Analysis and Control of Linear Systems - Chapter 9 pot

When the interferences are zero, Bp=0, the transfer function of the looped The bandwidth of a system is the interval of angular frequencies for which the module of open loop harmonic gai

System Control

Analysis by Classic Scalar Approach

9.1 Configuration of feedback loops

9.1.1 Open loop – closed loops

The block diagram of any closed loop control system (Figure 9.1) consists of an action chain and of a reaction (or feedback) chain which makes it possible to elaborate an error signal ε(t), the difference between the input magnitude e (t) and the measured output magnitude r (t) The output of the system is s (t)

Figure 9.1 Block diagram of a feedback control

When the system is subjected to interferences b(t), its general structure is

represented by Figure 9.2 by supposing that its working point in the direct chain is known We designate by E(p),R(p), ε(p) and S ( p) the Laplace transforms of the input, the measurement, difference and the output respectively (see Figure 9.2)

Chapter written by Houria SIGUERDIDJANE and Martial DEMERLÉ

Figure 9.2 General block diagram

The open loop transfer function of this chain is the product of transfer functions

of all its elements; it is the ratio:

)(

p E

p S

When the interferences are zero, B(p)=0, the transfer function of the looped

The bandwidth of a system is the interval of angular frequencies for which the

module of open loop harmonic gain is more than 1 in arithmetic value:

(j ) (j ) 1

Approximate trace

In order to simplify the determination of closed loop transfer functions, we can

use the approximations [9.10] and [9.11]:

If µ(jω)β(jω) >>1 then

)(

1)(

)(

ωβω

ω

j j

E

j S

If µ(jω)β(jω) <<1 then ( )

)(

j S

Figure 9.3 shows, in Bode plane, the approximate trace (in full line), of the gain curve of the closed loop frequency response

Figure 9.3 Approximate trace of the closed loop harmonic response

Point A, in particular for integrator systems, is often rejected at ω=0

9.1.2.1 Black-Nichols diagram

The Black-Nichols diagram makes it possible to graphically pass from the open loop system to the closed loop system This diagram corresponds to a unitary feedback The chart in Figure 9.4 is usable for open loop gains going from –40 dB to +40 dB and a phase difference between 0° and –360°

Figure 9.4 Black-Nichols diagram

When the feedback is not unitary, the transfer function can be re-written as a

unitary feedback gain transfer function that is divided by the return gain β because:

1( )

9.1.2.2 Estimation of closed loop time performances from the harmonic analysis

The closed loop (CL) frequency response is characterized by the quality

factorQ r, also called magnification Q, i.e the passage from the module through a

maximum to an angular frequency ω called resonance angular frequency r

The time response is characterized, for a step function input, by the time of the

first maximum t m and the overflow D, as indicated in Figure 9.5a, i.e

∞

∞

−

D ( max )/ where smax represents the maximum value obtained from the

output to instant t m and s∞ that obtained in permanent state The overflow is

expressed as a percentage

Figure 9.5 (a) Time response in CL and (b) frequency response in CL

When the system has good damping, let ξ be the value of the damping

coefficient delimited between 0.4 and 0.7, we have the relation ωc t m ≈3, where

c

ω represents the gap angular frequency It is the angular frequency for which the

open loop arithmetic gain is equal to the unit For a well damped system, the quality

factor Q r has a value of less than 3 dB

9.2 Stability

A looped system is called stable if its transfer function:

( )( )

does not have poles of positive or zero real part

In other words, the necessary and sufficient condition of stability of such a

system is that F ( p) has all its poles with a negative real part

When the denominator of F ( p) is a polynomial of order higher than 3 and does

not reveal any obvious root, the analytical calculation of the roots may be fastidious

To study the stability, we then use either the geometrical criterion called Nyquist,

where we reason only on the open loop in order to determine the stability of the

closed loop or the so-called Routh algebraic criterion where we reason on the F ( p)

specific equation without calculating its roots

We can firstly show that the stability of a linear closed loop control system is

connected to the diagrams of its open loop frequency response

The transfer functions µ( p) and β( p) are in general in the form of polynomials

inp:

)(1

)(1)

(

p D

p N

)(2

)(2)(

p D

p N

then:

)(2)(1)(2)(1

)(2)(1)

(

p N p N p D p D

p D p N p

F

+

The system’s characteristic equation is:

0)(2)(1)(2

(

The system is stable if equation [9.17] does not have zeros of positive or zero

real part

NOTE 9.1.– the methods presented in this chapter are valid when the open loop

transfer function µ( ) ( )p β p does not result from a set of transfer functions

presenting simplifications of the poles-positive real part zeros type

9.2.1 Nyquist criterion

This criterion is based on the traditional property of analytical functions and it

makes it possible to predict the behavior of a looped system by only knowing the

open loop To do this, we use the following Cauchy’s theorem

When a point M of affix p describes in the complex plane a closed contour C

(Figure 9.6a), clockwise, surrounding P poles and Z zeros of a function A ( p) of the

complex variable p, then the image of the point M through application A surrounds

