Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room


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1 Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5303B) 2nd class Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room
2 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.
3 Sensitivity as Feedback Performance in SISO Systems Disturbance Attenuation Openloop Closedloop : Sensitivity small: Good Feedback Performance 3
4 Sensitivity as Feedback Performance in SISO Systems Insensitivity to Plant Variations [SP05, p. 23] small : Good Feedback Performance (Absolute Value) MIMO? 4
5 Norm as System Gain System Gain [SP05] (p. 158) 1 2 [Ex.] : Proper stable system MATLAB Command hinfg = normhinf(g)  plot 5
6 Difference between the and norms [SP05, pp. 75, 159] Minimizing norm Minimizing norm (LQG) Push down peak of maximum singular value Worst direction, worst frequency Push down whole thing (all singular values over all frequencies) Average direction, average frequency Multiplicative property Multiplicative property? 6
7 Optimization in Feedback Control Feedback Performance = Sensitivity Sensitivity optimization with norm (System Gain) ( parameterization) Sensitivity from Reference to Error + 7
8 George Zames ( ) G.Zames, IEEE TAC, 26, 1981 Frustration with LQG control H control ( H 2 control) Formulation of the optimization problem not on time domain but on frequency domain 1939 The World War II occurred. Escaping to Europe through Lithuania Witnessed by Soviet s tank Through Russia, Siberia and Japan sea, 1941 Arrival in Kobe Sugihara Sempo Chiune, consular officer of Japan, helped him a lot. Leaving for Canada 8
9 Bode Sensitivity Integrals (Waterbed Effects) for Stable Plant [SP05, p. 167, p. 223] There exists a frequency range over which the magnitude of the sensitivity function exceeds 1 if it is to be kept below 1 at the other frequency range. [db] Dirt Frequency [rad/s] [SP05, Ex., p. 170] RHP(RightHalf Plane) Zero (unstable) 9
10 Weighted Sensitivity [SP05, p. 60] Small? 10 [db] SISO Case Frequency [rad/s] Waterbed Effects Small! Intractable Tractable! : Performance weight transfer function matrix [SP05, pp. 62, 80] How to specify? 10
11 Performance Weight [SP05, pp. 62, 80] Firstorder Performance Weight Feedback Effect : the frequency at which the asymptote of crosses 1, and the bandwidth requirement approximately : at high frequencies ( : Rule of thumb) : at low frequencies 11
12 Stabilization and Performance Unstable Plant [SP05 Sec 5.9] Real RHP Poles: Imaginary Poles: Complex RHP Poles: Stable Plant Firstorder System Secondorder System Rise time Rise time Rise time [QZ07] L. Qiu and K. Zhou (2007) Introduction to Feedback Control, Prentice Hall. 12
13 Nominal Performance (NP) [SP05, p. 81] 3 + Nominal Performance (NP) Test Given a controller, 13
14 Nominal Performance Test in SISO Systems [SP05, p. 60] (NP) [Ex.] 1) (NP) 2) (NP) : fast : small 3) (NP) : large 14
15 Nominal Performance Test in SISO Systems (NP) [SP05, p. 60] [Ex.] Norm Condition 1) (NP) : fast : small 2) (NP) 3) (NP) : large 15
16 [Ex.] Spinning Satellite: Performance Weight Performance Weight Specifications MATLAB Command Ms = 2; A = 1e2; wb = 11.5; wp = tf([1/ms wb], [1 wb*a]); WP = eye(2)*wp; figure sigma(wp) hold on; grid on; Poles on the imaginary axis 10 Gain crossover frequency 11.5 rad/s Phase stabilization [SP05, p. 194] System bandwidth of Actuator/Sensor/Controller 10 rad/s 11.5rad/s the steady state error 16
17 [Ex.] Plant Spinning Satellite: Nominal Performance Controller: Inversebased Controller 20dB/dec [rad/s] 40dB/dec Target Loop Transfer Function (Output) Sensitivity Function 21rad/s NP MATLAB Command KI = inv(pnom)*tf([1],[1 30 0])*diag([ ]); FI = loopsens(pnom,ki); sigma(fi.so) ; hinfso = normhinf(wp*fi.so) 11.5rad/s 17
