1 Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE/MATH 332 Introduction to Control (Preliminary) Supplemental Lecture Notes Serdar Yuksel¨ April 3, 2021 2 Contents 1 Introduction ....................................................................................1 1.1 Introduction................................................................................1 1.2 Linearization...............................................................................4 2 Systems ........................................................................................7 2.1 System Properties...........................................................................7 2.2 Linear Systems.............................................................................8 2.2.1 Representation of Discrete-Time Signals in terms of Unit Pulses..............................8 2.2.2 Linear Systems.......................................................................8 2.3 Linear and Time-Invariant (Convolution) Systems................................................9 2.4 Bounded-Input-Bounded-Output (BIBO) Stability of Convolution Systems........................... 11 2.5 The Frequency Response (or Transfer) Function of Linear Time-Invariant Systems..................... 11 2.6 Steady-State vs. Transient Solutions............................................................ 11 2.7 Bode Plots................................................................................. 12 2.8 Interconnections of Systems................................................................... 13 2.9 Feedback Control Systems.................................................................... 13 2.10 State-Space Description of Linear Systems...................................................... 14 2.10.1 Principle of superposition.............................................................. 14 2.10.2 State-space description of input-output systems............................................ 14 2.10.3 Stability of linear systems described by state equations...................................... 15 2.11 Exercises.................................................................................. 15 3 Frequency Domain Analysis of Convolution Systems ................................................ 19 3.1 Transfer Functions for Convolution Systems..................................................... 20 3.2 Bode Plots................................................................................. 21 3.3 Exercises.................................................................................. 21 4 Contents 4 The Laplace and Z-Transformations .............................................................. 25 4.1 Introduction................................................................................ 25 4.1.1 The Two-sided Laplace Transform....................................................... 25 4.1.2 The Two-sided Z-Transform............................................................ 25 4.1.3 The One-sided Laplace Transform....................................................... 26 4.1.4 The One-sided Z-Transform............................................................ 26 4.2 Properties.................................................................................. 26 4.2.1 Linearity............................................................................ 26 4.2.2 Convolution.......................................................................... 26 4.2.3 Shift Property........................................................................ 26 4.2.4 Converse Shift Property................................................................ 27 4.2.5 Differentiation Property (in time domain)................................................. 27 4.2.6 Converse Differentiation............................................................... 27 4.2.7 Scaling.............................................................................. 29 4.2.8 Initial Value Theorem................................................................. 29 4.2.9 Final Value Theorem.................................................................. 29 4.3 Computing the Inverse Transforms............................................................. 30 4.4 Systems Analysis using the Laplace and the Z Transforms......................................... 31 4.5 Causality (Realizability), Stability and Minimum-Phase Systems.................................... 31 4.6 Initial Value Problems using the Laplace and Z Transforms........................................ 32 4.7 Exercises.................................................................................. 32 5 Control Analysis and Design through Frequency Domain Methods ................................... 35 5.1 Transfer Function Shaping through Control: Closed-Loop vs. Open-Loop............................ 35 5.2 Bode-Plot Analysis.......................................................................... 35 5.3 The Root Locus Method...................................................................... 35 5.4 Nyquist Stability Criterion.................................................................... 38 5.4.1 System gain, passivity and the small gain theorem.......................................... 40 5.5 A Common Class of Controllers: PID Controllers................................................ 41 5.6 Exercises.................................................................................. 42 6 Realizability and State Space Representation ....................................................... 43 6.1 Realizations: Controllable, Observable and Modal Forms.......................................... 44 6.1.1 Controllable canonical realization....................................................... 44 6.1.2 Observable canonical realization........................................................ 45 Contents 5 6.1.3 Modal realization..................................................................... 46 6.2 Zero-State Equivalence and Algebraic Equivalence............................................... 47 6.3 Discretization............................................................................... 48 7 Stability and Lyapunov’s Method ................................................................. 49 7.1 Introduction................................................................................ 49 7.2 Stability Criteria............................................................................ 49 7.2.1 Linear Systems....................................................................... 49 7.3 A General Approach: Lyapunov’s Method....................................................... 51 7.3.1 Revisiting the linear case............................................................... 52 7.4 Non-Linear Systems and Linearization.......................................................... 53 7.5 Discrete-time Setup.......................................................................... 55 7.6 Exercises.................................................................................. 56 8 Controllability and Observability ................................................................. 61 8.1 Controllability.............................................................................. 61 8.2 Observability............................................................................... 64 8.3 Feedback and Pole Placement................................................................. 65 8.4 Observers and Observer Feedback............................................................. 66 8.5 Canonical Forms............................................................................ 67 8.6 Using Riccati Equations to Find Stabilizing Linear Controllers [Optional]............................ 68 8.6.1 Controller design via Riccati equations................................................... 68 8.6.2 Observer design via Riccati equations.................................................... 69 8.6.3 Putting controller and observer design together............................................ 69 8.6.4 Continuous-time case.................................................................. 69 8.7 Applications and Exercises................................................................... 70 9 Performance Analysis and Limitations: Robustness, Tracking and Disturbance Rejection ............... 75 9.1 Robustness to Load (System) Noise and Measurement Noise and Internal Stability..................... 75 9.2 Fundamental Limitations and Bode’s Sensitivity Integral........................................... 76 9.3 Feedforward Design......................................................................... 78 A Signal Spaces and Representation of Signals ....................................................... 79 A.1 Normed Linear (Vector) Spaces and Metric Spaces............................................... 79 A.2 Hilbert Spaces.............................................................................. 83 A.3 Why are we interested in Hilbert Spaces?....................................................... 83 6 Contents A.4 Separability................................................................................ 85 A.4.1 Separability and Existence of Basis...................................................... 85 A.4.2 Fourier Series:........................................................................ 87 A.4.3 Approximation:....................................................................... 87 A.4.4 Other expansions:....................................................................