Mathematical Models of Control Systems Dr. Mohammed Abdulrazzaq October 2016-2017

YA ALLAH ! ADVANCE ME IN KNOWLEDGE AND TRUE UNDERSTANDING بسم الله الرحمن الرحيم (قَالَ رَبِّ اشْرَحْ لِي صَدْرِي وَيَسِّرْ لِي أَمْرِي وَاحْلُلْ عُقْدَةً مِّن لِّسَانِي يَفْقَهُوا قَوْلِي)  صدق الله العلي العظيم سورة طه 25-28 YA ALLAH ! ADVANCE ME IN KNOWLEDGE AND TRUE UNDERSTANDING

Introduction 1- Why？ 1) Easy to discuss the full possible types of the control systems —only in terms of the system’s “mathematical characteristics”. 2) The basis of analyzing or designing the control systems. 2- What is ？ Mathematical models of systems — the mathematical relation- ships between the system’s variables. 3- How get？ 1) theoretical approaches 2) experimental approaches 3) discrimination learning

Non-Linear Models Tractable non-linearity Intractable non-linearity Equation may be transformed to a linear model. Intractable non-linearity No linear transform exists

1) Differential equations 2) Transfer function 4- types 1) Differential equations 2) Transfer function 3) Block diagram、signal flow graph 4) State variables The input-output description of the physical systems — differential equations The input-output description—description of the mathematical relationship between the output variable and the input variable of physical systems.

Contents Introduction Differential Equations of Physical Systems The Laplace Transform Transfer Function of Linear Systems Block Diagram

Tractable Non-Linear Models Several general Types Polynomial Power Functions Exponential Functions Logarithmic Functions Trigonometric Functions

Polynomial Models Linear Parabolic Cubic & higher order polynomials All may be estimated with OLS – simply square, cube, etc. the independent variable.

Power Functions Simple exponents of the Independent Variable Estimated with

Exponential and Logarithmic Functions Common Growth Curve Formula Estimated with Note that the error terms are now no longer normally distributed!

Trigonometric Functions Sine/Cosine functions Fourier series

Introductions A mathematical model is a set of equations (usually differential equations) that represents the dynamics of systems. In practice, the complexity of the system requires some assumptions in the determination model. The equations of the mathematical model may be solved using mathematical tools such as the Laplace Transform. Before solving the equations, we usually need to linearize them.

Differential Equations Examples: How do we obtain the equations? Physical law of the process  Differential Equation Mechanical system (Newton’s laws) Electrical system (Kirchhoff’s laws)

Differential Equations Example 1: Springer-mass-damper system Assumption: Wall friction is a viscous force. The time function of r(t) sometimes called forcing function Linearly proportional to the velocity

Differential Equations Example 2: Springer-mass-damper system Newton’s 2nd Law:

Example 3 : A passive circuit define: input → ur output → uc。 we have：

Example 4 : A mechanism Define: input → F ，output → y. We have: Compare with example 4: uc→y, ur→F---analogous systems

Differential Equations Example 5: RLC Circuit

Differential Equations Example 6: An operational amplifier (Op-amp) circuit Input →ur output →uc

Differential Equations Example 7 : A DC motor Input → ua， output → ω1

The Laplace Transform The differential equations are transformed into algebraic equations, which are easier to solve. The Laplace transformation for a function of time, f(t) is: If, , then, Similarly, Thus,

The Laplace Transform Example 8: Spring-mass-damper dynamic equation Laplace Transform for the equation above: When r(t)=0, y(0)= y0 and (0)=0:

The Laplace Transform Example 9: Spring-mass-damper dynamic equation Some Definitions q(s) = 0 is called characteristic equation (C.E.) because the roots of this equation determine the character of the time response. The roots of C.E are also called the poles of the system. The roots of numerator polynomial p(s) are called the zeros of the system.