18 Sensitivity Minimization Optimal Sensitivity Problem Find a stabilizing controller make smaller minimize Intractable which Sensitivity Minimization Problem Given, find all stabilizing controllers such that iteration + Linear Fractional Transformation (LFT) Control no db Parameterization 18
19 Sensitivity Minimization Optimal Sensitivity Problem Find a stabilizing controller make smaller tractable which Sensitivity Minimization Problem Given, find all stabilizing controllers such that iteration Linear Fractional Transformation (LFT) Control no 1 Parameterization 19
20 Sensitivity for MIMO Systems [SP05, p. 70] Sensitivity to Output Disturbance Output Sensitivity Function: Sensitivity to Input Disturbance Input Sensitivity Function: For SISO Systems but for MIMO Systems Good disturbance rejection at output does not always mean good rejection at input 20
21 Standard Feedback Configuration with Weights [SP05, p. 363] Sensitivity Minimization Problem 21
22 Remarks on Fundamental Limitations Gunter Stein Respect the unstable Bode lecture, CDC, 1989 Control Systems Magazine, 23(4):1225, Im unstable zero unstable pole Re Unstable Zero Time Delay Wrong Sensor Placement 22
23 SISO Loop Shaping [SP05, pp. 41, 42, 343] Performance Unstable Pole: Unstable Zero: Time Delay: 4 5 Robust Stability (+ Rolloff) Loop Shaping gives us graphical interpretation Bode Plot System Gain 23
24 RHP Poles/Zeros, Time Delays and Sensitivity in SISO Systems For systems with a RHP pole and RHP zero (or a time delay ), any stabilizing controller gives sensitivity functions with the property 6 7 The zero and the pole must be sufficiently far apart The product of RHP pole and time delay must be sufficiently small S( jω) T ( jω) 25
25 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 Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, [SP05, Sec. 6.2]
26 3. Robustness and Uncertainty 3.1 Why Robustness? [SP05, Sec , 7.1, 9.2] 3.2 Representing Uncertainty 3.3 Uncertain Systems [SP05, Sec. 7.2, 7.3, 7.4] [SP05, Sec. 8.1, 8.2, 8.3] 3.4 Systems with Structured Uncertainty [SP05, Sec. 8.2] Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, 2005.
27 Norm [SP05, A.5] Key properties 1. Nonnegative 2. Positive iff 3. Homogeneous, :scalar 4. Triangle inequality Vector Norm [Ex.] 1 (Euclidean Vector Norm) (Induced) Matrix Norm [Ex.] 27
28 Norm [SP05, A.5] Signal Norm [Ex.] Energy of signal ( norm, : Lebesgue space) Integral absolute error Energy Area 2 maximum value over time System Norm MIMO (System Gain) 33
29 Nominal Performance in SISO Systems 3 Nyquist Plot [SP05, p. 281] 1 1 should be away from by 29
30 Fundamental Limitations Bound on the Crossover Frequency RHP (Right halfplane) Zero [SP05, pp. 183] 4 Fast RHP Zeros ( large): Loose Restrictions Slow RHP Zeros ( small): Tight Restrictions Im worse better Time Delay 0 z Unstable zero Step Response Re Frequency [rad/s] Time [s] 30
31 Fundamental Limitations Bound on the Crossover Frequency RHP (Right halfplane) Pole Slow RHP Poles ( Fast RHP Poles ( Loose Restrictions large): Tight Restrictions small): [SP05, pp. 192, 194] 5 Frequency [rad/s] Poles on imaginary axis Im better worse 0 p Re Unstable pole 31
32 RHP Poles/Zeros, Time Delays and Sensitivity in SISO Systems Allpass system ( ) 6 Phase [deg] Phase [deg] Frequency [rad/s] Frequency [rad/s] or The zero and the pole must be sufficiently far apart The product of RHP pole and time delay must be sufficiently small 34
33 Fundamental Limitations: Sensitivity in MIMO Systems [SP05, Sec. 6.2] Algebraic Constraint 7 is large if and only if is large Fundamental Limitations: Bounds on Peaks in MIMO Systems, One RHP Pole and One RHP Zero [SP05, Sec. 6.3] :Pole and Zero Direction [SP05, 4.4, 4.5 ] 33
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