Approximation of L\"owdin Orthogonalization to a Spectrally Efficient Orthogonal Overlapping PPM Design for UWB Impulse Radio

In this paper we consider the design of spectrally efficient time-limited pulses for ultrawideband (UWB) systems using an overlapping pulse position modulation scheme. For this we investigate an orthogonalization method, which was developed in 1950 by Per-Olov L\"owdin. Our objective is to obtain a set of N orthogonal (L\"owdin) pulses, which remain time-limited and spectrally efficient for UWB systems, from a set of N equidistant translates of a time-limited optimal spectral designed UWB pulse. We derive an approximate L\"owdin orthogonalization (ALO) by using circulant approximations for the Gram matrix to obtain a practical filter implementation. We show that the centered ALO and L\"owdin pulses converge pointwise to the same Nyquist pulse as N tends to infinity. The set of translates of the Nyquist pulse forms an orthonormal basis or the shift-invariant space generated by the initial spectral optimal pulse. The ALO transform provides a closed-form approximation of the L\"owdin transform, which can be implemented in an analog fashion without the need of analog to digital conversions. Furthermore, we investigate the interplay between the optimization and the orthogonalization procedure by using methods from the theory of shift-invariant spaces. Finally we develop a connection between our results and wavelet and frame theory.Comment: 33 pages, 11 figures. Accepted for publication 9 Sep 201

Frequency response modeling and control of flexible structures: Computational methods

The dynamics of vibrations in flexible structures can be conventiently modeled in terms of frequency response models. For structural control such models capture the distributed parameter dynamics of the elastic structural response as an irrational transfer function. For most flexible structures arising in aerospace applications the irrational transfer functions which arise are of a special class of pseudo-meromorphic functions which have only a finite number of right half place poles. Computational algorithms are demonstrated for design of multiloop control laws for such models based on optimal Wiener-Hopf control of the frequency responses. The algorithms employ a sampled-data representation of irrational transfer functions which is particularly attractive for numerical computation. One key algorithm for the solution of the optimal control problem is the spectral factorization of an irrational transfer function. The basis for the spectral factorization algorithm is highlighted together with associated computational issues arising in optimal regulator design. Options for implementation of wide band vibration control for flexible structures based on the sampled-data frequency response models is also highlighted. A simple flexible structure control example is considered to demonstrate the combined frequency response modeling and control algorithms

Sampling from a system-theoretic viewpoint: Part II - Noncausal solutions

This paper puts to use concepts and tools introduced in Part I to address a wide spectrum of noncausal sampling and reconstruction problems. Particularly, we follow the system-theoretic paradigm by using systems as signal generators to account for available information and system norms (L2 and L∞) as performance measures. The proposed optimization-based approach recovers many known solutions, derived hitherto by different methods, as special cases under different assumptions about acquisition or reconstructing devices (e.g., polynomial and exponential cardinal splines for fixed samplers and the Sampling Theorem and its modifications in the case when both sampler and interpolator are design parameters). We also derive new results, such as versions of the Sampling Theorem for downsampling and reconstruction from noisy measurements, the continuous-time invariance of a wide class of optimal sampling-and-reconstruction circuits, etcetera

Robustification and Optimization in Repetitive Control For Minimum Phase and Non-Minimum Phase Systems

Repetitive control (RC) is a control method that specifically aims to converge to zero tracking error of a control systems that execute a periodic command or have periodic disturbances of known period. It uses the error of one period back to adjust the command in the present period. In theory, RC can completely eliminate periodic disturbance effects. RC has applications in many fields such as high-precision manufacturing in robotics, computer disk drives, and active vibration isolation in spacecraft. The first topic treated in this dissertation develops several simple RC design methods that are somewhat analogous to PID controller design in classical control. From the early days of digital control, emulation methods were developed based on a Forward Rule, a Backward Rule, Tustin’s Formula, a modification using prewarping, and a pole-zero mapping method. These allowed one to convert a candidate controller design to discrete time in a simple way. We investigate to what extent they can be used to simplify RC design. A particular design is developed from modification of the pole-zero mapping rules, which is simple and sheds light on the robustness of repetitive control designs. RC convergence requires less than 90 degree model phase error at all frequencies up to Nyquist. A zero-phase cutoff filter is normally used to robustify to high frequency model error when this limit is exceeded. The result is stabilization at the expense of failure to cancel errors above the cutoff. The second topic investigates a series of methods to use data to make real time updates of the frequency response model, allowing one to increase or eliminate the frequency cutoff. These include the use of a moving window employing a recursive discrete Fourier transform (DFT), and use of a real time projection algorithm from adaptive control for each frequency. The results can be used directly to make repetitive control corrections that cancel each error frequency, or they can be used to update a repetitive control FIR compensator. The aim is to reduce the final error level by using real time frequency response model updates to successively increase the cutoff frequency, each time creating the improved model needed to produce convergence zero error up to the higher cutoff. Non-minimum phase systems present a difficult design challenge to the sister field of Iterative Learning Control. The third topic investigates to what extent the same challenges appear in RC. One challenge is that the intrinsic non-minimum phase zero mapped from continuous time is close to the pole of repetitive controller at +1 creating behavior similar to pole-zero cancellation. The near pole-zero cancellation causes slow learning at DC and low frequencies. The Min-Max cost function over the learning rate is presented. The Min-Max can be reformulated as a Quadratically Constrained Linear Programming problem. This approach is shown to be an RC design approach that addresses the main challenge of non-minimum phase systems to have a reasonable learning rate at DC. Although it was illustrated that using the Min-Max objective improves learning at DC and low frequencies compared to other designs, the method requires model accuracy at high frequencies. In the real world, models usually have error at high frequencies. The fourth topic addresses how one can merge the quadratic penalty to the Min-Max cost function to increase robustness at high frequencies. The topic also considers limiting the Min-Max optimization to some frequencies interval and applying an FIR zero-phase low-pass filter to cutoff the learning for frequencies above that interval

