A_5_Specification of Control Systems

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Specification of Control Systems1. Introduction

Generally, control system specifications can be devided into two categories,performance and robustness specifications. The performance specification describe

the desired response of the nominal system to command inputs. The robustness

specifications limit the degradation in performance due to variations in the system and

disturbances.

2. Specifications for SISO systems.

2.1 Transient response specifications.

In many practical cases, the desired performance characteristics of control

systems are specified in terms of time-domain quantities, and frequently in terms of

the transient and steady state response to a unit step input. The unit step signal is often

used because there is a close correlations between a system response to a unit step

input and systems stability to perform under normal operating conditions.

Theoretically, if the response to a unit step input is known, it is possible to compute

the response to any other input. The transient response of a controlled system often

exhibits damped oscillations before reaching steady state. It is thus common to

specify the system by the following quantities:

1. Rise time ( rt ): time to rise from 10% to 90%

2. Percent overshoot (PO)

3. Peak time ( pt ): time to reach the first peak.

4. Settling time ( st ): time required for the response to settle within a certain

percentage of its final value.

5. Half delayed time ( ht ): time required for the response to reach 50% of its finalvalue.

The following figure shows a typical unit step response of a second order system:

2

2 2( )

2

n

n n

G ss s

=

+ +

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For this second-order system, we have the following peoperties:

2/ 1PO e

= , 21p

n

t

=

,

4(2% criterion)

3= (5% criterion)

n

s

n

t

==

A prices formula for rise time and half-delayed time in terms of damping and

natural frequency cannot be found. But useful approximations are:2 21.1 0.125 0.469 1 0.4167 2.917

;h r

n n

t t

+ + + =

Notice that the above expressions are only accurate for a second order system. Many

systems are more complicated than the pure second order system. If the values of

, , , , ands h r p

t t t t PO are specified, the shape of the response curve is virtually

determined. However, not all these specifications necessarily apply to any given case.

Quite often, the transient response requirements are described in terms of pole-zero

specifications instead of step response specifications. For example, poles are specified

in the shaded area of the following figure. In general, we may have the following

constraints on the response due to the specifications on the transient response:

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n

1.8

0.6(1 )

n

d d

r r r

d d

ds

n d

s

t t t

PO PO PO

l Ft t

t

The above specifications and constraints forms the shaded area in the following

figure.

The contours of constant n are circles in the complex plane, while the contours of

constant are straight lines as shown in the following figure.

High order systems can always be decomposed into subsystems of 1 st order or of 2nd

order, slowest subsystems usually determines the settling time of the system.

The poles and zeros of a system have direct effects on the steady-state performance. It

can be shown that the following identities hold for LTI systems:

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10

2

0

( ) lim (0)

ln ( )( )

s

s

e S

d T se

ds

=

=

where, S(s) and T(s) are sensitivity and complementary sensitivity function of

the system. If ( )T s is given by the following,

1

1

( )

( )

( )

m

i

i

n

j

j

k s z

T s

s p

=

=

+=

+

it can be shown that:

21 1

1 1

( )

n m

j ij ie p z

= = = The above equation is known as the Truxals formula.

In some cases, where the systems may not have dominant poles, the time domain

requirements on the transient and steady-state performance may be specified as

bounds on the command step response as shown in the next figure.

2.2 Stead-state specifications

Let ( )o

G s represents the open-loop transfer function of the closed-loop system in the

above figure so that:

G(s)

H(s)

y(s)R(s)

+-

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( )( )

( )

n

o N

d

kp sG s

s p s= , where

(0)

(0)

n

d

p

pare finite.

The system will then be called as type-N system. For example, when N=0, it is a type-

0 system, N=1 as type-1 system, and so on.

The error of the system can be written as:

1 1( ) ( ) ( )

1 1o o

e s R s R sG G

= =+ +

When R is a unit step change, then according to the final value theorem, the steady-

state error becomes:

0

1 1 1( ) lim

1 (0) 1 (0)so o

e t sG s G

= = + +

If we assume that (0) 1H = , we have:

0

1 1 1 1( ) lim

1 (0) 1 (0) 1so o P

e t sG s G K

= = =

+ + +

where, PK is known as the position error constant. So, if zero steady-state error is

required, we should let (0)oG , in other words, the ( )oG s should be, at least, oftype-1 system.

Similarly, if the system is required to have zero offset to a ramp input, we should let

( )o

sG s because of the following:

20 0

0

1 1 1 1 1( ) lim lim

1 (0) (0) lim (0)s so o o v

s

e t sG s sG sG K s

= = = =

+ +

And, VK is known as the velocity error constant. In order to have zero offset, ( )oG s

has to be of at least type-2.

