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Simultaneous interconnection and damping assignment passivity-based control of mechanical systems using dissipative forces
Systems & Control Letters 94 (2016) 118–126
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Simultaneous interconnection and damping assignment passivity-based control of mechanical systems using dissipative forces A. Donaire a,b,∗ , R. Ortega c , J.G. Romero d a
PRISMA Lab, University of Naples Federico II, Italy
b
School of Engineering, The University of Newcastle, Australia
c
Laboratoire des Signaux et Systèmes, CNRS-SUPELEC, Plateau du Moulon, 91192 Gif-sur-Yvette, France
d
Departamento Académico de Sistemas Digitales, Instituto Tecnológico Autónomo de México-ITAM, Río Hondo No.1, 01080, Distrito Federal, Mexico
article
info
Article history: Received 16 June 2015 Received in revised form 1 April 2016 Accepted 19 May 2016
Keywords: Stability of nonlinear systems Passivity-based control Mechanical systems
abstract To extend the realm of application of the well known controller design technique of interconnection and damping assignment passivity-based control (IDA-PBC) of mechanical systems two modifications to the standard method are presented in this article. First, similarly to Batlle et al. (2009) and Gómez-Estern and van der Schaft (2004), it is proposed to avoid the splitting of the control action into energy-shaping and damping injection terms, but instead to carry them out simultaneously. Second, motivated by Chang (2014), we propose to consider the inclusion of dissipative forces, going beyond the gyroscopic ones used in standard IDA-PBC. The contribution of our work is the proof that the addition of these two elements provides a non-trivial extension to the basic IDA-PBC methodology. It is also shown that several new controllers for mechanical systems designed invoking other (less systematic procedures) that do not satisfy the conditions of standard IDA-PBC, actually belong to this new class of SIDA-PBC. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Stabilization of underactuated mechanical systems shaping their potential energy function, and preserving the systems structure, is a simple, robust and highly successful technique first introduced in [1]. To enlarge its realm of application it has been proposed to modify the kinetic energy of the system as well. This idea of total energy shaping was first introduced in [2] with the two main approaches being now: the method of controlled Lagrangians [3] and Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) [4], see also the closely related work [5]. In both cases stabilization (of a desired equilibrium) is achieved identifying the class of systems – Lagrangian for the first method and Hamiltonian for IDA-PBC – that can possibly be obtained via feedback. The conditions under which such a feedback law exists are identified by the so-called matching equations, which are a set of quasi-linear partial differential equations (PDEs), that
∗ Correspondence to: PRISMA Lab, Dipartimento di Ingegneria Elettrica e Tecnologie dell’Informazione, Università degli Studi di Napoli Federico II, via Claudio 21, 80125, Naples, Italy. E-mail addresses: [email protected] (A. Donaire), [email protected] (R. Ortega), [email protected] (J.G. Romero). http://dx.doi.org/10.1016/j.sysconle.2016.05.006 0167-6911/© 2016 Elsevier B.V. All rights reserved.
are naturally split into kinetic energy (KE-PDE) and potential energy (PE-PDE). Although a lot of research effort has been devoted to the solution of the matching equations – see [6,7] for a recent survey of the existing results – this task remains the main stumbling block for the application of these methods. The solution of the KE-PDE is simplified by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the matching equation that reduces the number of PDEs to be solved. Due to its Hamiltonian formulation, this term is intrinsic in IDA-PBC, and was added to the original controlled Lagrangian method of [3,8] – for the first time in [9] – and adopted later in [10], as recognized in its introduction. In [9] it is shown that the PDEs of the (extended) controlled Lagrangian method and IDA-PBC are the same, see also [10]. Recently, in [11] it has been proposed to consider a more general form of gyroscopic forces, relaxing the skew-symmetry condition. However, it is shown in [6] that the inclusion of these forces does not reduce the number of KE-PDEs. One of the objectives of this paper is to show that, even though the number of PDEs is not reduced, the inclusion of dissipative forces effectively extends the realm of application of IDA-PBC. A second modification to IDA-PBC proposed in the paper is to simultaneously carry out the energy shaping and damping injection steps—instead of doing them as
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
separate steps. This modification has been previously reported in [12,13], where it is shown that the partition into two steps of the design procedure induces some loss of generality. In particular, it is shown in [12] that (two-step) IDA-PBC is not applicable for the induction motor, while SIDA-PBC does apply; and the result in [13] suggests that SIDA-PBC may be needed for stabilization of underactuated mechanical systems with damping. In our work, the idea of performing energy shaping and damping injection simultaneously is tailored to PBC design of mechanical systems that, together with the use of dissipative forces, provides a nontrivial extension of the basic IDA-PBC methodology. In the paper we also show that several recent controller designs that do not fit in the standard IDA-PBC paradigm, actually belong to this new class of SIDA-PBC with dissipative forces. In this way, it is shown that these controllers, that were derived invoking less systematic procedures, are obtained following the well-established SIDA-PBC methodology. The remaining of the paper is organized as follows. Section 2 briefly recalls the IDA-PBC methodology. Section 3 contains the main result, which is the definition of SIDA-PBC by using dissipative forces. Two recently reported controller design techniques are shown to belong to this class in Section 4. The paper is wrapped-up with concluding remarks in Section 5. Notation. In is the n × n identity matrix and 0n×s is an n × s matrix of zeros, 0n is an n-dimensional column vector of zeros. Given ai ∈ R, i ∈ n¯ := {1, . . . , n}, we denote with col(ai ) the n-dimensional column vector with elements ai . For any matrix A ∈ Rn×n , (A)i ∈ Rn denotes the ith column, (A)i the ith row and (A)ij the ij-th element. ei ∈ Rn , i ∈ n¯ , are the Euclidean basis vectors. For x ∈ Rn , S ∈ Rn×n , S = S ⊤ > 0, we denote the Euclidean norm |x|2 := x⊤ x, and the weighted-norm ∥x∥2S := x⊤ Sx. Given a function f : Rn → R we define the differential operators
∇x f :=
∂f ∂x
⊤
,
∇xi f :=
∂f ∂ xi
⊤
(∇ g1 )⊤ ∇ g := ... ,
• Md : Rn → Rn×n is positive definite, • J2 : Rn × Rn → Rn×n fulfills the skew-symmetry condition J2 (q, p) = −J2⊤ (q, p).
(3)
n
• Vd : R → R verifies q⋆ = arg min Vd (q),
(4)
and the minimum is isolated, that satisfies the PDEs G⊥ ∇q H − Md M −1 ∇q Hd + J2 Md−1 p = 0,
(5)
where G⊥ : Rn → Rs×n , s := n − m is a full rank left annihilator of G, i.e., G⊥ G = 0s×m and rank(G⊥ ) = s. Then, (i) (Energy shaping) The system (1) in closed-loop with the control law u = uES (q, p), where uES : Rn×n → Rm is defined as uES = (G⊤ G)−1 G⊤ ∇q H − Md M −1 ∇q Hd + J2 Md−1 p ,
(6)
can be written as a pH system
0n×n q˙ = Σd : p˙ −Md (q) M −1 (q)
M −1 (q) Md (q) J2 (q, p)
× ∇ Hd (q, p)
(7) n
n
with the new total energy function Hd : R × R → R, 1 ⊤ −1 p Md (q) p + Vd (q). (8) 2 Therefore (q⋆ , 0) is a stable equilibrium point of (7) with Lyapunov function Hd . (ii) (Damping injection) Consider the mapping u = uDI (q, p), where uDI : Rn×n → Rm is defined as (9)
m×m
where gi : Rn → R is the ith element of g. When clear from the context the subindex in ∇ will be omitted. To simplify the expressions, the arguments of all mappings will be omitted, and will be explicitly written only the first time that the mapping is defined. 2. Standard IDA-PBC for mechanical systems To make the paper self-contained a brief review of IDA-PBC is presented in this section. IDA-PBC was introduced in [4] to control underactuated mechanical systems described in port-Hamiltonian (pH) form by In 0n×n
0 ∇ H (q, p) + n×m G(q)
u,
(1)
where q, p ∈ Rn are the generalized position and momenta, respectively, u ∈ Rm is the control, G : Rn → Rn×m with rank(G) = m < n, the function H : Rn × Rn → R, H (q, p) :=
Proposition 1 ([4]). Consider the system (1). Assume that there exist the following
uDI = −Kp G⊤ Md−1 p,
⊤
q˙ 0 Σ: = n×n p˙ −I n
objective is to generate a state-feedback control that assigns to the closed-loop the stable equilibrium (q, p) = (q⋆ , 0), q⋆ ∈ Rn . This is achieved in IDA-PBC via a two step procedure. The first step is called energy shaping and the second step is called damping injection. The following proposition summarizes these two steps.
