53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
Construction of Lyapunov Functions for Homogeneous Second-Order Systems Francisco Lopez-Ramirez, Tonametl Sanchez and Jaime A. Moreno Abstract— Finding an explicit Lyapunov function for stability analysis of a given dynamical system entails the nontrivial task of solving a partial differential inequality. Although many methods for finding Lyapunov functions are available, much remains to be done in this regard since there isn’t a universal constructive method for finding simple, explicit Lyapunov functions for dynamical systems with stable equilibria. Homogeneity properties of systems may be used to address this problem since they are capable of reducing the complexity of the equations involved. The present work outlines a method to obtain homogeneous Lyapunov functions for homogeneous second-order systems. In comparison with previous results, the method described here provides explicit Lyapunov functions for a larger set of dynamical systems and greatly reduces the sign-definiteness analysis of the underlying equations.
I. INTRODUCTION Lyapunov’s direct method (LDM) is a fundamental control theory tool conceived by A.M. Lyapunov in 1892 to prove the stability of dynamic systems. The method is based on the existence of a positive definite (pd) scalar function V , known as Lyapunov function (LF), such that its time-derivative V˙ along the trajectories of the system is a negative semidefinite (nsd) scalar function. To find an explicit form of V not only allows to conclude stability, it provides direct means to study other dynamic properties such as finite-time stability, transition processes, attraction region estimation, robust control design and reaching-time estimates. Despite the advantages that LDM offers to study dynamic systems, the method is useful only if function V can be found and do not address the problem of finding such function at all. This means that, in the general case, the nontrivial task of solving a partial differential equation (PDE) must be performed in order to find V . Since the second half of the XX century to date, many efforts have been made to design methods for obtaining explicit Lyapunov functions. Zubov’s method [1] and the Variable Gradient Method [2] stand as some of the first and most important methods available. The former relies in a specific form of Partial Differential Equation (PDE), derived from LDM, from which a Lyapunov Function can be determined uniquely by proposing a quadratic pd function; the latter assumes that the Lyapunov function V is the resulting pd gradient of a function g and works by integrating this function along any path joining the origin and the point x. Shortly after these methods were published, another Francisco Lopez-Ramirez (
[email protected]), Tonametl Sanchez (tonametl
[email protected]) and Jaime A. Moreno-P´erez (
[email protected]) are with the Instituto de Ingenier´ıa, Universidad Nacional Aut´onoma de M´exico (UNAM), 04510 Mexico D.F., Mexico.
978-1-4673-6090-6/14/$31.00 ©2014 IEEE
important method appeared, namely Krasovskii’s Method [3], which draws an explicit Lyapunov function provided the system’s Jacobian fulfills the algebraic Lyapunov equation. Although these methods are widely used, the problem of finding simple, explicit Lyapunov functions is far from being closed for many classes of nonlinear systems. In recent years, homogeneity properties have been used to address stability analysis in systems where conventional methods can’t be used [4], [5], [6], [7], [8]. In these works, homogeneity is used either to obtain LF or to directly draw conclusions about the system’s stability, however, none of them exploits the specific properties of homogeneous secondorder systems (HSOS). The purpose of this article is to present a method to obtain homogeneous Lyapunov functions for HSOS. This constructive method is based in LDM and uses homogeneity properties to perform a variable reduction, simplifying thus the underlying equations. This idea has already been presented in [9], nonetheless, the approach presented here uses homogeneity in a different manner as Zubov, leading the method to attain explicit Lyapunov functions for a broader set of systems and to greatly reduce the sign-definiteness analysis of the underlying equations. Moreover, we consider not only systems with degree of homogeneity greater than zero, but also with degree of homogeneity smaller than zero. Although not formalized here, our method may also be extended for discontinuous systems such as (homogeneous) second-order sliding mode algorithms. This fact is illustrated in Example 2 for the Super–Twisting Algorithm [10]. This article is organized as follows. Section II states some preliminaries regarding LDM and homogeneous functions. In III, we briefly present the main idea behind [9] for comparison purposes. We next present our method in IV and illustrate its most significant features with some examples in V. Finally, in Section VI some conclusions are drawn. II. PRELIMINARIES A. Lyapunov’s Direct Method The classical concept of LDM for second-order systems may be stated as follows. Theorem 1: [11][Lyapunov’s Direct Method] Consider the second-order system x˙ 1 = f1 (x1 , x2 ) x˙ 2 = f2 (x1 , x2 ).
