53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
A Constructive Lyapunov function design method for a class of homogeneous systems Ton´ametl S´anchez and Jaime A. Moreno Abstract— In this paper a method to design homogeneous Lyapunov functions for a class of homogeneous dynamical systems is proposed. The method is constructive and simple. The procedure exploits homogeneity and polynomial-like properties of the class of functions considered for the vector fields and for the Lyapunov function candidates. It also makes strong use of an algebraic result known as P´olya’s Theorem to establish the required positive and negative definite properties of the Lyapunov function and its derivative. We illustrate the method by designing smooth Lyapunov functions for two systems. The first one is a Second Order Sliding Mode algorithm known as Super Twisting. We propose here for the first time a smooth Lyapunov function for it. The second example corresponds to a double integrator controlled by a continuous homogeneous state feedback, whose origin is finite time stable.
I. I NTRODUCTION Lyapunov’s direct method is one of the most powerful tools in control systems analysis and design. However, its main disadvantage is that, in general, it is not a trivial task to design a Lyapunov function. Over time, many efforts have been made in order to provide methodologies to generate Lyapunov functions. In the first place it is the method to design quadratic Lyapunov functions for Linear systems, consisting in solving the well known Algebraic Lyapunov Equation. However, for nonlinear systems, the things are not so simple. We can mention for example the very general Zubov’s method [8]. This consists in the not easy task of solving a partial differential equation. Since the problem of finding a constructive method to design Lyapunov functions is practically unsolvable for the general non-linear case, the emerging methods are conceived for particular classes of systems. For instance in [4] a method is given to compute polynomial Lyapunov functions for systems with polynomial vector fields. In [6] and [7] two distinct methods have been provided for specific classes of High Order Sliding Mode algorithms. Two issues are central in the designing of a Lyapunov function. The first one consists in finding the appropriate structure of the Lyapunov function candidate. The second difficult issue is to determine whether the Lyapunov function and its time derivative are positive or negative definite, respectively. For polynomial systems in [4] the natural choice is the polynomial class of Lyapunov function candidates. Checking positive (negative) definiteness can be effectively Ton´ametl S´anchez (
[email protected]) and Jaime A. Moreno (
[email protected]) are with the Instituto de Ingenier´ıa, Universidad Nacional Aut´onoma de M´exico (UNAM), 04510 M´exico D.F., Mexico.
978-1-4673-6090-6/14/$31.00 ©2014 IEEE
done through a LMI formulation of the Sum of Squares decomposition (SOS). In this paper we propose a constructive method to design Lyapunov functions for homogeneous systems, defined by a class of polynomial-like multivariable functions, which we will call Generalized Forms (GF). This class of functions have some of the properties of polynomials, but not all. For example, if the Lyapunov function candidate is chosen in the class of GF, their derivative is also a GF. Nevertheless, the problem of determining the positive definiteness of a GF remains. Unfortunately the SOS properties for homogeneous polynomials is not valid for GF, at least directly. Although it is possible to apply indirectly the SOS analysis, in this paper we use an alternative to determine the positiveness of a GF, it is a procedure based on an algebraic result for homogeneous polynomials known as P´olya’s Theorem. The method is effective, simple, and is able to provide more general results than SOS analysis. The present work is organized as follows. In Section II the class of systems, we are going to deal with, is set up. In the same Section P´olya’s Theorem is recalled, and it is shown how it can be used for positiveness analysis of a GF. The procedure to design Lyapunov functions is described in Section III. Sections IV and V are devoted to exemplify the construction of Lyapunov functions for two different systems. Finally, Section VI contains some conclusions. II. P RELIMINARIES Before setting up the class of systems we are going to deal with, let us define the following notation which will be used along the whole paper: d·cρ = | · |ρ sign(·), note for example that dxc0 = sign(x) and dxc1 = dxc = x. A. A class of homogeneous systems Consider the following homogeneous1 dynamical system of degree k with the weights (r1 , . . . , rn ) ∈ Rn+ : x˙ = f (x) ,
(1)
n
where x ∈ R is the state, and the vector field f (x) = [f1 (x), . . . , fn (x)]T is such that each component is a Generalized Form (GF), this means that, it is the sum of a finite number of monomials with rational exponents and real coefficients, so that it takes the following form: fi (x) =
n X j=1
κj dxj c%j +
s X k=1
λk
n Y
dxj c%k,j ,
(2)
j=1
1 In this work homogeneous refers to weighted homogeneous (see for example the definitions in [3]) unless it be specified the classical sense.
