Mathematical Models of Systems Objectives We use quantitative mathematical models of physical systems to design and analyze control systems. The dynamic behavior is generally described by ordinary differential equations. We will consider a wide range of systems, including mechanical, hydraulic, and electrical. Since most physical systems are nonlinear, we will discuss linearization approximations, which allow us to use Laplace transform methods. We will then proceed to obtain the input–output relationship for components and subsystems in the form of transfer functions. The transfer function blocks can be organized into block diagrams or signal-flow graphs to graphically depict the interconnections. Block diagrams (and signal-flow graphs) are very convenient and natural tools for designing and analyzing complicated control systems
Six Step Approach to Dynamic System Problems Introduction Six Step Approach to Dynamic System Problems Define the system and its components Formulate the mathematical model and list the necessary assumptions Write the differential equations describing the model Solve the equations for the desired output variables Examine the solutions and the assumptions If necessary, reanalyze or redesign the system
Differential Equation of Physical Systems
Differential Equation of Physical Systems Electrical Inductance Describing Equation Energy or Power Translational Spring Rotational Spring Fluid Inertia
Differential Equation of Physical Systems Electrical Capacitance Translational Mass Rotational Mass Fluid Capacitance Thermal Capacitance
Differential Equation of Physical Systems Electrical Resistance Translational Damper Rotational Damper Fluid Resistance Thermal Resistance
Differential Equation of Physical Systems
Differential Equation of Physical Systems
Differential Equation of Physical Systems
Differential Equation of Physical Systems
Linear Approximations
Linear Approximations Linear Systems - Necessary condition Principle of Superposition Property of Homogeneity Taylor Series http://www.maths.abdn.ac.uk/%7Eigc/tch/ma2001/notes/node46.html
Linear Approximations – Example 2.1
The Laplace Transform Historical Perspective - Heaviside’s Operators Origin of Operational Calculus (1887)
Historical Perspective - Heaviside’s Operators Origin of Operational Calculus (1887) v = H(t) Expanded in a power series (*) Oliver Heaviside: Sage in Solitude, Paul J. Nahin, IEEE Press 1987.
The Laplace Transform
The Laplace Transform
The Laplace Transform
The Laplace Transform
The Laplace Transform The Partial-Fraction Expansion (or Heaviside expansion theorem) Suppose that The partial fraction expansion indicates that F(s) consists of a sum of terms, each of which is a factor of the denominator. The values of K1 and K2 are determined by combining the individual fractions by means of the lowest common denominator and comparing the resultant numerator coefficients with those of the coefficients of the numerator before separation in different terms. F s ( ) z1 + p1 p2 × or K1 K2 Evaluation of Ki in the manner just described requires the simultaneous solution of n equations. An alternative method is to multiply both sides of the equation by (s + pi) then setting s= - pi, the right-hand side is zero except for Ki so that Ki pi s = - pi
The Laplace Transform
The Laplace Transform Useful Transform Pairs
The Laplace Transform Consider the mass-spring-damper system
The Laplace Transform
Transfer function Circuit Systems Mechanical Systems Transfer Function Modern Control Systems
Circuit Systems Inductor Capacitor Resistor ．(Kirchhoff Current Law) ．(Kirchhoff Voltage Law) ．(Ohm’s Law) Modern Control Systems
Example ： RC Series Circuit Circuit Systems Example ： RC Series Circuit By Kirchhoff and Ohm’s Law Fig. 2.1 Example 2.2：By Kirchhoff Current Law (Node Analysis) Fig. 2.2 Modern Control Systems
Mechanical Systems (Translational Motion) Spring : spring constant Damper : Velocity : viscous friction Constant : Viscous Friction Force Modern Control Systems
Mechanical Systems (Translational Motion) Mass : Mass : inertia force ：acceleration Modern Control Systems
Mechanical Systems (Translational Motion) Example : Mass-Spring-Damper Form Newton’s Second Law Under ZIC, take Laplace transform both sides (Transfer Function) Fig. 2.3 Note: ZIC=Zero Initial Condition Modern Control Systems
Mechanical Systems (Translational Motion) Example ：(Suspension system) Mathematical Model: Fig. 2.4 Tire spring constant Modern Control Systems
：angular acceleration Mechanical Systems (Rotational Motion) Spring T：torque : angle Damper ：angular velocity Inertia ：angular acceleration Modern Control Systems
Gear train (1) (2) (3) (4) no energy loss Modern Control Systems
Mechanical Systems (Rotational Motion) Example : Mass－Spring－Damper (Rotational System) Form Newton’s Second Law Under ZIC, take Laplace transform both sides Fig. 2.5 Modern Control Systems
Transfer Function Definition ZIC: Zero Initial Condition Modern Control Systems
Transfer Function: Gain that depends on the frequency of input signal When input Under ZIC, the steady state output (2.1) where Special Case: is also called the DC gain. Conclusion: Under ZIC, for sinusoidal input, the steady state Output is also a sinusoidal wave. Modern Control Systems
With reference to (2.1), we know Transfer Function Example Find the output y(t) ? With reference to (2.1), we know and Fig. 2.6 A=0.707 <1, Attenuation!