The Laplace Transform Transform table: Impulse function Step function f(t) F(s) 1. δ(t) 1 2. u(t) 3. t u(t) 4. tn u(t) 5. e-at u(t) 6. sin t u(t) 7. cos t u(t) Impulse function Step function Ramp function

The Laplace Transform Transform Properties

The Laplace Transform Example: Find the Laplace Transform for the following. Unit function: Ramp function: Step function:

The Laplace Transform Transform Theorem Differentiation Theorem Integration Theorem: Initial Value Theorem: Final Value Theorem:

The Laplace Transform The inverse Laplace Transform can be obtained using: Partial fraction method can be used to find the inverse Laplace Transform of a complicated function. We can convert the function to a sum of simpler terms for which we know the inverse Laplace Transform.

The Laplace Transform We will consider three cases and show that F(s) can be expanded into partial fraction: Case 1: Roots of denominator A(s) are real and distinct. Case 2: Roots of denominator A(s) are real and repeated. Case 3: Roots of denominator A(s) are complex conjugate.

The Laplace Transform Case 1: Roots of denominator A(s) are real and distinct. Example: Solution: It is found that: A = 2 and B = -2

The Laplace Transform Case 1: Roots of denominator A(s) are real and distinct. Problem: Find the Inverse Laplace Transform for the following.

The Laplace Transform Case 2: Roots of denominator A(s) are real and repeated. Example: Solution: It is found that: A = 2, B = -2 and C = -2

The Laplace Transform Case 3: Roots of denominator A(s) are complex conjugate. Example: Solution: It is found that: A = 3/5, B = -3/5 and C = -6/5

The Laplace Transform Case 3: Roots of denominator A(s) are complex conjugate. Example: Solution:

The Laplace Transform Problem: Find the solution x(t) for the following differential equations.

The Transfer Function Transfer Functions and Linear Systems in which a, b, c are given constants and f(t)is a given function. In this arena f(t)is often called the input signal or forcing function and the solution y(t)is often called the output signal. We shall assume that the initial conditions are zero (in this case y(0)= 0, y’ (0)= 0). Now, taking the Laplace transform of the differential equation, gives:

in which we have used y(0)= y'(0)= 0 and where we have designated L{y(t)} = Y (s)and L{f(t)} = F(s). We define the transfer function of a system to be the ratio of the Laplace transform of the output signal to the input signal with the initial conditions as zero. The transfer function (a function of s), is denoted by H(s). In this case Now, in the special case in which the input signal is the delta function f(t) = δ(t)we have F(s)= 1 and so,

Modelling of Linear Systems by Transfer Functions::: - To begin, we can imagine a differential equation: It is the system that changes the input signal into the output signal. This is easy to describe pictorially. in the t−domain After the Laplace transform of the differential equation is taken the differential equation is transformed into : in the s−domain

- Consider a spring-mass-damper dynamic equation with initial zero condition. The transfer function is given by the following. Y(s) R(s)

The Transfer Function Electrical Network Transfer Function Component V-I I-V V-Q Impedance Admittance

The Transfer Function Problem: Obtain the transfer function for the following RLC network. Answer:

The Transfer Function Problem: Obtain the transfer function for the following RC network.

The Transfer Function Mechanical System Transfer Function Problem: Find the transfer function for the mechanical system below. The external force u(t) is the input to the system, and the displacement y(t) of the mass is the output. The displacement y(t) is measured from the equilibrium position. The transfer function of the system.

Block Diagram Feedback Control System Therefore, The negative feedback of the control system is given by: Ea(s) = R(s) – H(s)Y(s) Y(s) = G(s)Ea(s)

Transfer Function G(s) Block Diagram A block diagram of a system is a practical representation of the functions performed by each component and of the flow of signals. Cascaded sub-systems: Transfer Function G(s) Output Input

Block Diagram Feedback Control System

Block Diagram Reduction Rules

Block Diagram Reduction Rules

Block Diagram Problem:

Block Diagram Problem:

“The whole of science is nothing more than a refinement of everyday thinking…” The End…