(1)
Let x = 0 be an equilibrium point for x˙ = f (x) and D ⊂ R2 be a domain containing x = 0. Let V : D → R and
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W : D → R be continuously differentiable functions such that V (0, 0) = 0 and
V (x1 , x2 ) > 0
in D − {0} (2)
W (x1 , x2 ) ≤ 0 ∂V ∂V V˙ = f1 (x1 , x2 ) + f2 (x1 , x2 ) = W ∂x1 ∂x2 Then, x = 0 is stable. Moreover, if W (0, 0) = 0 and W (x1 , x2 ) < 0 Then, x = 0 is asymptotically stable.
in
This last theorem guarantees the existence of a strict homogeneous Lyapunov function for any continuous, homogeneous system with an asymptotically stable equilibrium point.
(3) in D (4)
D − {0} (5)
A continuously differentiable function V satisfying (2) and (4) is called a Lyapunov function. If furthermore, (5) holds, then V is commonly named a strict Lyapunov function. Further development in Lyapunov’s theory has shown that the stability analysis can be completely based in the monotonic decrease of V [6], removing the continuous differentiability restriction and allowing non-smooth or even discontinuous forms of V [12].
C. Variable Reduction Given the multiplicative scaling behavior of homogeneous functions, it is always possible to express an homogeneous function H : Rn → R by two functions H+ , H− with one variable less. To illustrate this for homogeneous functions in R2 let us choose the family of dilations δεr (x) = (εr1 x1 , εr2 x2 ), which substituted in Definition 2 gives H(εr1 x1 , εr2 x2 ) = εm H(x1 , x2 ) ε > 0. 1
Setting ε = |x1 |− r1 , transforms the last equation into H sign(x1 ),
B. Weighted Homogeneity An homogeneous function is a function that exhibits multiplicative scaling behavior, i.e., if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. Let us state this in a more precise fashion 1 Definition 1: [13][Dilations] Fix a set of coordinates (x1 , ..., xn ) in Rn . Set r = (r1 , ..., rn ) be a n-uplet of positive real numbers. The one-parameter family of dilations (δεr )ε>0 (associated with r) is defined by δεr (x) := (εr1 x1 , ..., εrn xn ), ∀x = (x1 , ..., xn ) ∈ Rn , ∀ε > 0,
(6)
the numbers ri are the weights of the coordinates. Definition 2: [13][Homogeneous Function] A function H : Rn → R is said to be δ r − homogeneous of degree m (m ∈ R) if H(δεr (x)) = εm H(x) ∀x ∈ Rn , ∀ε > 0. (7) Definition 3: [13][Homogeneous Vector Field] A vector field f = [f1 (x), ..., fn (x)]T is said to be δ r -homogeneous of degree k if the component fi is δ r -homogeneous of degree k + ri for each i; that is, fi (εr1 x1 , ..., εrn xn ) = εk+ri fi (x), ∀x ∈ Rn , ∀ε > 0, ∀i ∈ [[1, n]]. (8) Theorem 2: [13][Homogeneous Lyapunov Function] Let f be a continuous vector field on Rn such that the origin is a locally asymptotically stable equilibrium point. Assume that f is δ r -homogeneous of degree k for some r ∈ (0, +∞)n . Then, for any p ∈ N − {0} and any m > p · maxi {ri }, there exists a strict Lyapunov function V , which is δ r homogeneous of degree m and of class C p . As a direct P n ∂V fi (x) is δ r consequence, the time-derivative V˙ = i=1 ∂x i homogeneous of degree m + k. 1 Analogous
(9)
x2 |x1 |
r2 r1
m
= |x1 |− r1 H(x1 , x2 ), x2
By choosing a new variable z = terms, Equation (10) becomes
|x1
r2 | r1
m
H(x1 , x2 ) = |x1 | r1 H± (z),
x1 6= 0.
(10) and rearranging
x1 6= 0,
(11)
where H± (z) indicates the existence of two functions in the new variable z, H+ (z) and H− (z), for the positive and negative part of sign(x1 ), respectively. Note that H± (z) may be homogeneous or not. With this change of variable, the study of the limits limx1 →0+ H+ (z) and limx1 →0− H− (z) becomes relevant. Rearranging these limits in terms of the new variable z gives the following four limits lim
z→−∞
lim
z→+∞
H− (z) m r2
|z| H− (z) |z|
m r2
= L− −∞
,
= L− +∞
,
lim
z→−∞
lim
z→+∞
H+ (z) m
|z| r2 H+ (z) |z|
m r2
= L+ −∞ (12) = L+ +∞ (13)
which, depending on the order 2 of function H± (z) as z → ∞, accept only three cases, namely m 0, if H± (z) ∈ o(|z| r2 ) ± m H (z) ± = L±∞ , if H± (z) ∈ Θ(|z| r2 ) (14) lim m z→±∞ |z| r2 m ±∞, if H± (z) ∈ ω(|z| r2 ) with L± ±∞ ∈ R. 1) Continuity of H along the x2 -axis: From the analysis above, it is clear that for H to be continuous along the x2 + − + axis, L− −∞ = L−∞ and L+∞ = L+∞ . and for this to happen m
H± (z) ∈ O(|z| r2 ).
definitions for homogeneous differential inclusions can be 2 The
found in [4]
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asymptotic notation used here may be consulted in [14].
(15)
2) Positive Definiteness of H: For H(x1 , x2 ) to be a positive function, it is clear from (11) that H± (z) > 0 ⇔ H(x1 , x2 ) > 0.
(16)
The conditions over H± (z) that render H(x1 , x2 ) a pd function are not as straightforward as the previous case. Let us start by assuming that H± (z) > 0 and H± (0) 6= 0. We first consider the case when x2 = 0. From (11), this results in m
H(x1 , 0) = |x1 | r1 H± (0).
(17)
Since H± (0) 6= 0, H(x1 , 0) is a pd function in x1 . Considering next, the case when x1 = 0, we have from (11) in terms of z that m
H(0, x2 ) = |x2 | r2
lim
z→±∞
H± (z) m
|z| r2
.
(18)
±
m
H (z) If H± (z) ∈ o(|z| x2 ), limz→±∞ |z| m/r2 = 0 and H(0, x2 ) = ± 0 regardless of the value of x . 2 However, if H (z) ∈ m r2 Θ(|z| ), Equation (18) becomes m
H(0, x2 ) = |x2 | r2 L± ±∞ , a pd function in x2 . From this considerations we can now state the conditions over H± (z) that render H(x1 , x2 ) pd as m
H± (z) > 0, H± (0) 6= 0, H± (z) ∈ Θ(|z| r2 ) ⇒ H(0, 0) = 0, H(x1 , x2 ) > 0 in D − {0}.
Equations (4) and (22), form a system of differential equations that may be presented in the following matrix form ∂V " # W ∂x1 f1 f2 = , where A = A . r1 x1 r2 x2 ∂V mV ∂x2 (23) Assuming invertibility of matrix A, Equation (23) can be expressed as ∂V " W # ∂x1 1 r2 x2 −f2 = , ∂V r2 x2 f1 − r1 x1 f2 −r1 x1 f1 mV ∂x2 (24) leading to the following equations mf2 r2 x2 W ∂V + V = (25) ∂x1 r2 x2 f1 − r1 x1 f2 r2 x2 f1 − r1 x1 f2 ∂V mf1 r1 x1 W − V =− . (26) ∂x2 r2 x2 f1 − r1 x1 f2 r2 x2 f1 − r1 x1 f2 As we can see, using the generalized Euler vector, the author has accomplished the decoupling of a second order PDE into two separate first order PDE that can be treated as ordinary differential equations (ODE). From here, there are two ways of finding function V : a) Solve both ordinary differential equations (25) and (26) and equal the solutions in order to find the function that satisfies both equations; b) Solve either equation (25) or (26), substitute the solution in the opposite equation and solve it. Using the latter approach we arrive to the general form
(19) R x1
Note that in this conditions, equality between the limits ± L± +∞ and L−∞ is not required. Note also that the signdefiniteness analysis of H, a two-variable function, rests entirely in H± (z), a single-variable function. Note finally that the entire analysis of this subsection may have also been 1 done by choosing ε = |x2 |− r2 .
