Lyapunov Exponents of Second Order Linear Systems

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1 Reent Researhes in Computational Intelligene and Information Seurity Lyapunov Exponents of Seond Order Linear Systems ADAM CZORNIK and ALEKSANDER NAWRAT Department of Automati Control Silesian Tehnial University Aademia 6, 44- Gliwie POLAND Abstrat: - In this paper we desribe the imal and the imal values of Lyapunov exponents for seond order disrete time invariant linear system perturbed by time varying bounded perturbation. The results may be interpreted in terms of generalized spetral radius and applied to obtained formulas for robust stability problem. Key-Words: - Time varying disrete linear systems, Lyapunov exponents, robust stability, generalized spetral radius Introdution Sine the thesis of Lyapunov [5] more than one hundred years ago appeared Lyapunov exponents have been used to haraterize growth rates of dynamial systems. In partiular, they have been widely applied to stability analysis. Many theoretial properties of this harateristi have been obtained and it has been notied that it is in generally impossible to obtain these uantities analytially. In pratial appliations the parameters of the system are nown only approximately and robust analysis is reuired. In this paper we investigate robust properties of Lyapunov exponents for the seond order disrete time invariant linear system perturbed by time varying perturbation. The deteration of the movability boundary of Lyapunov exponents under various perturbations is an important problem of the modern theory of harateristi exponents. This problem has been solved for ontinuous-time systems for many lasses of perturbations. For example, upper bound for the higher exponent under small perturbations, the soalled entral exponent, was onstruted in [9], p. 4. This problem has also been investigated in [] for ontinuous linear systems with perturbations dereasing at infinity at various rates. Later in [7] perturbations infinitesimal in mean with a weight funtion have been investigated. Similar problems for disrete-time systems have been investigated in [3]. None of the results however enable alulation of the hanges to the Lyapunov exponents aused by the time varying perturbations norm bounded by a given onstant. The disussed above results inlude as a speial ase the problem of robust stability. In this problem we have a system with negative Lyapunov exponents and we would lie to desribe the perturbations that preserve the negativity of Lyapunov exponents. In the past, onsiderable researh efforts were devoted to study of robust stability of disrete linear systems (see, e.g. [], [], [3], [], [3], [3], [3], [3] and the referenes therei. A partiular ase of this problem is the uestion about stability radius. The uestion is how large the perturbation may be without destroying stability assug that the original system is stable. More preisely we are looing for the largest bound r suh that stability is preserved for all perturbations of norm stritly less than r in a given normed perturbation set. This largest bound is alled the stability radius. First time the problem was formally formulated for the ontinuous time invariant system in the famous paper of Hinrihsen and Prithard [9] and sine then many results have been obtained []. For the time invariant system and time invariant perturbations the problem is ompletely solved in the ited papers even in more general formulation. The review of the results in time varying situation may be found in [8]. The available results provide formulas for the stability radii whih are diffiult to alulate and therefore in pratie we have to use numerial methods that give only ertain approximation of the stability radii. Therefore it would be onvenient to have a benhmar to evaluate suh a numerial proedure. The obtained results may serve as suh a benhmar. The main result of this paper is desription of possible hanges of imal Lyapunov exponents of seond order disrete time invariant linear system perturbed by time varying ISBN:

