Asymptotic stabilization with locally semiconcave control Lyapunovfunctions on general manifoldsHisakazu Nakamura a,∗, Takayuki Tsuzuki b, Yoshiro Fukui c, Nami Nakamura a
a Department of Electrical Engineering, Faculty of Science and Technology, Tokyo University of Science, Yamazaki 2641, Noda, Japanb Interdisciplinary Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu, Matsue, Japanc Ritsumeikan Global Innovation Research Organization (R-GIRO), Ritsumeikan University, 1-1-1 Noji-higashi, Kusatsu, Shiga, Japan
a r t i c l e i n f o
Article history:Received 24 January 2010Received in revised form22 May 2013Accepted 18 June 2013Available online 3 August 2013
Asymptotic stabilization on noncontractible manifolds is a difficult control problem. If a configurationspace is not a contractible manifold, we need to design a time-varying or discontinuous state feedbackcontrol for asymptotic stabilization at the desired equilibrium.
For a system defined on Euclidean space, a discontinuous state feedback controller was proposed byRiffordwith a semiconcave strict control Lyapunov function (CLF). However, it is difficult to apply Rifford’scontroller to stabilization on general manifolds.
In this paper, we restrict the assumption of semiconcavity of the CLF to the ‘‘local’’ one, and introducethe disassembled differential of locally semiconcave functions as a generalized derivative of nonsmoothfunctions. Further, we propose a Rifford–Sontag-type discontinuous static state feedback controller forasymptotic stabilization with the disassembled differential of the locally semiconcave practical CLF (LS-PCLF) bymeans of sample stability. The controller does not need to calculate limiting subderivative of theLS-PCLF.
Moreover, we show that the LS-PCLF, obtained by the minimum projection method, has a special ad-vantage with which one can easily design a controller in the case of the minimum projection method.Finally, we confirm the effectiveness of the proposed method through an example.
Asymptotic stabilization on noncontractible manifolds, a diffi-cult control problem [1–3], has been studied by a few researchers[1,2,4–7]. The main problem is that a noncontractible manifold, asa configuration space, never has a continuous asymptotically sta-bilizing static state feedback control at any desired equilibrium.Hence one needs to design a discontinuous or time-varying sta-bilizing controller [3].
Control Lyapunov functions (CLFs) play an important role infeedback control design [3,8,9]. In particular, semiconcave strictCLFs enable designing discontinuous asymptotic stabilizing con-trollers [10]. Rifford proposed a discontinuous controller, definedon Euclidean space, based on semiconcave strict CLFs [10]. How-ever, Rifford’s controller cannot be directly applied to stabilizationon manifolds or systems with unbounded inputs.
In this paper, we introduce the framework of a locally semi-concave practical CLF for stabilization on manifolds. We consider
the disassembled differential instead of the limiting subderiva-tive of a locally semiconcave function. Then, we show that thedirectional subderivative used in the definition of the practicalCLF is replaced with the disassembled differential. Further, wepropose a Rifford–Sontag-type discontinuous asymptotically sta-bilizing static state feedback controller with the disassembled dif-ferential of the locally semiconcave practical control Lyapunovfunction (LS-PCLF) by means of sample stability.
For general differentiable manifolds, we proposed the mini-mum projection method to design a locally semiconcave strict CLF[11,12], but we did not show how to stabilize the origin of controlsystems defined on manifolds with the LS-PCLFs. In this paper, weshow that the locally semiconcave CLF, obtained by the minimumprojectionmethod, is particularly advantageous for calculating thedisassembled differential. Therefore, one can easily design a con-troller when the LS-PCLF is obtained by the minimum projectionmethod.
2. Preliminaries
2.1. Differentiable manifolds
A brief introduction of differentiable manifolds is necessary todiscuss the control systems defined on manifolds [13,14]. In this
H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909 903
paper,X denotes an n-dimensional smoothmanifold, TxX a vectorspace called the tangent space to X at x, and an element of TxX atangent vector at x. T ∗
x X denotes the dual space to TxX, called thecotangent space at x, and an element of T ∗
x X a cotangent vector (ora differential 1-form) at x. A subsetM ⊂ X is said to be precompactin X, if its closure in X is compact. For each x ∈ X, there exists alocal chart (W, η) such that W ⊂ X and η : W → Y = Im(η) ⊂
Rn is a homeomorphism. Then, η(x) is called the local coordinaterepresentation of xwith the chart (W, η).
Consider a function V : X → R and a chart (W, η) for X. Then,the function VW : Y → R, defined by VW (ξ) = V η−1(ξ), iscalled the coordinate representation of V . Note that VW is definedon a subset of Rn. Therefore, addition and scalar multiplication aredefined as usual. By the same manner, fW denotes the local coor-dinate representation of f ∈ TxX. Let V : X → R be a smoothfunction. The mapping dV : X → T ∗
x X denotes the differentialof V . Let (ξ1, . . . , ξn) be local coordinates of X with a local chart(W, η). Then, the mapping dVW can be defined by
dVW (η(x)) =
ni=1
∂VW
∂ξi(η(x))dξi. (1)
The natural pairing ⟨dV , f ⟩ between a cotangent vector and a tan-gent vector is defined by Lie derivative as follows: ⟨dV , f ⟩ = Lf V .In local coordinates,
⟨dV (x), f (x)⟩W =
ni=1
∂VW
∂ξifW i(η(x)). (2)
If X and X are smooth manifolds and φ : X → X is a smoothmapping, for each x ∈ X, the mapping φ∗ : TxX → Tφ(x)X de-notes the differential (or the pushforward) of φ. Let (ξ1, . . . , ξn)and (ξ1, . . . , ξn) be local coordinates of X and X at φ(x) and xwith local charts (W, η) and (W, η), respectively. Then, the map-ping φ∗W,W : Y → Y can be defined by
φ∗W,W
∂
∂ξi
=
nj=1
∂(η φ η−1)j
∂ξi
∂
∂ξj(1 ≤ i ≤ n). (3)
A function V : X → R is called locally semiconcave (with linearmodulus [15]) if for any chart (W, η) and compact set M ⊂ W ,there exists C > 0 such that
V (x)+ V (y)− 2VW
12(η(x)+ η(y))
≤ C∥η(x)− η(y)∥2 (4)
for all x, y ∈ M satisfying (η(x)+ η(y))/2 ∈ η(M). Note that, theexistence of C does not depend on the choice of charts.
2.2. Control systems defined on differentiable manifolds
We consider the following nonlinear control system on a finite-dimensional arc-connected C1-differentiable manifold X:
x = f (x, u), (5)
where x ∈ X, u ∈ U ⊂ F(R,Rm); t → u(t) ∈ U ⊂ Rm, andwhere F(R,Rm)denotes a set ofmappings fromR toRm.Moreover,a mapping f : X × U → TxX is assumed to satisfy f (0, 0) = 0,where 0 ∈ X, called the origin, is the desired equilibrium, andlocally Lipschitz continuous with respect to x; i.e., for a fixed u0 ∈
U, a local chart (W, η) and a compact set M ⊂ W , there exists Lsuch that
for all x, y ∈ M.A function k : X → U is called a static state feedback (or sim-
ply feedback). The objective of the paper is to develop an asymp-totically stabilizing static state feedback controller u = k(x) at theorigin of (5).
We consider the sample-and-hold solution, defined as follows,as solutions of (5).
Definition 1 (Partition [5,16]). Any infinite sequence π = ti ∈
R≥0i∈Z≥0 consisting of numbers 0 = t0 < t1 < t2 < · · · withlimi→∞ ti = +∞ is called a partition and the number d(π) :=
supi∈Z≥0(ti+1 − ti) its diameter.
Definition 2 (Sample-and-Hold Solution [5,16,17]). Let u = k(x) bea given feedback, π a partition, and x ∈ X an initial state. Thesample-and-hold solution ψ(t, x, k(x)) : R≥0 × X × U → X for(5) is defined as the mapping such thatψ(t, x, k(x)) = x(t), wherex(t) is a continuous mapping obtained by recursively solving
x(t) = f (x(t), k(x(ti))) (7)
from the initial time ti to the maximal time
si = max ti, sups ∈ [ti, ti+1]|x(·) is defined on [ti, s) , (8)
with x(0) = x.
The feedback u = k(x) implicitly determines the control u(t) =
k(x(t)) by the sample-and-hold solution. Note that every sample-and-hold solution is absolutely continuous. Then, the followinglemma holds:
Lemma 1. Consider a diffeomorphism φ : X → M, where M ⊂ X.Then, φ−1(ψ(t, x, k(x))) is a sample-and-hold solution of ˙x = φ−1
∗
f (φ(x), k(φ(x))) if and only if ψ(t, x, k(x)) is a sample-and-hold so-lution of (5) on M.
We define sample stability as follows [5, s-stability]:
Definition 3 (Sample Stability). Consider system (5).P denotes theset of all open precompact subset of X containing the origin.
A feedback k : X → U is said to sample stabilize the origin ofthe system (5) if the following holds for arbitrary sets R1,R2 ∈ P
such that R1 ⊂ R2.
