Stabilization of Hybrid Systems using a Min-Projection Strategy

Transcription

1 Stabiization of Hybrid Systems usin a Min-Proection Stratey Stefan Pettersson, Bent Lennartson Contro and Automation Lab, Chamers University of Technooy S Götebor, Sweden stp@s2chamersse, b@s2chamersse Abstract This paper describes a method how to stabiize a system consistin of severa subsystems The subsystems are described by noninear modes with different vector fieds The method is denoted min-proection stratey, since the vector fied associated with the smaest (skew) proection is seected for each state Conditions are iven uaranteein (exponentia) stabiity It is aso shown how these conditions can be formuated as a noninear optimization probem, or, for a pre-determined proection matrix, a inear matrix ineuaity (LMI) probem Sidin motions may occur in the basic form of the stratey However, it is shown how this behavior can be avoided by introducin hysteresis around the switch surfaces, sti preservin stabiity of the cosedoop hybrid system Two exampes are iven to motivate and exempify the stratey Keywords: Hybrid systems, Switched systems, Exponentia stabiity, Stabiization, Linear matrix ineuaities, LMIs Introduction This paper addresses issues concernin the stabiization of systems that switches amon different system structures The motivation is the are number of systems controed by different controer structures or contro-aws, see [2, 8, 4, 9, 5] and the references therein The stabiity of the cosed-oop controed system depends in enera of the switchin stratey Switchin between stabe system structures not necessariy impy a stabe cosedoop behavior [3, 4] Contrary, by desinin a proper switch stratey, unstabe system structures can be stabiized [5] Durin the ast decade, severa stabiity resuts of switched and hybrid systems have been proposed in the iterature [3, 6, 3, 5] Unfortunatey, these resuts are not constructive how to desin the switch stratey, but merey ive sufficient conditions to uarantee stabiity of a cosed-oop system usin a proper switchin stratey There exist constructive approaches to desin a switchin stratey The approach proposed by Wicks et a [9] describes how to (asymptoticay) stabiize a system consistin of two inear subsystems for which there exists a stabe convex combination Another approach is iven by Mambor et a [] In this approach, it is assumed that each controer (vector fied) has an associated Lyapunov function for which the enery decreases in a certain reion, and these reions toether cover the state space Furthermore, it is assumed that the correspondin Lyapunov functions are eua at every chane of controer Mambor et a propose a min-switch stratey to satisfy this assumption, which impies that the controer correspondin to the smaest Lyapunov function shoud be seected With the assumptions made, the minswitch stratey impies a stabe cosed-oop system A simiar approach is suested by Bishop [] where the individua Lyapunov functions are normaized to impicity enerate the switchin surfaces Another approach to desin stabiizin contro aws is iven by Xu et a [2] for second-order switched systems Since they restrict themseves to two dimensions, they can ive sufficient as we as necessary conditions for the stabiization of the switched system The ideas of the stratey proposed in this paper oriinay appeared in our work [6] However, we have refined the method to avoid sidin modes, if desirabe Furthermore, conditions uaranteein stabiity is proposed and formuated as a noninear optimization probem, which, for a specific desin parameter, resuts in a inear matrix ineuaity (LMI) probem [2] Two exampes are proposed to iustrate the stratey The detais of the stratey toether with the compete proofs of the stabiity resuts in this paper and a detaied comparison with (some of) the above approaches can be found in [4] The outine of the paper is as foows: The min-proection stratey is proposed in the next section toether with an iustratin exampe Conditions uaranteein stabiity appyin the min-proection stratey are iven in Section 3 The min-proection stratey with hysteresis is described in Section 4 foowed by an exampe Some concusions end the paper

2 i i 2 Desin stratey The probem considered in this paper is as foows: Let the dynamic evoution of the switched system be described by a differentia euation of the form where "!$#&%( and )+*-, The desin task is to seect the index function at each time instant to stabiize from the index set % (in the sense of Lyapunov) the overa system Even thouh the index function miht ust be dependent on, we wi suest a feedback desin stratey where is a function of the state, or a hybrid feedback with memory in the oop to avoid sidin motions described ater on To compicate the situation, it is possibe to restrict a specific vector fied to be seected ony in parts of the state space Therefore, define / by / 2 3-4% 65 3 can be seected in 7! which denotes the set of a vector fieds that are aowed to be seected for the continuous state For a we-formuated probem, it is assumed that the set / is non-empty at every state in the reion of vaidity of the desin, so at east some vector fied can be chosen at each state There may be severa reasons for not aowin a vector fied to be seected at certain states For instance, there may perhaps be reuirements on the cosed-oop system that obviousy are not fufied for certain vector fieds at certain states Furthermore, the seection of a certain vector fied may indirecty ead to a cosed-oop system that does not satisfy the reuirements, which is prevented by considerin it as non-aowabe or forbidden at certain states Physica restrictions may aso impy that certain vector fieds do not exist at certain continuous states It is aso possibe to force a vector fied to be chosen at a certain state by aowin ony that one to be seected This may be usefu in case when it is known a priori that a certain vector fied best satisfies the reuirements in a specific reion The desin probem in this paper can be formuated as foows: For each state, choose one vector fied from / such that the cosed-oop system becomes (exponentiay) stabe (if possibe) 2 Min-proection stratey The proposed desin stratey can be formuated as foows: Let 8 9*-,:;, be a matrix For a specific (in the reion of vaidity), choose the vector fied accordin to the criterion < >=@? A BDCFE GIH JKML@NO JK QP 8 R@ This stratey is denoted the min-proection stratey and is very appeain from an enineerin point of view, since it is easy to appy (as we as understand) The intuition behind the stratey is that the cosed-oop system shoud become stabe if it is aways possibe to seect a vector fied that points in a (proected) direction such that the traectories approach the euiibrium point (see Theorem ), cf Fiure By seectin the vector fied correspondin to the smaest such proection, there is therefore a ood chance to obtain a stabe cosed-oop system PSfra repacements { S$T UWVMX;Y ` ZFa Z [-\^]_ Fiure : The distance P 8 3 Before ivin conditions uaranteein stabiity of the stratey, we iustrate the stratey in the foowin exampe 22 Exampe Consider the probem of switchin between the foowin two inear vector fieds (the same vector fieds as in []): bdc feh )k b$ men Appyin the min-proection stratey, with 8 identity matrix, impies that b c is seected for c po b is seected for c p +k eua to the which resuts in a obay exponentiay stabe cosed-oop system (which can be verified by the stabiity resuts and optimization formuations described ater on in this paper) In fact, the min-proection stratey coincides in this exampe with the time-optima contro to reach the oriin Some typica traectory simuations of the obtained cosed-oop behavior are pictured in Fiure 2 3 Conditions for stabiity There are exampes for which the appication of the minproection stratey resuts in unstabe cosed-oop systems However, the intuition behind the min-proection stratey is that the cosed-oop system shoud resut in a stabe system if it is aways possibe to seect a vector fied that points in a (proected) direction such that the traectories approach the euiibrium point The foowin theorem verifies this fact THEOREM If 8, and there exist a constant r such that for a states (in the reion of vaidity) at east

3 & & 8 o The proof foows by showin (by contradiction) that the conditions in Theorem are vaid [4] 5 32 Linear vector fieds The conditions in Lemma (Theorem ) can be verified by the foowin optimization probem in case of inear vector fieds (sti assumin that a vector fieds are seectabe) repacements c PROBLEM If there is a soution to B^=,+ r subect to r 2 - #, $ / o 8 % ( c # b P 8 * 8, 4%, - b o ( c # r % Fiure 2: Appyin the min-proection stratey eads to a obay exponentiay stabe system some / satisfies P 8 3 o r P then the min-proection stratey resuts in cosed-oop system where the oriin is exponentiay stabe in the sense of Lyapunov In this theorem, it is assumed that the possibe sidin motion dynamics is iven by Fiippov s convex combination definition [7] The proof of this theorem foows then by usin the Lyapunov function 3h P 8, whose time derivative satisfies 3 o r P for a traectories [4] If the conditions in Theorem are