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MEE3165: CONTROL SYSTEMS ENGINEERING: Handout 2
1. Mathematical Modelling of Dynamic Systems The control systems can be represented with a set of mathematical equations known as mathematical model. These models are useful for analysis and design of control systems. Analysis of control system means finding the output when we know the input and mathematical model. Design of control system means finding the mathematical model when we know the input and the output. The following mathematical models are mostly used. Differential equation model Transfer function model State space model 1.1. The Laplace Transform and Transfer Functions The Laplace Transform converts an equation from the time-domain into the so- called "S-domain", or the Laplace domain, or even the "Complex domain". The Transform can only be applied under the following conditions:
- The system or signal in question is analog.
- The system or signal in question is Linear.
- The system or signal in question is Time-Invariant.
- The system or signal in question is causal. The transform is defined as such: This operation can be performed using this MATLAB command: laplace This operation can be performed using this MATLAB command (Inverse Laplace Transform) : ilaplace By NGENDAHAYO Aimable
A table of important Laplace transform pairs is given in Table below By NGENDAHAYO Aimable
Most of the electrical systems can be modelled by three basic elements : Resistor, inductor, and capacitor. Circuits consisting of these three elements are analysed by using Kirchhoff's Voltage law and Current law. (a) Resistor(R): Block diagram
R
V ( t )= i ( t ) R (^) ; i ( t )= V ( t ) R (b) Inductor(L): Block diagram
L
V ( t )= L di ( t ) dt (c) Capacitor: Block diagram
L
V ( t )= 1 C
∫ idt
Class example: Obtain the relation between the applied voltage and the current in the form of (a) Differential equation (b) Transfer function By NGENDAHAYO Aimable
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Common Uses of Dashpots(EXAMPLES): Door Stoppers Vehicle Suspension Bridge Suspension Flyover Suspension By NGENDAHAYO Aimable
F = B ( x ˙ 1 − x ˙ 2 )
Spring: The third element which opposes motion is the spring. The restoring force of a spring is proportional to the displacement. F ( t )= Kx ( t ) X 1 -displacement of end B X 2 - displacement of end A If force applied at end B, f=K(X 1 - X 2 ) If force applied at end A, f=K(X 2 - X 1 ) Where K is known as the stiffness of the spring or simply spring constant Note: viscous and elastic forces always oppose motion 1.5. Time Response Analysis of Control Systems The input can be a time varying function which may be described by known mathematical functions or it may be a random signal Thus it is customary to subject the control system to some standard input test signals which strain the system very severely. These standard input signals are : an impulse, a step, a ramp and a parabolic input. Analysis and design of control systems are carried By NGENDAHAYO Aimable
Parabolic signal Here is an image of the various system metrics, acting on a system in response to a step input: By NGENDAHAYO Aimable
The target value is the value of the input step response. The rise time is the time at which the waveform first reaches the target value. The overshoot is the amount by which the waveform exceeds the target value. The settling time is the time it takes for the system to settle into a particular bounded region. This bounded region is denoted with two short dotted lines above and below the target value. Class examples 1: Find the transfer function of the following Spring-mass-damper Notice:- D’Alembert’s principle is a slightly modified form of Newton’s second law of motion and can be stated as follows: For anybody, the algebraic sum of the externally applied forces and the forces resisting motion in any given direction is zero. -Newton’s second law summing the forces as shown in the equation below: ∑ F^ y =^ F^ ( t^ )− B^ y ¿
− ky = M y
¿⋅¿
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For zero initial conditions, the transfer function between Y(s) and F(s) is obtained by taking the Laplace transform on both sides of Eq. (above) with zero initial conditions: Time domain step response for Equation above is calculated using MATLAB: clear all clc K=1; M=1; B=1; t=0:0.02:30; num= [1]; den = [M B K] ; G = tf(num,den); y1 =step(G,t); plot(t,y1); xlabel( 'Time (Second) ' ) ;ylabel( 'Step Response' ) title( 'Response of the system to step input' ) By NGENDAHAYO Aimable
Class examples 2. get Laplace transform of, f(t)= 30t^2 + 20t ii) ROTATIONAL SYSTEMS The rotational motion of a body can be defined as motion about a fixed axis. The extension of Newton's law of motion for rotational motion states that the algebraic sum of moments or torque about a fixed axis is equal to the product of the inertia and the angular acceleration about the axis. Or ∑ torques = Jαα where J denotes the inertia and a is the angular acceleration. The other variables generally used to describe the motion of rotation are torque T, angular velocity ω^ , and angular displacement θ^. The elements involved with the rotational motion are as follows: Inertia. Inertia, J , is considered a property of an element that stores the kinetic energy of rotational motion. The inertia of a given element depends on the geometric composition about the axis of rotation and its density. T ( t )= Jα d 2 θ ( t ) dt 2 = Jαα ( t ) By NGENDAHAYO Aimable
proportional to the angular velocity of the body. Assume the moment of inertia and elasticity are negligible. The damping or frictional torque which opposes the rotational motion is given by T ( t )= B dθ ( t ) dt = Bω ( t )
- Component input: torque T(t)
- Component output: angle θ(t) CLASS EXAMPLE 3: Obtain the transfer function of mechanical network shown bellow https://books.google.rw/books? id=1Bmxdk6E08sC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v =onepage&q&f=false Solution: Free body diagram By NGENDAHAYO Aimable
For J 1 : T(t)-K( θ 1 − θ )= Jα 1 θ ´ 1 J 2 : K( θ 1 − θ )+ B θ ´+ Jα 2 θ ´ 1 =
2. GEAR TRAINS A gear train, lever, or timing belt over a pulley is a mechanical device that transmits energy from one part of the system to another in such a way that force, torque, speed, and displacement may be altered. These devices can also be regarded as matching devices used to attain maximum power transfer. Two gears are shown coupled together in Figure below By NGENDAHAYO Aimable
The torque equation on the side of gear 1 is Using Equation below The following equation is converted to The above equation indicates that it is possible to reflect inertia, friction, compliance, torque, speed, and displacement from one side of a gear train to the other. The following quantities are obtained when reflecting from gear 2 to gear 1: By NGENDAHAYO Aimable
- Now substituting equation below Into we get Where 3. MATHEMATICAL MODELLING OF FLUID SYSTEM AND THERMAL SYSTEMS By NGENDAHAYO Aimable
