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## Chapter 2 Mathematical Foundation

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**Chapter 2Mathematical Foundation**Automatic Control Systems, 9th Edition F. Golnaraghi & B. C. Kuo**0, p. 16**Main Objectives of This Chapter • To introduce the fundamentals of complex variables • To introduce frequency domain analysis and frequency plots • To introduce differential equations and state space systems • To introduce the fundamentals of Laplace transforms • To demonstrate the applications of Laplace transforms to solve linear ordinary differential equations • To introduce the concept of transfer functions and how to apply them to the modeling of linear time-invariant systems • To discuss stability of linear time-invariant systems and the Routh-Hurwitz criterion • To demonstrate the MATLAB tools using case studies**1, pp. 16-17**2-1 Complex-Variable Concept • Rectangular form: • Euler formula: • Polar form: (2-1) magnitude of z (2-3) (2-2) phase of z (2-4) (2-5) (2-6)**1, p. 17**Conjugate of the Complex Number • Conjugate (2-7) (2-8) (2-9)**1, p. 18**Basic Mathematical Properties**Example 2-1-1Find j3 and j4.**Example 2-1-2Find zn using Eq. (2-6). 1, p. 18 Examples 2-1-1 & 2-1-2 (2-10)**1, p. 19**Complex s-plane**1, p. 19**Function of a Complex Variable • G(s) = Re[G(s)] + jIm[G(s)] (2-11) • Single-valued function: • Not single-valued function: (2-12) one-to-one mapping**1, p. 20**Analytic Function • A function G(s) of the complex variable s is called an analytic function in a region of the s-plane if the function and its all derivatives exist in the region. • G(s) = 1/[s(s+1)] is analytic at every point in the s-plane except at the points s = 0 and s = 1. • G(s) = (s+2) is analytic at every point in the s-plane.**1, p. 20**Poles and Zeros of a Function • A pole of order r at s = pi if the limit has a finite, nonzero value. • A zero of order r at s = zi if the limit has a finite, nonzero value. • Example: (2-13)**1, p. 22**Polar Representation Polar representation of G(s) in Eq. (2-12) at s = 2j. (2-15) (2-16) (2-17)**1, pp. 22-23**Example 2-1-3 Polar representation of for s = j, = 0 .**1, p. 23**Example 2-1-3 (cont.) • Graphical representation of G(j)**1, p. 24**Example 2-1-3 (cont.) • The magnitude decreases as the frequency increases. • The phase goes from 0º to 180º.**1, p. 24**Example 2-1-3 (cont.) • Alternative approach**1, p. 24**Example 2-1-3 (cont.)**1, p. 26**Frequency Response • G(j) is the frequency response function of G(s) . • Second order system: • General case:**2, p. 26**2-2 Frequency-Domain Plots • Frequency-domain plots of G(j) versus : • Polar plot • Bode plot • Magnitude-phase plot phase magnitude**2, p. 27**Polar Plots • A plots of the magnitude of G(j) versus the phase of G(j) on polar coordinates as is varied from zero to infinity.**2, p. 27**Example 2-2-1 Consider the function , where T > 0.**2, p. 28**Example 2-2-2 Consider the function , where T1, T2 > 0.**2, p. 28**General Shape of Polar Plot • The behavior of the magnitude and phase of G(j) at = 0and = . • The intersections of the polar plot with the realand imaginary axes, and the values of at the intersections. • Intersections with the real axis: Im[G(j)] = 0 • Intersections with the imaginary axis:Re[G(j)] = 0**2, p. 30**Example 2-2-3 Consider the transfer function • The magnitude and phase of G(j) at = 0and = • Determine the intersections, if any. Re[G(j)] = 0 = , G(j) = 0; Im[G(j)] = 0 = **2, p. 30**Example 2-2-3 (cont.)**2, p. 31**Example 2-2-4 Consider the transfer function • The magnitude and phase of G(j) at = 0and = • Determine the intersections, if any. Re[G(j)] = 0 = , G(j) = 0 Im[G(j)] = 0 = rad/sec,**2, p. 31**Example 2-2-4 (cont.)**2, p. 32**Bode Plot • Bode plot of function G(j) is composed of two plots: • The amplitude of G(j) in dB versus log10 or • The phase of G(j) in degree as a function of log10 or • Bode plot corner plot or asymptotic plot • For constructing Bode plot manually, G(s) is preferably written in the form of (2-61).**2, p. 32-33**Magnitude and Phase of G(j): Example • Magnitude of G(j) in dB: • Phase of G(j):**2, p. 34**Five Simple Types of Factors • Constant factors: K • Poles or zeros at the origin of order p: (j)p • Poles or zeros at s = 1/T of order q: (1+jT)q • Complex poles and zeroes of order r: (1+j2/n2/2n)r • Pure time delay:**2, p. 34**Real Constant K: Magnitude**2, p. 34**Real Constant K: Phase**2, p. 34**Poles and Zeros at the Origin: Magnitude Slope:**2, p. 36**Poles and Zeros at the Origin: Phase**2, p. 35**Decade versus Octave**2, p. 37**Simple Zero, 1+jT Consider the functionG(j) = 1+jT (2-74) • Magnitude: • At very low frequencies, T << 1: • At very high frequencies, T >> 1: • Intersection of (2-76) and (2-77): = 1/T (2-78) • Phase:**2, p. 38**Values of (1+jT) versus T**2, p. 39**Simple Pole, 1/(1+jT) For the functionG(j) = 1/(1+jT)(2-80) • Magnitude: • Phase:**2, p. 38**Bode plots of G(s)=1+Ts & G(s)=1/(1+Ts)**2, p. 39**Quadratic Poles and Zeros: Magnitude Consider the second-order transfer function • Magnitude: • At very low frequencies, T << 1: • At very high frequencies, T >> 1:**2, p. 40**Quadratic Poles and Zeros: Magnitude**2, p. 40**Quadratic Poles and Zeros: Phase • Phase:**2, p. 42**Pure Time Delay • Magnitude: • Phase:(2-90) • For the transfer function(2-91)**2, pp. 42-43**Example 2-2-5 Consider the function corner frequencies: =2, 5, and 10 rad/sec**2, p. 43**Example 2-2-5 (cont.)**2, pp. 44-45**Magnitude-Phase Plot Example 2-2-6: Magnitude-phase plot Polar plot**2, p. 46**Gain- and Phase-Crossover Points • Gain-crossover point: the point at which • Gain-crossover frequency g • Phase-crossover point: the point at which • Phase-crossover frequency p**2, p. 47**Minimum- & Nonminimum-Phase • Minimun-phase transfer function:no poles or zeros in the right-half s-plane • Unique magnitude-phase relationship • For G(s) with m zeros and n poles, excluding the pole at s = 0, when s = j and = 0 , the total phase variation of G(j) is (nm)/2. • G(s) 0, if 0. • Nonminimun-phase transfer function:haseither a pole or a zero in the right-half s-plane • a more positive phase shift as = 0**2, p. 47**Example 2-2-7 Minimum-phase transfer function Nonminimum-phase transfer function**3, p. 49**2-3 Introduction to Differential Eqs. • Linear ordinary differential equation: • Coefficients a0, a1, …, an1 are not function of y(t) • First-order linear ODE: • Second-order linear ODE: • Nonlinear differential equation:**3, p. 50**State Equations: Second-Order • Differential equation of a series electric RLC network: • State variables: • State equations: