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 26 th CML Meeting UC Berkeley 1

Introduction UC Berkeley In disk drive, performance is of great importance. The control algorithm, while handling different uncertainties, should not be conservative Uncertainty in the plant’s dynamic is inevitable for HDDs due to UNCERTAINTY 7 A   ( ) i G s Parametric  2     Manufacturing Tolerances 2 2 • s s  1 i i i i (real) • Batch Product • Different Environmental Condition • Time    d ( ) (s)( ( ) ) G s G I W s Dynamic 0 Frequency Dependent 2

Introduction and Objective UC Berkeley Robust control designed for track ‐ following servo systems controller is well studied in literature. However all the proposed methods are based on classical deterministic optimization of “ worst ‐ case design ”. This optimization problem is computationally complex and will lead to a conservative control design . Objective • Design a controller that remains robust against parametric and dynamic variations. • Reduce computational complexity. • Reduce conservatism.

Deterministic Robust Control Design based on Linear Matrix Inequality UC Berkeley • 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. A B B B � �  0 u   � � 0 C D D   � �    0 u   C D D D    � 0 00 0   � Dynamic uncertainties 0 0   C D   0 y y     [ ,..., , ] ฀ diag 1 n V Parametric uncertainties 2 / H H    A B B B  0   u    Z    0    C D D        0  u   Z    0   0   C D D D       0 00 0    A B B B  y   u  0 0  C D     0 0 y y u   0 C D D    min max || ( , ) ||    0 G K u  2   z   0 0 C D D D K    0 0 0 0   0 0   C D   0 y y 4

Deterministic Robust Control Design based on Linear Matrix Inequality UC Berkeley     || ( , ) ||  G k Subject to   2 min  min max || ( , ) || z p G K 0 0   2 z     0 0 || ( , ) || 1 K  G k    z p v v     [ ,..., ] diag 1 p n W,P,K,G,   minimize   (W,P,K,G, , ) M 0 S.T  i     2   (W, P,K, G, , ) ( (W)) M trace i     P 0 K G 0 ( K , ) ( , ) B A i cl i   W 0 ( K ,  ) G cl   C  i  * G T P 1 G 0  cl      * G G P T     * * I       P 0 K G 0 ( K , ) ( , ) B 0 A i cl i cl    T 1 T 0  * G P G G ( K , ) 0 C     i cl  * * 0 I   * * *     I 5

Limitation of the Conventional Robust approach: UC Berkeley  Mixed 2 / H H   Robust H 2 • The original non-linear parametric uncertainty is embedded into a larger affine structure. In other words, the original uncertain system is changed to 7 A   ( ) polytopic uncertain system. i G s      2 2 VCM 2 s s  1 i i i i • Design of a globally optimal full order output feedback controller for polytopic uncertain system is NP ‐ HARD. Damping Ratios ( ζ ) Resonance Modes ( ω ) Example 20% 15% Robust H 2 Problem – Computational complexity increase exponentially with number of uncertainties. We should solve 2 N 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. 6

Probabilistic Robust Control (Randomized Algorithms) 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 & Randomized Algorithms UC Berkeley 10 ‐ 6 in our case Accuracy study                *       M( , * ) 0 1 1 j M( , ) 0 1    * PR PR M( , ) 0 PR  (1... ( ))   N k 10 ‐ 4 in our case Confidence 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. • estimate this probability using randomization because of introducing randomization 9

Violation Function 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        ( , ) ( ( *, *)) f M max [1]. G. Calafiore and B. T. Polyak, “Stochastic algorithms for exact and approximate feasibility of robust LMIs,” Automatic Control, IEEE Transactions on , 2001. 10

Design Procedure UC Berkeley Choose an initial condition (solution to nominal case) and Initialization set iteration counters to zero 1 1    ( ) (ln 1.11ln(k) 2.27) N k    1 ln 1   θ seq     Oracle 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 Update Rule exit with current design parameters; if not, update the current design parameters based on cutting plane algorithm (next step). Outer Iteration

Update the rule (Cutting plane method) UC Berkeley 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. f   ( , ) From convexity of and the de fi nition of sub ‐ gradient it holds that             ( , ) ( , ) ( , )( ) i i i f f f  k k k  f    i ( , i ) 0 where is the point for which (the violation certi fi cate obtained from the probabilistic oracle)            { : ( , ) ( , ) } � ��� i i H f f k   k k k � ��� we can conclude that the solution is not in the intersection � � of the current polytope L and the half space and it can H k k be cut from the solution set. � � � � � , � � � � � � ��� � , � � �� � � �             ( , i ) [ T ( ) ,...., T ( ) ] f G G 12  max 1 max max max k n

Test using a HDD Plant model with uncertainty UC Berkeley UC Berkeley A commercial Drive 2TB capacity Track pitch 100nm Rotational speed 7200 rpm 4 platters 13

Identification of Disturbance: UC Berkeley 14

Simulation results Probabilistic H H / 2 ∞ UC Berkeley H / H  2 Closed loop sensitivity plots of 500 random realizations Closed loop sensitivity plots of 500 random for the designed probabilistic controller realizations for the nominal H 2 controller resulted from h2hinfsyn command in MATLAB Closed loop eigenvalues plot with MATLAB controller Closed loop eigenvalues plot with controller designed 15 designed using h2hinfsyn for 500 random samples using probabilistic framework for 500 random sample

Implementation results UC Berkeley Experimental and simulated closed loop Experimental Setup transfer function for the designed controller Experimental and simulated step response of Experimental and simulated sensitivity transfer 50 nm with corresponding control input function for the designed controller 16