Design of a Probabilistic Robust Track-Following Controller for Hard - - PowerPoint PPT Presentation

Design of a Probabilistic Robust Track-Following Controller for Hard Disk Drive Servo Systems

• E. Keikha, B. Shahsavari, F. Zhang, O. Bagherieh,
• R. Horowitz

26th CML Meeting

1

UC Berkeley

UC Berkeley

• Manufacturing Tolerances
• Batch Product
• Different Environmental Condition
• Time

UNCERTAINTY

Introduction

2

Uncertainty in the plant’s dynamic is inevitable for HDDs due to In disk drive, performance is of great

• importance. The control algorithm, while

handling different uncertainties, should not be conservative

Parametric (real) Dynamic Frequency Dependent

7 2 2 1

( ) 2  



  



i i i i i

A G s s s ( ) (s)( ( ) )   d G s G I W s

UC Berkeley

Robust control designed for track‐following servo systems controller is well studied in literature. However all the proposed methods are based

• n classical deterministic optimization of “worst‐case design”. This
• ptimization problem is computationally complex and will lead to a

conservative control design.

Introduction and Objective

• Design a controller that remains robust against parametric

and dynamic variations.

• Reduce computational complexity.
• Reduce conservatism.

Objective

UC Berkeley Parametric uncertainties Dynamic uncertainties

• The performance criteria is to

minimize RMS of PES signal in time domain while disturbance applied to the system is assumed to be Gaussian white noise.

Deterministic Robust Control Design based

• n Linear Matrix Inequality

4 1

[ ,..., , ]    ฀

n V

diag 

• 2 /

H H

00 u u y y

A B B B C D D C D D D C D

      

             

0 0 u u y y

A B B B C D D C D D D C D

      

             

2

min max || ( , ) ||

z K

G K

  



UC Berkeley

Deterministic Robust Control Design based on Linear Matrix Inequality

5

2

min max || ( , ) ||

z K

G K

  



min

2

|| ( , ) || || ( , ) || 1

 





        

v v

z p z p

G k G k

Subject to

1

[ ,..., ]    

p n

diag

cl cl

M trace ( , ) B ( , ) B

2 1 1

(W, P,K, G, , ) ( (W)) K G ( , ) ( , ) * * * * K G ( , ) * ( , ) * * * * *

i i i T i i i T T i

P K W K G G P G G G P P K G P G G K  

 

                                           

cl cl T cl cl

A C I A C I I

M (W,P,K,G, , )

i

  

minimize

W,P,K,G, 

S.T

 Mixed  Robust

• The original non-linear parametric uncertainty is

embedded into a larger affine structure. In other words, the original uncertain system is changed to polytopic uncertain system.

Example

Robust H2 Problem

– Computational complexity increase exponentially with number of uncertainties. – We should solve 2N BMIs; where N is the number of uncertainties. – Considering 12 parametric uncertainties, we should solve 4096 BMIs!!!! – It can just solve the problem for polytopic uncertain system.

• Design of a globally optimal full order output feedback

controller for polytopic uncertain system is NP‐HARD.

Limitation of the Conventional Robust approach:

6

2 /

H H

2

H

UC Berkeley

Damping Ratios (ζ) Resonance Modes (ω) 20% 15%

7 2 2 1

( ) 2

i VCM i i i i

A G s s s   



  



UC Berkeley

• In

the proposed design, by allowing a small probability (in the order of 10‐6 ) that the objective being violated, the designed controller is less conservative and the system performance improves drastically.

• The design is performed in a probabilistic framework

where the uncertain parameters are treated as random variables and the design specification is met with a given probability level.

7

Probabilistic Robust Control (Randomized Algorithms)

UC Berkeley

– Convex optimization in design parameter space. – Randomization in uncertainty space to estimate the probability of violation.

• The computational complexity does not depend on the number of

uncertain parameters (breaking the curse of dimensionality).