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for some κj , λk ∈ R, %j , %k,j ∈ Q and s ∈ N. An important difference with classical polynomials is that the exponents are rational numbers instead of integer numbers. More important is the fact that the power of the real variable dxcρ = |x|ρ sign(x) is signed, that is dxcρ has the same sign as x, in contrast to the classical case where x2 is always non negative, in spite of the sign of x. Note that the sum of GFs is a GF, the product of GFs is a GF, and the partial derivative of a GF is also a GF. However, the composition of two GFs is in general not a GF. A large class of systems can be described by GFs. They appear in finite-time observers and controllers, and also in High Order Sliding Mode Controllers. One good example of this kind of systems is the well known Super–Twisting Algorithm [2] given by 1
x˙ 2 = −k2 dx1 c0 ,
x˙ 1 = −k1 dx1 c 2 + x2 ,
(3)
where the state x = [x1 , x2 ]T ∈ R2 and 0 < k1 , k2 ∈ R. System (3) is homogeneous of degree k = −1 with the weights (r1 , r2 ) = (2, 1). Another example is the double integrator:
thus, according to the theorem, it is sufficient that a < 1 in order to guarantee positiveness of F . It is important to mention that this process is iterative and nothing tells us the minimum value of p for which P´olya’s Theorem is accomplished. The previous process is restricted to forms with non negative indeterminates. However, Generalized Forms (with rational powers) can always be transformed, by quadrants, into classical homogeneous polynomials with non negative indeterminates. The next example illustrates this process. Let g : R2 → R be an homogeneous function of degree m = 5, with weights (r1 , r2 ) = (3, 2), given by: 5
x˙ 2 = u ,
(4)
where the state x = [x1 , x2 ]T ∈ R2 , and the controller u = u(x) ∈ R is given by: 1
2
u(x) = −k1 dx1 c 2 − k2 dx2 c 3 ,
0 < k1 , k2 ∈ R .
•
Thus the closed loop (4), (5) is an homogeneous system of degree k = − 12 with the weights (r1 , r2 ) = (2, 3/2). •
for some G with strictly positive coefficients. Note that the existence of p such that G has only positive coefficients is a necessary and sufficient condition to F (z) > 0 for all z ∈ Pn . Thus, the last theorem can be used to determine the positiveness of a form F (z) by looking for a p such that G has only positive coefficients. Let us show this fact through the following example. Consider F as follows F (z) = z12 − az1 z2 + z22 ,
z ∈ Pn ,
0 < a ∈ R,
for p = 1 it is obtained (z1 + z2 )F (z) = z13 + (1 − a)z12 z2 + (1 − a)z1 z22 + z23 , 2 For
this theorem, [1] can also be consulted.
(6)
For the quadrant {x1 > 0, x2 > 0}, define (z1 , z2 ) = 1/21 1/14 x1 , x2 , and with this we get:
For the quadrant {x 1 > 0, x2 < 0}, define (z1 , z2 ) = 1/21 x1 , (−x2 )1/14 , and with this we get: g(z1 , z2 ) = z135 − az16 z229 + z235 .
B. P´olya’s Theorem
(z1 + z2 + · · · + zn )p F (z) = G(z),
5
g(z1 , z2 ) = z135 + az16 z229 + z235 .
