Feedback Stabilization of Metriplectic Systems

1st IFAC Workshop on Thermodynamic Foundations of Mathematical Systems Theory July 13-16, 2013. Lyon, France

Feedback Stabilization of Metriplectic Systems Nicolas Hudon ∗ Denis Dochain ∗ Martin Guay ∗∗ ∗

ICTEAM, Universit´e Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium ({nicolas.hudon,denis.dochain}@uclouvain.be) ∗∗ Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada, K7L 3N6 ([email protected]) Abstract: This paper considers the problem of stabilizing control affine systems where the drift dynamics is generated by a metriplectic structure. These systems can be viewed as an extension of generalized (or dissipative) Hamiltonian systems, where two potentials, interpreted as generalized energy and entropy, are generating the dynamics. The proposed approach consists in generating, by homotopy centered at an equilibrium of the system, a mixed potential for the metriplectic system, and in using the obtained potential to construct a damping state feedback controller. Stability of the closed-loop system is then considered. Keywords: Nonlinear control, feedback stabilization, metriplectic systems. 1. INTRODUCTION

Systems of the form (2) are known in the literature as metriplectic systems (Kaufman, 1984; Morrison, 1986; Guha, 2007). These systems are generated by a conserved quantity E(x) (the generalized energy) and a metric quantity S(x) (the generalized entropy). Under the degeneracy conditions (3), they can be expressed as dissipative Hamiltonian systems of the form

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 where the drift vector field X0 (x) is of the form X0 (x) = J(x)∇T E(x) + R(x)∇T S(x),

x˙ = [J(x) − R(x)] ∇T H(x) + g(x)u, studied extensively in the literature, for example in (van der Schaft, 2000; Ortega et al., 2002; Cheng et al., 2002), if one pick H(x) to be the free energy at unit temperature, H(x) = E(x) − S(x) (Favache et al., 2010). The interest for systems of the form (2) can be justified from the point of view of irreversible thermodynamic systems, since the so-called GENERIC formulation of thermodynamics ¨ ¨ proposed in (Grmela and Ottinger, 1997; Ottinger and Grmela, 1997) is based on the development of metriplectic systems proposed originally in (Kaufman, 1984; Morrison, 1986). It should be noted that under the degeneracy conditions given above, the quantity F (x) = E(x) − S(x) (or more generally, F (x) = E(x) − T S(x), where T is the temperature) can be interpreted as generalized free energy (Morrison, 1986; Guha, 2007). Moreover, this class of systems is related to the representation of smooth nonlinear dynamical systems as the sum of a gradient system, given by the metric part of the system generated by the generalized entropy, and (n − 1) Hamiltonian systems, given by the symplectic part of the system generated by the generalized energy, as presented for example in Roels (1974); Kiehn (1974); Villa Mendes and Taborda Duarte (1981). It can also be related to a recent construction presented in (Guay et al., 2012), where nonlinear control systems are decomposed using the Hodge decomposition theorem (Morita, 2001; Warner, 1983) to obtain potential-driven representations, such as gradient systems, generalized Hamiltonian systems, and

(2)

under degeneracy constraints J(x) · ∇T S(x) = R(x) · ∇T E(x) = 0,

(3)

T

and such that J(x) = −J (x). We assume no a priori structure or sign for R(x). We assume that the generating functions E(x) and S(x) are of class C k (Rn ; R), with k ≥ 2. Under those conditions, it can be shown (Morrison, 1986) that dE dS = 0, ≥ 0, dt dt along the trajectories of the drift vector field. In this note, we assume that X0 (x) has an isolated equilibrium x∗ , and the task is to design a damping feedback control u(x) such that x∗ is stabilized in closed-loop. More precisely, following the original contribution in (Jurdjevic and Quinn, 1978) and the recent monograph (Malisoff and Mazenc, 2009), the objective of this paper is to construct a potential ψ(E(x), S(x)) > 0, with ψ(x∗ ) = 0, and a damping feedback law of the form uk (x) = −(Xk · ∇ψ)(x), k = 1, . . . , m such that the equilibrium x stable in closed-loop. 978-3-902823-40-3/2013 © IFAC

