# Robust Control. 1st class. Spring, 2017 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 11th April, 2017, 10:45~12:15, S423 Lecture Room

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1 Robust Control Spring, 2017 Instructor: Prof. Masayuki Fujita (S5-303B) 1st class Tue., 11th April, 2017, 10:45~12:15, S423 Lecture Room

2 Reference: [H95] R.A. Hyde, Aerospace Control Design: A VSTOL Flight Application, Springer, Harrier Jump Jet 2

3 Robust Control for Flight Control Process Control Automotive Control Mechatronics Smart Grid 3

4 Motivating Example: Spinning Satellite s Attitude Control JAXA: ETS-VIII Spinning Satellite Yaw =10rad/s Inputs: Outputs: Torque Angular velocity Roll Pitch Multi-Input Multi-Output System (MIMO System) Single-Input Single-Output System (SISO System) 4

5 Multivariable Plants 古典制御の時代が最初に壁にぶつかったのが 多変数 の問題である [Tsien54] H. S. Tsien:Engineering Cybernetics, McGraw-Hill, 1954 [ 木村 89] 木村 : 制御技術と制御理論, システム / 制御 / 情報,33(6) 257/263, 1989 Spinning Satellite Transfer Function Matrix Interaction (Coupling) 1 State Space Representation Unified treatment for SISO/MIMO 5

6 Control of Multivariable Plants [SP05, pp ] 1. Diagonal Controller (Decentralized Control) Controller Interaction (Coupling) 0 - MATLAB Command P11 = tf([1-100],[ ]) ; K = pidtune( P11, PID ) ; 6

7 Control of Multivariable Plants [SP05, pp ] 2. Dynamic Decoupling Loop Shaping Design Target Loop (Desired Loop) Inverse-based Controller dB/dec 30 Stabilization Delay 48 [rad/s] -40dB/dec? 7

8 Control of Multivariable Plants [SP05, pp ] Inverse-based Controller Controller Uncertainty 0 - Uncertainty 8

9 Control of Multivariable Plants 3. Robust Controller Robust Controller Uncertainty Uncertainty 9

10 Robust Control Instructor: Prof. Masayuki Fujita (S5-303B) Schedule: Units: 11 th, 18 th, 25 th April, 2 nd, 9 th, 16 th, 23 rd, 30 th May 1 unit Teaching Assistants (TA): Riku Funada, Made Widhi Surya Atman (S5-303A) Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, [ZD97] K. Zhou and J. C. Doyle, Essentials of Robust Control, Prentice Hall, [M17] Robust Control Toolbox User s Guide R2017a, MathWorks, Grading: Reports on 2nd (15%), 4th (30%) and 6 th (55%) classes ( MATLAB: Robust Control Toolbox)

11 1. Multivariable Feedback Control and Nominal Stability 1.1 Multivariable Feedback Control [SP05, Sec. 3.5] 1.2 Multivariable Frequency Response Analysis [SP05, Sec. 3.3, A.3, A.5] 1.3 Internal Stability [SP05, Sec. 4.1, 4.7] 1.4 All Stabilizing Controllers [SP05, Sec. 4.8] Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, 2005.

12 Frequency Response for SISO Systems [Ex.] Bode Plot (Gain) 12

13 Frequency Response for MIMO Systems [Ex.] SISO MIMO? 13

14 Singular Value Decomposition [SP05, Ex. 3.3] (p. 74) [SP05, A.3] svd(g) A. J. Laub Minor Axis Major Axis :Unitary Matrices Singular Values : -th eigenvalue Maximum Singular Value Minimum Singular Value 14

15 -plot SISO: MIMO: [SP05, p. 79] Absolute value Singular value plot [Ex.] -plot of Extension of Bode gain plot to MIMO Systems MATLAB Command num = { [10 10], 1; [1 2], [5 5] }; den = { [ ], [1 1]; [ ], [1 5 6] }; G = tf( num, den ); figure sigma(g) 15

16 Motivating Example for Internal Stability in SISO Systems [SP05, Ex. 4.16] (p. 144) ー Closed Loop Transfer Function Stable? Another Closed Loop Transfer Function C.A. Desoer Unstable!! 3 5 C.A. Desoer and W.S. Chan, Journal of the Franklin Institute, 300 (5-6) , 1975 Why? Unstable Pole/Zero Cancellation 16

17 Gang of Four (SISO) In order to avoid pole/zero cancellation, consider input injection & output measurement for each dynamic block. ー [AM08] K. J. Astrom and R. Murray, Feedback Systems, Princeton University Press, 2008 Sensitivity Complementary Sensitivity Load Sensitivity Noise Sensitivity 17