In the method outlined in [9], the author makes use of the following definition and theorem in order to simplify (4): Definition 4: [13][Euler Vector Field] The generalized Euler vector field associated with the family of dilations (δεr ) is defined by n X ∂ (20) e= ri xi ∂x i i=1 Theorem 3: [13] Let (δεr )ε>0 and e be as in Definition 4. Let V be a function of class C 1 in Rn , and let m ∈ R. Then V is δ r -homogeneous of degree m if, and only if,
r1 x1
∂V ∂V + r2 x2 = mV. ∂x1 ∂x2
a1 (ν,x2 )dν
hZ
x1
−
a2 (ν, x2 )e
Rν
x∗ 1
a1 (η,x2 )dη
dν+
x∗ 1
(27)
where a1 (x1 , x2 ) =
mf2 (x1 , x2 ) r2 x2 f1 (x1 , x2 ) − r1 x1 f2 (x1 , x2 )
and
III. V.I. Z UBOV H OMOGENEITY A PPROACH
e · V = mV. Rewriting (21) for n = 2 gives
V =e i c(x2 ) ,
x∗ 1
(21)
(22)
−r2 x2 W (x1 , x2 ) . r2 x2 f1 (x1 , x2 ) − r1 x1 f2 (x1 , x2 ) Since equation (27) must also satisfy (26) we have dc = c1 (x2 )c + c2 (x2 ) dx2 whose general solution is "Z # R x2 R x2 c1 (ν)dν − xν∗ c1 (η)dη x∗ 2 c(x2 ) = e 2 e c2 (ν)dν + γ . a2 (x1 , x2 ) =
x∗ 2
Constant γ may be uniquely determined by finding the initial conditions, x∗1 and x∗2 such that V (0, 0) = 0. By relying in the Euler vector field, this method imposes two important restrictions: a) V must be a continuously differentiable function, b) matrix A in (23) must be invertible. Also, only systems with positive homogeneous degree are considered in [9].
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IV. M AIN R ESULT Consider the second-order homogeneous system (1) with homogeneity degree k ∈ R and an equilibrium point at the origin. Consider also the homogeneous pd functions V (x1 , x2 ) and W (x1 , x2 ), such that, following Section IIC, can be expressed as m
V (x1 , x2 ) = |x1 | r1 V ± (z), W (x1 , x2 ) = |x1 |
m+k r1
±
W (z),
x1 6= 0,
(28)
x1 6= 0,
(29)
where m ∈ R+ represents the homogeneity degree of function V and r1 stands for the homogeneity weight of coordinate x1 . Since functions f1 and f2 are also homogeneous by hypothesis, it is also possible to express them in terms of the new variable z f1 (x1 , x2 ) = |x1 | f2 (x1 , x2 ) = |x1 |
r1 +k r1 r2 +k r1
φ± 1 (z),
x1 6= 0,
(30)
φ± 2 (z),
x1 6= 0.
