Synthesis of absolutely stabilising controllers

Although the dynamic characteristics of the baroreflex system have been described by baroreflex transfer functions obtained from open-loop analysis, the predictability of time-series output dynamics from input signals, which should confirm the accuracy of system identification, remains to be elucidated. Moreover, despite theoretical concerns over closed-loop system identification, the accuracy and the predictability of the closed-loop spontaneous baroreflex transfer function have not been evaluated compared with the open-loop transfer function. Using urethane and α-chloralose anaesthetized, vagotomized and aortic-denervated rabbits (n = 10), we identified open-loop baroreflex transfer functions by recording renal sympathetic nerve activity (SNA) while varying the vascularly isolated intracarotid sinus pressure (CSP) according to a binary random (white-noise) sequence (operating pressure ± 20 mmHg), and using a simplified equation to calculate closed-loop-spontaneous baroreflex transfer function while matching CSP with systemic arterial pressure (AP). Our results showed that the open-loop baroreflex transfer functions for the neural and peripheral arcs predicted the time-series SNA and AP outputs from measured CSP and SNA inputs, with r2 of 0.8 ± 0.1 and 0.8 ± 0.1, respectively. In contrast, the closed-loop-spontaneous baroreflex transfer function for the neural arc was markedly different from the open-loop transfer function (enhanced gain increase and a phase lead), and did not predict the time-series SNA dynamics (r2; 0.1 ± 0.1). However, the closed-loop-spontaneous baroreflex transfer function of the peripheral arc partially matched the open-loop transfer function in gain and phase functions, and had limited but reasonable predictability of the time-series AP dynamics (r2, 0.7 ± 0.1). A numerical simulation suggested that a noise predominantly in the neural arc under resting conditions might be a possible mechanism responsible for our findings. Furthermore, the predictabilities of the neural arc transfer functions obtained in open-loop and closed-loop conditions were validated by closed-loop pharmacological (phenylephrine and nitroprusside infusions) pressure interventions. Time-series SNA responses to drug-induced AP changes predicted by the open-loop transfer function matched closely the measured responses (r2, 0.9 ± 0.1), whereas SNA responses predicted by closed-loop-spontaneous transfer function deviated greatly and were the inverse of measured responses (r, −0.8 ± 0.2). These results indicate that although the spontaneous baroreflex transfer function obtained by closed-loop analysis has been believed to represent the neural arc function, it is inappropriate for system identification of the neural arc but is essentially appropriate for the peripheral arc under resting conditions, when compared with open-loop analysis.

This chapter studies the robust synthesis problem for strictly positive real (SPR) transfer functions. The concepts of SPR regions and weak SPR regions are introduced. By using the complete discrimination system (CDS) for polynomials, complete characterization of the (weak) SPR regions for transfer functions in coefficient space is given. It is shown that the weak monic SPR region associated with a fixed polynomial is bounded and the intersection of several weak monic SPR regions associated with different polynomials cannot be a single point. Furthermore, we show how to construct a point in the SPR region from a point in the weak SPR region. Based on these theoretical development, we propose an algorithm for robust design of SPR transfer functions. This algorithm works well for both low-order and high-order polynomial families. Especially, the derived conditions are necessary and sufficient for robust SPR design of polynomial segment or low-order (n ≤ 4) interval polynomials. Illustrative examples are provided to show the effectiveness of this algorithm.

There are some different statements for necessary and sufficient conditions of strict positive realness in the control literatures. These statements are usually extracted by using state space approaches. In this paper, unlike the other works the frequency domain tools is used to prove the necessary and sufficient conditions for test strict positive realness property of real rational transfer function. The first main advantages of this approach is develop a new sufficient condition for strict positive realness and the second is prove necessary and sufficient conditions for strict positive realness property of real rational transfer function directly by using original definition in the frequency domain and without any restriction on the relative degree of transfer function by default. In this paper, the original definition of strict positive realness in the frequency domain is interpreted as a mapping from S-plane to G(s)-plane. Then the maximum modulus principle and the Taylor expansion which are the fundamental tools in the complex analysis is used to extract a new sufficiency condition for strictly positive real (SPR) transfer functions. This geometry interpretation implies a new more accurate definition for SPR transfer functions. Finally more completed and simplified version of necessary and sufficient conditions for strict positive realness is proved based on the complex analysis

