Fr Frequency equency Shaping Shaping fo for Pe Performance Enhancem - PowerPoint PPT Presentation

Fr Frequency equency Shaping Shaping fo for Pe Performance Enhancem Enhancemen ent of of Slid Slidin ing Mo Mode Con Control fo for Har Hard Di Disk sk Dri Drives es Minghui Zheng Xu Chen Masayoshi Tomizuka CML SPONSORS’ MEETING 2014

Outline 1.Motivation 2.Basic Idea 3.Control Scheme 3.0 Design Considerations for Frequency ‐ shaped SMC 3.1 System Description 3.2 Controller Design 3.3 Filter Design Single ‐ peak Filter & Multi ‐ peak Filter 4.Simulation 5.Adaptive Filter 6.Summary

1. Motivation Audio Vibrations 0.14 Audio Vib 1 Audio Vib 2 0.12 Audio Vib 3 0.1 Magnitude 0.08 0.06 0.04 0.02 0 0 1000 2000 3000 4000 Freq / Hz Cause Significant Servo Performance Degradation Motivation for frequency ‐ shaped sliding mode control: performance enhancement at the frequencies where the performance is degraded.

2. Basic Idea e 0.14 2 Sliding Surface High ‐ frequency 0.12 � � � peaks exist in e 1 or e 0.1 some vibrations. f Amplitude 0.08 0 0.06 Approaching Dynamics 0.04 0.02 � � � � � � � , � � � ��� � � � � � , � � 0 0 1000 2000 3000 4000 Freq / Hz 1. In conventional Sliding Mode Control (SMC), 2. In Frequency ‐ shaped SMC, sliding sliding surface is defined based on � � and � � . surface is defined based on � � and � � . � � : PES; � � : derivative of PES; � � �1, � � � � � : To increase control effort at preferred frequencies

3.0 Design Considerations for Frequency ‐ shaped SMC Sliding Surface e 1. Controller design to obtain 2 � � � desired approaching dynamics e f 0 2. Filter design to obtain desired sliding surface Approaching Dynamics Vibration Reference PES Part II: Frequency Part I: Shaping Position Frequency ‐ Hard Disk Shaped Drive SMC Differentiator

3.1 System Description Plant Model Enlarged System    e      1      e Ae B u t ( ( ) d t ( )) B v t ( )         e Ae B u t ( ( ) d t ( )) B v t ( ) a a    e a a 2 where � � is the position error signal (PES); where � � is the velocity error signal; | � � | � � is     A B 0 e the input disturbance; |� � ���| � � � is the    w w w      e , A ,   0 A e   audio vibration.       0 0 � � Realization         B , B a B B        e A e B e a w w w w 1    e Q { } e C e D e f f 1 w w w 1

3.2 Controller Design Controller                 1  u t ( ) ( HB ) [ HAe t ( ) qs t ( ) sgn( ( )) s t ( HBD HB V )sgn( ( ))] s t a a For Known Dynamics For Unknown Disturbance � � � � � � � � , � � 0, 1 ≫ � � 0 where � Approaching Dynamics        s t ( ) qs t ( ) ( ( ))sgn( ( )) t s t    s t ( ) He t ( ) where :               ( ) t HBD HBV HBd t ( )sgn( ( )) s t HB v sgn( ( )) s t 0 a a a e 2 Sliding Surface This controller can guarantee that ���� � � � � will converge to the sliding e f surface . 0 Approaching Note: Controller design in discrete ‐ time domain involves Dynamics more complex analysis, which is included in Reference [1] .

3.3 Filter Design (Frequency Shaping ) 3.3.1 Design Objective 3.3.2 Single ‐ peak Filter 3.3.3 Multi ‐ peak Filter    Sliding Surface s Q { } e h e 0 f 1 2 2 Design filter � � to guarantee that � → 0 implies � � → 0 , and � � → 0 , i.e., the sliding surface is stable; 1. 2. Error dynamics on the sliding surface ( � � 0 ) has desired frequency properties. B p ( )  Q f A p ( ) 1 B p ( ) 1   1 0 The sliding surface is stable if and only if roots of have negative real parts. h A p p ( ) 2 Note: filter design starts from continuous ‐ time system and extends to discrete ‐ time system for direct implementation [1].

