ECE3330A, Control Systems Week 2 Material PDF

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EffusiveElPaso6495

Uploaded by EffusiveElPaso6495

The University of Western Australia

2022

Mehrdad R. Kermani

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control systems L-transformations engineering mathematics

Summary

This document is a week 2 material for ECE3330A, Control Systems. It introduces L-transformations and their applications in control systems engineering. The document also contains examples of various functions and their L-transformations.

Full Transcript

The University of Western Ontario Faculty of Engineering Department of Electrical and Computer Engineering ECE3330A, Control Systems Week 2 Material Last updated in September 2022 Professor: Mehrdad R. K...

The University of Western Ontario Faculty of Engineering Department of Electrical and Computer Engineering ECE3330A, Control Systems Week 2 Material Last updated in September 2022 Professor: Mehrdad R. Kermani 1 Introduction The fist step in every control design case is developing a mathemati- cal model for the system. To this effect, fundamental physical laws of science and engineering are applied to the system to obtain a differ- ential equation that can describe the relationship between the input and output of the system. Ohm’s and Kirchhoff’s laws are examples that are used to model electrical networks. Example 1. Use your knowledge of physics and mechanics to obtain a differential equation governing the rotation angle of the system. fig_04_21 Figure 1: A rotational mechanical system 2 L-transformation Although a differential equation represents the relationship between the input and output of a system, it is not a satisfying representation from a system perspective, since the input and output as well as the coefficients of the equation (system parameters) appear throughout the equation. We would prefer a mathematical representation such as that shown in 2 where the input, output and system are separate parts; in other words to represent the system as a block diagram. One ECE3330A, Control Systems 2 main advantage of such a representation is simple representation of the interconnection of several subsystems, being cascade or parallel. Using Laplace transform we can achieve this objective. Figure 2: The input r(t) stands for reference input and the output c(t) for controlled output The L-transformation is defined as, Z ∞ L[f (t)] = F (s) = f (t)e−st dt t=0− where s = σ + jω is a complex variable representing the frequency fig 02 01 fig_02_01 domain variable. The lower limit of the Laplace transform means that even if f (t) is discontinuous at t = 0, we can still obtain trans- formation so long as the integral converges. When f (t) is also defined for t < 0, i.e., f (t) ̸= 0, the two-sided (as apposed to one-sided) L-transform can be defined as follows, Z ∞ L[f (t)] = f (t)e−st dt t=−∞ In this course, unless stated otherwise, all transformations are one- sided. Example 2. Obtain the L-transform of impulse function defined as, ∞, 0− < t < 0+  δ(t) = 0, otherwise Example 3. Obtain the L-transform of step function defined as,  1, t ⩾ 0 u(t) = 0, t < 0 ECE3330A, Control Systems 3 Example 4. Obtain the L-transform of ramp function defined as,  t, t ⩾ 0 r(t) = 0, t < 0 Example 5. Obtain the L-transform of exponential function defined as, Ae−at , t ⩾ 0  f (t) = 0, t

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