Continuous Control Systems: a review

1

Continuous Control Systems: A Review


The aim of this chapter is to review briefly the main concepts of continuous control systems. The presentation is such that it will permit at a later stage an easy transition to digital control systems.
The subject matter handled relates to the description of continuous-time models in the time and frequency domains, the properties of closed-loop systems and PI and PID controllers.

1. Continuous-time Models

1. 
   Time Domain
   

Equation 1.1.1 gives an example of a differential equation describing a simple dynamic system:

^dy   ¹ y(t)  ^G u(t)

(1.1.1)

dt T T

In Equation 1.1.1 u represents the input (or the control) of the system and y the output. This equation may be simulated by continuous means as illustrated in Figure 1.1.

The step response illustrated in Figure 1.1 reveals the speed of the output variations, characterized by the time constant T, and the final value, characterized by the static gain G.
1

d_{_}y _y
dt _y
t _t


Figure 1.1. Simulation and time responses of the dynamic system described by Equation

1. 
   (I - integrator)
   

Using the differential operator p = d/dt, Equation 1.1.1 is written as

( p  ¹ ) y(t)  ^G u(t) ; p  ^d

(1.1.2)

T T dt

For systems described by differential equations as in Equation 1.1.1 we distinguish three types of time response:

1. 
   The “free” response: it corresponds to the system response starting with an initial condition y(0)=y₀ and for an identically zero input for all t (u = 0, 
   

t).

2. 
   The “forced” response: it corresponds to the system response starting with an identically zero initial condition y(0) = 0 and for a non-zero input u(t) for all t  0 (u(t) = 0, t < 0 ; u(t)  0, t  0 and y(t) = 0 for t  0).
   
3. 
   The “total” response: it represents the sum of the “free” and “forced” responses (the system being linear, the superposition principle applies).
   

Nevertheless later we will consider separately the “free” response and the “forced” response.

2. 
   Frequency Domain
   

The characteristics of the models in the form of Equation 1.1.1 can also be studied in the frequency domain. The idea is then to study the system behavior when the input u is a sinusoidal or a cosinusoidal input that varies over a given range of frequencies.

Remember that

e ^{j t}  cos t  j sin  t

(1.1.3)

and, consequently, it can be considered that the study of the dynamic system described by an equation of the type 1.1.1, in the frequency domain, corresponds to the study of the system output for inputs of the type u(t) = e^jt.

Since the system is linear, the output will be a signal containing only the frequency ω, the input being amplified or attenuated (and possibly a phase lag will appear) according to ω; i.e. the output will be of the form

y(t)  H ( j )e ^{j t}

Figure 1.2 illustrates the behavior of a system for an input u(t) = e^jt.

(1.1.4)

However there is nothing to stop us from considering that the input is formed by damped or undamped sinusoids and cosinusoids, which in this case are written as

_{u(t)  e} t _e j t _{ e}(  j )t _{ e}st _;
s    j
(1.1.5)

where s is interpreted as a complex frequency. As a result of the linearity of the system, the output will reproduce the input signal, amplified (or attenuated), with a phase lag or not, depending on the values of s; i.e. the output will have the form

y(t)  H (s) e^st
(1.1.6)

and it must satisfy Equation 1.1.1 for u(t) = e^st1.

u(t) = e^{j t}
u(t) = e^st
y(t) = H(j)e^{j t}