## A Mathematical Introduction to Control TheoryStriking a careful balance between mathematical rigor and engineering-oriented applications, this textbook aims to maximize the readers' understanding of both the mathematical and engineering aspects of control theory. The bedrock elements of classical control theory are comprehensively covered: the Routh-Hurwitz theorem and applications, Nyquist diagrams, Bode plots, root locus plots, the design of controllers (phase-lag, phase-lead, lag-lead, and PID), and three further advanced topics: non-linear control, modern control and discrete-time control. A Mathematical Introduction to Control Theory will be an invaluable book for junior and senior level university students in engineering, particularly electrical engineering. Students with a good knowledge of algebra and complex variables will also find many interesting applications in this volume. |

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### Contents

Mathematical Preliminaries | 1 |

Transfer Functions | 35 |

FeedbackAn Introduction | 61 |

The RouthHurwitz Criterion | 75 |

The Principle of the Argument and Its Consequences | 91 |

The Root Locus Diagram | 131 |

Cancellation | 159 |

Compensation | 167 |

Some Nonlinear Control Theory | 203 |

An Introduction to Modern Control | 227 |

Control of Hybrid Systems | 251 |

Answers to Selected Exercises | 295 |

345 | |

### Common terms and phrases

amplifier angle approximately assume attenuation bilinear transform Bode plots branches calculate circuit Compensated System consider a system Consider the system control theory define denominator describing function technique differential discrete-time e~Ts encirclements equation exponential feedback system finite Frequency rad/sec frequency response gain margin Gc(s given in Figure Gp(s imaginary axis infinity initial conditions integral inverse Laplace transform lag-lead compensator leave the real left half-plane let Gp(s Let us consider limit cycle linear low-pass mapped Matlab matrix matrix exponential negative nonlinear element Note number of zeros Nyquist plot origin oscillations output phase margin phase-lag compensator phase-lead PID controller polynomial Problem Qi(s rational function real coefficients region right half-plane root locus diagram Routh array Routh-Hurwitz rule samples semi-circle sequence shown in Figure signal solutions Step Response Suppose system is stable system of Figure transfer function unit circle unstable Vin(s Vin(t Vo(s voltage z-transform