Asymptotic Stability Analysis of 2-D Discrete State Space Systems with Singular Matrix

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1 Asymptotic Stability Analysis of 2-D Discrete State Space Systems with Singular Matri GUIDO IZUTA Department of Social Infmation Science Yonezawa Women s Juni College Toi Machi, Yonezawa, Yamagata JAPAN izuta@yone.ac.p Abstract: - The aim of this paper is to establish, on the basis of Lagrange method f solving partial difference equations, conditions under an asymptotic stability analysis procedure to investigate conditions f the eistence of a solution to 2-D (two dimensional) discrete system whose state space representation is composed by a non-singular matri. To accomplish it, the concept of generalized inverse of matrices and Jdan canonical transfmation are applied on the iginal system and then Lagrange solutions to the transfmed systems are pursued. Once the conditions are determined on the grounds of the transfmed system and the eistence conditions of solutions f this system is accomplished, the conditions f the iginal system is obtained by back transfmation. A numerical eample is given to show how the procedure wks. Key-Wds: - 2-D systems, analysis, asymptotic stability, Lagrange method 1 Introduction The pioneering research on the stability of 2-D discrete systems described by the state space state space model paralleling the dinary 1-D discrete system model that appears in engineering goes back to the early 1960s when the concept of multivariable positive real functions was introduced to study the stability of electrical netwks with varying parameters [1]. This framewk was then further etended and generalized to allow one to investigate me methodically the stability of these kinds of systems with two as well as me indices [2]. In this scope, basically two state space models, which are equivalent to each other, stood out in automatic control, digital signal processing and other related fields of engineering as frame of references f studying the stability and control design of 2-D discrete systems; namely, FM [3] and Roesser [4] models. As far as the tools f delving into the stability of 2-D discrete systems are concerned, z-transfm was, and still is to some etent the most widely adopted fmalism [5] [6]. As f the dinary 1-D systems case, this approach has provided a wide range of techniques and apparatus f testing the stability of systems. However, when there are multiple state variables composing the state space model that depend on the same inde, say i, the analysis is in general neither straightfward eplicitly achievable. To cope with it the energy methods [7], which is essentially a generalization of Lyapunov based procedures developed in the ambit of 1-D systems, was intensively used in the 1980s. These in turn eventually evolved into linear matri inequalities methods in the 1990s and early 2000s (see f eample [8] and references therein). Powerful mechanisms in their own rights, they provide only sufficient conditions to inspect the stability of systems. In der to deal with this shtcoming, Izuta [9-12] have investigate the solutions to 2-D discrete control systems of Roesser type from the Lagrange solution method perspective. Briefly, the philosophy is to transfm the system matri into a diagonal matri by means of control feedback design and then determine the stability conditions under which the overall feedback control systems is asymptotically stable in the sense of Lagrange method f solving partial difference equations. This scheme makes it clear not only the role of the eigenvalues of the system matri but also provides an eplicit solution, when it eists. Yet, recently the auths [13] focused on 2-D systems with singular system matri and established asymptotic stability conditions f the case in which the eigenvalues of the non-singular matri block composing the iginal system matri are all mutually different. Motivated by the investigations so far, this paper aims to etend the previous paper [13] in the sense that, we aim here to establish an analysis method to check the eistence of a solution. In addition, unlike considering the similarity transfmation, the Jdan E-ISSN: Volume 12, 2017

