Pablo Borja

Rafael Cisneros

Romeo Ortega
email: ortega]@lss.supelec.fr

Shaping the Energy of Port-Hamiltonian Systems Without Solving PDE's

Keywords: Passivity, nonlinear systems, passivity-based control, Hamiltonian systems

Equilibrium stabilisation of nonlinear systems via energy shaping is a well-established, robust, passivity-based controller design technique. Unfortunately, its application is often stymied by the need to solve partial differential equations. In this paper a new, fully constructive, procedure to shape the energy for a class of port-Hamiltonian systems that obviates the solution of partial differential equations is proposed. Proceeding from the well-known passive, power shaping output we propose a nonlinear static state-feedback that preserves passivity of this output but with a new storage function. This function contains some tuning gains used to ensure it is positive definite, hence a suitable Lyapunov function for the closed-loop. Connections with other standard passivity-based controllers are indicated and it is shown that the new controller design is applicable to two benchmark examples.

I. INTRODUCTION

Control of physical systems via energy shaping is a wellestablished technique whose roots can be traced back to Lagrange's and Dirichlet's work [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications[END_REF]. In the control community this fundamental concept was first introduced by Takegaki and Arimoto in [START_REF] Takegaki | A New Feedback Method for Dynamic Control of Manipulators[END_REF] for mechanical systems and by Jonckheere in [START_REF] Jonckheere | Lagrangian Theory of Large Scale Systems[END_REF] for electromechanical systems. In [START_REF] Takegaki | A New Feedback Method for Dynamic Control of Manipulators[END_REF] the Hamiltonian formalism is used to describe the system dynamics while the Euler-Lagrange formulation is used in [START_REF] Jonckheere | Lagrangian Theory of Large Scale Systems[END_REF]. To date, these two mathematical descriptions of the system dynamics are favoured to carry out the energy-shaping task with a lot of research devoted to the particular case of mechanical systems, see e.g., [START_REF] Bloch | Controlled Lagrangians and the Stabilization of Mechanical Systems I. The First Matching Theorem[END_REF], [START_REF] Fujimoto | Canonical Transformation and Stabilization of Generalized Hamiltonian Systems[END_REF], [START_REF] Ortega | Stabilization of a Class of Underactuated Mechanical Systems Via Interconnection and Damping Assignment[END_REF] and references therein.

One of the central features of existing energy shaping techniques is the preservation in closed-loop of the original systems structure-either Lagrangian or Hamiltonian-and a key step is the identification of the energy (or Lagrangian) functions that can be assigned via feedback. This assignable energy functions are the solutions of the so-called matching equations, which are a set of partial differential equations (PDEs) that need to be solved to complete the controller design. A lot of research effort has been devoted to the solution of the matching equations-see e.g., [START_REF] Acosta | Interconnection and Damping Assignment Passivity-based control of Mechanical Systems with Underactuation Degree One[END_REF], [START_REF] Auckly | On the λ-Equations for Matching Control Laws[END_REF], [START_REF] Blankenstein | The Matching Conditions of Controlled Lagrangians and IDA-Passivity Based Control[END_REF], [START_REF] Lewis | Notes on Energy Shaping[END_REF], [START_REF] Mahindrakar | Further Constructive Results on Interconnection and Damping Assignment Control of Mechanical Systems: the Acrobot Example[END_REF], [START_REF] Nunna | Constructive Interconnection and Damping Assignment for Port-Controlled Hamiltonian Systems[END_REF], [START_REF] Viola | Total Energy Shaping Control of Mechanical Systems: Simplifying the Matching Equations Via Coordinate Changes[END_REF]. Also, there is a large list of applications where it has been possible to solve these equations, including (almost) all basic pendular systems considered in the literature, motors, generators, power systems, power converters, level control systems, etc.-see [START_REF] Ortega | Interconnection and Damping Assignment Passivity-Based Control: A Survey[END_REF] for a partial list. In spite of that, this difficult key step remains the main stumbling block for the wider application of these methods.

In the recent paper [START_REF] Donaire | Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations[END_REF] it has been proposed for mechanical systems to abandon the objective of structure preservation and attention has been concentrated on the energy shaping objective only. That is, we look for a static statefeedback that stabilizes the desired equilibrium assigning to the closed-loop a Lyapunov function of the same form as the energy function of the open-loop system but with new, desired inertia matrix and potential energy function. However, it was not required that the closed-loop system is a mechanical system with this Lyapunov function qualifying as its energy function. In this way, the need to solve the matching equations is avoided.

In this paper we pursue the same research line considering the more general case of port-Hamiltonian (pH) systems [START_REF] Duindam | Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach[END_REF] of the form ẋ = F (x)∇H(x) + g(x)u, where F (x) + F ⊤ (x) ≤ 0 and F (x) is full rank. The starting point of the design is the well-known power shaping output [START_REF] Ortega | Power shaping: A New Paradigm for Stabilization of Nonlinear RLC Circuits[END_REF], which is known to define a passive output for the pH system with storage function its energy function H(x). Assuming that this output is "integrable", the next step is the design of a nonlinear static state-feedback that preserves this passive output but now with a new storage function. The latter is constructed as the weighted sum of the original energy function and a quadratic term of the shifted "integral" of the power shaping output. The weighting gains and the aforementioned shifting constant are the degrees of freedom introduced to ensure positive definiteness of the new energy function, which then qualifies as a Lyapunov function for the closed-loop system. The condition of integrability of the power shaping output boils down to a classical integrability condition of the vector fields F -1 (x)g i (x), with g i (x) the columns of the matrix g(x), hence it can be readily verified.

The remaining of the paper is organized as follows. Section II presents the problem formulation and main assumptions on the pH system. The design of the energy shaping feedback is carried out in Section III. Section IV contains the main stabilization result. Section V explores the relationship between the new controller design and the two well-known passivity-based control (PBC) techniques of energy balancing (EB-PBC) [START_REF] Ortega | Putting Energy Back in Control[END_REF], [START_REF] Ortega | Control by (State-Modulated) Interconnection of Port-Hamiltonian Systems Nonlinear Control Systems[END_REF] and interconnection and damping assignment (IDA-PBC) [START_REF] Ortega | Interconnection and Damping Assignment Passivity-Based Control: A Survey[END_REF]. In Section VI we show that the method is applicable to two benchmark examples and, to study the limitations of the technique, consider the case of LTI systems. In particular, we prove that, unlike general IDA-PBC of LTI systems but similarly to the case of IDA-PBC of mechanical systems [START_REF] Ortega | Control via Interconnection and Damping Assignment of Linear Time-Invariant Systems: A Tutorial[END_REF], stabilisability of the LTI system is not enough to ensure that the proposed controller yields a stable closed-loop system. The paper is wrapped-up with concluding remarks and future research in Section VII.

Notation For x ∈ R n , S ∈ R n×n , S = S ⊤ > 0, we denote the Euclidean norm |x|2 := x ⊤ x, and the weightednorm x 2 S := x ⊤ Sx. For a scalar function H : R n → R we denote ∇ x H(x) = ∂H(x) ∂x ⊤ . For vector functions C : R n → R m , we define its (transposed) Jacobian matrix ∇C(x) = [∇C 1 (x), . . . , ∇C m (x)]. For any mapping F : R n → R p×q and the distinguished element x * ∈ R n we denote the constant matrix F * := F (x * ). We define the pseudoinverse of a full-rank matrix G ∈ R n×m as G † := (G ⊤ G) -1 G ⊤ . When clear from the context the subindex of the operator ∇ and the arguments of the functions will be omitted. All the functions in the paper are assumed sufficiently smooth.

II. PROBLEM FORMULATION AND MAIN ASSUMPTIONS

The standard representation of pH systems is of the form

ẋ = [J (x) -R(x)]∇H(x) + g(x)u, (1)
where x ∈ R n is the state vector, u ∈ R m , m ≤ n, is the control vector, H : R n → R is the systems stored energy, J , R : R n → R n×n , with J (x) = -J ⊤ (x) and R(x) = R ⊤ (x) ≥ 0, are the interconnection and damping matrices, respectively, and g : R n → R n×m is the input matrix, which is full rank. To simplify the notation in the sequel we define the matrix

F : R n → R n×n F (x) := J (x) -R(x).
The control objective is to stabilise an equilibrium x * , which is an element of the set of assignable equilibria given by

E := x ∈ R n | g ⊥ (x)F (x)∇H(x) = 0 . (2)
The following assumptions identify the class of pH systems for which the proposed control strategy is applicable.

Assumption 1: The matrix F (x) is full rank. Assumption 2: The vector fields F -1 (x)g i (x), with g i (x), i = 1, . . . , m, the columns of the matrix g(x), are gradient vector fields. That is,

∇ F -1 (x)g i (x) = [∇ F -1 (x)g i (x) ] ⊤ .
If Assumption 1 holds, it is possible to define the power shaping output as follows

y PS := -g ⊤ (x)F -⊤ (x) [F (x)∇H(x) + g(x)u] . (3)
As shown in [START_REF] Ortega | Power shaping: A New Paradigm for Stabilization of Nonlinear RLC Circuits[END_REF], [START_REF] Ortega | Control by (State-Modulated) Interconnection of Port-Hamiltonian Systems Nonlinear Control Systems[END_REF] y PS is a cyclo-passive output for the pH system (1) with storage function H(x). More precisely, the following dissipation inequality holds

Ḣ ≤ u ⊤ y PS . (4)
Noting that y PS may be written as

y PS = -g ⊤ (x)F -⊤ (x) ẋ (5)
and recalling Poincare's Lemma it is easy to see that Assumption 2 ensures the existence of a function γ :

R n → R m such that γ = (∇γ) ⊤ ẋ = y PS , (6)
with y PS defined in (3).

III. ENERGY SHAPING

In this section we define a static state-feedback such that the system (1) in closed-loop with this control preserves passivity of the mapping v → y PS but with a suitably modified storage function.

