President UniversityErwin SitompulModern Control 1/1 Dr.-Ing. Erwin Sitompul President University Lecture 1 Modern Control

President UniversityErwin SitompulModern Control 1/2 Textbook : “Linear System Theory and Design”, Third Edition, Chi-Tsong Chen, Oxford University Press, 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 Textbook and Syllabus Modern Control

President UniversityErwin SitompulModern Control 1/3 Grade Policy Modern Control Final Grade = 10% Homework + 20% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra Points Homeworks will be given in fairly regular basis. The average of homework grades contributes 10% of final grade. Homeworks are to be submitted on A4 papers, otherwise they will not be graded. Homeworks must be submitted on time, one day before the schedule of the lecture. Late submission will be penalized by point deduction of –10·n, where n is the total number of lateness made. There will be 3 quizzes. Only the best 2 will be counted. The average of quiz grades contributes 20% of the final grade.

President UniversityErwin SitompulModern Control 1/4 Grade Policy Modern Control Midterm and final exams follow the schedule released by AAB (Academic Administration Bureau). Make up for quizzes must be requested within one week after the date of the respective quizzes. Make up for mid exam and final exam must be requested directly to AAB. Heading of Homework Papers (Required) Modern Control Homework 4 Ronald Andre March 2021 No.1. Answer:

President UniversityErwin SitompulModern Control 1/5 Grade Policy Modern Control To maintain the integrity, the maximum score of a make up quiz or exam can be set to 90. Extra points will be given if you solve a problem in front of the class. You will earn 1 or 2 points. Lecture slides can be copied during class session. It also will be available on internet around 1 days after class. Please check the course homepage regularly. The use of internet for any purpose during class sessions is strictly forbidden. You are expected to write a note along the lectures to record your own conclusions or materials which are not covered by the lecture slides.

President UniversityErwin SitompulModern Control 1/6 Chapter 1 Introduction Modern Control

President UniversityErwin SitompulModern Control 1/7 Classical Control and Modern Control Chapter 1Introduction Classical Control SISO (Single Input Single Output) Low order ODEs Time-invariant Fixed parameters Linear Time-response approach Continuous, analog Before 80s 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 80s and after The difference between classical control and modern control originates from the different modeling approach used by each control. The modeling approach used by modern control enables it to have new features not available for classical control.

President UniversityErwin SitompulModern Control 1/8 Signal Classification Chapter 1Introduction Continuous signal Discrete signal

President UniversityErwin SitompulModern Control 1/9 Classification of Systems Systems are classified based on: The number of inputs and outputs: single-input single-output (SISO), multi-input multi-output (MIMO), MISO, SIMO. 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. RLC- circuit. 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. Dimensionality: the dimension of system can be finite (lumped) or infinite (distributed). Linearity: superposition of inputs yields the superposition of outputs. Time-Invariance: the characteristics of a system with the change of time. Chapter 1Introduction

President UniversityErwin SitompulModern Control 1/10 Classification of Systems Chapter 1Introduction Finite-dimensional system (lumped-parameters, described by differential equations Linear and nonlinear systems Continuous- and discrete-time systems Time-invariant and time-varying systems Infinite-dimensional system (distributed-parameters, described by partial differential equations) Heat conduction Power transmission line Antenna Fiber optics

President UniversityErwin SitompulModern Control 1/11 Linear Systems Chapter 1Introduction A system is said to be linear in terms of the system input u(t) and the system output y(t) if it satisfies the following two properties of superposition and homogeneity. Superposition Homogeneity

President UniversityErwin SitompulModern Control 1/12 Example: Linear or Nonlinear Check the linearity of the following system. Let, then Thus  The system is nonlinear Chapter 1Introduction

President UniversityErwin SitompulModern Control 1/13 Example: Linear or Nonlinear Check the linearity of the following system (governed by ODE). Let Then Chapter 1Introduction  The system is linear

President UniversityErwin SitompulModern Control 1/14 Properties of Linear Systems Chapter 1Introduction For linear systems, if input is zero then output is zero. A linear system is causal if and only if it satisfies the condition of initial rest:

President UniversityErwin SitompulModern Control 1/15 Time-Invariance Chapter 1Introduction A system is said to be time-invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal. Time-invariant system A system is said to be time-invariant if its parameters do not change over time.

