79 Defining of Lyapunov Functions for the Generalized Linear Dynamical Object Roman Voliansky1, Oleksander Sadovoi1, Yuliia Sokhina1, Nina Volianska2 1. Department of Electrotechnic and Electromechanic, Dniprovsk State Technical University, Ukraine, Kamyanske, 2 Dniprobudivska str., email: voliansky@ua.fm 2. Heat Transfer Department, Dniprovsk State Technical University, Ukraine, Kamyanske, 2 Dniprobudivska str., email: ninanin@ua.fm Abstract: The paper deals with the developing method function only as functions on parameters of the considered of determination of Lyapunov functions for a generalized object. linear dynamical object. This method is based on the shift Our paper is organized as follows: first of all we consider and the rotation coordinate transformation which the transformation of a generalized linear object into parallel translates a motion of the considered object in a new form. Secondly we define Lyapunov function for the virtual state space. One can perform such sort of transformed object. Thirdly, we perform inverse transformation by using partial fraction decomposition. It transformation and write down Lyapunov function which is easy to define Lyapunov function in a new state space depends only on parameters and coordinates of dynamical and then do inverse transformation. The proposed object. Lastly, we show the example of using proposed method can be used for a determination of signed approach and make a conclusion. Lyapunov functions, which can be used as basis for the analysis of system dynamic and the synthesis of desired II. USAGE OF PARALLEL MODEL FOR motions. Keywords: stability analysis, Lyapunov function, SIMULATION AND ANALYSIS OF DYNAMICAL coordinate transformation, dynamical system, partial SYSTEM fraction decomposition. A. Representation of object’s dynamic in parallel way I. INTRODUCTION Let us consider a linear single-input dynamical object which dynamic is given as follows Nowadays Lyapunov functions are widely used to solve n ∑ bij xi + mnU , different control problems. One can find these functions very usable while synthesis and analysis problems are being sx j = (1) solved. The great interest to Lyapunov functions can be i= explained by their unique properties and strong mathematical where s = d / dt is a derivative operator, xi , x j are state background which allows getting mathematically valid variables, U is a control input, bij , mn are coefficients, n is results. One can easily use these results while optimization problems are formulated for various dynamical systems. an order of dynamical object. Equations (1) can be rewriting into matrix form Although one can find a lot of publications in recent scientific periodicals which describe definitions of non- sX = BX + MU , (2) quadratic Lyapunov functions [1], quadratic forms are still where X = (x1 x2  xn T ) is a state space vector, commonly used for defining candidate to Lyapunov function [2-4]. This fact can be simply explained by physical meaning M = (0 0  mn ) is a n-th sized vector of input T of these quadratic functions which have an energetic coefficients, and B is a n-th sized square matrix of background and show redundant energy of dynamical object. coefficients  b11 b12  b1n  Due to Lyapunov’s theorem about stability of motion   corresponding Lyapunov function must satisfy Sylvester  b21 b22  b2 n  B= . (3) criterion [5]. Since, there is an infinity number of quadratic        functions which satisfy this criterion, their definition is a b  nontrivial problem of the control theory. This problem is  n1 bn 2  bnn  solved by using Riccati and/or Lyapunov equations, which in One can find a matrix transfer function of the control common case depend on some cost function and can be object (1) in an easy way by using equation (2) solved only numerically [6,7]. W(s ) = (sE − B )−1 M , (4) In order to avoid above-mentioned drawbacks of Lyapunov here E is an identity matrix. function’s we suggest define them by developing analytical We assume that this matrix B has only real eigenvalues. method