2 canonical fm is eploited to embrace a larger class of matri blocks. In a few wds, the iginal system is re-arranged in a suitable fmat such that there will be a non-singular matri block at the upper left cner of the system matri; then, Jdan canonical transfmation is applied on the subsystem defined by this matri block. Thus since this procedure allows us to establish the conditions f the eistence of a Lagrange solution to the transfmed sub-system, this solution, if any, will in turn lead to an asymptotically stable solution to the iginal system. Finally, this paper is ganized as follows. Section 2 states the definitions relied on in the sequel, section 3 writes down the mathematical tools needed to unfold the suggested theetical reasoning, and section 4 presents the results, and section 5 shows a numerical eample. 2 Preliminaries and Problem In this section the definitions and concepts required hereafter are presented. Firstly, consider the system as follows. Definition 1. Let the 2-D discrete system be described by the following real valued state space model with indices being natural numbers. 1(i + 1, ) A nn 1(i, ) k (i + 1, ) a 11 a 1n = k (i, ) (1) k+1 (i, + 1) a n1 a nn k+1 (i, ) n (i, + 1) n (i, ) rank (A nn ) = p, 1 p n where the over-braced A n n stands f sht notation of the matri under it. Secondly, the definition of stability given net is the multidimensional version of the one used to study systems of difference equations [14]. Definition 2. System (1) is asymptotically stable as far as its solutions * (i, ) s (f all * ) fulfill the following condition. lim (i+) (i, ), f all = 1,, n (2) Thirdly, the theetical framewk relies on the Lagrange method f solving partial difference equations, which consists basically in defining candidate solutions and then testing them f stability conditions. If those are satisfied then the candidates are in fact a solution to the system. This cresponds to the eponential functions that are assumed as solutions to partial differential equations in continuous settings. Definition 3. A general fm of a non-null Lagrange candidate solution is given by the following equation. n (i, ) = k=1 I k α i k β k (3) I k s stand f the initial values and α k s and β k s are non-null real valued numbers. Taking these into consideration, the problem to be tackled in this paper is as follows. Problem. To establish, on the basis of Lagrange method f solving partial difference equations, conditions under which there eists an asymptotically stable solution to 2-D discrete systems. 3 Mathematical Facts In this section, we present the mathematical tools that will be used thereafter. First of all, note that algebra of matrices allows one to rewrite systems of equations with singular matri in such a fm that the top left matri block of the resulting matri is non-singular so that the system described by equation (1) can be written to have its non-singular matri ption epressed as a matri block as follows. uu(ii + 11, + 11) vv (ii + 11, + 11) A AA pp BB p (n p) CC (n p)p CC (n p)p AA 1 uu(ii, ) (4) pp BB p (n p) vv (ii, ) in which A pp is a non-singular matri block composing matri A nn and such that rank AA pp = p, 1 p n holds and the vects are defined as given hereafter. 1(i + 1, ) uu (ii + 11, + 11) r (i + 1, ) r+1 (i, + 1) p (i, + 1) 1(i, ) r (i, ) uu (ii, ) r+1 (i, ) p (i, ) E-ISSN: Volume 12, 2017

3 p+1(i + 1, ) vv (ii + 11, + 11) s (i + 1, ) s+1 (i, + 1) s (i, + 1) p+1(i, ) vv (ii, ) s (i, ) s+1 (i, ) s (i, ) It is wth noting that nevertheless, f the sake of clarity and without loss of generality, the vect u (i+1,+1) has the entries defined by the state variables written sequentially from 1 (i+1,) to r (i+1,) followed by variables from r+1 (i,+1) to p (i,+1), in practice the computations lead in general to a different dering with the state variables appearing in a mied up fashion rather than aggregated as written here. In addition, should it be the case, the state variables composing the entries of vect u (i,) will also follow the cresponding arrangement. The same