Robust Control, Spring, 2019

(Course Number: SCE.C402)


Prof. Masayuki Fujita

Department of Systems and Control Engineering, Tokyo Tech

Office: S5-303B

Lecture Time and Place

Tuesday 3-4 periods (10:45-12:15), S423 Lecture Room


Unit: 1-0-0

Teaching Assistants

Hayato Dan, Made Widhi Surya Atman (S5-204A)

Email :

Please send e-mail to above address if you have any questions.

Office Hour

Course Learning Outcomes

1. An ability to formulate, analyze and design fundamental control problems

2. An ability to use a numerical computing environment, such as MATLAB, to solve engineering problems


Reports on 4th(45%) and 6th(55%) classes.


The course will follow closely:

[SP05] S. Skogestad and I. Postlethwaite, "Multivariable Feedback Control: Analysis and Design, Second Edition," Wiley, 2005 (ISBN: 978-0-470-01167-6).
Available at the library.

Reference Books

Other recommended texts are:

[M19] "Robust Control Toolbox Documentation(R2019a)," MathWorks, 2019.
[ZD97] K. Zhou and J. C. Doyle, "Essentials of Robust Control," Prentice Hall, 1997 (ISBN: 0-13-525833-2).
[H17] M. Hirata, "Practical Robust Control(実践ロバスト制御)," Corona Publishing Co., LTD., 2017 (ISBN: 978-4-339-03311-3).

Tokyo Tech OCW

Check here

Computer Access and MATLAB

The students can install MATLAB R2018b (MATLAB, Robust Control Toolbox) in your computer. Please see the GSIC's installation guide for more details. The students can also use MATLAB R2017a at GSIC.

Collaboration Policy

Collaboration on homework assignment is encouraged. But it is forbidden to copy solutions from other students.

Lecture Schedule

Lecture Date Topics Reading Assignment Handouts Report
1st4/9 1. Multivariable Feedback Control
and Nominal Stability

  1.1 Multivariable Feedback Control
  1.2 Multivariable Frequency Response Analysis
  1.3 Internal Stability
  1.4 All Stabilizing Controllers
Sec. 3.5
Sec. 3.3, A.3, A.5
Sec. 4.1, 4.7
Sec. 4.8
2nd4/16 2. Nominal Performance
  2.1 Weighted Sensitivity
  2.2 Nominal Performance
  2.3 Sensitivity Minimization
  2.4 Remarks on Fundamentals Limitations
Sec. 2.8, 3.3, 4.10, 6.2, 6.3
Sec. 2.8, 3.2, 3.3
Sec. 3.2, 3.3, 9.3
Sec. 6.2

3rd4/23 3. Robustness and Uncertainty
  3.1 Why Robustness?
  3.2 Representing Uncertainty
  3.3 Uncertain Systems
  3.4 Systems with Structured Uncertainty
Sec. 4.1.1, 7.1, 9.2
Sec. 7.2, 7.3, 7.4
Sec. 8.1, 8.2, 8.3
Sec. 8.2
4th5/7 4. Robust Stability and Loop Shaping
  4.1 Robust Stability and Robust Stabilization
  4.2 Mixed Sensitivity and Loop Shaping
  4.3 1st Report
Sec. 7.5, 8.4, 8.5
Sec. 2.6, 2.8, 9.1

5th5/14 5. H∞ Control
  5.1 General Control Problem Formulation
  5.2 H∞ Control Problem and DGKF Solutions
  5.3 Structure of H∞ Controllers
Sec. 3.8
Sec. 9.3

6th5/21 6. Design Example 1
  6.1 Spinning Satellite: H∞ Control
  6.2 2nd Report
Sec. 3.7

