Construction of Control Lyapunov Functions for ... - Semantic Scholar
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
FrC01.5
Construction of Control Lyapunov Functions for Damping Stabilization of Control Affine Systems N. Hudon and M. Guay Abstract— This paper considers the construction of control Lyapunov functions (CLF) for the stabilization of nonlinear control affine systems that satisfy Jurdjevic-Quinn conditions. First, we obtain a one-form for the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we obtain a control Lyapunov function for the closed-loop system. Under Jurdjevic-Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics, which is associated with the structure of the anti-exact part. An example is presented to illustrate the method.
I. INTRODUCTION We consider control affine systems of the form x˙ = X0 (x) +
m X
uk Xk (x)
(1)
k=1
with states x ∈ Rn , and control u = (u1 , . . . , um ) ∈ Rm . We assume that the vector fields Xk are of class C ∞ (Rn ; Rn ) and that Xk (0) = 0, for k = 0, 1, . . . , m. It is known since the seminal works of [9] and [10], that under certain conditions, and if a Lyapunov function V (x) is known for the system, that a feedback law uk = −Xk · ∇V, k = 1, . . . , m
(2)
stabilizes the system to the origin (see for example [2], [15], [4], and [11] and references therein). A well-known limitation of this approach is that there is no systematic way to build the required Lyapunov function a priori. A connection between mechanical systems and the construction of control Lyapunov functions for this damping feedback approach was pointed out in [12] using a Hamiltonian function (see also [11]). As illustrated in [13], to determine an admissible Hamiltonian function for a general nonlinear system is still a challenging problem. In [17], approximate dissipative Hamiltonian realization techniques were developed with connections to stabilization. Precisely, it was shown how k-th degree approximate dissipative Hamiltonian systems can be used to solve the realization problem, and how an N. Hudon and M. Guay are with the Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada.
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
associated k-th degree approximate Lyapunov function can be used to study the stability of such systems. Parts of their argument were based on a decomposition of the dynamics into a gradient part (generated by a potential) and a tangential part. This observation motivates the approach considered in the present paper to construct a local dissipative function V0 (x) to stabilize control affine nonlinear systems of the form (1) using a damping feedback controller of the form uk = −Xk · ∇V0 (x), k = 1, . . . , m. In [8], it was shown that a radial homotopy operator can be used to decompose the drift dynamics into a dissipative (exact) part and a non-dissipative (anti-exact) one. More precisely, a one-form for the system was obtained by taking the interior product of a non vanishing two-form with respect to the drift vector field. A radial homotopy operator centered at the desired equilibrium point for the system was designed. Applying this linear operator on the aforementioned oneform for the system, an exact one form generated by the desired potential function and an anti-exact form that generates the tangential dynamics were obtained. In the present paper, it is shown how this procedure, using the interior product of a non-vanishing two-form with the drift vector field X0 (x) of (1), leads to the construction of a first function V0 (an auxiliary scalar field following the nomenclature proposed in [11] and [6]) can be used locally by a second application of the decomposition method to obtain a control Lyapunov function for stability characterization. The idea of using an auxiliary scalar function V0 (x), with the property that ∇V0 (x) · X0 < 0, to obtain a control Lyapunov function for the control affine system (1) under damping feedback control was studied in [6] for homogeneous system (see for example [7]) using the flow of a vector field constructed using a linear combination of elements of {adjX0 Xk , j ∈ N, k = 1, . . . , m}. Another solution to the problem was presented in [12]. The existence of CLF for these closed-loop systems was reported in the literature (see for example [16]). The paper is organized as follows. Section II presents the mathematical background for the approach considered, including definitions of the radial homotopy operator that is used in the sequel. The main result of the paper, which is the construction of a suitable control Lyapunov function for control affine system based on a dissipative potential for the drift vector field, is presented in Section III. An illustration of the proposed approach is given in Section IV. Conclusion and future areas for investigation are outlined in Section V.
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FrC01.5 II. PRELIMINARIES In this section, basic elements of exterior calculus on Rn are reviewed. Of interest are the properties of the exterior derivative, the interior product, Lie derivative of exterior forms, and Cartan’s formula. We then proceed to present the properties of the radial homotopy operator. A complete account of exterior calculus and homotopy operators can be found in [5]. A. Exterior Calculus We denote a smooth vector field X ∈ Γ∞ (Rn ) as a smooth map n
n
X : R → TR ,
X|x =
n X
v i (x)∂xi |x ,
(3)
i=1
i.e. a map taking a point x ∈ Rn and assigning a tangent vector X|x ∈ Tx Rn . The cotangent (dual) space Tx∗ Rn is the set of all linear functionals on Tx Rn , Tx∗ Rn = {ω|x : Tx Rn → R}
(4)
2. d(α ∧P β) = dα ∧ β + (−1)deg α α ∧ dβ. ∂f )dxi , ∀f (x) ∈ Λ0 (Rn ). 3. df = i ( ∂x i 4. d ◦ dα = 0. A k-form α is said to be closed if dα = 0. It is said to be exact if there exists a (k − 1)-form β such that dβ = α. The interior product y is a map y : Γ∞ (Rn ) × Λk (Rn ) → Λk−1 (Rn ),
L : Λk (Rn ) → Λk (Rn ), (5)
The standard basis of Tx∗ Rn is given by {dx1 , . . . , dxn }, where dxi (∂xj ) = δji , δji being the Kronecker delta. An element ω|x in the cotangent space Tx∗ Rn can be written as n X ω|x = ωi dxi , ωi ∈ R. (6) i=1
In the sequel, differential one-forms are used. They are generated the following way. A differential one-form ω on Rn is a smooth map taking a point x ∈ Rn and assigning an element of its dual space Tx∗ Rn . We write ω=
n X
ωi (x)dxi ,
(7)
i=1
where ωi are smooth functions on Rn . The exterior (wedge) product ∧ is defined on Λ1 (Rn ) × Λ1 (Rn ) by the requirements dxi ∧ dxj dxi ∧ f (x)dxj
L[X1 ,X2 ] α = (LX1 LX2 − LX2 LX1 )α.
with the following properties: 1. d(α + β) = dα + dβ.