3.3 Filter Design (Frequency Shaping ) 3.3.1 Design Objective 3.3.2 Single ‐ peak Filter 3.3.3 Multi ‐ peak Filter Single ‐ peak Filter   2 2 B p ( ) p 2 bw p w      ( � � : peak frequency) d d Q (0 a b 1)   f 2 2 A p ( ) p 2 aw p w d d Root ‐ loci Analysis Method Im 1 B p ( ) 1   ( � � varies from �∞ to 0) 1 0 h A p p ( ) 2 Re   bw aw Nice Property: ALWAYS STABLE d d regardless of what the peak frequency Open-loop Poles � � is, and what � � �� 0� is. Open-loop Zeros Design Flexibility in stability preservation Root loci ( single ‐ peak filter case)

3.3 Filter Design (Frequency Shaping ) 3.3.1 Design Objective 3.3.2 Single ‐ peak Filter 3.3.3 Multi ‐ peak Filter Usually there are more than one peak in audio vibrations. Multi ‐ peak Filter n B p ( ) B p ( )    i Q f A p ( ) A p ( )  i 1 i where    2 2 A p ( ) p 2 aw p w i di di    2 2 B p ( ) p 2 bw p w i di di Note: although the stability analysis is more involved, the analysis based on root locus method provides design flexibility, intuitive design and easy analysis Root loci (one three ‐ peak filter case)

4. Simulation Vibrations Filters Audio Vibration 1 Audio Vibration 2 Audio Vibration 3 Three kinds of vibrations Three peak filters Bode Diagram 0.14 25 Audio Vib 1 Audio Vib 2 20 0.12 Qf_1 Audio Vib 3 Magnitude (dB) 15 Qf_2 Qf_3 0.1 10 Magnitude 5 0.08 0 360 0.06 270 0.04 Phase (deg) 180 0.02 90 0 0 0 1000 2000 3000 4000 1 2 3 4 5 10 10 10 10 10 Freq / Hz Frequency (Hz) Note: Frequency ‐ shaped SMC implementation is in the discrete time domain, which requires some additional analysis in discrete time [1].

4. Simulation Vibrations Filters Audio Vibration 1 Audio Vibration 2 Audio Vibration 3 0.1 SMC 3  = 0.37447 Freq SMC 3  = 0.31265 0.08 Nomalized Amplitude 0.06 � ��% overall 3 ‐ Sigma value of PES reduction � ��% amplitude reduction at specific frequency 0.04 0.02 0 0 1000 2000 3000 4000 5000 6000 7000 8000 Freq / Hz

4. Simulation Vibrations Filters Audio Vibration 1 Audio Vibration 2 Audio Vibration 3 0.05 SMC 3  = 0.36601 Freq SMC 3  = 0.32399 0.04 Normalized Amplitude � ��% overall 3 ‐ Sigma value of 0.03 PES reduction � ��% amplitude reduction at 0.02 specific frequency 0.01 0 0 1000 2000 3000 4000 5000 6000 7000 8000 Freq / Hz

4. Simulation Vibrations Filters Audio Vibration 1 Audio Vibration 2 Audio Vibration 3 0.04 SMC 3  = 0.19932 Freq SMC 3  = 0.15796 Normalized Amplitude 0.03 � ��% overall 3 ‐ Sigma value of PES reduction 0.02 � ��% amplitude reduction at specific frequency 0.01 0 0 1000 2000 3000 4000 5000 6000 7000 8000 Freq / Hz

5. Adaptive Filter (part of future work) Usually the peak frequency ( � � ) of PES is unknown. (a)A notch filter can be used to identify the peak frequency:    1 2 2 z 2 b cos(2 T z ) b   Q z ( ) n d n or Q n    n Q ( ) z 2 2 z 2 a cos(2 T z ) a f n d n 1  k 2 (b) The frequency is identified by optimization: min e i ( ) p 1 2 w d Optimization over � � � � Notch Filter � � Reference PES Peak Filter � � � � Frequency ‐ Position Hard Disk Shaped Drive SMC Differentiator

6. Summary 1. Design method for frequency shaped sliding mode controllers has been presented. 2. Simulation study has been performed to demonstrate: (a) reduction of the overall 3 � value of PES; (b) reduction of the amplitude of PES spectrum at specific frequencies; (c) nearly no performance sacrifice at other frequencies. 3. Future work (a) Adaptive filter (for both single ‐ peak filter and multi ‐ peak filter) (b) Nonlinear sliding surface design [1] Minghui Zheng, Xu Chen, Masayoshi Tomizuka, “Discrete ‐ time Frequency ‐ shaped Sliding Mode Control for Audio ‐ Vibration Rejection in Hard Disk Drives,” submitted for IFAC 2014