for defining form and coefficients of Lyapunov In this case its characteristic polynomial can be written thus ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 80 n n D(s ) = det (sE − B ) = ∏ (s + λi ) , (5) x j = ∑ y ji , (12) i =1 i =1 where λi are the eigenvalues of matrix B . where Now we suggest to simplify transfer function (4) in the y ji = W ji (s )U , (13) following way here ai is the i-th component of matrix A , W(s ) = ∏ n 1 A, (6) ( ) W ji (s ) = α j ai / s + λ j . (14) i =1 (s + λi ) Expression (12) allows us make the following statement. where Statement 1. For state variable x j inverse coordinate A = adj (sE − B )M , (7) here adj (sE − B ) is the matrix which is adjunct to matrix transformation from virtual phase space into normal is performed by summing all relevant virtual space variable sE − B y ji . It is clear that first cofactor of expression (6) contain n-fold multiplication of elementary fractions. This multiplication Now we consider determination of y ji coordinates while can be replaced with a sum of some elementary fractions as direct transformation is being performed. follows [8] Let us take into account equation (15) and complete n 1 n αi equation (12) with its first n-1 derivatives. In such a way we ∏ =∑ , (8) i =1 (s + λi ) i = 1 + λi ) (s get a n-th order system of linear equations with n unknown where variables y ji n n  i −1 n  n ∑ α i = 0; ∑ α i  ∑ λ j + ∑ λ j  = 0 Lkf x j = ∑ Lkf y ji , k = 0 ,...,n − 1 , (15) i =1 i =1  j =1 j = 1+ 1  i =1  (9) where Lkf x j , Lkf y ji are k-th order Lie derivatives. n  i −1 n  Solution of this system for unknown virtual state variable ∑ ai  ∏ λ j  ∏ λ j  = 1. y ji allows us define them in such a way i =1  j =1 j =i +1  n Expression (9) is obtained by using only characteristic y ji = ∑ γ kji xk + κ jiU , (16) polynomial (5) and it is independent of selected component k =1 of matrix A . This fact allows to rewrite matrix transfer where γ kji , κ ji are some numbers. function (6) thus Expressions (12) and (16) allow us claim the following. n αi W(s ) = ∑ A. (10) Statement 2. Direct and inverse transformations are i = 1(s + λi ) described with simple algebraic expressions and it is defined Thereby, we replace the series transfer function (6) where with some family of shift and rotation transformations. the calculation can be performed only by using consecutive That is why the above-given transformation can be use for calculations with the parallel one (10) which can be simplification of a dynamical system description and calculated in a parallel way. One of the benefits of such an performing some actions like stability analysis. approach is increasing of the calculation speed while parallel C. Stability analysis simulation is implemented. It is clearly understood that one can use expressions (12) B. Direct and inverse coordinate transformations and (16) for stability analysis. This analysis after performing Apart from above-mentioned calculation advantage, the transformation (8) comes down to analysis of stability every proposed approach has significant methodological values. transfer function (14). This fact allows us formulate the One can find these methodological benefits while following statement. performing stability analysis and considering energy Statement 3. Considered dynamical object has stable transformation. In this case the proposed approach allows dynamic on condition that every parallel channels has stable us to perform some coordinate transformation from dynamics as well. In this case we can claim that all normal phase space to some virtual one and in such a way components of vector Y are bounded. simplify Lyapunov function. One can perform stability analysis for each channel in a Let us consider these transformations in detail. different way. The simplest one is analysis of λi eigenvalues. First of all, we define square matrix Y as follows The more complex one is based on usage of Lyapunov  y11 y12  y1n  functions. In spite of its complexity Lyapunov function   allows us not only