Proposition 1: Suppose Assumptions 1 and 2 hold. Define the mapping u BC :

R n → R m u BC (x) := [k e I -k a g ⊤ (x)F -⊤ (x)g(x)] -1 [v- K I (γ(x) + C) + k a g ⊤ (x)F -⊤ (x)F (x)∇H(x) (7)
where 1 ∇γ(x

) := -F -1 (x)g(x) (8)
and k a , k e ∈ R + , C ∈ R m , K I ∈ R m×m with K I = K ⊤ I > 0 are free parameters. The system (1) in closed-loop with the control u = u BC (x) defines a cyclo-passive mapping v → y PS with storage function

H d (x) = k e H(x) + 1 2 γ(x) + C 2 K I . (9) Proof: From (9) we get Ḣd = k e Ḣ + y ⊤ PS K I (γ + C) ≤ y ⊤ PS [k e u + K I (γ + C)] = y ⊤ PS v + k a g ⊤ F -⊤ F ∇H + k a g ⊤ F -⊤ gu = y ⊤ PS (v -k a y PS ) ≤ y ⊤
PS v, where we used [START_REF] Ortega | Stabilization of a Class of Underactuated Mechanical Systems Via Interconnection and Damping Assignment[END_REF] in the first equality, (4) in the first inequality and ( 7) and (3) for the second and third equality, respectively.

From [START_REF] Ortega | Stabilization of a Class of Underactuated Mechanical Systems Via Interconnection and Damping Assignment[END_REF] we have that

γ(x(t)) = t 0 y PS (x(τ ))dτ + γ(x(0)),
where we recall that y PS is defined in [START_REF] Jonckheere | Lagrangian Theory of Large Scale Systems[END_REF]. Hence, the second term in the new storage function ( 9) may be interpreted as the integral of y PS . This establishes a connection with the PI-like controllers proposed for mechanical systems in [START_REF] Donaire | Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations[END_REF].

IV. STABILIZATION

From Proposition 1 it is clear that if the new storage function H d (x) is positive definite (with respect to the desired equilibrium x * ) it qualifies as a bona fide Lyapunov function for the closed-loop system (with v = 0) that ensures stability of x * . This fact is enunciated in the proposition below where we also give easily verifiable conditions for .

Proposition 2: Consider the system (1), verifying Assumptions 1 and 2, in closed-loop with the control u = u BC (x), with v = 0, where u BC (x) is given by [START_REF] Acosta | Interconnection and Damping Assignment Passivity-based control of Mechanical Systems with Underactuation Degree One[END_REF].

(i) If x * ∈ E and

x * = arg min H d (x), (10)
y PS (t) ≡ 0 ⇒ lim t→∞ x(t) = x * .
(ii) Condition ( 10) holds if the following inequality is satisfied

(∇ 2 H d ) * > 0. (11) Proof: The proof of (i) follows from the fact that H d (x) is positive definite with Ḣd = -k a |y PS | 2 ≤ 0,
and classical Lyapunov theory [START_REF] Khalil | Nonlinear Systems[END_REF].

To establish (ii) first notice that from ( 5) and ( 6) the control (7) reduces to

u = - 1 k e [K I (γ + C) + k a y PS ] . (12)
Since at the equilibrium y PS * = 0, then [START_REF] Nunna | Constructive Interconnection and Damping Assignment for Port-Controlled Hamiltonian Systems[END_REF] becomes

u * = - 1 k e K I (γ * + C) (13)
also, from (1)

(∇H) * = -F -1 * g * u * = 1 k e F -1 * g * K I (γ * + C) (14)
where the last equation was obtained from the substitution of ( 13). On the other hand,

(∇H d ) * = k e (∇H) * + (∇γ) * K I (γ * + C) = k e (∇H) * -F -1 * g * K I (γ * + C). (15
) Replacing ( 14) in [START_REF] Donaire | Shaping the Energy of Mechanical Systems Without Solving Partial Differential Equations[END_REF] we have (∇H d ) * = 0 which ensures that x * is a critical point of the closed-loop system. The proof is completed noting that, the inequality [START_REF] Mahindrakar | Further Constructive Results on Interconnection and Damping Assignment Control of Mechanical Systems: the Acrobot Example[END_REF] implies that (10) holds.

Remark 1: From (15) it is clear that-for the purpose of stabilization-the constant vector C, which ensures x * is an equilibrium point of the closed-loop, is uniquely defined by

C := k e K -1 I g † * F * (∇H) * -γ * . (16)
V. RELATION WITH CLASSICAL PBCS

In this section we discuss the relationship between the new controller and the classical PBC techniques of EB and IDA. 2It should be underscored that, in contrast with these PBC design techniques, the proposed method does not preservein general-the pH structure in closed-loop, but we will show that for some particular choice of the tuning gains it does.

A. Energy-balancing PBC

The basic idea of EB-PBC (with the output y PS ) is to look for state feedbacks u EB : R n → R m such that Ḣa = -u ⊤ EB y PS , for some "added" energy function H a : R n → R. In this case, setting u = u EB (x) transforms the cyclo-pasivity inequality (4) into Ḣ + Ḣa ≤ 0, and if H(x) + H a (x) is positive definite the closed-loop system will have a stable equilibrium at x * . The following proposition states that, for a suitable choice of the tuning gains, the new controller. Proposition 3: Consider the pH system (1) verifying Assumptions 1 and 2. Fix the gains of the mapping u BC (x) as k e = 1 and k a = 0. Then, the control u = u BC (x), with v = 0, is an EB-PBC with added energy function On the other hand, from [START_REF] Ortega | Power shaping: A New Paradigm for Stabilization of Nonlinear RLC Circuits[END_REF] we have

H a (x) := 1 2 γ(x) + C 2 K I . (17
Ḣa = y ⊤ PS K I (γ + C) = -y ⊤ PS u BC , completing the proof.

