La Place Transforms
Transcript of La Place Transforms
![Page 1: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/1.jpg)
NTTF
CONTROL SYSTEMS
LAPLACE TRANSFORMS
![Page 2: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/2.jpg)
Control Systems - Laplace transforms 2NTTF
Introduction• A linear real-time control system can be replaced
with a mathematical model for the purpose of analysis.
• The model is in the form of linear differential equations.
• Solutions of these differential equations completely describe the control system characteristics, including the transient response.
• Several techniques, available in calculus for obtaining the solutions of these differential equations, some of which are quite demanding.
![Page 3: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/3.jpg)
Control Systems - Laplace transforms 3NTTF
Introduction• Laplace transformation is a method that allows the
solutions of linear differential equations to be obtained without much complexity.
• In this section, the concept of Laplace transforms is introduced, and some of the transform properties are discussed.
• A rigorous treatment of transforms is not intended. The emphasis is on using a transform table to perform forward and inverse transformations for determining the response of control systems.
![Page 4: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/4.jpg)
Control Systems - Laplace transforms 4NTTF
Transformations• The Laplace transform is a method of operational
calculus that takes a function of time (time domain) and converts it into a function of complex variable s (frequency domain, or s domain).
• In some ways, using a Laplace transform is analogous to logarithmic transformation.
• Before the advent of calculators and computers, logarithms were used for performing multiplication, division, and exponential calculations.
• These two processes are fairly comparable.
![Page 5: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/5.jpg)
Control Systems - Laplace transforms 5NTTF
Logarithmic Process• Logarithm tables are used to convert numerical
terms.
• Arithmetic operations are carried out to reduce the expression to a single numerical term.
• Logarithm tables are used to find antilogarithms, yielding the desired result.
![Page 6: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/6.jpg)
Control Systems - Laplace transforms 6NTTF
Laplace Transformation Process• Laplace transform tables are used to transform
a differential equation.
• The transformed equation is simplified to isolate the desired variable.
• Inverse transformation is applied to obtain the desired time equation.
![Page 7: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/7.jpg)
Control Systems - Laplace transforms 7NTTF
Laplace Transform• Laplace transforms are useful in control system
analysis. • The time response of a control system can be
obtained by first applying a Laplace transform and then taking its inverse transform.
• Because the forward transformation leads into a frequency domain, the frequency response of a control system can be obtained directly from the transformed expression.
• The transfer function of a control system is defined in s domain and provides valuable information about stability and performance of a closed-loop system.
![Page 8: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/8.jpg)
Control Systems - Laplace transforms 8NTTF
Laplace TransformThe Laplace transform of a function of time, f(t), is defined as the integral
This function is defined for every s, which results in convergence of the integral. The variable s is a complex variable (s =σ+ jw).
![Page 9: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/9.jpg)
Control Systems - Laplace transforms 9NTTF
Laplace Transforms• Forward Laplace Transformation
– The process of converting a time domain function into s domain is known as forward Laplace transformation, or simply forward transformation.
• Inverse Laplace Transformation– The process of converting an s-domain function
back into a time-domain function is called the inverse Laplace transformation, or simply inverse transformation.
![Page 10: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/10.jpg)
Control Systems - Laplace transforms 10NTTF
Transform Notation• The forward Laplace transformation process is
indicated by the letter L;– for example, L(f(t)) = F(s).
• The inverse Laplace transformation process is indicated by L -1; – for example, f(t) ==L–1 (F(s)).
• Lowercase letters with or without a t in parentheses are used to indicate functions of time; – for example, f(t), x(t), y(t), f, x, and y.
![Page 11: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/11.jpg)
Control Systems - Laplace transforms 11NTTF
Transform Notation• Uppercase letters with an s in parentheses are
used to indicate transformed functions; – for example, F(s), X(s), and Y(s).
• Uppercase letters without a t or an s in parentheses are used to indicate constants; – for example, F, X, and Y.
![Page 12: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/12.jpg)
Control Systems - Laplace transforms 12NTTF
Examples
The following are some examples of constant, time-domain, and s-domain terms.
a. Y constantb. x(t) time-domain functionc. v time-domain functiond. X(s) s-domain functione. E constantf. Z(s) s-domain function
![Page 13: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/13.jpg)
Control Systems - Laplace transforms 13NTTF
Rules of Transformation• A number of rules have been developed to aid
in performing forward and inverse Laplace transformation.
![Page 14: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/14.jpg)
Control Systems - Laplace transforms 14NTTF
Rule 1• Multiplication (or Division) by a Constant
– When a function is multiplied by a constant, the Laplace transformation is the product of the original transform and the constant.
– Let K be a constant and let L(f(t)) = F(s), then
L[Kf(t)] = KF (s)
• Similarly, if C is a constant and L[f(t)] = F(s), then
![Page 15: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/15.jpg)
Control Systems - Laplace transforms 15NTTF
Rule 2• Sum (or Difference) of Two Functions
– The transform of a sum of two time functions is equal to the sum of their individual transforms.