Derivation of T.F. from Differential Equation Transfer Function Derivation of T.F. from Differential Equation A Second-Order Example Set I.C. =0 and Take L.T. both sides (Transfer Function from r to y) Modern Control Systems
Example : A second-order Circuit Transfer Function Example : A second-order Circuit From Kirchihoff Voltage Law, we obtain Fig. 2.7 (Transfer Function from ) Modern Control Systems
Example : Mass-Spring-Damper Transfer Function Example : Mass-Spring-Damper Form Newton’s Second Law Take Laplace Transfrom both sides Fig. 2.8 Transfer Function Modern Control Systems
Transfer Function Modern Control Systems
Transfer Function Example Under ZIC, take L.T., we get the transfer function Poles： Zeros： Char. Equation:
Transfer Function Time Constant of a first-order system Consider : (Time Constant)
Transfer Function Example The output voltage For RC=1 is called step-function. When A=1, it is called unit step function. The output voltage For RC=1 Modern Control Systems
Time Constant: Measure of response time of a first-order system 2.3 Transfer Function Time Constant: Measure of response time of a first-order system A 1 2 3 4 5 gain-time constant form pole-zero form Modern Control Systems
Types of Systems Static System: If a system does not change with time, it is called a static system. Dynamic System: If a system changes with time, it is called a dynamic system.
Dynamic Systems A system is said to be dynamic if its current output may depend on the past history as well as the present values of the input variables. Mathematically, Example: A moving mass M y u Model: Force=Mass x Acceleration
Mechanical Systems Translational Mass Spring Damper/dashpot Rotational Inertia Damper
Mechanical System Building Blocks Basic building block: spring, dashpots, and masses. Springs represent the stiffness of a system Dashpots/Damper represent the forces opposing motion, for example frictional or damping effects. Masses represent the inertia or resistance to acceleration. Mechanical systems does not have to be really made up of springs, dashpots, and masses but have the properties of stiffness, damping, and inertia. All these building blocks may be considered to have a force as an input and displacement as an output.
Mass The mass exhibits the property that the bigger the mass the greater the force required to give it a specific acceleration. The relationship between the force F and acceleration a is Newton’s second law as shown below. Energy is needed to stretch the spring, accelerate the mass and move the piston in the dashpot. In the case of spring and mass we can get the energy back but with the dashpot we cannot. Force Acceleration Mass
Translational Spring, k (N) x(t) Fa(t)
Dashpot/Damper The dashpot block represents the types of forces experienced when pushing an object through a fluid or move an object against frictional forces. The faster the object is pushed the greater becomes the opposing forces. The dashpot which represents these damping forces that slow down moving objects consists of a piston moving in a closed cylinder. Movement of the piston requires the fluid on one side of the piston to flow through or past the piston. This flow produces a resistive force. The damping or resistive force is proportional to the velocity v of the piston: F = cv or F = c dv/dt.
Rotational Damper, Bm (N-m-sec/rad) Fa(t) Bm
Rotational Spring, ks (N-m-sec/rad) Fa(t) ks
Mechanical Building Blocks Equation Energy representation Translational Spring F = kx E = 0.5 F2/k Damper F = c dx/dt P = cv2 Mass F = m d2x/dt2 E = 0.5 mv2 Rotational T = k E = 0.5 T2/k T = c d/dt P = c2 Moment of inertia T = J d2/dt2 P = 0.5 J2
Building Mechanical Blocks Output, displacement Mass Mathematical model of a machine mounted on the ground Ground Input, force
Building Mechanical Blocks Moment of inertia Mathematical model of a rotating a mass Torque Torsional resistance Block model Shaft Physical situation
Example 1 Consider the following system Free Body Diagram M
Example 1 Differential equation of the system is: Taking the Laplace Transform of both sides and ignoring Initial conditions we get
Example 1 if
Example 2 Find the transfer function of the mechanical translational system given in Figure-1. Free Body Diagram M Figure-1
Ways to Study a System System Experiment with actual System Experiment with a model of the System Physical Model Mathematical Model Analytical Solution Simulation Frequency Domain Time Domain Hybrid Domain
Model A model is a simplified representation or abstraction of reality. Reality is generally too complex to copy exactly. Much of the complexity is actually irrelevant in problem solving.