B. Interconnection and damping assignment PBC

In IDA-PBC we fix desired interconnection and damping matrices, hence, fix a matrix F d : R n → R n×n such that for some energy function H IDA : R n → R, which has a minimum at the desired equilibrium. It is easy to show that the assignable energy functions H IDA (x) are characterised by the solutions of the following PDE

F d (x) + F ⊤ d (x) ≤ 0,
g ⊥ (x) [F d (x)∇H IDA (x) -F (x)∇H(x)] = 0, (18)
and the control is univocally defined as

u IDA (x) := g † (x) [F d (x)∇H IDA (x) -F (x)∇H(x)] (19)
The proposition below establishes the relation between IDA-PBC and Proposition 1.

Proposition 4: Consider the pH system (1) verifying Assumptions 1 and 2. Fix the gain of the mapping u BC (x) as k a = 0 and select the desired interconnection and damping matrices as

F d (x) = 1 k e F (x).
Then, the energy function H d (x) defined in ( 9) and the control u = u BC (x), with v = 0, given in ( 7) satisfy the IDA-PBC equations ( 18) and ( 19), respectively. Proof: Replacing the gradient of H d (x), given by

∇H d (x) = k e ∇H(x) -F -1 (x)g(x)K I (γ(x) + C),
in the PDE (18) we get

g ⊥ F d k e ∇H -F -1 gK I (γ + C) -F ∇H = g ⊥ 1 k e F k e ∇H -F -1 gK I (γ + C) -F ∇H = 0.
On the other hand, the control law ( 7) is given by

u BC (x) = - 1 k e K I (γ(x) + C), (20)
furthermore

u IDA = g † F d k e ∇H -F -1 gK I (γ + C) -F ∇H = g † 1 k e F k e ∇H -F -1 gK I (γ + C) -F ∇H = - 1 k e g † gK I (γ + C) (21)
which coincides with [START_REF] Ortega | Control via Interconnection and Damping Assignment of Linear Time-Invariant Systems: A Tutorial[END_REF].

VI. EXAMPLES

In this section we apply the proposed controller to two physical systems and investigate, with the example of LTI systems, the limitations of the method.

A. Micro electro-mechanical optical switch [START_REF] Borovic | Control of a MEMS Optical Switch[END_REF] Consider the optical switch system with pH model

ẋ =   0 1 0 -1 -b 0 0 0 -1 r   ∇H(x) +   0 0 1 r   u
and energy function

H(x) = 1 2m x 2 2 + 1 2 a 1 x 2 1 + 1 4 a 2 x 4 1 + 1 2c 1 (x 1 + c 0 ) x 2 3 ,
where a 1 > 0, a 2 > 0 are the spring constants, b > 0, r > 0 are resistive elements, c 0 > 0, c 1 > 0 are constants that determine the capacitance function and, finally, m > 0 denotes the mass of the actuator. It is important to underscore the physical constraint x 1 > 0. The set of equilibria for this system is

x 2 * = 0, x 3 * = (c 0 + x 1 * ) 2c 1 x 1 * (a 1 + a 2 x 2 1 * )
and the goal is to stabilize at x 1 * > 0.

Some simple calculations prove that y PS = ẋ3 , therefore γ(x) = x 3 . Also, F is clearly full rank. Hence, Assumptions 1 and 2 hold. It only remains to show that (ii) of Proposition 2 holds. Since at the equilibrium

(∇ 2 H d ) * =k e   a 1 + 3a 2 x 2 1 * + d 2 1 d 2 0 -d 1 d 2 0 1 m 0 -d 1 d 2 0 d 2   + K I diag(0, 0, 1)
where

d 1 := 2c 1 x 1 * (a 1 + a 2 x 2 1 * ), d 2 := 1 c 1 (c 0 + x 1 * ) ,
then for all K I > 0, k e > 0 the condition (11) holds. Hence, x * is a stable equilibrium for the closed-loop system.

B. Magnetic levitation system [START_REF] Astolfi | Nonlinear Adaptive Control with Applications[END_REF] In this example we show that, including coordinate changes-as proposed in [START_REF] Fujimoto | Canonical Transformation and Stabilization of Generalized Hamiltonian Systems[END_REF]-it is possible to add a new degree of freedom to the design. First, we will prove that in the natural cordinates considered in [START_REF] Astolfi | Nonlinear Adaptive Control with Applications[END_REF] the proposed method is not applicable. The pH model is given by

ẋ =     0 1 0 0 -1 0 0 0 0 0 -r 2 1 c 0 0 -1 c -1 r1c 2     ∇H(x) +     0 0 0 1 r1c     u,
with energy function

H(x) = 1 2m x 2 2 + mgx 1 + 1 2n x 2 3 (1 -x 1 ) + c 2 x 2 4 ,
where m > 0 is the mass of the ball, r 1 > 0 and c > 0 are the parasite resistance and capacitance, respectively; n > 0 is constant which depends on the number of coil turns, r 2 > 0 is the coil resistance and g is the gravitational acceleration.

