Modern Control Lecture 1 Dr Ing Erwin Sitompul

Modern Control Lecture 1 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com 2 0 1 3 President University Erwin Sitompul Modern Control 1/1

Textbook and Syllabus Textbook: “Linear System Theory and Design”, Third Edition, Chi-Tsong Chen, Oxford University Press, 1999. Syllabus: Chapter 1: Introduction Chapter 2: Mathematical Descriptions of Systems Chapter 3: Linear Algebra Chapter 4: State-Space Solutions and Realizations Chapter 5: Stability Chapter 6: Controllability and Observability Chapter 7: Minimal Realizations and Coprime Fractions Chapter 8: State Feedback and State Estimators Additional: Optimal Control President University Erwin Sitompul Modern Control 1/2

Multivariable Calculus Grade Policy Final Grade = 10% Homework + 20% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra Points n Homeworks will be given in fairly regular basis. The average of homework grades contributes 10% of final grade. n Homeworks are to be submitted on A 4 papers, otherwise they will not be graded. n Homeworks must be submitted on time. If you submit late, < 10 min. No penalty 10 – 60 min. – 20 points > 60 min. – 40 points n There will be 3 quizzes. Only the best 2 will be counted. The average of quiz grades contributes 20% of the final grade. n Midterm and final exam schedule will be announced in time. n Make up of quizzes and exams will be held one week after the schedule of the respective quizzes and exams. President University Erwin Sitompul Modern Control 1/3

Multivariable Calculus Grade Policy n The score of a make up quiz or exam, upon discretion, can be multiplied by 0. 9 (i. e. , the maximum score for a make up is then 90). n Extra points will be given if you solve a problem in front of the class. You will earn 1, 2, or 3 points. n You are responsible to read and understand the lecture slides. I am responsible to answer your questions. Modern Control Homework 4 Ronald Andre 009201700008 21 March 2021 No. 1. Answer: . . . . n Heading of Homework Papers (Required) President University Erwin Sitompul Modern Control 1/4

Chapter 1 Introduction Classical Control and Modern Control Classical Control • SISO (Single Input Single Output) • Low order ODEs • Time-invariant • Fixed parameters • Linear • Time-response approach • Continuous, analog • Before 80 s Modern Control • MIMO (Multiple Input Multiple Output) • High order ODEs, PDEs • Time-invariant and time variant • Changing parameters • Linear and non-linear • Time- and frequency response approach • Tends to be discrete, digital • 80 s and after n The difference between classical control and modern control originates from the different modeling approach used by each control. n The modeling approach used by modern control enables it to have new features not available for classical control. President University Erwin Sitompul Modern Control 1/5

Chapter 1 Modeling Approach Laplace Transform Approach RLC Circuit Input variables: • Input voltage u(t) Output variables: • Current i(t) President University Resistor Erwin Sitompul Inductor Capacitor Modern Control 1/6

Chapter 1 Modeling Approach Laplace Transform Approach Current due to input Current due to initial condition n For zero initial conditions (v 0 = 0, i 0 = 0), where Transfer function President University Erwin Sitompul Modern Control 1/7

Chapter 1 Modeling Approach State Space Approach n Laplace Transform method is not effective to model time-varying and non-linear systems. n The state space approach to be studied in this course will be able to handle more general systems. n The state space approach characterizes the properties of a system without solving for the exact output. n Let us now consider the same RLC circuit and try to use state space to model it. President University Erwin Sitompul Modern Control 1/8

Chapter 1 Modeling Approach State Space Approach RLC Circuit State variables: • Voltage across C • Current through L • We now have two first-order ODEs • Their variables are the state variables and the input President University Erwin Sitompul Modern Control 1/9

Chapter 1 Modeling Approach State Space Approach n The two equations are called state equations, and can be rewritten in the form of: n The output is described by an output equation: President University Erwin Sitompul Modern Control 1/10

Chapter 1 Modeling Approach State Space Approach n The state equations and output equation, combined together, form the state space description of the circuit. n In a more compact form, the state space can be written as: President University Erwin Sitompul Modern Control 1/11

Chapter 1 Modeling Approach State Space Approach n The state of a system at t 0 is the information at t 0 that, together with the input u for t 0 ≤ t < ∞, uniquely determines the behavior of the system for t ≥ t 0. n The number of state variables = the number of initial conditions needed to solve the problem. n As we will learn in the future, there are infinite numbers of state space that can represent a system. n The main features of state space approach are: n It describes the behaviors inside the system. n Stability and performance can be analyzed without solving for any differential equations. n Applicable to more general systems such as non-linear systems, time-varying system. n Modern control theory are developed using state space approach. President University Erwin Sitompul Modern Control 1/12

Chapter 2 Mathematical Descriptions of Systems Classification of Systems n Systems are classified based on: n The number of inputs and outputs: single-input single-output (SISO), multi-input multi-output (MIMO), MISO, SIMO. n Existence of memory: if the current output depends on the current input only, then the system is said to be memoryless, otherwise it has memory purely resistive circuit vs. RLCcircuit. n Causality: a system is called causal or non-anticipatory if the output depends only on the present and past inputs and independent of the future unfed inputs. n Dimensionality: the dimension of system can be finite (lumped) or infinite (distributed). n Linearity: superposition of inputs yields the superposition of outputs. n Time-Invariance: the characteristics of a system with the change of time. President University Erwin Sitompul Modern Control 1/13

Chapter 2 Mathematical Descriptions of Systems Linear System n A system y(t) = f(x(t), u(t)) is said to be linear if it follows the following conditions: n If then n If and then n Then, it can also be implied that President University Erwin Sitompul Modern Control 1/14

Chapter 2 Mathematical Descriptions of Systems Linear Time-Invariant (LTI) System n A system is said to be linear time-invariant if it is linear and its parameters do not change over time. President University Erwin Sitompul Modern Control 1/15

Chapter 2 Mathematical Descriptions of Systems State Space Equations n The state equations of a system can generally be written as: are the state variables are the system inputs • State equations are built of n linearly-coupled first -order ordinary differential equations President University Erwin Sitompul Modern Control 1/16

Chapter 2 Mathematical Descriptions of Systems State Space Equations n By defining: we can write President University State Equations Erwin Sitompul Modern Control 1/17

Chapter 2 Mathematical Descriptions of Systems State Space Equations n The outputs of the state space are the linear combinations of the state variables and the inputs: are the system outputs President University Erwin Sitompul Modern Control 1/18

Chapter 2 Mathematical Descriptions of Systems State Space Equations n By defining: we can write President University Output Equations Erwin Sitompul Modern Control 1/19

Chapter 2 Mathematical Descriptions of Systems Example: Mechanical System State variables: Input variables: • Applied force u(t) Output variables: • Displacement y(t) President University State equations: Erwin Sitompul Modern Control 1/20

Chapter 2 Mathematical Descriptions of Systems Example: Mechanical System n The state space equations can now be constructed as below: President University Erwin Sitompul Modern Control 1/21

Chapter 2 Mathematical Descriptions of Systems Homework 1: Electrical System n Derive the state space representation of the following electric circuit: Input variables: • Input voltage u(t) Output variables: • Inductor voltage v. L(t) President University Erwin Sitompul Modern Control 1/22

Chapter 2 Mathematical Descriptions of Systems Homework 1 A: Electrical System n Derive the state space representation of the following electric circuit: Input variables: • Input voltage u(t) Output variables: • Inductor voltage v. L(t) President University Erwin Sitompul Modern Control 1/23