• The non‐linear parametric uncertainty is treated as it is (reducing

conservatism).

8

Probabilistic Robust Control (Randomized Algorithms)

UC Berkeley

• estimate this probability using randomization because of introducing

randomization

9

Probabilistic Robust & Randomized Algorithms

*

M( , )   

 

 

(1... ( ))

*

M( , ) 1 1   

 

     

N k

j

PR PR

Confidence

10‐6 in our case study

 

*

M( , ) 1  



    PR

Accuracy

10‐4 in our case study

• The aim is to find design parameter which satisfies the LMI constraint.

*



• Shift the problem to probabilistic space. It requires the computation of

multidimensional integrals associated with the probability.

UC Berkeley

• We want to solve a set of LMIs, hence, we need an scalar function which

indicates if the LMI is violated for the particular design and uncertain parameters.

• It is a non-negative function.
• It is positive if and only if performance function is violated.
• we use the maximum eigenvalue as indicator function which is convex in

for fixed

10

[1]. G. Calafiore and B. T. Polyak, “Stochastic algorithms for exact and approximate feasibility of robust LMIs,” Automatic Control, IEEE Transactions on, 2001.

Violation Function

max

( , ) ( ( *, *))        f M

 

UC Berkeley Initialization Oracle Update Rule Outer Iteration θseq

Design Procedure

1 1 ( ) (ln 1.11ln(k) 2.27) 1 ln 1 N k            

Choose an initial condition (solution to nominal case) and set iteration counters to zero Extract a pre‐specified number of samples from the uncertainty set and check if violation function is zero for all of them (Monte Carlo Simulation). If it is zero, then exit with current design parameters; if not, update the current design parameters based on cutting plane algorithm (next step).

UC Berkeley

12

• Update the rule (Cutting plane method)

( , ) ( , ) ( , )( )

i i i k k k

f f f



            { : ( , ) ( , ) }

i i k k k k

H f f

 

           , ,

From convexity of and the definition of sub‐ gradient it holds that

( , ) f  

where is the point for which (the violation certificate obtained from the probabilistic oracle)

i

 ( , )

i

f   

• Cutting plane method is is a localization based method

which at each iteration tries to shrink the volume of a polytope containing the solution set.

max 1 max max max

( , ) [ ( ) ,...., ( ) ]

i T T k n

f G G



           

• we can conclude that the solution is not in the intersection
• f the current polytope

and the half space and it can be cut from the solution set.

k

H

k

L

UC Berkeley

Test using a HDD Plant model with uncertainty

13

UC Berkeley A commercial Drive 2TB capacity Track pitch 100nm Rotational speed 7200 rpm 4 platters

Identification of Disturbance:

14

UC Berkeley

UC Berkeley

Simulation results

15 Closed loop sensitivity plots of 500 random realizations for the designed probabilistic controller Closed loop sensitivity plots of 500 random realizations for the nominal H2controller resulted from h2hinfsyn command in MATLAB Closed loop eigenvalues plot with controller designed using probabilistic framework for 500 random sample Closed loop eigenvalues plot with MATLAB controller designed using h2hinfsyn for 500 random samples

2

H / H

2 ∞

H / H

Probabilistic

UC Berkeley

Implementation results

16

Experimental and simulated sensitivity transfer function for the designed controller Experimental and simulated closed loop transfer function for the designed controller Experimental and simulated step response of 50 nm with corresponding control input Experimental Setup

Conclusion

• A robust controller was designed which minimizes the worst case root mean square (RMS)

value of the desired output subject to the closed‐loop stability in the presence of parametric and dynamic uncertainties.

• The computational complexity of the algorithm does not depend on the dimension of the

uncertainty set

• Is less conservative dealing with parametric uncertainty.
• The proposed controller is implemented on a HDD and shows drastic improvement in the

track following performance compared to the classical robust approaches.

17

UC Berkeley

Any Question ?

18

UC Berkeley

Thank you.