18 Internal Stability of Multivariable Feedback Systems Nominal Stability [SP05, Fig. 4.3] (p. 145) ー : Transfer function matrices Well-posedness: (Gang of Four: well-defined and proper) : Vectors [SP05, Theorem 4.6] (p. 145) Nominal Stability(NS) Test Assume contain no unstable hidden modes. Then, the feedback system in the figure is internally stable if and only if all four closed-loop transfer matrices are stable. 18

19 Internal Stability of Multivariable Feedback Systems Nominal Stability [SP05, Fig. 4.3] (p. 145) ー State-space representation: [SP05, p. 124] [ZD97, Theorem 5.5](p. 70) Nominal Stability(NS) Test The system is internally stable iff is stable [ZD97] K. Zhou and J. C. Doyle, Essentials of Robust control, Prentice Hall,

20 Youla-parameterization (Q-parameterization) Stable Plant Plant : Proper Stable Transfer Function Matrices [SP05, p. 148] Gang of Four Model All Stabilizing Controllers Surprising Fact: Necessary and Sufficient Internally stable Internally stable 20

21 Youla-Kucera-parameterization Unstable Plant Left Coprime Factorization (can be also on the right) [SP05, p. 149] [SP05, p. 122] M. Vidyasagar, The MIT Press,1985 Coprime: No common unstable zeros iff (Bezout Identity) : Stable coprime transfer funcion matrices All Stabilizing Controllers : Stable transfer function matrix satisfying 21

22 Youla-Kucera-parameterization (Unstable Plants) [SP05, Ex. 4.1] [SP05, p. 149] (*) :(*) Bezout Identity A Stabilizing Controller! Stable Plant Case All Stabilizing Controllers! 22

23 State-Space Computation of All Stabilizing Controllers State Space Representation [SP05, p. 124] 6 All Stabilizing Controllers Let matrices, be such that, are stable Matrix Computation System Structure on Controllers If, then is State Feedback + Observer 23

24 Completion of Linear Feedback System Theory A stabilizing controller State feedback/observer All stabilizing controllers (Youla) Parametrization Transfer Function Pole/Zero Structure Controllability, Observability State Space Form (Data Structure) State - 24

25 1. Multivariable Feedback Control and Nominal Stability 1.1 Multivariable Feedback Control [SP05, Sec. 3.5] 1.2 Multivariable Frequency Response Analysis [SP05, Sec. 3.3, A.3, A.5] 1.3 Internal Stability [SP05, Sec. 4.1, 4.7] 1.4 All Stabilizing Controllers [SP05, Sec. 4.8] Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, 2005.

26 2. Nominal Performance 2.1 Weighted Sensitivity [SP05, Sec. 2.8, 3.3, 4.10, 6.2, 6.3] 2.2 Nominal Performance [SP05, Sec. 2.8, 3.2, 3.3] 2.3 Sensitivity Minimization [SP05, Sec. 3.2, 3.3, 9.3] 2.4 Remarks on Fundamental Limitations 2.5 1st Report [SP05, Sec. 6.2] Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, 2005.

27 Relative Gain Array [SP05, Sec. 3.4] [SP05, Ex. 3.9] (pp. 85) Transfer Function Matrix 1 Relative Gain Array element wise multiplication Pairing rule 1 Prefer paring on RGA elements close to 1 Use to control and use to control Pairing rule 2 Avoid pairing on negative RGA elements Pairing rule 2 is satisfied for this choice Rule 1 Rule 2 27

28 Control of Multivariable Plants Steady-State Decoupling Controller [SP05, pp ]

29 Poles [SP05, 4.4] [SP05, Theorem 4.4] (p. 135) The pole polynomial corresponding to a minimal realization of a system with transfer function is the least common denominator of all non-identically zero minors of all orders of. [SP05, Ex. 4.10] (pp. 136, 139) 3 The minors of order 1 The minors of order 2 The least common denominator of all the minors Poles 29

30 Zeros [SP05, Sec. 4.5] [SP05, Theorem 4.5] (p. 139) The zero polynomial, corresponding to a minimal realization of the system, is the greatest common divisor of all the numerators of all order- minors of, where is the normal rank of, provided that these minors have been adjusted in such a way as to have the pole polynomial as their denominator. [SP05, Ex. 4.10] (pp. 136, 139) (Cont.) 4 Normal rank: 2 The minors of order 2 The greatest common divisor of numerator Zeros 30

31 Pole/Zero Cancellation [SP05, Sec. 4.5] 5 Poles Poles Poles of and : Poles Poles of, is cancelled 31

32 Two degrees of freedom Controller [SP05, p. 147] 6 Parameterize : Stable matrix 32

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