Passivity based control of a class of Hamiltonian systems with ...

Passivity based control of a class of Hamiltonian systems with nonholonomic constraints ? Kenji Fujimoto a , Satoru Sakai b , Toshiharu Sugie c a b

Nagoya University, Nagoya, Japan

Shinshu University, Nagano, Japan c

Kyoto University, Kyoto, Japan

Abstract This paper is concerned with state and output feedback stabilization of a class of port-Hamiltonian systems with nonholonomic constraints. First we study canonical forms for port-Hamiltonian systems with nonholonomic constraints. Second, we give a new state feedback stabilization method by using non-smooth Hamiltonian functions via generalized canonical transformations. Third, we propose a dynamic output feedback stabilization method without measuring the velocity based on the corresponding state feedback result. Numerical examples demonstrate the effectiveness of the proposed method. Key words: port-Hamiltonian systems; nonholonomic systems; stabilization.

1

Introduction

Port-Hamiltonian systems [32,33,16] are introduced as generalization of conventional Hamiltonian systems [21]. They can describe electric circuits [18], fluid systems [26] and their combinations [15] as well as conventional mechanical ones. It is also shown that mechanical systems with a class of nonholonomic constraints can be modeled by reduced order port-Hamiltonian systems without any constraints [12,31]. For port-Hamiltonian systems, several control strategies are investigated. The authors propose generalized canonical transformations [7] as generalization of the well-known canonical transformations in classical mechanics which can design feedback controllers using the energy conservation law. This method is also applied to trajectory tracking control [6], path following control [29], and iterative learning control [8] as well. Alternative approaches to stabilization of portHamiltonian systems can be found in [31,27] which are based on the interconnection and the Casimirs generation. IDA-PBC method [25] employs partial differential ? This paper was partly presented at the 38th IEEE Int. Conf. CDC and the XVIth IFAC world congress. Corresponding author S. Sakai. Tel. +81-26-269-5171. Fax +8126-269-5145. Email addresses: [email protected] (Kenji Fujimoto), [email protected] (Satoru Sakai), [email protected] (Toshiharu Sugie).

Preprint submitted to Automatica

equations to derive stabilizing controllers in a similar manner to the generalized canonical transformation approach and it can take care of underactuated mechanical systems [24]. All of those results are natural generalizations of passivity based control for conventional mechanical systems, e.g., [28,23,1]. Mechanical systems with nonholonomic constraints (nonholonomic systems) are often modeled by driftless systems (see, e.g., [19,14]). In order to stabilize them asymptotically, several strategies have been proposed. Some typical ones among them employ discontinuous feedbacks [4,2,13,30]. Most of the existing stabilization methods for driftless systems are based on special techniques peculiar to them. On the other hand, backstepping techniques [5,9] can be used to derive a stabilizer for a nonholonomic drift system such as a Lagrangian system or a port-Hamiltonian system (typically consisting of a driftless system and an integrator) based on a stabilizer for the corresponding driftless one. In general, however, backstepping techniques derive complicated stabilizers and, moreover, it enlarges the discontinuity of the original ones. This often causes the resultant feedback system to be very sensitive to external noises. Hence, in order to solve all of the above problems, this paper investigates a stabilization problem for a class of port-Hamiltonian systems with nonholonomic constraints based on generalized canonical transformations

3 July 2012

port-Hamiltonian systems. In such a case, generalized canonical transformations are useful [7,6].

with the techniques developed in [31,7]. Since this strategy is a natural generalization of the passivity based control for conventional mechanical systems, the result in this paper can be regarded as a passivity based control for a class of nonholonomic systems. First we will investigate canonical forms for port-Hamiltonian systems. In particular, how to transform a given port-Hamiltonian system with nonholonomic constraints into a certain canonical form such as a chained form will be shown. Second, we will explain how to assign a non-smooth Hamiltonian function via a generalized canonical transformation which makes the resultant feedback system asymptotically stable. An explicit procedure to design a state feedback stabilizer is also given. Third, we will clarify an equivalence between a dynamic output feedback stabilization and the corresponding state feedback stabilization. An output feedback stabilization method is derived based on this equivalence. 2

A generalized canonical transformation is a set of trans¯ x) = (H + U )(Φ−1 (¯ formations x ¯ = Φ(x), H(¯ x)), y¯ = y + α(x) and u ¯ = u + β(x) which preserves the structure of a time-varying port-Hamiltonian system. Since we construct time invariant feedbacks in this paper, a simplified time-invariant version is explained. Any function Φ with U and β defines a generalized canonical transformation with a certain α if and only if J(x)∇U+K(x)∇(H + U )+ g(x)β(x) = 0

holds for some skew-symmetric matrix K(x) ∈ Rn×n . Using a generalized canonical transformation, we can change the property of the system without changing the inherent port-Hamiltonian structure with passivity, and we can convert the system into several convenient forms. If we convert a given port-Hamiltonian system into another port-Hamiltonian one which has a positive definite Hamiltonian function and is zero-state detectable, then the transformed system can be easily stabilized asymptotically by the simple feedback of the form (2) [7,6].

Port-Hamiltonian systems and stabilization

This section reviews the preliminary results on portHamiltonian systems [16,31] and generalized canonical transformations [7]. Consider a port-Hamiltonian system with a Hamiltonian function H(x) ≥ 0 described by

3

  x˙ = J(x)∇H(x) + g(x)u

3.1

(1)

 y = g(x)T ∇H(x)

u = −C(x) y,

C(x) ≥ εI > 0 ∈ R

Nonholonomic constraints

q

  y = B(q)T ∇p H(q, p)       0 = A(q)T ∇ H(q, p) p

(2)

called damping injection yields dH ≤ −y T C(x) y ≤ 0 dt

Canonical forms of port-Hamiltonian systems with nonholonomic constraints

Let us consider a mechanical system with nonholonomic constraints described by a conventional Hamiltonian system with Lagrangian multipliers [33] as follows.   q˙ = ∇p H(q, p)        p˙ = −∇ H(q, p) + A(q)λ + B(q)u

where x ∈ Rn is the state, u, y ∈ Rm are the input and the output. The matrix J(x) is skew-symmetric, i.e., −J(x) = J(x)T 1 . One of the most important properties of the port-Hamiltonian system is passivity. It is passive with respect to the storage function H(x), i.e., (dH/dt) ≤ y T u holds. Therefore, the feedback m×m

(4)

Here x = (q, p) ∈ Rl × Rl is the state, u, y ∈ Rm are the input and the output, the Hamiltonian function H(q, p) := (1/2)pT M (q)−1 p describes the kinetic energy with the symmetric inertia matrix M (q) > 0, and A(q)λ with the Lagrangian multiplier λ and B(q)u denote the constraint force and the external force, respectively. We assume that (A(q), B(q)) is nonsingular for A(q) ∈ Rl×(l−m) and B(q) ∈ Rl×m . The symbol ∇q denotes the (partial) gradient with respect to the variable q. Local controllability of this system is also assumed.

(3)

which implies that the input and the output satisfy u(t) → 0, y(t) → 0 as t → ∞. Furthermore if H(x) is positive definite and the system is zero-state detectable, then the origin is stabilized asymptotically by the feedback (2) (see, e.g., [17][31]). The positive definiteness of the Hamiltonian and the zero-state detectability do not always hold for general

It was shown in [33] that the Hamiltonian system defined above can be described by a port-Hamiltonian system by the following procedure. First we choose a matrix J12 (q) ∈ Rl×m in such a way that A(q)T J12 (q) =

1

J can be replaced by a negative semidefinite matrix. See [31,6].

2

Hamiltonian system

0 holds and that (A(q), J12 (q)) is nonsingular. Using the matrix J12 (q), we can define the following coordinate transformation p˜ := (˜ p1 , p˜2 ) = (J12 (q)T p, A(q)T p). Then ∇p˜2 H ≡ 0 holds where p˜2 coordinate does not affect the input-output behavior. Consequently the constrained system can be described by the reduced order port-Hamiltonian system

" # " #" # " # ¯12 (¯ ¯  ˙ q ¯ 0 J q ) ∇ H 0  q ¯   = + u   p¯˙ ¯ ¯ q) −J¯12 (¯ q )T J¯22 (¯ q , p¯) ∇p¯H G(¯ | {z }   ¯ q ,p) ≡J(¯ ¯     y = G(¯ ¯ q )T ∇p¯H(¯ ¯ q , p¯)

" # " #" # " #  0 J12 (q) ∇q H 0  q˙   = + u   p˜˙1 −J12 (q)T J22 (q, p˜1 ) ∇p˜1 H G(q) . {z } |  1)  ≡J(q, p ˜     y = G(q)T ∇ 1 H(q, p˜1 ) p˜ (5) 1T ˜ −1 1 Here the Hamiltonian function is H = (1/2)˜ p M (q) p˜ ˜ (q) ∈ with a symmetric and positive definite matrix M m×m T R . The matrix G(q) := J12 (q) B(q) ∈ Rm×m is nonsingular since the matrices B(q) and J12 (q) i,j have full column rank. The (i, j)-th element J22 (q, p˜1 ) 1 of the skew symmetric matrix J22 (q, p˜ ) is given by i,j j i i J22 (q, p˜1 ) := −˜ pT [ J12 , J12 ](q) where J12 (q) denotes the i-th column of J12 (q) matrix and the operation [ ·, · ] is the Lie bracket with respect to q. The state of this system is denoted by x := (q, p˜1 ) ∈ Rn with n := l + m. 3.2

(9) ¯ x) = (1/2) p¯T M ¯ (¯ q )−1 p¯, with the Hamiltonian function H(¯ where J¯12 := (∇q Ψ)T J12 N T J¯22 := ∇q (N T p˜1 ) (J12 N ) − N T J T ∇q (N T p˜1 ) + N T J22 N 12

¯ := N T M ˜ N, G ¯ := N T G. M

Note that the converted system (9) has the “canonical” ¯ q ) matrix if the corresponding driftless system (7) has J(¯ the canonical 2 J¯12 (¯ q ).

PROOF. The set of transformations (8) is a generalized canonical transformation since Equation (4) holds for β = 0 and K = 0. The claim can be proved by a direct calculation. (Q.E.D.)

Canonical forms

Y

The dynamics of q in Equation (5) can be rewritten as the velocity-input driftless system

u2 q

q˙ = J12 (q) v

q¯˙ = J¯12 (¯ q ) ζ.

p¯

=

Ψ(q) N (q)T p˜1

¯ x) := H(Φ−1 (¯ H(¯ x))

X

Fig. 1. A rolling coin

Example 2 We consider a rolling coin on the horizontal plane as depicted in Figure 1. Let q1 denote the heading angle of the coin, and (q2 , q3 ) denote the position of the contact point of the coin in Cartesian coordinates. Furthermore let p1 be the angular momentum with respect to q1 ; p2 be the momentum of the coin; u1 and u2 be control input with respect to p1 and p2 , respectively. Also, let all physical parameters be unity for simplicity, then the behavior of the coin is expressed as the port-Hamiltonian system (5) with

(7)



Then the following set of transformations Ã

q2

0

Proposition 1 Suppose that there exists a pair of a coordinate transformation q¯ = Ψ(q) and an input transformation v = N (q)ζ with a nonsingular matrix N (q) which converts the driftless system (6) into

x ¯ :=

q1

(6)

where v := ∇p˜1 H is regarded as the input. This system can be transformed into canonical forms (e.g., a chained form, a power form, etc) by an appropriate coordinate and input transformation [3]. The following proposition derives a canonical form for the port-Hamiltonian system (5) based on generalized canonical transformations.

à ! q¯

u1

1

0



  1  J12 (q) =   0 cos q1  , J22 (q, p˜ ) = 0, G(q) = I (10)

! =: Φ(x)

0 sin q1 (8)

2

The word ‘canonical’ means that the corresponding driftless system has a canonical form such as the chained form, the power form, etc.

converts the system (5) into the following port-

3

T ˜ = I. and H = (1/2)˜ p1 p˜1 with p˜1 = (p1 , p2 ) and M The velocity-input driftless system (6) corresponding to this system can be transformed into a chained form (7) by the following pair of transformations: q¯ = Ψ(q) := [q2 , tan q1 , q3 ] and v = N (q)ζ with

" N (q) :=

0

p

1/(1 + tan2 q1 )

1 + tan2 q1

ˆ x) positive to make the new Hamiltonian function H(¯ definite. It can be readily checked that the functions U (¯ q ) and β(¯ q ) thus defined with K(¯ x) = 0 satisfy the condition (4) for generalized canonical transformations. Hence, the transformed system is described by the portHamiltonian system with the new Hamiltonian function ˆ x) as follows. H(¯

# .

" # " #" # " # ¯12 (¯ ˆ  ˙ q ¯ 0 J q ) ∇ H 0  q ¯  + u ¯ = ˆ ¯ q) p¯˙ −J¯12 (¯ q )T J¯22 (¯ q , p¯) ∇p¯H G(¯    y¯ = G(¯ ¯ q )T ∇p¯H ˆ (14) Equation (12) implies that the new Hamiltonian funcˆ x) is positive definite if U (¯ tion H(¯ q ) is positive definite with respect to q¯. Then, we can apply the feedback

0

Therefore the generalized canonical transformation (8) with the functions Ψ and N transforms the coin system into the canonical form (9) where 

1 0



   ¯ q) = J¯12 (¯ q) =   0 1  , G(¯ q¯2 0

"

0

# p 1 + q¯22

1/(1 + q¯22 )

(11)

0

u ¯ = −C(¯ x)¯ y , C(¯ x) ≥ εI > 0

¯ = diag(1 + q¯2 , 1/(1 + q¯2 )2 ). It is verified that the and M 2 2 J¯12 matrix has the chained form. 4

as in Equation (2) to make the input and the output converge to zero u(t) → 0, y(t) → 0 as t → ∞ based on the passivity of the system. Summarizing the procedure, we can make the input and the output signals of the system (9) converge to zero by the following feedback