(5)
With the aim to recall P´olya’s theorem, let Pn = {z ∈ Rn | z 6= 0, zi ≥ 0, i = 1, 2, . . . , n} with n ∈ {1, 2, . . .}. Suppose that F : R → Rn is a (classical) polynomial with real coefficients and indeterminates z = (z1 , z2 , . . . , zn ) ∈ Pn , also suppose that F is an homogeneous function (in the classical sense) of degree q. Thus we can state the following strong version of P´olya’s theorem2 [5]. Theorem 2.1: Let F be as above. If F (z) > 0 for all z ∈ Pn then for any sufficiently large p ∈ {1, 2, . . .},
29
The objective is turning into integers the exponents of each term in g, what can be done by a change of variable. Note that the least common multiple of the denominators of all exponents that correspond to x1 is 21. For the case of x2 it is 14. Now, the change of variables must be done by quadrants in order to maintain the positiveness of the variables. •
x˙ 1 = x2 ,
2
g(x1 , x2 ) = |x1 | 3 + adx1 c 7 dx2 c 14 + |x2 | 2 .
(7)
For the quadrant {x1 < 0, x2 < 0}, define (z1 , z2 ) = (−x1 )1/21 , (−x2 )1/14 , and with this we get: g(z1 , z2 ) = z135 + az16 z229 + z235 .
•
For the quadrant {x 1 < 0, x2 > 0}, define (z1 , z2 ) = 1/14 1/21 (−x1 ) , x2 , and with this we get: g(z1 , z2 ) = z135 − az16 z229 + z235 .
Note that in any quadrant the resulting polynomial g(z) is classically homogeneous of degree 35 with strictly positive indeterminates. Thus, the problem of positiveness of (6) can be solved by using P´olya’s theorem over g(z) as it was done in the former example. Note that due to the symmetry, there are only two polynomials, and it is only necessary to analyze one, polynomial (7), if a > 0. III. LYAPUNOV FUNCTION DESIGN For a system with components that are GF it is natural to look for Lyapunov functions in the same class of GFs, since, due to the properties of GFs, if the Lyapunov function candidate V (x) is a GF and the system is a GF then the derivative V˙ is also a GF. In the next paragraphs the procedure that we propose for the design of an homogeneous Lyapunov function for (1) is described.
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Design procedure: n
S1 Let V : R → R be an m−degree homogeneous GF given by: n X
V (x) =
m
ηi |xi | ri +
i=1
q X k=1
θk
n Y
dxj cρk,j ,
(8)
j=1
for some q ∈ N, and ρk,j ∈ Q. For positiveness of V it is necessary that 0 < ηi ∈ R for all i, and for homogeneity, ρk,j must be such that r1 ρk,1 + r2 ρk,2 + · · · + rn ρk,n = m .
i
and ρk,j ≥ 1, ∀j, k .
IV. LYAPUNOV FUNCTION FOR THE S UPER T WISTING A LGORITHM
(9)
S2 Additionally, in order to guarantee differentiability of V , the homogeneity degree m and the exponents ρk,j must be restricted to: m > max(ri ),
monomials and increase its number when they are not sufficient. Another difficulty in the procedure is that the value of the power p in Polya’s Theorem is unknown, and there is no indication when to stop the searching process. So, in practice the method becomes a sufficient condition for checking the positive definiteness of V and W . In order to clarify the general procedure described above, we use the method in two (non trivial) examples in the next section.