satisfied, the traectory converes accordin to 7 o c where c and, and is the initia state If, and J denote the smaest respectivey the arest eienvaue of 8, then it can be shown that the -parameters may be taken as ;c! and " [4] 3 A vector fieds seectabe If a vector fieds may be seected in the entire reion of vaidity, then the foowin emma uarantees that the minproection stratey resuts in a stabe cosed-oop system # %$ LEMMA If 8 ( c, %, such that # and ) c, and there exist constants r and # P 8 * r P o () then the min-proection stratey resuts in a cosed-oop system where the oriin is exponentiay stabe in the sense of Lyapunov then the min-proection stratey resuts in a cosed-oop system where the oriin is exponentiay stabe in the sense of Lyapunov The unknown # variabes in this optimization probem are the r and the :s Furthermore, if the matrix 8 is not decided a priori, we have to search for this variabe as we If 8 is decided a priori, there is no need to reuire 8 in the first condition However, if 8 is unknown and it is a part of the desin stratey to seect a 8 such that stabiity is uaranteed (if possibe), then the probem is scaed in such a way that a soution 8 % is souht Without such a scain, the optima soution woud otherwise be obtained for r and infinitey are If 8 is decided a priori, Probem is an LMI probem [2] and can easiy be soved by existin software [8] However, if 8 is not iven # a priori, the compication is that the unknown variabes and 8 are mutipied, impyin that the ineuaity in the second condition is noninear Hence, the probem in this case is not an LMI probem However, it may be soved as a noninear optimization probem Conditions and 2 are euivaent to the noninear constraints / CFA3 8, CFAQ 8 2 CFA - ( c # b P 8 * 8 % o b 2* r % o where CA43 denotes the eienvaues of 3, which are rea numbers since the matrices are symmetric Hence, the optimization probem of findin the arest vaue of the inear obective function r satisfyin noninear ineuaity constraints is a standard optimization probem which for instance may be soved by the routine constr or attoa in the Optimization Toobox by MATLAB [] In practice, the success of findin the optima soution to a noninear optimization probem depends strony on, o %

4 8 & & 8 o whether the initia start variabes are cose to the optima soution or not, since noninear optimization probems in enera are not convex optimization probems (which LMI probems are) The practica conseuence is that even if there is a soution to the optimization probem where a constraints are satisfied, the numerica routines may find a soution correspondin to a oca optimum, which hopefuy satisfies the constraints Any soution satisfyin the constraints in Probem uarantees stabiity appyin the min-proection stratey, which is of primary concern, so it is usuay not crucia if a oca optimum is found instead of the oba one However, it is worse if no soution is found at a 33 Spit optimization probem In the case 8 is not decided a priori, it is possibe to spit Probem in two ess compex optimization probems in such a way that the possibiity to find a vaid soution increases If there exists a soution to Probem, then it is necessary that CA3 # b * % o # ) c is satisfied for some, %, and, where denotes the rea part of a compex number The reason is that 8 is a Lyapunov function for the matrix iven by b ( c # b impyin that the rea part of a eienvaues of this matrix must be (stricty) neative Hence, if no stabe convex combination of matrices b exists, then there is no soution satisfyin the conditions in Probem This impies without oss of eneraity that it is possibe to first search for a stabe convex combination b and, if one is found, then to find a uadratic Lyapunov function for the stabe convex combination The first optimization probem may formay be formuated as (unknown and # :s): PROBLEM 2 BD= + subect to $ #, $, 4%, - ) c # 2 CFA - ) c # b * % o This noninear optimization probem is in enera easier to sove than the previous one, since it does not contain the unknown matrix 8, impyin that the number of unknown variabes is consideraby reduced in case of a hih dimension Note that a soution to Probems and 2 exists if some of the matrices b are stabe The probem of findin a Lyapunov function verifyin stabiity is an LMI probem in the unknown matrix 8, and can be formuated as (unknown 8 and r ): (2) LMI PROBLEM If there is a soution to B^=,+ r subect to r 2 - / o 8 % ( c # b P 8 * 8 b o r % then the min proection stratey resuts in a cosed-oop system where the oriin is exponentiay stabe in the sense of Lyapunov The obtained