Course Description This is an advanced course in which we explore the field of Statistical Optics. Topics covered include such subjects as the statistical properties of natural (thermal) and laser light, spatial and temporal coherence, effects of partial coherence on optical imaging instruments, effects on imaging due to randomly inhomogeneous media, and a statistical treatment of the detection of light. Development of this more comprehensive model of the behavior of light draws upon the use of tools traditionally available to the electrical engineer, such as linear system theory and the theory of stochastic processes.

http://ieeexplore.ieee.org/iel5/3/23094/01073023.pdf

Statistical optics

The main stream research on fault detection and fault tolerant control has been focused on model based methods. As far as a model is concerned, changes therein due to faults have to be extracted from measured data. Generally speaking, existing approaches process measured inputs and outputs either by a filter designed based on a known model (e.g. for additive faults), or by an identification scheme to estimate the changed model parameters (e.g. due to multiplicative faults). Since the classical system identification methods are usually more involved than solving a linear least-squares problem, they can hardly be implemented online. The contribution of this thesis is hence the development of reliable numerical methods that try to extract fault information from data, in a manner as easy as solving a linear least-squares problem. These methods include “data driven” controller reconfiguration (CR) and fault detection and identification (FDI), and online experiment design to ensure that the “data” are informative enough to discover the changed system properties due to faults. The focus of Chapter 2 is on the development of the “data driven” controller reconfiguration approach, i.e. the closed-loop subspace predictive control (SPC). The key step in this approach is identifying a future output predictor (in terms of Markov parameters), which maps the past I/Os and future inputs to the future outputs of a system. A predictive controller is then parameterized by the identified predictor. The closed-loop SPC skips the realization of the system model and relies only on its Markov parameters; and can therefore be easily implemented online to adaptively accommodate multiplicative faults. It has been proven that with infinitely long measurement and prediction horizons, the closed-loop SPC is equivalent to the classical LQG design for LTI systems. In the case of a limited number of noisy data samples, the identified Markov parameters are biased and noisy, which then lead to a stochastic uncertainty in the output predictor of the SPC. This chapter develops a probabilistic robust solution in an explicit form, which is “cautious” to this uncertainty. The closed-loop SPC for LTI systems is extended to linear parameter varying systems. Chapter 3 focuses on developing Fault detection and Identification approaches Connected to Subspace Identification (FICSI) for both LTI and LPV systems. FICSI avoids projecting a residual vector onto the left null space of the extended observability matrix, as the classical parity space approach (PSA) requires. The advantage of avoiding this projection is the preservation of the degree of freedom in the residual vector, and consequently, the fault information contained therein. The thesis proves that FICSI produces residualsmore sensitive to faults than PSA does. It then proves the asymptotic unbiasedness of the FICSI fault estimation schemes. The novelty of the FICSI algorithms compared to the existing fault detection and estimation approaches based on PSA or unknown input observer (UIO), can also be attributed to the fact that FICSI requires only a sequence of Markov parameters mapping the I/O measurements to the residual. As a consequence, the FICSI detector and estimation filters can be directly identified from I/O measurements in a closed-loop plant, without modeling its dynamics in a state-space form. Both the SPC and FICSI algorithms rely on a sequence of Markov parameters identified from data. To identify these parameters, the inputs to the plant have to be “informative” enough; i.e. they must persistently excite the system to a sufficient order. Classical experiment design and the recent literatures on identification for control parameterize the spectrum of input signals in frequency domain. The persistency of excitation condition is ensured, if the input autocorrelation coefficients are nonzero up to a sufficient order. In recently developed identification for control approaches, optimal input signals (in terms of robust closed-loop performance achievable by the identified model) need to be searched on a grid of frequency points. This is unfortunately not suitable for online implementations. This thesis hence develops in Chapter 4 an online input design approach in time domain, which avoids parameterizing the input spectrum. This approach requires constraining input signals by a set of LMIs, and can be incorporated into a convex optimization problem. The combination of these input constraints with the SPC approach leads to an adaptive predictive control scheme with guaranteed parametric convergence. Due to its convexity and the recursive nature, this online design approach is more suitable than the classical approaches to capture the abruptly changed system conditions due to faults in a timely manner.