State feedback stabilization

u = −β(¯ q ) − C(¯ x)¯ y −1 ¯ ¯ ¯ q )T M ¯ (¯ = −G(¯ q ) J12 (¯ q )T ∇q¯U (¯ q ) − C(¯ x)G(¯ q )−1p¯.(16)

This section discusses state feedback stabilization of the port-Hamiltonian system (9) in the case (l, m) = (3, 2). Section 4.1 exhibits the main result using the idea of conventional passivity based control. In order to obtain stabilizing controllers for nonholonomic systems treated in this paper, we need to employ discontinuous feedbacks which are achieved by adopting non-smooth potential functions. The reason why non-smooth functions are needed is explained in Sections 4.2 and 4.3. The following sections 4.4 and 4.5 discuss how to design the feedback controllers. 4.1

This method is called passivity based control. Here β(¯ q) and C(¯ x)¯ y can be regarded as the generalized versions of the P-gain term and the D-gain term of conventional PD controllers, respectively. According to the procedure stated in Section 2, if the converted system (14) is zero-state detectable, then the feedback (15) stabilizes it asymptotically. In order to stabilize the nonholonomic system (9) by the passivity based control (16), we need to employ a non-smooth (non-differentiable) potential function U (¯ q ). How to choose such a function is stated in the main theorem below. The following assumptions are adopted in order to prove the main theorem. Here we use the shorthand notation q¯12 := (¯ q1 , q¯2 )T .

Main result

Let us consider the stabilization problem of the portHamiltonian system (9). As stated in Section 2, if the Hamiltonian function is positive definite and the system is zero-state detectable, then the feedback (2) stabilizes the system asymptotically. Since the Hamiltonian function of the system (9) is not positive definite, we employ a generalized canonical transformation with a potential function 3 U (¯ q) ∈ R 1 ˆ x) = H(¯ ¯ x) + U (¯ ¯ (¯ H(¯ q ) = p¯T M q )−1 p¯ + U (¯ q) 2 −1 T ¯ q ) J¯12 (¯ u = − G(¯ q ) ∇q¯U (¯ q ) +¯ u, y¯ = y | {z }

(15)

Assumption 3 The matrix J¯12 (¯ q ) ∈ R3×2 satisfies " J¯12 (¯ q) =

(12)

I T q¯12 S2 (¯ q)

# S1 (¯ q)

(17)

with S1 (¯ q ), S2 (¯ q ) ∈ R2×2 where S1 (¯ q ) is nonsingular and

(13)

β(¯ q)

4 det S2 (¯ q ) − (trS2 (¯ q ))2 > 0

3

This name is derived from the potential functions of conventional mechanical systems.

holds for all q¯12 ∈ R2 \{0} and all q¯3 ∈ R.

4

(18)

Assumption 4 The function U (¯ q ) ∈ R is given by U (¯ q ) = V (k¯ q12 k, q¯3 )

equation. µ

(19)

∇q¯U (¯ q ) = diag

where V : R+ ×R → R is a continuous function satisfying the following properties: (i) There exists a class-K function µ satisfying

¶ q¯12 , 1 ∇(s,r) V (k¯ q12 k, q¯3 ). k¯ q12 k

(25)

Therefore the parameter 1 in the case q¯12 = 0 and q¯3 6= 0 is 1 = limk¯q12 k→0 (¯ q12 /k¯ q12 k). Note also that Assumption 5 guarantees its existence and boundedness.

V (s, r) ≥ µ(k(s, r)k). Please also note that there are many functions V (s, r) satisfying Assumptions 4 and 5. A more concrete procedure to choose those functions will be explained in Sections 4.4 and 4.5. The following Sections 4.2 and 4.3 are devoted to stability analysis of the feedback system.

(ii) The function V (s, r) is differentiable for all s > 0 and all r ∈ R and satisfies ∇(s,r) V (s, r) 6= 0, ∀s > 0, ∀r ∈ R.

(20)

4.2

Assumption 5 The limit lims→+0 ∇(s,r) V (s, r) of the function V (s, r) in Assumption 4 exists for all r ∈ R and satisfies ν(r) := lim ∇s V (s, r) < 0, ∀r ∈ R\{0}. s→+0

Let us consider the feedback system consisting of the plant (9) and the passivity based controller (16). If the plant system is a conventional mechanical system, i.e., l = m and J¯12 (¯ q ) = I hold, then it is zero-state detectable for any positive definite potential function U (¯ q) and consequently the feedback (16) stabilizes the plant (9) asymptotically. In the nonholonomic case where l > m, however, the system (9) is not zero-state detectable even if the potential function U (¯ q ) is positive definite. The passivity based controller (16) makes the state of the feedback system converge to the invariant set Ω defined below instead of the origin as in the case l = m. Let Ω ⊂ Rn denote the invariant set of the feedback system contained in the output nulling set

(21)

Using those assumptions, we can state one of the main results of this paper as follows. Theorem 6 Consider the port-Hamiltonian system (9) under Assumptions 3-5. Then the feedback ¯ q )−1 J¯12 (¯ ¯ TM ¯ (¯ u = −G(¯ q )T γ(¯ q ) − C(¯ x)G(q) q )−1 p¯ (22)   ∇ U (¯ q) (¯ q12 6= 0)   q¯ q12 = 0, q¯3 6= 0) γ(¯ q ) = diag(1,1) lim ∇(s,r)V (s, q¯3 ) (¯ s→+0    0 (¯ q12 = 0, q¯3 = 0)

Ωy = {(¯ q , p¯) | y = 0}. The state converges to Ω ⊂ Ωy due to the passivity property in Equation (3). It is easy to observe that the invariant set Ω of the system (9) with the feedback (16) is described by

(23) makes the state converge to the origin, where 1 ∈ R2 is a vector satisfying 1T 1 = 1. Furthermore if there exists a class-K function µ such that V (s, r) ≤ µ(k(s, r)k)

Invariant set

¯ © ª Ω = (¯ q , p¯) ¯ J¯12 (¯ q )T ∇q¯U (¯ q ) = 0, p¯ = 0 .

(26)

This equation is directly obtained by a standard calculation of invariant sets (see, e.g., Section 4.2 of [11]).

(24)

holds, then the origin is asymptotically stable in the Lyapunov sense.

In order to achieve asymptotic stability, we want to make the invariant set Ω as small as possible because the system is asymptotically stable if we can achieve Ω = {0}. Since Ω depends on the potential function U (¯ q ) and the matrix J¯12 (¯ q ) as in Equation (26), the following section discusses how to make Ω small by selecting the parameter functions U and J¯12 .

Since the proof of this theorem needs several lemmas, it will be proved later in Section 4.4. Note that the feedback controller (22) proposed in the theorem coincides with the passivity based one (16) if q¯12 6= 0. The controller (22) is a generalized version of (16) to take care of the non-smoothness of the potential function U (¯ q ) on the region {¯ q | q¯12 = 0}.