soution is a soution satisfyin the conditions in Probem If desirabe, the found 8 matrix uarantees stabiity usin the min-proection stratey with an estimate of " ( r if 8 is unknown and restricted by % ) Furthermore, it is aways possibe to use the obtained soution as initia vaues to the oriina optimization probem in Probem to find a better vaue of r, if feasibe It shoud finay be pointed out that there are systems for which it does not exist a stabe convex combination but sti resuts in a stabe cosed-oop system appyin the minproection stratey [4] In this case, stabiity has to be checked afterwards It may for instance be verified by simuations or throuh a more forma anaysis, for instance by searchin for a piecewise (mutipe) uadratic Lyapunov functions usin inear matrix ineuaities [5, 4] 4 Min-proection stratey with hysteresis The direct appication of the min-proection stratey may resut in a cosed-oop system where sidin motions occur, due to the fact that the seected vector fied is ony a function of the continuous state and not the used vector fied The infinitey fast switchins in finite time between the vector fieds and R 3 (or possibe further vector fieds) occur at states satisfyin P 8 3( P 8 KR as a resut of appyin the min-proection stratey if the smaest vector fied proection on each side of this surface points towards it [7, 7] If sidin motions are undesirabe, the vector fied chanes cannot occur exacty at the coincidin switchin surface but have to be adusted such that the switch sets become separated This can be achieved by introducin hysteresis around the surfaces where sidin motions occur, meanin that the seected vector fied wi be a function of both the continuous state and the vector fied around the sidin motion surfaces Hence, a hybrid feedback with memory in the oop is obtained If Lemma is satisfied, it is aways possibe to adust the switch surfaces such that sidin motions are avoided and sti uarantee stabiity of the cosed-oop system, at a cost of havin a ower estimate of the converence rate For

5 r " a vaue r satisfyin P 8 %* r P o r, the reions of states where are satisfied for the different vector fieds wi stricty overap each other, since a boundary state of the reion of states satisfyin P 8 * is an interior point of the set of states satisfy- r P o Hence, by instead switchin r P o,, sidin r P o in P 8 %* vector fieds at the boundary of P 8 3 * that is, at states satisfyin P 8 * r P motions are avoided with the disadvantae that instead of previousy " The arer a vaue of r that can be obtained and the smaer a vaue of r that is accepted, the arer the hysteresis reion becomes A vaue of ony uarantees stabiity However, it is most often r the case that r as we as r are uite conservative estimates, and the traectories usuay converes much faster If it is desirabe to avoid sidin motions, the min-proection stratey is adusted in the foowin way Initiay, the first vector fied to be seected is the one correspondin to the smaest proection Denote this vector fied by The vector fied does not chane at the states where another proected vector fied becomes smaer if sidin motions wi be the resut, but remains the same unti some state satisfyin P 8 3 * r P > is reached, in which case the vector fied correspondin to the smaest proection PSfra repacements is seected This adusted stratey wi be referred to the minproection stratey with hysteresis Since it may be difficut to decide whether sidin motions wi occur in hiher dimensions, a chanes of vector fieds may occur at surfaces P 8 * r P 5 Exampes To iustrate the min-proection stratey and the minproection stratey with hysteresis, consider the probem of stabiizin the cosed-oop system by switchin amon the foowin three individuay unstabe inear vector fieds: b c i;f i i i Q CA3 b c CA3 b$ CA3 b F@ i i; i i There does not exist a stabe convex combination for any two matrices so the system cannot be stabiized by switchin ony between two of the inear systems However, there is a stabe contro-aw switchin between the three inear systems By sovin Probem 2 and LMI probem and usin the obtained # vaues as initia data to Probem, the resut is a soution c # i,, #, and r i@i ii F@ F@ i Hence, appyin the min-proection stratey with this 8 matrix resuts in a obay exponentiay stabe cosed-oop system Sidin motions occur (defined accordin to Fiippov s convex combination definition) but can be avoided if desirabe by appyin the min-proection stratey with hysteresis Traectory simuations in both cases are shown in Fiure 3 Note that a continuous states convere to the oriin - - This impies that Fiure 3: Traectory simuations appyin the min-proection stratey (above) without hysteresis and (beow) with hysteresis for a vaue of!