/pdf/data-driven-fault-tolerant-control-a-subspace-approach-3s71ewcxv8.pdf

Data Driven Fault Tolerant Control: A Subspace Approach

In this paper, we consider the identification of matrix state-space models (MSSM) of the following form: \begin{align*} \mathbf {X}(k+1) &= \mathbf {A}_2 \mathbf {X}(k) \mathbf {A}_1^T + \mathbf {B}_2 \mathbf {U}(k) \mathbf {B}_1^T \\ \mathbf {Y}(k) &= \mathbf {C}_2 \mathbf {X}(k) \mathbf {C}_1^T + \mathbf {E}(k) \end{align*} for all time dependent quantities and matrices of appropriate dimensions. Due to the large size of these matrices, vectorization does not allow the use of standard multivariable subspace methods such as N4SID or MOESP. In this paper, the resulting Kronecker structure that appears in the system matrices due to vectorization is exploited for developing a scalable subspace-like identification approach. This approach consists of first estimating the Markov parameters associated to the MSSM via the solution of a regularized bilinear least-squares problem that is solved in a globally convergent manner. Second, a bilinear low-rank minimization problem is tackled which allows to write a three-dimensional low-rank tensor and consequently to estimate the state sequence and the lower-dimensional matrices $\mathbf {A}_1,\mathbf {A}_2,\mathbf {B}_1,\mathbf {B}_2,\mathbf {C}_1,\mathbf {C}_2$ . A numerical example on a large-scale adaptive optics system demonstrates the ability of the algorithm to handle the identification of state-space models within the class of Kronecker structured matrices in a scalable manner which results in more compact models.

/pdf/k4sid-large-scale-subspace-identification-with-kronecker-4oli2orgyr.pdf

K4SID: Large-Scale Subspace Identification With Kronecker Modeling

A new prototype adaptive deformable mirror for future AO-systems is presented that consists of a thin continuous membrane on which push-pull actuators impose out-of-plane displacements. Each actuator has ±10?m stroke, nanometer resolution and only mW’s heat dissipation. The mirror’s modular design makes the mechanics, electronics and control system extendable towards large numbers of actuators. Models of the mirror are derived that are validated using influence and transfer function measurements. First results of a prototype with 427 actuators are also presented.