(1) There exists a set M ⊂ X depending only upon R2 and twopositive numbersΩ, T > 0depending onR1 andR2 such that,for any initial value x ∈ R2, for any partition π of the diame-ter less than Ω , the corresponding sample-and-hold solutionψ(t, x, k(x)) satisfies the following conditions:(a) ψ(t, x, k(x)) ∈ R1 for all t ≥ T ,(b) ψ(t, x, k(x)) ∈ M for all t ≥ 0.
(2) for each E ∈ P, there exists a set P ∈ P such that if R2 ⊂
P ,M in (1) can be chosen satisfying M ⊂ E .
3. Locally semiconcave control Lyapunov functions
3.1. Locally semiconcave practical control Lyapunov functions
Strict CLFs are commonly used for the development of anasymptotically stabilizing controller. For discontinuous control de-sign, semiconcave strict CLF was introduced by Rifford [10]. Thelocally semiconcave strict CLF is defined as follows.
Definition 4 (Locally Semiconcave Strict CLF).A global locally semi-concave strict control Lyapunov function for system (5) is a locallysemiconcave function V : X → R such that the following proper-ties hold:
(A1) V is proper; that is, the set x ∈ X|V (x) ≤ L is compact forevery L > 0.
(A2) V is positive definite; that is, V (0) = 0, and V (x) > 0 for allx ∈ X \ 0.
904 H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909
(A3) there exists a control u admissible for x, a continuous positivedefinite function Q : X → R, and a local chart (W, η) suchthat
DVW (η(x); fW (x, u)) < −Q (x), ∀x ∈ X \ 0, (9)
where the directional subderivative DVW of the semiconcavefunction VW is defined as follows [18]:
DVW (η(x); vW ) = limt↓0
VW (η(x)+ tvW )− V (x)t
. (10)
Remark 1. Note that DVW is well-defined for every locally semi-concave function by [15, Theorem 3.2.1]. Moreover, if a directionalvector v is defined on the tangent space TxX, DVW (η(x); vW ) doesnot depend on choices of local coordinates (W, η).
However, strict CLFs are not appropriate for noncompact U. Seethe following example.
Example 1. Consider x = u, where x ∈ R and U = R, as also thefollowing function V : R → R:
V (x) = minx2, (x − 1)2 + 1. (11)
The function V is a locally semiconcave strict CLF; however, theredoes not exist a compact set U ⊂ R and a continuous positivedefinite function Q such that
minu∈U
DV (x; u) < −Q (x) (12)
in the neighborhood of x = 1.
Hence, we introduce the locally semiconcave practical controlLyapunov function (LS-PCLF) as follows:
Definition 5 (LS-PCLF). A locally semiconcave practical controlLyapunov function (LS-PCLF) for system (5) is a locally semiconcavefunction V : X → R such that (A1), (A2) and the followingproperty hold:
(A3′) for arbitrary R1, R2 ∈ R>0 such that R2 > R1 > 0, there exista compact set U ⊂ U, a positive real constant Q , and a localchart (W, η) such that
minu∈U
DVW (η(x); fW (x, u)) < −Q ,
∀x ∈ x ∈ X|R1 ≤ V (x) ≤ R2. (13)
3.2. Disassembled differential
The limiting subderivative of the semiconcave strict CLF hasbeen employed in the Rifford controller [10], and proximal sub-derivative becomes necessary for calculating the limiting sub-derivative. However, the proximal subderivativemay be empty fora semiconcave function [15]. This implies that the computationof the limiting subderivatives is difficult in many cases. That ex-plains the need to introduce the disassembled differential of a lo-cally semiconcave function.
Reverting Euclidean spaces, locally semiconcave functions havethe following good property [15, Theorem 3.4.2]:
Theorem 1 ([15]). Let V : Rn→ R be a locally semiconcave
function. Then, V can be locally written as the minimum of class C2
functions. More precisely, for any compact M ⊂ Rn, there exist acompact set S ⊂ R2n and a family of functions Vss∈S such that eachVs : M → R is C2 with respect to x and
V (x) = mins∈S
Vs(x), ∀x ∈ M. (14)
The theorem claims that every locally semiconcave function canbe considered as the minimum of smooth functions. Of course, thefollowing corollary holds for manifolds:
Corollary 1. Let V : X → R be a locally semiconcave function.For any compact M ⊂ W with a local chart (W, η), there exist acompact set S ⊂ R2n and a family of functions Vss∈S such that eachVs : M → R is C2 with respect to x and
V (x) = mins∈S
Vs(x), ∀x ∈ M. (15)
Hence, in this paper, the derivatives of the smooth functions havebeen used, and disassembled differential of a locally semiconcavefunction is defined as follows:
Definition 6 (Disassembled Differential). Suppose that V : X → Ris a locally semiconcave function. Then, the following set-valuedmap DV : X → 2T∗
x X is said to be a disassembled differential of V :
DV (x) =
dVs(x)
s ∈ s ∈ S|V (x) = Vs(x). (16)
The disassembled differential can be written as follows in the localchart (W, η):
DVW (x) =
∂ VsW
∂ξ(η(x))dξ
s ∈ s ∈ S|V (x) = Vs(x)
. (17)
The disassembled differential DV , the reachable differential (theset of reachable gradients) D∗V [15] and the limiting subdifferen-tial ∂LV [18] are similar concepts. Note that, the following relationholds for locally semiconcave functions.
