Characteristics of first- and second-order systems

In an earlier section we noticed that the poles of the system determined its modes - and thus the form of the transient response. We will now examine how the pole locations determine the characteristics of that response. It is important to note that, by understanding the nature of the transient responses for simple first- and second-order systems, it is possible to predict the characteristics of the dynamic behaviour of higher-order systems.

2.8.1 First-order system

The transfer function of the first-order system has the forms:

; (2.22) T is called the time constant of the system, U(s) and Y(s) are its input and output signals, respectively.

The response to a step input, , is ,

or . (2.23)

The time-domain response is shown if Figure 2.5. Note the following important prop- erties of the first-order system:

• At a time t equal to the time constant, , the term in the above

response is , i.e. this term has decayed to 36.8% of

its initial value, . The value of the response, however, is , i.e. 63.2% of its final value.

• After a time equal to four time constants, the response lies within 2% ¹ of the final value, units, i.e. in effect, the transient response has completely decayed away. This time is known as the “2% settling time”, . Similarly, a “5% settling time” is often quoted for which .

1. Actual value at four time constants is 1.83%; for three time constants it is 4.98%.

C s  = G s E s  , E s  = R s –H s C s 

W s = C s R s  = G s 1+G s H s 

Y s  U s  --- 1

1+sT --- a

s a+

--- where a 1 T---

= = =

U s  = A s Y s  a s a+ --- A

---s

 1

---s 1 s a+ ---

 – 

 A

= =

y t  = A1–e^–at y t 

t = T = 1a Ae^–at Ae^–at = Ae^–aa = Ae^–1 = 0.368A

Ae^–at_t=0 = A y t = T  = 0.632A

y t 

A Ae^–at

t_s = 4T t_s = 3T

40 Control systems techniques Ch. 2 From the denominator of (2.22) we note there is a real pole at Np/s. The associated term in the transient response is . Thus a real pole is associated with an exponentially decaying modein the response if is negative.

Figure 2.5 Time response of a first-order system to a step input of magnitude units.

2.8.2 The second-order system

2.8.2.1 The characteristics of the second-order system

The typical form of the differential equation of a second-order system is:

.

Having taken the Laplace Transform and assuming zero initial conditions, we can express the transfer function in the following two forms:

if . (2.24)

The second form is the ideal or classical form of the second-order transfer function which has complex poles. The parameters and in (2.24), and associated quantities marked as important (*), are defined below:

• * is the undamped natural frequency (rad/s), ;

• * is the damping ratio, ; .

The poles of the above transfer function are complex when and are of the form where

s = –a K e^–at

a –

0 T=1/a 2T 3T

y(t)

time (s)

1 1+sT --- U s 

A

0.5A Y(s)

A

p²+a₁p a+ ₀

 y t  = b₀u t 

Y s  U s 

--- b₀ s²+a₁s a+ ₀

--- _n² s²+2_ns+_n² ---

= = a₀ = b₀

 _n

_n _n = a₀

 01  = a₁2_n

01 s_{1 2} = j_d

Sec. 2.8 Characteristics of first- and second-order systems 41

• * (rad/s) is the frequency of the damped oscillations in the transient response,

• * is the damping constant (Neper/s), and

• * the relation between radian frequency and frequency f in Hertz (Hz) is . The characteristic equation for the system of (2.24) is

, with roots .

Since the damping ratio lies in the range the complex poles of the second-order transfer function are:

. (2.25)

Based on the above definitions the frequency of the damped oscillations and the damping con- stant are given by rad/s and Np/s, respectively. Solving for from the latter two relations it is found that the damping ratio is:

if .

The time-domain response of the second-order transfer function (2.24) to a step input of magnitude units is:

.

The associated time response consists of two terms, the steady-state term of value and a transient component (an exponentially decaying sinusoid); it is:

, , (2.26)

where and .

It is often useful to refer to the characteristics of the step response of the ideal second-order system with a complex pole pair. The damped oscillatory response is shown in Figure 2.6 in which some meaningful measures that characterize the response are defined.

_d



 = 2f

s²+2_ns+_n² = 0 s_{1 2} = –_n _n²–_n² 01

s_{1 2} = j_d = –_nj_n 1–²

_d = _n 1–²  = –_n 

 = – ²+_d²–_d 0.3_d

A

Y s  _n² s²+2_ns+_n² --- A

---s

= 

A

y t  A A 1–²

---e^–ntsin_dt+ –

= 01

cos =  sin = 1–²

42 Control systems techniques Ch. 2

Figure 2.6 Characteristics of the response to a unit step input of the ideal second-order system with a complex pole-pair.

The frequency in Hertz of the damped oscillation is: Hz.

The period of the oscillation is: s.

The time to the first peakis 1/2 of the period: s.

For a unit step input the peak overshoot occurs at and is:

.

As already stated, the settling time is the time for envelope to decay to a value of 2% (ac- tually 1.8%) of the final value of and is equal to four time constants:

i.e. s.¹

The useful reference family of normalised-time responses for a step input is shown in Figure 2.7 for values of between 0.1 and 1. The figure can be interpreted as follows. If, say, rad/s and then the first peak in the transient occurs at 3.2 s, however, if the first peak is reached at 1.6 s, etc.

1. A 5% settling time is equal to three time constants

0 5 10 15 20 25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Response y(t)

Time (s) period = 2π/ω_d

envelope e^−αt peak overshoot

tp t

s

2% settling time

f_d = _d2 t_f = 1f_d = 2  _d

t_p=   _d =   _n 1–² t = t_p

M₀ = y t _p –1 = e^tp = e^{– 1–2}

t_s y t 

t_s = –4 = 4 _n



_n = 1  = 0.1

_n = 2

Sec. 2.8 Characteristics of first- and second-order systems 43

Figure 2.7 Response of the second-order transfer function to a unit step input as a func- tion of normalised time, , for damping ratios between 0.1 and 1.

2.8.2.2 Implications for the dynamic performance of power systems

Certain electro-mechanical modes of oscillation, associated with the rotors of generators, are typically complex and lightly damped and of the form . The important features of these modes to which frequent reference will be made are listed in Section 2.8.2.1.

In the analysis of power system dynamic performance the damping ratio, is used in sev- eral contexts, e.g. a criterion for the dynamic performance of the system is that the damping ratio for all rotor modes should be better than, say, 0.05 (or 5%). For the second-order trans- fer function given in (2.24), the complex poles vary as shown in Figure 2.8 for . A line of constant damping ratio makes an angle with the negative real axis such that . Note that the loci of the poles in the complex s-plane is along a semi-circle of radius . The nature of the time-domain response for a mode located on the semi-circle can be ascertained directly by inspection.

0 5 10 15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Response y(t)

Normalised Time

ξ = 0.10 0.15 0.20 ξ = 0.25 0.30 0.40 ξ = 0.50 0.71 1.0

_nt

j_d



j_d 0



= cos

_n

44 Control systems techniques Ch. 2

Figure 2.8 Trajectory of the complex poles of the second-order transfer function, (2.24), for the damping ratio .