(10)
S3 By taking the derivative of (8) along the trajectories of (1), we obtain the GF W : Rn → R such that: ∂V (x) f (x) = −W (x) . V˙ = ∂x S4 For W (x) to be positive it is necessary that it has monomials of every single variable with a positive coefficient. This imposes some restrictions on the possible values of θk and ρk,j . S5 Perform a quadrant analysis for V and W by using Polya’s Theorem (as described in Section II), with the aim to find conditions that guarantee positive definiteness of both V and W . Those conditions are a set of inequalities on ηi , θk and the system’s parameters κi , λk . It is important to mention that such inequalities are linear in the Lyapunov function’s coefficients and also linear in the system’s parameters. But they are in general bilinear in both of them. This is analog to the situation of analysis and design for LTI systems by means of LMI’s. Similarly to that case it is sometimes possible (when the gains of a controller want to be designed) to define new variables to render the inequalities obtained from P´olya’s Theorem linear in the design parameters. Our method relies on the following facts. Fact 1: Consider (1) with f a GF as in (2). If the function V : Rn → R is a GF as in (8) then the function W : Rn → R given by W (x) = −∇V (x) · f (x) is also a GF. Fact 2: Suppose we have a GF system and we propose a GF Lyapunov function candidate V as in (8). If there exists a Lyapunov function for (1) with the same structure as V , then there exists a power p such that the positive definiteness of V and W can be determined by using P´olya’s Theorem. Fact 2 shows that the proposed method can be a powerful tool for the design of Lyapunov functions for the class of systems considered. There are however, some drawbacks. First it is not clear which monomials should be taken when selecting the Lyapunov function candidate. Although the method is not restricted to few monomials, the complexity of the calculations clearly grows with the number of them. So probably one should start with a minimal number of
Consider the Super Twisting Algorithm given by (3), and recall that it is a GF system, homogeneous of degree k = −1 with the weights (r1 , r2 ) = (2, 1). According to the procedure described in the last section, we propose the following homogeneous GF function of degree m as a Lyapunov function candidate for (3): m
¯ 2 dx1 cρ1 dx2 cρ2 + α3 |x2 |m . V (x) = α1 |x1 | 2 + α It is necessary that α1 , α3 > 0 to guarantee that V is a positive definite function. For homogeneity of V we have from (9) that: ⇔
2ρ1 + ρ2 = m
ρ2 = m − 2ρ1 .
Furthermore, in order to guarantee differentiability of V , it is clear from (10) that: m > 2, ρ1 ≥ 1 and ρ2 = m − 2ρ1 ≥ 1, note that from the last inequality it is sufficient that m ≥ 3. Thus, if it is considered m = 3 then we get the following function: 3
V (x) = α1 |x1 | 2 + α ¯ 2 dx1 cρ1 dx2 c3−2ρ1 + α3 |x2 |3 .
(11)
Taking the derivative of (11) along the trajectories of (3), we get that V˙ = −W (x) where: W (x) =
1 3α1 k1 3α1 |x1 | − dx1 c 2 dx2 c+ 3α3 k2 dx2 c2 dx1 c0 2 2 +¯ α2 k2 (3 − 2ρ1 )|x1 |ρ1 |x2 |3−2ρ1 −1
+¯ α2 k2 ρ1 dx1 c
2ρ1 −1 2
dx2 c3−2ρ1
−¯ α2 ρ1 |x1 |ρ1 −1 |x2 |3−2ρ1 +1 .
(12)
Since (12) has to be a positive definite function, it must have positive terms in each variable x1 and x2 . The first term of (12) provides a positive term in x1 . Note that a positive term in x2 can be obtained from the last term of W if we choose ρ1 = 1 and −¯ α2 = α2 > 0. By making this selection we obtain from (11) and (12) the following functions: V (x) W (x)
3
= α1 |x1 | 2 − α2 dx1 cdx2 c + α3 |x2 |3 , =
where
5502
w1 =
w1 |x1 | + w2 x22 − +w4 dx2 c2 dx1 c0 , 3 α1 k1 − α2 k2 , 2
(13)
1 2
w3 dx1 c dx2 c (14) w2 = α2 ,
3 α1 + α2 k2 , and w4 = 3α3 k2 . 2 Again, it is obvious that w1 and w2 must be positive, therefore, the following inequality has to be fulfilled: w3 =
3α1 k1 > 2α2 k2 .