4.3

Remark Note that the gradient in the definition of the function γ(¯ q ) in Equation (23) satisfies the following

An advantage of passivity based control (16) for conventional mechanical systems is the freedom in choosing

5

Designing the invariant set

Then S˜2 (0) is the skew-symmetric part of S2 (0) and

the potential function U since one can use it as a design parameter. For example, obstacle avoidance control and anti-windup control can be realized by selecting U appropriately (see, e.g., [1]). In the nonholonomic case, however, the invariant set Ω varies when we change the potential function U . This is quite inconvenient when we use the function U as a design parameter. If J¯12 (¯ q) satisfies Assumption 3 and U (¯ q ) is chosen in such a way that Assumption 4 holds, then the invariant set Ω is the same for any positive definite function V in Equation (19). The following lemma describes this fact.

1 S˜2 (0) = 2

0

[S2 (0)]12 − [S2 (0)]21

[S2 (0)]21 − [S2 (0)]12

0

since [S2 (0)]12 6= [S2 (0)]21 . Therefore, the condition (18) holds in a neighborhood of the origin. Example 8 (continued) The matrices S2 ’s corresponding to the matrices J¯12 (¯ q )’s in Equations (10) and (11) are computed as follows, respectively. "

(27)

The proof is in Appendix A. It is noted that there always exist a matrix J¯12 (¯ q ) satisfying the Assumption 3 and the corresponding transformation in Equation (8) in a neighborhood of the origin if the plant system is locally controllable as explained in the following remark.

"

¡

1 ¯2 1 ¯2 J¯12 , J12 , [J¯12 , J12 ]

¢

0

0

#

" , S2 (¯ q) =

0

p¯1

#

" T 1

= N (q) p˜ :=

1/(1 + tan2 q1 )

p¯2

00

# .

10

0

0

#

p p˜1 2 1 + tan q1

for which the corresponding S1 and S2 matrices are "

(28)

S1 = I,

2 1 (¯ q )) and the Lie bracket (¯ q ), J¯12 with J¯12 (¯ q ) = (J¯12 [·, ·] with respect to the variable q¯. This implies that [S2 (0)]1,2 6= [S2 (0)]2,1 since

10

0 tan q1 /q1

So, both matrices do not satisfy the condition (18). On the other hand, we can choose a new state q¯ = Ψ(q) := T [tan q1 , q2 , 2q3 − q2 tan q1 ] and

Remark Let us consider the system (9) with Equation ¯ (¯ (17), M q ) = I and S1 (q) = I without loss of generality and assume the controllability of the system characterized by



.

4detS˜2 (0) − (trS˜2 (0))2 = ([S2 (0)]12 − [S2 (0)]21 )2 > 0

S2 (¯ q) =

¡ 1 ¢ 2 1 ¯2 rank J¯12 (0), J¯12 (0), [J¯12 , J12 ](0) = 3

#

Substituting this for S2 in Equation (18) we obtain

Lemma 7 Consider the port-Hamiltonian system (9) with the feedback (22), subject to Assumptions 3 and 4. Then, the following property holds for the invariant set Ω in (26) Ω ⊂ Ω0 := { (¯ q , p¯) | q¯12 = 0 , p¯ = 0 }.

"

S2 =

0 1 −1 0

# (29)

satisfying the condition (18). We will use this realization for stabilization in what follows.



4.4

   (0)= 0 0 1  0 0 [S2 (0)]12 − [S2 (0)]21

Non-smooth potential functions

The previous section shows that the passivity based controller (22) with any positive definite function U satisfying Assumption 4 (as in Lemma 7) will make the state converge to the subset Ω0 of the state space defined in Equation (27). Remember that a non-smooth function U is needed to derive a discontinuous state feedback for asymptotic stabilization and that the property (27) of the invariant set Ω also holds for a non-smooth function U as far as it satisfies Assumption 4. This implies that we can select a non-smooth function U which is not differentiable on the set Ω0 .

where [S2 (0)]ij denotes the (i, j) element of the matrix S2 (0). Take a coordinate transformation to remove the symmetric part of S2 (0) as T ˜ q˜ = φ(q) := (¯ q1 , q¯2 , q¯3 − (1/2) q¯12 S2 (0)¯ q12 )T .

Then the dynamics can be described using q˜ as "

# I q˜˙ = p¯ q˜T S˜2 (˜ q)

Theorem 6 claims that any potential function U which is not differentiable on Ω0 satisfying Assumptions 4 and 5 yields asymptotic stability. The following lemma reveals the property of the feedback system caused by the nonsmoothness of the potential function U .

12

˜ S˜2 (φ(q)) = S2 (¯ q ) − (1/2)(S2 (0) + S2 (0)T ).

6

10

Proposition 10 Consider the port-Hamiltonian system (9), subject to Assumption 3. Given a smooth positive definite function Vs (k¯ q12 k, q¯3 ) such that Assumption 4 holds. Then the feedback (22) with V = Vs +Vn (k¯ q12 k, q¯3 ) and finite X |r|γi +δi (33) Vn (s, r) = (ai |s| + bi |r|)δi i

10

8

8

6

6

4

4

2

2

0 -5-4 -3-2 4 5 -1 2 3 0 1 q 01 2 1 2 3 q12 3 34 5 5 4

0 -5-4 -3-2 -1 0 1

(a) example 1

0 23 2 1 45 5 4 3 q q 3 12

5 3 4 1 2

(ai , bi > 0, γi ≥ 0, δi ≥ 2) makes the state converge to the origin. Furthermore if mini γi > 0 holds, then the origin of the feedback system is asymptotically stable in the Lyapunov sense. If mini γi = 0, C(0) > 0 and (∇s Vs (s, r))(0, 0) > 0 hold, then the convergence rate is exponential, that is, for any given initial condition x ¯(t0 ), there exist constants k1 , k2 > 0 such that

(b) example 2

Fig. 2. Non-smooth functions V ’s

Lemma 9 Consider the port-Hamiltonian system (9) with the feedback (22), subject to Assumptions 3-5. Suppose that x ¯(t1 ) ∈ Ω0 \{0} at the time t = t1 . Then the state x ¯ goes out of Ω0 \{0} immediately, that is, there exists a positive constant ² > 0 such that x ¯(t) 6∈ Ω0 \{0}, ∀t ∈ (t1 , t1 + ²).

k¯ x(t)k ≤ k1 e−k2 t , ∀t ≥ t0 .

(30) The proof of this proposition is provided in Appendix A. Proposition 10 gives an explicit procedure of constructing stabilizing controllers. It can be readily confirmed that the function in Fig. 2 (a) defined by Equation (31) has the form of Equation (33) in Proposition 10.

The proof of this lemma is given in Appendix A. This lemma implies that the state does not converge to Ω0 \{0} and, consequently, Lemma 7 shows that the feedback (22) with a non-smooth function U selected as in the lemma makes the invariant set Ω coincide with {0}. Hence asymptotic stability is achieved. This is the main idea of the proof of Theorem 6 which is presented in Appendix A.