0 ≤ k ≤ n − 1,
(12)
(13)
B. Homotopy Operator
(8)
for all α, β, γ ∈ T ∗ Rn . If α ∈ Λk (Rn ), then we write deg α = k. Notice that Λ1 (Rn ) = T ∗ Rn and Λ0 (Rn ) = C ∞ (Rn ). The differential operator d is the unique operator on Ln Λ(Rn ) = k=0 Λk (Rn ), d : Λk (Rn ) → Λk+1 (Rn ),
(11)
We also define ad0X1 X2 = X2 and the k iterates for k = k 1, 2, . . . by adk+1 X1 X2 = [X1 , adX1 X2 ]. We complete this review with Cartan’s identity, which relate the Lie derivative of a differential form to exterior differentiation: LX ω = Xydω + d(Xyω).
for all smooth functions f (x) and by α ∧ (β + γ) = α ∧ β + α ∧ γ,
0 ≤ k ≤ n,
with the following properties: 1. LX f = Xydf . 2. LX (α + β) = LX α + LX β. 3. LX (α ∧ β) = (LX α) ∧ β + α ∧ (LX β). 4. LX dα = d(LX α). 5. Lf ·X α = f · LX α + df ∧ (Xyα). 6. LX1 +X2 α = LX1 α + LX2 α. 7. LX1 (X2 yα) = LX1 (X2 yα) − X2 y(LX1 α). The action of the Lie derivative for X1 , X2 ∈ Γ∞ (Rn ) is of course given by the Lie bracket LX1 X2 = [X1 , X2 ] and satisfies
= −dxj ∧ dxi = f (x)dxi ∧ dxj
(10)
with the following properties ∀α, β ∈ Λk (Rn ), ∞ n 0 n ∀X, X1 , X2 ∈ Γ (R ) and ∀f, g ∈ Λ (R ): 1. Xyf = 0. 2. Xyω = ω(X), ∀ω ∈ Λ1 (Rn ). 3. Xy(α + β) = Xyα + Xyβ. 4. (f X1 + gX2 )yα = f · (X1 yα) + g · (X2 yα). 5. Xy(α ∧ β) = (Xyα) ∧ β + (−1)deg(α) α ∧ (Xyβ). 6. X1 y(X2 yα) = −X2 y(X1 yα). The last property leads to the following composition rule for interior product: Xy(Xyα) = 0. The Lie derivative L is a map
where each ω|x is linear, i.e. (aω1 |x + bω2 |x )(X|x ) = aω1 |x (X|x ) + bω2 |x (X|x ).
0 ≤ k ≤ n,
(9)
In this section, we show how to construct a homotopy operator H, i.e., a linear operator on elements of Λ(Rn ) that satisfies the identity ω = d(Hω) + Hdω,
(14)
for a given differential form ω ∈ Λ(Rn ). The first step in the construction of a homotopy operator is to define a star-shaped domain on Rn . An open subset S of Rn is said to be star-shaped with respect to a point p0 = (x01 , . . . , x0n ) ∈ S if the following conditions hold: 0 • S is contained in a coordinate neighborhood U of p .
8009
FrC01.5 The coordinate functions of U assign coordinates (x01 , . . . , x0n ) to p0 . • If p is any point in S with coordinates (x1 , . . . , xn ) assigned by functions of U , then the set of points (x + λ(x − x0 )) belongs to S, ∀λ ∈ [0, 1]. A star-shaped region S has a natural associated vector field X, defined in local coordinates by
III. MAIN RESULT
•
X(x) = (xi − x0i )∂xi , ∀x ∈ S.
(15)
In this paper, we consider, for simplicity, the case where the star-shaped domain is centered at the origin, hence X(x) = xi ∂xi . For a differential form ω of degree k on a star-shaped region S centered at the origin, the homotopy operator is defined, in coordinates, as Z 1 (Hω)(x) = X(x)yω(λx)λk−1 dλ, (16)
We now present the main contribution of the paper. First, we recall the classical result on damping feedback stabilization from [10] following the treatment in [4] (see also [15] and [11]). Then, we restate the damping feedback controller construction in terms of differential forms and show under what conditions a dissipative potential function for the drift dynamics can be deformed to obtain a control Lyapunov function using a radial homotopy operator. The main features of the proposed method is that it results in a constructive algorithm for local CLF for systems satisfying the JurdjevicQuinn conditions and it leads to an explicit construction of the invariant set for the dynamics. A. Stabilization by Damping Feedback Consider a control affine system
0
x˙ = X0 (x) +
where ω(λx) denotes the differential form evaluated on the star-shaped domain in the local coordinates defined above. The important properties of the homotopy operator that are used in the sequel are the following: 1. H maps Λk (S) into Λk−1 (S) for k ≥ 1 and maps Λ0 (S) identically to zero. 2. dH + Hd = identity for k ≥ 1 and (Hdf )(x) = f (x) − f (x0 ) for k = 0. 3. (HHω)(xi ) = 0, (Hω)(x0i ) = 0. 4. XyH = 0, HXy = 0. The first part of the right hand side of (14), d(Hω), is obviously a closed form, since d ◦ d(Hω) = 0. Since by property (1) of the homotopy operator, for ω ∈ Λk (S), we have (Hω) ∈ Λk−1 (S), d(Hω) is also exact on S. We denote the exact part of ω by ωe = d(Hω) and the anti-exact part by ωa = Hdω. It is possible to show that ω vanishes on Rn if and only if ωe and ωa vanish together [5]. From the decomposition outlined above, we have ω − ωa = ωe .
(17)
Taking the exterior derivative on both sides and using the fact that ωe is closed, we have d(ω − ωa ) = d(ωe ) = 0.
(18)
In the sequel, we apply the homotopy operator on oneforms. Since in our applications, ωe is an exact one-form, (Hωe ) computed by homotopy is a dissipative potential for the system. We show in the sequel under what conditions this potential can be used to construct a damping feedback controller and under what conditions a second application of the decomposition leads to a control Lyapunov function stabilizing the origin. A non-dissipative potential is associated with the anti-exact part, but on the star-shaped domain S, ωa does not contribute to the dissipative part of the system. In other words, ωa belongs to the kernel of H, which can be seen by applying property (3) from above to the definition of ωa .
m X
uk Xk (x), ∀(x, u) ∈ Rn × Rm ,
(19)
k=1
for some X0 , · · · , Xm ∈ C ∞ (Rn ; Rn ) and assume that Xk (0) = 0, k = 0, . . . , m. Definition 3.1 (Assignable Lyapunov Functions): Let V0 be a positive definite and radially unbounded function. We say that a continuous u : Rn → Rm assigns V0 to be a Lyapunov function for the closed loop system (19) if the derivative of V0 along the trajectories of the closed loop system is negative definite, i .e., that for all x ∈ Rn \ {0}, LX0 · V0 +
m X
uk LXk · V0 < 0.
(20)
k=1
First, we recall the statement of Artstein’s theorem [1]. Theorem 3.2: Let V (x) be a positive radially unbounded function. There exists a continuous feedback that assigns V (x) to be a Lyapunov function for the closed loop system (19) if and only if (a) It is a control Lyapunov function, i .e., for all x ∈ Rn \ {0}, LXk V (x) = 0 ⇒ LX0 V (x) < 0, k = 1, . . . , m. (21) (b) It satisfies the small control property, i .e., for any > 0, there is a δ > 0 such that for all x ∈ Rn \ {0}, ( kuk < kxk < δ ⇒ ∃u (22) Pm LX0 + k=1 uk LXk V (x) < 0. We consider stabilization by damping feedback. Consider the feedback law u = (u1 , . . . , um )T defined by uk = −Xk (x) · ∇V0 (x), ∀ k ∈ 1, . . . , m.
