Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization A.2 Sensitivity and Feedback Performance A.3 Loop Shaping [SP05, Sec. 3.2, 4.1.5, 4.7, 4.8] [SP05, Sec. 2.2, 5.2] [SP05, Sec. 2.4, 2.6] Reference: [SP05] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control; Analysis and Design, Second Edition, Wiley, 2005.
Internal Stability Gang of Four [AM08, p. 317] Sensitivity Complementary Sensitivity Load Sensitivity Noise Sensitivity [AM08] K.J. Astrom and R.M. Murray, Feedback Systems, Princeton University Press, 2008 2
Internal Stability [SP05, Ex. 4.16] (p. 144) Sensitivity Load Sensitivity Comp. Sensitivity Noise Sensitivity Stable? Step Response Unstable Time [s] Time [s] Time [s] Time [s] 3
Internal Stability [SP05, Theorem 4.6] (p. 145) The feedback system in the above figure is internally stable if and only if all Gang of Four ( ) are stable Well-posedness: (Gang of Four: well-defined and proper) C. Desoer C.A. Desoer and W.S. Chan, Journal of the Franklin Institute, 300 (5-6) 335-351, 1975 4
Youla Parameterization ( Parameterization) Case 1: Stable Plant [SP05, p. 148] All Stabilizing Controllers: Internal Model Control (IMC) Structure -parameter Gang of Four : Proper Stable Transfer Function 5
Youla Parameterization Case 2: Unstable Plant Coprime Factorization [SP05, p. 122] [SP05, p. 149] Coprime: No common right-half plane(rhp) zeros : Proper Stable Transfer Functions [SP05, Ex. 4.1] (*) Bezout Identity : Coprime : Proper Stable Transfer Functions [SP05, Ex.] : (*) [Ex.] : Integer : Integer 6
Youla Parameterization Case 2: Unstable Plants [SP05, p. 149] A Stabilizing Controller [SP05, Ex.] All Stabilizing Controllers Gang of Four Affine Functions of 7
Sensitivity and Feedback Performance Disturbance Attenuation Open-loop Closed-loop : Sensitivity small: good Feedback Performance 8
Insensitivity to Plant Variations [SP05, p. 23] small : good Feedback Performance 9
Benefits of Feedback Disturbance Attenuation Insensitivity to Plant Variations Stabilization (Unstable Plant) Linearizing Effects Reference Tracking : small Two-degrees-of-freedom Control Feedback + Feedforward 10
Waterbed Effects [SP05, p. 167] 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] 10 0 10 10 0 10 1 [SP05, Ex., p. 170] Frequency [rad/s] (unstable) 11
Maximum Peaks of Sensitivity and [SP05, p. 36] Complementary Sensitivity : Maximum Peak Magnitude of : Maximum Peak Magnitude of : Bandwidth Frequency of : Bandwidth Frequency of 12
Loop Shaping Loop Transfer Function Sensitivity: Comp. Sensitivity: + Constraint large Loop Shaping small Closed-loop Open Loop small small Stability, Performance, Robustness 13
Loop Transfer Function [SP05, Ex. 2.4] (p. 34) Gain Crossover Frequency Stability Margins [SP05, p. 32] Gain Margin Phase Margin Time Delay Margin Stability Margin [SP05, Ex. 2.4] (p. 34) 14
Frequency Domain Performance [SP05, Ex. 2.4] (p. 34) Maximum Peak Criteria [SP05, p. 36] [Ex.] [Ex.] 15
Bode Gain-phase Relationship [SP05, p. 18] (minimum phase systems) Slope of the Gain Curve at Steep Slope: Small Phase Margin [SP05, Ex., p. 20] 0-2 -1-2 16
Fundamental Limitations Bound on the Crossover Frequency [SP05, pp. 183] RHP (Right half-plane) Zero 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]
Fundamental Limitations Bound on the Crossover Frequency RHP (Right half-plane) Pole [SP05, pp. 192, 194] Slow RHP Poles ( small): Loose Restrictions Fast RHP Poles ( large): Tight Restrictions Frequency [rad/s] Poles on imaginary axis Im better worse 0 p Re Unstable pole
SISO Loop Shaping [SP05, pp. 41, 42, 343] Performance Robust Stability (+ Roll-off) Loop Shaping Specifications Gain Crossover Frequency Shape of System Type, Defined as the Number of Pure Integrators in Roll-off at Higher Frequencies 19
Step response analysis/performance criteria Rise time Settling time Peak time Overshoot Error tolerance First-order System 0.9 1 0.1 0 Second-order System Rise time Settling time Rise time Settling time Overshoot Overshoot Peak Time [QZ07] L. Qiu and K. Zhou (2007) Introduction to Feedback Control, Prentice Hall. 20
Design Relations Maximum Peak Magnitude of Complementary Sensitivity Phase Margin Bandwidth if if if : Maximum Peak Magnitude of : Bandwidth Frequency of [FPN09] G.F. Franklin, J.D. Powell and A. E.-Naeini (2009) Feedback Control of Dynamic Systems, Sixth Edition, Prentice Hall. 21
Controllability analysis with SISO feedback control [SP05, pp. 206-209] Margin to stay within constraints Margin for performance Margin because of RHP-pole Margin because of RHP-zero Margin because of frequency where plant has phase lag Margin because of delay Typically, the closed-loop bandwidth of the spacecraft is an order of magnitude less than the lowest mode frequency, and as long as the controller does not excite any of the flexible modes, the sampling period may be selected solely based on the closed-loop bandwidth. [Le10] W.S. Levine (Eds.) (2010) The Control Handbook, Second Edition: Control System Fundamentals, Second Edition, CRC Press. 22
RHP Poles/Zeros, Time Delays and Sensitivity p z M S For systems with a RHP pole and RHP zero (or a time delay τ ), any stabilizing controller gives sensitivity functions with the property = sup S( jω) ω p p + z z M T = supt ( jω) ω e pτ RHP pole and zero and time delay significantly limit the achievable performance of a system S( jω) M S p p + z z M T T ( jω) pτ e 9
RHP Poles/Zeros, Time Delays and Sensitivity All-pass system( p =1, z = b, τ ) sτ b s e P ap ( s) = P ( ) = s ap s 1 s 1 z or 6 < z / p pτ < 0. 3 / p <1/ RHP pole/zero pair 6 The zero and the pole must be sufficiently far apart P allowable phase lag of at : ap ω RHP pole and time delay The product of RHP pole and time delay must be sufficiently small gc ϕ l = 90 10