Example 11 (continued) Let us apply Theorem 6 to the system (9) with J¯12 (¯ q ) satisfying Equations (17) and (29). Figures 3(a), 3(c) and 3(e) show the responses on the X-Y plane with the parameters a1 = b1 = γ1 = 1, C = 5I, 2I and (1/2)I, from the initial conditions q(0) = (0, 0, 1)T and p˜1 (0) = (0, 0)T . All trajectories converge to the origin and when C = (1/2)I the trajectory oscillates while there is no big difference as far as we choose the gain C greater than 2I. Figures 3(b), 3(d) and 3(f ) show the time responses of q where the solid, dashed and dashed-dotted lines denote q1 , q2 and q3 respectively. Every state converges to the origin smoothly but now the convergence rates are clearly different between the cases of C = 2I and C = 5I. If Vn = 0, the closed-loop system is just a linear second-order system whose damping takes the critical at C = 2I. One can observe that the responses with non-zero Vn behave similarly to the linear case. This fact suggests a gain-tuning guideline to select the critical C as the initial gain and then to adjust it.

Rough pictures of the functions V ’s satisfying Equation (21) in Assumption 5 are shown in Figures 2(a) and 2(b). They are defined by 1 |¯ q 3 |3 V (k¯ q12 k, q¯3 ) = q¯T q¯ + 2 (k¯ q12 k + |¯ q3 |)2 ( V (k¯ q12 k, q¯3 ) = max k¯ q12 k2+|¯ q3 |,

(31) )

3

|¯ q3 | 2 1

1

q3 | 2 k¯ q12 k 2 + |¯

.(32)

The second choice in Equation (32) is proposed in [20]. Since their actual pictures are of 4 dimensions, we only show their reduced dimensional sketches by setting q¯1 = q¯2 . It is observed that V is not smooth for q¯12 = 0. The ridge like shape at q¯12 = 0 suggests that the set where the function is non-smooth has the property in Lemma 9. There exist many functions V ’s satisfying the conditions in Theorem 6. In fact, if a given function V has the ridge like shape whose ridge-line lies in Ω0 and sgn(¯ q3 )V (0, q¯3 ) is strictly increasing, then it satisfies the condition (21). Hence we have a sufficiently broad freedom in choosing the function V , and can utilize it for controller design as in conventional passivity based control [28,23,1]. 4.5

(34)

5

Output feedback stabilization

This section is devoted to output feedback stabilization of the port-Hamiltonian system (9). Here we suppose that the state q¯ is measured but the momentum (or velocity) p¯ is not measured. Since the state feedback controller (22) depends on the variable p¯, the objective of this section is to derive a dynamic output feedback controller instead of (22) using the information of q¯ only. To this end, it is shown how to derive dynamic passivity based controllers using generalized canonical transformations. Then the equivalence is clarified between

Controller design

The following proposition provides candidate potential functions V ’s satisfying the assumptions in Theorem 6.

7

1.5

1.5

1

1 0.5

0

q

q 3

0.5

-0.5

0 -0.5

-1

-1

-1.5 -1.5

-1

-0.5

0 q 2

0.5

1

-1.5

1.5

0

(a) C = (1/2)I

time [s]

100

1.5

1

150

0.5 q

0

-0.5

with any constant and positive definite matrix D > 0. Then the system (35) is converted into a portHamiltonian system        ˆ¯  ∇qˆH 0 J¯12 0 0 0 " # qˆ˙     ˆ        ˆ¯ + I 0  u  pˆ˙  =  −J¯T J¯22 −J¯T   ∇pˆH   12 12          u ˆo ˆ ˙rˆ ¯ ¯ 0 J12 0 ∇rˆH 0 I  " # " #   ˆ¯  yˆ ∇pˆH    = .  ˆ¯ yˆo ∇rˆH (37)

q 1 q 2 q3

1

0.5 q 3

50

(b) C = (1/2)I

1.5

0 -0.5

-1

-1

-1.5 -1.5

-1

-0.5

0 q 2

0.5

1

-1.5

1.5

0

(c) C = 2I

50

time [s]

100

150

(d) C = 2I

1.5

1.5

1

q1 q2 q3

1

0.5

0.5

0

q

q 3

[ˆ q , pˆ, rˆ] = [¯ q , p¯, q¯ − r] ˆ¯ T ¯ q , p¯) + 1 (¯ H = H(¯ q − r) 2 q − r) D(¯ " # " #" # " # ¯ −T 0 yˆ G y 0 (36) = + yˆo 0 −I yo D(¯ q − r) " # " #" # " # T ¯ u ˆ G 0 u J¯12 D(¯ q − r) = + u ˆo 0 −I uo 0

q1 q2 q3

-0.5

0 -0.5

-1

PROOF. The set of transformations (36) is a generalized canonical transformation since Equation (4) is satisfied. Then the system (37) is a port-Hamiltonian system. This can be confirmed by a direct calculation. (Q.E.D.)

-1

-1.5 -1.5

-1

-0.5

0 q2

0.5

1

(e) C = 5I

1.5

-1.5

0

50

time [s]

100

150

(f) C = 5I

Since rˆ is a function of both q¯ and r, a feedback interconnection between the integrator (the dynamic controller) and the plant is achieved via a generalized canonical transformation by adding a potential function of rˆ as in the next section.

Fig. 3. Responses on the X-Y plane

asymptotic stability of a dynamic output feedback system and that of the corresponding state feedback one. Finally, an output feedback stabilizer is derived based on Theorem 6.

5.2 5.1

Dynamic extension

Output feedback stabilization

Now we have the following result based on Theorem 6 in the case (l, m) = (3, 2).

Consider the port-Hamiltonian system (9) with an integrator. Then the whole system can be described by another port-Hamiltonian system

Theorem 13 Consider the systems (9) and (37), subject to Assumptions 3-5. Then the following two conditions are equivalent under the assumption of the existence and uniqueness of the piecewise smooth trajectories of both closed-loop systems. (i)The equilibrium set of the closed-loop system of (9) and the stabilizer

       ¯ q¯˙ 0 J¯12 0 ∇q¯H 0 0 " #        u  p¯˙  =  −J¯T J¯22 0   ∇p¯H ¯ + G ¯     12    0 uo ¯ r˙ 0 0 0 ∇r H 0 I  " # " #   ¯ T ∇p¯H ¯  y G    =  ¯ yo ∇r H (35) where r ∈ Rl is the state of the integrator which is not connected to the plant yet. The next lemma shows how to connect them by a generalized canonical transformation.         

T ¯ −1 (¯ u = −G q )J¯12 (¯ q )γ(¯ q ) − Cs (¯ q , p¯)y

(38)

with any positive definite matrix Cs ≥ εI > 0 only contains the origin and it is asymptotically stable. (ii)The equilibrium set of the closed-loop system of (37) and the stabilizer " # " # T u ˆ −J¯12 (ˆ q )γ(ˆ q) = (39) u ˆo −Co (ˆ q )ˆ yo

Lemma 12 Consider the system (35) and the following set of transformations

8

1.5

 


 


1

  q3

0

-0.5



-1












(a) q(0) = (0 0 1)T

 

 

 

 



-1.5 -1.5

-1

-0.5

0 q2

0.5

1





 

 




1.5

(b) q(0) = (0 0 1)T















 

 

  





















(c) Co = (1/3)I




 





 





(d) Co = I(X-Y plane)

Fig. 5. Responses on the X-Y plane

5.3

Transient behavior

For comparison, the response of the corresponding state feedback system is also depicted by the dashed line in Figure 4(b) for which we choose Cs = 3I2 and the same potential function U (q) and the same initial condition. This figure tells us that it is important to improve the transient behavior of the proposed output feedback system.