(31)
Substituting Equations (28) to (31) in LDM (Equation 4), leads to r d m+k 2 ± V ± (z) + |x1 | r1 − z sign(x1 )φ± (z) + φ (z) 1 2 r1 dz m+k m m+k ± r1 |x1 | r1 W ± (z). sign(x1 )φ± 1 (z)V (z) = |x1 | r1 Taking the negative and positive part of function sign(x1 ) gives the following first-order linear differential equations d + dz V (z)
+
m r1
r
φ+ 1
+ − r2 zφ+ 1 +φ2
V + (z) =
1
d − dz V (z)
−
r2 r1
φ− 1 − zφ1 +φ− 2 m r1
W+ r + − r2 zφ+ 1 +φ2
(32)
1
V − (z) =
r2 r1
W− −, zφ− 1 +φ2
(33)
whose general solutions are Z R R + − α+ (z)dz α+ (z)dz + V (z) = e e β (z)dz + C (34) Z R − R − V − (z) = e− α (z)dz e α (z)dz β − (z)dz + D , (35) with α+ (z) = β + (z) =
m + φ (z) r1 1 , r2 + − r1 zφ1 (z) + φ+ 2 (z)
W + (z) , − φ+ 2 (z)
r2 zφ+ 1 (z) r1
α− (z) = β − (z) =
− rm1 φ− 1 (z) r2 zφ− 1 (z) r1
− φ− 2 (z)
W − (z) . + φ− 2 (z)
r2 zφ− 1 (z) r1
To complete the procedure of finding V , one must transform (34) and (35) to the original coordinates (x1 , x2 ) and find the conditions that render the union of V + and V − continuous along the x2 -axis. Function V , if positive definite, represents then the whole family of continuous, homogeneous Lyapunov functions whose derivative along the trajectories of (1) equals W . Note that Equations 34 and 35 allow the proposal of single-variable negative functions W ± (z), homogeneous or not, in order to find function V . Here, by means of homogeneous properties, we have reduced the original second-order problem of finding V (x1 , x2 ) into two first-order problems for finding V ± (z).
Even if no explicit form of V ± (z) can be obtained, V is in fact a LF provided it fulfills (19). It is also important to note that the sign-definiteness analysis gets also reduced into a single variable problem since, as mentioned in Section II-C.2, the sign of V ± (z) implies the sign of V ± (x1 , x2 ). As with any other ordinary differential equation, if no explicit general solution can be found in (34) nor in (35), it is possible to propose a power series in z that satisfies (32) and (33), potentially reducing the problem of finding V ± (z) into a system of linear algebraic equations. 1 Note that, in (10) we may also have chosen ε = |x2 |− r2 , which leads to the variable change z = x1r1 and analogous |x2 | r2
equations in our method. Depending on the homogeneity weights, choosing one of these produces simpler functions in the variable z. With the main layout covered, we may present our method in the following constructive fashion. A. Constructive Procedure Consider again, the HSOS (1) with an equilibrium point at the origin. In order to find a Lyapunov function that proves the origin’s stability: 1) Fix an homogeneity degree m ∈ R+ for the Lyapunov function to be obtained3 . 2) To obtain a Lyapunov function, propose negative functions W ± (z). To obtain a strict Lyapunov function, propose negative functions W ± (z) such that W ± (0) 6= 0 and W ± (z) ∈ Θ(|z|(m+k)/r2 ). 3) Express functions f1 (x1 , x2 ), f2 (x1 , x2 ) of system (1) in terms of variable z by following the procedure described in Section II-C. 4) Obtain functions V ± (z) using Equations (34), (35). If no explicit general solution can be obtained using these equations, use another method to obtain a particular solution of the linear ODE (32), (33), e.g., the power series method. If still no solution can be found, fix a different m and/or propose different functions W ± (z). 5) Test V ± (z) for pd conditions (19). If the tests succeeds, proceed with the following step. 6) Transform to original coordinates (x2 , x2 ) by findm ing m V + (x1 , x2 ) = |x1 | r1 V + (z) and V − (x1 , x2 ) = |x1 | r1 V − (z). 7) Find the conditions that render V + and V − continuous along the x2 −axis by equating these functions. 8) Compose the resulting Lyapunov function V as V + for x1 > 0 and V − for x1 < 0. V. E XAMPLES Example 1: [A Comparative Example] Consider the nonlinear, homogeneous system x˙ 1 = x2 − x31 x˙ 2 = −x51 ,
(36)
3 From Theorem 2, an educated guess regarding m may be drawn, i.e., for a class C p Lyapunov function m > p · maxi {ri }, p ∈ N − {0}.