Hd (q, p) :=
,
where xi ∈ Rp is an element of the vector x. For a mapping g : Rn → Rm , its Jacobian matrix is defined as
(∇ gm )
119
1 ⊤ −1 p M (q) p + V (q) 2
(2)
is the total energy with M : Rn → Rn×n , the positive definite inertia matrix and V : Rn → R the potential energy. The control
with Kp ∈ R positive definite. Then, the system (1) in closedloop with the control u = uES (q, p)+ uDI (q, p) has an asymptotic stable equilibrium (q⋆ , 0) provided that the output G⊤ Md−1 p is detectable. Proof. We show here a sketch of the proof in [4]. First, to prove (i), we equate the right-hand sides of (1) and (7) to obtain the so-called matching equations
∇q H − G u = Md M −1 ∇q Hd − J2 Md−1 p.
(10)
These equations are equivalent to the solution of the PDEs (5) and the (univocally defined) control (6). Since, by assumption, the functions H, M, Hd , Md and J2 satisfy (5), then the closed-loop can be written in the form (7). To prove stability, we take Hd as a Lyapunov candidate function and we compute its time derivative along the trajectories of (7), which takes the form
˙ d = p⊤ Md−1 J2 Md−1 p ≡ 0. H By adding the damping injection term in the controller as in (ii), we obtain that
˙ d = −∥G⊤ Md−1 p∥2K ≤ 0, H p which ensures asymptotic stability if the output G⊤ Md−1 p is detectable [14].
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A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
Remark 1. Notice that, as shown in [4], the PDEs (5), which are expanded here ⊥
G
1
∇q V (q) + ∇q [p⊤ M −1 (q)p] − Md (q) M −1 (q) ∇q Vd (q) 2
1
− Md (q) M
(and partial derivatives of its components). Hence the number of free elements on the right-hand-side of (14) entirely determines the number of KE-PDE’s to be solved. It is shown in [6] that this number equals 1
s (s + 1) (s + 2). (15) 6 Also, contrary to the claim in [11], the explicit formula (14) – given in a different form also in [15] – shows that there is no ansatz for the determination of J2 in IDA-PBC.
(q) ∇q [p Md (q)p] 2 −1 + J2 (q, p) Md (q) p = 0s , −1
−1
⊤
are equivalent to the solution of the (p-dependent) KE-PDE
3. Simultaneous IDA-PBC with dissipative forces
G⊥ ∇q (p⊤ M −1 p) − Md M −1 ∇q (p⊤ Md−1 p) + 2 J2 Md−1 p
In this paper, motivated by [11], we investigate the possibility to extending the realm of application of IDA-PBC by considering more general external forces. The inclusion of dissipative forces is also motivated by the fact that the use of gyroscopic (workless) forces1 is done with a loss of generality. This is clearly illustrated by the Example 4.2.4 of [10], where it is shown that the system
= 0s ,
(11)
and the (p-independent) PE-PDE G⊥ ∇ V − Md M −1 ∇ Vd = 0s .
(12)
The success of IDA-PBC relies on the possibility of solving the PDEs (11) and (12). As shown below, the inclusion of dissipative forces affects only the KE-PDE (11), therefore in the sequel we concentrate our attention on the KE-PDE (11). In [6] a more explicit expression for this equation is obtained as follows. First, note that to be consistent with (11), whose remaining terms are quadratic in p, the free matrix J2 must be linear in p. Hence, without loss of generality we can take J2 of the form n
J2 (q, p) =
−1 e⊤ i Md p Ui (q),
(13)
i =1
where Ui : Rn → Rn×n verify Ui (q) = −Ui⊤ (q). To streamline the presentation of the result of [6] we denote the columns of G⊥ as
G (q) =: ⊥
v1⊤ (q) .. , .
where vk : R → R , k ∈ s¯ := {1, . . . , s} is given by vk := col(vki ). Also, we introduce the mappings n
Ak : Rn → Rn×n ,
Γkj : R → R, n
Ak := Md
n
Wk : Rn → Rn×n
Γkj :=
vki ∇qi M
−1
Md ,
k ∈ s¯
vki (Md M
Bk := Md
n
−1
)ij ,
k ∈ s¯, j ∈ n¯
x21 x2
x1 −x2
can be seen as a
= 0.
ΣT :
0n×n M −1 (q) Md (q) q˙ = p˙ −Md (q) M −1 (q) 0n×n 0 × ∇ Hd (q, p) + , C (q, p)
(16)
Γki ∇qi Md
Md ,
(17)
Since ΣT and Σd coincide for the particular choice C = J2 Md p, it is clear that considering these more general forces enlarges the set of desired closed-loop dynamics. The matching equation now takes the form
−1
k ∈ s¯
1
⊤ ⊤ ⊤ vk U 1 vk U1 .. .. Wk := . + . , vk U n
∇ H Fg = x1 x22
⊤
−1
i=1
⊤
However, we notice that Fg = x1 −x2 generalized dissipative force that satisfies
p⊤ Md−1 C ≤ 0.
i=1
2 x˙ 1 x x 0 = J (x) 12 2 + u. x˙ 2 1 x1 x2
where C : Rn × Rn → Rn is a mapping to be defined, and which we refer to as dissipative forces. Notice that, to ensure Hd is a Lyapunov ˙ d ≤ 0 – the mapping C should function of the closed-loop – i.e., H satisfy
i=1 n
is conservative with storage function H = 12 x21 x22 , and that there does not exist a 2 × 2 skew-symmetric matrix J (x), depending smoothly on x, such that
Bk : Rn → Rn×n ,
as
y = x21 x2
We notice that any conservative system can be written using dissipative forces but may not be possible using gyroscopic forces. In [11] it is proposed to replace the target dynamics Σd in (7) by
vs⊤ (q) n
0 x1 x˙ 1 + , = −x 2 1 x˙ 2
− ∇q (p⊤ M −1 p) − ∇ V + G u 2 1 = −Md M −1 ∇q (p⊤ Md−1 p) + ∇ Vd + C ,
k ∈ s¯.
(18)
2
vk Un ⊤
the KE-PDE (11) becomes
The proof of the lemma below is given in [6].
G⊥ ∇q (p⊤ M −1 p) − Md M −1 ∇q (p⊤ Md−1 p) + 2 C = 0s ,
Lemma 1. The KE-PDE (11) is equivalent to the PDEs
(19)
(14)
while the PE-PDE (12) remains unchanged. Stemming from the equation above we have two important observations regarding C .
Note that the left-hand-side of (14) is a function of the unknown matrix Md (and partial derivatives of its components), while the right-hand-side of (14) is independent of the unknown matrix Md
1 It is clear that the skew-symmetry condition (3) makes gyroscopic forces workless.
Bk (q) − Ak (q) = Wk (q),
k ∈ s¯.
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
O1. Since C (q, 0) = 0n must be satisfied, C can always be expressed in the form C (q, p) = Λ(q, p)Md−1 (q)p, for some mapping Λ : Rn × Rn → Rn×n . O2. C must be quadratic in p—this in contrast to the case of J2 that is linear in p. For convenience, and without loss of generality, we take it of the form 2 C (q, p) =
n
p⊤ Md−1 (q) Qi (q) Md−1 (q)p ei
i=1
with Qi : Rn → Rn×n free matrices. Consequently, we have
Λ(q, p) :=
n 1
2 i=1
ei p Md (q) Qi (q). ⊤
−1
(20)
Two consequences of the remarks above are, on one hand, that the target dynamics ΣT can be written in the familiar form
0n×n q˙ = ΣT : p˙ −Md M −1
M
−1
Md
Λ
∇ Hd ,
(21)
and the stability condition (17) now becomes p⊤ Md−1 (q)Λ(q, p)Md−1 (q)p ≤ 0.