DV ⊃ D∗V = ∂LV (18)
3.3. Local semiconcavity and control Lyapunov functions
In this subsection, we consider the locally semiconcave CLF inthe context of the disassembled differential. For locally semicon-cave functions defined on Euclidean spaces, we can use the follow-ing lemma by direct deduction of [15, Theorems 3.3.6 and 3.4.4]:
Lemma 2.
DVRn(η(x); v) = minp∈coDVRn (x)
⟨p, v⟩, (19)
where co denotes the convex hull.
The mapping co depends on the choice of coordinates. How-ever, the following fact holds according to Carathéodory theorem[19, p. 62]:
Fact 1. DVW (η(x); v) < −Q (x) if and only if there exists a mappingp : X → T ∗
x X such that p(x) ∈ DV (x) and ⟨p(x), v⟩ < −Q (x).
Then, the following theorem holds.
Theorem 2. Consider the control system (5) and a locally semicon-cave function V : X → R satisfying (A1) and (A2). Then, functionV is a practical control Lyapunov function if and only if the followingcondition (A3′′) is satisfied:
(A3′′) for arbitrary R1, R2 ∈ R>0 such that R2 > R1 > 0, there exista compact set U ⊂ U, a positive real constant Q , a mappingp : X → T ∗
x X such that p(x) ∈ DV (x) such that
minu∈U
⟨p(x), f (x, u)⟩ < −Q ,
∀x ∈ x ∈ X|R1 ≤ V (x) ≤ R2. (20)
H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909 905
4. Asymptotic stabilization with locally semiconcave controlLyapunov functions
4.1. Sample stabilization with locally semiconcave control Lyapunovfunctions
In this section, we consider an asymptotically stabilizing con-troller design problem for the following control-affine system:
x = f (x)+ g(x)u (21)
:= f (x)+
mi=1
gi(x) · ui, (22)
where x ∈ X and u ∈ F(R,Rm); t → u(t) ∈ Rm. Moreover, map-pings f , gi : X → TxX are assumed to be locally Lipschitz contin-uous with respect to x for all i ∈ 1, . . . ,m, and satisfies f (0) = 0.
We introduced the disassembled differential for locally semi-concave functions and comment on some properties of thedisassembled differential. In this subsection, we design an asymp-totically stabilizing controller with an LS-PCLF by means of samplestability.
The following theorem regarding the Rifford–Sontag-type [9,10] controller design is the main theorem of the paper:
Theorem 3. Assume that V is a locally semiconcave practical controlLyapunov function for (21). Consider a static state feedback controlu = k(x) such that
ki(x) =
−
⟨p, f ⟩ +
⟨p, f ⟩2 +
mi=1
⟨p, gi⟩22
mi=1
⟨p, gi⟩2⟨p, gi⟩
mi=1
⟨p, gi⟩2 = 0
0
mi=1
⟨p, gi⟩2 = 0
,
(23)
where p(x) ∈ DV : X → T ∗x X is a mapping satisfying condi-
tion (A3′′).Then, if d(π) is sufficiently small, (23) sample stabilizes the origin
of the control system (21).
4.2. Proof of Theorem 3
To prove Theorem 3, we prove the following three lemmas.
Lemma 3. Let R2 > R1 > 0 be constants. Then, there exists G > 0such that
mi=1 k
2i (x) < G for all x ∈ R = x ∈ X|R1 ≤ V (x) ≤ R2.
Proof. Taking into account that R is compact, it is sufficient toprove that there exists no convergent sequence xjj∈N such thatlimj→∞
mi=1 k
2i (xj) = +∞.