∗

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is locally asymptotically

12

10.3182/20130714-3-FR-4040.00015

2013 IFAC TFMST July 13-16, 2013. Lyon, France

systems described by Brayton–Moser equations. To the knowledge of the authors, this class of system has never been considered from a control design point of view. However, it should be noted that asymptotic stability of metriplectic systems was studied in (Birtea et al., 2007) using Lasalle’s invariance principle. The contribution by Birtea et al. (2007) considered the class of metriplectic systems originally proposed in (Kaufman, 1984) where the metric part of the system is generated by Casimir functions.

A smooth vector field in Γ∞ (Rn ) is denoted by

As stated above, the objective of this paper is to construct a mixed potential ψ(E(x), S(x)) to construct damping feedback stabilizing feedback for (2). Obviously, mixed potentials, that is, potentials combining energy and entropy, are known from classical thermodynamics (Callen, 1985), see for example the availability potential used in (Ydstie and Alonso, 1997) in the context of dissipative systems theory and control of process systems. In this note, we depart from this approach and consider instead a local approach based on a decomposition of the drift vector field, proposed in Hudon et al. (2008), to generate a potential ψ(E(x), S(x)) for stabilization purposes, extending the results given in (Hudon and Guay, 2009). In Hudon et al. (2008), 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 one-form for the system, an exact one form generated by the desired potential function and an antiexact form that generates the tangential dynamics were obtained. In (Hudon and Guay, 2009), it was shown that this approach leads to the construction of a first function ψ(·) (an auxiliary scalar field following the nomenclature proposed by Malisoff and Mazenc (2009)) and that a second application of the decomposition method to obtain a control Lyapunov function for stability characterization. In the present contribution, we seek to exploit the structure of the drift vector field (2) in the application of these ideas.

where v i (x) and ωi (x) are smooth functions on Rn . The standard basis for vectors in Γ∞ (Rn ) and one-forms in ∂ and dxi , respectively. If α ∈ Λ1 (Rn ) are denoted by ∂x i k n Λ (R ), then we write deg α = k. Note that in this context, the space of 0-forms, Λ0 (Rn ) = C ∞ (Rn ), the space of smooth functions on Rn .

X(x) =

n X

v i (x)

i=1

∂ , ∂xi

and a smooth differential one-form in Λ1 (Rn ) by ω(x) =

n X

ωi (x)dxi ,

i=1

The wedge product ∧ is defined on Λ1 (Rn ) × Λ1 (Rn ) by dxi ∧ dxj = −dxj ∧ dxi dxi ∧ f (x)dxj = f (x)dxi ∧ dxj for all smooth functions f (x) and by α ∧ (β + γ) = α ∧ β + α ∧ γ, for all α, β, γ ∈ Λ(Rn ), where Λ(Rn ) denotes Ln the space of differential forms of all degrees on Rn , k=0 Λk (Rn ). The differential operator d is the unique operator on Λ(Rn ), d : Λk (Rn ) → Λk+1 (Rn ), with the following properties: 1. 2. 3. 4.

0 ≤ k ≤ n − 1,

d(α + β) = dα + dβ; d(α ∧ β) = dα ∧ β + (−1)deg α α ∧ dβ; P ∂f df = i ( ∂x )dxi , ∀f (x) ∈ Λ0 (Rn ); i 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

The paper is organized as follows. In Section 2, mathematical background for the approach considered, including definitions of the radial homotopy operator that is used in the sequel, is reviewed. The main result of the paper, which is the construction of a stabilizing damping state feedback control law for the system (2) is presented in Section 3. An illustration of the proposed approach is given in Section 4. Conclusion and future areas for investigation are outlined in Section 5.

y : Γ∞ (Rn ) × Λk (Rn ) → Λk−1 (Rn ), 0 ≤ k ≤ n, with the following properties for all α, β ∈ Λk (Rn ), X, X1 , X2 ∈ Γ∞ (Rn ) and f, g ∈ Λ0 (Rn ): 1. 2. 3. 4. 5. 6.