(22)
A sufficient, but not necessary, condition for (22) to hold is clearly
Λ + Λ⊤ ≤ 0 . Notice that, in contrast with the two step design procedure of standard IDA-PBC, in this new formulation the energy shaping and the damping injection are carried out simultaneously. This is in the spirit of [12] where it is shown that the partition into two steps of the design procedure induces some loss of generality. On the other hand, it is easy to see (see [6]), that new KE-PDE becomes n ⊤ (vk Md M −1 ei ) ∇qi Md − (vk⊤ ei ) Md ∇qi M −1 Md i=1
=−
n
ei vk⊤ Qi (q),
(23)
i =1
with k ∈ s¯, and the control law takes the form u = (G⊤ G)−1 G⊤ ∇q H − Md M −1 ∇q Hd + Λ Md−1 p .
(24)
Similarly to classical IDA-PBC, the presence of the matrices Qi allows us to reduce the number of PDEs to be solved. Interestingly, this is equal to (15), that is, the number of PDEs of IDA-PBC; see [11]. In spite of this fact, we show in the next section – via a series of examples – that SIDA-PBC with dissipative forces is applicable to a larger class of systems than standard IDA-PBC. We wrap-up this section with a simple proposition that summarizes the developments presented above and whose proof follows verbatim the proof of stability of standard IDA-PBC [4], which is sketched in proof of Proposition 1. Proposition 2. Consider the underactuated mechanical system (1) in closed-loop with the control (24) verifying the following conditions. (i) Hd and Λ are given by (8) and (20), respectively. (ii) Md and Λ satisfy (22). (iii) Md , Vd and Q verify the matching equations (12) and (23). (iv) Md is positive definite and Vd satisfies (4). The closed-loop system takes the form (21) and it has a stable equilibrium at the desired point (q, p) = (q⋆ , 0), with Lyapunov function Hd . The equilibrium is asymptotically stable if 1
yD := (Λ + Λ⊤ ) 2 Md−1 p is a detectable output of the closed-loop system. In addition, the stability properties are global provided that Vd is a proper function.
121
4. Examples of SIDA-PBC with dissipative forces In this section we prove that several stabilizing controllers for mechanical systems – that have been derived invoking other considerations – actually belong to the class of SIDA-PBC, where the pH closed loop is written in terms of the dissipative forces as presented in the previous section. More precisely, we prove that replacing the aforementioned state-feedback laws in the system (1) yields the desired target dynamics (21), i.e., that the matching equation (18) holds. The definition below is instrumental to articulate our results. Definition 1. A state-feedback control law u : Rn × Rn → Rm for the mechanical system (1) is said to be a SIDA-PBC with dissipative forces if the following identity holds true
−∇q H (q, p) + G(q)u(q, p) = −Md (q)M −1 (q)∇q Hd (q, p) + Λ(q, p)Md−1 (q)p
(25)
where Hd is of the form (8), for some positive definite Md and Vd , Λ verifying (4) and (22), respectively. Such controllers ensure that the closed-loop system takes the pH form (21) and verify the conditions of Proposition 2. 4.1. Energy-shaping without solving PDEs: the controller of [7] In [7] a static state-feedback that assigns the Lyapunov function Hd (8) for a class of mechanical systems was given. This control law does not satisfy the matching equation (10), therefore is not an IDA-PBC. However, we show in this subsection that it does satisfy (25)—proving that it belongs to the class of SIDA-PBC with dissipative forces. The design of [7] proceeds in two steps, first, a partial feedback linearization inner loop is applied to transform the system into Spong’s Normal Form [16]. Invoking Proposition 7 of [17], conditions on M and V are imposed to ensure the partially linearized system is still a mechanical system. A consequence of the latter is the identification of two new cyclo-passive outputs based upon which the controller is designed in a second step. The derivations in [7] are done in the Lagrangian form, to fit it into the framework of this paper, we present below its pH formulation. Consider a mechanical system (1) with input matrix of the form
G=
Im 0s×m
.
Partition the generalized coordinates as q = col(qa , qu ), with qa ∈ Rm and qu ∈ Rs , which correspond to the actuated and unactuated coordinates, respectively. The inertia matrix is conformally partitioned as M (q) =
maa (q) m⊤ au (q)
mau (q) , muu (q)
(26)
where maa : Rn → Rm×m , mau : Rn → Rm×s and muu : Rn →∈ Rs×s . In Proposition 7 of [17] it is shown that the mechanical structure is preserved after partial feedback linearization if the following conditions are satisfied. A1. The inertia matrix depends only on the unactuated variables qu , i.e., M (q) = M (qu ). A2. The sub-matrix maa of the inertia matrix is constant. A3. The potential energy can be written as V (q) = Va (qa )+ Vu (qu ). A4. The columns of the matrix mau (qu ) satisfy
∇quj (mau )k = ∇quk (mau )j ,
∀j ̸= k, j, k ∈ s¯,
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A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
or, equivalently,2 the rows of mau (qu ) satisfy
∇(mau )i = [∇(mau )i ]⊤ ,
with KP > 0 ensures that the closed-loop system has a stable equilibrium at the desired point (q, p) = (q⋆ , 0) with Lyapunov function
∀i ∈ {1, 2, . . . , m}.
That is the rows of mau (qu ) are gradient vector fields, which is equivalent to the existence of a function VN : Rs → Rm such that V˙ N = −mau (qu )˙qu .
(27)
Under these conditions the system (1) in closed-loop with the static state-feedback control law u = uPL (q, p) + v,
(28)
where uPL : Rn × Rn → Rm is the partially linearizing feedback given in [16], see also Section VII of [7], takes the pH form
0 q˙ ˙ = −I n p
0 In ∇ H˜ + ˜ v 0 G(qu )
(29)
˜ (qu ) = M
0 , muu (qu )
Im 0
p=
pa pu
with Md−1 and Vd given by (32) and (33), respectively. The equilibrium is asymptotically stable if 1 yN := ka pa − ku mau (qu )m− uu (qu )pu
is a detectable output of the closed-loop system. The stability properties are global if Vd is proper. Now, we proceed to prove that the control (34) is a SIDA-PBC with dissipative forces. Towards this end, we first notice that the matching equation (25) for the system (29) reduces to 1
1 ˜ −1 ∇q (p⊤ Md−1 p) − Md M ˜ −1 ∇q Vd + ΛMd−1 p. = − Md M
(36) 2 Hence, we must prove that (34) verifies (36) for some Λ satisfying (22). This fact is stated in the proposition below whose proof involves a series of long computations, therefore, it is given in Appendix A.
Im . −m ⊤ au (qu )
(30)
Notice that we have defined a new momenta via
Proposition 4. Consider the underactuated mechanical system (29) with mau , muu and Vu satisfying Assumptions A4 and A5. The control (34) is a SIDA-PBC with dissipative forces and
˜ (qu )˙q. := M
To complete the controller design the following additional assumption is made in [7].
Λ(q, p) =
× (qu )m⊤ (31) au (qu ). (b) The matrix k k I + k2a Kk X(qu ) Md−1 (qu ) := e a m⊤ , (32) X (qu ) Y(qu ) 1 with X(qu ) = −ka ku Kk mau (qu )m− uu (qu ) and Y (qu ) = −1 2 −1 ⊤ 1 ke ku muu (qu ) + ku muu (qu )mau (qu )Kk mau (qu )m− uu (qu ), is positive definite and the function 1 (33) Vd (q) := ke ku Vu (qu ) + ∥ka qa + ku VN (qu )∥2KI , 2 satisfies condition (4), and the minimum is isolated. The following proposition is the main stabilization result of [7]. Proposition 3. Consider the underactuated mechanical system (29) with mau , muu and Vu satisfying Assumptions A4 and A5. The control v : Rn × Rn → Rm given by
1 v(q, p) = −K −1 ku Kk mau m− uu ∇qu Vu ku 2
1 Kk mau m− uu
1 −1 −1 × ∇q⊤u [m− uu pu ]pu + ku Kk ∇q [mau muu pu ]muu pu 1 − KP K ⊤ (ka pa − ku mau m− uu pu ),
(34)
2 The authors thank Dr A. Satici for the fruitful discussions on this equivalence.
1 2
˜ −1 p ] Md − Md−1 ∇q⊤ [M
˜ −1 ∇q⊤ [Md−1 p] − Md−1 GK ˜ −1 +M .. −1 ⊤ −1 × 0 . ku Kk mau muu ∇qu [muu pu ] Md 1 −1 − 2ku Kk ∇q [mau m− uu pu ]muu
A5. There exist constants ke , ka , ku ∈ R, Kk , KI ∈ Rm×m , Kk , KI ≥ 0 such that the following hold. (a) det[K (qu )] ̸= 0, ∀qu ∈ Rs , where K : Rs → Rm×m is defined as 1 K (qu ) := ke Im + ka Kk + ku Kk mau (qu )m− uu
+ KI (ka qa + ku VN ) −
(35)
2
and
˜ (qu ) := G
1 ⊤ −1 p Md p + Vd (q), 2
˜ −1 p) − ∇q V + G˜ v − ∇q (p⊤ M
˜ (q, p) = 1 p⊤ M ˜ −1 (qu )p + Vu (qu ) H 2 where
Hd (q, p) =
− G˜ (q)KP G˜ ⊤ (q).