P

Z

N= − times the origin in the same direction We suppose that there is no

singularity on C

If we take, for example, Z =2 and P=3, then N =−1, the point M makes 1

tour around the origin, in counterclockwise direction (Figure 9.6b)

Figure 9.6 Plane of the complex variable p and plane of A ( p)

The application to the Nyquist criterion leads to consider the transformation

)

( p

A as being the denominator of the transfer function of the looped system We

want this transfer function not to have poles of positive real part and hence its

denominator:

)()(1

)

not to have positive real part zeros We then choose as contour C a semicircle of

infinite radius in the complex semi-plane on the right of the imaginary axis C is

called the Nyquist contour The image of C through A ( p) transformation must thus

surround the origin, in a counterclockwise direction, as many times as the number of

unstable poles of equation [9.18] and hence of µ( ) ( )p β p

Figure 9.7 Contour and image of the Nyquist curve

Contour C is chosen in such as way as to surround the poles of possible zeros of

)

(

)

(

1+µ p β p with strictly positive real part If C contains, for example, Z=1 zero

and P=3 poles (Figure 9.7a), the Nyquist diagram will make N =−2 circuits

around the origin in a clockwise direction and it will go around twice in a

counterclockwise direction (Figure 9.7b)

To be stable in closed loop, it is necessary that z = 0, the image of C must then

make N = – P circuits around the origin If the open loop transfer function µ(p)β(p)

is stable, we have P=0, the image of C through the transformation 1+µ(p)β(p)

should not surround the origin However, the number of circuits made around the

origin in the transformation 1+µ(p)β(p) is equal to the number of circuits made

around the critical point –1 in the transformation µ(p)β(p)

In what follows, we will deal only with this latter transformation and we will

study the case of open loop stable systems, that of integrator systems and finally the

case of open loop unstable systems

Case 1 Open loop stable system: Nyquist diagram

Let us take the example:

(p a)(p b)(p c)

K p

p

+++

=)(

)

( β

where a, b and c are positive Contour C and the Nyquist image corresponding curve

are given by the graphs in Figure 9.8

Figure 9.8 Contour and image of Nyquist curve

The image of the semicircle of infinitely high radius, through this transformation,

is reduced to a point, which in general is the origin (the systems that can be

physically created always have zero gains for infinite angular frequencies)

The image of the radius ]0+,+∞[ is the curve in Figure 9.8b, corresponding to

the trace of the open loop frequency response and described in the direction of

increasingω The image of the ray ]−∞,0−[ is the symmetric curve of Nyquist

place, with respect to the axis of real numbers We can easily show this symmetry

from the expression of the open loop transfer function In fact, based on the

definition:

∫

=0)()(

−

=0

)sin(

)cos(

)()(

)

)()()

or:

Case 2 Open loop integrator system

Let us take the example:

)()

(

)

(

a p p

K p

p

+

=β

where a is positive

Contour C is chosen in such a way as to exclude the origin as indicated by Figure

9.9a This contour does not contain the unstable poles(P=0) Let Cε be the circle

of radiusε , p=εe jθ (Figure 9.9c)

The image of the ray ]0+,+∞[ is the trace of the frequency response in open

loop covered in the direction of increasing ω When ω→ 0+, the gain of

µ and the phase is of −π /2 When ω→+∞, the gain →0 and the

phase is of − The image of the ray π ]−∞,0`−[ is the symmetric curve, with

respect to the axis of real numbers The image of the semicircle of radius ε is an arc

of circle of radius∞ When p→0, µ(p)β(p)→K/p=K/εe jθ If ω increases

from 0− to 0+, θ varies from −π/2 to +π /2 and the open loop gain is infinite

Figure 9.9 The choice of the contour of C excludes the origin (a), the transform

of the contour C (b), Cε circle of radius ε, p = εe jθ (c)

The image of the semicircle of an infinitely big radius is the origin of the

complex plane The transform of contour C, representing the complete Nyquist

place, is represented by Figure 9.9b

NOTE 9.2 – when the open loop transfer function µ(p)β(p) has terms of the form

/ k

K p with k>1, θ varies from −kπ/2 to +kπ/2 and µ(p)β(p) describes

infinite k semicircles in a clockwise direction

Statement of Nyquist criterion

A looped system is stable (bounded input-bounded output) if and only if the

Nyquist place of its open loop transfer function µ(p)β(p), which is described in

the direction of increasing angular frequencies, does not go through the critical point

– 1 and makes around it a number of circuits in the counterclockwise direction equal

to the number of unstable poles of µ(p)β(p)