If limj→∞
mi=1⟨p(xj), gi(xj)⟩
2= 0, limj→∞
mi=1 k
2i (xj) <
+∞.The other case is limj→∞
mi=1⟨p(xj), gi(xj)⟩
2= 0. According to
the assumption that V is an LS-PCLF, for each j ∈ N
minu∈U
⟨p(xj), f (xj)⟩ +
mi=1
⟨p(xj), gi(xj)⟩ui
< −Q . (24)
Therefore, ⟨p(xj), f (xj)⟩ < 0 as j → ∞. In this case, there exists aconstant J > 0 such that ⟨p(xj), f (xj)⟩ < 0 for all j > J .
Note that
⟨p, f ⟩2 +
mi=1
⟨p, gi⟩22
≤
|⟨p, f ⟩| +
mi=1
⟨p, gi⟩22
, (25)
and
|⟨p, f ⟩| ≤
⟨p, f ⟩2 +
mi=1
⟨p, gi⟩22
≤ |⟨p, f ⟩| +
mi=1
⟨p, gi⟩2. (26)
Hence, for each x ∈ R, there exists H ∈ [0, 1] such that⟨p, f ⟩2 +
mi=1
⟨p, gi⟩22
= |⟨p, f ⟩| + Hmi=1
⟨p, gi⟩2. (27)
Then, there exists a function h : R → [0, 1] such that
ki(xj) = −h(xj)⟨p(xj), gi(xj)⟩mi=1
⟨p(xj), gi(xj)⟩2 = 0 and j > J
.
(28)
According to the semiconcavity of V , limj→∞
mi=1 k
2i (xj) <
+∞.
Lemma 4. There exist a constant P > 0 and a mapping p : X →
T ∗x X such that p(x) ∈ DV (x) satisfying
⟨p(x), f (x)+ g(x)k(x)⟩ < −P (29)
for all x ∈ R = x ∈ R|R1 ≤ V (x) ≤ R2.
Proof. Since V is an LS-PCLF, there exist amapping p and a positiveconstant Q such that
⟨p(x), f (x)⟩ + minu∈U
mi=1
⟨p(x), gi(x)⟩ui < −Q (30)
for all x ∈ R. This implies that if ⟨p(x), f (x)⟩ ≥ −Q/2,
minu∈U
mi=1
⟨p(x), gi(x)⟩ui < −Q/2. (31)
Hence, there exists a constant S > 0 such thatmi=1
⟨p(x), gi(x)⟩2 ≥ S (32)
for all x ∈ x ∈ R|⟨p(x), f (x)⟩ ≥ −Q/2. Then,
⟨p(x), f (x)+ g(x)k(x)⟩ ≤ −
⟨p, f ⟩2 +
mi=1
⟨p, gi⟩22
(33)
≤ −minQ/2, S. (34)
Therefore, let P = minQ/2, S, and we have
⟨p(x), f (x)+ g(x)k(x)⟩ < −P (35)
for all x ∈ R.
Lemma 5. There exists a positive constant Ω such that the followinginequality holds uniformly on R:
V (ψ(t, x, k(x)))− V (x) ≤ −Ωt (36)
for all t ≥ 0.
906 H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909
Proof. Since the set R is compact, R is covered by finitely manycharts (Wj, ηj) (j ∈ J := 1, . . . , J) [14, Proof of Lemma 1.6]. Thisdeduces that for each x ∈ R, there exist a chart (Wj, ηj) and D > 0such that y|∥ηj(y)− ηj(x)∥ ≤ D ⊂ Im(ηj) [20, Lemma 4.21].
By local Lipschitz continuity of f and g , there exists L such that
∥fWj(ξ)+ gWj(ξ)k(x)− fWj(ηj(x))− gWj(ηj(x))k(x)∥
< L∥ξ − ηj(x)∥ (37)
for all ξ ∈ Im(ηj) and j ∈ J. Moreover, there existsM such that
∥fWj(ηj(x))+ gWj(ηj(x))k(x)∥ < M (38)
for all x ∈ Wj and j ∈ J according to Lemma 3. Then, for eachx ∈ R, there exists j ∈ J such that
ψ(t, x, k(x)) ∈ Wj (39)
for all t ∈ [0,D/M].According to the local semiconcavity of V and [15, Proposition
3.3.1], the following holds for all t ∈ [0,D/M]:
V (ψ(t, x, k(x)))− V (x) ≤ ⟨p(x), ψ(t, x, k(x))− x⟩
+ C∥ηj(ψ(t, x, k(x)))− ηj(x)∥2. (40)
The mean value theorem implies that there exists t∗ ∈ [0, t] suchthat
V (ψ(t, x, k(x)))− V (x)
≤ ⟨p(x), f (ψ(t∗, x, k(x)))+ g(ψ(t∗, x, k(x)))k(x)⟩t
+ C∥ηj(ψ(t, x, k(x)))− ηj(x)∥2. (41)
Since V is locally Lipschitz, there exists a Lipschitz constant K suchthat
|V (x)− V (y)| ≤ K∥ηj(x)− ηj(y)∥. (42)
Then by the same discussion as [21], we obtain the following in-equality:
V (ψ(t, x, k(x)))− V (x) (43)
≤ ⟨p(x), f (x)+ g(x)k(x)⟩t
+ ⟨p(x), f (ψ(t∗, x, k(x)))+ g(ψ(t∗, x, k(x)))k(x)
− f (x)− g(x)k(x)⟩t + C∥ηj(ψ(t, x, k(x)))− ηj(x)∥2 (44)
Therefore, for d(π) satisfying d(π) ≤ minP/(2M(KL + CM)),D/M,
V (ψ(t, x, k(x)))− V (x) ≤ −12Pt (47)
for all t ≥ 0. WithΩ = P/2, the lemma was proved.
