Mathematical Models and Block Diagrams of Systems Regulation And Control Engineering

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

 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

 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.

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

 Example: 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

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

 Example: RLC Circuit

 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,

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

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

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

 Transform Properties

 Example: Find the Laplace Transform for the following. i. Unit function: ii. Ramp function: iii. Step function:

 Transform Theorem i. Differentiation Theorem ii. Integration Theorem: iii. Initial Value Theorem: iv. Final Value Theorem:

 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.

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

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

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

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

 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

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

 Problem: Find the solution x(t) for the following differential equations. i. ii.

 The transfer function of a linear system is the ratio of the Laplace Transform of the output to the Laplace Transform of the input variable.  Consider a spring-mass-damper dynamic equation with initial zero condition.

 The transfer function is given by the following. Y(s)R(s)

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

 Problem: Obtain the transfer function for the following RC network.

 Problem: Obtain the transfer function for the following RLC network.  Answer:

 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.

 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) OutputInput

 Feedback Control System

Therefore, The negative feedback of the control system is given by: E a (s) = R(s) – H(s)Y(s) Y(s) = G(s)E a (s)

 Reduction Rules

 Problem:

 Chapter 2 i. Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed), Prentice Hall. ii. Nise N.S. (2004). Control System Engineering (4th Ed), John Wiley & Sons.

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