(23)
With this feedback, one has for all x ∈ Rn \ {0} m
X dV0 = X0 (x) · ∇V0 (x) − (Xk (x) · ∇V0 (x))2 < 0. (24) dt k=1
n
Therefore, 0 ∈ R is a stable point for the closed-loop system x˙ = X(x, u(x)). By Lasalle’s invariance principle,
8010
FrC01.5 0 ∈ Rn is globally asymptotically stable if for every x(t) ∈ C ∞ (R; Rn ), for all t ∈ R, x(t) ˙ Xk (x(t)) · ∇V0 (x(t))
= X0 (x(t)), =
(25)
0, ∀k ∈ 0, . . . , m, (26)
one obtains x(t) = 0. An alternate formulation of this result is given by the following theorem (a proof of this statement can be found in [4]). Theorem 3.3: Given the smooth control affine system (19) and a function V0 such that LX0 V0 < 0 for every x ∈ R\{0}. Suppose moreover that span{X0 (x), adkX0 Xi (x)
: i = 1, . . . , m, k ∈ N} = R
n
(27)
on Rn \ {0}. Then the feedback law uk = −LXk V0 , ∀i ∈ 1, . . . , m
(28)
globally asymptotically stabilizes the control affine system (19). As noted in [4], the function V0 (x) is not a CLF in general. We refer to [6] and [12] for other CLF construction methods based on the prior knowledge of a function V0 satisfying the weak Jurdjevic-Quinn conditions (see [11, Definition 2.2]). In the next section, we show under which conditions those results can be obtained using the tools presented in Section (II). B. Damping Feedback and Deformation using Homotopy Operator First, we define a non-vanishing closed two-form Ω on Rn as X Ω= dxi ∧ dxj . (29)
arguments. The positive closed one-forms, i .e., hω0 , X0 i > 0, was generated as ω0 = −dV0 . In our case, it will be clear when the negative sign should be used, depending on the orientation of the basis for (29). We assume that V0 (x), obtained after application of the homotopy operator (i .e., V0 = (Hω0,e )), is such that LX0 V0 < 0 for x ∈ Rn \ {0}. If it is not the case, it is always possible to apply a change of coordinates (see for example [2]) to ensure that the above contraction with respect to the drift vector field generates a one-form with the desired properties. We construct the damping feedback controller uk (x) = −LXk V0 .
By the first property of the Lie derivative from Section (II-A), we have LXk V0 = Xk ydV0 , which is given as Xk yω0,a − Xk yω0 by the homotopy decomposition. We now consider the affine system under the feedback law, i .e., we consider the system X0 −
The orientation of the two-form will be fixed, if necessary, by checking the sign of the obtained dissipative function. We obtain a first one-form associated to the system by contracting this two-form with respect to the drift vector field, ω0 = X0 yΩ.
(31)
From Section III-B, we know that we can locally construct a homotopy operator on Rn such that ω0 = ω0,e + ω0,a . Since ω0,e is exact, it is given as the exterior derivative of a potential function and we rewrite ω0 = −dV0 + ω0,a .
(32)
Remark 3.4: The negative sign for −dV0 is used to comply with the notation introduced in [3], where a known closed positive one-form ω0 (i .e., ω0,a = 0) was used for stability
LXk V0 · Xk .
(34)
Taking now the interior product of this closed-loop vector field with respect to the non-vanishing two-form Ω, we obtain the one-form ! m X ω = X0 − (Xk yω0,a − Xk yω0 )Xk yΩ. (35) k=1
Since (Xk yω0,a − Xk yω0 ) ∈ Γ∞ (Rn ) and by the property (4) of the interior product, we can re-write this one-form as ω
= X0 yΩ −
m X
(Xk yω0,a − Xk yω0 ) · (Xk yΩ)(36)
k=1
= ω0 +
In the present paper, the non-vanishing two-form Ω is not necessarily defined in a canonical way, since the objective is ultimately to compute an admissible Lyapunov function (and not a minimal one). For example, if n = 3, we would have, (30)
m X k=1
1≤i<j≤n
Ω = dx1 ∧ dx2 + dx1 ∧ dx3 + dx2 ∧ dx3 .
(33)
= ω0 −
m X
(Xk yω0,e )(Xk Ω)
i=1 k X
fk (Xk yΩ),
(37)
(38)
k=1
where the fk ’s account for the deformation of the potential function with respect to the controlled vectors. The one-form ω will serve as the basis for the computation of a local CLF using the homotopy operator defined in Section III-B. To simplify the computations and to follow the construction from [3], we will assume in the sequel that the dissipative potential V0 (x) computed above is such that ω0,a ≡ 0. This is equivalent to assume that V0 (x) is a first integral for the system. We can now restate a local form of the result from Section III-A in terms of differential forms. Theorem 3.5: Assume that for all x ∈ Rn \ {0}, one can construct V0 such that ω0 is exact, i .e., ω0 = −dV0 6= 0 and that dV0 vanishes at the origin. Then, the damping feedback uk = −Xk ydV0 locally asymptotically stabilizes the system at the origin. Moreover, a local CLF for the closed-loop system can be computed using V = (Hω), with ω given by (38), if the anti-exact part ωa = Hdω vanishes only at the origin.
8011
FrC01.5 Proof: The first part of the theorem is a re-statement of the original result. Consider the closed loop system under damping feedback control with V0 as required: X0 −
m X
LXk V0 · Xk .
(39)
k=1
Then V˙ 0 = X0 · ∇V0 −
m X
(Xk · ∇V0 )2 .
(40)
k=1
In terms of differential forms, we have V˙ 0 = X0 ydV0 −
m X
2
(Xk ydV0 ) .
(41)
k=1
The second term of the right hand side is obviously negative definite. Since X0 ydV0 = X0 y(ω0,a − ω0 ), which can be decomposed by the properties of the interior product as X0 ydV0
Since by assumption, ω0 = −dV0 vanishes at the origin, we are left with n X X0 yXk yΩ = 0, x = 0, (51)
=
X0 yω0,a − X0 yω0
(42)
=
X0 yω0,a − X0 yX0 yΩ
(43)
= X0 yω0,a .
(44)
k=1
which is guaranteed by the original assumption on the vector fields Xk (0) = 0, k = 0, 1, . . . , m. Under the assumption that dV0 6= 0, ∀ x ∈ Rn \ {0} and that ωa vanishes only at the origin, the maximal invariant set is therefore {0}. By the usual Lasalle’s arguments, damping feedback stabilizes asymptotically the origin of the system. It remains to show that V is a CLF on the star-shape domain. Restating condition (21), we have that V (x) is a CLF provided that ∀ x ∈ Rn \ {0}, Xk ydV = 0 ⇒ X0 ydV < 0, k = 1, . . . , m. Consider V˙ =
V˙ 0 = −
2
(Xk ydV0 ) < 0,
∀ x ∈ Rn \ {0},
k X
(Xk yω0,a − Xk yω0 ) · (Xk yΩ).
