Third Edition Chi Tsong Chen Oxford University Press 1999 Syllabus Chapter 1 Introduction Chapter 2 Mathematical Descriptions of Systems Chapter 3 Linear Algebra ID: 798797
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Slide2Textbook:“Linear System Theory and Design”, Third Edition, Chi-Tsong Chen, Oxford University Press, 1999.Syllabus: Chapter 1: IntroductionChapter 2: Mathematical Descriptions of SystemsChapter 3: Linear AlgebraChapter 4: State-Space Solutions and RealizationsChapter 5: StabilityChapter 6: Controllability and ObservabilityChapter 7: Minimal Realizations and Coprime FractionsChapter 8: State Feedback and State EstimatorsAdditional: Optimal Control
Textbook and Syllabus
Slide3Grade PolicyMultivariable CalculusFinal Grade = 10% Homework + 20% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra PointsHomeworks 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. If you submit late, < 10 min. No penalty 10 – 60 min. –20 points > 60 min. –40 pointsThere will be 3 quizzes. Only the best 2 will be counted. The average of quiz grades contributes 20% of the final grade.Midterm and final exam schedule will be announced in time.Make up of quizzes and exams will be held one week after the schedule of the respective quizzes and exams.
Slide4Grade PolicyMultivariable CalculusThe 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).Extra points will be given if you solve a problem in front of the class. You will earn 1, 2, or 3 points.You are responsible to read and understand the lecture slides. I am responsible to answer your questions.Heading of Homework Papers (Required)Modern ControlHomework 4Ronald Andre00920170000821 March 2021
No.1. Answer: . . . . . . . .
Slide5Classical Control and Modern ControlChapter 1IntroductionClassical Control SISO (Single Input Single Output)Low order ODEsTime-invariantFixed parametersLinearTime-response approachContinuous, analogBefore 80sModern Control MIMO(Multiple Input Multiple Output)High order ODEs, PDEsTime-invariant and time variantChanging parametersLinear and non-linearTime- 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.
Slide6Chapter 1Laplace Transform ApproachRLC CircuitInput variables: Input voltage u(t)Output variables: Current i(t)ResistorInductorCapacitor
Modeling Approach
Slide7Chapter 1Current dueto input
Current due to
initial condition
For zero initial conditions (
v
0
= 0,
i
0
= 0),
where
Transfer function
Laplace Transform Approach
Modeling Approach
Slide8State Space ApproachChapter 1Modeling ApproachLaplace 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.
Slide9Chapter 1Modeling ApproachRLC CircuitState Space ApproachState variables: Voltage across CCurrent through L
We now have two first-order ODEs
Their variables are the state variables and the input
Slide10State Space ApproachThe two equations are called state equations, and can be rewritten in the form of:Chapter 1Modeling ApproachThe output is described by an output equation:
Slide11In a more compact form, the state space can be written as:State Space ApproachChapter 1Modeling ApproachThe state equations and output equation, combined together, form the state space description of the circuit.
Slide12The 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 ApproachChapter 1Modeling ApproachThe 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.
Slide13Chapter 2Mathematical Descriptions of SystemsClassification of SystemsSystems 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.
Slide14Chapter 2Mathematical Descriptions of SystemsLinear SystemA 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
Slide15Linear Time-Invariant (LTI) SystemA system is said to be linear time-invariant if it is linear and its parameters do not change over time.Chapter 2Mathematical Descriptions of Systems
Slide16Chapter 2Mathematical Descriptions of SystemsState Space EquationsThe state equations of a system can generally be written as:are the state variablesare the system inputsState equations are built of n linearly-coupled first-order ordinary differential equations
Slide17Chapter 2Mathematical Descriptions of SystemsState Space EquationsBy defining:we can writeState Equations
Slide18Chapter 2Mathematical Descriptions of SystemsState Space EquationsThe outputs of the state space are the linear combinations of the state variables and the inputs:are the system outputs
Slide19By defining:we can writeOutput EquationsChapter 2Mathematical Descriptions of SystemsState Space Equations
Slide20Chapter 2Mathematical Descriptions of SystemsExample: Mechanical SystemInput variables: Applied force u(t)Output variables: Displacement y(t)State variables:
State
equations
:
Slide21Chapter 2Mathematical Descriptions of SystemsExample: Mechanical SystemThe state space equations can now be constructed as below:
Slide22Chapter 2Mathematical Descriptions of SystemsHomework 1: Electrical SystemDerive the state space representation of the following electric circuit:Input variables: Input voltage u(t)Output variables: Inductor voltage vL(t)
Slide23Chapter 2Mathematical Descriptions of SystemsHomework 1A: Electrical SystemDerive the state space representation of the following electric circuit:Input variables: Input voltage u(t)Output variables: Inductor voltage vL(t)