(15)
Up to now, we have only defined the exponents of V and some of its coefficients, but we have not assured the positiveness of (13) and (14). In order to find the conditions that guarantee this, we will perform a quadrant analysis as stated in S5 in Section III.
Now we apply the same procedure to (17) to determine the conditions for its positiveness. Consider: (z1 + z2 )p W (z) = Hp (z), For p = 1 we get that: H1 (z) = w1 z13 + (w2 + w4 − w3 )z1 z22 + (w1 − w3 )z12 z2 +(w2 + w4 )z23 , thus, it is straightforward from Theorem 2.1 that (17) is positive if the following inequalities are fulfilled: w2 + w4 > w3 ,
Quadrant analysis In order to be able to apply P´olya’s Theorem, we can get from V and W homogeneous polynomials with positive indeterminants by performing the following change of coordinates: |x1 | = z12 , |x2 | = z2 . Note that since V and W are symmetric functions with respect to the origin then the analysis can be reduced to only two sets. Thus, we get the following: • For the set {x1 x2 > 0}
•
V (z)
=
α1 z13 − α2 z12 z2 + α3 z22 ,
W (z)
=
w1 z12 − w3 z1 z2 + (w2 + w4 )z22 . (17)
2α2 +6α3 k2 > 2α2 k2 +3α1 ,
W (z)
=
w1 z12
+(w1 + w2 + w4 − 2w3 )z12 z22 + (w2 + w4 )z24 , hence, we get the following inequalities: 2w1 > w3 ,
+ w3 z1 z2 + (w2 −
(18) w4 )z22
w2 > w4
⇔
α2 > 3α3 k2 .
(20)
(z1 + z2 ) V (z) = Gp (z),
(23) Considering (15), (20), (21), and (23) the following system of inequalities is obtained: 3α1 k1 α2 α3 α1 4α2 + 12α3 k2 3α1 k1 + 6α3 k2 + 2α2
z > 0,
and in order to determine that V (z) is positive it is necessary find some p such that Gp (z) is a polynomial with positive coefficients. For p = 1: G1 = α1 z14 + (α1 − α2 )z13 z2 − α2 z12 z22 + α3 z1 z23 + α3 z24 , note that not all coefficients can be positive since α2 > 0. Then for p = 2: G2 (z)
=
α1 z15 + (2α1 − α2 )z14 z2 + (α3 − α2 )z12 z23 +(α1 − 2α2 )z13 z22 + 2α3 z1 z24 + α3 z25 ,
and every coefficient of G2 can be positive if the following set of inequalities is fulfilled: 2α1 − α2 > 0,
α3 − α2 > 0,
α1 − 2α2 > 0 .
These inequalities can be reduced to: α3 > α2 ,
α1 > 2α2 .
(21)
2w2 + 2w4 > w3 ,
3α1 k1 > 2α2 k2 , 4α2 + 12α3 k2 > 3α1 + 2α2 k2 , and 3α1 k1 + 6α3 k2 + 2α2 > 6α1 + 6α2 k2 .
So we have to apply P´olya’s Theorem only to (16) and (17) in order to determine the positiveness conditions. From Theorem 2.1 we have the equation: p
w1 + w2 + w4 > 2w3 ,
that are equivalent to:
. (19)
Note that in the set {x1 x2 < 0} function V is positive. Function W is positive if and only if
3α1 k1 > 4α2 k2 +3α1 . (22)
H2 (z) = w1 z14 + (2w1 − w3 )z13 z2 + (2w2 + 2w4 − w3 )z1 z23
(16)
= α1 z13 + α2 z12 z2 + α3 z22 ,
w1 > w3 , ⇔
Thus, to guarantee that (13) is a Lyapunov function for (3) it is sufficient to fulfill the system of inequalities conformed by (15), (20), (21) and (22). Unfortunately, there is no solution for this system. In order to find a system of inequalities with solution, we can repeat P´olya’s analysis for (17) with p = 2. Thus, (z1 + z2 )2 W (z) = H2 (z) where:
For the set {x1 x2 < 0} V (z)
z > 0.