An explicit expression of the dynamic stabilizer is r˙ = Co (¯ q )D(¯ q − r) −1 ¯ ¯ u = −G(¯ q ) J12 (¯ q )T [D(¯ q − r) + γ(¯ q )] .






 

 






 


The proof of this theorem is shown in Appendix A. Note that yˆ of the system (37) depends on the unmeasured signal p¯ hence it is not used for the stabilizer (39). Theorem 13 implies that we can derive the dynamic output feedback stabilizers using the state feedback ones. This means that the system (37) is stabilized by using any function U which was used in the stabilizer (22) in Theorem 6.



(b) Co = 3I

 

with any positive definite matrix Co (ˆ q ) ≥ εI > 0 only contains the origin and it is asymptotically stable.




(a) C0 = I

Fig. 4. Responses on the X-Y plane

(

 




0.5

 

 

 

 

 

(40)

Note that we only need the information of the matrices ¯ q ) of the plant model (9) for controller J¯12 (¯ q ) and G(¯ design, similarly to the state feedback case where the ¯ q ). The controller (22) depends only on J¯12 (¯ q ) and G(¯ other matrices and function D, Co and U are the design parameters.

So we discuss how to select the gain parameters. We start with a linear mechanical system described by " # q˙

" =

p˙ Example 14 The coin example is considered again. We skip the discussion of the existence and uniqueness of the trajectories. We choose the function U of the stabilizer (40) as in Equation (31) and all the parameters as unity for simplicity. Figure 4(a) shows the response of q from the initial condition q(0) = (0, 0, 1)T and p(0) = (0, 0)T . In this figure, the solid, dashed and dashed-dotted lines denote q1 , q2 and q3 , respectively. In Figure 4(b), the solid line depicts the response on the X-Y plane from the same initial condition. Here the gains are selected as Co = 3I3 and D = I3 . The turns of the coin are automatically generated and every state converges to the origin. These figures show the effectiveness of the proposed method.

0 1

#" # q

−K 0

+

" # 0

p

u

1

and the proposed dynamic output feedback stabilizer (

r˙ = −Co Dr + Co Dq u = Dr − Dq

.

The spring constant K corresponds to the function U . Let us regard only Co as a controller gain, then the loop transfer function Co·D(s2 + K)/(s(s2 + K + D)) has the poles and zeros on the imaginary axis. It follows from the root locus theory that there exists a certain optimal gain Co maximizing the convergence rate of the closed loop system. In fact we can find the optimal gain Co minimizing the real part of the dominant poles as

Remark Note that the stabilizer (40) is a generalization of PD controllers using the dirty derivative proposed in [10,22]. This fact can be observed easily by the linear case in the next subsection. By replacing J¯12 by I, we can construct an output feedback stabilizer for conventional mechanical systems as well.

√ Clinear =

9

K 1 + √ D 2 K

which yields the least oscillation. Hence we propose a gain-tuning guideline to select the optimal gain Co = Clinear as the initial gain and then to adjust it. On the other hand, if we regard only D as a controller gain instead of Co , then the loop transfer function does not have zeros on the imaginary axis which implies that the convergence rate grows monotonically as we increase the parameter D.

[2] A. Astolfi. Discontinuous control of nonholonomic systems. Systems & Control Letters, 27:37–45, 1996. [3] A. M. Bloch. Nonholonomic mechanics and control. SpringerVerlag, 2003. [4] A. M. Bloch, M. Reyhanoglu, and N. H. McClamroch. Control and stabilization of nonholonomic dynamic systems. IEEE Trans. Autom. Contr., AC-37(11):1746–1757, 1992. [5] R. Fierro and F. L. Lewis. Control of a nonholonomic mobile robot: backstepping kinematics into dynamics. In Proc. 34th IEEE Conf. on Decision and Control, pages 3805–3810, 1995.

Example 15 (continued) We show the result of gaintuning of Co . We employ the function U in Equation (31) which corresponds to K = 1. For this parameter K, we select big enough D = 5. Then Equation (41) tells us the optimal gain Co = Clinear = 0.7. Figures 5(a),5(b) and 5(c) show the responses of q in the cases Co = I3 , 3I3 and (1/3)I3 , respectively. The best transient behavior is achieved at Co = I3 which is the closest to the initial gain 0.7I3 . This indicates that our initial gain based on Equation (41) might be reasonable. Figure 5(d) shows the responses on the X-Y plane in the case Co = I3 . The solid line depicts the response from the initial condition q(0) = (0, 0, 1)T and p(0) = (0, 0)T which is the same as in Figure 4(b) where the solid line depicts the output feedback case with Co = 3I3 and the dashed one depicts the state feedback case with Cs = 3I2 . We see that the transient behavior in Figure 5(d) is much better than that in Figure 4(b). In Figure 5(d), the dashed line depicts the response from another initial condition q(0) = (1, 1, 1)T and p(0) = (0, 0)T which shows that good transient behavior is generated. These facts suggest that the proposed gain-tuning guideline is reasonable. 6

[6] K. Fujimoto, K. Sakurama, and T. Sugie. Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica, 39(12):2059–2069, 2003. [7] K. Fujimoto and T. Sugie. Canonical transformation and stabilization of generalized hamiltonian systems. Systems & Control Letters, 42(3):217–227, 2001. [8] K. Fujimoto and T. Sugie. Iterative learning control of Hamiltonian systems: I/O based optimal control approach. IEEE Trans. Autom. Contr., 48(10):1756–1761, 2003. [9] Z.-P. Jiang and H. Nijmeijer. Tracking control of mobile robots: a case study in backstepping. Automatica, 33:1393– 1399, 1997. [10] R. Kelly. A simple set point robot controller by using only position measurements. In Proc. 12th IFAC World Congress, volume 6, pages 173–176, 1993. [11] H. K. Khalil. Nonlinear Systems. Macmillan Publishing Company, New York, third edition, 1996. [12] H. Khennouf, C. Canudas de Wit, A. J. van der Schaft, and J. Abraham. Preliminary results on asymptotic stabilization of Hamiltonian systems with nonholonomic constraints. In Proc. 34th IEEE Conf. on Decision and Control, pages 4305– 4310, 1995. [13] H. Khennouf and C. Canudas de Wit. Quasi-continuous exponential stabilizers for nonholonomic systems. In Proc. IFAC 13th World Congress, volume 2b, page 174, 1996.

Conclusion

[14] I. Kolmanovsky and N. H. McClamroch. Developments in nonholonomic control problems. IEEE Control Systems Magazine, 15(6):20–36, 1995.

This paper discusses asymptotic stabilization of a class of port-Hamiltonian systems with nonholonomic constraints. First, it is shown that a port-Hamiltonian system with nonholonomic constraints is transformed into a certain canonical form such as a chained form via a generalized canonical transformation. Second, a novel state feedback stabilization method is proposed by assigning a non-smooth Hamiltonian function. Moreover, a sufficient condition for the non-smooth Hamiltonian functions to stabilize the system asymptotically is derived by analyzing the invariant set. Third, we prove an equivalence between the stability of the state feedback system and that of the corresponding dynamic output feedback system. A discontinuous output feedback stabilizer for the port-Hamiltonian system is derived based on the equivalence. Finally, some numerical examples show the effectiveness of the proposed method.