(46)
k=1
By assumption ω0,a = 0 and we can re-express the second term of the right hand side as (−Xk yω0 ), which is given, for all k = 1, . . . , m, as −Xk yω0
=
−Xk yX0 yΩ
(47)
=
X0 yXk yΩ.
(48)
Hence, we have ω = ω0 −
k X
(X0 yXk yΩ) · (Xk yΩ).
(49)
dV = ω0 −
m X
X0 yXk yΩ = 0, x = 0.
· ∇V.
(53)
m X
(LXk V0 ) · Xk ydV.
(54)
If Xk ydV = 0 for all k = 1, . . . , m, we observe that since we use the same non-vanishing two-form in both steps of the construction, we have no deformation. Hence V obtained by application of the homotopy operator is the same as V0 , and by the above discussion, we have that X0 ydV = X0 ydV0 < 0. As a consequence, V˙ < 0 on Rn \ {0} and V is a CLF for the system. Remark 3.6: The exactness condition on ω0 could be relaxed, provided some convexity conditions on (Hω0 ), which is possible to compute only if a structure is assumed on the drift vector field. The same remark on “classical” JurdjevicQuinn conditions were discussed in [6]. As mentioned earlier, restating the results from Section III-A using differential forms is practical in the sense that locally, one can obtained a CLF by application of the homotopy operator on the one-form constructed using the closed-loop system. The structure of ωa leads to an expression of the invariants set for the closed-loop dynamics. An application of this construction is presented in the next Section to illustrate these observations.
k=1
Applying the homotopy operator on ω, one obtains ω = ωe + ωa . From Section II-B, we have that ω = 0 if and only if ωa and ωe vanish at the same time. By assumption, ωa vanishes only at the origin, at least on the star-shaped domain where the homotopy operator is defined. Hence, we have to show that the exact part of ω, i .e., dV vanishes at the origin. We have the requirement
LXk V0 · Xk
k=1
(45)
under the assumption that dV0 6= 0, ∀ x ∈ Rn \ {0}. We now need to shown that the origin is the only invariant for the closed-loop system. We consider the deformation of V0 to compute a CLF using the homotopy operator on a one-form ω. The one-form ω is given by
!
In terms of interior products, V˙ = X0 ydV −
k=1
ω = ω0 −
X0 −
m X k=1
By assumption, ω0,a ≡ 0, and we obtain m X
(52)
IV. APPLICATION EXAMPLE We apply the above construction to a controlled pendulum example, taken from [15]. Consider the control affine system x˙ 1
=
x˙ 2
= − sin(x1 ) + u.
x2
(55) (56)
Setting Ω = dx1 ∧dx2 , a first dissipative potential is obtained with ω0 = X0 yΩ, given as
(50)
k=1
8012
ω0 = sin(x1 )dx1 + x2 dx2 .
(57)
FrC01.5 Applying the radial homotopy operator defined in Section II-B, we obtain 1 V0 = (Hω) = x22 + (1 − cos(x1 )), (58) 2 which is of the desired form locally. In this particular case ω0 is already exact, hence ωa = ω0 − ωe = 0. The obtained function V0 is positive definite, but it is not a CLF (it is in fact a first integral, i .e., LX0 V0 ≡ 0). The damping feedback controller u(x) is given as −X1 ydV0 = −X1 yωe = −x2 . Applying the radial homotopy operator on the new one-form, one has ω = x2 dx1 − (x2 + sin(x2 ))dx2 which results after integration leads to the local CLF 1 V = 1 − cos(x1 ) + (x1 x2 + x22 ). 2 Computing ∇V · X(x, u(x)), we have 1 V˙ = − x22 + x1 x2 + x1 sin(x1 ) < 0, kxk < 1. 2 Figures 1 and 2 present the locally obtained V (x) and in a neighborhood of the origin.
Fig. 1.
V (x) for the mechanical example
Fig. 2.
V˙ (x) for the mechanical example
V. CONCLUSION
(61)
In this paper, we considered the problem of control Lyapunov functions construction for the stabilization of nonlinear control affine systems satisfying Jurdjevic-Quinn conditions. Provided that we can obtain a positive definite function (a dissipative potential) by taking the interior product of a nonvanishing two-form with respect to the drift vector field, we showed how a CLF can be computed for the closed-loop vector field subject to damping feedback control under some conditions on the anti-exact part of a one-form obtained for the system. The proposed method is local since the approach relies on a homotopy operator centered at a desired equilibrium point (herein, at the origin). Further research will focus on systematic computation of the domain of attraction. Another possible investigation is the characterization of the non-exact part, ωa , when it is not vanishing only at the origin. One problem is to decide if the non-exact part contains information about stable complex patterns, for example limit cycles.
V˙ (x)
R EFERENCES
(59)
(60)
One important aspect to note here is that the one-form ω is not exact. In fact, after the second application of the homotopy operator, we have 1 ωa = (x2 dx1 − x1 dx2 ) . (62) 2 This non-exact part vanishes only at the origin, hence the largest invariant set for the dynamics, as outlined by the nonexact one-form is the origin {0}.