2. PRELIMINARIES

Xyf = 0; Xyω = ω(X), ∀ω ∈ Λ1 (Rn ); Xy(α + β) = Xyα + Xyβ; (f X1 + gX2 )yα = f · (X1 yα) + g · (X2 yα); Xy(α ∧ β) = (Xyα) ∧ β + (−1)deg(α) α ∧ (Xyβ); X1 y(X2 yα) = −X2 y(X1 yα).

The last property leads to the following composition rule for interior product: Xy(Xyα) = 0. For a vector field X and a k-form α, the (k − 1)-form obtained by taking the interior product Xyα can be viewed as a covariant tensor. The exterior derivative and the interior product can be used together to determines the Lie derivative of a k-form α along a vector field X by using the Cartan’s formula:

2.1 Elements of Exterior Calculus Background geometric elements and definitions for the main constructions presented in the paper are given, expressed in the formalism of exterior calculus and differential forms on Rn . For a complete review on the topic, the reader is referred to the references (Edelen, 2005; Flanders, 1989).

LX α = Xydα + d(Xyα). 13

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2013 IFAC TFMST July 13-16, 2013. Lyon, France

A star-shaped region S has a natural associated vector field X, defined in local coordinates by

In particular, for a zero-form f , since Xyf = 0, the Lie derivative of f along X can be computed using interior products as LX f = Xydf .

X(x) = (xi − x∗i )

The Hodge star operator ? : Λk (Rn ) → Λn−k (Rn ) has the following properties (Morita, 2001, Chapter 4). For any f and g in C ∞ (Rn ) and for any α and β in Λk (Rn ), 1. 2. 3. 4. 5.

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

?(f α + gβ) = f ? α + g ? β; ? ? α = (−1)k(n−k) α; α ∧ ?β = β ∧ ?α = hα, βiµ; ?(α ∧ ?β) = ?(β ∧ ?α) = hα, βi; h?α, ?βi = hα, βi.

Z

!] ωi (x)dxi

=

i=1

n X

ωi (x)

i=1

i=1

Xi (x)

∂ ∂xi

![ =

n X

where ω(x∗ +λ(x−x∗ )) denotes the differential form evaluated on the star-shaped domain in the local coordinates defined above. The important properties of the homotopy operator used in the present context 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)(x∗i ) = 0; 4. XyH = 0, HXy = 0.

∂ ∂xi

The first part of the right hand side of (7), d(Hω), is obviously a closed form, since by definition of the exterior derivative, d ◦ d(Hω) = 0. By the first property of the homotopy operator, for ω ∈ Λk (S), we have (Hω) ∈ Λk−1 (S), hence 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 (Edelen, 2005).

Xi (x)dxi .

i=1

The divergence of the vector field X is given as div X = ?d ? ωX .

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If the one-form ω is closed on a star-shaped domain in Rn , i .e., if dω = 0,it is also locally exact by virtue of the Poincar´e Lemma. However, if the one-form is not closed, ω can be expressed as the sum of an exact component and an anti-exact component using a homotopy decomposition, and the divergence of the drift vector field is encoded in the anti-exact part. A locally-defined homotopy operator is presented in the next section, based on the discussion in (Edelen, 2005, Chapter 5).

In the sequel, the homotopy operator is used to construct a locally-defined potential ψ(E(x), S(x)) to be used for damping feedback stabilization of the system (1) with a drift given by (2). 3. STABILIZATION OF METRIPLECTIC SYSTEMS 3.1 Construction of a Mixed Potential The first step in the proposed construction is to compute, by homotopy, a locally-defined potential ψ(E(x), S(x)) for the drift vector field X0 (x) given by (2). Using the notation from Section 2, the drift vector field (2) is given as ! n n X X ∂E(x) ∂S(x) ∂

2.2 Homotopy Operator A homotopy operator H, is a linear operator on elements of Λk (Rn ) that satisfies the identity ω = d(Hω) + Hdω, for a given differential form ω ∈ Λk (Rn ).