(37)
Application to the inverted pendulum on a cart To illustrate Proposition 4 we consider here the controller for classical cart-pendulum example reported in [7]. This is a 2-dof system with potential energy given by V (qu ) = mg ℓ cos(qu ), mass matrix M (qu ) =
Mc + m mℓ cos(qu )
mℓ cos(qu ) , mℓ2
and the input matrix is G = col(1, 0), where qa is the position of the car and qu denotes the angle of the pendulum with respect to the up-right vertical position. The parameter Mc is the mass of the car, m is the mass of the pendulum and ℓ its length. The control objective is to stabilize the up-right vertical position of the pendulum. The system satisfies assumptions A1–A4, thus, after using a partial-feedback linearizing control (28), the dynamics can be written as in (29), with momentum vector p = col(pa , pu ) = col(˙qa , m1ℓ2 q˙ u ), muu = mℓ2 and mau = mℓ cos(qu ). In [7] Proposition 3 was used to derive the (locally stabilizing) controller
1 2 v = −ku Kk m sin(qu ) pu − g cos(qu ) K (qu ) m 2 ℓ3 ku − Kp K (qu ) pa − cos(qu )pu ℓ 1
(38)
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
where KI = 0, ka = 1 and
Now, define the vector
K (qu ) = ke + Kk + ku Kk m cos (qu ) 2
ke + Kk Md (qu ) = ku Kk − cos(qu ) ℓ Vd (qu ) = ke ku mg ℓ cos(qu ).
−
C (q, p) := g (q) + f (q, p) + Gu +
ku Kk
ke ku mℓ
2
cos(qu )
ℓ
−1
+
k2u Kk
ℓ
2
cos2 (qu )
ku < 0,
(39)
ke + Kk + ku Kk m cos (qu ) < 0,
2
Md
−
−
0 ka ku Kk ml3
sin(qu )pu
1 K 1 −mℓ cos(qu ) P
2ka ku Kk ml3
ka ku Kk ml3
sin(qu )pu
sin(qu )pa
−mℓ cos(qu ) .
Md
(40)
Several works have proposed an approach using direct Lyapunov method for control design of underactuated mechanical systems (see e.g. [18–20]). In the following, we summarize the main idea proposed in these works. Consider a mechanical system with dynamics as follows q˙ = M −1 (q)p
p˙ = g (q) + f (q, p) + Gu.
(41)
This dynamics could result from a change of coordinate or a preliminary feedback (or change of coordinates) on the mechanical system (1) that may not preserve either Lagrangian or Hamiltonian structure. Notice that the system (41) coincides with the standard mechanical system (1) if g (q) + f (q, p) ≡ −∇q H (q, p),
(42)
and M is the inertia matrix. To proceed with the design, the Lyapunov function candidate 1 ⊤ −1 p Md (q)p + Vd (q), 2
(43)
p⊤ Md−1 Λ(q, p)Md−1 p ≤ 0.
(45)
Proposition 5. The control law obtained via the so-called direct Lyapunov approach is a SIDA-PBC with dissipative forces. Proof. The proof follows noting that, from the derivations above, the control law should verify g (q) + f (q, p) + Gu 1 −1 ⊤
= − Md M ∇q [Md−1 p]p − Md M−1 ∇ Vd + ΛMd−1 p 2
(46)
which coincides with the matching equation (25), if we consider a more general class of open-loop dynamics for the momenta. This matching equation together with the stability condition (45) shows that the controller is a SIDA-PBC with dissipative forces, and the closed-loop takes the form (21). Application to the ball and beam system We present here the 2-dof example of the ball and beam solved in [18] using the direct Lyapunov method, and show that the resulting controller is a SIDA-PBC with dissipative forces. The design in [18] first applies a partial-feedback linearizing control and a change of coordinate that allows us to write the dynamics of the system as follows
1 q˙ a 2(ϵ + q2u ) = q˙ u
0
pa
0
p˙ a = p˙ u
qu p2a 2(ϵ +
q2u
)
− δ pu
(47)
pu
1
0
+
0 + G u, − sin(qa )
(48)
where qa is the angle of the beam and qu is the position of the ball on the beam. The momentum vector is defined as p = M (qu )˙q, with M (qu ) = diag( 2(ϵ + q2u ), 1), and G = col(1, 0). The control objective is to stabilize the equilibrium q⋆ = (0, 0). The Lyapunov function candidate has the form (43) with
with Md > 0 and q⋆ = arg min Vd (q) is proposed. The control law is computed to ensure that the time derivative of (43) along the dynamics (41) is negative semidefinite. That is,
2ϵ + q2u
Md−1 (qu ) =
− ϵ + q2u
− ϵ + q2u , 2ϵ + q2u
and
H˙ d = p⊤ Md−1 [g (q) + f (q, p) + Gu]
√
Vd (q) = ϵ 2[1 − cos(qa )] +
1
+ ∇q⊤ [p⊤ Md−1 p]M−1 p + ∇ ⊤ Vd M−1 p 2 1 ⊤ −1 = p Md g (q) + f (q, p) + Gu + Md M−1 2 × ∇q⊤ [Md−1 p]p + Md M−1 ∇ Vd ≤ 0.
Replacing the equations above in (44) yields
= −Md M−1 ∇q Hd + Λ(q, p)Md−1 p,
4.2. Lyapunov approach for control of underactuated mechanical systems
Hd (q, p) =
C (q, p) = Λ(q, p)Md−1 p.
2
Λ(q, p) =
Md M−1 ∇q⊤ [Md−1 p]p
and, recalling observation O1, rewrite it (without loss of generality) as
for all qu ∈ (− π2 , π2 ). Some lengthy, but straightforward, calculations show that the control law (38) satisfies the matching condition (36) with Λ, derived from (37), given by 1
1 2
+ Md M−1 ∇ Vd
The conditions of Proposition 3 are satisfied if the controller gains verify ke > 0,
123
K 2
1 qa − √ sinh−1 2
qu
√
2ϵ
.
The controller proposed in [18] is as follows 2ϵ + q2u 1 u = − sin(qa ) + ∇qu Vd − ca pa − cu pu 2 ϵ + qu ϵ + q2u
(44)
−
δ + KP 2ϵ +
q2u
pa + KP
ϵ + q2u pu ,
(49)
124
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
−
0
1 Λ(q, p) := − Md 2
qu pu
−
qu pu
ϵ + q2u
2ϵ + q2u qu pa
ϵ + q2u
+
2ϵ + q2u qu pa
(ϵ + q2u ) 0
(ϵ + q2u ) δ ϵ + q2u 1 δ 2ϵ + q2u + ϵ KP . − ϵ δ ϵ + q2u δ 2ϵ + q2u
Md
(50)
Box I.
with the functions qu pu qu pa cu (q, p) := − + 2(ϵ + q2a ) 2 2ϵ + q2u ϵ + q2u qu pa ca (q, p) := − , 2 2ϵ + q2u ϵ + q2u and parameters K , Kp and ϵ positive constants to be chosen. We show in Appendix B that the controller (49) satisfies the matching equation (46) with Eq. (50) given in Box I. We now verify that the matrix Λ defined in (50) satisfies the stability condition (45). For, we notice that the first matrix in (50) is skew symmetric. Now, factoring the term ϵδ , the second matrix can be partitioned as
2ϵ + q2u + ϵ KP
ϵ + q2u 2ϵ + q2u = ϵ + q2u
ϵ + q2u
2ϵ + q2u
ϵ+
q2u
2ϵ +
q2u
+
˜= Md−1 G
ka Im
1 ⊤ −ku m− uu mau
K.