[1] Z. Artstein (1983). Stabilization with Relaxed Controls. Nonlinear Analysis, 7(11):1163–1173. [2] A. Bacciotti (1992). Local Stabilizability of Nonlinear Control Systems. Series on Advances in Mathematics for Applied Sciences Volume 8, World Scientific, Singapore. [3] C. I. Byrnes (2007). Differential Forms and Dynamical Systems. In Modeling, Estimation and Control, Lecture Notes in Control and Information Sciences Volume 364, Springer-Verlag, Berlin, 35–44. [4] J.-M. Coron (2007). Control and Nonlinearity. Mathematical Surveys and Monographs Volume 136, American Mathematical Society, Providence, RI. [5] D. G. B. Edelen (1985). Applied Exterior Calculus. John Wiley and Sons, New York, NY. [6] L. Faubourg and J. B. Pomet (2000). Control Lyapunov Functions for Homogeneous Jurdjevic-Quinn Systems. ESAIM Control, Optimisation and Calculus of Variations, 5:293–311. [7] H. Hermes (1991). Asymptotically Stabilizing Feedback Controls. Journal of Differential Equations, 92:76–89. [8] N. Hudon, K. H¨offner and M. Guay (2008). Equivalence to Dissipative Hamiltonian Realization. In Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, 3163–3168. [9] D. H. Jacobson (1977). Extensions of Linear Quadratic Control, Optimization and Matrix Theory. Academic Press, London. [10] V. Jurdjevic and J. F. Quinn (1978). Controllability and Stability. Journal of Differential Equations, 28:381–389. [11] M. Malisoff and F. Mazenc (2009). Constructions of Strict Lyapunov Functions. Communications and Control Engineering, SpringerVerlag, London. [12] F. Mazenc and M. Malisoff (2006). Further Constructions of ControlLyapunov Functions and Stabilizing Feedbacks for Systems Satisfying the Jurdjevic-Quinn Conditions. IEEE Transactions on Automatic Control, 51(2):360–365. [13] R. Ortega, A. van der Schaft, B. Maschke and G. Escobar (2002). Interconnection and Damping Assignment Passivity-based Control of Port-controlled Hamiltonian Systems. Automatica, 38:585–596. [14] R. Outbib and J. C. Vivalda (1999). On Feedback Stabilization of Smooth Nonlinear Systems. IEEE Transactions on Automatic Control, 44(1):200–202. [15] E. D. Sontag (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd Ed. Texts in Applied Mathematics Volume 6, Springer-Verlag, New York, NY. [16] J. Tsinias (1991). Existence of Control Lyapunov Functions and Applications to State Feedback Stabilization of Nonlinear Systems. SIAM Journal on Control and Optimization, 29(2):457–473. [17] Y. Wang and D. Cheng and S. S. Ge (2007). Approximate Dissipative Hamiltonian Realizations and Construction of Local Lyapunov Functions. Systems and Control Letters, 56:141–149.
8013
FrC01.5
Construction of Control Lyapunov Functions for Damping Stabilization of Control Affine Systems N. Hudon and M. Guay Abstract— This paper considers the construction of control Lyapunov functions (CLF) for the stabilization of nonlinear control affine systems that satisfy Jurdjevic-Quinn conditions. First, we obtain a one-form for the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we obtain a control Lyapunov function for the closed-loop system. Under Jurdjevic-Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics, which is associated with the structure of the anti-exact part. An example is presented to illustrate the method.
I. INTRODUCTION We consider control affine systems of the form x˙ = X0 (x) +
m X
uk Xk (x)
(1)
k=1
with states x ∈ Rn , and control u = (u1 , . . . , um ) ∈ Rm . We assume that the vector fields Xk are of class C ∞ (Rn ; Rn ) and that Xk (0) = 0, for k = 0, 1, . . . , m. It is known since the seminal works of [9] and [10], that under certain conditions, and if a Lyapunov function V (x) is known for the system, that a feedback law uk = −Xk · ∇V, k = 1, . . . , m
(2)
stabilizes the system to the origin (see for example [2], [15], [4], and [11] and references therein). A well-known limitation of this approach is that there is no systematic way to build the required Lyapunov function a priori. A connection between mechanical systems and the construction of control Lyapunov functions for this damping feedback approach was pointed out in [12] using a Hamiltonian function (see also [11]). As illustrated in [13], to determine an admissible Hamiltonian function for a general nonlinear system is still a challenging problem. In [17], approximate dissipative Hamiltonian realization techniques were developed with connections to stabilization. Precisely, it was shown how k-th degree approximate dissipative Hamiltonian systems can be used to solve the realization problem, and how an N. Hudon and M. Guay are with the Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada.
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
associated k-th degree approximate Lyapunov function can be used to study the stability of such systems. Parts of their argument were based on a decomposition of the dynamics into a gradient part (generated by a potential) and a tangential part. This observation motivates the approach considered in the present paper to construct a local dissipative function V0 (x) to stabilize control affine nonlinear systems of the form (1) using a damping feedback controller of the form uk = −Xk · ∇V0 (x), k = 1, . . . , m. In [8], it was shown that a radial homotopy operator can be used to decompose the drift dynamics into a dissipative (exact) part and a non-dissipative (anti-exact) one. More precisely, a one-form for the system was obtained by taking the interior product of a non vanishing two-form with respect to the drift vector field. A radial homotopy operator centered at the desired equilibrium point for the system was designed. Applying this linear operator on the aforementioned oneform for the system, an exact one form generated by the desired potential function and an anti-exact form that generates the tangential dynamics were obtained. In the present paper, it is shown how this procedure, using the interior product of a non-vanishing two-form with the drift vector field X0 (x) of (1), leads to the construction of a first function V0 (an auxiliary scalar field following the nomenclature proposed in [11] and [6]) can be used locally by a second application of the decomposition method to obtain a control Lyapunov function for stability characterization. The idea of using an auxiliary scalar function V0 (x), with the property that ∇V0 (x) · X0 < 0, to obtain a control Lyapunov function for the control affine system (1) under damping feedback control was studied in [6] for homogeneous system (see for example [7]) using the flow of a vector field constructed using a linear combination of elements of {adjX0 Xk , j ∈ N, k = 1, . . . , m}. Another solution to the problem was presented in [12]. The existence of CLF for these closed-loop systems was reported in the literature (see for example [16]). The paper is organized as follows. Section II presents the mathematical background for the approach considered, including definitions of the radial homotopy operator that is used in the sequel. The main result of the paper, which is the construction of a suitable control Lyapunov function for control affine system based on a dissipative potential for the drift vector field, is presented in Section III. An illustration of the proposed approach is given in Section IV. Conclusion and future areas for investigation are outlined in Section V.
8008
FrC01.5 II. PRELIMINARIES In this section, basic elements of exterior calculus on Rn are reviewed. Of interest are the properties of the exterior derivative, the interior product, Lie derivative of exterior forms, and Cartan’s formula. We then proceed to present the properties of the radial homotopy operator. A complete account of exterior calculus and homotopy operators can be found in [5]. A. Exterior Calculus We denote a smooth vector field X ∈ Γ∞ (Rn ) as a smooth map n
n
X : R → TR ,
X|x =
n X
v i (x)∂xi |x ,
(3)
i=1
i.e. a map taking a point x ∈ Rn and assigning a tangent vector X|x ∈ Tx Rn . The cotangent (dual) space Tx∗ Rn is the set of all linear functionals on Tx Rn , Tx∗ Rn = {ω|x : Tx Rn → R}
(4)
2. d(α ∧P β) = dα ∧ β + (−1)deg α α ∧ dβ. ∂f )dxi , ∀f (x) ∈ Λ0 (Rn ). 3. df = i ( ∂x i 4. d ◦ dα = 0. A k-form α is said to be closed if dα = 0. It is said to be exact if there exists a (k − 1)-form β such that dβ = α. The interior product y is a map y : Γ∞ (Rn ) × Λk (Rn ) → Λk−1 (Rn ),
L : Λk (Rn ) → Λk (Rn ), (5)
The standard basis of Tx∗ Rn is given by {dx1 , . . . , dxn }, where dxi (∂xj ) = δji , δji being the Kronecker delta. An element ω|x in the cotangent space Tx∗ Rn can be written as n X ω|x = ωi dxi , ωi ∈ R. (6) i=1
In the sequel, differential one-forms are used. They are generated the following way. A differential one-form ω on Rn is a smooth map taking a point x ∈ Rn and assigning an element of its dual space Tx∗ Rn . We write ω=
n X
ωi (x)dxi ,
(7)
i=1
where ωi are smooth functions on Rn . The exterior (wedge) product ∧ is defined on Λ1 (Rn ) × Λ1 (Rn ) by the requirements dxi ∧ dxj dxi ∧ f (x)dxj
L[X1 ,X2 ] α = (LX1 LX2 − LX2 LX1 )α.
with the following properties: 1. d(α + β) = dα + dβ.