Case 3 Unstable system in open loop

EXAMPLE 9.1.– the Nyquist trace is given in Figure 9.10

( p)

p

K p

p

21)(

Figure 9.10 Nyquist trace, open loop unstable system

We have P = 1, this system, which is unstable in open loop, is equally unstable in

closed loop because the Nyquist place does not surround once the critical point –1

p

1.012

1)

The Nyquist trace is given in Figure 9.11a

Figure 9.11 Nyquist trace, open loop stable system (a), closed loop stable system (b)

We have P=0; this system, which is stable in open loop, is equally stable in

closed loop because the Nyquist place does not surround the critical point – 1

EXAMPLE 9.3

)1.01)(

51(2

)2.01)(

1()(

)

(

p p

p

p p

K p

p

++

++

=

β

The Nyquist trace is given in Figure 9.11b We haveP=0, the stability of the

closed loop depends on the value of K Let α be the meeting point of the curve with

the axis − (as indicated in the figure) If π −1<α <0, the looped system is stable

because Nyquist place does not surround the point –1, if α <−1, the looped system

is unstable

Simplified Nyquist criterion: reverse criterion

A simplified criterion can be deduced from the previous criterion

A system, stable in open loop, is stable in closed loop if, covering the Nyquist

place of the open loop in the direction of the increasing ω , leaves the critical point

on the left (Figure 9.12a) If it leaves the critical point on the right, it is unstable

(Figure 9.12b) If the gain curve µβ j goes through the critical point, the system ( ω)

is oscillating (Figure 9.12c)

Figure 9.12 Stable system (a), unstable system (b), oscillating system (c)

9.2.2 Routh’s algebraic criterion

This criterion formulates a necessary and sufficient condition so that any n

degree polynomial has all its roots of strictly negative real part

We re-write the characteristic equation [9.17] in the polynomial form:

Then, we create Table 9.1 (with n + 2 rows)

Table 9.1 Routh’s table

For a regular system, the number of non-zero terms decreases with 1 every 2

rows and we stop as soon as we obtain a row consisting only of zeros The first

column of Routh’s table has n + 1 non-zero elements for a characteristic equation of

n order The roots of this equation are of strictly negative real part if and only if the

terms of this first column of the table have the same sign and are not zero

Statement of Routh’s criterion

A system is stable in closed loop if and only if the elements of the first column of

Routh’s table have the same sign

EXAMPLE 9.4.– let us consider again example 9.1 :

)21()

(

)

(

p p

K p

p

−

=β

The characteristic equation is p(1−2p)+K=0 or −2p2+p+K=0 The first

two coefficients (which are the first two elements of the first column of Routh’s

table) are of opposite signs, the looped system is unstable

Let us now take example 9.2:

)1.01(2)1()(

)

(

p p

p K p

p

+

+

=β

The characteristic equation is p2(1+0.1p)+K(1+p)=0 or by developing

01

The system is stable in closed loop if K>0

NOTE 9.3 – a necessary condition to have negative real part roots is that all

i

a coefficients have the same sign

9.2.3 Stability margins

The physical systems are represented by mathematical models which are

generally not very exact The stability of the mathematical model does not

necessarily entail that of the physical system Consequently, in order to take into

account the uncertainties of the model, security margins must be defined during the

theoretical study in order to ensure a satisfactory behavior to the looped system,

especially when the Nyquist place of the harmonic response in open loop is near the

critical point

a) Phase margin – gain margin

The phase margin is obtained by calculating the difference between the system

phase considered and –180° to the gap angular frequency The gain margin is

obtained by calculating the difference between the system gain and 0 dB to the

angular frequency where the phase reaches –180° These margins, noted by ∆ φ

and G∆ , are represented in Bode, Nyquist and Black-Nichols planes in Figure 9.13

Figure 9.13 Bode plane (a), Nyquist plane (b), Black-Nichols plane (c)

In Bode and Black-Nichols planes:

)180(− o

When a system is at a minimal phase difference, i.e when all its zeros are of

negative real part and of low-pass type, the only consideration of the phase margin is

enough in general to ensure a convenient damping

b) Delay margin – module margin

The delay margin, for a stable system with a phase margin∆ is defined as φ

being the ratio:

c

φ

where ω is the gap angular frequency The delay margin represents the maximum c

allowed delay value leading exactly to the cancellation of the phase margin

The module margin represents the shortest geometrical distance between curve

µβ and point –1

Figure 9.14 Nyquist plane

c) Degree of stability of a second order system

Let us consider a second degree system whose closed loop transfer function, with

a unitary feedback, has the form:

2 0 0 2

2 0

2)

(

ωξω

ω++

=

p p

p

where ξ represents the damping coefficient and ω is the system’s angular 0

frequency If ξ is low, the unit-step response is oscillating, if ξ is high, the

response is strongly damped and the transient state is long

The magnification Q r, i.e the maximal gain of the closed loop module curve,

which can be measured directly in the Black-Nichols plane (Figure 9.15), is related

to the damping coefficient by the relation:

2

12

1ξ

ξ −

=

r