(A.1)
The proof of Proposition 4 is divided in two parts. First, we verify that (34) satisfies the matching equation (25). Second, we prove that Λ given in (37) satisfies the stability condition (22). The matching equation (25) for the system (29) is equivalent to 1 ˜ −1 p] − Md−1 ∇q V + Md−1 G˜ v − Md−1 ∇q [p⊤ M 2
1 −1 ˜ −1 ∇q Vd + Md−1 ΛMd−1 p. ˜ ∇q [p⊤ Md−1 p] − M =− M
2 Then, the control law should satisfy
˜ as in (32) and (30) , Lemma 2. Given the matrices Md−1 and G respectively. The following relation holds
ϵ KP 0
˜v = Md−1 G
0 . 0
This matrix is positive definite because δ , KP and ϵ are positive constants and the determinant of the first right hand matrix equals ϵ . Therefore, the control law (49) is a SIDA-PBC, and the closed-loop dynamics can be written in the form (21). 5. Conclusions An extension to the well known IDA-PBC method for mechanical systems has been reported. It essentially consists of two parts: (i) allowing the presence in the target dynamics of forces, which are more general than the usual gyroscopic ones, and (ii) the proposition of simultaneously carrying out the energy shaping and damping injection steps—instead of doing them as separate steps. These two modifications have been previously reported in [11,12], respectively. It has been shown that several recent controller designs that do not fit in the standard IDA-PBC paradigm, actually belong to this new class of SIDA-PBC with dissipative forces. In this way, it is shown that these controllers, that were derived invoking less systematic procedures, are obtained following the wellestablished SIDA-PBC methodology. In future research we will focus on the development of systematic procedures to choose the free parameters of SIDA-PBC to avoid the task of solving the PDEs that result from matching the open and target closed-loop dynamics. Appendix A. Proof of Proposition 4
2
˜ −1 p]p + Md−1 ∇q V − Md−1 ∇q⊤ [M
1 2
˜ −1 ∇q⊤ [Md−1 p]p M
˜ −1 ∇q Vd + Md−1 ΛMd−1 p −M =
1 2
˜ −1 p]p + Md−1 ∇q V − Md−1 ∇q⊤ [M
1 2
˜ −1 ∇q⊤ [Md−1 p]p M
1 ⊤ −1 ˜ −1 ku Kk mau m− ˜ −1 ∇q Vd − 1 Md−1 GK −M uu ∇qu [muu pu ] 2 1 −1 − 2ku Kk ∇q [mau m− uu pu ]muu pu 1 ˜ −1 ∇q⊤ [Md−1 p]p ˜ −1 p]p + 1 M − Md−1 ∇q⊤ [M 2
2
˜ P G˜ ⊤ Md−1 p − Md−1 GK ˜ −1 ∇q Vd − Md−1 GK ˜ P G˜ ⊤ Md−1 p = Md−1 ∇q V − M 1 1 ⊤ −1 ˜ −1 ku Kk mau m− − Md−1 GK uu ∇qu [muu pu ] 2 1 −1 − 2ku Kk ∇q [mau m− uu pu ]muu pu 1 −ka ku Kk mau m− uu = ∇qu V 1 2 −1 ⊤ −1 ke ku m− uu + ku muu mau Kk mau muu k a Im 0m×s − −1 ∇qu V − −1 ⊤ ke ku muu −ku muu mau ˜ −1 × KI (ka q1 + ku VN ) − Md−1 GK
ku 2
1 ⊤ −1 Kk mau m− uu ∇qu [muu pu ]pu
−1 ˜ 1 −1 ˜ −1 ku Kk ∇q [mau m− ˜ ⊤ −1 + Md−1 GK uu pu ]muu pu − Md GKP G Md p k a Im −1 =− 1 ⊤ ku Kk mau muu ∇qu V −k u m − uu mau ka Im −1 ˜ −1 − 1 ⊤ KI (ka q1 + ku VN ) − Md GK −ku m− uu mau
× We state first the following lemma, whose proof is established using straightforward calculations, that will be used below.
1
ku 2
−1 ˜ −1 1 ⊤ −1 Kk mau m− ku Kk ∇q uu ∇qu [muu pu ]pu + Md GK
−1 ˜ 1 −1 ˜ ⊤ −1 × [mau m− uu pu ]muu pu − Md GKP G Md p
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
1 ⊤ −1 −1 −1 −ka ku Kk mau m− uu ∇q (muu pu ) − 2∇q (mau muu pu )muu 1 ⊤ −1 ⊤ −1 −1 −1 k2u m− uu mau Kk mau muu ∇q (muu pu ) − 2∇q (mau muu pu )muu 1 ⊤ −1 ka ku Kk mau m− uu ∇q (muu pu ) 1 ⊤ −1 2 −1 ⊤ −1 ⊤ −1 −ke ku m− uu ∇q (muu pu ) − ku muu mau Kk mau muu ∇q (muu pu )
0m×m
∆1 = ∆2 =
125
0s×m
0m×m 0s×m
(A.3)
(A.4)
0m×m
0m×s 1 ⊤ −1 ⊤ −1 ⊤ −1 −ka ku m− uu ∇q (muu mau Kk pa ) + ke ku muu ∇q (muu pu ) . 2 −1 ⊤ −1 ⊤ −1 + ku muu ∇q (muu mau Kk mau muu pu )
∆3 = 1 ⊤ −1 −ka ku m− uu ∇ (Kk mau muu pu )
(A.5)
Box II.
1 ⊤ −1 − ∇q⊤ (m− uu mau Kk mau muu pu )
1 ˜ −1 ku Kk mau m− = −Md−1 GK uu ∇qu V + KI (ka q1 + ku VN )
−
ku 2
1 ⊤ −1 −1 −1 Kk mau m− uu ∇qu [muu pu ]pu + ku Kk ∇q [mau muu pu ]muu pu
˜ P G˜ ⊤ Md−1 p − p⊤ Md−1 GK ˜ P G˜ ⊤ Md−1 p, = −p⊤ Md−1 GK which shows that the condition (22) is satisfied since KP > 0.
1 ˜ P K ⊤ (ka pa − ku mau m− − Md−1 GK uu pu ).
Appendix B. Matching equation (46) for the Ball and Beam
Finally,
From the matching equation (46), we obtain that the control law should satisfy the following equation
1 ˜ v = Md−1 G˜ −K −1 ku Kk mau m− Md−1 G uu ∇qu V
+KI (ka q1 + ku VN ) −
ku 2
Gu = −g (q) − Md M −1 ∇ Vd − f (q, p) −
1 Kk mau m− uu
1 −1 −1 × ∇q⊤u [m− uu pu ]pu + ku Kk ∇q [mau muu pu ]muu pu
=
−1
∆1 = −Md GK −1 ˜
−1
0m×m
.. .
−1
⊤
−1
ku Kk mau muu ∇qu [muu pu ]
+
=
=
pa pu
0m×m 1 ⊤ 1 −ka ku m− ∇ (Kk mau m− uu uu pu )
δ
0 sin(qa )
−
˜ −1 ∇q⊤ [Md−1 p] ∆3 = M
˜ P G˜ ⊤ Md−1 ]p (37) = p⊤ [∆1 + ∆2 + ∆3 − Md−1 GK
0
p−
1 2
×
pa ˜ P G˜ ⊤ Md−1 p − p⊤ Md−1 GK pu
= 2ka ku pa Kk ∇q (mau muu pu )muu pu −1 −1 − p⊤ a ∇q (Kk mau muu pu )muu pu ⊤ −1 −1 −1 − k2u p⊤ u muu 2∇q (mau muu pu )Kk mau muu pu
⊤
−1
2(ϵ + q2u )
0
p
2ϵ + q2u 1 − ∇ V + sin ( q ) qu d a − ϵ + q2u ϵ + q2u sin(qa )
0 qu pa
0
p + 0
ϵ+
2
2ϵ + qu p a
q2u
0
δ
0
0 2(ϵ + q2u ) qu p a
1
(ϵ + q2u ) δ + KP 2ϵ + q2u
q2u
−
p qu pa
(ϵ +
q2u
)
+
qu pu
ϵ+
q2u
2ϵ +
q2u
0
p
−KP ϵ + q2u p 0 δ 2ϵ + q2u 1 ∇qu Vd − sin(qa ) = ϵ + q2u ϵ + q2u −
0
−
0
Md M−1 ∇q⊤ [Md−1 p]p + Λ(q, p)Md−1 p
δ + KP 2ϵ + q2u
−KP ϵ + q2u
0 1 −1 2ka ku Kk ∇ ⊤ (mau m− uu pu )muu 2 −1 ⊤ ⊤ −1 1 −2ku muu mau Kk ∇q (mau muu pu )m− uu −1 ⊤ −1 ⊤ −ka ku muu ∇q (muu mau Kk pa ) 1 ⊤ −1 ⊤ −1 +k2u m− uu ∇q (muu mau Kk mau muu pu )
0 qu pa
+
from which we obtain Eqs. (A.3)–(A.5) given in Box II. Now, we compute (22) using (37) and (A.3)–(A.5)
0 0
1 −1 −2ku Kk ∇q [mau m− uu pu ]muu
˜ −1 p] ∆2 = −Md−1 ∇q⊤ [M
⊤
0 − Md M−1 ∇ Vd − sin(qa )
(A.2)
where we used the definition of Λ in the second equality, and the relation (A.1) in the sixth equality. The control law (34) exactly coincides with the term in curly brackets hence it satisfies (A.2) and, therefore, the matching equation (25). Now, to prove that Λ given in (37) satisfies Eq. (22), we compute first some terms of Λ as follows
Md M−1 ∇q⊤ [Md−1 p]p
+ Λ(q, p)Md−1 p
+ KP K (ka pa − ku mau muu pu ) , ⊤
1 2
qu pa
+ 2 ϵ +
p
0
q2u
2ϵ +
q2u
−
qu pa 2(ϵ +
q2u
)
0
+
qu pu
2 ϵ+
q2u
0
2ϵ + q2u 1 ∇ V − sin ( q ) ca a = ϵ + q2u qu d − ϵ + q2u 0 0
−1
−
δ + KP 2ϵ + q2u 0
−KP ϵ + q2u 0
p,
which is satisfied by the control law (49).