0 ≤ k ≤ n − 1,
(12)
(13)
B. Homotopy Operator
(8)
for all α, β, γ ∈ T ∗ Rn . If α ∈ Λk (Rn ), then we write deg α = k. Notice that Λ1 (Rn ) = T ∗ Rn and Λ0 (Rn ) = C ∞ (Rn ). The differential operator d is the unique operator on Ln Λ(Rn ) = k=0 Λk (Rn ), d : Λk (Rn ) → Λk+1 (Rn ),
(11)
We also define ad0X1 X2 = X2 and the k iterates for k = k 1, 2, . . . by adk+1 X1 X2 = [X1 , adX1 X2 ]. We complete this review with Cartan’s identity, which relate the Lie derivative of a differential form to exterior differentiation: LX ω = Xydω + d(Xyω).
for all smooth functions f (x) and by α ∧ (β + γ) = α ∧ β + α ∧ γ,
0 ≤ k ≤ n,
with the following properties: 1. LX f = Xydf . 2. LX (α + β) = LX α + LX β. 3. LX (α ∧ β) = (LX α) ∧ β + α ∧ (LX β). 4. LX dα = d(LX α). 5. Lf ·X α = f · LX α + df ∧ (Xyα). 6. LX1 +X2 α = LX1 α + LX2 α. 7. LX1 (X2 yα) = LX1 (X2 yα) − X2 y(LX1 α). The action of the Lie derivative for X1 , X2 ∈ Γ∞ (Rn ) is of course given by the Lie bracket LX1 X2 = [X1 , X2 ] and satisfies
= −dxj ∧ dxi = f (x)dxi ∧ dxj
(10)
with the following properties ∀α, β ∈ Λk (Rn ), ∞ n 0 n ∀X, X1 , X2 ∈ Γ (R ) and ∀f, g ∈ Λ (R ): 1. Xyf = 0. 2. Xyω = ω(X), ∀ω ∈ Λ1 (Rn ). 3. Xy(α + β) = Xyα + Xyβ. 4. (f X1 + gX2 )yα = f · (X1 yα) + g · (X2 yα). 5. Xy(α ∧ β) = (Xyα) ∧ β + (−1)deg(α) α ∧ (Xyβ). 6. X1 y(X2 yα) = −X2 y(X1 yα). The last property leads to the following composition rule for interior product: Xy(Xyα) = 0. The Lie derivative L is a map
where each ω|x is linear, i.e. (aω1 |x + bω2 |x )(X|x ) = aω1 |x (X|x ) + bω2 |x (X|x ).
0 ≤ k ≤ n,
(9)
In this section, we show how to construct a homotopy operator H, i.e., a linear operator on elements of Λ(Rn ) that satisfies the identity ω = d(Hω) + Hdω,
(14)
for a given differential form ω ∈ Λ(Rn ). The first step in the construction of a homotopy operator is to define a star-shaped domain on Rn . An open subset S of Rn is said to be star-shaped with respect to a point p0 = (x01 , . . . , x0n ) ∈ S if the following conditions hold: 0 • S is contained in a coordinate neighborhood U of p .
8009
FrC01.5 The coordinate functions of U assign coordinates (x01 , . . . , x0n ) to p0 . • If p is any point in S with coordinates (x1 , . . . , xn ) assigned by functions of U , then the set of points (x + λ(x − x0 )) belongs to S, ∀λ ∈ [0, 1]. A star-shaped region S has a natural associated vector field X, defined in local coordinates by
III. MAIN RESULT
•
X(x) = (xi − x0i )∂xi , ∀x ∈ S.
(15)
In this paper, we consider, for simplicity, the case where the star-shaped domain is centered at the origin, hence X(x) = xi ∂xi . For a differential form ω of degree k on a star-shaped region S centered at the origin, the homotopy operator is defined, in coordinates, as Z 1 (Hω)(x) = X(x)yω(λx)λk−1 dλ, (16)
We now present the main contribution of the paper. First, we recall the classical result on damping feedback stabilization from [10] following the treatment in [4] (see also [15] and [11]). Then, we restate the damping feedback controller construction in terms of differential forms and show under what conditions a dissipative potential function for the drift dynamics can be deformed to obtain a control Lyapunov function using a radial homotopy operator. The main features of the proposed method is that it results in a constructive algorithm for local CLF for systems satisfying the JurdjevicQuinn conditions and it leads to an explicit construction of the invariant set for the dynamics. A. Stabilization by Damping Feedback Consider a control affine system
0
x˙ = X0 (x) +
where ω(λx) denotes the differential form evaluated on the star-shaped domain in the local coordinates defined above. The important properties of the homotopy operator that are used in the sequel are the following: 1. H maps Λk (S) into Λk−1 (S) for k ≥ 1 and maps Λ0 (S) identically to zero. 2. dH + Hd = identity for k ≥ 1 and (Hdf )(x) = f (x) − f (x0 ) for k = 0. 3. (HHω)(xi ) = 0, (Hω)(x0i ) = 0. 4. XyH = 0, HXy = 0. The first part of the right hand side of (14), d(Hω), is obviously a closed form, since d ◦ d(Hω) = 0. Since by property (1) of the homotopy operator, for ω ∈ Λk (S), we have (Hω) ∈ Λk−1 (S), d(Hω) is also exact on S. We denote the exact part of ω by ωe = d(Hω) and the anti-exact part by ωa = Hdω. It is possible to show that ω vanishes on Rn if and only if ωe and ωa vanish together [5]. From the decomposition outlined above, we have ω − ωa = ωe .
(17)
Taking the exterior derivative on both sides and using the fact that ωe is closed, we have d(ω − ωa ) = d(ωe ) = 0.