/pdf/validation-of-a-new-adaptive-deformable-mirror-concept-3ufre2qbze.pdf

Validation of a new adaptive deformable mirror concept

Refractive index variations in the earth's atmosphere cause wavefront aberrations and limit thereby the resolution in ground-based telescopes With Adaptive Optics (AO) the temporally and spatially varying wavefront distortions can be corrected in real time Most implementations in a ground based telescope include a WaveFront Sensor, a Deformable Mirror and a real time wavefront control system The largest optical telescopes built today have a ~ 1 Om primary mirror Telescopes with more collecting area and higher resolution are desired ELTs are currently designed with apertures up to 42m For these telescopes serious challenges for all parts of the AO system exist This thesis addresses the challenges for the DM An 8m class telescope on a representative astronomical site is the starting point The atmosphere is characterized by the spatial and temporal spectra of Kolmogorov turbulence and the frozen flow assumption The wavefront fitting error, caused by a limited number of actuators and the temporal error, caused by a limited control bandwidth, are the most important for the DM design It is shown that ~5000 actuators and 200Hz closed loop bandwidth form a balanced choice between the errors and correct an 8m wavefront in the visible to nearly diffraction limited An actuator stroke of ~56J!m and ~036J!m inter actuator stroke is thereby needed Together with the nm's resolution, low power dissipation, no hysteresis and drift, these form the main DM requirements The design, realization and tests of a new DM that meets these requirements and is extendable and scalable in mechanics, electronics and control to suit further Extremely Large Telescopes (ELTs) is presented In the DM a few layers are distinguished: a continuous mirror facesheet, the actuator grid and the base frame In the underlying layer - the actuator grid - low voltage electromagnetic push-pull actuators are located Identical actuator modules, each with 61 actuators, hexagonally arranged on a 6mm pitch can be placed adjacent to form large grids The base frame provides a stable and stiff reference A thin facesheet is needed for low actuator forces and power dissipation, whereby its lower limit is set by the facesheets inter actuator deflection determined by gravity or wind pressure For both scaling laws for force and dissipation are derived Minimum power dissipation is achieved when beryllium is used for the mirror facesheet Pyrex facesheets with 100J!m thickness are chosen as a good practical, alternative in the prototype development Struts (001 x 8mm) connect the facesheet to the actuators and ensure a smooth surface over the imposed heights and allow relative lateral movement of the facesheet and the actuator grid Measurements show 3nm RMS surface unflattness from the glued attachment The stiffness of the actuators form the out-of-plane constraints for the mirror facesheet and determine the mirrors first resonance frequency and is chosen such that the resonance frequency is high enough to allow the high control bandwidth but not higher that needed to avoid excessive power dissipation and fix points in the surface in case of failure The electromagnetic variable reluctance actuators designed, are efficient, have low moving mass and have suitable stiffness Other advantages are the low costs, low driving voltages and negligible hysteresis and drift The actuators consist of a closed magnetic circuit in which a PM provides static magnetic force on a ferromagnetic core that is suspended in a membrane This attraction force is increased of decreased by a current through a coil The actuators are free from mechanical hysteresis, friction and play and therefore have a high positioning resolution with high reproducibility The actuator modules are build in layers to reduces the number of parts and the complexity of assembly and to improve the uniformity in properties Dedicated communication and driver electronics are designed FPGA implemented PWM based voltage drivers are chosen because of their high efficiency and capability to be implemented in large numbers with only a few electronic components A multidrop LVDS based serial communication is chosen for its low power consumption, high bandwidth and consequently low latency, low communication overhead and extensive possibilities for customization A flat-cable connects up to 32 electronics modules to a custom communications bridge, which translates the ethernet packages from the control PC into LVDS Two DMs prototypes were successfully assembled: a 050mm DM with 61 actuators and a 0l50mm DM with 427 actuators In the second prototype modularity is shown by the assembly of seven identical grids on a common base The dynamic performance of each actuator is measured, including its dedicated driver and communication All actuators were found to be functional, indicating that the manufacturing and assembly process is reliable A nonlinear mathematical model of the actuator was derived describing both its static and dynamic behavior based on equations from the magnetic, mechanic and electric domains The actuator model was linearized, leading to expressions for the actuator transfer function and properties such as motor constant, coil inductance, actuator stiffness and resonance frequency From frequency response function measurements these properties showed slight deviations from the values derived from the model, but the statistical spread for the properties was small, stressing the reliability of the manufacturing and assembly process The mean actuator stiffness and resonance frequency were 047kN/m and 18kHz respectively, which is close to their design values of 500N/m and 19kHz The time domain response of an actuator to a 4Hz sine voltage was used to determine hysteresis and semi-static nonlinear response of the actuator This showed the first to be negligible and the second to remain below 5% for ±10J!m stroke Measurements showed that in the expected operating range, the total power dissipation is dominated by indirect losses in FPGAs The static DM performance is validated using interferometric measurements The measured influence matrix is used to shape the mirror facesheet into the first 28 Zernike modes, which includes the piston term that represents the best flat mirror The total RMS error is ~25nm for all modes The dynamic behavior of the DM is validated by measurements A laser vibrometer is used to measure the displacement of the mirror facesheet, while the actuators are driven by zero-mean, bandlimited, white noise voltage sequence Using the MOESP system identification algorithm, high-order black-box models are identified with VAF values around 95% The first resonance frequency identified is 725Hz, and lower than the 974Hz expected from the analytical model This is attributed to the variations in actuator properties, such as actuator stiffness The power dissipation in each actuator of the 050mm mirror to correct a typical Von Karmann turbulence spectrum is ~ 15m W

Adaptive deformable mirror : based on electromagnetic actuators

Adaptive optics (AO) is a technique to actively correct the wavefront distortions introduced in a light beam as it propagates through a turbulent medium. Nowadays, it is commonly applied in ground-based telescopes to counteract the devastating effect of atmospheric turbulence. This thesis focuses on the control aspects of AO. Whereas most AO systems use a simple control law derived from physical insights, this thesis addresses the AO control problem from a control engineering perspective. The main objective is to show that the performance of the current generation of AO systems can be improved by applying advanced control strategies that account for the AO system dynamics and the spatio-temporal correlation in the wavefront. To this end a data-driven optimal control is proposed. It consists of a dedicated subspace-identification algorithm to identify an atmospheric disturbance model from open-loop wavefront sensor data, followed by an optimal control design. By an efficient implementation, this approach can be used to design a full multi-variable controller for AO systems with up to a few hundred sensors and actuators, without assuming any form of decoupling. The data-driven optimal control strategy is thoroughly tested both in simulations and on an AO laboratory setup. The experiments show, that compared to the standard approach, optimal control is able to improve the wavefront suppression performance, especially under low light level conditions and rough turbulence. The gain in performance is explained the improved ability to anticipate future distortions when accounting for the spatio-temporal correlation in the wavefront.

/pdf/data-driven-optimal-control-for-adaptive-optics-e3e8tal4vh.pdf

K.J.G. Hinnen

Papers

Data-Driven Optimal Control for Adaptive Optics