Textbook and Syllabus Textbook: Syllabus: Modern Control 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

Modern Control Grade Policy 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.

Modern Control Grade Policy 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. Modern Control Homework 4 Ronald Andre 009201700008 21 March 2021 No.1. Answer: . . . . . . . . Heading of Homework Papers (Required)

Modern Control Grade Policy 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. http://zitompul.wordpress.com 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.

Modern Control Chapter 1 Introduction

Classical Control and Modern Control 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 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.

Signal Classification Chapter 1 Introduction Signal Classification Continuous signal Discrete signal

Classification of Systems Chapter 1 Introduction 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.

Classification of Systems Chapter 1 Introduction Classification of Systems 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

Chapter 1 Introduction Linear Systems 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

Example: Linear or Nonlinear Chapter 1 Introduction Example: Linear or Nonlinear Check the linearity of the following system. Let , then Let , then Thus  The system is nonlinear

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

Properties of Linear Systems Chapter 1 Introduction Properties of Linear Systems 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:

Chapter 1 Introduction Time-Invariance 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.

Laplace Transform Approach Chapter 1 Introduction Laplace Transform Approach RLC Circuit Resistor Inductor Capacitor Input variables: Input voltage u(t) Output variables: Current i(t)

Laplace Transform Approach Chapter 1 Introduction Laplace Transform Approach Current due to input Current due to initial condition For zero initial conditions (v0 = 0, i0 = 0), where Transfer function

Chapter 1 Introduction State Space Approach In Laplace Transform, the input-output model is assumed to be time-invariant, linear, and causal. Besides, this method is not effective to model time-varying and non-linear systems. The input-output model regards a system as a black box, with certain inputs and certain outputs. What is happening inside the box is not clear. Inputs Outputs Black box 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. Inputs Outputs State Space States

Chapter 1 Introduction State Space Approach In state space approach, the knowledge of the states is enough to express anything inside the model box. With all states known, the future outputs can be determined based on the future inputs that will be given to a system.

State Space Equations as Linear System Chapter 1 Introduction 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 and then If then Then, it can also be implied that

Chapter 1 Introduction State Space Approach Let us now consider the same RLC circuit and try to use state space to model it. 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

Chapter 1 Introduction State Space Approach The two equations are called state equations, and can be rewritten in the form of: The output is described by an output equation:

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

Chapter 1 Introduction State Space Approach The state of a system at t0 is the information at t0 that, together with the input u for t0 ≤ t < ∞, uniquely determines the behavior of the system for t ≥ t0. 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. 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.

Mathematical Descriptions of Systems Modern Control Chapter 2 Mathematical Descriptions of Systems

Chapter 2 Mathematical 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

State Space Equations By defining: we can write State Equations Chapter 2 Mathematical Descriptions of Systems State Space Equations By defining: we can write State Equations A is called system matrix B is called input matrix

Chapter 2 Mathematical 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

State Space Equations By defining: we can write Output Equations Chapter 2 Mathematical Descriptions of Systems State Space Equations By defining: we can write Output Equations C is called output matrix D is called feedthrough matrix

Block Diagram of a State Space Chapter 2 Mathematical Descriptions of Systems Block Diagram of a State Space

Example: Mechanical System Chapter 2 Mathematical Descriptions of Systems Example: Mechanical System State variables: Input variables: Applied force u(t) Output variables: Displacement y(t) State equations:

Example: Mechanical System Chapter 2 Mathematical Descriptions of Systems Example: Mechanical System The state space equations can now be constructed as below:

Homework 1: Electrical System Chapter 2 Mathematical 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 vL(t)

Homework 1A: Electrical System Chapter 2 Mathematical 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 vL(t)