X(x∗ + λ(x − x∗ ))yω(x∗ + λ(x − x∗ ))λk−1 dλ, (8)

0

and the dual one-form to a vector field Xi is given by n X

1

(Hω)(x) =

The musical isomorphisms sharp ] : Λ1 (Rn ) → Γ∞ (Rn ) and flat [ : Γ∞ (Rn ) → Λ1 (Rn ) (Cort´es et al., 2005) are used in the present paper. For a vector field X, consider ωX , the one-form obtained by ωX = X [ (v) = g(X, v). In particular, for the choice of metric g = dx1 ⊗ dx1 + . . . + dxn ⊗ dxn (the standard metric (Banaszuk and Hauser, 1996)), the dual vector field to a one-form ω is given by n X

∂ , ∀x ∈ S. ∂xi

X0 (x) =

(7)

Jij (x)

i=1

The first step in the construction of a homotopy operator is to define a star-shaped domain on Rn .

j=1

∂xj

+ Rij (x)

∂xj

∂xi

.

(9)

Following the approach proposed in (Guay et al., 2012), we first construct a differential one-form ω ∈ Λ1 (Rn ) for X0 (x). To simplify the notation, we assume without lost of generality that the isolated equilibrium is located at the origin, possibly through a change of variable, and that ∇T E(x∗ ) = ∇T S(x∗ ) = 0n . The procedure used in the present paper relies on the canonical Riemannian metric in Rn , given as g = dx1 ⊗ dx1 + . . . + dxn ⊗ dxn with its associated volume form in Λn (Rn ), expressed as µ = dx1 ∧ dx2 ∧ . . . ∧ dxn . A (n − 1) differential form j is obtained by taking the interior product of the volume µ with respect to the drift vector field X0 (x), i .e.,

An open subset S of Rn is said to be star-shaped with respect to a point p∗ = (x∗1 , . . . , x∗n ) ∈ S if the following conditions hold: 1. S is contained in a coordinate neighborhood U of p∗ ; 2. The coordinate functions of U assign coordinates (x∗1 , . . . , x∗n ) to p∗ ; 3. If p is any point in S with coordinates (x1 , . . . , xn ) assigned by functions of U , then the set of points (x∗ + λ(x − x∗ )) belongs to S, for all λ ∈ [0, 1]. 14

2013 IFAC TFMST July 13-16, 2013. Lyon, France

j= =

assume that there exists an isolated equilibrium x∗ for the drift vector field, X0 (x∗ ) = 0. Assume moreover that for every x ∈ O \ {x∗ }, where O is a neighborhood of the equilibrium O ⊂ Rn , we have

! ∂ X0,i (x) yµ ∂xi i=1

n X

n X c i ∧ . . . ∧ dxn , (−1)i−1 X0,i (x)dx1 ∧ . . . ∧ dx

span{X0 , ad0X0 Xk , . . . , adpX Xk , p ∈ N, k = 1, . . . , m} = Rn ,(12)

i=1

0

c i denotes a removed element such that j is a where dx (n − 1) form. Taking the exterior derivative of j, and by the property of the wedge product that dxi ∧ dxi = 0, we obtain (Lee, 2006), dj =

n X i=1

where, by convention, ad0X h i 0 Xk (x) = [X0 , Xk ] (x) and p p−1 adX0 Xk (x) = X0 , adX0 Xk (x) for p ∈ N. We first define the concept of a weak Jurdjevic–Quinn function, following (Malisoff and Mazenc, 2009, Chapter 2). Definition 1. A weak Jurdjevic–Quinn function, W (x), is a smooth function such that

∂X0,i (x)dx1 ∧ . . . ∧ dxn = divX0 (x)µ. ∂xi

The desired one-form ω is obtained by using the Hodge star operator ? of the (n − 1) form j, i .e.,

1. W (x) > 0 for all x ∈ X \ {x∗ }, 2. W (x∗ ) = 0 and (LX0 W )(x∗ ) = 0, 3. (LX0 W )(x) < 0 for all x in a neighborhood O ⊂ Rn \ {x∗ }.

