Robust Control. 1st class. Spring, 2017 Instructor: Prof. Masayuki Fujita (S5303B) Tue., 11th April, 2017, 10:45~12:15, S423 Lecture Room


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1 Robust Control Spring, 2017 Instructor: Prof. Masayuki Fujita (S5303B) 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: ETSVIII Spinning Satellite Yaw =10rad/s Inputs: Outputs: Torque Angular velocity Roll Pitch MultiInput MultiOutput System (MIMO System) SingleInput SingleOutput 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, McGrawHill, 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([1100],[ ]) ; K = pidtune( P11, PID ) ; 6
7 Control of Multivariable Plants [SP05, pp ] 2. Dynamic Decoupling Loop Shaping Design Target Loop (Desired Loop) Inversebased Controller dB/dec 30 Stabilization Delay 48 [rad/s] 40dB/dec? 7
8 Control of Multivariable Plants [SP05, pp ] Inversebased 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 (S5303B) 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 (S5303A) 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 (56) , 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 Wellposedness: (Gang of Four: welldefined 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 closedloop transfer matrices are stable. 18
19 Internal Stability of Multivariable Feedback Systems Nominal Stability [SP05, Fig. 4.3] (p. 145) ー Statespace 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 Youlaparameterization (Qparameterization) 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 YoulaKuceraparameterization 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 YoulaKuceraparameterization (Unstable Plants) [SP05, Ex. 4.1] [SP05, p. 149] (*) :(*) Bezout Identity A Stabilizing Controller! Stable Plant Case All Stabilizing Controllers! 22
23 StateSpace 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 SteadyState 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 nonidentically 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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