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with homogeneous degree k = 2, homogeneity weights (r1 , r2 ) = (1, 3), and an equilibrium point at the origin. Consider also the negative semidefinite function W (x1 , x2 ) = −x8 , with homogeneity degree n = 8. To use the procedure described in [9] involves solving the term Z x1 2ν 3 −3x2 −2 √ √ arctan −3x2 ν 8 3 3x2 dν, 4 e (2x22 − 3ν 3 x2 + ν 6 ) 3 x∗ 1 which does not posses an explicit representation and function V must remain an integral expression, not useful for stability analysis. By applying the method described above, we first fix a desired homogeneous degree m = 6 for the Lyapunov function V to be obtained, such that V (x1 , x2 ) = |x1 |6 V ± (z) with z = x2 /|x1 |3 . Next, we propose the negative functions W + (z) = W − (z) = −1. Transforming f1 and f2 in (36) into functions in the variable z gives
important second-order sliding mode algorithm widely used for control [15], [16], observation [17] and exact differentiation [18]. The unperturbed STA is described as follows 1
x˙ 1 = −k1 |x1 | 2 sign(x1 ) + x2 x˙ 2 = −k2 sign(x1 ),
(39)
where ki are constants used as gains. The algorithm has homogeneous degree k = −1 and homogeneity weights (r1 , r2 ) = (2, 1). We start by fixing m = 2 and, for illustrative purposes, we hold the proposal of a negative function W ± (z). Performing the change of variable for f1 , f2 in (39) gives 1
f1 (x1 , x2 ) = |x1 | 2 (z − k1 sign(x1 )) f2 (x1 , x2 ) = −k2 sign(x1 ).
f1 (x1 , x2 ) = |x1 |3 (sign(x1 ) − z) f2 (x1 , x2 ) = −|x1 |5 sign(x1 ),
The linear ordinary differential equations that satisfy all homogeneous Lyapunov functions for the STA take the following form
± ± Substituting φ± in (34), (35) gives the explicit 1 , φ2 and W solutions 1 z2 + V + (z) = 6 2√ √ −2 3 arctan( 3(2z−1)) −Ce (−3z 2 + 3z − 1) 2 1 z V − (z) = + 6 2√ √ 2 3 arctan( 3(2z+1)) +De (3z 2 + 3z − 1).
and their solutions are
Both positive functions for C, D ≥ 0. Note that, since condition (19) is met for V ± (z), V (x1 , x2 ) will be pd and we can already conclude that the origin of (36) is stable. Transforming to the original coordinates we obtain
where ρ+ (z) = −z 2 + k1 z − 2k2 and ρ− (z) = −z 2 − k1 z − 2k2 . We now propose the functions W ± (z) = −(1+z 2 ), which substituted in equations (40), (41) gives the explicit forms
x61 x2 + 2 + ω1 (x1 , x2 ) 6 2 x61 x22 − V (x1 , x2 ) = + + ω2 (x1 , x2 ) 6 2 V + (x1 , x2 ) =
where
√ √ x −2 3 arctan( 3(2 |x2 | −1))
ω1 = −Ce and
1
(37) (38)
d V + (z) + (z − k1 )V + (z) = W + (z) (− 21 z 2 + 21 k1 z − k2 ) dz d ( 21 z 2 + 12 k1 z + k2 ) dz V − (z) − (z + k1 )V − (z) = W − (z),
−
V + (z) = e
−
V − (z) = e
R
√ √ x −2 3 arctan( 3(2 |x2 | +1)) 1
dz
Z
dz + C (40) m(z+k1 ) dz ρ− (z) dz + D , (41)
R m(z−k1 ) ρ+ (z)
dz
2k2 + 1 1 + k12 + 2k2 2 − 2z + z + Ω1 (z) k1 2k1 k2 1 + k12 + 2k2 2 2k2 + 1 + 2z + z + Ω2 (z), V − (z) = k1 2k1 k2 with
(−3x22 + 3|x1 |3 x2 − x61 )
1 √ −2k 2
arctan
−k1 +8k2
!
√ k1 −2z 2
−k1 +8k2
(−z 2 +k1 z−2k2 )C
and
(3x22 + 3|x1 |3 x2 − x61 ).
In order to find the values of C and D that render function V continuous along the x2 -axis, we equate (37) and (38). In this example, V is continuous only for C = D = 0, so that the Lyapunov function V for (36) finally results in V =
ρ+ (z)
V + (z) =
Ω1 (z) = −e
ω2 = De
W+ e ρ+ (z) Z R m(z+k1 ) dz W− ρ− (z) e − ρ (z)
R m(z−k1 )
x2 x61 + 2. 6 2
Ω2 (z) = e
1 √ −2k 2
arctan
−k1 +8k2
√ k1 +2z 2
!