2 Reent Researhes in Computational Intelligene and Information Seurity perturbation matrix whose elements are bounded by given onstant. We olso interpret our result in terms of generalized spetral radius and subradius. Consider the linear disrete time-invariant two dimensional system x( n + ) Ax(, n () where A is a two by two matrix in Jordan anonial form a A () a or a A (3) a where a, a are positive. Let a ( a( ) be a seuene of real numbers. Together with () we onsider the following disturbed system y ( n + ) ( A + Q( ) y(, (4) where (Q ), ( ( ( Q ( ( ( is a bounded seuene of two by two matries suh that ij ( (5) for all i, j, and all n,,.... By denote the Eulidean norm in R and the indued operator norm. The numbers (or the symbol ± ) defined as λ( a) lim sup a( n are alled the harateristi exponent of the seuene ( a( ). For x R, x the Lyapunov exponent λ (x ) of () is defined as the harateristi exponent of ( x( n, x ) ), where ( n, x ) is the solution of () with initial ondition x n x, x ) x, that is λ( x ) limsup x( n, x ). In ( n n the same way we define Lyapunov exponents of (4). It is well nown [4] that the Lyapunov exponents of system () are a, a and the set of Lyapunov exponents of system (4) ontains at most elements, say λ ( A + Q ) λ ( A + Q). In this note we desribe the influene of the perturbation ( Q( ), on the harateristi exponents of (). We will investigate the following ounties λ ( A, λ ( A + Q), λ A, λ ( A + ), ( Q where the ima and ima are taen over all perturbation seuenes ( Q( ) satisfying (5). Main results The following two theorems ontain the main results of this paper. Theorem We have ( A, ) λ a + ( ) + a + + a a 4 if A is given by () and λ ( A, a + + ( ) + ( ) a + a a 4 + if A is given (3). Proof: Consider first the ase when A is given by (). Then (4) has the following form ( a ( ) y( ( y ( ), (6) ( y ( ( a ( ) y ( ), (7) y ( n + ) + + n y n + ) + + ( n [ where y y ( y ( )] T. Consider initial ( n ondition [ ] T where y ) y () y () suh that ( ( ) y y (), ( a a ) a a and define n in the following way n sup{ n N : y( y ( for all,..., n } If n is finite we define n in the following way n sup{ n N : y ( > y ( for all n,..., n }. n nl If we already defined,..., and n l is finite then we define n l + sup{ n N : y( y ( for all n l,..., n } and similarly, if we already defined n,...,nl+ and n l+ is finite then we define n sup { n N : y ( ) y ( l + > ) for all nl+,..., n }. By definition of it is lear that n y ( y ( (8) for,..., n. By the last ineuality and by (7) we obtain ISBN:

3 Reent Researhes in Computational Intelligene and Information Seurity and therefore for for ( + a + y ( ) (9) y( y ( y( y ( y,..., n. By definition of n we have ( + a + () () y ( < y( n,..., n and by (6) we get y ( < y( a + + y( ) n a + + y( n ). From (6) we have y n ) a + y ( n ) + y ( ( ) ) ( n applying (8) leads to y n ) a + + ( ) y ( ) ( n and by () we obtain y n ) a a + n ( )( ) (). ( y Using the last ineuality in ( ) we have n () () y ( < y( a + + * n * ( a + + ( + a + y () (3) for n,..., n. The definition of implies that a a + γ, where ( a a ) γ a + a and therefore by (3) y ( < y( γ y () (4) for n,..., n. Finally () and (4) imply { y (, y ( } γ () y for,..., n. By straightforward indution we an show that the above ineuality is true for all natural, what implies λ ( A, γ. Finally observe that for Q ( we have λ ( A + Q) γ. Consider now the ase when A is given by (3). Then (4) has the following form y ( n + ) a + ( y ( + ( y ( y ( ) (5) ( n + ) + ( ( n )) y ( + ( a + ( ) y (. (6) Consider initial ondition suh that y () y (), where [ y () y ( ] T y ) ) ( a + ( a ) + 4( + ) ( ) a a + and define seuene ( n l ) l N exatly in the same way as previously. By definition of n it is lear that y( y ( (7) for,..., n. By the last ineuality and by (6) we obtain y( y ( (( + + a + y ( ) and therefore y ( y ( + + a + () (( ) ) (8) y for,..., n. By definition of n we have y ( < y ( for n,..., n and by (5) we get y ( < y( a + + y( ) n a + + y( n ). From (5) we have y n ) a + y ( n ) + y ( n ( ) ) ( applying (7) leads to ( n ) a + + y ( n ( ) ) y and by (8) we obtain y ( n ) ( a + + * n (( + ) + a + (). * y Using the last ineuality in (9) we have y ( < y( a + + n + + a + y ( (( ) ) ) * n + * (9) for n,..., n. The definition of implies that a + + ( + ) + + γ a, where ( a a ) + 4( ). γ a + a The ineuality λ ( A, γ we get now in the same way as in the previous ase. Finally observe that for Q ( ISBN:

