Time-optimal output transitions for minimum phase systems: A frequency domain approach to post-actuation

The settling problem in hard disk drives (HDD) is an output transition problem - given an initial state x(0), the goal is to bring the output to a desired value y(tf) = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> y and hold it constant after the end of settling, t > tj. This article develops the direct solution to the optimal output transition (OOT) problem for seek settling, and develops a method to use post-actuation inputs to optimize the seek settling cost. The approach is illustrated with simulations on an example dual-stage system model. Simulations are also used to illustrate the reduction in the computational burden for online implementation by using linear combinations of precomputed inputs.

This paper finds minimum-energy inputs to change the position of an output point, on a flexible structure, from one value to another. Current methods transform the output-transition problem into a state-to-state transition problem by constraining the initial and final states of the output transition, for example, to be rest (rigid-body) configurations of the flexible structure. However, the choice of the initial and final states can be ad hoc, and the resulting output-transition cost (input energy) might not be minimum. The contribution of this paper is the direct solution of the optimal output-transition problem; the problem is posed and solved for general linear discrete-time systems. The novelty of the proposed approach is that inputs are not applied just during the output-transition time interval; rather, inputs are also applied outside the output-transition time interval, that is, before the beginning of and after the end of the output-transition time interval. (These inputs are called pre- and postactuation.) The implications of using pre- and postactuation are illustrated by using an example of discrete-time flexible structure model, which consists of a flexible rod connecting two masses. Simulation results are presented, which show substantial reduction of output-transition costs with the use of the proposed method when compared to the use of the standard state-to-state transition approach.

The issue of optimal output transition control for nonlinear differential flat systems with constraints is investigated. Of special interest is the generation of a state reference trajectory and a feedforward input, which are essential to the two-degree-of-freedom design. The proposed approach is to transfer the transition problem in the real output space into a trajectory planning issue in the flat output space. The distinguishing feature of our approach is the generation of a noncausal trajectory for the flat output. This approach is shown to be highly effective in creating performance improvements. It should also be noted that the proposed methodology guarantees that the planned trajectories are feasible for all nonlocal transitions. This allows the application of stable inversion in the planning of optimal output transitions. The proposed method is illustrated on a benchmark system, the isothermal continuous stirred tank reactor, although its applicability is much wider.

We discuss the concept of `minimum phase' for scalar semi Hurwitz transfer functions. The latter are rational functions where the denominator polynomial has its roots in the closed left half complex plane. In the present note, minimum phase is defined in terms of the derivative of the argument function of the transfer function. The main tool to characterize minimum phase is the Hurwitz reflection. The factorization of a weakly stable transfer function into an all-pass and a minimum phase system leads to the result that any semi Hurwitz transfer function is minimum phase if, and only if, its numerator polynomial is semi Hurwitz. To characterize the zero dynamics, we use the Byrnes-Isidori form in the time domain and the internal loop form in the frequency domain. The uniqueness of both forms is shown. This is used to show in particular that asymptotic stable zero dynamics of a minimal realization of a transfer function yields minimum phase, but not vice versa.

Optimal output-transitions for linear systems

We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse is that the system has ‘strict and non-positive relative degree’. We present two main results: first, a so-called ‘zero dynamics form’ is derived; this form is—within the class of system equivalence—a simple form of the DAE; it is a counterpart to the well-known Byrnes–Isidori form for ODE systems with strictly proper transfer function. The ‘zero dynamics form’ is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A, E, B)-invariant subspaces. Secondly, it is shown that the ‘funnel controller’ (that is a static non-linear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.

A fundamental problem in the design of data transmission systems is the synthesis of pulse-shaping networks which satisfy specifications in both the time and frequency domains. This paper considers the problem of designing a network to shape an arbitrary input pulse into a band-limited pulse having minimum intersymbol interference. The design procedure uses the zeros of the network transfer function to achieve the band-limiting properties (using a modified Temes and Gyi constraint) while the transfer function poles are optimized with a computer to give the desired time response. By limiting the specifications on the shaped pulse to an absolute minimum, very accurate results are achieved with simple networks. Some sample designs and experimental results are included. For example, an 11th order transfer function is designed to shape rectangular pulses for a synchronous baseband pulse amplitude modulation system. The shaped pulses have a bandwidth 20 percent in excess of the Nyquist bandwidth and a theoretical worst-case distortion of 2.1 percent. An active realization of this transfer function achieved a worst-case distortion of about 2.5 percent.