President UniversityErwin SitompulModern Control 1/16 Chapter 1 Laplace Transform Approach RLC Circuit Input variables: Input voltage u(t) Output variables: Current i(t) ResistorInductorCapacitor Introduction

President UniversityErwin SitompulModern Control 1/17 Chapter 1 Current due to input Current due to initial condition For zero initial conditions (v 0 = 0, i 0 = 0), where Transfer function Laplace Transform Approach Introduction

President UniversityErwin SitompulModern Control 1/18 State Space Approach Chapter 1Introduction Laplace Transform method is not effective to model time-varying and non-linear systems. The state space approach to be studied in this course will be able to handle more general systems. The state space approach characterizes the properties of a system without solving for the exact output. Let us now consider the same RLC circuit and try to use state space to model it.

President UniversityErwin SitompulModern Control 1/19 State Space Equations as Linear System A system y(t) = f(x(t),u(t)) is said to be linear if it follows the following conditions: If then If and then Then, it can also be implied that Chapter 1Introduction

President UniversityErwin SitompulModern Control 1/20 Chapter 1Introduction RLC Circuit State Space Approach 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 UniversityErwin SitompulModern Control 1/21 State Space Approach The two equations are called state equations, and can be rewritten in the form of: Chapter 1Introduction The output is described by an output equation:

President UniversityErwin SitompulModern Control 1/22 In a more compact form, the state space can be written as: State Space Approach Chapter 1Introduction The state equations and output equation, combined together, form the state space description of the circuit.

President UniversityErwin SitompulModern Control 1/23 The main features of state space approach are: It describes the behaviors inside the system. Stability and performance can be analyzed without solving for any differential equations. Applicable to more general systems such as non-linear systems, time-varying system. Modern control theory are developed using state space approach. State Space Approach Chapter 1Introduction 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. The number of state variables = the number of initial conditions needed to solve the problem. As we will learn in the future, there are infinite numbers of state space that can represent a system.

President UniversityErwin SitompulModern Control 1/24 Chapter 2 Mathematical Descriptions of Systems Modern Control

President UniversityErwin SitompulModern Control 1/25 Chapter 2Mathematical Descriptions of Systems State Space Equations 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 UniversityErwin SitompulModern Control 1/26 Chapter 2Mathematical Descriptions of Systems State Space Equations By defining: we can write State Equations

President UniversityErwin SitompulModern Control 1/27 Chapter 2Mathematical Descriptions of Systems State Space Equations The outputs of the state space are the linear combinations of the state variables and the inputs: are the system outputs

President UniversityErwin SitompulModern Control 1/28 By defining: we can write Output Equations Chapter 2Mathematical Descriptions of Systems State Space Equations

President UniversityErwin SitompulModern Control 1/29 Chapter 2Mathematical Descriptions of Systems Example: Mechanical System Input variables: Applied force u(t) Output variables: Displacement y(t) State variables: State equations:

President UniversityErwin SitompulModern Control 1/30 Chapter 2Mathematical Descriptions of Systems Example: Mechanical System The state space equations can now be constructed as below:

President UniversityErwin SitompulModern Control 1/31 Chapter 2Mathematical Descriptions of Systems Homework 1: Electrical System Derive the state space representation of the following electric circuit: Input variables: Input voltage u(t) Output variables: Inductor voltage v L (t)

President UniversityErwin SitompulModern Control 1/32 Chapter 2Mathematical Descriptions of Systems Homework 1A: Electrical System Derive the state space representation of the following electric circuit: Input variables: Input voltage u(t) Output variables: Inductor voltage v L (t)