Then, we can prove Theorem 3 using the proof of Theorem 21in [16].
Proof of Theorem 3. In the set R, Lemmas 4 and 5 hold. Then, fort ≥ (R2 − R1)/Ω, V (ψ(t, x, k(x))) < R1. The rest of the proof isclear.
Remark 2. The proposed controller is different from [10]. This isbecause Rifford’s controller requires some extra conditions for Vand for avoiding computation of proximal subgradient of an LS-PCLF V . If V satisfies Rifford’s condition, Rifford’s controller is alsovalid for practical stabilization at the origin of (21).
5. Practical stabilization with minimum projection method
According to the preceding discussion,we can design an asymp-totic stabilizing controllerwith an LS-PCLF bymeans of sample sta-bility. To design an LS-PCLF onmanifolds, we can use theminimumprojection method. In this section, we show how the LS-PCLF gen-erated by the method has an advantage in controller design.
5.1. Minimum projection method for strict control Lyapunov functiondesign
In this subsection, we introduce the minimum projectionmethod [11,12] for CLF design on noncontractible manifolds. Themethod considers a single manifold X different from the originalmanifold X.
Consider the control-affine system (21) defined onX. Then, theminimum projection method is summarized as follows:
(M1) choose amanifold X associatedwith a surjectionφ such thatthe continuous mapping φ∗ : TxX → Tφ(x)X is bijective forall x ∈ X and φ(0) = 0.
(M2) design a strict control Lyapunov function V on X for asymp-totic stabilization of the origin of the following control sys-tem:
˙x = f (x)+ g(x) · u := φ−1∗
f (φ(x))+ φ−1∗
g(φ(x)) · u. (48)
(M3) the following function is an LS-PCLF on X:
V (x) = minx∈φ−1(x)
V (x). (49)
With the method, the following theorem was proved [11,12].
Theorem 4. If V defined on X is a locally semiconcave practical con-trol Lyapunov function with respect to x ∈ X, the function V definedby (49) is a locally semiconcave practical control Lyapunov functionwith respect to x ∈ X.
In this paper, we suppose that an LS-PCLF V on another manifoldis sufficiently smooth. That is, we assume that V satisfies thefollowing condition in every local chart:
infu
Lf V + Lg V · u
< 0. (50)
Remark 3. The multilayer minimum projection method that is ageneralization of the ‘‘single-layer’’ minimum projection methodintroduced earlier. However, we do not use themethod to simplifythe discussion in this section. Note that, our discussion in this sec-tion can be applied to themultilayer minimum projectionmethod.
5.2. Stabilization with locally semiconcave control Lyapunov func-tions via minimum projection method
The LS-PCLF generated by the minimum projection method hasthe following special properties.
Lemma 6. Assume that V is a locally semiconcave practical con-trol Lyapunov function for (21) obtained by the minimum projec-tion method with smooth V . Consider a set argminx∈φ−1(x)V (x) :=
x|V (x) = miny∈φ−1(x) V (y).
H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909 907
Then,
dV (x) :=
dV (x)
x ∈ argminx∈φ−1(x)
V (x)
⊂ DV (x). (51)
Lemma 7. Assume that V satisfies the conditions of Lemma 6. Con-sider a mapping p : X → T ∗
x X such that p(x) ∈ dV (x) at x. Then,⟨p, f ⟩ < 0 for all x ∈ X when ⟨p, gi⟩ = 0 for all i ∈ 1, . . . ,m.