> > > > > >
2α2 k2 3α3 k2 α2 . 2α2 3α1 + 2α2 k2 6α1 + 6α2 k2
(24)
Therefore, if k1 , k2 , α1 , α2 , and α3 are such that (24) is fulfilled, then (13) is a Lyapunov function for (3). For example, the values α1 = 4.1, α2 = 2, α3 = 5, k1 = 2, and k2 = 0.1 satisfy (24). Figure 1 shows an example plot of V with these values. Remark 1. Note that, if the gains k1 and k2 are given then the Lyapunov function designing procedure is used only for stability analysis, but if k1 and k2 are not given then this procedure can be used to design such gains. Remark 2. It is important to mention that (24) is linear in α1 , α2 , and α3 . Note that also is linear in k1 and k2 , but it is bilinear in both. However, if we define as new variables β1 = α1 k1 , β2 = α2 k2 and β3 = α3 k2 then (24) becomes linear in β1 , β2 , β3 , α1 , α2 (when k2 is assumed positive). Once one has obtained the solutions for these new variables it is possible to compute the values of the original variables.
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additionally, we choose m = 7/2 then ρ1 = 1, and thus we get the following GFs: 7
7
V (x)
= α1 |x1 | 4 + α2 dx1 cdx2 c + α3 |x2 | 3 , (27)
W (x)
= w1 |x1 | 2 + w2 x22 − w3 dx1 c 4 dx2 c
3
3
1
4
2
+w4 dx1 c 2 dx2 c 3 + w5 dx1 cdx2 c 3 ,
(28)
where w1 = α2 k1 ,
w4 = 73 α3 k1 ,
w3 = 47 α1 , x2
x1
w2 = 73 α3 k2 − α2 , w5 = α2 k2 .
Since W must be positive, it is necessary that w2 > 0, thus:
0
7α3 k2 > 3α2 . Fig. 1.
Example plot for (13)
The next step in the designing process is to make a sector analysis, using P´olya’s theorem, as stated in S5.
Notice that V (x) is a differentiable Lyapunov function for the Super-Twisting Algorithm. This is the first time a smooth function for this algorithm is obtained, since the ones known in the literature are not differentiable. V. T HE D OUBLE I NTEGRATOR WITH HOMOGENEOUS STATE FEEDBACK
Now we apply the procedure described in Section II to build a Lyapunov function for the closed loop system described by (4), (5), that we rewrite here as follows: x˙ 1 = x2 ,
1
2
x˙ 2 = −k1 dx1 c 2 − k2 dx2 c 3 .
Quadrant analysis As in the previous example, note that V and W are symmetric functions with respect to the origin, then the analysis is reduced to only two quadrants. For the positiveness analysis of V and W we choose the following change of variables: |x1 | = z14 , |x2 | = z23 . If it is applied to V and W we get the following: • For the set {x1 x2 > 0}
(25)
Recall that (25) is an homogeneous GF system of degree k = −1/2 with the weights (r1 , r2 ) = (2, 3/2). According to S1 we propose the following GF as a Lyapunov function candidate: V (x) = α1 |x1 |
m 2
ρ1
+ α2 dx1 c dx2 c
ρ2
+ α3 |x2 |
2m 3
V (z)
= α1 z17 + α2 z14 z23 + α3 z27 ,
W (z)
= w1 z16 + w2 z26 − w3 z13 z23 + w4 z12 z24 +w5 z14 z22 .
•
V (z)
,
W (z)
= α1 z17 − α2 z14 z23 + α3 z27 , =
In S1, it is required for differentiability of V that: m > 2, ρ ≥ 1, and ρ2 = (2m − 4ρ1 )/3 ≥ 1.
m−2 2
+ w2 z26 −w5 z14 z22 .