to do stability analysis but consider energy  y 21 y 22  y 2 n  Y= (11)     conversion while dynamical object is operating, also define    algorithms and structure of controller for the considered y   n1 y n 2  y nn  object. and set the following interrelation ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 81 D. Lyapunov function dedetrmination wii = kij γ kji , i = 1,..., n; wi (n +1) = kijκ ji ; The simplest Lyapunov function is the following quadratic wij = 2k jiγ kjiγ mji , i , m = 1,..., n; (28) expression V ji = k ji y 2ji , (17) wij = 2k jiγ kjiκ ji , (i = n )or ( j = n ) The function (25) is a j-th component of matrix Lyapunov where k ji is a positive number. function So, while analysis of stability is being performed, one can V = (V1 V2  Vn ) , (29) use expression (12) and determine the following Lyapunov which describes redundant energy stored in each channel. function One can use Lyapunov function defined in such a way for n stability analysis and design of closed-loop control system. V j = ∑ k ji y 2ji . (18) Now let us consider example of defining Lyapunov functions i =1 for the speed and current loops of DC motor. The function (18) can be written down as matrix expression III. EXAMPLE USE FOR PROPOSED APPROACH V j = Y j K j YTj , (19) Let us consider a dynamic of DC motor given by the where following equations Y j = ( y1 y2  yn ) , (20) sx1 = b12 x2 ; sx2 = b21 x1 + b22 x2 + m2U , (30)  k j1 0  0  where coefficients aij are defined thus    0 k j2  0  1 1 1 JR L K j = b12 = ; b21 = b22 = − ; m2 = ;Tm = 2 ;Te = , (31)    . (21)   Tm Te Te c R    0 0  k jn  here J is a rotor inertia, R is a armature resictance, c is a  back-emf constant, L is an armature inductance, y1 , y 2 are Quadratic form (19) depends only on diagonal matrix (21). It is quite clear that determinant of matrix (21) and its DC rotor speed and current respectively. diagonal minors can be easily defined thus We assume that the rotor inertia has significant value and the following condition is true ( ) n det K j = ∏ k ji (22) Tm > 4Te . (32) i =1 In this case both of eigenvalues of the characteristic k polynomial M k = ∏ k ji , k = 1, ,n . (23) i =1 D(s ) = s 2 − b22 s − b12 b21 . (33) Analysis of expressions (22) and (23) allows us to are negative formulate the following statement. Statement 4. According to Sylvester criterion [6] Lyapunov λ1,2 = 0.5b22 ± 0.5 4b12b21 + b22 2 . (34) function (19) is positive if k ji coefficients are positive. This fact allows us represent dynamic DC motor in form (7) Positiveness of j-th Lyapunov function (19) means stability  x (s ) / U (s )  W(s ) =  1 of j-th channel dynamic. A  = , (35) One can substitutes interrelation (16) info function (18)  2 x (s ) / U (s )  s 2 − b22 s − b12b21 and write down following expression for Lyapunov function where in real state variables 2 A = (m2 b12 m2 s )T (36) n  n  V j = ∑ k ji  ∑ γ kji xk + κ jiU  . (24) or i =1  k =1  T   W(s ) =  Lyapunov function (24) is a positive function as well due to m2 b12 m2 s  . (37)  s2 − b s − b b 2 − −  positivenes of function (18).  22 12 21 s b22 s b 12 21  b One can open brackets in function (24) and transform it Now we transform transfer function (35) into the form (10) into modification of well-known quadratic Lyapunov A α1 α2 function = + , (38) s − b22 s − b12 b21 s − λ1 s − λ2 2 V j = ZWZT , (25) where where T T Z = (x1 x2  xn U ) ,  b m m2λ1  −b m − m2λ2  (26) α1 =  12 2  ; α2 =  12 2  . (39)     w11  w1(n + 1)   λ1 − λ2 λ1 − λ2   λ1 − λ2 λ1 − λ2    The matrices (39) allow us write the following equations W=    , (27) sy11 = λ1 y11 + α 11U ; sy12 = λ2 y12 + α 21U ;  w(n + 1)1  w(n + 1)(n + 1)  (40)   sy 21 = λ1 y 21 + α 12 U ; sy 22 = λ2 y 22 + α 22 U . here ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic 82 These equations allow us rewrite interrelations (12) and (16) to zero and coefficients w13 , w23 , w33 in expression (47) can between real and virtual coordinates if the output variable is be simplified as follows the variable x1 w11 = 2 2b21 ; w22 = (λ2 - b22 )2 + (λ1 - b22 )2 ; λ1 λ2 α 11 + α 21 x1 = y11 + y12 ; x2 = y11 + y12 + U (41) (λ1 − λ2 ) 2 (λ1 − λ2 )2 b12 b12 b12 b (λ - b ) b21 (λ1 - b22 ) - 2m2b21 and if the output variable is the variable x2 w12 = − 21 2 22 − ; w13 = ; (48) (λ1 − λ2 ) 2 (λ1 − λ2 ) 2 (λ1 − λ2 )2 x2 = y 21 + y 22 ; 2m22 - m2 (λ1 + λ2 - 2b22 ) λ1 − b22 λ2 − b22 α 12 + α 22 − m2 (42) w33 = ; w23 = . x1 = y 21 + y 22 + U. (λ1 − λ2 ) 2 (λ1 − λ2 )2 b21 b21 b21 We consider expression (41) and (42) as inverse coordinate This fact allows us formulate following statement. transformation. The expressions for the direct transformation Statement 5. One should define Lyapunov function (25) in can be obtained as solution equations (41) and (42) for state an extended n+1-th order space state with space vector (26) if variables y11 , y12 , y21 , y21 output variable is described with the differential equation which contains control input. It can be defined in a normal n- - λ2 x1 - x2b12 + U (α 11 + α 21 ) th order state space otherwise. y11 = ; (43) λ1 - λ2 IV. CONCLUSION - x2b12 + λ1 x1 + U (α 11 + α 21 ) y12 = The proposed approach based on decomposition of transfer λ1 - λ2 function of linear dynamical object with elementary fractions b x + (- λ2 + b22 )x2 − (α 12 + α 22 - m2 )U can be used for simulation of considered object, the study of y21 = 21 1 ; λ1 - λ2 its dynamics and synthesis of its control system. This (44) - b21 x1 + (λ1 - b22 )x2 + (α 12 + α 22 - m2 )U approach simplifies mathematical model of a linear object y22 = . and transform this model into some virtual state space. The λ1 - λ2 mentioned transformation allows us define Lyapunov One can use coordinates (43) and (44) to determine function in an easy way. This function can be defined for both Lyapunov functions. The simplest ones can be written down object and linear closed-loop control system. In this case its for the speed loop coefficients depend only on parameters of dynamical system. V1 = y11 2 + y12 2 = w11 x12 + 2 w12 x1 x2 + w22 x22 + REFERENCES (45) + 2 w13 x1U + 2 w23 x2U + w33U , 2 [1] Hu Tingshu, et.al Non-quadratic Lyapunov functions for where performance analysis of saturated systems, Proceedings λ12 + λ22 − b (λ + λ ) of 44th IEEE conference on Decision and Control, w11 = ; w12 = 12 1 2 2 ; (λ1 − λ2 ) 2 (λ1 − λ2 ) pp.8106-8111, 2005 [2] T.L. Vu , K. Turitsyn Lyapunov Functions Family 2 2b12 2b12 (α 11 + α 21 ) Approach to Transient Stability Assessment IEEE w22 = ; w23 = ; (46) (λ1 − λ2 ) 2 (λ1 − λ2 )2 Transactions on Power Systems, Volume: 31, Issue: 2, pp 1269 — 1277, 2015 2(α 11 + α 21 )2 2(λ1 + λ2 )(α 11 + α 21 ) [3] A. Bacciotti, L. Rosier, Liapunov functions and stability w33 = ; w13 = . (λ1 − λ2 )2 (λ1 − λ2 )2 in control theory. Berlin: Springer, 2005, 237p. Lyapunov function for the current loop can be defined in a [4] M. Malisoff, F. Mazenc, Constructions of strict Lyapunov similar way but it has different coefficients Functions. London: Springer, 2009, 386p. [5] G.Giorgio, Various proofs of the Sylvestr criterion for w11 = 2 2b21 (λ2 - b22 )2 + (λ1 - b22 )2 ; ; w22 = quadratic forms, Journal of Mathematical Researches, (λ1 − λ2 )2 (λ1 − λ2 )2 (47) Vol. 9(6), pp.55-66, 2017 b21 (λ2 - b22 ) b21 (λ1 - b22 ) [6] T. Penzi, Numerical solution of generalized Lyapunov w12 = − − ; equations, Advanced in Comp. Math., Vol.8 pp.33-48, (λ1 − λ2 )2 (λ1 − λ2 )2 1898 - 2b21 (α 12 + α 22 - m2 ) 2(α 12 + α 22 − m2 )2 [7] W.F. Arnold and A.J.Laub Generaliztion eigenproblem w13 = ; w33 = ; (λ1 − λ2 )2 (λ1 − λ2 )2 algorithms and software for algebraic Riccati equations, Proceeding of the IEEE, Vol. 72, pp1746-1754, 1984 (λ + λ2 - 2b22 )(α 12 + α 22 - m2 ) . w23 = 1 [8] R. Witula and D. Slota Partial fractions decompositions of (λ1 − λ2 )2 some rational functions, Appl. Math. Comput., Vol. 197, If one takes into account matrices (39) and performs analysis pp.328-336, 2008. of coefficients (46) and (47), it still can be possible define that coefficients w13 , w23 , w33 in expression (46) are equal ACIT 2018, June 1-3, 2018, Ceske Budejovice, Czech Republic