On the local chart (W, η), we can calculate dVW as follows:
dVW =
∂ VW
∂ξ
∂η φ η−1
∂ξ
−1
dξ
η−1(ξ )
∈ argminx∈φ−1η−1(ξ)
V (x)
. (52)
Moreover, the pairing ⟨p, f (x)⟩ and ⟨p, g(x)⟩ in (23) can be chosenas follows.
⟨p, f (x)⟩ = Lf V (x), ⟨p, gi(x)⟩ = Lgi V (x), (53)
where x ∈ argminx∈φ−1(x)V (x).The LS-PCLF generated by the minimum projection method has
an advantage, according to which we can obtain the followingtheorem:
Theorem 5. Assume that V is a locally semiconcave practical controlLyapunov function for (21) obtained by the minimum projectionmethod with smooth V . Consider a static state feedback control u =
k(x) such that
ki(x) =
−
⟨p, f ⟩ +
⟨p, f ⟩2 +
mi=1
⟨p, gi⟩22
mi=1
⟨p, gi⟩2⟨p, gi⟩
mi=1
⟨p, gi⟩2 = 0
0
mi=1
⟨p, gi⟩2 = 0
,
(54)
where p : X → T ∗x X is a mapping such that p(x) ∈ dV (x) at x ∈ X.
Then, (54) sample stabilizes the origin of the control system (21).
Remark 4. Note that the requirements in p in (54) are much morerelaxed than those in (23), and dV (x) can be easily calculated in thecase of the minimum projection method.
6. Example: obstacle avoidance of a mobile robot
In accordance with the preceding discussion, we can design asample stabilizing controller for control system (21) defined onmanifold X. In this section, we show an example of the proposedmethod.
We consider the following mobile robot system discussed in[11]:x = u1
y = u2,(55)
where (x, y) ∈ R2 is a state and (u1, u2) ∈ R2 is an input. Alsoconsidered is a disk-shaped obstacle with radius 1 centered at(−2, 0).
The problem here is to design a sample stabilizing controller atthe origin (x, y) = (0, 0) taking into account the obstacle avoid-ance. In this case, the configuration space is X = (x, y) ∈ R2
|(x+
2)2 + y2 > 1 ≃ S1. Note that X is neither contractible nor com-plete metric space. Using the notation [x, y]T = g1u1 + g2u2 for(55) here, equation (55) can be regarded as (21) with f (x) = 0.
We consider the following two mappings φ1 : R2→ R2 and
φ2 : R2→ R2:
φ1 :
x = r cos θ − 2y = r sin θ,
(56)
φ2 :
r =
r + 2 (r ≥ 0)2π
tan−1π2r
+ 2 (r < 0)
θ = θ ,
(57)
where we restrict image of tan−1 to (−π/2, π/2).Then, the compositemappingφ = φ1φ2 : R2
→ X; (r, θ ) →
(x, y) satisfies conditions for φ in (M1) in the minimum projectionmethod. Note that X = R2. Moreover, control system on R2 is ob-tained as follows:
˙r =
u1 cos θ + u2 sin θ (r ≥ 0)
u1cos θ
1 +π r/2
2 + u2sin θ
1 +π r/2
2 (r < 0),(58)
˙θ =
−u1sin θr + 2
+ u2cos θr + 2
(r ≥ 0)
−u1sin θ
2 tan−1(π r/2)/π + 2
+ u2cos θ
2 tan−1(π r/2)/π + 2(r < 0).
(59)
Then, the function
V =12r2 +
12θ2 (60)
is a smooth control Lyapunov function on X = R2.We can obtain the following LS-PCLF V by the minimum pro-
jection method (Fig. 1):
V (x, y) = min(r,θ )∈φ−1(x,y)
V (r, θ ) (61)
=
12
(x + 2)2 + y2 − 2
2+
12(arg(x + 2 + iy))2
(x + 2)2 + y2 ≥ 2
12
2π
tanπ2
(x + 2)2 + y2 − π
2
+12(arg(x + 2 + iy))2
1 <(x + 2)2 + y2 < 2
,
(62)
where the image of arg is restricted to (−π, π]. Function V is notdifferentiable at each (x, y) ∈ (x, y)|x < −2, y = 0.
Recall that the space X is not complete. In this case, Lyapunovstability may not be defined [22]; however, we can easily designan LS-PCLF.