A physical constraint of this model is that 1 > x 1 > 0. The set of equilibria is given by

x 2 * = 0, x 3 * = 2d 0 , x 4 * = r 2 (1 -x 1 * ) 2mg n
where d 0 = nmg and the control goal is to stabilize the system at an assignable equilibrium with a desired ball position, x 1 * , satisfying the constraints.

In this example

y PS = 1 r ( ẋ3 + r 2 c ẋ4 ), (22)
whose time integration defines γ(x) as follows

γ = 1 r (x 3 + r 2 cx 4 )
where r = r 1 + r 2 . The Hessian of the energy function ( 9) is, then, given by

∇ 2 H d =k e     0 0 -1 n x 3 0 0 1 m 0 0 -1 n x 3 0 1 n (1 -x 1 ) 0 0 0 0 c     + K I r2     0 0 0 0 0 0 0 0 0 0 1 r 2 c 0 0 r 2 c r 2 2 c 2    
which clearly is not positive definite, consequently the proposed method is not applicable to this system. The source of the problem stems from the fact that with the power shaping output given by ( 22) the respective γ cannot shape the energy in the first coordinate, which is needed to ensure [START_REF] Mahindrakar | Further Constructive Results on Interconnection and Damping Assignment Control of Mechanical Systems: the Acrobot Example[END_REF]. To overcome this problem we propose the coordinate change

z = col(x 1 , x 2 , x 3 √ 1 -x 1 , x 4 ). (23)
The system in the new coordinates is given by

ż = F ∇ H(z) + ḡu
where ḡ := col(0, 0, 0, 1 r1c ) and

F :=      0 1 0 0 -1 0 z3 2(1-z1) 0 0 -z3 2(1-z1) -r 2 (1 -z 1 ) √ 1-z1 c 0 0 - √ 1-z1 c -1 r1c 2      .
Its energy function and power shaping output are, respectively,

H(z) = 1 2m z 2 2 + mgz 1 + 1 2n z 2 3 + c 2 z 2 4 , ȳPS = 1 r 1 2 z 3 (1 -z 1 ) 3 2 ż1 + 1 √ 1 -z 1 ż3 + r 2 c ż4 .
Since Assumptions 1 and 2 are invariant under the change of coordinates [START_REF] Astolfi | Nonlinear Adaptive Control with Applications[END_REF], they are still satisfied. Furthermore,

γ(z) = 1 r z 3 √ 1 -z 1 + r 2 cz 4 .
Notice that it is possible to shape the energy in the first coordinate since the second partial derivative of γ with respect to z 1 is different to zero. The set of equilibria in the new coordinates is z 2 * = 0,

z 3 * = x 3 * 1 -z 1 * , z 4 * = r 2 (1 -z 1 * ) 2mg n .
Some simple computations prove that

(∇ 2 H d ) * = k e diag(0, 1 m , 1 m , c) + K I r2       d1 (1-z1 * ) 2 0 d2 (1-z1 * ) 3 2 d3 1-z1 * 0 0 0 0 d2 (1-z1 * ) 3 2 0 1 1-z1 * r2c √ 1-z1 * d3 1-z1 * 0 r2c √ 1-z1 * r 2 2 c 2      
where we defined

d1 := 3 4 r√ 2d0 + 1 2 d0, d2 := 1 2 r + d0 2 , d3 := r2c d0 2 .
It is easy to see that, with an appropriate selection of ke and KI , condition [START_REF] Mahindrakar | Further Constructive Results on Interconnection and Damping Assignment Control of Mechanical Systems: the Acrobot Example[END_REF] holds and so, the stabilisation objective is achieved with the proposed controller.

C. LTI systems: Stabilisability is not enough

In the important paper [START_REF] Prajna | An LMI Approach to Stabilization of Linear Port-Controlled Hamiltonian Systems[END_REF] it was shown that IDA-PBC for LTI systems is a universal stabiliser, in the sense that it is applicable to all stabilisable systems. On the other hand, it was shown in [START_REF] Ortega | Control via Interconnection and Damping Assignment of Linear Time-Invariant Systems: A Tutorial[END_REF] that stabilisability is not enough for IDA-PBC of mechanical system, but a stronger condition must be imposed on the system for stabilisation with IDA-PBC-see Proposition 4.1 in [START_REF] Ortega | Control via Interconnection and Damping Assignment of Linear Time-Invariant Systems: A Tutorial[END_REF].

The difference between these two cases is that, while for general IDA-PBC there is no constraint on the structure of the desired energy function, for mechanical systems a particular structure is imposed to it. Since in the methodology proposed in this paper there is also a constraint on the desired energy function, namely [START_REF] Blankenstein | The Matching Conditions of Controlled Lagrangians and IDA-Passivity Based Control[END_REF], it is expected that a condition stronger than stabilisability should be imposed for the method to apply. Now, recall that for LTI systems the energy function is of the form H(x) = 1 2 x ⊤ Qx, the matrices F and g are constant and, without loss of generality, we can take x * = 0. Therefore, the control (7) becomes a simple linear, state-feedback of the form uBC(x) = Kx with

K := keI -kag ⊤ F -⊤ g -1 (kag ⊤ F -⊤ F Q + KI g ⊤ F -⊤ ), (24)
where we have set the vector C = 0. To prove the aforementioned conjecture we will construct an LTI, stabilisable pH system for which the controller (24) yields an unstable closed-loop system for all values of the tuning gains ke, ka, KI . It is important to note that the Lyapunov stability test utilised in Proposition 2 is sufficient, but not necessary-even for LTI systems. Therefore, instability must be proved checking directly the closed-loop system matrix. Also, the sign constraints imposed to the tuning gains, which are required to ensure positivity of the shaped energy function, need not be imposed in the LTI case where, as indicated above, a stability analysis-other than Lyapunov-will be carried out.

