2020 Fundamentals of Dynamical Systems

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Academic unit or major
Undergraduate major in Systems and Control Engineering
Imura Jun-Ichi 
Course component(s)
Mode of instruction
Day/Period(Room No.)
Tue1-2(S421)  Fri1-2(S421)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

Control Engineering is decomposed into three topics, i.e., mathematical modeling, system analysis, and control system design. This course treats the first two topics among these topics. In particular, the mathematical modeling in the first half of this course includes topics on state equations and transfer functions of dynamical system, which are mathematical models suited for the control design, and further Euler-Lagrange motion equations for deriving the state-equations. In the second half, as a part of the system analysis, students learn how to analyze system behavior such as transient/steady-state responses and the stability of dynamical systems from the perspectives of time responses.
 The aim of the course is that students will have the basic knowledge on control engineering, including state equations/transfer functions, system response and stability, to understand control system design theory based on state equations and transfer functions of the systems.

Student learning outcomes

By completing this course, students will be able to:
1) Understand the outline of control engineering.
2) Derive a state equation/transfer function based on mathematical models such as equations of motion for a given controlled plant of mechanical systems and electric circuits.
3) Analyze time responses of dynamical systems such as transient response and stability.


Dynamical system, static system, input/output/state, controller, sensor, feedback control, feed-forward control, state equation, equation of motion, Euler-Lagrange equation, linearized system, equilibrium point, transfer function, block diagram, input-output response, transient response, steady state response, initial value response, step response, impulse response, convolution integral, pole, zero, system degree, relative degree, first order system, second order system, gain, time constant, damping coefficients, natural angular frequency, principal pole, stability, internal stability, input-output stability, characteristic polynomial, characteristic equation, Hurwitz stability criterion, Routh stability criterion

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Basically, writing on the blackboard and explaining are taken turns. It is very important to make your own notes, which can provide systematic knowledge on modeling and analysis of dynamical systems at the end of this course. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purposes.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Dynamical system and control Explain control engineering, system representation, dynamical systems, and pros/cons of feedback control.
Class 2 State equation representation Explain the definition and significance of state equation.
Class 3 State equation representation of mechanical systems: Euler-Lagrange equation of motions Understand a basic modeling method via Euler-Lagrange motion equation for deriving equations of motion of mechanical systems.
Class 4 Exercise on state equation representation of simple mechanical systems with translational/rotational motions Derive equations of motion and state equations of 1-DOF rotational motions/ 2-DOF translational motions of mechanical systems.
Class 5 State equation representation of 2-DOF systems including rotational motions Derive equations of motion and state equations composed of 2-DOF translational/rotational motions of mechanical systems.
Class 6 State equation representation of electric systems and fluid systems Derive state equations of electric systems and fluid systems.
Class 7 State equation representation of complex systems Derive equations of motion and state equations of complex systems.
Class 8 Linearization Understand a linearization method of nonlinear systems.
Class 9 Transfer function and block diagram Explain the notion of transfer functions and its relation to the state equation, and derive a transfer function from a block diagram.
Class 10 System response: Impulse response and step response Understand the definition of impulse and step responses, and explain the notion of time constant and gain of the 1st order systems.
Class 11 System response of 2nd order systems Describe the relation between system response and three parameters; damping coefficients, natural angular frequency, gain.
Class 12 System response: pole and zero Describe the relation between system response and poles/zeros of a transfer function in high-order systems.
Class 13 Stability analysis Understand a method to determine the stability of transfer functions and apply it to some examples.
Class 14 Learning look back via Practice Look back on modeling and system analysis of dynamical systems in total.


Toshiharu Sugie and Masayuki Fujita. An Introduction to Feedback Control, CORONA PUBLISHING CO., LTD. ISBN 978-4339033038 (in Japanese)

Reference books, course materials, etc.

Tsuneo Yoshikawa, Jun-ichi Imura, Modern Control Theory, CORONA PUBLISHING CO., LTD. ISBN 978-4339032123 (in Japanese)
Hiroshi Kogou, Tsutomu MIta, An Introduction to System Control Theory, JIKYOU PUBLISHING CO., LTD. ISBN 978-4407022056 (in Japanese)

Assessment criteria and methods

1) Students will be assessed on their basic understanding of derivation of motion equations, the representation of state equations and transfer functions, system response.
2) Students’ course scores are based on small exercises (48%) and final exercise (52%).

Related courses

  • SCE.C202 : Feedback Control
  • SCE.C301 : Linear System Theory
  • SCE.A201 : Mathematics for Systems and Control A

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed SCE.A201 Mathematics for Systems and Control A or have equivalent knowledge.

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