cu p 0
2ϵ +
q2u
p
126
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
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Contents lists available at ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Simultaneous interconnection and damping assignment passivity-based control of mechanical systems using dissipative forces A. Donaire a,b,∗ , R. Ortega c , J.G. Romero d a
PRISMA Lab, University of Naples Federico II, Italy
b
School of Engineering, The University of Newcastle, Australia
c
Laboratoire des Signaux et Systèmes, CNRS-SUPELEC, Plateau du Moulon, 91192 Gif-sur-Yvette, France
d
Departamento Académico de Sistemas Digitales, Instituto Tecnológico Autónomo de México-ITAM, Río Hondo No.1, 01080, Distrito Federal, Mexico
article
info
Article history: Received 16 June 2015 Received in revised form 1 April 2016 Accepted 19 May 2016
Keywords: Stability of nonlinear systems Passivity-based control Mechanical systems
abstract To extend the realm of application of the well known controller design technique of interconnection and damping assignment passivity-based control (IDA-PBC) of mechanical systems two modifications to the standard method are presented in this article. First, similarly to Batlle et al. (2009) and Gómez-Estern and van der Schaft (2004), it is proposed to avoid the splitting of the control action into energy-shaping and damping injection terms, but instead to carry them out simultaneously. Second, motivated by Chang (2014), we propose to consider the inclusion of dissipative forces, going beyond the gyroscopic ones used in standard IDA-PBC. The contribution of our work is the proof that the addition of these two elements provides a non-trivial extension to the basic IDA-PBC methodology. It is also shown that several new controllers for mechanical systems designed invoking other (less systematic procedures) that do not satisfy the conditions of standard IDA-PBC, actually belong to this new class of SIDA-PBC. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Stabilization of underactuated mechanical systems shaping their potential energy function, and preserving the systems structure, is a simple, robust and highly successful technique first introduced in [1]. To enlarge its realm of application it has been proposed to modify the kinetic energy of the system as well. This idea of total energy shaping was first introduced in [2] with the two main approaches being now: the method of controlled Lagrangians [3] and Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) [4], see also the closely related work [5]. In both cases stabilization (of a desired equilibrium) is achieved identifying the class of systems – Lagrangian for the first method and Hamiltonian for IDA-PBC – that can possibly be obtained via feedback. The conditions under which such a feedback law exists are identified by the so-called matching equations, which are a set of quasi-linear partial differential equations (PDEs), that
∗ Correspondence to: PRISMA Lab, Dipartimento di Ingegneria Elettrica e Tecnologie dell’Informazione, Università degli Studi di Napoli Federico II, via Claudio 21, 80125, Naples, Italy. E-mail addresses: [email protected] (A. Donaire), [email protected] (R. Ortega), [email protected] (J.G. Romero). http://dx.doi.org/10.1016/j.sysconle.2016.05.006 0167-6911/© 2016 Elsevier B.V. All rights reserved.
are naturally split into kinetic energy (KE-PDE) and potential energy (PE-PDE). Although a lot of research effort has been devoted to the solution of the matching equations – see [6,7] for a recent survey of the existing results – this task remains the main stumbling block for the application of these methods. The solution of the KE-PDE is simplified by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the matching equation that reduces the number of PDEs to be solved. Due to its Hamiltonian formulation, this term is intrinsic in IDA-PBC, and was added to the original controlled Lagrangian method of [3,8] – for the first time in [9] – and adopted later in [10], as recognized in its introduction. In [9] it is shown that the PDEs of the (extended) controlled Lagrangian method and IDA-PBC are the same, see also [10]. Recently, in [11] it has been proposed to consider a more general form of gyroscopic forces, relaxing the skew-symmetry condition. However, it is shown in [6] that the inclusion of these forces does not reduce the number of KE-PDEs. One of the objectives of this paper is to show that, even though the number of PDEs is not reduced, the inclusion of dissipative forces effectively extends the realm of application of IDA-PBC. A second modification to IDA-PBC proposed in the paper is to simultaneously carry out the energy shaping and damping injection steps—instead of doing them as
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
separate steps. This modification has been previously reported in [12,13], where it is shown that the partition into two steps of the design procedure induces some loss of generality. In particular, it is shown in [12] that (two-step) IDA-PBC is not applicable for the induction motor, while SIDA-PBC does apply; and the result in [13] suggests that SIDA-PBC may be needed for stabilization of underactuated mechanical systems with damping. In our work, the idea of performing energy shaping and damping injection simultaneously is tailored to PBC design of mechanical systems that, together with the use of dissipative forces, provides a nontrivial extension of the basic IDA-PBC methodology. In the paper we also show that several recent controller designs that do not fit in the standard IDA-PBC paradigm, actually belong to this new class of SIDA-PBC with dissipative forces. In this way, it is shown that these controllers, that were derived invoking less systematic procedures, are obtained following the well-established SIDA-PBC methodology. The remaining of the paper is organized as follows. Section 2 briefly recalls the IDA-PBC methodology. Section 3 contains the main result, which is the definition of SIDA-PBC by using dissipative forces. Two recently reported controller design techniques are shown to belong to this class in Section 4. The paper is wrapped-up with concluding remarks in Section 5. Notation. In is the n × n identity matrix and 0n×s is an n × s matrix of zeros, 0n is an n-dimensional column vector of zeros. Given ai ∈ R, i ∈ n¯ := {1, . . . , n}, we denote with col(ai ) the n-dimensional column vector with elements ai . For any matrix A ∈ Rn×n , (A)i ∈ Rn denotes the ith column, (A)i the ith row and (A)ij the ij-th element. ei ∈ Rn , i ∈ n¯ , are the Euclidean basis vectors. For x ∈ Rn , S ∈ Rn×n , S = S ⊤ > 0, we denote the Euclidean norm |x|2 := x⊤ x, and the weighted-norm ∥x∥2S := x⊤ Sx. Given a function f : Rn → R we define the differential operators
∇x f :=
∂f ∂x
⊤
,
∇xi f :=
∂f ∂ xi
⊤
(∇ g1 )⊤ ∇ g := ... ,
• Md : Rn → Rn×n is positive definite, • J2 : Rn × Rn → Rn×n fulfills the skew-symmetry condition J2 (q, p) = −J2⊤ (q, p).
