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

• f Slid

Slidin ing Mo Mode Con Control fo for Har Hard Di Disk sk Dri Drives es

CML SPONSORS’ MEETING 2014

Minghui Zheng Xu Chen Masayoshi Tomizuka

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

Motivation for frequency‐shaped sliding mode control: performance enhancement at the frequencies where the performance is degraded. Audio Vibrations Cause Significant Servo Performance Degradation

1000 2000 3000 4000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Freq / Hz Magnitude

Audio Vib 1 Audio Vib 2 Audio Vib 3

• 2. Basic Idea
• 2. In Frequency‐shaped SMC, sliding

surface is defined based on and .

• 1. In conventional Sliding Mode Control (SMC),

sliding surface is defined based on and . :PES; : derivative of PES; 1,

: To increase control effort at preferred frequencies

1 or f

e e

2

e

Sliding Surface Approaching Dynamics

1000 2000 3000 4000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Freq / Hz Amplitude

High‐frequency peaks exist in some vibrations. , ,

3.0 Design Considerations for Frequency‐shaped SMC

Part II: Frequency Shaping Differentiator Hard Disk Drive Position PES Reference Vibration

• 1. Controller design to obtain

desired approaching dynamics

• 2. Filter design to obtain desired

sliding surface

f

e

2

e

Sliding Surface Approaching Dynamics

Part I: Frequency‐ Shaped SMC

3.1 System Description

Plant Model

1 1 1

{ }

w w w w f f w w w

e A e B e e Q e C e D e       where is the position error signal (PES); is the velocity error signal; | | is the input disturbance; ||

is the

audio vibration.

Realization Enlarged System

where

1 2

( ( ) ( )) ( )

a a

e e Ae B u t d t B v t e               ( ( ) ( )) ( )

a a

e Ae B u t d t B v t          

, , ,

w w w a a

A B e e A A e B B B B                                  

3.2 Controller Design

1

( ) ( ) [ ( ) ( ) sgn( ( )) ( )sgn( ( ))]

a a

u t HB HAe t qs t s t HBD HB V s t 



               This controller can guarantee that will converge to the sliding surface.

For Known Dynamics For Unknown Disturbance

Note: Controller design in discrete‐time domain involves more complex analysis, which is included in Reference [1] .

( ) ( ) ( ( ))sgn( ( )) s t qs t t s t       

Approaching Dynamics Controller

where

• , 0, 1 ≫ 0

where :

f

e

2

e

Sliding Surface Approaching Dynamics

( ) ( ) s t He t    ( ) ( )sgn( ( )) sgn( ( ))

a a a

t HBD HBV HBd t s t HB v s t              

3.3 Filter Design (Frequency Shaping )

Sliding Surface

1 2 2

{ }

f

s Q e h e    Design filter to guarantee that 1. →0 implies →0, and →0, i.e., the sliding surface is stable;

• 2. Error dynamics on the sliding surface ( 0) has desired frequency properties.

3.3.2 Single‐peak Filter 3.3.3 Multi‐peak Filter 3.3.1 Design Objective The sliding surface is stable if and only if roots of have negative real parts.

2

1 ( ) 1 1 ( ) B p h A p p   ( ) ( )

f

B p Q A p 

Note: filter design starts from continuous‐time system and extends to discrete‐time system for direct implementation [1].

3.3 Filter Design (Frequency Shaping )

Nice Property: ALWAYS STABLE regardless of what the peak frequency is, and what 0 is.

2 2 2 2

2 ( ) (0 1) ( ) 2

d d f d d

p bw p w B p Q a b A p p aw p w         

Single‐peak Filter

Root‐loci Analysis Method (: peak frequency)

3.3.2 Single‐peak Filter 3.3.3 Multi‐peak Filter 3.3.1 Design Objective

Open-loop Poles

d

bw 

d

aw 

Re Im

Open-loop Zeros

2

1 ( ) 1 1 ( ) B p h A p p  

Design Flexibility in stability preservation

Root loci ( single‐peak filter case) ( varies from ∞ to 0)

3.3 Filter Design (Frequency Shaping )

Multi‐peak Filter where

Note: although the stability analysis is more involved, the analysis based on root locus method provides design flexibility, intuitive design and easy analysis

1

( ) ( ) ( ) ( )

n i f i i

B p B p Q A p A p



 

2 2 2 2

( ) 2 ( ) 2

i di di i di di

A p p aw p w B p p bw p w      

3.3.2 Single‐peak Filter 3.3.3 Multi‐peak Filter 3.3.1 Design Objective

Usually there are more than one peak in audio vibrations.

Root loci (one three‐peak filter case)

• 4. Simulation

Audio Vibration 3 Audio Vibration 1 Audio Vibration 2 Vibrations Filters

1000 2000 3000 4000 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Freq / Hz Magnitude

Audio Vib 1 Audio Vib 2 Audio Vib 3

5 10 15 20 25 Magnitude (dB) 10

10

10

10

10

90 180 270 360 Phase (deg) Bode Diagram Frequency (Hz)

Qf_1 Qf_2 Qf_3

Note: Frequency‐shaped SMC implementation is in the discrete time domain, which requires some additional analysis in discrete time [1].

Three kinds of vibrations Three peak filters

• 4. Simulation

1000 2000 3000 4000 5000 6000 7000 8000 0.02 0.04 0.06 0.08 0.1 Freq / Hz Nomalized Amplitude SMC 3 = 0.37447 Freq SMC 3 = 0.31265

% overall 3‐Sigma value of PES reduction % amplitude reduction at specific frequency

Audio Vibration 3 Audio Vibration 1 Audio Vibration 2 Vibrations Filters

• 4. Simulation

1000 2000 3000 4000 5000 6000 7000 8000 0.01 0.02 0.03 0.04 0.05 Freq / Hz Normalized Amplitude SMC 3 = 0.36601 Freq SMC 3 = 0.32399

% overall 3‐Sigma value of PES reduction % amplitude reduction at specific frequency

Audio Vibration 3 Audio Vibration 1 Audio Vibration 2 Vibrations Filters

• 4. Simulation

1000 2000 3000 4000 5000 6000 7000 8000 0.01 0.02 0.03 0.04 Freq / Hz Normalized Amplitude SMC 3 = 0.19932 Freq SMC 3 = 0.15796

% overall 3‐Sigma value of PES reduction % amplitude reduction at specific frequency

Audio Vibration 3 Audio Vibration 1 Audio Vibration 2 Vibrations Filters

• 5. Adaptive Filter (part of future work)

(a)A notch filter can be used to identify the peak frequency: (b) The frequency is identified by optimization:

2 2 2 2

2 cos(2 ) 2 cos(2 )

n d n n n d n

z b T z b Q z a T z a       

2 1

1 min ( ) 2

d

k p w

e i



1 ( ) ( )

n f

Q z Q z 

• r

Peak Filter Differentiator Frequency‐ Shaped SMC Hard Disk Drive Position PES Reference Notch Filter

• Optimization over

Usually the peak frequency () of PES is unknown.

• 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