ω = ?j = ?(X0 (x)yµ) n X = (−1)n−1 X0,i (x)dxi .

Consider the feedback law u = (u1 , . . . , um )T defined by

i=1

For the metriplectic drift vector field (9), we thus obtain a differential one-form given by: ! n n X X ∂S(x) ∂E(x) ω = (−1)n−1

Jij (x)

i=1

j=1

+ Rij (x)

∂xj

∂xj

uk (x) = −(Xk ydW )(x), ∀ k ∈ 1, . . . , m, with W (x) a weak Jurdjevic–Quinn function for the system (1). With this feedback, one has for all x ∈ Rn \ {x∗ }

dxi . (10)

m

X dW (Xk ydW )2 (x) < 0. (x) = (X0 ydW )(x) − dt

By application of the homotopy operator centered at the origin (assumed to be the equilibrium of the drift system), we have

k=1

Therefore, the equilibrium x stable in closed-loop.

ψ(E(x), S(x)) = Hω n−1

Z

1

(λx) ×

= (−1)

X n  n X

0

i=1

∂S(λx) + Rij (λx) ∂xj



j=1

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Using the damping feedback controller

In what follows, to keep the discussion as general as possible, we keep this locally-defined potential in this integral form and denote it by ψ(x). It should be noted however that if both generating potentials E(x) and S(x) are quadratic with constant structure matrices J and R, the obtained potential ψ(x) is again quadratic.

uk (x) = −κk (Xk ydψ)(x), with κk a controller gain, the metriplectic vector field is given by

X(x) =

Following (Hudon and Guay, 2009), we obtain the exact part of the dynamics ωe (x), by differentiation of the obtained potential ωe (x) =

n X ∂ψ(x) i=1

∂xi

∈ Rn is asymptotically

We now use a Jurdjevic–Quinn controller to stabilize the metriplectic system (2) and show stability of the closedloop using a Lyapunov function derived from the potential ψ(x).

∂E(λx) Jij (λx) ∂xj

dλ.

∗

n  n X X i=1

j=1

−

m X

∂E(x) ∂S(x) Jij (x) + Rij (x) ∂xj ∂xj

2 κk Xik

k=1

∂ψ(x) ∂xi



∂ . ∂xi

!

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To show the stability of the closed-loop, we consider the Lyapunov function candidate

dxi ,

which is used in the sequel to study the stability of the closed-loop system.

1 2 ψ (x). (14) 2 Along the trajectories of the closed-loop system, the timederivative of V (x) is given by V (x) =

3.2 Damping Stabilization of Metriplectic Systems We now consider the problem of stabilizing an isolated equilibrium of the drift dynamics X0 (x) using the damping feedback design procedure proposed originally in (Jurdjevic and Quinn, 1978) and developed in (Malisoff and Mazenc, 2009). For a control affine system of the form (1),

V˙ = ψ(x) × or, equivalently, by 15



 ∂ψ · X(x) , ∂x

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2013 IFAC TFMST July 13-16, 2013. Lyon, France

V˙ = ψ(x)(Xyωe )

from which it is possible to recover a metriplectic representation (2), by setting S(x) = φ(l2 (x)), with φ(·), an arbitrary function, and " #

= ψ(x)(X0 yωe ) − ψ(x)

n m X X i=1

k=1

= ψ(x)(X0 yωe ) − ψ(x)

∂ψ(x) 2 κk Xik (x) ∂xi

n m X X i=1

= ψ(x)

n X

X0i

i=1

∂ψ(x) − ∂xi

κk



! y



Xik (x)

k=1 m X

κk



Xik (x)

k=1

∂ψ(x) dxi ∂xi

∂ψ(x) ∂xi

2

2

!

∂ψ(x) ∂xi



0 1 0 J = −1 0 0 0 0 0

!

