 1

L(ω) = [20 log Tdω−20 logp

T²ω²+ 1]

Phase diagram ϕ(ω) = arctg 1

T ω =π

2 − arctg(T ω)

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Second Order Lag Element

Figure 34 : Second Order Underdmped Lag: a) pneumatic positioner, b) RLC network

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a) Pneumatic positioner

md²y (t)

dt² + Bdy (t)

dt + Cy (t) = Ap(t) m

C d²y (t)

dt² +B C

dy (t)

dt + y (t) = A Cp(t)

Equation of motion

T²d²y (t)

dt² + 2ξdy (t)

dt + y (t) = ku(t)

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Second Order Lag Element

b) RLC network

U₃(t) = I (t)R U4(t) = LdI (t)

dt I (t) = CdU2(t)

dt

U1(t) = U2(t) + U3(t) + U4(t) LCd²U₂(t)

dt² + RCdU₂(t)

dt + U₂(t) = U₁(t)

Equation of motion

T²d²y (t)

dt² + 2ξdy (t)

dt + y (t) = ku(t)

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Equation of motion

T²d²y (t)

dt² + 2ξdy (t)

dt + y (t) = ku(t) 1

ω²₀ d²y (t)

dt² +2ξ ω0

dy (t)

dt + y (t) = ku(t) d²y (t)

dt² + 2ξω0

dy (t)

dt + ω²₀y (t) = kω²₀u(t) where: 0 < ξ < 1 - damping ratio, ω0- natural frequency.

Static characteristic

y = ku

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Second Order Lag Element

Transfer function

G (s) = Y (s)

U(s) = k

T²s²+ 2ξTs + 1

G (s) = Y (s)

U(s) = kω₀² s²+ 2ξω₀s + ω₀² Transient response

y (t) = L⁻¹

ust

1 s

kω²₀ s²+ 2ξω₀s + ω₀

= ku_st

"

1 − 1

p1 − ξ²e^−ξω⁰^tsin ω₀p

1 − ξ²t + φ

#

φ = arctg

p1 − ξ² ξ

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Figure 35 : Transient response of Second Order Lag

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Second Order Lag Element

Figure 36 : Influence of the values of damping ratio ξ on transient response of Second Order Lag

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G (j ω) = kω²₀[(ω²₀− ω²) − j 2ξω0ω]

(ω₀²− ω²)²+ (2ξω₀ω)² P(j ω) = kω₀²[(ω₀²− ω²)]

(ω²₀− ω²)²+ (2ξω0ω)² Q(j ω) = − k[2ξω₀³ω]

(ω₀²− ω²)²+ (2ξω0ω)² Amplitude diagram

M(ω) = kω₀²

(ω²₀− ω²)²+ (2ξω0ω)² L(ω) =

20 log kω₀²− 20 logq

(ω₀²− ω²)²+ (2ξω₀ω)²

Phase diagram

ϕ = −arctg 2ξω0ω ω²₀− ω²

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Second Order Lag

Figure 37 : Nyquist plot Figure 38 : Bode plots

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Figure 39 : Delay - belt conveyor system

where: Q₁, Q₂- stream of mass, at the beginning and at the end of the conveyor.

Q2(t) = Q1(t − T0), T0= L v

Equation of motion

y (t) = u(t − T₀)

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Delay

Dynamics equation y (t) = u(t − T0) where: T0- transport delay.

Static characteristic y = u Transfer function

G (s) = Y (s)

U(s) = e^−T⁰^s

Figure 40 : Static characteristic of delay

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Transient response

y (t) = L⁻¹[u_st1

se^−T⁰^s] = u_st1(t − T₀)

Figure 41 : Transient response of delay

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Delay

G (j ω) = e^−jT⁰^ω P(ω) = cos (−T0ω) Q(ω) = sin (−T0ω)

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Amplitude diagram

M(ω) = 1, L(ω) = 0 Phase diagram

ϕ(ω) = −T0ω

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