Consider the following controllable, LTI system

ẋ = 0 1 a1 1 -a1 x + 0 1 u. (25)
Some simple calculations show that it admits a pH representation

ẋ = F Qx + gu (26)
with g := col(0, 1),

F := -1 a1 1 2 a1 -a 2 1 , Q := - 2 a 2 1 a 2 1 a1 a1 1 -a 1 2 , (27)
which satisfies F + F ⊤ < 0 and Assumption 1. 3 Proposition 5: Consider the LTI, pH system (26), (27) in closed-loop with the controller [START_REF] Prajna | An LMI Approach to Stabilization of Linear Port-Controlled Hamiltonian Systems[END_REF]. For all values of the controller gains ke, ka and KI the closed-loop system is unstable.

Proof: The closed-loop system is given by

ẋ = 0 1 a1 -a1 k 1 -a1 -k x where k := 2 a 2 1 ke + 2ka a 2 1 -1
(KI + ka).

3 Assumption 2 is always satisfied for single input LTI systems.

Clearly, the closed-loop system matrix is Hurwitz if and only if the following conditions can be satisfied a1 -a1 k < 0, 1 -a1 -k < 0.

If the system is open-loop stable, that is, a1 < 0, these inequalities are equivalent to k < 1, k > 1 -a1, Since 1-a1 > 1 the inequalities cannot be simultaneously satisfied.

VII. CONCLUDING REMARKS AND FUTURE RESEARCH

We have presented in this paper a new energy shaping method to stabilize pH systems that, in contrast with the classical PBC methods, does not require the solution of PDEs. The key modification introduced here is to abandon the objective of preservation in closed-loop of the pH structure, which is the condition that gives rise to the PDEs.

The class of systems for which the method is applicable is identified by Assumptions 1 and 2, which can be easily verified from the systems data. The invertibility Assumption 1 is rather weak, and is satisfied in many practical examples. Notice that if it does not hold then there exists equilibria for the open-loop system, which are not extrema of the energy function-a situation that its not reasonable in physical systems. The integrability Assumption 2 is a technical condition needed to create the term added to the open-loop energy function [START_REF] Blankenstein | The Matching Conditions of Controlled Lagrangians and IDA-Passivity Based Control[END_REF]. As indicated at the end of Section III, this term may be interpreted as an integral term on the power shaping output. Unfortunately, besides this nice interpretation, we don't have at this point any physical, nor practical motivation, for Assumption 2. The controller design parameters are introduced to ensure that H d (x) is positive definite, hence, it is a Lyapunov function candidate-with the vector C (essentially) needed to make (∇H d ) * = 0.

A lines of research that is currently being pursued is to base the construction of the controller on other passive outputs, besides yPS. It was recently revealed in [START_REF] Ortega | New Results on Control by Interconnection and Energy-Balancing Passivity-based control of Port-Hamiltonian Systems[END_REF], [START_REF] Venkatraman | Energy Shaping of Port-Hamiltonian Systems by Using Alternate Passive Input-Output Pairs[END_REF] that there exists a large class of passive outputs that can be used for this purpose.

  ) Proof: For v = 0, k a = 0 and k e = 1 the mapping u BC (x) reduces tou BC (x) = -K I [γ(x) + C].

  and look for a control u = u IDA (x) such that the closed-loop has the form ẋ = F d (x)∇H IDA (x);

  with H d (x) defined in[START_REF] Blankenstein | The Matching Conditions of Controlled Lagrangians and IDA-Passivity Based Control[END_REF], then x * is stable (in the sense of Lyapunov) with Lyapunov function H d (x). It is asymptotically stable If y PS , defined in (3) is detectable, that is, if the following implication is true

Notice that the existence of γ(x) is ensured by Assumption

The interested reader is referred to[START_REF] Ortega | Putting Energy Back in Control[END_REF],[START_REF] Ortega | Control by (State-Modulated) Interconnection of Port-Hamiltonian Systems Nonlinear Control Systems[END_REF] for further details on EB-PBC and IDA-PBC.