In system identification and adaptive control, the problem of designing strictly positive real (SPR) transfer functions in the presence of uncertain adaptation parameters is essential for stability and convergence in a group of parameter adaptation algorithms. This paper proposes a convex optimization approach to address the robust SPR problem. Besides achieving the robust SPR condition, the presented solution is optimal in the sense of minimizing the distance from the transfer function to unity. Such consideration is important for parameter convergence in practical applications. New topics such as minimum-order compensator and minimum high-frequency magnitude design are also introduced.Copyright © 2012 by ASME

Necessary and sufficient conditions for strictly positive realness of discrete-time interval functions, which may be associated in parameter space by a multidimensional boxed domain, are provided. Positive realness of interval functions are studied, and both necessary and sufficient conditions are provided for such functions to be strictly positive real (SPR) in the presence of interval variations of the function's parameters corresponding to a Kharitonov-type box domain. The number of checking functions that have to be tested for SPR property in order to establish the interval SPR property grows with the order of the interval function. These tests give both necessary and sufficient conditions for SPR. The approach presented for interval functions does not require the checking of all the vertices for unit circle positive realness. >

A unified theory of identification in a feedback system is presented from the viewpoint of inverse problem on reactor noise. A power reactor described by a 4 block feedback model is expressed in terms of equivalent correlation functions as a 5 block feedback model with an additional loop, if one of subsystems of the reactor becomes unstable, however, the total system is stable by negative feedback effects, and if noise sources are not independent. The addition of a loop in the feedback system leads errors in estimation of open loop transfer functions. In the case, modified open loop transfer functions are identified instead of original ones. From the viewpoint of reactor diagnosis, notes on system identification are discussed on an open loop transfer function in a low power reactor and on effects of diagonalization in covariance space of noise sources.

Self–excited combustion oscillations arise from a coupling between unsteady combustion and acoustic waves, and can cause structural damage to many combustion systems. Active control provides a way of extending the system's stable operating range by interrupting the damaging thermoacoustic interaction. The active controller considered injects some fuel unsteadily into the burning region, thereby altering the heat–release rate, in response to an input signal detecting the oscillation. Although the feasibility of such a control configuration was demonstrated on laboratory–scale experiments over 15 years ago, the triple requirement for full–scale applications is to adapt the controller response to varying operating conditions and to guarantee that the controller will cause no harm, while relying as little as possible on a particular combustion model. As a first step towards meeting these requirements, a self–tuning regulator (STR) is proposed in this paper for a class of combustion systems that satisfy some fairly non–restrictive assumptions. An open–loop transfer function from the controller output voltage driving the actuator to the pressure signal detecting the oscillation is derived, and is shown to be the product of a time delay τ tot and a transfer function that satisfies certain structural properties. Using the open–loop transfer function, it is shown that the stabilizing STR structure is that of a simple phase–lead compensator if τ tot=0, and the same compensator combined with a Smith controller if τ tot=0. The adaptive rule for the adjustment of the STR parameters is derived from a Lyapunov stability analysis. Promising results are obtained on a simulation based on a nonlinear premixed flame model and on an experimental set–up which consists of a laminar flame burning in a Rijke tube.

Identification of transfer function matrices in a feedback system is discussed by using the singular value decomposition of Hankel matrix from the viewpoint of inverse problems. A method of model reduction is considered, and selection criteria are proposed for identification of them. Transformation formula between open loop and closed loop transfer function matrices are determined from the feedback loop structure, and they are needed for identification of open loop transfer function matrices under such a condition where the feedback system is in a minimum phase. Though the identifiability of open loop transfer function matrices can be examined in the framework of innovation model equivalent to the feedback system, there are pole-zero cancellations in the identification of them. The method to reduce a model order of an open loop transfer function is discussed by using the singular value decomposition of a gramian given by the open loop transfer function with higher degree. To check reliability of the present algorithm, a simulation study is performed for an example.

