Robust Control System Design Applied To A DVD Video Player (Radial Loop)

1. Robust Control System Design Applied To A DVD Video Player(Radial Loop) Politehnica University of Bucharest Institute National Polytechnique de Grenoble

2. Outline • Motivation of the research • General Description of a DVD Video-Player • Control problem formulation • The analysis of the radial actuator and the state of the art concerning its control system • The design of a new set of robust controllers • Robustness analysis under the presence of uncertainties • Conclusions and perspectives

3. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

4. ● Strict performance Tracking improvement specifications ● High rotational speed and A more precise control system increased storage capacity for the laser position ● The validation of the actual Reference point in the plant control solutions and a precisemodeling and future analysis knowledge of the process dynamics ● Model based controller designCombined pole placement and sensitivity function shaping Norm-based robust control design

5. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

6. Reflective layer Pits Track pitch 60 mm Pit Pit Land Pit length Pit Land Laser beam Protective layer Substrate Metalic layer Laser Beam 2.1. Physical structure of the disk Wavelength of the laser beam: Distance between two subsequent track locations: (track pitch)

7. 2.2. DVD Player Architecture

8. 2.3. Optical Description

9. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

10. 3.1. The Control Loops

11. Focus loop • Noise caused by the photodetector • Internal disturbances • (Caused by the spindle • rotational frequency - eccentricity • and vertical deviations) ● External disturbances: shocks and vibrations • Nonlinearities in generating the • the error signal Radial loop • disk imperfections: • (scratches, dust, fingerprints • impressed pit and land • imperfections) • Noise A/D & D/A • Actuator nonlinearities & • coupling effects 3.2. Disturbances

12. 3.2. Disturbances Front view Disk eccentricity Lateral view Power spectrum of the radial error

13. 3.3. Control Problem Description ● - laser’s spot position determinedby the displacement of the objective lens – must coincide every moment with the center of the track ● - actual position on the track, which can be considered as a reference for the control system, it cannot be determined from measurementsand can be considered as a disturbance signalwhich acts at the output of The reference for the considered control systemis given by the displacement errorand as we want its steady state to be zero, in the regular control structure we would have had a set-point of 0.

14. 3.3. Control problem description ● The disturbancecan be considered as a superpositionofa known part - the track reference positionand an unknown part - disturbance due to non-perfectly spiral-shaped tracksor eccentric rotation of the disk. For the radial loop, can be modeled as a superposition of a ramp and a disturbance – the slope of the rampisso smallsuch that it can be neglected, when the behavior of the servo system is analyzed around a track location. The rotational frequency of the disc is not constant since the Constant Linear Velocity Method (CLV)is employed for data read-out: , where ◦ – over-speed factor ◦ – constant scanning velocity ◦ – distance between the disk hole center and the falling laser beam – desired reference position

15. 3.3. Control Problem Description • Radial servo specifications, N=1 • Steady-state error: Maximum radial spot position error: Template for the inverse of the output sensitivity function • Lowest Corner Frequency: • Highest Corner Frequency: • Closed-loop bandwidth: • Minimum required sensitivity: • Rise Time:

16. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

17. 4.1. Physical Models The actuator model contains a mechanical part and an electrical part. By using the data fromSanyo – the system has a pair ofcomplex polespositioned very close to the origin, characterized by , and a real pole given by which characterizes the electrical part of the system.

18. 4.2. Actual control solutions used in the industry ● Standard ECMA-350, 1st Edition – The radial actuator is approximated by a double integrator: , and the considered control solution is a phase-lead one: ,with: - frequency where the absolute value of the open-loop transfer function is 1. - constant whose square illustrates the high-frequency gain of the controller. By using the pole-zero methodwe have proposed a controllerthat assures the same performancesas the phase-lead solution, but for the real model of the radial actuator. Trade-off: We have obtained a 4th order controller.

19. 4.2. Actual control solutions used in the industry ● STMicroelectronics Grenoble – lag-lead controller, designed on the basis of the real model of the actuatorintroduced in section 4.1. ◦ Poleat - reject low-frequency disturbancesacting on a range bounded by the frequency of the zero . ◦ A new zero atfor assuring an acceptable phase margin ◦ Pole at high frequencyto reduce the effect of electro-mechanic resonance and measurement noise.

20. 4.2. Actual control solutions used in the industry ● Industrial Implementation: It is possible to offer a global characterization of therobustness and stability of the closed-loop systemthanks to the generalized stability margin. We will start from the matrix of the closed-loop system sensitivity functions: After SVD, the modulus ofis given by the highest singular value: ,and

21. 4.2. Actual control solutions used in the industry The generalized stability margin is defined as: As the generalized stability margin decreases, the closed-loop system will be closer toinstabilityand will become less robust to the uncertainties from the nominal transfer function of the plant. For the solution offered by STMicroelectronics, we have obtained:

22. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

23. 5.1. A new control system for the DVD-Player Process: Controller: With: ◦ - fixed partsof the controller. ◦ - polynomials resulting from Bezout Equation: represents the characteristic polynomial of the system (closed-loop poles)

24. These sensitivity functions play acrucial role in the robustness analysis of the closed-loop system with respect to modeling. They arecalibratedsuch that they guaranteenominal performances in the rejection of disturbancesand system stability underthe presence of uncertainties. Modeling of the repetitive disturbance in the radial loop ● The repetitive disturbance is caused by the time-varying displacement. Its amplitude, called the disc eccentricity is defined by . ● It can be observed that the amplitude of the displacement is constantfrom the beginning to the end of the data zone. ● So the disc displacement on the spindle can beapproximated by the following equation: The data amount increases proportionally with the radius of the disc, Therefore, the disturbance rejection performances are imperative as the data density on the disc increases.