[15] A. Macchelli and C. Melchiorri. Control by interconnection of mixed port Hamiltonian systems. IEEE Trans. Automatic Control, 50(11):1839–1844, 2005. [16] B. M. J. Maschke, R. Ortega, and A. J. van der Schaft. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. In Proc. 37th IEEE Conf. on Decision and Control, pages 1498–1502, 1998. [17] B. M. J. Maschke and A. J. van der Schaft. Port-controlled Hamiltonian systems: modeling origins and system-theoretic properties. In IFAC Symp. Nonlinear Control Systems, pages 282–288, 1992. [18] B. M. J. Maschke, A. J. van der Schaft, and P. C. Breedveld. An intrinsic Hamiltonian formulation of the dynamics of LCcircuits. IEEE Trans. Circ. Syst., CAS-42:73–82, 1995. [19] R. M. Murray and S. S. Sastry. Nonholonomic motion planning: steering using sinusoids. IEEE Trans. Autom. Contr., 38:700–716, 1993. [20] H. Nakamura, Y. Yamashita, H. Nishitani, and H. Yamamoto. Discontinuous control of nonholonomic systems using nondifferentiable lyapunov functions. In Proc. SICE Annual Conference 2003, pages 1967–1970, 2003.

References [1] S. Arimoto. Control Theory of Non-linear Mechanical Systems: Passivity-based and Circuit-theoretic Approach. Clarendon Press, Oxford, 1996.

[21] H. Nijmeijer and A. J. van der Schaft. Nonlinear Dynamical Control Systems. Springer-Verlag, New York, 1990.

10

lemma.

[22] R. Ortega, A. Loria, R. Kelly, and L. Praly. On passivitybased output feedback global stabilization of Euler-Lagrange systems. Int. J. Robust Nonlinear Contr, 5(4):313–325, 1995.

q¯12 6= 0 ⇒ ∇q˜¯V 6= 0 ⇒ det T 6= 0 ⇒ (∇q¯U )T J¯12 6= 0

[23] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez. Passivity-based Control of Euler-Lagrange Systems. SpringerVerlag, London, 1998.

This relation follows from (A.1), Assumption 4-(ii) and "

[24] R. Ortega, M. W. Spong, F. G´ omez-Estern, and G. Blankenstein. Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Autom. Contr., 47(8):1218–1233, 2002.

det T = ∇q˜¯V

[25] R. Ortega, A. J. van der Schaft, B. M. J. Maschke, and G. Escobar. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica, 38(4):585–596, 2002.

"

0 1

1

trS2 /2

trS2 /2 det S2

#"

1 ˜1 q¯

0

0 1

# ∇q˜¯V

1

trS2 (¯ q )/2

# >0

which is equivalent to (18) in Assumption 3. This completes the proof. (Q.E.D.) Proof of Lemma 9. First of all, it is noted that the feedback system does not have any deadlock phenomena (which will be proved later). The lemma is proved by contradiction. Suppose that there does not exist a constant ² satisfying the claim of the lemma (30). Then

[28] M. Takegaki and S. Arimoto. A new feedback method for dynamic control of manipulators. Trans. ASME, J. Dyn. Syst., Meas., Control, 103:119–125, 1981. [29] M. Taniguchi and K. Fujimoto. Time-varying path following control for port-hamiltonian systems. In Proc. 48th IEEE Conf. on Decision and Control, pages 3323–3328, 2009.

q¯˙12 (t) = q¨¯12 (t) = p¯˙(t) = 0, ∀t ∈ [t1 , t1 + ²)

[30] K. Tsuchiya, T. Urakubo, and K. Tsujita. A motion control of a two-wheeled mobile robot. In Proc. IEEE SMC, pages 690–696, 1999.

(A.3)

for a small ² > 0, because x ¯ stays in Ω0 \{0}. On the other hand, the dynamics (9) with the feedback (22) implies that

[31] A. J. van der Schaft. L2 -Gain and Passivity Techniques in Nonlinear Control. Springer-Verlag, London, 2000. [32] A. J. van der Schaft and B. M. J. Maschke. On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics, pages 225–233, 1994.

x ¯ ∈ Ω0 \{0} ⇒ q¯12 = p¯ = 0 ¯ −1 p¯ = 0 ⇒ q¯˙ = J¯12 (¯ q )M  ¯ ¯ T ¯ −1 ¯ ¯ ¯ ¯T ¯T ˙    p¯ = −J12 ∇q¯H + J22 ∇p¯H − J12 γ − GC G M p¯   T  = −J¯12 γ(¯ q) ⇒ T  = −S1 (I S2T q¯12 )diag(1, 1) lim ∇(s,r) V (s, q¯3 )   s→+0    T = −S1 (¯ q ) 1 ν(¯ q3 ) (A.4) ( d ¯ (¯ ¯ (¯ q¨¯12 = dt (S1 (¯ q )M q )−1 p¯) = S1 (¯ q )M q )−1 p¯˙ ⇒ ¯ (¯ = −S1 (¯ q )M q )−1 S1 (¯ q )T 1 ν(¯ q3 ) (A.5)

[33] A. J. van der Schaft and B. M. J. Maschke. Mathematical modeling of constrained Hamiltonian systems. In Proc. 3rd IFAC Symp. Nonlinear Control Systems, 1995.

Proofs

This section provides the proofs of the theorems, the lemmas and the propositions in this paper. Proof of Lemma 7. In order to prove (27), we shall show the property q¯12 6= 0 ⇒ (∇q¯U )T J¯12 (¯ q ) 6= 0 using (26). Define a vector q˜ ¯ = (q˜ ¯1 , q˜ ¯2 ) := (k¯ q12 k, q¯3 ). Then the function U can be described as U (¯ q ) = V (q˜ ¯) and Assumption 3 gives

The last equation in (A.5) contradicts with Equation (A.3) since ν(¯ q3 ) 6= 0, which implies that the claim (30) is true. What we need to show for completing the proof is that the feedback system does not have any deadlock phenomena. Using Equations (A.5) and (25), q¨¯2 at the time t = t1 + 0 is described by

(A.1)

with T (¯ q ) := (1/q˜¯1 )∇q˜¯1 V (q˜ ¯)I + ∇q˜¯2 V (q˜ ¯)S2 (¯ q ).

#"

trS2 (¯ q )/2 det S2 (¯ q)

[27] S. Stramigioli, B. M. J. Maschke, and A. J. van der Schaft. Passive output feedback and port interconnection. In Proc. 4th IFAC Symp. Nonlinear Control Systems, pages 613–618, 1998.

T (∇q¯U )T J¯12 (¯ q ) = q¯12 T (¯ q ) S1 (¯ q)

0

because both S1 (¯ q ) and T (¯ q ) are nonsingular and

[26] P. Ramkrishna and Arjan van. der. Schaft. A portHamiltonian approach to modeling and interconnections of canal systems. In Proc. 17th MTNS, page WeA08, 2006.

