Frequency Response

Control Concepts

8.1 DEFINITION OF TERMS

8.1.5 Frequency Response

System analysis in the frequency domain is a convenient method of establishing stability margins. By considering the frequency characteristics of the elements in a loop, i.e. their response to a sinusoidal input stimulus, the characteristics of the overall system can be determined without the necessity of solving differential equations, frequently of higher orders.

The response of an element or system to a sinusoidal input stimulus is shown in Fig. 8.7. The phase shift 4> between the input and output and the ratio of the output to input amplitude

i8J8d

fully define its characteristics at any particular frequency. The frequency response characteristics of the elements in a loop can only be utilized for system modelling if the frequency spectrum over a sufficiently wide range of frequencies is known.

The frequency characteristics can be obtained either from the transfer function by making the substitution for the Laplace operator s

=

jw and thence treating the transfer function as a function of a complex variable or, empirically, by measuring the phase angle and amplitude of the output relative to the input. As an example, consider the transfer function of the

48 HYDRAULIC AND ELECTRO-HYDRAULIC CONTROL SYSTEMS

Fig. 8.7 Response to sinusoidal input stimulus.

spring mass system given in eqn (8.4) as a function of s. Expressing the coefficients of the characteristic equation in terms of the steady-state gain, K, natural frequency, w0 , and the damping factor, C

(Jo { K

(Ji (s) = KG s) =

(s

/w~) +

(2C/w₀)s

+

1 ^(8.5) where K = ajb, w0 = j(S/m) and C = D/(2j(S/m)).

Then making the substitution, s = jw,

(8.6)

The phase shift,¢, is given by the argument of the function of the complex variable, or

¢(jw) = -tan 2C(w/wo)

1-(w²/w~) (8.7)

and the ratio of output to input amplitude is given by the modulus, or

!Bj()i(jw)l = j((l-

(w ² /w~)~ ⁺

^(2C(w/w^{0 )) 2)} ^(8.8)

It is generally more convenient to represent the transfer function logarithmically, and since communication engineers played a leading part in the development of feedback control theory, the decibel was adopted as the logarithmic unit for the modulus. Equation (8.8) then becomes

(8.9)

CONTROL CONCEPTS 49 Open and closed loop transfer functions of some typical hydraulic transmissions are tabulated in Table 8.1. Overall system transfer functions are usually of a higher order, since they contain additional elements which influence the performance of the electro-hydraulic control system. More complex systems will be analysed in subsequent chapters.

When the open loop transfer function is represented in its general form as 8Jc. = KG(s), the closed loop transfer function is given by the expression

eo

^KG(s)

ei

= l

+

KG(s) (8.10)

System (i) is an idealized velocity control system in which all dynamic effects have been omitted. It can be seen from eqn (8.12) in Table 8.1 that when the loop is closed, the output quantity approaches the reference input, provided the loop gain K is substantially larger than unity.

System (ii) is a velocity control system in which dynamic effects have been taken into account.

In the position control system (iii) the actuator is assumed to be a pure integrator, whereas in system (iv) dynamic effects have been included.

All the transfer functions given in Table 8.1 can be expressed as a function of a complex variable by making the substitutions= jw. In Fig. 8.8 the transfer functions are represented on a complex plane over the entire frequency range, and in Fig. 8.9 the phase angle and log modulus are plotted separately versus the frequency. The former frequency response contours are called Nyquist diagrams, the latter Bode diagrams.

At low frequencies, the log modulus of the G function of both the open and closed loop transfer functions of the velocity control system (ii), represented by eqns (8.13) and (8.14) respectively, approach zero. As the Bode diagrams plotted in Fig. 8.9 include the loop gain K, the log modulus of the open loop system approaches this value at the low end of the frequency spectrum, while the closed loop system approaches a constant value K/(1

+

^K).At the high end of the frequency spectrum, both open and losed loop contours are asymptotic to a slope of - 12 dB per octave, i.e.

ioubling of frequency.

At the resonant frequency of w = w0 for the open loop and w =

j(l +

K)w0 for the closed loop system, the phase shift is -90°, reaching a maximum value of -180° at high frequencies.

