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Aalborg Universitet Fault Tolerant Control View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by VBN Aalborg Universitet Fault Tolerant Control: A Simultaneous Stabilization Result Stoustrup, Jakob; Blondel, V.D. Published in: IEEE Transactions on Automatic Control DOI (link to publication from Publisher): 10.1109/TAC.2003.822999 Publication date: 2004 Document Version Tidlig version også kaldet pre-print Link to publication from Aalborg University Citation for published version (APA): Stoustrup, J., & Blondel, V. D. (2004). Fault Tolerant Control: A Simultaneous Stabilization Result. IEEE Transactions on Automatic Control, 49(2), 305-310. https://doi.org/10.1109/TAC.2003.822999 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. ? Users may download and print one copy of any publication from the public portal for the purpose of private study or research. ? You may not further distribute the material or use it for any profit-making activity or commercial gain ? You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us at [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from vbn.aau.dk on: December 25, 2020 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 2, FEBRUARY 2004 305 [2] , “A generalized entropy criterion for Nevanlinna–Pick interpola- and isolation block in the control system. Whenever a fault is detected tion with degree constraint,” IEEE Trans. Automat. Contr., vol. 46, pp. and isolated, a supervisory system takes action, and modifies the struc- 822–839, June 2001. ture and/or the parameters of the feedback control system. In contrast, [3] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1992. in the passive fault tolerant control approach, a fixed compensator is [4] B. A. Francis, A Course in Control Theory. New York: Springer- designed, that will maintain (at least) stability if a fault occurs in the Verlag, 1987, Lecture Notes in Control and Information Sciences. system. [5] J. W. Helton and O. Marino, Classical Control Using Methods. This note will only discuss the passive fault tolerant control ap- Philadelphia, PA: SIAM, 1998. [6] H. Kwakernaak, “Robust control and -optimization-Tutorial proach, also sometimes referred to as reliable control. This approach paper,” Automatica, vol. 29, no. 2, pp. 255–273, 1993. has mainly two motivations. First, designing a fixed compensator can [7] D. J. N. Limebeer and B. D. O. Anderson, “An interpolation theory ap- be made in much simpler hardware and software, and might thus be ad- proach to controller degree bounds,” Linear Alg. Applicat., vol. missible in more applications. Second, classical reliability theory states 98, pp. 347–386, 1988. that the reliability of a system decreases rapidly with the complexity [8] R. Nagamune, “A shaping limitation of rational sensitivity functions with adegree constraint,” IEEE Trans. Automat. Contr., vol. 49, pp. of the system. Hence, although an active fault tolerant control system 296–300, Feb. 2004. might in principle accomodate specific faults very efficiently, the added [9] , “Robust control with complexity constraint: A Nevanlinna–Pick complexity of the overall system by the fault detection system and the interpolation approach,” Ph.D. dissertation, Dept. Math., Royal Inst. supervisory system itself, might in fact sometimes deteriorate plant re- Technol., Stockholm, Sweden, 2002. [10] , “A robust solver using a continuation method for Nevanlinna–Pick liability. interpolation with degree constraint,” IEEE Trans. Automat. Control, In [10], a fault tolerant control problem has been addressed for sys- vol. 48, pp. 113–117, Jan. 2003. tems, where specific sensors could potentially fail such that the corre- [11] J. L. Walsh, Interpolation and Approximation by Rational Functions in sponding outputs were unavailable for feedback, whereas other outputs the Complex Domain. Providence, RI: AMS, 1956, vol. 20. were assumed to be available at all times. [12] K. Zhou, Essential of Robust Control. Upper Saddle River, NJ: Pren- tice-Hall, 1998. In [11, Sec. 5.5], the question of fault tolerant parallel compensation has been discussed, i.e., whether it is possible to design two compen- sators such that any of them alone or both in parallel will internally stabilize the closed loop system. The existence results given in [10] and [11] can be considered to be special cases of the main results of this note. Fault Tolerant Control: A Simultaneous In this note, we shall consider systems for which any sensor (or in the Stabilization Result dual case any actuator) might fail, and we wish to determine for