Nadine Martin
email: nadine.martin@lis.inpg.fr

Twelfth International Congress on Sound and Vibration A CRITERION FOR DETECTING NONSTATIONARY EVENTS

This paper tackles the problem of detecting nonstationary events in a signal. An ergodic process is stationary if its law is invariant whatever the time translation is. So the control of the invariance of the law parameters can lead to a stationarity measure. We propose such a test stated from a time-frequency representation so as to localize both in time and frequency nonstationary events. At each frequency we define two observation subsets. The first one is the subset of time frequency points, whose process is random and stationary eventually added with a deterministic process. The second one gathers all time-frequency points, whose process is non stationary. The problem is thus simplified into two competing sets or hypotheses between which we have to choice. A binary hypothesis test is stated from the Neyman-Pearson criterion. We construct a decision rule to have maximum probability of detection while not allowing the probability of false alarm to exceed a given value. Given that no a priori information is known, a recursive algorithm is performed in order to estimate the unknown parameters of the decision rule. The test converges when the subsets become steady. To be independent of the choice of the false alarm probability, the decision rule is applied for different values of this probability. To initialize the algorithm, we only assume that a time stationary part exists at each frequency. Under all these assumptions, all types of nonstationarities can be detected. The proposed detector is a post-processing to a timefrequency estimator. In this paper, we use a spectrogram or a gliding correlogram, which sets the size of the nonstationary events to detect. The detector could be adapted to any other time-frequency estimator if its statistical law is known. Applications on real signals are carried out and show that the proposed method performs well.

INTRODUCTION

To detect nonstationary occurrences in a signal is a current problem in a great deal of domains. Natural sources are indeed nonstationary rather than stationary. Acoustic signals such as speech, breath sounds, bioacoustic sounds emitted by whales and dolphins, seismic signals such as those emitted by avalanches or earthquakes contain occurrences that can be of interest to detect. Engineering systems also produce nonstationary signals such as cavitation signals, transient signals got from shocks on a structure, vibratory signals in rotating machine or sensor signals. The processing of these signals is not trouble-free. Each possible method takes accounts of some particular characteristics of the signal.

In this paper, we do not made any specific assumption about the nonstationarities of the signal. We propose a method that is suitable to all signals, whatever the variations might be, and that is of interest whenever a simple Cusum test or an energy detector does not perform well. This method only depends on a timefrequency representation (TFR), which must afford an adequate resolution. Under this assumption, the method can be widely applied to the detection of transients, frequency modulated signals, frequency jumps, births or deaths of deterministic or random components. Only amplitude modulations will be badly detected. As is to be expected if any a priori information is used, the method is not optimal but robust and automatic.

PRINCIPLE OF THE METHOD

An ergodic random process is completely stationary if its probability density function (pdf) remains unaltered under a time shift. This definition is a severe requirement and is often relax by considering a stationarity up to a given order n. Under this weaker condition, the moments of process should be identical up to order n only. These definitions imply the ergodicity of the process, that is to say the equality between time and ensemble averages. We a priori assume this property, so that the moments estimated with observations from a single realization converges in mean square to the corresponding ensemble average. Furthermore, if the process is Gaussian, the stationarity up to order 2 necessarily implies complete stationarity [PRI 81].

In this paper, we are dealing with nonstationary random signals, whose probability density function changes over time. Consequently, the mean and the variance change over time and are thus functions of time. The detection scheme implies the knowledge of the probability density function of the process. The detector is here designed for a Gaussian noise, but it could be extended to other types of random process.

In this framework, a measure of the stationarity results in the time invariance control of the statistical properties of the process. Such an approach requires a time observation scale and a way of measuring the moment invariance. In addition, we aim at estimating both the time and the frequency locations of each nonstationary occurrence. As we already said in the introduction, we design the detector in the time-frequency plane to cope with this problem. The TFR is estimated with a spectrogram or a gliding correlogram, whose time and frequency resolutions are given a priori. The detector is then a post-processing, whose properties depend on the TFR properties.

STATISTICAL TEST IN THE TIME-FREQUENCY PLANE

One of important statistical tools in signal theory detection is hypothesis testing. The hypotheses are statements of the possible decisions that are being considered. Among possible criterion for making decisions, the Bayes criterion is optimal because the average cost of all possible decision or risk is minimized In this paper, we propose a detector, which takes advantage of the locality of the TFR as well. But we do not want using any a priori information on the signal because many applications require this context. We cannot thus determine the Bayes risk and we will consider the Neyman-Pearson criterion, which involves neither a priori probabilities nor cost estimates. In comparison with previous works, theis detector assigns a label to each time frequency point without defining regions or contours of these regions [HOR 02], [MAR 03]. Let

[ ]

x n be a discrete time signal. The observation set [ ] (

x k L is a subset of so that 2 R [ ] ( ) [ ] [ ] { } 2 , / , x x k n k S n k fora givenx n = ∈ ∃ L R , (1)
H [ ] 0 H k L is defined as [ ] ( ) [ ] [ ] ( ) { } 0 0 0 , / , x H x S H k n k k p p et E S n k = ∈ = L L m i n , (2)
n x S n ) Var , k of the observation [ ] ( ) [ ] ( ) [ ] ( ) ( ) 2 Var , 2 Varn , = r , x x x S n k S n k E S n k = = .
(3)

The parameter α is defined as

[ ] ( ) , b α E S k r ν = , (4)
where [ ] (5)

, b S n k is the RTF of [ ] b n . If a deterministic signal [ ] d n is present, [ ]
Whether the RTF estimator is either a spectrogram or a correlogram, the normalized variance depends on the method parameters and mostly on the average rate. So the degree r is nearly equal to the degree r. ' Knowing 0 p , we can write at each frequency k the decision rule, ,

[ ] [

1 0 , H x P H S n k k η > < ] f a
which allows making a decision between the two hypothesis H 0 and H 1 The probability of false alarm is given by the user and, therefore, permits the calculation of the detection threshold We note that the normalized variance in (5) is always lower than the normalized variance in (3). So, if we use the gamma distribution instead of the non-central chisquare distribution, the effective will be lower. Pfa

