Politehnica University of BucharestInstitute National Polytechnique de Grenoble

Student: Valentin Borsu

Supervisors: Prof. Ph. D. Eng. Dumitru Popescu

Prof. Ph. D. Eng. Alina Voda-Besancon

Robust Control System Design

Applied To A DVD Video Player(Radial Loop)

Outline

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

● 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 precise modeling and future analysis knowledge of the process dynamics

● Model based controller design Combined pole placement and sensitivity function shaping Norm-based robust control

design

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

2.1. Physical structure of the disk

Wavelength of the laser beam: Distance between two subsequent track locations: (track pitch)

Reflective layer

PitsTrack pitch

LandPitPitPit length

Substrate

Metalic layer

Protective layer

Pit Land

Laser Beam

74.0

60 mm

Laser beam

2.2. DVD Player Architecture

2.3. Optical Description

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

3.1. The Control Loops

Focus loop

Radial loop

• Noise caused by the photodetector

• Internal disturbances (Caused by the spindle rotational frequency - eccentricityand vertical deviations)● External disturbances: shocks and vibrations

• disk imperfections: (scratches, dust, fingerprints impressed pit and land imperfections)

•Noise A/D & D/A

• Actuator nonlinearities & coupling effects

• Nonlinearities in generating the

the error signal

3.2. Disturbances

3.2. Disturbances

Lateralview

Frontview

Disk eccentricity

Power spectrum of the radial error

3.3. Control Problem Description

● - laser’s spot position determined by 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 measurements and can be considered as a disturbance signal which acts at the output of

The reference for the considered control system is given by the displacement error and

as we want its steady state to be zero, in the regular control structure we would have had a set-point of 0.

3.3. Control problem description

● The disturbance can be considered as a superposition of a known part - the track reference position and an unknown part - disturbance due to non-

perfectly spiral-shaped tracks or 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 ramp is so small such 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

3.3. Control Problem Description

Radial servo specifications, N=1

Steady-state error: Maximum radial spot positionerror: Template for the inverse of the output sensitivity function

Lowest Corner Frequency:

Highest Corner Frequency:

Closed-loop bandwidth:

Minimum required sensitivity:

Rise Time:

)()(1)( 1 sHsCgsS optyp

Lf Hf

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. 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 controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

4.1. Physical Models

The actuator model contains a mechanical part and an electrical part.By using the data from Sanyo – the system has a pair of complex poles positioned very

close to the origin, characterized by , and a real pole given by which characterizes the electrical part of the system.

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 method we have proposed a controller that assures the same performances as the phase-lead solution, but for the real model of the radial actuator.

Trade-off: We have obtained a 4th order controller.

4.2. Actual control solutions used in the industry

● STMicroelectronics Grenoble – lag-lead controller, designed on the basis of the real model of the actuator introduced in section 4.1.

◦ Pole at - reject low-frequency disturbances acting on a range bounded by the frequency of the zero . ◦ A new zero at for assuring an

acceptable phase margin ◦ Pole at high frequency to

reduce the effect of electro-mechanic resonance and

measurement noise.

4.2. Actual control solutions used in the industry

● Industrial Implementation:

It is possible to offer a global characterization of the robustness and stability of the closed-loop system thanks to the generalized stability margin.

We will start from the matrix of the closed-loop system sensitivity functions:

After SVD, the modulus of is given by the highest singular value:

,and

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 to instability and will become less robust to the uncertainties from the nominal transfer function of the plant.

For the solution offered by STMicroelectronics, we have obtained:

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust

controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

5.1. A new control system for the DVD-Player

Process: Controller:

With: ◦ - fixed parts of the controller. ◦ - polynomials resulting from Bezout Equation:

represents the characteristic polynomial of the system (closed-loop poles)

These sensitivity functions play a crucial role in the robustness analysis of the closed-loop system with respect to modeling.

They are calibrated such that they guarantee nominal performances in the rejection of disturbances and system stability under the 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 constant from the beginning to the end of the data zone.

● So the disc displacement on the spindle can be approximated 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.

We have imposed a slight modification of 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 poles near 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 poles to ensure

disturbance rejection in the low frequency range.◦ - a real zero to lower the magnitude of the input

sensitivity function at high frequency.

The controller was used with the help of the software ppmaster.

We have obtained a 5th order controller, whose order has been reduced by using a balanced reduction method.

The controller order reduction must aim in keeping as much as possible the properties imposed on the closed-loop system.

A balanced realization represents an asymptotically stable minimal realization in which the observabillity and controllability gramians are equal and diagonal.

In a balanced realization each Hankel singular value is associated to a state of the balanced system.

By applying this method we have obtained a 4th order controller which maintains the disturbance rejection performances as the controller we started with.

Also, for obtaining lower order controllers we have used ppmaster with inferior specifications than the presented ones and we obtained two controllers of order 3, respectively 2.

5.2. Simulation Results

5.2. Simulation Results

We obtain better disturbance rejection by employing the controllers designed with this method.

5.3. Controller Design with the Method

P(s) is usually defined as:

The closed-loop transfer function w → z is given by the linear fractional transformation(TLF):

, where:

GS/KS Configuration

Problem Statement. Find a stabilizing controller which minimizes:

Find controllers which satisfy:

, and if this is valid

it can be shown that:

Application to the DVD-Player

The control structure used 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:

Results

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

Results

● Only the RS4 controller verifies 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.

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

6.1. Nominal and uncertainty model (unstructured uncertainties)

Nominal plant:

Uncertainty model:

Relative Error Function:

The weighting function has the following form:

Uncertainty weight

6.2. Robust Stability (RS) and Robust Performance (RP) Analysis

Open-loop transfer function:

Robust Stability:

Robust Performance:

6.2. A new method for RS analysis

Normalized difference between two transfer functions:

The normalized distance or Vinnicombe distance is defined as:

New RS Condition: The controller K which stabilized the model G1, will stabilize the model G2, if:

6.3. Simulation Results

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:

6.4. Structured Uncertainties – μ-Analysis

The structured singular value of a generic matrix , having the same dimension as with respect to the set , is defined as:

By assuming that the diagonal block of the uncertainties satisfies we have:

The way of choosing and representing the uncertainties (DVD Case)

State-space representation of the process:

Uncertainty modeling

The matrix N is given by the equations:

Results

1. Motivation of the research2. General Description of a DVD Video-Player3. Control problem formulation4. The analysis of the radial actuator and the state

of the art concerning its control system5. The design of a new set of robust controllers6. Robustness analysis under the presence of

uncertainties7. Conclusions and perspectives

7. Conclusions and perspectives

● The radial actuator dynamics is appropriately described 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.

THANK YOU VERY MUCH FOR YOUR ATTENTION!

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Questions