No AccessEngineering NoteAdaptive Suppression of Linear Structural Vibration Using Control Moment GyroscopesQuan Hu, Yinghong Jia and Shijie XuQuan HuSchool of Astronautics, Beihang University, 100191 Beijing, People's Republic of China, Yinghong JiaSchool of Astronautics, Beihang University, 100191 Beijing, People's Republic of China and Shijie XuSchool of Astronautics, Beihang University, 100191 Beijing, People's Republic of ChinaPublished Online:25 Apr 2014https://doi.org/10.2514/1.62267SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations About References [1] D'Eleuterio G. M. T. and Hughes P. C., "Dynamics of Gyroelastic Continua," Journal of Applied Mechanics, Vol. 51, No. 2, June 1984, pp. 412–422. JAMCAV 0021-8936 Google Scholar[2] D'Eleuterio G. M. T. and Hughes P. C., "Dynamics of Gyroelastic Spacecraft," Journal of Guidance, Control, and Dynamics, Vol. 10, No. 4, 1987, pp. 401–405. doi:https://doi.org/10.2514/3.20231 JGCDDT 0162-3192 LinkGoogle Scholar[3] Peck M., "Practicable Gyroelastic Technology," 27th Annual AAS Guidance and Control Conference, AAS Paper 2004-023, Feb. 2004. Google Scholar[4] Damaren C. J. and D'Eleuterio G. M. T., "Controllability and Observability of Gyroelastic Vehicles," Journal of Guidance, Control, and Dynamics, Vol. 14, No. 5, 1991, pp. 886–894. doi:https://doi.org/10.2514/3.20728 JGCDDT 0162-3192 LinkGoogle Scholar[5] Damaren C. J. and D'Eleuterio G. M. T., "Optimal Control of Large Space Structures Using Distributed Gyricity," Journal of Guidance, Control, and Dynamics, Vol. 12, No. 5, 1989, pp. 723–731. JGCDDT 0162-3192 LinkGoogle Scholar[6] Yang L. F., Mikulas M. M. and Park K. C., "Slewing Maneuvers and Vibration Control of Space Structures by Feedforward/Feedback Moment-Gyro Controls," Journal of Dynamic Systems, Measurement, and Control, Vol. 117, No. 3, 1995, pp. 343–351. doi:https://doi.org/10.1115/1.2799125 JDSMAA 0022-0434 CrossrefGoogle Scholar[7] Shi J. F. and Damaren C. J., "Control Law for Active Structural Damping Using a Control Moment Gyro," Journal of Guidance, Control, and Dynamics, Vol. 28, No. 3, 2005, pp. 550–553. doi:https://doi.org/10.2514/1.11269 JGCDDT 0162-3192 LinkGoogle Scholar[8] Sobel K., Kaufman H. and Mabus L., "Adaptive Control for a Class of MIMO Systems," IEEE Transactions on Aerospace, Vol. 18, No. 2, 1982, pp. 576–590. CrossrefGoogle Scholar[9] Barkana I., Kaufman H. and Balas M., "Model Reference Adaptive Control of Large Structural Systems," Journal of Guidance, Control, and Dynamics, Vol. 6, No. 2, 1983, pp. 112–118. JGCDDT 0162-3192 LinkGoogle Scholar[10] Balas M., "Direct Model Reference Adaptive Control in Infinite-Dimensional Linear Spaces," Journal of Mathematical Analysis and Applications, Vol. 196, No. 1, 1995, pp. 153–171. doi:https://doi.org/10.1006/jmaa.1995.1403 JMANAK 0022-247X CrossrefGoogle Scholar[11] Barkana I. and Ben-Asher J. Z., "Simple Adaptive Control Applications to Large Flexible Structures," Journal of Guidance, Control, and Dynamics, Vol. 34, No. 6, 2011, pp. 1929–1932. doi:https://doi.org/10.2514/1.54217 JGCDDT 0162-3192 LinkGoogle Scholar[12] Livneh R. and Slater G. L., "Some Conditions for Strictly Positive Real and Almost Strictly Positive Real Structures," IEEE International Conference on Systems Engineering, Vol. 1, IEEE Publications, Piscataway, NJ, 1990, pp. 81–85. Google Scholar[13] Mehiel E. A. and Balas M. J., "Determining Strictly Positive Realness from System Modal Characteristics," Journal of Guidance, Control, and Dynamics, Vol. 30, No. 5, 2007, pp. 1539–1542. doi:https://doi.org/10.2514/1.28240 JGCDDT 0162-3192 LinkGoogle Scholar[14] Hu Q., Jia Y. and Xu S., "Recursive Dynamics Algorithm for Multibody Systems with Variable-Speed Control Moment Gyroscopes" Journal of Guidance, Control, and Dynamics, Vol. 35, No. 5, 2013, pp. 1388–1398. JGCDDT 0162-3192 LinkGoogle Scholar[15] Damaren C. J., "Optimal Control Formulation for Lightly Damped, Gyroelastic Continua," M.S. Thesis, Inst. for Aerospace Studies, Univ. of Toronto, Downsview, ON, Canada, 1987. Google Scholar[16] Wen J. T., "Time Domain and Frequency Domain Conditions for Strict Positive Realness," IEEE Transactions on Automatic Control, Vol. 33, No. 10, 1988, pp. 988–992. doi:https://doi.org/10.1109/9.7263 IETAA9 0018-9286 CrossrefGoogle Scholar[17] Anderson B. D. O. and Vongpanitlerd S., Network Analysis and Synthesis, Prentice–Hall, Englewood Cliffs, NJ, 1973, pp. 215–220. Google Scholar[18] Broussard J. R. and O'Brien M. J., "Feedforward Control to Track the Output of a Forced Model," IEEE Transactions on Automatic Control, Vol. 25, No. 4, 1980, pp. 851–853. doi:https://doi.org/10.1109/TAC.1980.1102409 IETAA9 0018-9286 CrossrefGoogle Scholar Previous article Next article