Multi-Input Multi-Output Repetitive Control Theory And Taylor Series Based Repetitive Control Design

Repetitive control (RC) systems aim to achieve zero tracking error when tracking a periodic command, or when tracking a constant command in the presence of a periodic disturbance, or both a periodic command and periodic disturbance. This dissertation presents a new approach using Taylor Series Expansion of the inverse system z-transfer function model to design Finite Impulse Response (FIR) repetitive controllers for single-input single-output (SISO) systems, and compares the designs obtained to those generated by optimization in the frequency domain. This approach is very simple, straightforward, and easy to use. It also supplies considerable insight, and gives understanding of the cause of the patterns for zero locations in the optimization based design. The approach forms a different and effective time domain design method, and it can also be used to guide the choice of parameters in performing in the frequency domain optimization design. Next, this dissertation presents the theoretical foundation for frequency based optimization design of repetitive control design for multi-input multi-output (MIMO) systems. A comprehensive stability theory for MIMO repetitive control is developed. A necessary and sufficient condition for asymptotic stability in MIMO RC is derived, and four sufficient conditions are created. One of these is the MIMO version of the approximate monotonic decay condition in SISO RC, and one is a necessary and sufficient condition for stability for all possible disturbance periods. An appropriate optimization criterion for direct MIMO is presented based on minimizing a Frobenius norm summed over frequencies from zero to Nyquist. This design process is very tractable, requiring only solution of a linear algebraic equation. An alternative approach reduces the problem to a set of SISO design problems, one for each input-output pair. The performances of the resulting designs are studied by extensive examples. Both approaches are seen to be able to create RC designs with fast monotonic decay of the tracking error. Finally, this dissertation presents an analysis of using an experiment design sequence for parameter identification based on the theory of iterative learning control (ILC), a sister field to repetitive control. This is suggested as an alternative to the results in optimal experiment design. Modified ILC laws that are intentionally non-robust to model errors are developed, as a way to fine tune the use of ILC for identification purposes. The non-robustness with respect to its ability to improve identification of system parameters when the model error is correct is studied. It is demonstrated that in many cases the approach makes the learning particularly sensitive to relatively small parameter errors in the model, but sensitivity is sometimes limited to parameter errors of a specific sign

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Examination of Bandwidth Enhancement and Circulant Filter Frequency Cutoff Robustification in Iterative Learning Control