remarks apply to vects v (i+1,+1) and v (i,). Apropos, another mathematical concept that plays a key role in this paper is that of generalized inverse of matrices [15], which assigned on matri Ā in equation (4) renders straightfwardly the generalized inverse matri G fulfilling the equality ĀGĀ=Ā and whose eplicit epression is given by the following equation. G = A 1 pp 0 p (n p) (5) 0 (n p)p 0 (n p)(n p) On focusing on this generalized inverse matri, system in equation (4) is solvable if and only if the following equality stands up. A G uu(ii + 11, + 11) vv (ii + 11, + 11) = uu(ii + 11, + 11) (6) vv (ii + 11, + 11) From which it is a simple matter of algebraic calculation to reach to the equation given by vv (ii + 11, + 11) = CA 1 uu (ii + 11, + 11) (7) which means that the vect v (i,) is determined as far as the vect u (i,) is established. As a matter of fact, in der to carry out this task, note first that a particular solution to system (4) is accomplished by the following system. G uu(ii + 11, + 11) vv (ii + 11, + 11) = uu(ii, ) (8) vv (ii, ) thus it turns out that u (i,) is computed from uu (ii + 11, + 11) = A uu (ii, ) (9) In other wds, equations (7) and (9) provide a particular solution to system (4), which will allow one to draw conclusions regarding the eistence of asymptotically stable Lagrange solutions. 4 Results In this section, we present the main results of this paper. In fact, they are gathered in the following theem. Theem 1. Consider system (1) and its particular solution given by equations (7) and (9). In addition, assume that matri A in equation (9) can be transfmed into a Jdan canonical fm. Then there eists an asymptotically stable Lagrange solution to system (1) if the initial values of the solutions to the system can be set to meet certain conditions as well as the eigenvalues of matri A are all non-null numbers with absolute values less than unit Proof. Since on solving equation (9) f u (i,), v (i,) is promptly obtained and consequently a solution to system (1) is achieved, we focus specifically on this problem in what follows. F, consider the transfmation applied on (9) given by zz(ii, ) = T 1 uu (ii, ) (10) T: a non-singular matri composed by generalized eigenvects of matri A. This procedure leads to the Jdan canonical fm as follows. T 1 uu (ii + 11, + 11) = T 1 ATT 1 uu (ii, ) (11) which by means of (10) reads J pp zz(ii + 11, + 11) = J 1 zz(ii, ) (12) J t in which matri J pp is square matri of rank p and its composing Jdan block J i is a matri given by J i = λ i 1 J i = λ i (13) λ i 1 λ i (14) E-ISSN: Volume 12, 2017

4 Note that equation (14) has multiple rows and the last one is the diagonal entry alone and the previous entries have always two values: an eigenvalue on the diagonal line and number 1 at the right column. Despite a little simplified but without loss of generality, a solution to (12) is established by basically considering the following cases. Case 1: let J i be given by λ i then the possible combinations are as follows. z s (i + 1, ) = λ s z s (i, ) (15) z t (i, + 1) = λ t z t (i, ) (16) to which the Lagrange candidate solutions are taken to be z s (i, ) = α s i β s (17) and z t (i, ) = K t α t i β t (18) with and K t being the initial conditions. Now substituting (17) into (15) and (18) into (16) lead to and α s i+1 β s from which we obtain = λ s α s i β s K t α t i β t +1 = K t λ t α t i β t (19) (20) α s = λ s, and β t = λ t (21) meaning that (15) and (16) are asymptotically stable as far as λ s and λ t are non-null real numbers with absolute values less than unit and α s and β t are chosen in the same way. Case 2: let J i a non-diagonal Jdan matri with the diagonal entry given by λ i and (12) have the following entries. z s (i + 1, ) = λ s z s (i, ) + z r (i, ) (22) z r (i + 1, ) = λ r z r (i, ) (23) z r (i, + 1) = λ r z r (i, ) (24) From case 1, z r is given by z r (i, ) = K r λ r i β r z r (i, ) = K r α r i λ r Meover, assuming the Lagrange solution equation (22) becomes z s (i, ) = α s i β s (25) (26) (27) α s i+1 β s α s i+1 β s which can be written as = α s i β s = α s i β s (αα s 1)α s i β s (αα s 1)α s i