Continuous Control Systems: a review

Figure 1.8. Frequency characteristic of a first-order system

The relation

 ()  (n  m)  90

(1.1.19b)

gives the asymptotic phase deviation.

Note that the rise time (t_R) for a system depends on its bandwidth (_B). We have the approximate relation

_{t } 2 to 3
R _
(1.1.20)

B

6. 
   Study of the Second-order System
   

The normalized differential equation for a second-order system is given by:

_d ² y(t) _{ 2}
^dy(t)  _ ² y(t)  _ ² u(t)

(1.1.21)

dt^2
0 _dt 0 0

Using the operator p = d/dt, Equation 1.1.21 is rewritten as

( p²  2 ₀ p  _ ²) y(t)  _ ² u(t)

(1.1.22)

0 0

Letting u(t) = e^st in Equation 1.1.21, or p = s in Equation 1.1.22, the normalized transfer function of a second-order system is obtained:

H (s) 
2


0

0
_s²  2 _0 s  _ ²

(1.1.23)

in which

• 
  ₀ : natural frequency in rad/s (₀ = 2  f₀)
  
• 
   : damping factor
  

The roots of the transfer function denominator (poles) are

1. 
    < 1, complex poles (oscillatory response) : s_1,2 = -  ₀ ± j ₀
   

(1.1.24a)

(₀

2. 
   
   

is called “damped resonance frequency”).
 1, real poles (aperiodic response) :

s_1,2 = -  ₀ ± ₀
(1.1.24b)

The following situations are thus obtained depending on the value of the damping factor :

• 
   > 0 : asymptotically stable system
  
• 
   < 0 : unstable system
  

These different cases are summarized in Figure 1.9.



Figure 1.9. The roots of the second-order system as a function of  (for ||  1)



0
The step response for the second-order system described by Equation 1.1.21 is given by the formula (for   1)

y(t)  1 
¹ _e^0t (sin 
1   ² t   )


(1.1.25)

in which

 = cos^-1  (1.1.26)

Figure 1.10 gives the normalized step responses for the second-order system. This diagram makes it possible to determine both the response of a given second-order system and the values of ₀ and   in order to obtain a system having a given rise

(or settling) time and overshoot.
To illustrate this, consider the problem of determining ₀ and  so that the rise time (0 to 90% of the final value) is 2.75s with a maximum overshoot  5%. From Figure 1.10, it is seen that in order to ensure an overshoot  5% we must choose
 = 0.7. The corresponding normalized rise time is: ₀ t_M  2.75. It can be
concluded that to obtain a rise time of 2.75s, ₀ = 1 rad/sec must be taken.

y
1,4

1,2

1

0,8

0,6

0,4

0,2

0
0 1 2 3 4 5 6 7 8 9 10

Figure 1.10. Normalized frequency responses of a second-order system to a step input

100,00
90,00

80,00
70,00
60,00
50,00
40,00
30,00
20,00
10,00
0,00

M %


0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

 ₀ ^t _R

4

3,5

3

2,5

2

1,5

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Figure 1.11. Second-order system: a) maximum overshoot M as a function of the damping factor  ; b) normalized rise time as a function of 

In order to make easier the determination of ₀ and  for a given rise time t_R and a given maximum overshoot M, the graph of M as a function of  and the graph of
₀ t_R as a function of  have been represented in Figure 1.11a, b.
The curve given in Figure 1.11a allows choosing the damping factor  for a given maximum overshoot M. Once the value of  chosen, the Figure 1.11b gives the corresponding value of ₀ t_R. This allows one to determine ₀ for a given rise time t_R.
The functions omega_damp.sci (Scilab) and omega_damp.m (MATLAB^®) allow one to obtain the values of ₀ and  directly from the desired overshoot and
rise time³.


2^nd Order Systems : Bode Magnitude Diagram
20

10

0



Magnitude (dB)
-10

-20

-30
-4010^-1 10⁰ 10¹
Frequency (/₀)
Figure 1.12. Normalized frequency responses of a second-order system (gain)

The settling time t_S, for different values of  and of the tolerance zone around the final value, can be determined from the normalized responses given in Figure 1.10.

Figure 1.12 gives the normalized frequency responses for a second-order system.


³ To be downloaded from the book website.

7. 
   Systems with Time Delay
   

Many industrial processes exhibit a step response of the form shown in Figure

1.13. The period of time during which the output does not react to the input is called time delay (denoted by .
A first-order dynamic system with a time delay  is described by the following differential equation:

^dy   ¹ y(t)  ^G u(t   )

(1.1.27)

dt T T

where the argument of u(t - ) reflects the fact that the input will act with a time delay of . Equation 1.1.27 is to be compared with Equation 1.1.1.

The corresponding transfer function is


_Ges

H (s) 

1  s T

(1.1.28)

in which e^-s represents the transfer function of the time delay .

y(t)
u(t)


_ t
Figure 1.13. Step response of a system with time delay

Equations 1.1.27 and 1.1.28 can be straightforwardly extended to high-order systems with time delay.

Note that for the systems with time delay the rise time t_R is generally defined from t = .
The frequency characteristics of the time delay are obtained by replacing s=j
in e^-s. We then obtain


H_delay (j ) = e^{- j} = | 1 | .  ^ (1.1.29)

with


 ^  -   (rad) (1.1.30)

Therefore a time delay does not modify the system gain, but it introduces a phase deviation proportional to the frequency.

8. 
   Non-minimum Phase Systems
   

For continuous time systems (exclusively), non-minimum phase systems have one or more unstable zeros. In the continuous time case, the main effect of unstable zeros is the appearance of a negative overshoot at the beginning of the step response, as it is shown for example in Figure 1.14. The effect of the unstable zeros cannot be offset by the controller (one should use an unstable controller).

1
0.8
0.6
0.4
0.2
0
-0.2

-0.4

0 1 2 3 4 5 6
Time (s)

Figure 1.14. Step responses of a non-minimum phase system (H(s)=(1-sa)/(1+s)(1+0.5s), a=1,0.5)

As an example consider the system

H (s) 
1  sa


(1  s)(1  0.5s)

with a  1 and 0.5 . Figure 1.14 represents the step response of the system.

2. Closed-loop Systems

Figure 1.15 shows a simple control system. y(t) is the plant⁴ output and represents

the controlled variable, u(t) is the input (control signal) applied to the plant by the controller (manipulated variable) and r(t) is the reference signal.





Figure 1.15. Control system

The control systems have a closed-loop structure (the control signal is a function of the difference between the reference and the measured value of the controlled variable) and contains at least two dynamic systems (the plant and the controller).

We shall examine in this section the computation of the closed-loop transfer function, the steady-state error with respect to the reference signal, the rejection of disturbances and stability of the closed-loop systems.

1.