In the position control system (iii) which represents a pure integrator, the open loop transfer function attenuates at 6 dB per octave at a constant phase shift of -90°. When the loop is closed, the log modulus at low frequencies approaches 0 dB while at high frequencies it is asymptotic with

System (i) Velocity Servo (ii) Velocity Servo (iii) Position Servo ~ (iv) Position Servo

~ ~

^POSITION

SU'IO

+

Table 8.1 Table of transfer functions Open Loop Transfer Function

~=K

e Oo K s2 2(s 2+-+1 Wo Wo 00 K -rs Oo K (s• Z(s -) s 2+-+1 Wo Wo

~ L___._J.

VELOCITY SEIIVO

+

(8.11) (8.13) (8.15) (8.17)

Closed Loop Transfer Function 00 K 1 9;' = 1 +K = 1 +(1/K) 00 K K

o;-=~

2+--!:K+1 Wo Wo =

o

+ KJ ~-r __ s2~ + 2cs + 1] (K+l)w~ (K+1lwo Oo 0; = -rs+ 1 00 K 0; = s3 2(s2 2+-+s+K Wo Wo s3 2(s2 s

--+--+-+1 Kw~ KwK 0

(8.12) (8.14) (8.16) (8.18)

CONTROL CONCEPTS

(a) REPRESENTED ON COMPLEX PLANE

~ -c.,

"'-- ><

...

LOCUS OF OPEN LOOP VELOCITY SERVO

REAL AXIS

..

LOCUS OF OPEN LOOP 0 POSITION SERVO (PUI\E IIITEGIIATOI\)

RECIPROCAL OF GAIN MARGIN

REAL AXIS

SYSTEM (ii)

-I

SYSTEM (iii)

SYSTEM (iV)

..

LOCUS OF CLOSED lOOP VELOCITY SERVO

REAL AXIS

LOCUS Of CLOSED LOOP POSITION SERVO (PUIIE INTEGIIATOP.)

LOCUS OF OPEN LOOP LOCUS OF CLOSED LOOP

POSITION SERVO POSITION SERVO

Fig. 8.8 Nyquist diagrams.

(b) PLOTTED AS LOG MODULUS AND ANGLE VERSUS LOG OF FREQUENCY

K ^K

L., "'i+'K

dB ASYMPTOTIC dB

T0-12d8 PER L.,

OCTAVE

o•

¢J -90°

-18 -180°

OPEN LOOP VELOCITY SERVO CLOSED LOOP VELOCITY SERVO SYSTEM (ii)

ASYMPTOTIC TO -6 dB PER OCTAVE

dB dB

0 0

L.,

"'=+

^L.,

o't---..::::--+---t/>

-45°1---"'----"k-:~---

-9011---+-_;::::,--- o"t---t/>

-9o⁰

t---OPEN lOOP POSITION SERVO (PURE INTEGRATOR) CLOSED LOOP POSITION SERVO (PURE INTEGRATOR)

I

SYSTEM (iii) ASYMPTOTIC TO -6 dB PER OCTAVE

or---~+---ASYMPTOTIC TO lm

-18 dB PER OCTAVE

-9o't---... ; : - t - - ----

+-IB00'1----4---OPEN LOOP POSITION SERVO SYSTEM (iV) CLOSED LOOP POSITION SERVO

Fig. 8.9 Bode diagram.

CONTROL CONCEPTS 53 a -6 dB per octave slope. At the break or corner frequency of w

=

1/r, the phase lag is 45°, reaching a maximum value of 90° at high frequencies.

The open loop log modulus of the position control system (iv) approaches a low frequency asymptote attenuating at 6 dB per octave and a high frequency asymptote attenuating at 18 dB per octave. The two asymptotes intersect at the natural frequency w0 , corresponding to a phase lag of 180°. At low frequencies, the integration in the loop causes a phase lag of 90° which increases to a maximum value of 270° at high frequencies.

Closing the loop removes the effect of the integration at low frequencies.

The frequency at peak amplitude is called the resonant frequency of the system.

Dalam dokumen Hydraulic and Electric-Hydraulic Control Systems (Halaman 58-64)

Control Concepts

8.1 DEFINITION OF TERMS

8.1.5 Frequency Response

i8J8d

=

(s

/w~) +

+

(w 2 /w~)~ +

eo

ei

+

+

j(l +

~ ~

+

~=K

~ L___._J.

+

o

...

..

..

"'=+

I

=

(w ² /w~)~ ⁺