– Let L[f1(t)] = F1(s),and L[f2(t)] = F2(s),then
– L[ f1(t) + f2(t)] = L[ f1(t)] + L[ f2(t)]
– = F1(s) + F2(S)
![Page 16: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/16.jpg)
Control Systems - Laplace transforms 16NTTF
Rule 3• Derivative of a Function
– First Derivative• The transform of the first derivative of a time function
is given as
where F(s) =L[f(t)] and the term f(O) is the value of function f(t) at time t = 0 (also known as the initial value).
![Page 17: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/17.jpg)
Control Systems - Laplace transforms 17NTTF
Rule 3• Second Derivative
– The transform of the second derivative of a time function is given as
where F(s) = L[f(t)], f(O) is the value of function f(t) at time t =0 (initial value), and df(O)/dt is the value of the first derivative of function f(t) at time t = o.
![Page 18: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/18.jpg)
Control Systems - Laplace transforms 18NTTF
Note on rules
For a function with zero initial values, the transformation simplifies to
![Page 19: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/19.jpg)
Control Systems - Laplace transforms 19NTTF
Example• The current i(t) flowing into a 1-µF capacitor
is related to capacitor voltage v through the following equation.
• Determine the Laplace transform of the current. Initially, the capacitor has no voltage across it (v(O) = 0, zero initial condition).
![Page 20: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/20.jpg)
Control Systems - Laplace transforms 20NTTF
Solution• Take the Laplace transform of both sides of
the equation
• Because the initial value of capacitor voltage v(O) is zero, the final expression is
1(s) = sV(s)
![Page 21: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/21.jpg)
Control Systems - Laplace transforms 21NTTF
Example• Repeat above example, assuming that the
capacitor was initially (at time t = 0) charged, with the voltage across capacitor being 1.5 V.
![Page 22: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/22.jpg)
Control Systems - Laplace transforms 22NTTF
SolutionFrom the previous example, the transformed equation is
I(s) = sV(s) – v(O)
Substituting the value of v(O) (initial condition),
I(s) = sV(s) – 1.5
![Page 23: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/23.jpg)
Control Systems - Laplace transforms 23NTTF
Rule 4• Integral of a Function
– The transform of the first integral of a time function is given as
– where F(s) = L[f(t)], f(O)is the value of function f(t) at time t =0 (initial value), and ∫f(O)dt is the value of integral of function at time t = 0 (initial value). For functions with zero initial values, the transformation simplifies to
![Page 24: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/24.jpg)
Control Systems - Laplace transforms 24NTTF
Example• Find the Laplace transform of the following.
Assume zero initial conditions.– a. ∫ i(t)dt– b. 10∫ i(t)dt
![Page 25: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/25.jpg)
Control Systems - Laplace transforms 25NTTF
Solutiona. Here i(t) is the time function. The Laplace transform of i(t) is
![Page 26: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/26.jpg)
Control Systems - Laplace transforms 26NTTF
Solutionb. Because 10 is a constant, it is transparent to the transformation process (rule 2).
![Page 27: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/27.jpg)
Control Systems - Laplace transforms 27NTTF
Rule 5 • Initial Value Theorem
– The initial value (t → 0) of a time function f(t) whose transform is F(s) is given by the following limit:
– This theorem is useful in determining the initial value of the function f(t) from F(s) without performing the inverse Laplace transformation.
![Page 28: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/28.jpg)
Control Systems - Laplace transforms 28NTTF
Example• Current through a series RL circuit, when subjected to an
applied voltage, is given by , the following equation
– Where E= Battery voltage (10V)
– R=series resistance (100Ω)
– L= series inductance (10mH) • Determine the (instantaneous) value of current immediately
after the application of voltage. Assume that before the voltage was applied, no current was flowing in the circuit and that there was no magnetic field present across the coil.
![Page 29: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/29.jpg)
Control Systems - Laplace transforms 29NTTF
Solution Because the Laplace transform of the current l(s) is given, the initial value of the current can be obtained simply by applying the initial value theorem.
This is the expected result, because inductance offers almost an infinite resistance to current buildup from zero (initial state) value.
![Page 30: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/30.jpg)
Control Systems - Laplace transforms 30NTTF
Rule 6 • Final Value Theorem
– The final value (t → ∞) of a time function f(t) with transform F(s) is given by the following limit:
– This theorem is useful in determining the final value (steady state) of the function f(t) from F(s) without performing the inverse Laplace transformation.
![Page 31: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/31.jpg)
Control Systems - Laplace transforms 31NTTF
Example• . Determine the final (steady-state) value of
current in last example
![Page 32: La Place Transforms](https://reader034.fdocuments.us/reader034/viewer/2022042613/5469ffb3b4af9fe0338b4ead/html5/thumbnails/32.jpg)
Control Systems - Laplace transforms 32NTTF
Solution
The final value of current in the circuit (time t → ∞) can be obtained by applying the final value theorem.