908 H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909
Discontinuity in V (x) can be found in θ = −π and +π .Then,
dV (x) =
(x + 2)
(x + 2)2 + y2 − 2
(x + 2)2 + y2
−y arg(x + 2 + iy)(x + 2)2 + y2
,y(x + 2)2 + y2 − 2
(x + 2)2 + y2
+(x + 2) arg(x + 2 + iy)
(x + 2)2 + y2
(x + 2)2 + y2 ≥ 2, arg(x + 2 + iy) = +π
2(x + 2) sin
π(x + 2)2 + y2/2 − π
π(x + 2)2 + y2 cos3
π(x + 2)2 + y2/2 − π
−
y arg(x + 2 + iy)(x + 2)2 + y2
,
2y sinπ(x + 2)2 + y2/2 − π
π(x + 2)2 + y2 cos3
π(x + 2)2 + y2/2 − π
+(x + 2) arg(x + 2 + iy)
(x + 2)2 + y2
1 <
(x + 2)2 + y2 < 2, arg(x + 2 + iy) = +π
x + 4,
π
x + 2
,
x + 4,−
π
x + 2
(x ≤ −4, arg(x + 2 + iy) = +π)
2 sin (πx/2)π cos3 (πx/2)
,π
x + 2
,
2 sin (πx/2)π cos3 (πx2x)
,−π
x + 2
(−3 < x < −4, arg(x + 2 + iy) = +π).
(63)
We choose p(x) as follows:
p(x) =
(x + 2)(x + 2)2 + y2 − 2
(x + 2)2 + y2
−y arg(x + 2 + iy)(x + 2)2 + y2
,
y(x + 2)2 + y2 − 2
(x + 2)2 + y2
+(x + 2) arg(x + 2 + iy)
(x + 2)2 + y2
(x + 2)2 + y2 ≥ 2
2(x + 2) sin
π(x + 2)2 + y2/2 − π
π(x + 2)2 + y2 cos3
π(x + 2)2 + y2/2 − π
−
y arg(x + 2 + iy)(x + 2)2 + y2
,
2y sinπ(x + 2)2 + y2/2 − π
π(x + 2)2 + y2 cos3
π(x + 2)2 + y2/2 − π
+(x + 2) arg(x + 2 + iy)
(x + 2)2 + y2
1 <
(x + 2)2 + y2 < 2
(64)
Note that ⟨p, f ⟩ = 0, and we can finally obtain the practical stabi-lizing controller (54) as follows:
u1 = −⟨p, g1⟩ (65)
Fig. 1. CLF: mobile robot.
Fig. 2. Simulation result: state.
=
−
(x + 2)(x + 2)2 + y2 − 2
(x + 2)2 + y2
−y arg(x + 2 + iy)(x + 2)2 + y2
−
(x + 2)2 + y2 ≥ 2
−
2(x + 2) sinπ(x + 2)2 + y2/2 − π
π(x + 2)2 + y2 cos3
π(x + 2)2 + y2/2 − π
−
y arg(x + 2 + iy)(x + 2)2 + y2
1 <
(x + 2)2 + y2 < 2
(66)
u2 = −⟨p, g2⟩ (67)
=
−
y(x + 2)2 + y2 − 2
(x + 2)2 + y2
+(x + 2) arg(x + 2 + iy)
(x + 2)2 + y2
(x + 2)2 + y2 ≥ 2
−
2y sinπ(x + 2)2 + y2/2 − π
π(x + 2)2 + y2 cos3
π(x + 2)2 + y2/2 − π
+(x + 2) arg(x + 2 + iy)
(x + 2)2 + y2
1 <
(x + 2)2 + y2 < 2
(68)
We show the simulation results in Figs. 2–4, according to which,we can permit the mobile robot to arrive at the origin by avoidingthe obstacle. Fig. 3 illustrates that the input smoothly decreases tozero and remains at that level with passage of time.
H. Nakamura et al. / Systems & Control Letters 62 (2013) 902–909 909
0 5 10 15 20 25 30
0
0.5
1
1.5
2
Time
Input
u1
u2
Fig. 3. Simulation result: input.
Fig. 4. Simulation result: trajectory.
7. Conclusion
In this paper, we proposed a sample stabilizing controller for acontrol system defined on a general manifold with a locally semi-concave practical control Lyapunov function. We employed thedisassembled differential in the proposed controller, which en-abled us to easily design a sample stabilizing controller when thelocally semiconcave control Lyapunov function was designed bythe minimum projection method. Finally, we confirmed the effec-tiveness of the proposed method through an example.
We can now constructively design an asymptotically stabiliz-ing controller for a control system defined on a general manifold.However, two big problems remain to be solved. Optimality of theinput is one of them. Inverse optimal control is one of the benefitsof using control Lyapunov functions. However, we cannot identify
whether the controller proposed in the paper is inverse optimal.The other problem is the input-to-state stability [23]. These are thechallenges for future work.
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