which systems such (passive) fault tolerant compensators exist. The main re- Jakob Stoustrup and Vincent D. Blondel sults state that the only precondition for the existence of solutions to this fault tolerant control problem is just stabilizability from each input and detectability of the system from each output. Abstract—This note discusses the problem of designing fault tolerant Throughout this note, pm shall denote the set of proper, real- compensators that stabilize a given system both in the nominal situation, rational functions taking values in gpm,andpm shall denote as well as in the situation where one of the sensors or one of the actuators the set of strictly proper, real-rational functions taking values in gpm. has failed. It is shown that such compensators always exist, provided that pm the system is detectable from each output and that it is stabilizable. The rI shall denote the set of stable, proper, real-rational functions pm proof of this result is constructive, and a worked example shows how to taking values in g . The notation fs PCI X f@sAaHg will design a fault tolerant compensator for a simple, yet challenging system. be used as shorthand for zeros of f@ A on the positive real line. The A family of second order systems is described that requires fault tolerant lim f@sAaH compensators of arbitrarily high order. set includes the point at infinity if s3I . For matrices eY fY gY h of compatible dimensions, the expression Index Terms—Controller order, fault tolerant control, sensor faults, si- multaneous stabilization. e f q@sAa gh I. INTRODUCTION will be used to denote the transfer function q@sAag@ss eAIf C The interest for using fault tolerant controllers is increasing. A h. Real-rational functions will be indicated by their dependency of a number of theoretical results as well as application examples has now complex variable s (as in q@sAYu@sA), although the dependency of s been described in the literature; see, e.g., [1]–[9] to mention some of will be suppressed in the notation (as in qY u), where no misunder- the relevant references in this area. standing should be possible. The approaches to fault tolerant control can be divided into two main classes: Active fault tolerant control and passive fault tolerant control. II. PROBLEM FORMULATION In active fault tolerant control, the idea is to introduce a fault detection Consider asystem of the form x a ex C fu Manuscript received May 9, 2003; revised September 26, 2003. Recom- mended by Associate Editor F. M. Callier. yI a gIx J. Stoustrup is with the Department of Control Engineering, Institute of . Electronic Systems, Aalborg University, DK-9220 Aalborg, Denmark (e-mail: . [email protected]). V. D. Blondel is with the Department of Mathematical Engineering, Uni- yp a gpx (1) versité Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium (e-mail: n m [email protected]). where x PYu P Yyi PYiaIFFFYp and eY fY giYi a Digital Object Identifier 10.1109/TAC.2003.822999 IFFFYpare matrices of compatible dimensions. Each of the p measure- 0018-9286/04$20.00 © 2004 IEEE 306 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 2, FEBRUARY 2004 ments yiYiaIY FFFYp, is the output of a sensor, which can potentially We will need the following result (see [13, Th. 5.2, p. 106] or [11, fail. Cor. 6, p. 118]) on the strong stabilization problem, i.e., the problem of In this note, we will determine whether it is possible to design a finding a stable stabilizing compensator. feedback compensator that is guaranteed to stabilize a given system, Lemma 1: Let e@sAYf@sA be stable proper transfer functions. Then in case any sensor could potentially fail. To be more precise, we are there exists a stable proper transfer function @sA such that the function u a u@sAyY u Pmp looking for a dynamic compensator , with e@sACf@sA@sA the property, that each of the following feedback laws: is a unit in the ring of stable proper rational functions, if and only if u a u@sAyua u@sAyfYIY Yua u@sAyfYp e@zipA has constant sign for all zip Pfs PCI X f@sAaHg. yI H yI y y y P P P IV. MAIN RESULTS y a yfYI a Y YyfYp a (2) . In this section, we will present our main results which state that for yp yp H systems with several outputs, it is always possible to find a compen- are internally stabilizing, i.e., that both the nominal system as well as sator, that both stabilizes the nominal situation, as well as the situation each of the systems resulting from one of the sensors failing are all where any of the sensors fails. In a similar fashion, it is shown, that it stabilized by u@sA.
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