4 Reent Researhes in Computational Intelligene and Information Seurity we have λ ( A + Q) γ. ( A, The problem of evaluating λ ) seems to be muh harder. Some partiular ases are provided in the next Theorem. Theorem If A is given by () and 4 a a, then we have λ ( A, ) a 3 Conlusion + a + ( a a 4 a a In this paper we have onsidered seond order timeinvariant disrete linear systems with the matrix oeffiient in Jordan anonial form perturbed by oeffiient perturbation suh that absolute values of all elements of the perturbation matries are less or eual a given number. We obtain expliit formulas for the greatest, and in some speial ases for the smallest, possible value of the imal Lyapunov exponents of the perturbed systems. 4 Anowledgment The researh presented here was done as a part of researh and development projet O R 5 and have been supported by The National Centre for Researh and Development funds in the years -. Referenes: [] Amato, M. Mattei, A. Pironti, A note on uadrati stability of unertain linear disretetime systems, IEEE Trans. Automat.Control 43 () (998) [] Amato, M. Mattei, A. Pironti, A robust stability problem for disrete-time systems subjet to an unertain parameter, Automatia, 34 (4) (998) [3] Amato, R. Ambrosino, M. Ariola, A. Merola. A Proedure for Robust Stability Analysis of Disrete-Time Systems via Polyhedral Lyapunov Funtions, Proeedings of the 8 Amerian Control Conferene, Seattle, Washington, USA, June -3, 8 [4] Barreira, Ya. Pesin, Lyapunov exponents and Smooth Ergodi Theory, Univ. Leture Ser., vol. 3, Amer. Math So., Providene, RI,.. [5] A. Berger and Yang Wang, Bouned semigroups of matries, Linear Algebra Appl. 66 (99), -7. [6] V. D. Blondel, J. N. Tsitsilis, When is a pair of matries mortal?, Inform. Proess. Lett., 63 (997), [7] V. D. Blondel, J. N. Tsitsilis, Complexity of stability and ontrollability of elementary hybrid systems, Automatia, 35(999) [8] V. D. Blondel, J. N. Tsitsilis, Boundedness of all produts of a pair of matries is undeidable, Systems and Control Letters vol. 4 () pp. 35,. [9] B. F. Bylov, R. E. Vinograd, D. M. Grobman, and V. V. Nemytsii, Theory of Lyapunov Exponents and Its Appliations to Stability Theory, Mosow, Naua, 966.(Russia [] A. Czorni, On the generalized spetral subradius, Linear Algebra and Its Appliations 47 (-3), pp. 4-48, 5. [] A. Czorni, P. Jurgas, Some properties of the spetral radius of a set of matries, International Journal of Applied Mathematis and Computer Siene, 6; 6(): [] A. Czorni, P. Jurgas, Set of possible values of imal Lyapunov exponents of disrete timevarying linear system, Automatia 44 (), pp , 8. [3] A. Czorni, A. Nawrat, On the Sigma Exponent of Disrete Linear Systems, aepted for publiations in IEEE Transations on Automati Control [4] I. Daubehies, J. C Legarias, Sets of matries all infinite produts of whih onverge, Linear Algebra Appl. 6 (99), [5] I. Daubehies, J. C Legarias, Corrigendum/addendum to: Sets of matries all infinite produts of whih onverge, Linear Algebra Appl. 37 (), [6] L. Elsner, The generalized spetral radius theorem: An analyti-geometri proof, Linear Algebra Appl. (995), [7] G. Gripenberg, Computing the join spetral radius, Linear Algebra Appl. 34 (996), [8] L. Gurvits, Stability of disrete linear inlusion, Linear Algebra Appl. 3 (995), [9] D. Hinrihsen and A. J. Prihard, Stability radii of Linear Systems, Systems and Control Letters, vol. 7, pp. -, 986. [] D. Hinrihsen and A. J. Prihard, Mathematial systems theory. I, vol. 48 of texts in Applied Mathematis, Springer-Verlag, Berlin 5. [] N.A.Izobov, The Higher Exponent of a Linear System with Exponential Perturbations, ISBN:

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