β s + K r λ r i β r + K r α r i λ r = K r λ r i β r = K r α r i λ r (28) (29) (30) (31) and since (30) and (31) must hold f all the values of indices i and, we conclude that we get a solution if we can define non-null α s and β s with absolute values less than unit along with initial conditions such that the following are fulfilled. α s = λ s = 1 + K r, and β s = β r (32) α s = α r = 1 + K r, and β s = λ r (33) Case 3: dual equations to those given in case 2 are given by and z s (i, + 1) = λ s z s (i, ) + z r (i, ) (34) z r (i, + 1) = λ r z r (i, ) (35) z r (i + 1, ) = λ r z r (i, ) (36) As in the previous case, z r and z s are given by z r (i, ) = K r λ r i β r z r (i, ) = K r α r i λ r z s (i, ) = α s i β s (37) (38) (39) Thus, it turns out that equation (34) translates into the following epression. α s i β s +1 = Ks α s i β s α s i β s +1 = Ks α s i β s written the other way (ββ s 1)α s i β s (ββ s 1)α s i β s + K r α r i λ r + K r λ r i β r = K r α r i λ r = K r λ r i β r (40) (41) (42) (43) and analogous to the previous case, the conditions to be satisfied are E-ISSN: Volume 12, 2017

5 α s = α r, and β s = λ r = 1 + K r, (44) α s = λ r, and β s = β r = 1 + K r, (45) All in all, no matter the case, the Lagrange solution to equation (12) and the Jdan canonical transfmation (10) written as u (i,)=tz(i,) yield the solution u (i,), which has the general fmat epressed by the following equation. (i, ) = g I g i λ g β g + h I h i α h λ h (46) in which the terms I s are constant values resulting from the computations. Furtherme, (46) gives a solution to equation (7), and consequently the claim of theem follows. Remark 1: F the sake of simplicity, in cases 2 and 3, only Jdan block of size of 2 was considered. However care should be taken f blocks of greater sizes because, in general, they are solvable only if the entries of the non-diagonal Jdan canonical fm block with 1 s in its off diagonal positions, which are the rows of the Jdan block not including the last one, are given by either z r (i+1,) = λ r z r (i,) + z s (i,) along with z r (i+1,) = λ s z r (i,) + z t (i,), z r (i,+1) = λ r z r (i,) + z s (i,) along with z r (i,+1) = λ s z r (i,) + z t (i,). Yet if z r is the (n-1)-th row of the block, then the n-th row cresponding to z s (i,) can be a vect depending on either inde. 5 Numerical Eample Hereafter we present a numerical eample to show how the analysis procedure wks. Consider a randomly generated system be already written as in equation (4) and given by the following epression. 1 (i + 1, ) 3 (i + 1, ) 2 (i, + 1) = A C 4 (i, + 1) 1 (i, ) B CA 1 3 (i, ) (47) B 2 (i, ) 4 (i, ) in which A = B = (48) C = [ ] Note that matri A has only one eigenvalue (λ), namely λ=0.1, whose algebraic multiplicity is 3 and geometric multiplicity given by the dimension of kernel of matri (A-λI), which is given by the dimension of matri A minus the rank of (A-λI), is 1. Meover the geometric multiplicity says that there is only one Jdan block cresponding to this eigenvalue. Now as far as the eistence of a solution to (47) is concerned, it can be solved if and only if 1 (i, ) 4 (i, + 1) = CA 1 3 (i, ) (49) 2 (i, ) f a particular solution given by 1 (i + 1, ) 1 (i, ) 3 (i + 1, ) = A 3 (i, ). (50) 2 (i, + 1) 2 (i, ) Now, in der to obtain a solution to (50), we consider the linear space basis transfmation given by the following equation. z 1 (i, ) 1 (i, ) z 3 (i, ) = T 1 3 (i, ). (51) z 2 (i, ) 2 (i, ) with T being a matri composed by the generalized eigenvects of matri A and chosen here to be T = Thus, (50) and (51) lead to z 1 (i + 1, ) λ 1 0 z 1 (i, ) z 3 (i, + 1) = 0 λ 1 z 3 (i, ) (52) z 2 (i, + 1) 0 0 λ z 2 (i, ) so, assuming the following Lagrange solution f z 2 (i,) gives hence z 2 (i, ) = K 2 α 2 i β 2 K 2 α 2 i β 2 +1 = λ K 2 α 2 i β 2 (53) (54) β 2 = λ (55) which means that β 2 must be set to be λ, which in turn must be a non-null value of absolute value less than unit. As f z 3 (i,) we have that and letting z 3 (i + 1, ) = λ z 3 (i, ) + z 2 (i, ) (56) z 3 (i, ) = K 3 α 3 i β 3 which with equation (56) provides (57) E-ISSN: Volume 12, 2017