#"%$& $ The min-proection stratey and the min-proection stratey with hysteresis have been inspired by the work of Peeties, DeCaro and Wicks [9, 8], and there are simiarities but aso differences between the proposed desin methods Peeties, DeCaro and Wicks restrict the probem to switchin between two individuay unstabe inear vector fieds, and propose methods to stabiize the cosed-oop system when a stabe convex combination of the inear vector fieds exists We have no theoretica restriction on the number of vector fieds used Our stratey can be appied to inear as we as noninear vector fieds and we obtain estimates of the converence rate in the approach c c

6 6 Concusions This paper has suested a method to stabiize a system consistin of severa subsystems The method is denoted minproection stratey since the vector fied associated with the smaest (skew) proection is seected in each state Conditions uaranteein stabiity of the approach has been iven and it has been shown how to formuate these conditions as a noninear optimization probem, or, for a pre-determined proection matrix 8 as a inear matrix ineuaity (LMI) probem The noninear optimization probem has been spit into two ess compex optimization probems by first searchin for a convex combination of the inear matrices, which is a noninear optimization probem, and then findin the unknown positive definite matrix 8 by sovin an inear matrix ineuaity (LMI) probem It has been shown how sidin motions can be avoided by introducin hysteresis around the switch surfaces Two exampes have been iven to motivate and exempify the stabiization stratey Acknowedements This work has been financiay supported by the Swedish Research Counci for Enineerin Sciences (TFR) under the proect number References [] B E Bishop Lyapunov function normaization for controed switchin of hybrid systems In Proc of the American Contro Conference, paes , 999 [2] S Boyd, L E Ghaoui, E Feron, and V Baakrishnan Linear Matrix Ineuaities in System and Contro Theory SIAM, 994 [3] M S Branicky Stabiity of switched and hybrid systems In Proc of the 33rd IEEE Conference on Decision and Contro, paes , Lake Buena Vista, Forida, 994 [4] W P Dayawansa and C F Martin A converse Lyapunov theorem for a cass of dynamica systems which undero switchin IEEE Transaction on Automatic Contro, 44(4):75 76, 999 [5] R DeCaro, M Branicky, S Pettersson, and B Lennartson Perspectives and resuts on the stabiity and stabiizabiity of hybrid systems Proceedins of the IEEE, 88(7):69 82, 2 [6] M Doğrue and Ü Özüner Stabiity of hybrid systems In IEEE Internationa Symposium on Inteient Contro, paes 29 34, 994 [7] A F Fiippov Differentia Euations with Discontinuous Rihthand Sides Kuwer Academic Pubishers, 988 [8] P Gahinet, A Nemirovski, A J Laub, and M Chiai LMI Contro Toobox, For use with MATLAB The Math Works Inc, 995 [9] D Liberzon and A S Morse Basic probems in stabiity and desin of switched systems IEEE Contro Systems Maazine, 9(5):59 7, 999 [] J Mambor, B Bernhardsson, and K J Åström A stabiizin switchin scheme for muti controer systems In Proc of 3th IFAC, paes F: , 996 [] The Math Works, Inc, 24 Prime Park Way, Natick, MA 76, USA MATLAB Optimization Toobox, 996 [2] A S Morse Contro usin oic-based switchin In Trends in Contro; A European Perspective, paes 69 3 Spriner-Vera, 995 [3] P Peeties and R DeCaro Asymptotic stabiity of m-switched systems usin Lyapunov-ike functions In Proc of the American Contro Conference, paes , Boston, 99 [4] S Pettersson Anaysis and Desin of Hybrid Systems PhD thesis, Contro Enineerin Laboratory, Chamers University of Technooy, 999 [5] S Pettersson and B Lennartson Stabiity and robustness for hybrid systems In Proc of the 35th IEEE Conference on Decision and Contro, paes 22 27, Kobe, Japan, 996 [6] S Pettersson and B Lennartson Controer desin of hybrid systems In Oded Maer, editor, Lecture Notes in Computer Science 2, paes Spriner, 997 [7] V I Utkin Sidin Modes in Contro and Optimisation Spriner-Vera, 992 [8] M Wicks, P Peeties, and R DeCaro Switched controer synthesis for the uadratic stabiisation of a pair of unstabe inear systems European Journa of Contro, 4(2):4 47, 998 [9] M A Wicks, P Peeties, and R A DeCaro Construction of piecewise Lyapunov functions for stabiisin switched systems In Proc of the 33rd IEEE Conference on Decision and Contro, paes , 994 [2] X Xu and P J Antsakis Desin of stabiizin contro aws for second-order switched systems In Proc of 4th IFAC, paes C:8 86, 999