+
w3 z13 z23
(31) −
w4 z12 z24 (32)
(z1 + z2 )p V (z) = Gp (z),
All restrictions are fulfilled if m ≥ 7/2. Taking the derivative of V along the trajectories of (25), we get V˙ = −W (x) where: 1 − mα 2 dx1 c
w1 z16
On the set {x1 x2 > 0} function V is positive. So we have to analyze only (30), (31) and (32) to determine the positiveness conditions for (27). Let’s start with (31). From Theorem 2.1 we have the equation:
2ρ1 + 3ρ2 /2 = m ⇔ ρ2 = (2m − 4ρ1 )/3 .
=
(30)
For the set {x1 x2 < 0}
where α1 , α3 > 0, and from (9):
W (x)
(29)
dx2 c − ρ1 α2 |x1 |ρ1 −1 |x2 |ρ2 +1
z > 0.
Up to p = 5 there appear strictly negative terms in Gp (z), therefore we use the iteration with p = 6. So from G6 (z) (which is not written here due to lack of space) the following inequalities are obtained:
1
+ρ2 k1 α2 |x1 |ρ1 + 2 |x2 |ρ2 −1
20α1 − α2 > 0,
1
+ρ2 k2 α2 dx1 cρ1 dx2 cρ2 − 3 1
+ 2mk31 α3 dx1 c 2 dx2 c + 2mk32 α3 |x2 |
2m−1 3
.
15α1 − 6α2 > 0,
−20α2 + α1 > 0,
2m−3 3
Note from the third term of W that ρ2 = 1 and α2 > 0 are necessary conditions to assure a positive term in x1 . A positive term in x2 is already given by the last term. If
−15α2 + α3 > 0,
and
(26)
−15α2 + 6α1 > 0, 6α3 − 6α2 > 0,
15α3 − α2 > 0 .
These inequalities assure positiveness of (27), and can be reduced to:
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α1 > 20α2
and
α3 > 15α2 .
(33)
Now, for (30), we have from Theorem 2.1 that: (z1 + z2 )p W (z) = Hp (z),
z > 0,
and choosing p = 10 we compute H10 (z). We obtain from it the next set of inequalities that that guarantee the positiveness of (30): 10α2 k2 + 120α2 k1 > 74 α1 , 45α2 k2 + 73 α3 k1 + 210α2 k1 > 120α2 k2 + 252α2 k1 +
35 2 α1 , 70 315 3 α3 k1 > 4 α1 ,
x2
x1 0
210α2 k2 + 37 α3 k2 + 210α2 k1 + 105α3 k1 > α2 + 210α1 , 252α2 k2 +
70 3 α3 k2
+ 120α2 k1 + 280α3 k1 > 10α2 +
735 2 α1 ,
Fig. 2.
Example plot for (2)
210α2 k2 + 105α3 k2 + 45α2 k1 + 490α3 k1 > 45α2 + 441α1 , 120α2 k2 + 280α3 k2 + 10α2 k1 + 588α3 k1 > 120α2 + 735 2 α1 , 45α2 k2 + 490α3 k2 + α2 k1 + 490α3 k1 > 210α2 + 210α1 , 10α2 k2 + 588α3 k2 + 280α3 k1 > 252α2 + 490α3 k2 + α2 k2 + 105α3 k1 > 210α2 + 280α3 k2 + and
70 3 α3 k1
315 4 α1 , 35 2 α1 ,
> 120α2 + 74 α1 ,
105α3 k2 + 73 α3 k1 > 45α2 .
