[...]... 9.11 20 3 22 7 Introduction 22 7 The State Variables Formalism 22 7 Solving Matrix Differential Equations 22 9 The Significance of the Eigenvalues of the Matrix 23 0 Understanding Homogeneous Matrix Differential Equations 23 2 Understanding Inhomogeneous Equations 23 3 The Cayley-Hamilton Theorem 23 4 Controllability 23 5 Pole Placement 23 6 Observability 23 7 Examples 23 8 9.11.1 Pole Placement 23 8 9.11 .2 Adding an... 11.1.1 11.1 .2 11.1.3 11.1.4 Chapter 11 .2. 1 11 .2. 2 11 .2. 3 11 .2. 4 Chapter 11.3.1 11.3 .2 1 Problem Problem Problem Problem 2 Problem Problem Problem Problem 3 Problem Problem 1 3 5 7 1 3 5 7 1 3 25 1 25 1 25 2 25 3 25 7 25 8 26 0 26 1 26 1 26 2 26 6 26 8 26 9 27 1 27 3 27 6 27 8 27 8 27 9 28 4 28 5 28 8 28 9 29 1 29 5 29 5 29 5 29 6 29 7 29 8 29 8 29 8 29 9 300 301 303 303 304 xvi A Mathematical Introduction to Control Theory 11.3.3... Mathematical Preliminaries 1) We find that: 1 _ a_ b + S2~1~S-1 S + 1' A standard way of finding the coefficients a and 6 is to multiply both sides of the equation by s2 — 1 One finds that: 1 = a( s + 1) + b(s - 1) -1 = (a + b)s + (a - b) Equating coefficients of like powers of s, we find that a + b — 0, and a — b = 1 Adding the two equations we find that a = 1 /2 Clearly b = -1 /2 We find that: 1 S 2 -l... the generalized triangle inequality: / f(t)dt Ja < I \f{t)\dt, b >a Ja which says that the absolute value of the integral of a a function is less than or equal to the integral of the absolute value of the function Using the triangle inequality and the 11 Mathematical Preliminaries generalized triangle inequality, we find that: \h(s)-I2\ [A( e'st-l)f(t)dt < Jo + r(e-st-l)f(t)dt JA , A> 0 [A\ (e-st-l)f'(t)\dt... Function Concept 20 4 8 .2. 2 Predicting Limit Cycles 20 7 8 .2. 3 The Stability of Limit Cycles 20 8 8 .2. 4 More Examples 21 1 8 .2. 4.1 A Nonlinear Oscillator 21 1 8 .2. 4 .2 A Comparator with a Dead Zone 21 2 8 .2. 4.3 A Simple Quantizer 21 3 8 .2. 5 Graphical Method 21 4 Tsypkin's Method 21 6 The Tsypkin Locus and the Describing Function Technique 22 1 Exercises 22 3 An Introduction to Modern Control 9.1 9 .2 9.3 9.4 9.5 9.6... children, colleagues, and students, writing this book has been a pleasant and meaningful as well as an interesting and challenging experience Though all of the many people who have helped and supported me over the years have made their mark on this work I, stubborn as ever, made the final decisions as to what material to include and how to present that material The nicely turned phrase may well have been... = ~ f-5—J r) • Using the chain rule3 we find that: (!//(*))' = (-l/f2(s))f\s) Thus we find that £(fe~*sin(i))(s) is equal to: d f 1 \ _ -1 d ,2 " ^ VS2 + 2 s + 2 J = - ( s 2 + 2 s + 2 ) 2 ^ ( _ s v +2s + 2J 2s+ 2 ~ (s2 +2s + 2) 2 ' Often we need to calculate the Laplace transform of a function g(t) = f(at) ,a > 0 It is important to understand the effect that this "dilation" of the time variable has on... 10 .20 10 .21 10 .22 10 .23 10 .24 11 Introduction The Definition of the Z-Transform Some Examples Properties of the Z-Transform Sampled-data Systems The Sample -and- Hold Element The Delta Function and its Laplace Transform The Ideal Sampler The Zero-Order Hold Calculating the Pulse Transfer Function Using MATLAB to Perform the Calculations The Transfer Function of a Discrete-Time System Adding a Digital... Laplace transform that is used today is a "cousin" of Heaviside's operational calculus[Dea97] 1 2 A Mathematical Introduction to Control Theory From the definition of the Laplace transform, we find that: U(s) = C(u(t))(s) e-st-ldt = / Jo _ e~st °° ~s o , e~st = hm t-»oo — s 1 —S Denote the real part of s by a and its imaginary part by /? Continuing our calculation, we find that: Ms) = lim e'at- 1 P-j0t... PID Controller An Extended Example 7.7.1 The Attenuator 7.7 .2 The Phase-Lag Compensator 7.7.3 The Phase-Lead Compensator 7.7.4 The Lag-Lead Compensator 7.7.5 The PD Controller Exercises xiv 8 A Mathematical Introduction to Control Theory Some Nonlinear Control Theory 8.1 8 .2 8.3 8.4 8.5 9 9. 12 9.13 9.14 10 Introduction 20 3 The Describing Function Technique 20 4 8 .2. 1 The Describing Function Concept 20 4 . given system. In particular, we discuss phase-lag, phase- lead, lag-lead and PID (position integral derivative) controllers and how to use them. In the eighth chapter we discuss nonlinear systems,. fits into the worlds of mathematics and engineering. This book was written for students who have had at least one semester of complex analysis and some acquaintance with ordinary differential. presentation of the beautiful results herein described. I am happy to acknowledge Professor George Anastassiou&apos ;s support. Professor Anastassiou has both encouraged me in my efforts to have