The paper deals with the developing method of determination of Lyapunov functions for a generalized linear dynamical object. This method is based on the shift and the rotation coordinate transformation which translates a motion of the considered object in a new virtual state space. One can perform such sort of transformation by using partial fraction decomposition. It is easy to define Lyapunov function in a new state space and then do inverse transformation. The proposed method can be used for a determination of signed Lyapunov functions, which can be used as basis for the analysis of system dynamic and the synthesis of desired motions.
Nowadays Lyapunov functions are widely used to solve different control problems. One can find these functions very usable while synthesis and analysis problems are being solved. The great interest to Lyapunov functions can be explained by their unique properties and strong mathematical background which allows getting mathematically valid results. One can easily use these results while optimization problems are formulated for various dynamical systems.
Although one can find a lot of publications in recent scientific periodicals which describe definitions of nonquadratic Lyapunov functions [1], quadratic forms are still commonly used for defining candidate to Lyapunov function [2-4]. This fact can be simply explained by physical meaning of these quadratic functions which have an energetic background and show redundant energy of dynamical object.
Due to Lyapunov’s theorem about stability of motion corresponding Lyapunov function must satisfy Sylvester criterion [5]. Since, there is an infinity number of quadratic functions which satisfy this criterion, their definition is a nontrivial problem of the control theory. This problem is solved by using Riccati and/or Lyapunov equations, which in common case depend on some cost function and can be solved only numerically [6,7].
In order to avoid above-mentioned drawbacks of Lyapunov function’s we suggest define them by developing analytical method for defining form and coefficients of Lyapunov function only as functions on parameters of the considered object.
Our paper is organized as follows: first of all we consider the transformation of a generalized linear object into parallel form. Secondly we define Lyapunov function for the transformed object. Thirdly, we perform inverse transformation and write down Lyapunov function which depends only on parameters and coordinates of dynamical object. Lastly, we show the example of using proposed approach and make a conclusion.
II.
A. Representation of object’s dynamic in parallel way
Let us consider a linear single-input dynamical object which dynamic is given as follows
n
sx j = ∑ bij xi + mnU , (
i= where s = d / dt is a derivative operator, xi , x j are state variables, U is a control input, bij ,mn are coefficients, n is an order of dynamical object.
M = (0 0
X = (x1
coefficients, and coefficients x2 mn )T sX = BX + MU , xn )T
is a state space vector, is a n-th sized vector of input
b11 b21 B = bn1 b12 b22
bn2 b1n b2n .
bnn
W(s) = (sE − B)−1M ,
One can find a matrix transfer function of the control
object (
We assume that this matrix B has only real eigenvalues.
In this case its characteristic polynomial can be written thus
(
Now we suggest to simplify transfer function (
W(s) = ∏n 1 i =1(s + λi )
A ,
A = adj(sE − B)M , here adj(sE − B) is the matrix which is adjunct to matrix sE − B
It is clear that first cofactor of expression (
, n n i−1 ∑α i = 0; i∑=1α i j∑=1λ j + i=1 n ∑λ j = 0 j =1+1
Expression (9) is obtained by using only characteristic
polynomial (
W(s) = ∑n α i A . (10)
i=1(s + λi )
Thereby, we replace the series transfer function (
B. Direct and inverse coordinate transformations Apart from above-mentioned calculation advantage, the proposed approach has significant methodological values. One can find these methodological benefits while performing stability analysis and considering energy transformation. In this case the proposed approach allows us to perform some coordinate transformation from normal phase space to some virtual one and in such a way simplify Lyapunov function.
y11 y21 Y = yn1 y12 y22 yn2 y1n y2n
ynn (11) and set the following interrelation (12) (13) (14) (15) (16) n x j = ∑ y ji ,
i =1 y ji = W ji (s)U , here ai is the i-th component of matrix A ,
W ji (s) = α j ai / (s + λ j ).
Expression (12) allows us make the following statement. Statement 1. For state variable x j inverse coordinate transformation from virtual phase space into normal is performed by summing all relevant virtual space variable
Now we consider determination of y ji coordinates while direct transformation is being performed.
Let us take into account equation (15) and complete equation (12) with its first n-1 derivatives. In such a way we get a n-th order system of linear equations with n unknown variables y ji
n Lkf x j = ∑ Lkf y ji , k = 0,...,n − 1 ,
i =1 where Lkf x j ,Lkf y ji are k-th order Lie derivatives.
Solution of this system for unknown virtual state variable y ji allows us define them in such a way
n y ji = ∑γ kji xk + κ jiU ,
k =1 where γ kji ,κ ji are some numbers.
Statement 2. Direct and inverse transformations are described with simple algebraic expressions and it is defined with some family of shift and rotation transformations.
That is why the above-given transformation can be use for simplification of a dynamical system description and performing some actions like stability analysis.
C. Stability analysis
It is clearly understood that one can use expressions (12)
and (16) for stability analysis. This analysis after performing
transformation (
Statement 3. Considered dynamical object has stable dynamic on condition that every parallel channels has stable dynamics as well. In this case we can claim that all components of vector Y are bounded.
One can perform stability analysis for each channel in a different way. The simplest one is analysis of λi eigenvalues. The more complex one is based on usage of Lyapunov functions. In spite of its complexity Lyapunov function allows us not only to do stability analysis but consider energy conversion while dynamical object is operating, also define algorithms and structure of controller for the considered object.