(3)
n
• Vd : R → R verifies q⋆ = arg min Vd (q),
(4)
and the minimum is isolated, that satisfies the PDEs G⊥ ∇q H − Md M −1 ∇q Hd + J2 Md−1 p = 0,
(5)
where G⊥ : Rn → Rs×n , s := n − m is a full rank left annihilator of G, i.e., G⊥ G = 0s×m and rank(G⊥ ) = s. Then, (i) (Energy shaping) The system (1) in closed-loop with the control law u = uES (q, p), where uES : Rn×n → Rm is defined as uES = (G⊤ G)−1 G⊤ ∇q H − Md M −1 ∇q Hd + J2 Md−1 p ,
(6)
can be written as a pH system
0n×n q˙ = Σd : p˙ −Md (q) M −1 (q)
M −1 (q) Md (q) J2 (q, p)
× ∇ Hd (q, p)
(7) n
n
with the new total energy function Hd : R × R → R, 1 ⊤ −1 p Md (q) p + Vd (q). (8) 2 Therefore (q⋆ , 0) is a stable equilibrium point of (7) with Lyapunov function Hd . (ii) (Damping injection) Consider the mapping u = uDI (q, p), where uDI : Rn×n → Rm is defined as (9)
m×m
where gi : Rn → R is the ith element of g. When clear from the context the subindex in ∇ will be omitted. To simplify the expressions, the arguments of all mappings will be omitted, and will be explicitly written only the first time that the mapping is defined. 2. Standard IDA-PBC for mechanical systems To make the paper self-contained a brief review of IDA-PBC is presented in this section. IDA-PBC was introduced in [4] to control underactuated mechanical systems described in port-Hamiltonian (pH) form by In 0n×n
0 ∇ H (q, p) + n×m G(q)
u,
(1)
where q, p ∈ Rn are the generalized position and momenta, respectively, u ∈ Rm is the control, G : Rn → Rn×m with rank(G) = m < n, the function H : Rn × Rn → R, H (q, p) :=
Proposition 1 ([4]). Consider the system (1). Assume that there exist the following
uDI = −Kp G⊤ Md−1 p,
⊤
q˙ 0 Σ: = n×n p˙ −I n
objective is to generate a state-feedback control that assigns to the closed-loop the stable equilibrium (q, p) = (q⋆ , 0), q⋆ ∈ Rn . This is achieved in IDA-PBC via a two step procedure. The first step is called energy shaping and the second step is called damping injection. The following proposition summarizes these two steps.
Hd (q, p) :=
,
where xi ∈ Rp is an element of the vector x. For a mapping g : Rn → Rm , its Jacobian matrix is defined as
(∇ gm )
119
1 ⊤ −1 p M (q) p + V (q) 2
(2)
is the total energy with M : Rn → Rn×n , the positive definite inertia matrix and V : Rn → R the potential energy. The control
with Kp ∈ R positive definite. Then, the system (1) in closedloop with the control u = uES (q, p)+ uDI (q, p) has an asymptotic stable equilibrium (q⋆ , 0) provided that the output G⊤ Md−1 p is detectable. Proof. We show here a sketch of the proof in [4]. First, to prove (i), we equate the right-hand sides of (1) and (7) to obtain the so-called matching equations
∇q H − G u = Md M −1 ∇q Hd − J2 Md−1 p.
(10)
These equations are equivalent to the solution of the PDEs (5) and the (univocally defined) control (6). Since, by assumption, the functions H, M, Hd , Md and J2 satisfy (5), then the closed-loop can be written in the form (7). To prove stability, we take Hd as a Lyapunov candidate function and we compute its time derivative along the trajectories of (7), which takes the form
˙ d = p⊤ Md−1 J2 Md−1 p ≡ 0. H By adding the damping injection term in the controller as in (ii), we obtain that
˙ d = −∥G⊤ Md−1 p∥2K ≤ 0, H p which ensures asymptotic stability if the output G⊤ Md−1 p is detectable [14].
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A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
Remark 1. Notice that, as shown in [4], the PDEs (5), which are expanded here ⊥
G
1
∇q V (q) + ∇q [p⊤ M −1 (q)p] − Md (q) M −1 (q) ∇q Vd (q) 2
1
− Md (q) M
(and partial derivatives of its components). Hence the number of free elements on the right-hand-side of (14) entirely determines the number of KE-PDE’s to be solved. It is shown in [6] that this number equals 1
s (s + 1) (s + 2). (15) 6 Also, contrary to the claim in [11], the explicit formula (14) – given in a different form also in [15] – shows that there is no ansatz for the determination of J2 in IDA-PBC.
(q) ∇q [p Md (q)p] 2 −1 + J2 (q, p) Md (q) p = 0s , −1
−1
⊤
are equivalent to the solution of the (p-dependent) KE-PDE
3. Simultaneous IDA-PBC with dissipative forces
G⊥ ∇q (p⊤ M −1 p) − Md M −1 ∇q (p⊤ Md−1 p) + 2 J2 Md−1 p
In this paper, motivated by [11], we investigate the possibility to extending the realm of application of IDA-PBC by considering more general external forces. The inclusion of dissipative forces is also motivated by the fact that the use of gyroscopic (workless) forces1 is done with a loss of generality. This is clearly illustrated by the Example 4.2.4 of [10], where it is shown that the system
= 0s ,
(11)
and the (p-independent) PE-PDE G⊥ ∇ V − Md M −1 ∇ Vd = 0s .
(12)
The success of IDA-PBC relies on the possibility of solving the PDEs (11) and (12). As shown below, the inclusion of dissipative forces affects only the KE-PDE (11), therefore in the sequel we concentrate our attention on the KE-PDE (11). In [6] a more explicit expression for this equation is obtained as follows. First, note that to be consistent with (11), whose remaining terms are quadratic in p, the free matrix J2 must be linear in p. Hence, without loss of generality we can take J2 of the form n
J2 (q, p) =
−1 e⊤ i Md p Ui (q),
(13)
i =1
where Ui : Rn → Rn×n verify Ui (q) = −Ui⊤ (q). To streamline the presentation of the result of [6] we denote the columns of G⊥ as
G (q) =: ⊥
v1⊤ (q) .. , .
where vk : R → R , k ∈ s¯ := {1, . . . , s} is given by vk := col(vki ). Also, we introduce the mappings n
Ak : Rn → Rn×n ,
Γkj : R → R, n
Ak := Md
n
Wk : Rn → Rn×n
Γkj :=
vki ∇qi M
−1
Md ,
k ∈ s¯
vki (Md M
Bk := Md
n
−1
)ij ,
k ∈ s¯, j ∈ n¯
x21 x2
x1 −x2
can be seen as a
= 0.
ΣT :
0n×n M −1 (q) Md (q) q˙ = p˙ −Md (q) M −1 (q) 0n×n 0 × ∇ Hd (q, p) + , C (q, p)
(16)
Γki ∇qi Md
Md ,
(17)
Since ΣT and Σd coincide for the particular choice C = J2 Md p, it is clear that considering these more general forces enlarges the set of desired closed-loop dynamics. The matching equation now takes the form
−1
k ∈ s¯
1
⊤ ⊤ ⊤ vk U 1 vk U1 .. .. Wk := . + . , vk U n
∇ H Fg = x1 x22
⊤
−1
i=1
⊤
However, we notice that Fg = x1 −x2 generalized dissipative force that satisfies
p⊤ Md−1 C ≤ 0.
i=1
2 x˙ 1 x x 0 = J (x) 12 2 + u. x˙ 2 1 x1 x2
where C : Rn × Rn → Rn is a mapping to be defined, and which we refer to as dissipative forces. Notice that, to ensure Hd is a Lyapunov ˙ d ≤ 0 – the mapping C should function of the closed-loop – i.e., H satisfy
i=1 n
is conservative with storage function H = 12 x21 x22 , and that there does not exist a 2 × 2 skew-symmetric matrix J (x), depending smoothly on x, such that
Bk : Rn → Rn×n ,
as
y = x21 x2
We notice that any conservative system can be written using dissipative forces but may not be possible using gyroscopic forces. In [11] it is proposed to replace the target dynamics Σd in (7) by
vs⊤ (q) n
0 x1 x˙ 1 + , = −x 2 1 x˙ 2
− ∇q (p⊤ M −1 p) − ∇ V + G u 2 1 = −Md M −1 ∇q (p⊤ Md−1 p) + ∇ Vd + C ,
k ∈ s¯.
(18)
2
vk Un ⊤
the KE-PDE (11) becomes
The proof of the lemma below is given in [6].
G⊥ ∇q (p⊤ M −1 p) − Md M −1 ∇q (p⊤ Md−1 p) + 2 C = 0s ,
Lemma 1. The KE-PDE (11) is equivalent to the PDEs
(19)
(14)
while the PE-PDE (12) remains unchanged. Stemming from the equation above we have two important observations regarding C .
Note that the left-hand-side of (14) is a function of the unknown matrix Md (and partial derivatives of its components), while the right-hand-side of (14) is independent of the unknown matrix Md
1 It is clear that the skew-symmetry condition (3) makes gyroscopic forces workless.
Bk (q) − Ak (q) = Wk (q),
k ∈ s¯.