By the same reasoning, the system will have zero steady-state offset to a

parabolic input, ( )oG s should be at least of type-3, because:

3 2 2 20 0

0

1 2 1 1 1( ) lim lim

1 (0) (0) lim (0)s so ao o

s

e t sG Ks s s G s G

= = = = + +

where, aK is called as the acceleration error constant.

To summarize:

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1( ) for a unit step input

1

1( ) for a ramp input

1( ) for a parabolic input

P

V

a

eK

eK

eK

= +

=

=

2.3 Frequency-domain performance specifications

In control system design by means of frequency-domain methods, the following

specifications are often used:

1. Resonant peak ( PM )

2. Bandwidth ( b )

3. Cut-off rate

Notice that, referring to the following figure:

0

( )max ( ) ; ( )1 ( )

oP cl cl

o

G sM G j G sG s

>

= =+

{ }0

arg max ( )p cl

G j

>

=

In general, the magnitude of the maximum peak gives an indication of the

relative stability of a stable system. Normally, a large value of PM corresponds to a

large maximum overshoot of the step response in the time-domain. For most control

systems, it is desirable to have its value lying between 1.1 and 1.5.

The bandwidth is defined as the frequency at which the magnitude of closed-loop

system drops to 0.707nof its zero-frequency value, (in other words, the energy content

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is half of that of zero-frequency). In general, the bandwidth of a closed-loop system

gives a measure of the transient response properties, in that a large bandwidth

corresponds to a faster response. Conversely, a small bandwidth, the time response

will generally be slow and sluggish.

The cut-off rate is the slope of the closed-loop frequency response at high frequencies.

The cut-off rate together with bandwidth indicates the noise-filtering characteristics

and the robustness of the system.

For the standard second-order system, the resonant peak Mp, the resonant frequency

p, and the bandwidth b are uniquely related to and n of the system:

2

1 2 for 0.707P n =

2

1for 0.707

2 1 2P n

M

=

0.52 4(1 2 ) 4 4 2

b n = + +

Like the general envelope specification on a step response, the frequency-

domain requirements may also be given as constraint boundaries as shown in the

following figure.

The general feature of the open-loop transfer function are that the gain in low-

frequency region be large and, in high-frequency region be as low as possible. Near

the gain crossover frequency, the slope of log-magnitude curve in Bode plot should be

close to -20db/decade and approach -40db/decade at higher frequency region, if

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possible.

Based on the loop gain as shown in the previous figure, the complementary

sensitivity function and sensitivity function are given as follows:

2.4. Robustness specifications for SISO LTI systems

In control system design, besides the system being nominally stable with certain

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performance specifications, there needs also some guarantee of stability regarding to

the discrepancy between the nominal model and the real plant. In time domain,

relative stability is measured by parameters such as maximum overshoot and the

decay ratio. In frequency-domain, resonant peak can be used to indicate relative

stability. Besides, Gain margin and phase margin are two design criteria commonly

used to measure systems relative stability.

In the above figure, the open-loop transfer function, L(s) is considered for

defining the gain margin and phase margin. Referring to the following figure:

From the above figure, there are two related frequencies called as gain crossover

frequency, g , and phase crossover frequency, p . A general diagram to show the

definitions of these frequencies, and the definitions of GM or PM are given in the

following:f

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The gain margin indicates the amount of gain that can be inserted in the closed loop

before the closed-loop system reaches instability. The phase margin, on the other

hand, is the change of in open-loop phase shift required at unity gain to make the

closed-loop system unstable. The Nyquist plot showing the definitions of gain margin

and phase margin is given in the above figure. It should be noticed that neither the

gain margin or the phase margin alone gives a sufficient indication of the relative

stability. For an underdamped second-order system, the phase margin is in terms of

. This result can be obtained by considering the following two equivalent systems:

Thus, based on open loopG , one can obtaine the following:

2

1

4 2

2( ) , ( ) tan

(2 )

n n

CL CL

n

G j G j

= =+

R

0.54 24 1 2 ,c

c

n

cut off frequency

= + =

1

4 2

2. . tan

4 1 2P M

=+

In general,I

2

( 2 )

n

ns s

+

R

Ry

y

o p e n l G

111

2

( 2 )

n

ns s

+

R

Ry

y

CLG


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0.64 0 0.707

0.71 0 0.707

0.77 0 0.707

c

n

for

for

for

<


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1 min22

PM Sin

min1GM + , min

1 1

( ) ( )Min

T j Max T j

= =

2.5 Closed-loop Tracking performance Based on the frequency Response:

A correlation between the frequency and time responses of a system, leading to a

method of gain setting in order to achieve a specified closed-loop response has been

well developed. As has been shown previously, PM is correlated with and, hence,

the percent overshoot. The acceptable region of this PM is 1 1.4PM .

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The contours of constant values of PM drawing in the complex plane yield a

rapid means of determining the values of PM and r .

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A_5_Specification of Control Systems

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