AMJAD HASOON Process Control Lec4.

Transcription

1 Multiple Inputs Control systems often have more than one input. For example, there can be the input signal indicating the required value of the controlled variable and also an input or inputs due to disturbances which affect the system. The procedure to obtain the relationship between the inputs and the output for such systems is: 1. Set all inputs except one equal to zero 2. Determine the output signal due to this one non-zero input 3. Repeat the above steps for each of the remaining inputs in turn 4. The total output of the system is the algebraic sum (superposition) of the outputs due to each of the inputs. Example1: Find the output Y(s) of the block diagram in the figure below Exercise: Determine the output Y(s) of the following system. 1

2 Modelling This lecture we will concentrate on how to do system modeling based on the most commonly used technique In frequency domain using Transfer Function (TF) representation 1. Modelling and electrical systems Basic laws governing electrical circuits are Kirchhoff's current law and voltage law. Mathematical model of an electrical circuit can be obtained by applying one or both of Kirchhoff's laws to it. Example 1: find the transfer function of the following R-C circuit if you know the input is v(t) and the output is v c (t) then find the transfer function if the output is i(t) 2

3 H.W. : for the same previous example find V R (t)/v(t) Example 1: find the transfer function of the following R-L-C circuit if you know the the input is v(t) and the output is v c (t) 3

4 2. Modelling Physical System Example1: find the transfer function, X(S)/F(S) for the system of figure below: SOLUTION: Begin the solution by drawing the free-body diagram shown in Figure: 4

5 Taking the Laplace transform, assuming zero initial conditions, Exercise: In the mechanical system shown in the figure, m is the mass, k is the spring constant, b is the friction constant, u(t) is an external applied force and y(t) is the resulting displacement. 1) Find the differential equation of the system 2) Find the transfer function between the input U(s) and the output Y(s). 5

6 3. Modelling and Control of Water Tank Model The mathematical model of most of the physical processes is nonlinear in nature. On the other hand, most of the analysis like in simulation and design of the controllers, assumes that the process is linear in nature. In order to build this gap, the linearization of the nonlinear model is needed. This linearization is always done with respect to a particular operating point of the system. From the above figure it is understood that qi is the inflow rate and qo is the outflow rate (in m3/sec) of the tank, and h is the height of the liquid level of the tank at any time instant. And also assume that the cross sectional area of the tank be A. In steady state condition, both qi and qo are same, and the height h of the liquid level of the tank will be constant. 1. RESISTANCE OF LIQUID LEVEL SYSTEM The resistance for liquid flow in such a pipe or restriction is defined as the change in the level difference to a unit change in flow rate; that is, 2. CAPACITANCE OF LIQUID-LEVEL SYSTEMS The capacitance of a tank is defined to be the change in quantity of stored liquid necessary to cause a unity change in the potential (head). The potential (head) is the quantity that includes the energy level of the system. Capacitance (C) is nothing but is cross sectional area (A) of the tank. 6

7 Rate of change of fluid volume in tank = flow in flow out Since volume is (area x height) And cross sectional area can be replaced by capacitance Where the resistance R may be written as Then rearranging the equation (8) we get Substitute equation (9) in equation (7), we get After simplifying above equation the equation (10) becomes Taking Laplace transform considering initial conditions to zero RCSH (s)+ H(s) =R Q i (s) (12) The transfer function can be obtained as Where, C=cross sectional area of the tank 7

8 First Order System What is a first order system? It is a system whose dynamic behavior is described by a first order differential equation. Standard form of first order transfer functions The first order system has only one pole as shown The important characteristics of the standard form are as follows: The denominator must be of the form τs + 1 The coefficient of the s term in the denominator is the system time constant τ 8

9 The numerator is the steady-state gain K or (DC gain) the system ration between the input signal and the steady state value of output. Example1: For the first order system given below DC gain (K) is equal 10 Time Constant (T) is equal 3 Example2: For the first order system given below find the DC gain and time constant ( ) DC gain (k)= Time Constant (T) is equal There are two methods to analyze functioning of a control system that are time domain analysis and frequency domain analysis. In time domain analysis the response of a system is a function of time. Response of first order systems to some common forcing functions To get the response to any input follow the following steps (scheme) [Input] (t) LT (s) TF (s) L 1 (t)[output] 1. Step response ((t) = A (t); (t) =?? ) ( ) ( ) (s) 9

10 (s) Solving by partial fractions A (s) (t) = [1 e t / τ] The above equation is the general form of first order system response to step change. اهم ميزتين بالفيرست اورد سيستم ريسبونس هي: No overshoot No oscillations 10