ANALYSIS AND SYNTHESIS OF DISTURBANCE OBSERVER AS AN ADD-ON ROBUST CONTROLLER Hyungbo Shim (School of Electrical Engineering, Seoul National University, Korea) in collaboration with Juhoon Back, Nam Hoon Jo, and Youngjun Joo under the support from Hyundai Heavy Industries, Co., LTD. Oct. 15, 2010, at UCSB
Overview disturbance observer (DOB) dates back to the Japanese article (K. Ohnishi, 1987) has been of much interest in control application community, but did not draw much attention in control theory community 41 pub. record in IEEE Trans. Indus. Elec. (2003-2007) 8 in IEEE Trans. Contr. Sys. Tech. (2003-2007) 3 in IEEE Trans. Automat. Contr. (2003-2007) original idea is intuitively simple, but less analyzed rigorously This talk is about developing theoretical framework for DOB approach, and appreciation of it as a robust control tool presentation of robust stability condition and robust nominal performance recovery discussion about various aspects of DOB, and extension to nonlinear systems 2
Overview problem statement 3 Inner-loop dynamic controller: The same input-output behavior!
Overview problem statement 4 The same input-output behavior! not only just the same steady-state behavior, but also the same transient behavior (with the same IC between the nominal and the real) important feature required by industry where settling time, overshoot, etc., should be uniform among many product units a sharp contrast to other robust or adaptive redesign approach which possibly changes the transient behavior
Linear DOB: Intuition 5
Linear DOB: Intuition 6
Linear DOB: Intuitive justification 7
More Issues to be Solved 8 The argument so far does not present real power and limitations. robust stability is still of question what about transient performance recovery? unanswered observations from practice : It does not work for non-minimum phase plant. High bandwidth of Q(s) destabilizes the closed-loop. It cannot handle large parameter variation. Higher order Q(s) leads to instability. can we extend the DOB-based controller for uncertain nonlinear plants?
Our Starting Point 9
Normal form realization of Q(s) 10 Byrnes-Isidori Normal Form
Representation of P (nonlinear plant) 11 Byrnes-Isidori Normal Form
Representation of nominal plant Pn (with Q) 12
Summary: Problem Statement 13 State-space realization of DOB structure?
* (too brief) Review of Tikhonov s Theorem 14
Closed-loop system in the Standard Singular Perturbation Form 15 With coordinate change given by the overall closed-loop system is expressed by outer-loop controller C Robust stability of DOB structure with small = Robust stability of Boundary-Layer subsystem + Robust stability of Quasi-Steady-State subsystem
Quasi-Steady-State Subsystem 16 QSS subsystem: It is the same as the disturbance-free nominal closed-loop! z-dynamics becomes unobservable. z-dynamics should be ISS (minimum phase in linear case). implies (so Q-filter with the plant inverse acts as a state observer!)
Boundary-Layer Subsystem 17 Robust stability condition: Q-filter should be stable. No matter how large the uncertainty is, once bounded, there are s so that two polynomials are Hurwitz! proof Semi-global result for boundedness of
Summary: Robust Stability 18 Robust stability of the closed-loop with DOB is guaranteed if, on a compact set of the state-space, 1. zero dynamics of uncertain plant is ISS, 2. outer-loop controller stabilizes the nominal plant, 3. coefficients of Q-filter are chosen for the RS condition, 4. is sufficiently small.
Robust Steady-State Performance 19 Nominal closed-loop nominal plant outer-loop controller C (slow) transient steady-state Real closed-loop (fast) transient (slow) transient steady-state
Robust (slow) Transient Performance 20 I. Tikhonov s theorem guarantees that if II. However, (0) and»(0) become unbounded as! 0.
Peaking phenomenon reflected in coordinate change 21
Robust (slow) Transient Performance 22 I. Tikhonov s theorem guarantees that if II. However, (0) and»(0) become unbounded as! 0. Lesson: Due to peaking phenomenon, it is not true that conventional DOB recovers (slow) transient performance.
Saturating the Peaking Components Peaking components that affect slow dynamics: 23 [Esfandiari & Khalil, 1992] Replace with and
Stability Loss in BL Subsystem and its recovery by a dead-zone function 24 BL -dynamics becomes and loses GES. Add a dead-zone function: to recover GES.
Stability Analysis of BL Subsystem with saturation & deadzone 25
Application of Tikhonov s Theorem for the second interval 26
Summary: Robust Transient Performance Recovery 27 thanks to saturation & dead-zone functions.
Proposed Nonlinear DOB Structure 28 [Back/S, AUT, 08]
Example: Point Mass Satellite 29 [Back/S, TAC, 09]
Example: Point Mass Satellite 30 Nominal: Real: with DOB (solid=nominal) Real: w/o DOB:
31 Robust Stability Condition for Linear DOB back to linear presents simple linear robust stability condition working on frequency domain Shim & Jo, AUT, 2009
Problem Formulation 32 Standing Assumption: where is a collection of Ass: C(s) is designed for Pn(s). Problem: Design of Q(s) for robust stability
Characteristic equation with Q(s) 33 Q(s) = ( s) l + b l 1 ( s) l 1 + + b 1 ( s) + a 0 ( s) l+r + a l+r 1 ( s) l+r 1 + + a 1 ( s) + a 0 =: N Q(s; ) D Q (s; ) ; > 0
Robust Stability of the Overall System 34 [S/Jo, AUT, 09] Observation 1: Observation 2:
Design of Q(s): when Q(s) = a 0 ( s) r + a r 1 ( s) r 1 + + a 0 35 How to choose Q(s):
Discussion: Answers to the raised questions 36 1. It does not work for non-minimum phase plant. Correct, for small. 2. Small destabilizes the closed-loop. No, if Q(s) satisfies RS condition. It may be the effect of initial peakings but not a destabilization. 3. It cannot handle large parameter variation. No in general. By following the design guidelines, any bounded variation can be handled. 4. Higher order Q(s) leads to instability. No in general. But, for higher order Q(s), the selection of coefficients becomes more complicated.
Taken from Yi, Chang, & Shen, IEEE/ASME Trans. Mechatronics, 2009 37 Experiments robust stability condition in action
38
Step responses 39
Robust tracking control [Bang/S/Park/Seo, IEEE Trans. Industrial Elect., 2010] 40 Hyundai HS-165 (handling 165kg)
Robust tracking control 41 Experiment results
Summary 42 theoretical study on DOB robust stability condition (new even for classical linear DOB) transient performance recovery (thanks to saturation/dead-zone functions) DOB = Inner-loop controller = Your first AID kit!
THANK YOU 감사합니다 Any feedback or comments are welcome hshim@snu.ac.kr