6 K 3 (α 3 λ )α i 3 β 3 = K 2 α i 2 β 2 (58) since this equality must hold f all the indices i and, the cresponding terms must be equal. In other wds, α 3 =α 2, β 3 =β 2. Meover, there is a solution as far as β 3 given by the following equation has absolute value less than unit. α 3 = λ + K 2 K 3 (59) Finally, let us tackle z 1 (i,) which is given by z 1 (i + 1, ) = λ z 1 (i, ) + z 3 (i, ) (60) f which, similarly to the previous cases, the following Lagrange solution is chosen z 1 (i, ) = K 1 α 1 i β 1 Now from (57), (60) and (61) we have K 1 (α 1 λ )α 1 i β 1 = K 3 α 3 i β 3 (61) (62) and following the previous reasoning we conclude that α 1 =α 3, β 1 =β 3 along with α 1 = λ + K 3 K 1 (63) It turns out that since β 1 =β 3 =β 2 and β 3 is established by means of equation (59), the other β s are achieved in a quite natural way. On the other hand, f α s, despite their equality, they must satisfy equations (55) and (63), which lead to K 2 K 3 = K 3 K 1 (64) subect to the following condition in der to yield an asymptotically stable solution. K 2 = K 3 < 1 (65) K 3 K 1 which will hold as far as we can set adequately the initial values, which in practical terms are, if possible, carried out by means of a solution to (47). As a matter of fact, a solution to (52) is promptly accomplished by (53), (57) and (61) all with a common α as well as β followed by substitution into (51) not only to compute a particular solution to (47), but also to pursue 4 (i,) in equation (49). References: [1] H. Ozaki and T. Kasami, Positive real functions of several variables and their applications to variable netwk, IRE Trans. on Circuit They, CT-7, 1960, pp [2] H. G. Ansell, On certain two variable generalizations of circuit they, with applications to netwks of transmission lines and lumped reactances, IRE Trans. on Circuit They, CT-11, 1964, pp [3] E. Fnasini and G. Marchesini, Doubly indeed dynamical systems: State space models and structural properties, Math. Syst. Th., 12, 1978, pp [4] R. Roesser, A discrete state-space model f image processing, IEEE Trans. Automat. Contr., AC-20, 1975, pp [5] E. I. Juri, Stability of multidimensional scalar and matri polynomials, Proc. IEEE, 66, 1978, pp [6] N. K. Bose, Applied multdimensional systems they, Van Nostradand Reinhold Co., [7] W. S. Lu and A. Antoniou, Two-Dimensional Digital Filters, New Yk:Marcel Dekker, [8] G. Izuta, Stability and disturbance attenuation of 2-D discrete delayed systems via memy state feedback controller, Int. J. Gen. Systems, 36-3, 2007, pp [9] G. Izuta, Stability analysis of 2-D discrete systems on the basis of Lagrange solutions and doubly similarity transfmed systems, Proc. 35th conf. IEEE IES, 2010, pp [10] G. Izuta, On the asymptotic stability analysis of a certain type of discrete time 3-d linear systems, Proc. ICINCO 2014, pp [11] G. Izuta and T. Nishikawa, An observer controller design method f 2-D discrete control systems, Proc. IEEE Int. Conf. on Infmation and Automation, 2015, pp [12] G. Izuta, Eistence conditions of asymptotically stable 2-D feedback control systems on the basis of block matri diagonalization, Proc. ICINCO 2016, pp [13] G. Izuta, Asymptotic stability of partial difference equations systems with singular matri, Proc. 2 nd APAS, 2017, in press. [14] A. J. Jerri, Linear difference equations with discrete transfm methods, Kluwer Acad. Pub., [15] C. R. Rao, and S. K. Mitra, Generalized inverse of matrices and its applications, Wiley, [16] N. E. Mastakis, I. F. Gonos, and M. N. S. Swamy, Stability of multidimensional systems using genetic algithms, IEEE Transactions on Circuits and Systems I: Fundamental They and Applications, 2003, pp , Volume: 50, Issue: 7 [17] N. E. Mastakis; M. N. S. Swamy, A new method f computing the stability margin of two-dimensional continuous systems, IEEE Transactions on Circuits and Systems I: E-ISSN: Volume 12, 2017

7 Fundamental They and Applications, pp , Year: 2002, Volume: 49, Issue: 6 [18] N. E. Mastakis, New necessary stability conditions f 2-D systems, IEEE Transactions on Circuits and Systems I: Fundamental They and Applications, pp , Year: 2000, Volume: 47, Issue: 7 E-ISSN: Volume 12, 2017