What is Mathematical Model? A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system. What is a model used for? Simulation Prediction/Forecasting Prognostics/Diagnostics Design/Performance Evaluation Control System Design
Black Box Model When only input and output are known. Internal dynamics are either too complex or unknown. Easy to Model Input Output
Grey Box Model When input and output and some information about the internal dynamics of the system is known. Easier than white box Modelling. u(t) y(t) y[u(t), t]
White Box Model When input and output and internal dynamics of the system is known. One should know have complete knowledge of the system to derive a white box model. u(t) y(t)
Basic Elements of Electrical Systems The time domain expression relating voltage and current for the resistor is given by Ohm’s law i-e The Laplace transform of the above equation is
Basic Elements of Electrical Systems The time domain expression relating voltage and current for the Capacitor is given as: The Laplace transform of the above equation (assuming there is no charge stored in the capacitor) is
Basic Elements of Electrical Systems The time domain expression relating voltage and current for the inductor is given as: The Laplace transform of the above equation (assuming there is no energy stored in inductor) is
V-I and I-V relations Component Symbol V-I Relation I-V Relation Resistor Capacitor Inductor
Example#1 The two-port network shown in the following figure has vi(t) as the input voltage and vo(t) as the output voltage. Find the transfer function Vo(s)/Vi(s) of the network. C i(t) vi( t) vo(t)
Example#1 Taking Laplace transform of both equations, considering initial conditions to zero. Re-arrange both equations as:
Example#1 Substitute I(s) in equation on left
Example#1 The system has one pole at
Example#2 Design an Electrical system that would place a pole at -3 if added to another system. System has one pole at Therefore, C i(t) vi( t) v2(t)
Example#3 Find the transfer function G(S) of the following two port network. i(t) vi(t) vo(t) L C
Example#3 Simplify network by replacing multiple components with their equivalent transform impedance. I(s) Vi(s) Vo(s) L C Z
Transform Impedance (Resistor) iR(t) vR(t) + - IR(S) VR(S) ZR = R Transformation
Transform Impedance (Inductor) iL(t) vL(t) + - IL(S) VL(S) LiL(0) ZL=LS
Transform Impedance (Capacitor) ic(t) vc(t) + - Ic(S) Vc(S) ZC(S)=1/CS
Equivalent Transform Impedance (Series) Consider following arrangement, find out equivalent transform impedance. L C R
Equivalent Transform Impedance (Parallel)
Equivalent Transform Impedance Find out equivalent transform impedance of following arrangement. L2 R2 R1
Back to Example#3 I(s) Vi(s) Vo(s) L C Z
Example#3 I(s) Vi(s) Vo(s) L C Z
Operational Amplifiers
Example#4 Find out the transfer function of the following circuit.
Example#5 Find out the transfer function of the following circuit. v1
Example#6 Find out the transfer function of the following circuit. v1
Example#7 Find out the transfer function of the following circuit and draw the pole zero map. 10kΩ 100kΩ
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems Example 2.2
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
The Transfer Function of Linear Systems
Block Diagram Models
Block Diagram Models
Block Diagram Models Original Diagram Equivalent Diagram
Block Diagram Models Original Diagram Equivalent Diagram
Block Diagram Models Original Diagram Equivalent Diagram
Block Diagram Models
Block Diagram Models Example 2.7
Example 2.7 Block Diagram Models
Signal-Flow Graph Models For complex systems, the block diagram method can become difficult to complete. By using the signal-flow graph model, the reduction procedure (used in the block diagram method) is not necessary to determine the relationship between system variables.
Signal-Flow Graph Models
Signal-Flow Graph Models
Signal-Flow Graph Models Example 2.8
Signal-Flow Graph Models Example 2.10
Signal-Flow Graph Models
Design Examples
Design Examples Speed control of an electric traction motor.
Design Examples
Design Examples
Design Examples
Design Examples
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB error Sys1 = sysh2 / sysg4
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB error Num4=[0.1];
The Simulation of Systems Using MATLAB
The Simulation of Systems Using MATLAB
Sequential Design Example: Disk Drive Read System
Sequential Design Example: Disk Drive Read System
Sequential Design Example: Disk Drive Read System =
P2.11
P2.11 +Vd Vq Id Tm K1 K2 Km Vc -Vb K3 Ic
http://www.jhu.edu/%7Esignals/sensitivity/index.htm
http://www.jhu.edu/%7Esignals/