(34)
In the same manner, for (32) the following set of inequalities is obtained: 45α2 k1 > α2 k2 , 7 4 α1
+ 120α2 k1 > 10α2 k2 ,
35 7 2 α1 + 210α2 k1 > 45α2 k2 + 3 α3 k1 , 70 315 4 α1 + 252α2 k1 > 120α2 k2 + 3 α3 k1 , 7 3 α3 k2 + 210α1 + 210α2 k1 > 210α2 k2 + α2 + 105α3 k1 , 735 70 3 α3 k2 + 2 α1 + 120α2 k1 > 10α2 + 280α3 k1 + 252α2 k2 ,
VI. C ONCLUSIONS In this paper a new and constructive method to design Lyapunov functions, for the class of homogeneous systems described by Generalized Forms (GF), has been proposed. The usefulness of the method has been illustrated by two different examples. One of them is of great interest because it provides for the first time a smooth Lyapunov function for the well known Super–Twisting Algorithm, a basic system in second order sliding mode control. For the cases when the system’s parameters are given, the Lyapunov function design process is used only as an analysis method, but if the system’s parameters are not provided the Lyapunov function construction process can be used as a design method. Although, in this work the Lyapunov function design procedure is restricted to a certain class of homogeneous systems, it can be extended to the classical homogeneous polynomial systems.
105α3 k2 + 441α1 + 45α2 k1 > 210α2 k2 + 45α2 + 490α3 k1 ,
ACKNOWLEDGMENTS
280α3 k2 + 735 2 α1 + 10α2 k1 > 120α2 k2 + 120α2 + 588α3 k1 ,
The authors gratefully acknowledges the financial support from PAPIIT, UNAM, grant IN113614, Fondo de Colaboraci´on del II-FI, UNAM, IISGBAS-109-2013 and CONACyT CVU:371652.
490α3 k2 + 210α1 + α2 k1 > 45α2 k2 + 210α2 + 490α3 k1 , 315 4 α1 > 10α2 k2 + 280α3 k1 + 252α2 , 490α3 k2 + 35 2 α1 > 210α2 + α2 k2 + 105α3 k1 , 280α3 k2 + 74 α1 > 120α2 + 70 4 α3 k1 ,
588α3 k2 +
and
105α3 k2 > 45α2 + 73 α3 k1 .
R EFERENCES
(35)
Thus, if α1 , α2 , α3 , k1 , and k2 are chosen such that the inequalities (29), (33), (34) and (35) hold, then, (27) is a Lyapunov function for (25). For example, with the following constants all those inequalities are fulfilled: α1 = 24,
α2 = 1,
α3 = 17,
k1 = 2,
k2 = 6.
Figure 2 shows a plot of function (27) with these last values. Remark 3. Note that the higher the value of p is, the higher is the number of inequalities to be satisfied. However, it is very important to underline the fact that the Lyapunov function’s coefficients always appear linearly in those inequalities.
[1] Charles N. Delzell. Impossibility of extending P´olyas theorem to “forms” with arbitrary real exponents. Journal of Pure and Applied Algebra, pages 2612–2622, 2008. [2] A. Levant. Sliding order and sliding accuracy in Sliding Mode Control. International Journal of Control, 58(6):1247–1263, 1993. [3] A. Levant. Homogeneity approach to High–Order Sliding Mode design. Automatica, 41:823–830, 2005. [4] P. A. Parrilo. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, Pasadena, California, 2000. ¨ [5] G. P´olya. Uber positive Darstellung von Polynomen. Vierteljahrschrift Naturforschenden Ges, 73:141–145, 1928. [6] A. Polyakov and A. Poznyak. Lyapunov function design for finite-time convergence analysis: “Twisting” controller for second-order sliding mode realization. Automatica, 45(2):444–448, February 2009. [7] T. S´anchez and J. A. Moreno. Construction of Lyapunov Functions for a Class of Higher Order Sliding Modes algorithms. In 51st IEEE Conference on Decision and Control (CDC), pages 6454–6459, 2012. [8] V. I. Zubov. Methods of A. M. Lyapunov and their applications. Groningen: P. Noordho: Limited, 1964.
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