(18)
In the sequel, we apply the homotopy operator on oneforms. Since in our applications, ωe is an exact one-form, (Hωe ) computed by homotopy is a dissipative potential for the system. We show in the sequel under what conditions this potential can be used to construct a damping feedback controller and under what conditions a second application of the decomposition leads to a control Lyapunov function stabilizing the origin. A non-dissipative potential is associated with the anti-exact part, but on the star-shaped domain S, ωa does not contribute to the dissipative part of the system. In other words, ωa belongs to the kernel of H, which can be seen by applying property (3) from above to the definition of ωa .
m X
uk Xk (x), ∀(x, u) ∈ Rn × Rm ,
(19)
k=1
for some X0 , · · · , Xm ∈ C ∞ (Rn ; Rn ) and assume that Xk (0) = 0, k = 0, . . . , m. Definition 3.1 (Assignable Lyapunov Functions): Let V0 be a positive definite and radially unbounded function. We say that a continuous u : Rn → Rm assigns V0 to be a Lyapunov function for the closed loop system (19) if the derivative of V0 along the trajectories of the closed loop system is negative definite, i .e., that for all x ∈ Rn \ {0}, LX0 · V0 +
m X
uk LXk · V0 < 0.
(20)
k=1
First, we recall the statement of Artstein’s theorem [1]. Theorem 3.2: Let V (x) be a positive radially unbounded function. There exists a continuous feedback that assigns V (x) to be a Lyapunov function for the closed loop system (19) if and only if (a) It is a control Lyapunov function, i .e., for all x ∈ Rn \ {0}, LXk V (x) = 0 ⇒ LX0 V (x) < 0, k = 1, . . . , m. (21) (b) It satisfies the small control property, i .e., for any > 0, there is a δ > 0 such that for all x ∈ Rn \ {0}, ( kuk < kxk < δ ⇒ ∃u (22) Pm LX0 + k=1 uk LXk V (x) < 0. We consider stabilization by damping feedback. Consider the feedback law u = (u1 , . . . , um )T defined by uk = −Xk (x) · ∇V0 (x), ∀ k ∈ 1, . . . , m.
(23)
With this feedback, one has for all x ∈ Rn \ {0} m
X dV0 = X0 (x) · ∇V0 (x) − (Xk (x) · ∇V0 (x))2 < 0. (24) dt k=1
n
Therefore, 0 ∈ R is a stable point for the closed-loop system x˙ = X(x, u(x)). By Lasalle’s invariance principle,
8010
FrC01.5 0 ∈ Rn is globally asymptotically stable if for every x(t) ∈ C ∞ (R; Rn ), for all t ∈ R, x(t) ˙ Xk (x(t)) · ∇V0 (x(t))
= X0 (x(t)), =
(25)
0, ∀k ∈ 0, . . . , m, (26)
one obtains x(t) = 0. An alternate formulation of this result is given by the following theorem (a proof of this statement can be found in [4]). Theorem 3.3: Given the smooth control affine system (19) and a function V0 such that LX0 V0 < 0 for every x ∈ R\{0}. Suppose moreover that span{X0 (x), adkX0 Xi (x)
: i = 1, . . . , m, k ∈ N} = R
n
(27)
on Rn \ {0}. Then the feedback law uk = −LXk V0 , ∀i ∈ 1, . . . , m
(28)
globally asymptotically stabilizes the control affine system (19). As noted in [4], the function V0 (x) is not a CLF in general. We refer to [6] and [12] for other CLF construction methods based on the prior knowledge of a function V0 satisfying the weak Jurdjevic-Quinn conditions (see [11, Definition 2.2]). In the next section, we show under which conditions those results can be obtained using the tools presented in Section (II). B. Damping Feedback and Deformation using Homotopy Operator First, we define a non-vanishing closed two-form Ω on Rn as X Ω= dxi ∧ dxj . (29)
arguments. The positive closed one-forms, i .e., hω0 , X0 i > 0, was generated as ω0 = −dV0 . In our case, it will be clear when the negative sign should be used, depending on the orientation of the basis for (29). We assume that V0 (x), obtained after application of the homotopy operator (i .e., V0 = (Hω0,e )), is such that LX0 V0 < 0 for x ∈ Rn \ {0}. If it is not the case, it is always possible to apply a change of coordinates (see for example [2]) to ensure that the above contraction with respect to the drift vector field generates a one-form with the desired properties. We construct the damping feedback controller uk (x) = −LXk V0 .
By the first property of the Lie derivative from Section (II-A), we have LXk V0 = Xk ydV0 , which is given as Xk yω0,a − Xk yω0 by the homotopy decomposition. We now consider the affine system under the feedback law, i .e., we consider the system X0 −
The orientation of the two-form will be fixed, if necessary, by checking the sign of the obtained dissipative function. We obtain a first one-form associated to the system by contracting this two-form with respect to the drift vector field, ω0 = X0 yΩ.
(31)
From Section III-B, we know that we can locally construct a homotopy operator on Rn such that ω0 = ω0,e + ω0,a . Since ω0,e is exact, it is given as the exterior derivative of a potential function and we rewrite ω0 = −dV0 + ω0,a .
(32)
Remark 3.4: The negative sign for −dV0 is used to comply with the notation introduced in [3], where a known closed positive one-form ω0 (i .e., ω0,a = 0) was used for stability
LXk V0 · Xk .
(34)
Taking now the interior product of this closed-loop vector field with respect to the non-vanishing two-form Ω, we obtain the one-form ! m X ω = X0 − (Xk yω0,a − Xk yω0 )Xk yΩ. (35) k=1
Since (Xk yω0,a − Xk yω0 ) ∈ Γ∞ (Rn ) and by the property (4) of the interior product, we can re-write this one-form as ω
= X0 yΩ −
m X
(Xk yω0,a − Xk yω0 ) · (Xk yΩ)(36)
k=1
= ω0 +
In the present paper, the non-vanishing two-form Ω is not necessarily defined in a canonical way, since the objective is ultimately to compute an admissible Lyapunov function (and not a minimal one). For example, if n = 3, we would have, (30)
m X k=1
1≤i<j≤n
Ω = dx1 ∧ dx2 + dx1 ∧ dx3 + dx2 ∧ dx3 .
(33)
= ω0 −
m X
(Xk yω0,e )(Xk Ω)
i=1 k X
fk (Xk yΩ),
(37)
(38)
k=1
where the fk ’s account for the deformation of the potential function with respect to the controlled vectors. The one-form ω will serve as the basis for the computation of a local CLF using the homotopy operator defined in Section III-B. To simplify the computations and to follow the construction from [3], we will assume in the sequel that the dissipative potential V0 (x) computed above is such that ω0,a ≡ 0. This is equivalent to assume that V0 (x) is a first integral for the system. We can now restate a local form of the result from Section III-A in terms of differential forms. Theorem 3.5: Assume that for all x ∈ Rn \ {0}, one can construct V0 such that ω0 is exact, i .e., ω0 = −dV0 6= 0 and that dV0 vanishes at the origin. Then, the damping feedback uk = −Xk ydV0 locally asymptotically stabilizes the system at the origin. Moreover, a local CLF for the closed-loop system can be computed using V = (Hω), with ω given by (38), if the anti-exact part ωa = Hdω vanishes only at the origin.