The iterative learning control (ILC) problem considers control tasks that perform a specific tracking command, and the command is to be performed is many times. The system returns to the same initial conditions on the desired trajectory for each repetition, also called run, or iteration. The learning law adjusts the command to a feedback system based on the error observed in the previous run, and aims to converge to zero-tracking error at sampled times as the iterations progress. The ILC problem is an inverse problem: it seeks to converge to that command that produces the desired output. Mathematically that command is given by the inverse of the transfer function of the feedback system, times the desired output. However, in many applications that unique command is often an unstable function of time. A discrete-time system, converted from a continuous-time system fed by a zero-order hold, often has non-minimum phase zeros which become unstable poles in the inverse problem. An inverse discrete-time system will have at least one unstable pole, if the pole-zero excess of the original continuous-time counterpart is equal to or larger than three, and the sample rate is fast enough. The corresponding difference equation has roots larger than one, and the homogeneous solution has components that are the values of these poles to the power of k, with k being the time step. This creates an unstable command growing in magnitude with time step. If the ILC law aims at zero-tracking error for such systems, the command produced by the ILC iterations will ask for a command input that grows exponentially in magnitude with each time step. This thesis examines several ways to circumvent this difficulty, designing filters that prevent the growth in ILC. The sister field of ILC, repetitive control (RC), aims at zero-error at sample times when tracking a periodic command or eliminating a periodic disturbance of known period, or both. Instead of learning from a previous run always starting from the same initial condition, RC learns from the error in the previous period of the periodic command or disturbance. Unlike ILC, the system in RC eventually enters into steady state as time progresses. As a result, one can use frequency response thinking. In ILC, the frequency thinking is not applicable since the output of the system has transients for every run. RC is also an inverse problem and the periodic command to the system converges to the inverse of the system times the desired output. Because what RC needs is zero error after reaching steady state, one can aim to invert the steady state frequency response of the system instead of the system transfer function in order to have a stable solution to the inverse problem. This can be accomplished by designing a Finite Impulse Response (FIR) filter that mimics the steady state frequency response, and which can be used in real time. This dissertation discusses how the digital feedback control system configuration affects the locations of sampling zeros and discusses the effectiveness of RC design methods for these possible sampling zeros. The sampling zeros are zeros introduced by the discretization process from continuous-time system to the discrete-time system. In the RC problem, the feedback control system can have sampling zeros outside the unit circle, and they are challenges for the RC law design. Previous research concentrated on the situation where the sampling zeros of the feedback control system come from a zero-order hold on the input of a continuous-time feedback system, and studied the influence of these zeros including the influence of these sampling zeros as the sampling rate is changed from the asymptotic value of sample time interval approaching zero. Effective RC design methods are developed and tested based for this configuration. In the real world, the feedback control system may not be the continuous-time system. Here we investigate the possible sampling zero locations that can be encountered in digital control systems where the zero-order hold can be in various possible places in the control loop. We show that various new situations can occur. We discuss the sampling zeros location with different feedback system structures, and show that the RC design methods still work. Moreover, we compare the learning rates of different RC design methods and show that the RC design method based on a quadratic fit of the reciprocal of the steady state frequency response will have the desired learning rate features that balance the robustness with efficiency. This dissertation discusses the steady-state response filter of the finite-time signal used in ILC. The ILC problem is sensitive to model errors and unmodelled high frequency dynamics, thus it needs a zero-phase low-pass filter to cutoff learning for frequencies where there is too much model inaccuracy for convergence. But typical zero-phase low-pass filters, like Filtfilt used by MATLAB, gives the filtered results with transients that can destabilize ILC. The associated issues are examined from several points of view. First, the dissertation discusses use of a partial inverse of the feedback system as both learning gain matrix and a low-pass filter to address this problem The approach is used to make a partial system inverse for frequencies where the model is accurate, eliminating the robustness issue. The concept is used as a way to improve a feedback control system performance whose bandwidth is not as high as desired. When the feedback control system design is unable to achieve the desired bandwidth, the partial system inverse for frequency in a range above the bandwidth can boost the bandwidth. If needed ILC can be used to further correct response up to the new bandwidth. The dissertation then discusses Discrete Fourier Transform (DFT) based filters to cut off the learning at high frequencies where model uncertainty is too large for convergence. The concept of a low pass filter is based on steady state frequency response, but ILC is always a finite time problem. This forms a mismatch in the design process, and we seek to address this. A math proof is given showing the DFT based filters directly give the steady-state response of the filter for the finite-time signal which can eliminate the possibility of instability of ILC. However, such filters have problems of frequency leakage and Gibbs phenomenon in applications, produced by the difference between the signal being filtered at the start time and at the final time, This difference applies to the signal filtered for nearly all iterations in ILC. This dissertation discusses the use of single reflection that produced a signal that has the start time and end times matching and then using the original signal portion of the result. In addition, a double reflection of the signal is studied that aims not only to eliminate the discontinuity that produces Gibbs, but also aims to have continuity of the first derivative. It applies a specific kind of double reflection. It is shown mathematically that the two reflection methods reduce the Gibbs phenomenon. A criterion is given to determine when one should consider using such reflection methods on any signal. The numerical simulations demonstrate the benefits of these reflection methods in reducing the tracking error of the system

Lecture notes on the design of low-pass digital filters with wireless-communication applications

The low-pass filter is a fundamental building block from which digital signal-processing systems (e.g. radio and radar) are built. Signals in the electromagnetic spectrum extend over all timescales/frequencies and are used to transmit and receive very long or very short pulses of very narrow or very wide bandwidth. Time/Frequency agility is the key for optimal spectrum utilization (i.e. to suppress interference and enhance propagation) and low-pass filtering is the low-level digital mechanism for manoeuvre in this domain. By increasing and decreasing the bandwidth of a low-pass filter, thus decreasing and increasing its pulse duration, the engineer may shift energy concentration between frequency and time. Simple processes for engineering such components are described and explained below. These lecture notes are part of a short course that is intended to help recent engineering graduates design low-pass digital filters for this purpose, who have had some exposure to the topic during their studies, and who are now interested in the sending and receiving signals over the electromagnetic spectrum, in wireless communication (i.e. radio) and remote sensing (e.g. radar) applications, for instance. The best way to understand the material is to interact with the spectrum using receivers and or transmitters and software-defined radio development-kits provide a convenient platform for experimentation. Fortunately, wireless communication in the radio-frequency spectrum is an ideal application for the illustration of waveform agility in the electromagnetic spectrum. In Parts I and II, the theoretical foundations of digital low-pass filters are presented, i.e. signals-and-systems theory, then in Part III they are applied to the problem of radio communication and used to concentrate energy in time or frequency.Comment: Added Slepian ref. Added arXiv ID to heade