Continuous Control Systems: a review

Cascaded Systems

Figure 1.16 represents the cascade connection of two linear systems characterized by the transfer functions H₁(s) and H₂(s).

H ₂ (s)

H ₁ (s)
u₁(t) = e^st
y₁(t) = H₁(s) e^st = u₂
y₂ (t)

{
H(s) = H₁(s) H₂(s)


Figure 1.16. Cascade connection of two systems

If the input to H₁(s) is u₁ (t) = e^st the following relations are found:

u₂(t) = y₁(t) = H₁(s) e^st (1.2.1)
y₂(t) = H₂(s) u₂(t) = H₂(s) H₁(s) e^st = H(s) e^st (1.2.2)


⁴ the term “plant” defines the set : actuator, process to be controlled and sensor.

and we can conclude that the transfer function of two cascaded systems is


H(s) = H₂(s) H₁(s) (1.2.3)

or in the general case of n cascaded systems

H(s) = H_n(s) H_n-1(s).....H₂(s) H₁(s) (1.2.4)

2. 
   Transfer Function of Closed-loop Systems
   

Consider the closed-loop system represented in Figure 1.17.

r(t) = e ^st

H₂ (s)

H₁ (s)
+


u₁(t)
y (t) = H_CL(s) e^st

- y (t)
Figure 1.17. Closed-loop system

The output y(t) of the closed-loop system in the case of an external reference

r(t) = e^st is written as


y(t) = H_CL(s) e^st = H₂(s) H₁(s) u₁(t) (1.2.5)
But u₁(t) is given by the relation


u₁(t) = r(t) - y(t) (1.2.6)

Introducing this relation into Equation 1.2.5, one gets

[1 + H₂(s) H₁(s)] y(t) = H₂(s) H₁(s) r(t) (1.2.7)

from which


(1.2.8.)
The stability of the closed-loop system will be determined by the real parts of the roots (poles) of the transfer function denominator H_CL (s).

3. 
   Steady-state Error
   

When carrying out the synthesis of a closed-loop system, our aim is to obtain an asymptotically stable system having a given response time, a specified overshoot and ensuring a zero steady-state error with respect to the reference signal. In Figure 1.18, it is desired that, in steady-state, y(t) equals r(t), i.e. the steady-state gain of the closed-loop system between y(t) and r(t) must be equal to 1.

r(t) y(t)



Figure 1.18. Closed-loop system

In Figure 1.18 the global transfer function of the feedforward channel H_OL(s) is of the form

b₀  b₁ s  ...  b_m s^m
B(s)

H _OL (s)  a₀

• 
  a₁s  ...  a_n s ⁿ
  


A(s)
(1.2.9)

and the transfer function in closed-loop is given by

H (s)  ^{HOL (s)} 
B(s)

(1.2.10)

OL
^CL 1  H (s)
A(s)  B(s)

The steady-state corresponds to a zero frequency (s = 0). The steady-state gain is obtained by making s = 0 in the transfer function given by Equation 1.2.10.

y  H_CL

(0) r 

B(0) _{r }
A(0)  B(0)
^b0 _r ^a0 ^{ b}0
(1.2.11)

in which y and r represent the stationary values of the output and the reference.

To obtain a unitary steady-state gain (H_CL(0) = 1), it is necessary that

^b0  1  _a0  0
^a0 ^{ b}0
(1.2.12)

This implies that the denominator of the transfer function H(s) should be of the following form:

A(s)  s(a₁s  a₂ s ²  ...  a_n1s^n1)  s.A'(s) (1.2.13)
and, respectively:

H (s)  ¹  ^B(s)

(1.2.14)

OL _s
A' (s)

Thus to obtain a zero steady-state error in closed-loop when the reference is a step, the transfer function of the feedforward channel must contain an integrator.

This concept can be generalized for the case of time varying references as indicated below with the internal model principle: to obtain a zero steady-state error, H_OL(s) must contain the internal model of the reference r(t).
The internal model of the reference is the transfer function of the filter that

generates r(t) from the Dirac pulse. E.g.,
ramp  (1/ s² )  Dirac ^{). For more details see Appendix A.}
step  (1/ s)  Dirac ,

Therefore, for a ramp reference, H_OL(s) must contain a double integrator in order to obtain a zero steady-state error.

4. 
   Rejection of Disturbances
   

Figure 1.19 represents the structure of a closed-loop system in the presence of a disturbance acting on the controlled output. H_OL(s) is the open-loop global transfer function (controller + plant) and is given by Equation 1.2.9.

p(t) (disturbance)
Figure 1.19. Closed-loop system in the presence of disturbances

Generally, we would prefer that the influence of the disturbance p(t) on the system output be as weak as possible, at least in given frequency regions. In particular, we would prefer that the influence of a constant disturbance (step disturbance), often called “load disturbance”, be zero during in steady-state regime (t   , s  0) .

The transfer function between the disturbance and the output is written as:

^S yp
(s) 
1
1 H _OL

(s)
A(s)


A(s)  B(s)

(1.2.15)

S_yp(s) is called “output sensitivity function”.
The steady-state regime corresponds to s = 0.

y  _S
yp (0) p 
A(0) _{p }
A(0)  B(0)
a_{0 p}
a₀  b₀

(1.2.16)

in which y and p represent the steady-state values of the output and respectively of the disturbance.

S_yp(0) must be zero for a perfect rejection of the disturbance in steady-state regime. It follows (as in Section 1.2.3) that, in order to obtain the desired property, we must have a₀ = 0. This implies the presence of an integrator in the direct path
in order to have a perfect rejection of a step disturbance during steady-state regime (see previous section).
As a general rule, the direct path must contain the internal model of the disturbance in order to obtain a perfect rejection of a deterministic disturbance (see previous section).


Example: Sinusoidal Disturbance of Constant Frequency.

0
The internal model of the sinusoid is 1/(1  s ² /  ² )
(the transfer function of the

0
filter which, excited by a Dirac pulse, generates a sinusoid). For a perfect rejection (asymptotically) of this disturbance the controller must contain the transfer function 1/(1  s ² /  ² ) .
In general, we also have to check if there is not an amplification of the disturbance’s effect in certain frequency regions. That is why we must require that the modulus of |S_yp(j)| be inferior to a given value at all frequencies. A typical value for this condition is


| S_yp (j) | < 2 (6 dB) for all  (1.2.17)
We may also require that S_yp(j) introduces a given attenuation in a certain frequency range, if we know that a disturbance has its energy concentrated in this frequency range.

5. 
   Analysis of Closed-loop Systems in the Frequency Domain: Nyquist Plot and Stability Criterion
   

The transfer function of the open-loop H_OL(s) (Figure 1.19) can be represented in the complex plane when  varies from 0 to  as

H _OL ( j )  Re H _OL ( j )  j Im H _OL ( j )  H _OL ( j ) . ( )
(1.2.18)

The plot of the transfer function in this plane is graduated in frequencies (rad/s). This representation is often called a Nyquist plot (or hodograph).

Figure 1.20 shows the Nyquist plot for
H₁(s)  1/(1  s) and

H ₂ (s)  1/[s(1  s)] . Note that the plot of H₂(s) corresponds to the typical case where an integrator is present in the loop (to ensure a zero steady-state error).