−k1 +8k2
(z 2 + k1 z + 2k2 )D,
again positive functions for C, D ≥ 0. Transforming into the original coordinates we get 1 2k2 + 1 1 + k12 + 2k2 2 |x1 | − 2|x1 | 2 x2 + x2 + ω1 k1 2k1 k2 2 1 2k2 + 1 1 + k1 + 2k2 2 V− = |x1 | + 2|x1 | 2 x2 + x2 + ω2 , k1 2k1 k2 with
V+ =
Although the theory presented so far is valid for continuous systems, the next example illustrates that our method may also be extended for discontinuous systems. Example 2 (Lyapunov Functions For The STA): Let us consider now the Super Twisting Algorithm (STA), an
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ω1 = −e
1 √ −2k 2
arctan
−k1 +8k2
1
k1 x2 |x1 | 2 − 2|x1 |k2 )C
√
k1 −2x2 |x1 |
−1 2
2 +8k −k1 2
!
(−x22 +
and ω2 = e
1 √ −2k 2
arctan
−1 k1 +2x2 |x1 | 2
√
−k1 +8k2
2 +8k −k1 2
!
(x22 +
1
k1 x2 |x1 | 2 + 2|x1 |k2 )D. As in Example 1, function V is continuous along the x2 −axis only for C, D = 0. Thus, the continuous Lyapunov function that proves asymptotic stability of the STA results V =
1 1 + k12 + 2k2 2 2k2 + 1 |x1 | + 2|x1 | 2 x2 sign(x1 ) + x2 k1 2k1 k2
This result is consistent with the Lyapunov functions for the STA proposed in [19]. VI. C ONCLUSIONS A new method to obtain homogeneous Lyapunov functions for homogeneous second-order systems has been presented. This method allows the attainment of entire families of homogeneous Lyapunov functions and takes advantage of homogeneous properties to transform an original secondorder PDE into two single-variable linear ODE. Thus, to apply the method, it is only required to propose a desired homogeneity degree and a pair of single-variable functions. Conditions imposed in these single-variable functions in order to obtain a continuous pd Lyapunov function have also been presented. The fact that the differential equations involved are linear in one variable both allows to solve them using methods other than direct integration and significantly reduces its sign-definiteness analysis. Compared with other homogeneity approaches, we have shown that the method here presented draws simpler equations and attains explicit representations for Lyapunov functions for a broader set of systems.
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ACKNOWLEDGMENT The authors gratefully acknowledges the financial support from PAPIIT, UNAM, grant IN113614, Fondo de Colaboraci´on del II-FI, UNAM, IISGBAS-109-2013, CONACyT CVU:371652 and CVU: 488179. R EFERENCES [1] V. I. Zubov, Methods of AM Lyapunov and their Application. Noordhoff Groningen, 1964. 263pp. [2] D. Schultz and J. E. Gibson, “The variable gradient method for generating liapunov functions,” American Institute of Electrical Engineers, Part II: Applications and Industry, Transactions of the, vol. 81, no. 4, pp. 203–210, 1962. [3] N. N. Krasovskii, “Problems of the theory of stability of motion,” (Russian), 1959 English translation: Stanford University Press, 1963. [4] A. Levant, “Homogeneity approach to high-order sliding mode design,” Automatica, vol. 41, no. 5, pp. 823–830, 2005. [5] Y. Orlov, “Finite time stability and robust control synthesis of uncertain switched systems,” SIAM Journal on Control and Optimization, vol. 43, no. 4, pp. 1253–1271, 2004. [6] A. Polyakov and A. Poznyak, “Unified lyapunov function for a finitetime stability analysis of relay second-order sliding mode control systems,” IMA Journal of Mathematical Control and Information, vol. 29, no. 4, pp. 529–550, 2012. [7] T. Sanchez and J. A. Moreno, “Construction of lyapunov functions for a class of higher order sliding modes algorithms,” in Decision and Control (CDC), 2012 IEEE 51st Annual Conference on. IEEE, 2012, pp. 6454–6459.
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