First order system

The flow restrictor may be modeled, to a first approximation, as a linear relationship between flow and height of liquid in the tank [Eq. (40), where R is the valve resistance]. The material balance becomes Eq. (41). 

Since this is a self-regulating process, a steady state exists, governed by Eq. (42), where the subscript s denotes the steady-state value. Subtracting Eq. (42) from Eq. (41) yields Eq. (43). 

Since we are mainly only interested in deviations from the steady state, in this case we may define deviation variables by Eqs. (43), where u is the manipulated flow deviation variable and y is the controlled level deviation variable. 

In teams of the deviation variables, our mass balance becomes Eqs. (45). This is in the form of a first-order system [Eq. (46), where the time constant is given by Eq. (47) and the process gain by Eq. (4S). 

The dynamic behavior of a dynamic system can be well represented through the so-called step response . The dynamic evolution of the output variable can be monitored in response to a step-change of the input. We may ask how this system responds to a step-change of magnitude M in the input flow rate. The first-order system becomes that described by Eqs. (49), where H(t) is the Heaviside step function defined by Eq. (50). 

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Thus if the mass is negleeted, the system beeomes a first-order system. 

Time domain response of first-order systems 3.5.1 Standard form... 

Equation (3.23) is the standard form of transfer funetion for a first-order system, where K = steady-state gain eonstant and T = time eonstant (seeonds). 

Find an expression for the response of a first-order system to an impulse funetion of area A. 

Fig. 5.7 Root-locus diagram for a first-order system. Roots of characteristic equation...

Frequency response characteristics of first-order systems... 

These are conventional first-order systems where the phase of the output lags behind the phase of the input. 

As can be seen from equation (6.34), each time the frequency doubles (an increase of one octave) the modulus halves, or falls by 6dB. Or alternatively, each time the frequency increases by a factor of 10 (decade), the modulus falls by 10, or 20 dB. Hence the HF asymptote for a first-order system has a slope which can be expressed as —6 dB per octave, or —20 dB per decade. 

Sinee 1 / 2 is —3 dB, the exaet modulus passes 3 dB below the asymptote interseetion at /T rad/s. The asymptotie eonstruetion of the log modulus Bode plot for a first-order system is shown in Figure 6.8. 

The Bode phase plot for a first-order system is given in Figure 6.9. 

The step response of a first-order system (Example 3.5, Figure 3.13) is obtained using the step eommand... 

SIMULINK The Control System Toolbox does not possess a ramp eommand, but the ramp response of a first-order system (Example 3.6, Figure 3.15) ean be obtained using SIMULINK, whieh is an easy to use Graphieal User Interfaee (GUI). SIMULINK allows a bloek diagram representation of a eontrol system to be eonstrueted and real-time simulations performed. 

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