8011
FrC01.5 Proof: The first part of the theorem is a re-statement of the original result. Consider the closed loop system under damping feedback control with V0 as required: X0 −
m X
LXk V0 · Xk .
(39)
k=1
Then V˙ 0 = X0 · ∇V0 −
m X
(Xk · ∇V0 )2 .
(40)
k=1
In terms of differential forms, we have V˙ 0 = X0 ydV0 −
m X
2
(Xk ydV0 ) .
(41)
k=1
The second term of the right hand side is obviously negative definite. Since X0 ydV0 = X0 y(ω0,a − ω0 ), which can be decomposed by the properties of the interior product as X0 ydV0
Since by assumption, ω0 = −dV0 vanishes at the origin, we are left with n X X0 yXk yΩ = 0, x = 0, (51)
=
X0 yω0,a − X0 yω0
(42)
=
X0 yω0,a − X0 yX0 yΩ
(43)
= X0 yω0,a .
(44)
k=1
which is guaranteed by the original assumption on the vector fields Xk (0) = 0, k = 0, 1, . . . , m. Under the assumption that dV0 6= 0, ∀ x ∈ Rn \ {0} and that ωa vanishes only at the origin, the maximal invariant set is therefore {0}. By the usual Lasalle’s arguments, damping feedback stabilizes asymptotically the origin of the system. It remains to show that V is a CLF on the star-shape domain. Restating condition (21), we have that V (x) is a CLF provided that ∀ x ∈ Rn \ {0}, Xk ydV = 0 ⇒ X0 ydV < 0, k = 1, . . . , m. Consider V˙ =
V˙ 0 = −
2
(Xk ydV0 ) < 0,
∀ x ∈ Rn \ {0},
k X
(Xk yω0,a − Xk yω0 ) · (Xk yΩ).
(46)
k=1
By assumption ω0,a = 0 and we can re-express the second term of the right hand side as (−Xk yω0 ), which is given, for all k = 1, . . . , m, as −Xk yω0
=
−Xk yX0 yΩ
(47)
=
X0 yXk yΩ.
(48)
Hence, we have ω = ω0 −
k X
(X0 yXk yΩ) · (Xk yΩ).
(49)
dV = ω0 −
m X
X0 yXk yΩ = 0, x = 0.
· ∇V.
(53)
m X
(LXk V0 ) · Xk ydV.
(54)
If Xk ydV = 0 for all k = 1, . . . , m, we observe that since we use the same non-vanishing two-form in both steps of the construction, we have no deformation. Hence V obtained by application of the homotopy operator is the same as V0 , and by the above discussion, we have that X0 ydV = X0 ydV0 < 0. As a consequence, V˙ < 0 on Rn \ {0} and V is a CLF for the system. Remark 3.6: The exactness condition on ω0 could be relaxed, provided some convexity conditions on (Hω0 ), which is possible to compute only if a structure is assumed on the drift vector field. The same remark on “classical” JurdjevicQuinn conditions were discussed in [6]. As mentioned earlier, restating the results from Section III-A using differential forms is practical in the sense that locally, one can obtained a CLF by application of the homotopy operator on the one-form constructed using the closed-loop system. The structure of ωa leads to an expression of the invariants set for the closed-loop dynamics. An application of this construction is presented in the next Section to illustrate these observations.
k=1
Applying the homotopy operator on ω, one obtains ω = ωe + ωa . From Section II-B, we have that ω = 0 if and only if ωa and ωe vanish at the same time. By assumption, ωa vanishes only at the origin, at least on the star-shaped domain where the homotopy operator is defined. Hence, we have to show that the exact part of ω, i .e., dV vanishes at the origin. We have the requirement
LXk V0 · Xk
k=1
(45)
under the assumption that dV0 6= 0, ∀ x ∈ Rn \ {0}. We now need to shown that the origin is the only invariant for the closed-loop system. We consider the deformation of V0 to compute a CLF using the homotopy operator on a one-form ω. The one-form ω is given by
!
In terms of interior products, V˙ = X0 ydV −
k=1
ω = ω0 −
X0 −
m X k=1
By assumption, ω0,a ≡ 0, and we obtain m X
(52)
IV. APPLICATION EXAMPLE We apply the above construction to a controlled pendulum example, taken from [15]. Consider the control affine system x˙ 1
=
x˙ 2
= − sin(x1 ) + u.
x2
(55) (56)
Setting Ω = dx1 ∧dx2 , a first dissipative potential is obtained with ω0 = X0 yΩ, given as
(50)
k=1
8012
ω0 = sin(x1 )dx1 + x2 dx2 .
(57)
FrC01.5 Applying the radial homotopy operator defined in Section II-B, we obtain 1 V0 = (Hω) = x22 + (1 − cos(x1 )), (58) 2 which is of the desired form locally. In this particular case ω0 is already exact, hence ωa = ω0 − ωe = 0. The obtained function V0 is positive definite, but it is not a CLF (it is in fact a first integral, i .e., LX0 V0 ≡ 0). The damping feedback controller u(x) is given as −X1 ydV0 = −X1 yωe = −x2 . Applying the radial homotopy operator on the new one-form, one has ω = x2 dx1 − (x2 + sin(x2 ))dx2 which results after integration leads to the local CLF 1 V = 1 − cos(x1 ) + (x1 x2 + x22 ). 2 Computing ∇V · X(x, u(x)), we have 1 V˙ = − x22 + x1 x2 + x1 sin(x1 ) < 0, kxk < 1. 2 Figures 1 and 2 present the locally obtained V (x) and in a neighborhood of the origin.
Fig. 1.
V (x) for the mechanical example
Fig. 2.
V˙ (x) for the mechanical example
V. CONCLUSION
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In this paper, we considered the problem of control Lyapunov functions construction for the stabilization of nonlinear control affine systems satisfying Jurdjevic-Quinn conditions. Provided that we can obtain a positive definite function (a dissipative potential) by taking the interior product of a nonvanishing two-form with respect to the drift vector field, we showed how a CLF can be computed for the closed-loop vector field subject to damping feedback control under some conditions on the anti-exact part of a one-form obtained for the system. The proposed method is local since the approach relies on a homotopy operator centered at a desired equilibrium point (herein, at the origin). Further research will focus on systematic computation of the domain of attraction. Another possible investigation is the characterization of the non-exact part, ωa , when it is not vanishing only at the origin. One problem is to decide if the non-exact part contains information about stable complex patterns, for example limit cycles.
V˙ (x)
R EFERENCES
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(60)
One important aspect to note here is that the one-form ω is not exact. In fact, after the second application of the homotopy operator, we have 1 ωa = (x2 dx1 − x1 dx2 ) . (62) 2 This non-exact part vanishes only at the origin, hence the largest invariant set for the dynamics, as outlined by the nonexact one-form is the origin {0}.
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