I22 x22 + I32 x23 −I1 I2 x1 x2 −I1 I3 x1 x3 R = −I1 I2 x1 x2 I12 x21 + I32 x23 −I2 I3 x1 x3 . −I1 I3 x1 x3 −I2 I3 x1 x3 I12 x21 + I22 x22

"

.

"

ψ(x) = x23 I1 I3 x21 + I2 I3 x1 x2 − I12 x21 − I22 x22 + x1 x2 (I1 + 2I1 x1 x2 (I2 − I1 )) .

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u(x) = −2κx3 I1 (I3 − I1 )x21 + I2 I3 x1 x2 − I22 x22 ,(24) which is different from the controller proposed originally in (Bloch and Marsden, 1990). Figures 1 and 2 show the trajectory of the dynamics in closed-loop and the control action for the system with parameters I1 = 1/2, I2 = 1, I3 = 3/2, initial conditions x0 = (1, 1, 1)T , and a control gain κ = 1/2.

with H = 12 (x21 + x22 + x23 ), with an isolated equilibrium located at the origin. By homotopy, ψ(x) = H(x), which is positive definite, and taking V (x) = ψ 2 (x) leads to 2 X




A Jurdjevic–Quinn damping controller is obtained by taking the interior product of dψ with respect to the control vector field [0, 0, 1]T , leading to the feedback control

# " #! " # 0 −1 0 −1 1 0 0 T 1 0 0 + −1 −1 0 ∇ H(x) + 0 u, 0 0 0 0 0 −1 1

V˙ (x) = H(x)

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In the sequel, we consider the metriplectic system generated using J, R(x), E(x) and S(x) = l2 (x) given above. By homotopy centered at x∗ , we obtain the following potential for the system

In practice, if E(x) and S(x) are convex potentials and J and R are constant matrices, the picture is admittedly clearer, as shown in the following example: Example 2. Consider the port-Hamiltonian system given by # x˙1 x˙ 2 = x˙ 3

#

The objective is to stabilize an isolated equilibrium located at x∗ = (x∗1 , x∗2 , x∗3 )T = (x1d , 0, 0)T .

Depending of the sign of ψ(x), given by the integral equation (11), the task is therefore to select a set of gains κk such that the terms X0i ∂ψ(x) ∂xi are dominated to obtain ∗ ˙ V < 0 on x ∈ O \ {x }. Under the span conditions (12), such gains exist.

"

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−∇H · ∇T H − κx23 < 0.

i=1

Future work will focus on the application of the proposed approach for metriplectic systems in the context of irreversible thermodynamics, and in particular to cases where E(x) and S(x) are known (convex) potentials. An application of the proposed approach is given in the next section to illustrate the approach for cases where the structure matrices are not constant. 4. EXAMPLE To illustrate the above approach to the stabilization of metriplectic systems, we consider the stabilization of the rigid body, an example considered in (Morrison, 1986; Bloch and Marsden, 1990; Aeyels, 1992). The control system is given by (Morrison, 1986) x˙ 1 = (I2 − I3 )x2 x3

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x˙ 2 = (I3 − I1 )x1 x3

(17)

x˙ 3 = (I1 − I2 )x1 x3 + u,

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Fig. 1. Closed-loop state dynamics 5. CONCLUSION We considered the problem of stabilizing control affine systems where the drift is generated by two potentials, interpreted as generalized energy and generalized entropy. First, a mixed potential for the metriplectic system was derived using a homotopy operator centered at an isolated equilibrium of the system. Then, the obtained potential was used to construct a damping state feedback controller following the approach proposed originally by Jurdjevic and Quinn (1978). Stability of the closed-loop system was shown. Future investigations would consider applications where there exist multiple isolated equilibria, for example chemical reactors (Favache and Dochain, 2010).

where I1 < I2 < I3 are positive constants. Following the discussion in (Morrison, 1986), we have two conserved 1 E(x) = (I1 x21 + I2 x22 + I3 x23 ) 2 l2 (x) = x21 + x22 + x23 ,

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2013 IFAC TFMST July 13-16, 2013. Lyon, France

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