D. Lyapunov function dedetrmination
The simplest Lyapunov function is the following quadratic expression 2
V ji = k ji y ji , where k ji is a positive number.
So, while analysis of stability is being performed, one can use expression (12) and determine the following Lyapunov function
n V j = ∑ k ji y 2ji . (18)
i =1
The function (18) can be written down as matrix expression
Positiveness of j-th Lyapunov function (19) means stability of j-th channel dynamic.
One can substitutes interrelation (16) info function (18) and write down following expression for Lyapunov function in real state variables
n n 2 V j = ∑ k ji ∑γ kji xk + κ jiU . (24)
i=1 k =1 Lyapunov function (24) is a positive function as well due to positivenes of function (18).
One can open brackets in function (24) and transform it into modification of well-known quadratic Lyapunov function wii = kijγ kji , i = 1,...,n; wi(n+1) = kijκ ji ; wij = 2k jiγ kjiγ mji , i,m = 1,...,n; wij = 2k jiγ kjiκ ji , (i = n)or( j = n) (28)
The function (25) is a j-th component of matrix Lyapunov function
V = (V1 V2 Vn ) , (29) which describes redundant energy stored in each channel.
One can use Lyapunov function defined in such a way for stability analysis and design of closed-loop control system. Now let us consider example of defining Lyapunov functions for the speed and current loops of DC motor.
III.
Let us consider a dynamic of DC motor given by the following equations sx1 = b12 x2 ; sx2 = b21x1 + b22 x2 + m2U , (30) where coefficients aij are defined thus
1 1 1 JR L b12 = Tm ; b21 = b22 = − Te ; m2 = Te ;Tm = c2 ;Te = R here J is a rotor inertia, R is a armature resictance, c is a back-emf constant, L is an armature inductance, y1 , y2 are
Tm > 4Te . (32)
In this case both of eigenvalues of the characteristic polynomial , (31)
D(s) = s 2 − b22 s − b12b21 . are negative (17) (19) (20) (21) (33) (34) (36) (37) (38) (40) where here
V j = ZWZT , Z = (x1 w11 W = w(n+1)1 x2 xn
U ),
w1(n+1)
,
w(n+1)(n+1)
(25)
(26)
(27)
(
A = (m2b12 m2s)T W(s) = m2b12 s 2 − b22 s − b12b21
m2 s T .
s 2 − b22 s − b12b21 Now we transform transfer function (35) into the form (10)
A α1 α2 s2 − b22s − b12b21 =
+ s − λ1 s − λ2 The matrices (39) allow us write the following equations sy11 = λ1 y11 +α 11U ; sy12 = λ2 y12 +α 21U ; sy21 = λ1 y21 +α 12U ; sy22 = λ2 y22 +α 22U .
These equations allow us rewrite interrelations (12) and (16) between real and virtual coordinates if the output variable is the variable x1 and if the output variable is the variable x2 x1 = λ1 − b22 y21 + λ2 − b22 y22 + α 12 +α 22 − m2 U . (42) b21 b21 b21
We consider expression (41) and (42) as inverse coordinate transformation. The expressions for the direct transformation can be obtained as solution equations (41) and (42) for state y21 = y22 =
One can use coordinates (43) and (44) to determine Lyapunov functions. The simplest ones can be written down for the speed loop + 2w13 x1U + 2w23 x2U + w33U 2 , where w11 = w22 = λ12 + λ22 ; w12 = − b12 (λ1 + λ2 ) ; (λ1 − λ2 )2 (λ1 − λ2 )2
2b122 (λ1 − λ2 )2 ; w23 = 2(α 11 +α 21 )2 2(λ1 + λ2 )(α 11 +α 21 ) .
w33 = (λ1 − λ2 )2 ; w13 = (λ1 − λ2 )2 Lyapunov function for the current loop can be defined in a similar way but it has different coefficients
(λ2 - b2(λ21)2−+λ(2λ)12- b22 )2 ; w12 = − b21(λ2 - b22 ) − b21(λ1 - b22 ) ;
(λ1 − λ2 )2 (λ1 − λ2 )2 w13 = - 2b21 (α 12 +α 22 - m2 ) ; w33 = 2(α 12 +α 22 − m2 )2 (λ1 − λ 2 )2 (λ1 − λ 2 )2 ; (λ1 + λ 2 - 2b22 )(α 12 +α 22 - m2 ) . w23 = (λ1 − λ 2 )2 If one takes into account matrices (39) and performs analysis of coefficients (46) and (47), it still can be possible define that coefficients w13 , w23 , w33 in expression (46) are equal (43) (44) (45) (46) (47) to zero and coefficients w13 ,w23 ,w33 in expression (47) can be simplified as follows
(λ2 - b2(λ21)2−+λ(2λ)12- b22 )2 ; w33 = (λ1 − λ2 )2 ; w23 =
Statement 5. One should define Lyapunov function (25) in an extended n+1-th order space state with space vector (26) if output variable is described with the differential equation which contains control input. It can be defined in a normal nth order state space otherwise.
The proposed approach based on decomposition of transfer function of linear dynamical object with elementary fractions can be used for simulation of considered object, the study of its dynamics and synthesis of its control system. This approach simplifies mathematical model of a linear object and transform this model into some virtual state space. The mentioned transformation allows us define Lyapunov function in an easy way. This function can be defined for both object and linear closed-loop control system. In this case its coefficients depend only on parameters of dynamical system.