THRESHOLD OF THE DECISION RULE

A gamma variable is proportional to a chi-square variable 2 r χ . So, Eq. ( 7) can be written as

[ ] ( ) [ ] [ ] 2 1 r r Pfa Pfa Pfa k k k s Pfa g ds g u du α χ η λ η α α +∞ +∞ =   = =     ∫ ∫ 2 χ (8)
where is the pdf of a . Thus, we have

[ ] [ ] ( ) [ ] [ ] [ ] 0 0 / , 1 ˆ, b H b n S n k k H α k card k ∈ = ∑ L L Sn k r , (9)
where [ ] ˆk α is an estimation of α at frequency k .

Finally, taking (8) into account, the test (6) becomes

H is true S n k k x P f a H is true S n k k S n k k k λ α ∈ ∈ > × < L L The threshold [ ] (10) [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 0 0 , , , x H r H
( ) [ ] [ ] [ ] [ ] { } 0 1 , / , i i H x x P f a S n k S n k k k ν λ + = ≤ L α , (11) with [
0 0 0 0 1 I I H H H H when I ν ν ν - = L L L L ν = . ( ( ) ( ) ( ) ( )
) 12
We then deduce the complement and the would be lower than the desired value. But, in both cases, the estimation will be adjusted at the next iteration, the number of false alarms will decrease whereas the detection probability will increase. In addition, since we want the detector to be not sensitive to the choice of the , the decision rule is applied with several values of , which yields a classification of the detected points. The user has to choice this set of values.

[ ] [ ] 1 1 I H H k = L L k , (13)

Pfa

( ) [ ] [ ] [ ] [ ] [ ] [ ] 1 0 1 ˆĈlass = , / , c c c c I I x H x Pfa Pfa k Pfa Pfa Pfa S n k k k k S n k k λ α λ α + + ≤ < ∈ ≤ <       U L (

TRANSIENT DETECTION IN AN HYDRAULIC NOISE

We have tested the proposed detector for many vibratory and acoustic signals (ASpect data base). In this paper, we present the detection results of a vibratory signal provided by Electricité de France (EDF). The signal is a measure of an hydraulic noise in a primary pump of a nuclear reactor. The measure has the particularity to have recorded the shocks of a bolt against the walls of hydraulic circuits. Figure 1 shows the signal of 102 400 points sampled at 50 000 Hz. The RTF of this signal and the detection map are presented in Figure 2.

The criterion detects 4 main patterns corresponding to the shocks that we can identify by attentively listening the signal. The first three shocks are visible on the time representation between 0.9 and 1.1s. But, the last one is very difficult to recognize whereas the test detects it without any difficulty. We also observe that the hydraulic noise, which is stationary through all the observation time, is well not detected by the proposed test. Figure 3 shows projections of the detection map over the time axis and over the frequency axis. In addition to the detection map, this projections could be useful for defining simple tests for diagnostic or surveillance.

  distributed as gamma ( 2, ,0 Γ r α ), which defines 0 p [DUR 99]. The parameter r is an equivalent degree of freedom, which is derived from the normalized variance [ ]

  , inverted Eq. (8) gives the threshold, which depends on the given and on the parameter Pfa α . We have to estimate this last parameter defined in Eq. (4). The moment about zero of

  is a priori not known. We propose an iterative algorithm to be able to apply (10). At each iteration, threshold are updated. Let us define the partition set under at iteration i+1, 0 H

] ˆi α k the

  value of [ ] ˆk α at iteration i. The knowledge of the threshold at the previous iteration allows the determination of the partition sets. The test will converge when

  which is the desired set.The remaining problem is the initialization. We propose to initialize p is given a priori, which may induce a bias on the estimation of [ ] k and, subsequently, the will be higher than the desired value. On the other hand, if more than p % samples are nonstationary,

Figure 1 -Figure 2 -

 12 Figure 1 -Real vibratory signal (EDF), 102 400 samples(50 000 Hz)

Figure 3 -

 3 Figure 3 -Frequency and time average of

  [VTR 68], [WHA 71], [PRO 89]. Applied to signal processing, the hypotheses were first defined from time data [ARQ 82]. When data are nonstationary, it can be of interest to define the decision rule in a more adapted space. In 1980, Altès defined a detector from a spectrogram [ALT 80]. Kumar and Carroll extended it to a Wigner Ville distribution [KUM 84]. At that time, these authors agreed about the fact that the performance in a timefrequency space is identical to that obtained in a time space only. See also [KAY 85] and [FLA 86]. Marinovich decomposed the TFR in singular values in order to separate the signal components from the noise [MAR 91]. Later on, Lemoine showed the limits of an optimal detector [LEM 95]. Matz and Hlawatsch proposed a composite hypothesis test defined in a timefrequency region [MAT 99]. But, in this case, a training data set is necessary. Richard and Lengellé defined a performing linear detector, which suppresses the redundancy in a Wigner-Ville distribution [RIC 98]. Friedlander and Porat proposed a detector based on the maximum likelihood function applied on parts of the RTF only [FRI 89], [POR 92]. This algorithm is suitable to transient detection but, indeed, is not optimal.