A

1 ˜1 T q¯

1 ¯ −1 T 2 q3 ). ¯ −1 S T 1k (S1 M S1 ) 1 ν(¯ kS1 M 1 (A.6) The Filippov solution is characterized by the linear com-

(A.2)

q¨¯12 (t1 + 0) = −

The relationship (25) suggests that the following implication holds which is equivalent to the conclusion of the

11

ˆ = H ¯ + V bethere exists a time t = t1 such that H comes sufficiently small (to be compared with 1/bi ). Then |¯ q3 (t1 )| ¿ k¯ q12 (t1 )k has to hold because of the form of Vn (¯ q ) in (33). Moreover, asymptotic stability of the linearized system follows from the two facts that C(0) is positive definite and that the Hessian of Vs with respect to k¯ q12 k is positive definite in a neighborhood of the origin. This implies the local exponential convergence of q¯12 and p¯, i.e. there exist constants k¯1 , k¯2 > such that

bination of Equations (A.5) and (A.6) as ³ ¯ −1 S T κI + q¨ ¯12 = −S1 M 1

1−κ ¯ −1 S T ¯ −1 S T 1k S1 M 1 kS1 M 1

´ 1 ν(¯ q3 )

with 0 ≤ κ ≤ 1. Since the right hand side of the above equation is not zero for all κ and 1, the deadlock phenomenon will never occur. This completes the proof. (Q.E.D.) Proof of Theorem 6. Theorem is proved by contradiction in a way similar to the proof of the LaSalle’s invariance principle, e.g. in [11]. Lemma 7 implies that the state x ¯ of the feedback system converges to the set Ω0 defined in Equation (27). Suppose that the state from an initial condition x ¯(0) does not converge to the origin which implies that the state converges to Ω0 \{0}. Since ˆ x) ≥ 0 the Hamiltonian function has a lower bound as H(¯ and it is monotonically decreasing, it converges to a cerˆ x(t)) > 0. The state tain positive number c = limt→∞ H(¯ x ¯ converges to a constant x ¯? = (0, 0, q¯3? , 0, 0) ∈ Ω0 \{0} with q¯3? 6= 0 in a neighborhood of the origin, because the Hamiltonian function is a function only of q¯3 on Ω0 and it is positive definite. As the state approaches Ω0 and consequently p¯ → 0, it follows from Equations (25) and (A.4) that

¯ |¯ q3 (t)| ¿ k(¯ q12 (t), p¯(t))k ≤ k¯1 e−k2 t , ∀t ≥ t1 .

Then we have (34) holds for k1 = 2k¯1 and k2 = k¯2 . This completes the proof. (Q.E.D.) Proof of Theorem 13. (i)⇒(ii): First of all, we investigate the equilibrium sets. The equilibrium set of the system (9) with the feedback (38) is ¯ T Ωs = {(¯ q , p¯)¯J¯12 γ(¯ q ) = 0, p¯ = 0}. The equilibrium set of the system (37) with the feedback (39) is ¯ T Ωo = {(ˆ q , pˆ, rˆ)¯J¯12 γ(ˆ q ) = 0, pˆ = 0, rˆ = 0 ¯ T = {(¯ q , p¯, rˆ)¯J¯12 γ(¯ q ) = 0, p¯ = 0, rˆ = 0}.

p¯˙ → −S1 (0, 0, q¯3? )T 1 ν(¯ q3? ) 6= 0

Ωo is the origin if and only if Ωs is the origin. As for ˜ q , pˆ, rˆ) := the closed-loop system of (37) and (39), let H(ˆ ˆ ¯ + U , then the continuous function H ˜ monotonically H decreases when rˆ 6= 0. This property follows from the ˜˙ two facts that H(t) = −ˆ rT DT Co Dˆ r < 0 holds for qˆ12 := T (ˆ q1 , qˆ2 ) = q¯12 6= 0 and that the trajectory is piecewise smooth from the assumption. Here, we focus on a set q , pˆ, rˆ)|ˆ r = 0} which has the following vector Ωor = {(ˆ field at each point:    ¯ −1 pˆ qˆ˙ J¯12 M   ¤  £  pˆ˙ = −J¯T γ(ˆ ¯ −1 ¯ ¯ −1 pˆ. ¯T     12 q ) − (1/2)J12 ∇qˆ(M pˆ) − J22 M ¯ −1 pˆ rˆ˙ J¯12 M

with k1k = 1. But this property contradicts with p¯ → 0. Therefore the state converges to the origin which completes the proof. (Q.E.D.) Proof of Proposition 10. Only the case i = 1 is considered. The general case can be proved in the same way. As for the former part, we employ the notation q˜¯ = (q˜¯1 , q˜¯2 ) := (k¯ q12 k, q¯3 ) as in the proof of Lemma 7. First of all, it is proved that the potential V = Vs + Vn defined in (33) satisfies q˜¯1 6= 0

⇒

∇q˜¯V 6= 0

(A.7)

which implies (20). ∇q˜¯V can be computed as

In the case pˆ 6= 0, rˆ˙ 6= 0 holds because J¯12 is a full column matrix. In the case pˆ = 0, pˆ˙ 6= 0 holds except the origin because Ωs is the asymptotically stable origin and Ωo is ˜ the origin. These mean that Ωor \{0} = {(ˆ q , pˆ, rˆ)|H(t) = const.}\{0} is not an invariant set. Thus, the monoton˜ ically decreasing and the positive definite function H implies that the origin of the closed-loop system of (37) and (39) is asymptotically stable.

ai δi |q˜ ¯2 |γi +δi (ai |q˜ ¯1 | + bi |q˜ ¯2 |)δi +1 ˜ ai (γi + δi )|q¯1 | + bi γi |q˜ ¯2 | ∇q˜¯2 V = ∇q˜¯2 Vs + q˜ ¯2 |q˜ ¯2 |γi +δi −2 . (ai |q˜ ¯1 | + bi |q˜ ¯2 |)δi +1 ∇q˜¯1 V = ∇q˜¯1 Vs −

˜2 6= 0 ⇒ ∇q˜¯2 V 6= 0 because Therefore q˜¯1 6= 0, q¯ (∇q˜¯2 Vs )q˜¯2 ≥ 0. Also q˜ ¯1 6= 0, q˜ ¯2 = 0 ⇒ ∇q˜¯1 V = ∇q˜¯1 Vs 6= 0. These relations prove (A.7) which reduces to (20).

(i)⇐(ii): This can be proved in a similar way to the case (i)⇒(ii). Since the origin of the closed-loop system of ˆ (37) and (39) is asymptotically stable, {(¯ q , p¯)|H(t) = ¯ H + U = const.}\{0} is not an invariant set and the origin of the closed-loop system of (9) and (38) is asymptotically stable. (Q.E.D.)

The latter part is proved next. Lyapunov stability in γi > 0 directly follows from Theorem 6. The exponential convergence for γi = 0 is considered. It follows from Theorem 6 that, for any initial condition x(0),

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