Step response of First Order System I.
Objectives 1. To plot the response of first order liquid level system as a function of time and to evaluate the flow resistance R of a first order liquid level system.
II.
Materials/Equipment Needed PCT 9 process module PCT 10 electrical console
III.
Equipment Set Up
` Level Control Equipment IV.
Theory The step response of a system in a given initial state consists of the time evolution of its
outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behavior of the outputs of a general system when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter. From a practical standpoint, knowing how the system responds to a sudden input is important because large and possibly fast deviations from the long term steady state may have extreme effects on the component itself and on other portions of the overall system dependent on
this component. In addition, the overall system cannot act until the component's output settles down to some vicinity of its final state, delaying the overall system response. Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on
its
ability
to
reach
one
stationary
state
when
starting
from
another
(http://en.wikipedia.org/wiki/Step_response, 10/22/12). Time behavior of a system is important. In designing a system, the time behavior may well be the most important aspect of its behavior. How long it takes the temperature to reach a new steady state is important in a control system that’s controlling a temperature. The parameters you find in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs, we will still have the following quantities and system parameters: x(t) = Response of the System, u(t) = Input to the System, Ƭ = The System Time Constant, Gdc = The DC Gain of the System Every system will have an input which we can call u (t), and a response we will denote by x (t). Each system will also have a time constant and a DC gain. Ƭ, the time constant, will determine how quickly the system moves toward steady state. Gdc, the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value. Step Response of a First Order System: A standard first order linear system will satisfy this differential equation. dx(t) −x (t) = + Gdc U (t) dt Ƭ A first order linear system will almost always have this form - or can be put into this form. When the step is applied, the derivative of the output changes immediately. Unit-Step Response of First-Order Systems
o R(s) = 1/s, and therefore the unit-step response is: Y(s) =
o Expanding Y(s) into partial fractions: Y(s) =
1 T − s Ts+1
1 s ( Ts+1 )
=
1 1 − s 1 s+ T
o Take the inverse Laplace transform: y(t) = 1 – e –t/T , t ≥ 0 o
The solution has two parts: a steady-state response: y(t) = 1, and a transient response: y t (t) = e ,, which decays to zero as t
o
The steady-state error is the error after the transient response has decayed leaving only the continuous response. The error signal: e(t) = r(t) - y(t) = 1 - 1 + e-t/T = e-t/T
V.
Procedure Set up the PCT 9 with motorized valve fully open, solenoid valve 3, open, V3 and V4
closed. Adjust the flow meter to 0.50 L/min and record steady state height. This will take few minutes. When the height of the liquid is steady, adjust the flow meter to 0.70 L/min manually in the PCT 10 simultaneous with the timer. Record the time for every 5 mm of increase in liquid level. VI.
Results and Discussion Table 11.2. Liquid Level vs. Time Liquid Level, (mm) 15 20 25 30 35 40 45 50 55 60 65 70 75
Time, (s)
Table 11.2. Liquid Level vs. Time (Continuation) Liquid Level, (mm) 80 85 90 95 100 105 110 115 120 125 130 135
Time, (s)
insert Figure1.1 Plot of Liquid level vs. Time Elapsed VI.
Conclusion and Recommendation
VII.
References 1. Unit Operations of Chemical Engineering, 4th Edition (McCabe, W.L., et.al,) 2. Principles of Transport Processes and Separation Processes (Geankoplis, C.J.)
