Chapter 1

Study of Minimum and

Non-Minimum Phase Systems

1.1 Introduction

Mathematical models of most physical systems are characterized by differential equa-

tions [10]. The differential equation describing a linear time-invariant system (i.e., coef-

ficients of the describing differential equations are constants and not functions of time)

can be re-modeled into the transfer function representation for the transient response

or frequency response analysis of SISO linear systems. The transfer function of a linear

time-invariant system is defined to be the ratio of the Laplace transform of the output

variable to the Laplace transform of the input variable under the assumption that all

initial conditions are zero.Thus if C(s) is the Laplace transform of the output and R(s)

is the Laplace transform of the input, then the transfer function of the SISO system is

given by

C(s)

R(s)= G(s) =

b0sm + b1s

m−1 + ...+ bma0sn + a1sn−1 + ...+ an

;m < n (1.1)

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1.2 General Characteristics of Transfer Functions

1.2.1 Characteristics of Poles

The nature of g(t) is dependent on the poles of the transfer function G(s) which are the

roots of the characteristic equation.[11] These roots may be real and complex and may

have multiplicity of various orders. Certain observations can be made for the nature of

terms contributed by the types of roots with non zero real parts as follows: -

(a) All the roots which have nonzero real parts contribute response terms with a

multiplication factor of eσt. Also if σ < 0, i.e., the roots have negative real parts,

the response terms vanish as t and if σ > 0, i.e., the roots have positive real parts

the response terms increase without bound.

(i) For single root at s = σ, the nature of response is Aeσt.

(ii) For roots of multiplicity k at s = σ, the nature of response is (A1 + A2t +

...+ Aktk−1)eσt

(iii) For complex conjugate root pair at s = σ + /− jω, the nature of response is

Aeσtsin(ωt+ b)

(iv) For complex conjugate root pairs of multiplicity k at s = σ+/−jω, the nature

of response is [A1sin(ωt+ b1) + A2tsin(ωt+ b2)...+ Aktk−1sin(ωt+ bk)]e

σt

(b) The above observation leads to the general conclusion regarding stability: -

(i) If all the roots of the characteristic equation have negative real parts, then

the impulse response is bounded and eventually decreases to zero.

(ii) If any root of the characteristic equations has a positive real part, g(t) is

unbounded and the system is therefore unstable.

1.2.2 Phase Response Characteristics

The phase response of the system with transfer function G(s) will depend on both the

poles and zeros of the transfer function.[12]

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(a) Minimum phase systems are those whose transfer functions have all poles and

zeros in the left half of the s-plane i.e., the poles and zeros have negative real

parts. In this case, the phase response of the system is restricted within 0 degrees

to −90 degrees and the amplitude response is a constant or unique for a particular

frequency as ω is increased from 0 to inf. In other words the transfer function has

the least (minimum) phase angle range for a given magnitude curve and there is a

unique relationship between the phase and magnitude curves.

(b) All pass systems are those whose transfer functions are having a pole-zero pattern

which is anti-symmetric about the imaginary axis i.e., for every pole in the left

half of the s plane there is a zero in the mirror image position. In this case, the

phase response varies from 0 degrees to −180 degrees as ω is increased from 0 to

inf while the amplitude response is unity.

(c) Non minimum phase systems are those whose transfer functions have one or more

poles or zeros in the right half of the s plane.The range of phase angle of any non

minimum phase transfer function is greater than 90 degrees. Unlike a minimum

phase system where the transfer function can be uniquely determined from the

magnitude curve alone, the non minimum phase system transfer function cannot

be determined. Also, non minimum phase systems are slow in response because of

their faulty behavior at the start of the response.

1.3 Simulation Data and Results

In this work, the simulation parameters are assumed as follows:

(a) Minimum Phase Plant(MP): The transfer function of a second order system

with one zero in left hand plane is chosen as below:

p(s) =s+ 4

s2 + 2s+ 4

The step response plots and the bode plot of the above plant is as shown in Fig.2.1.

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−30

−20

−10

0

10

Magnitude (

dB

)

10−2

10−1

100

101

102

−90

−45

0

45

90

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

(a) Bode Plot for Min Phase Plant

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

time(s)

Pla

nt

ou

tpu

t

InputOutput

(b) Step Response of MP Plant

Figure 1.1: Characteristics of Minimum Phase Plant

(b) Non Minimum Phase Plant(NMP): The transfer function of a second order

system with one zero in right hand plane is chosen as below:

p(s) =−s+ 4

s2 + 2s+ 4

The step response plots and the bode plot of the above plant is as shown in Fig.2.2.

−30

−20

−10

0

10

Magnitude (

dB

)

10−2

10−1

100

101

102

−90

−45

045

90

135

180

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

(a) Bode Plot for NMP Plant

0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time(s)

Pla

nt

ou

tpu

t

InputOutput

(b) Step Response of NMP plant

Figure 1.2: Characteristics of Non Minimum Phase Plant

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1.4 Conclusion

In this chapter minimum phase and non minimum phase systems were studied and

simulations carried out to show their characteristics.

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