7th5/28 7. Robust Performance
  7.1 Robust Performance
  7.2 Structured Singular Value μ
  7.3 μ-Analysis and Synthesis
Sec. 7.6, 8.3, 8.4, 8.10
Sec. 8.5, 8.6, 8.8, 8.11
Sec. 7.6, 8.9, 8.10, 8.11
6/4 8. Design Example 2
  8.1 HiMAT: H∞ Control
Supplemental Topics Handouts
B. Robust Performance
  B.1 μ-synthesis and DK-iteration
  B.2 Design Example: Spinning Satellite
Related Book/Paper
B.1 [SP05] Sec. 8.12
C. H∞ Loop Shaping Design
  C.1 Perturbations of Coprime Factors
  C.2 Robust Stabilization
  C.3 Loop Shaping Design Procedure
  C.4 Design Example
Related Book/Paper
D.1 [SP05] Sec. 4.1.5, D.2 [SP05] Sec. 9.4.1, D.3 [SP05] Sec. 9.4.2
Case Studies Handouts
Q. Flexible Beam: Mixed Sensitivity
R. Flexible Beam: Signal-based H∞ Control
S. Magnetic Suspension System: μ-synthesis
T. Magnetic Bearing: Robust Performance


Students are expected to have taken feedback control, modern control and system control theory.

The following texts may be useful for some students.

[AM09] K. J. Astrom and R. M. Murray, "Feedback Systems," Princeton University Press, 2009 (ISBN: 978-0-691-13576-2).
[SF99] T. Sugie and M. Fujita, "Introduction to Feedback Control(フィードバック制御入門)," Corona Publishing Co., LTD., 1999 (ISBN: 4-339-03303-0).
Prerequisites Handouts
A1. SISO Feedback Control
  A1.1 Internal Stability and Youla Parameterization
  A1.2 Sensitivity and Feedback Perforemance
  A1.3 Loop Shaping
Related Book/Paper
A1.1 [SP05] Sec. 3.2, 4.1.5, 4.7, 4.8, A1.2 [SP05] Sec. 2.2, 5.2,
A1.3 [SP05] Sec. 2.4, 2.6
A2. SISO Loop Shaping
  A2.1 Computer Controlled System
  A2.2 Modeling
  A2.3 Example
Related Book/Paper
A2.1 [SP05] Sec. 3.7, 1.4, 1.5, A2.2 [SP05] Sec. 2.6, 5.6, 5.7, 5.9

Additional Materials

This page augments the main text with discipline-specific materials and additional information.

[DP05] G.E. Dullerud and F. Paganini, "A Course in Robust Control Theory: A Convex Approach," Text in Applied Mathematics, Springer, 2005 (ISBN: 978-1-4757-3290-0).
[BPT13] G. Blekherman, P. A. Parrilo and R. R. Thomas, "Semidefinite Optimization and Convex Algebraic Geometry," MOS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics (SIAM), 2013 (ISBN: 978-1-61197-228-3).
Additional Topics Handouts
D. Linear Matrix Inequality (LMI)
  D.1 Convex Optimization
  D.2 Linear Matrix Inequality (LMI)
  D.3 Control Design and LMI Formulation
Related Book/Paper
[SP05] Sec. 12
E. LPV System and Gain Scheduling
  E.1 Linear Parameter Varying(LPV) System
  E.2 Quadratic Stabilization
  E.3 Gain Scheduing
  E.4 Design Example
Related Book/Paper
[DP05] Sec. 11
F. Integral Quadratic Constraints(IQCs)
  F.1 Passivity and Dissipative System
  F.2 IQC Formulation
  F.3 Robust Stability via IQCs
  F.4 Design Examples
Related Book/Paper
[DP05] Sec. 10.1
G. Sum of Squares(SOS)
  G.1 SOS Program: SOS/PSD and SDP
  G.2 Duality, valid ineqalities and Cone
  G.3 Feasibility/Optimization and Ideal
  G.4 Exploiting Structure
Related Book/Paper
  • [BPT13] Chapter 3
  • SOSTOOLS User's guide
    H. System Level Synthesis(SLS)
    Related Book/Paper
  • System Level Synthesis
  • SLS Toolbox
  • wiki  
    I. Learning for Control
    Related Website/Papers
  • Workshop in CDC 2018
  • Systems and Control Research in the Era of ML and AI
  • Systems and Control for the future

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