25. We have imposed aslight modificationof the magnitude of the output sensitivity function for a better rejection of the repetitive disturbances ( ) Step 1. Step 2. Step 3. - linear interpolation between the two magnitudes introduced above. In order to respect the template modification, the controller was designed on the basis of following specifications: ◦ - a pair of complex polesnear the model’s slowest vibration frequency, but well damped - two multiple real poles for maintaining bounds on - a complex pole to restrain the controller action in higher frequencies where the gain of the system is low. ◦ - a pair of complex polesto ensuredisturbance rejection in the low frequency range. ◦ - a real zeroto lower the magnitude of the input sensitivity function at high frequency.

26. The controller was used with the help of the softwareppmaster. We have obtained a 5th order controller, whose order has been reduced by using a balanced reduction method.

27. The controller order reductionmust aim in keeping as much as possible the properties imposed on the closed-loop system. Abalanced realizationrepresents an asymptotically stable minimal realizationin which the observabillity and controllability gramians are equal and diagonal. In a balanced realization each Hankel singular valueis associated to a state of the balanced system. By applying this methodwe have obtained a 4th order controllerwhich maintains the disturbance rejection performances as the controller we started with. Also, for obtaining lower order controllers we have used ppmaster with inferior specificationsthan the presented ones and we obtained two controllers oforder 3, respectively 2.

28. 5.2. Simulation Results

29. 5.2. Simulation Results We obtain better disturbance rejectionby employing the controllers designed with this method.

30. 5.3. Controller Design with the Method P(s) is usually defined as: The closed-loop transfer functionw → z is given by the linearfractional transformation(TLF): , where:

31. GS/KSConfiguration Problem Statement.Find a stabilizing controllerwhich minimizes: Find controllers which satisfy: , and if this is valid it can be shown that:

32. Application to the DVD-Player The control structureused for the design of the controller applied to the DVD Player Performance Filters: ● - is used for imposing performance specifications to the sensitivity function . We have chosen: ● - is used for imposing a bound on the control signal according to the actuator limitations and equals:

33. Results We have obtained a 5th order controller, and after employing the balanced rezidualization technique we reduced it to a 3rd order one.

34. Results ● Only the RS4 controllerverifies the modified template structure of the output sensitivity function S. ● Although the controllers designed with the method have smaller phase and delay margins than those in the RS2-RS4 cases, they have got the highest generalized stability margins so they are more robust.

35. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

36. 6.1. Nominal and uncertainty model (unstructured uncertainties) Nominal plant: Uncertainty weight Uncertainty model: Relative Error Function: The weighting function has the following form:

37. 6.2. Robust Stability (RS) and Robust Performance (RP) Analysis Open-loop transfer function: Robust Stability: Robust Performance:

38. 6.2. A new method for RS analysis Normalized difference between two transfer functions: The normalized distance or Vinnicombe distanceis defined as: New RS Condition: The controller K which stabilized the model G1, will stabilize the model G2, if:

39. 6.3. Simulation Results

40. 6.4. Structured Uncertainties – μ-Analysis Complex uncertainties block Real uncertainties block The set of uncertainties which affect the model: The scheme is called the NΔ configuration and is useful for the analysis of RS and RP properties. The transfer matrix of the nominal system: ,and the closed-loop transfer function whose inputs are and outputs is:

41. 6.4. Structured Uncertainties – μ-Analysis The structured singular valueof a generic matrix , having the same dimension aswith respect to the set , is defined as: By assuming that the diagonal block of the uncertainties satisfieswe have: The way of choosing andrepresenting the uncertainties (DVD Case)

42. State-space representation of the process: Uncertainty modeling

43. The matrix N is given by the equations:

44. Results

45. 1. Motivation of the research 2. General Description of a DVD Video-Player 3. Control problem formulation 4. The analysis of the radial actuator and the state of the art concerning its control system 5. The design of a new set of robust controllers 6. Robustness analysis under the presence of uncertainties 7. Conclusions and perspectives

46. 7. Conclusions and perspectives ● The radial actuator dynamics is appropriatelydescribed by the 3rd order physical model. ● This complete model was used for the design of the new control laws. ● Performance and robustness analysis: ◦ show us that with the use of the new design methods: pole placement with sensitivity function shaping and design we obtain superior results than in the case of the actual industrial control solution. ◦ points out the importance of the continuous interaction between the design of the DVD-ROM-ului and the control law synthesis ● A phenomenon which was not considered in this work is linked to the influence of the radial deviations and their possible effects during the modeling procedure.

47. THANK YOU VERY MUCH FOR YOUR ATTENTION!

48. ? Questions