Strictly positive real (SPR) transfer functions are of importance in Adaptive systems theory. This paper addresses two issues related to finding a polynomial <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(s)</tex> such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(s)/a(s)</tex> is SPR for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a(s)</tex> belonging to a set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-degree Hurwitz polynomials <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\P}^n</tex> defined by the magnitude bounds on the coefficients of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a(s)</tex> . It is shown that irrespective of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> there exist four polynomials in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\P}^n</tex> such that if the ratios of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(s)</tex> with each of these is SPR then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(s)/a(s)</tex> is SPR for all members of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{\P}^n</tex> . A sufficient condition for the existence of this <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b(s)</tex> along with a method for its construction is outlined. The condition resembles the Routh array test for determining system stability.

The strict positive realness of transfer functions is a very crucial property to guarantee the global closed-loop asymptotic stability of such feed-forward systems under wide classes of nonlinear eventually time-varying controllers satisfying a certain Popov s type inequality. The stability is guaranteed for any feedback controller belonging to one such a class. Such a strong asymptotic stability property is known as the asymptotic hyperstability property. The paper investigates the strong / strict positive realness property of systems subject to internal point commensurate and incommensurate constant delays provided that the transfer function of their delay-free counterparts are strong/strict positive real.

This paper describes a prototype model of a phase-stabilized backward wave oscillator (BWO). The BWO is locked to a harmonic of a crystal oscillator by employing an automatic phase control loop. Tuning is accomplished by pulling the crystal oscillator and switching different crystals into the oscillator circuit. Power spectrum plots are taken in open-and closed-loop conditions, and methods of analyzing these plots are described. With the proper loop bandwidth, the BWO will assume the frequency stability of the reference oscillator. The phase-locked BWO is analyzed by a conventional feedback control technique. Each element in the loop is assigned a transfer function. From this the open-loop transfer function is determined. It is then shown that the improvement in short-term frequency stability is directly proportional to the open-loop transfer function and, hence, the open-loop gain. Utilizing plots of the absolute magnitude and phase angle of the open-loop gain vs. frequency, conclusions are drawn as to the maximum amount of gain which is possible while still maintaining system stability. Possible implementation of a series equalizer is considered in conjunction with this. Comments are also made on the relationship of the closed-loop bandwidth to the system lock-in range and system response to incremental changes in frequency.

Frequency domain conditions for strictly positive real (SPR) functions which appear in literature are often only necessary or only sufficient. This point is raised in [1], [2], where necessary and sufficient conditions in the s -domain are given for a transfer function to be SPR. In this note, the points raised in [1], I2] are clarified further by giving necessary and sufficient conditions in the frequency domain for transfer functions to be SPR. These frequency-domain conditions are easier to test than those given in the s -domain or time domain [1], [2].

Improved Decentralized PID Controller design for MIMO Processes

Sufficient conditions are derived for the existence of a fixed coefficients polynomial c*(s) such that (c*(s))/(a(s)) is a strictly positive real (SPR) transfer function, where a(s) belongs to the set of interval stable polynomials. The design procedure is described, when these conditions are satisfied, using an illustrative example. >