While the causality statement in the time domain is quite simple, the corresponding statement in the frequency domain involves Hilbert transform relations between the real and imaginary parts of the relevant transfer function. If the transfer function is minimum phase, one can go further and develop Hilbert transform relations between the amplitude and phase. Such relations are important and useful in at least two classes of problems: analysis of frequency‐sweep data and active control. In both problems, one usually wishes to simultaneously enforce the causality statement while achieving some other goals. This lecture will explore various ways of accomplishing this and will focus on implicitly causal expansions of transfer functions in the frequency domain. The expansion functions are built directly from Hilbert transform pairs and therefore the series is guaranteed to be causal. One example is a complex Fourier series in frequency in which the coefficients are required to be real valued. Examples will be...

Random switching logic (RSL) has been proposed as an efficient countermeasure to mitigate power analysis. The logic style equalizes the output transition probabilities using a random mask-bit. This manuscript, however, will show a successful attack against RSL. The single mask-bit can only add one bit of entropy to the information content of the overall power consumption variations and can very easily be deduced from the power consumption. Once the mask-bit is known, the a posteriori probabilities of the output transitions are not equal anymore and a power analysis can be mounted. A threshold filter suffices to remove the additional bit of information.

This paper presents transfer functions of a class of cyclic consensus system in closed forms. Four different kinds of network topologies; directed, undirected, one, two or full reference agents, are considered. Each agent of consensus systems is assumed to satisfy a first-order scalar dynamics which is driven by a common consensus protocol and an independent exogenous input. It is shown that every SISO transfer function between the exogenous input of one agent and the state of another generally different agent, is always minimum phase. In addition, poles and zeros, system degrees and relative degrees of such SISO transfer functions are specified. These results are interpretated in relation to the controllability and closed loop performance of a networked system having one leader agent. Furthermore, our transfer function representations are applied to an investigation of stability margins for a closed loop cyclic consensus system.

An optimal feedforward control design for the set-point following of MIMO processes

9 - Optimal and robust control system design

A method is presented for predicting the total response, in both frequency and time, of an unknown linear system when only the measured continuous wave (cw) magnitude is available. The approach is based on approximating the square of the measured magnitude by a rational function, from which various system transfer functions in terms of complex frequency are deduced. These transfer functions may or may not be at minimum phase. The corresponding impulse response is then obtained by taking the inverse Laplace transform of the transfer function. The impulse response of the minimum-phase case rises faster initially to its first maximum than the nonminimum-phase counterparts. This result confirms that, for the same cw magnitude response, the accumulative energy contained in the impulse response is the greatest when the transfer function is at minimum phase. Physical meaning of the energy content is also discussed.

Simplification of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transfer function is performed in the bilinear-transformed domain via approximating tangent phase function by a Padé technique. A modification has been given to the transformed transfer function so that the reduced <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transfer function has the same zero initial responses as the original system. The simplified <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -transfer function is always stable and minimum phase.

This paper illustrates the use of Prony signal analysis tools to derive a system model of the thyristor controlled series capacitor (TCSC) located at the Bonneville Power Administration's (BPA's) Slatt 500 KV substation near Arlington Oregon. The Slatt TCSC is part of EPRI's flexible AC transmission systems (FACTS) initiative. The analysis presented in this paper was carried out to find linear models for the TCSC system using Prony analysis and to determine the range of validity for these linear models. Prony analysis is performed on output data files generated by simulations using BPA's transient stability program (TSP) on the TCSC and surrounding system under both closed and open loop TCSC operation. TSP simulations of the power system are performed using a 2623 bus 455 machine stability model of the western power grid. Root loci of the open loop and closed loop identified systems were generated and the motion of the system's transfer function poles was compared to those obtained by analyzing the system responses for various TCSC modulation controller gains. Comparisons are also made to the system (via TSP simulations) and transfer function responses in the time domain for both the open and closed loop identified transfer functions. The results shown here illustrate the advantages of closed loop identification and the nonlinearity of the system.