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
O1. Since C (q, 0) = 0n must be satisfied, C can always be expressed in the form C (q, p) = Λ(q, p)Md−1 (q)p, for some mapping Λ : Rn × Rn → Rn×n . O2. C must be quadratic in p—this in contrast to the case of J2 that is linear in p. For convenience, and without loss of generality, we take it of the form 2 C (q, p) =
n
p⊤ Md−1 (q) Qi (q) Md−1 (q)p ei
i=1
with Qi : Rn → Rn×n free matrices. Consequently, we have
Λ(q, p) :=
n 1
2 i=1
ei p Md (q) Qi (q). ⊤
−1
(20)
Two consequences of the remarks above are, on one hand, that the target dynamics ΣT can be written in the familiar form
0n×n q˙ = ΣT : p˙ −Md M −1
M
−1
Md
Λ
∇ Hd ,
(21)
and the stability condition (17) now becomes p⊤ Md−1 (q)Λ(q, p)Md−1 (q)p ≤ 0.
(22)
A sufficient, but not necessary, condition for (22) to hold is clearly
Λ + Λ⊤ ≤ 0 . Notice that, in contrast with the two step design procedure of standard IDA-PBC, in this new formulation the energy shaping and the damping injection are carried out simultaneously. This is in the spirit of [12] where it is shown that the partition into two steps of the design procedure induces some loss of generality. On the other hand, it is easy to see (see [6]), that new KE-PDE becomes n ⊤ (vk Md M −1 ei ) ∇qi Md − (vk⊤ ei ) Md ∇qi M −1 Md i=1
=−
n
ei vk⊤ Qi (q),
(23)
i =1
with k ∈ s¯, and the control law takes the form u = (G⊤ G)−1 G⊤ ∇q H − Md M −1 ∇q Hd + Λ Md−1 p .
(24)
Similarly to classical IDA-PBC, the presence of the matrices Qi allows us to reduce the number of PDEs to be solved. Interestingly, this is equal to (15), that is, the number of PDEs of IDA-PBC; see [11]. In spite of this fact, we show in the next section – via a series of examples – that SIDA-PBC with dissipative forces is applicable to a larger class of systems than standard IDA-PBC. We wrap-up this section with a simple proposition that summarizes the developments presented above and whose proof follows verbatim the proof of stability of standard IDA-PBC [4], which is sketched in proof of Proposition 1. Proposition 2. Consider the underactuated mechanical system (1) in closed-loop with the control (24) verifying the following conditions. (i) Hd and Λ are given by (8) and (20), respectively. (ii) Md and Λ satisfy (22). (iii) Md , Vd and Q verify the matching equations (12) and (23). (iv) Md is positive definite and Vd satisfies (4). The closed-loop system takes the form (21) and it has a stable equilibrium at the desired point (q, p) = (q⋆ , 0), with Lyapunov function Hd . The equilibrium is asymptotically stable if 1
yD := (Λ + Λ⊤ ) 2 Md−1 p is a detectable output of the closed-loop system. In addition, the stability properties are global provided that Vd is a proper function.
121
4. Examples of SIDA-PBC with dissipative forces In this section we prove that several stabilizing controllers for mechanical systems – that have been derived invoking other considerations – actually belong to the class of SIDA-PBC, where the pH closed loop is written in terms of the dissipative forces as presented in the previous section. More precisely, we prove that replacing the aforementioned state-feedback laws in the system (1) yields the desired target dynamics (21), i.e., that the matching equation (18) holds. The definition below is instrumental to articulate our results. Definition 1. A state-feedback control law u : Rn × Rn → Rm for the mechanical system (1) is said to be a SIDA-PBC with dissipative forces if the following identity holds true
−∇q H (q, p) + G(q)u(q, p) = −Md (q)M −1 (q)∇q Hd (q, p) + Λ(q, p)Md−1 (q)p
(25)
where Hd is of the form (8), for some positive definite Md and Vd , Λ verifying (4) and (22), respectively. Such controllers ensure that the closed-loop system takes the pH form (21) and verify the conditions of Proposition 2. 4.1. Energy-shaping without solving PDEs: the controller of [7] In [7] a static state-feedback that assigns the Lyapunov function Hd (8) for a class of mechanical systems was given. This control law does not satisfy the matching equation (10), therefore is not an IDA-PBC. However, we show in this subsection that it does satisfy (25)—proving that it belongs to the class of SIDA-PBC with dissipative forces. The design of [7] proceeds in two steps, first, a partial feedback linearization inner loop is applied to transform the system into Spong’s Normal Form [16]. Invoking Proposition 7 of [17], conditions on M and V are imposed to ensure the partially linearized system is still a mechanical system. A consequence of the latter is the identification of two new cyclo-passive outputs based upon which the controller is designed in a second step. The derivations in [7] are done in the Lagrangian form, to fit it into the framework of this paper, we present below its pH formulation. Consider a mechanical system (1) with input matrix of the form
G=
Im 0s×m
.
Partition the generalized coordinates as q = col(qa , qu ), with qa ∈ Rm and qu ∈ Rs , which correspond to the actuated and unactuated coordinates, respectively. The inertia matrix is conformally partitioned as M (q) =
maa (q) m⊤ au (q)
mau (q) , muu (q)
(26)
where maa : Rn → Rm×m , mau : Rn → Rm×s and muu : Rn →∈ Rs×s . In Proposition 7 of [17] it is shown that the mechanical structure is preserved after partial feedback linearization if the following conditions are satisfied. A1. The inertia matrix depends only on the unactuated variables qu , i.e., M (q) = M (qu ). A2. The sub-matrix maa of the inertia matrix is constant. A3. The potential energy can be written as V (q) = Va (qa )+ Vu (qu ). A4. The columns of the matrix mau (qu ) satisfy
∇quj (mau )k = ∇quk (mau )j ,
∀j ̸= k, j, k ∈ s¯,
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A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
or, equivalently,2 the rows of mau (qu ) satisfy
∇(mau )i = [∇(mau )i ]⊤ ,
with KP > 0 ensures that the closed-loop system has a stable equilibrium at the desired point (q, p) = (q⋆ , 0) with Lyapunov function
∀i ∈ {1, 2, . . . , m}.
That is the rows of mau (qu ) are gradient vector fields, which is equivalent to the existence of a function VN : Rs → Rm such that V˙ N = −mau (qu )˙qu .
(27)
Under these conditions the system (1) in closed-loop with the static state-feedback control law u = uPL (q, p) + v,
(28)
where uPL : Rn × Rn → Rm is the partially linearizing feedback given in [16], see also Section VII of [7], takes the pH form
0 q˙ ˙ = −I n p
0 In ∇ H˜ + ˜ v 0 G(qu )
(29)
˜ (qu ) = M
0 , muu (qu )
Im 0
p=
pa pu
with Md−1 and Vd given by (32) and (33), respectively. The equilibrium is asymptotically stable if 1 yN := ka pa − ku mau (qu )m− uu (qu )pu
is a detectable output of the closed-loop system. The stability properties are global if Vd is proper. Now, we proceed to prove that the control (34) is a SIDA-PBC with dissipative forces. Towards this end, we first notice that the matching equation (25) for the system (29) reduces to 1
1 ˜ −1 ∇q (p⊤ Md−1 p) − Md M ˜ −1 ∇q Vd + ΛMd−1 p. = − Md M
(36) 2 Hence, we must prove that (34) verifies (36) for some Λ satisfying (22). This fact is stated in the proposition below whose proof involves a series of long computations, therefore, it is given in Appendix A.
Im . −m ⊤ au (qu )
(30)
Notice that we have defined a new momenta via
Proposition 4. Consider the underactuated mechanical system (29) with mau , muu and Vu satisfying Assumptions A4 and A5. The control (34) is a SIDA-PBC with dissipative forces and
˜ (qu )˙q. := M
To complete the controller design the following additional assumption is made in [7].
Λ(q, p) =
× (qu )m⊤ (31) au (qu ). (b) The matrix k k I + k2a Kk X(qu ) Md−1 (qu ) := e a m⊤ , (32) X (qu ) Y(qu ) 1 with X(qu ) = −ka ku Kk mau (qu )m− uu (qu ) and Y (qu ) = −1 2 −1 ⊤ 1 ke ku muu (qu ) + ku muu (qu )mau (qu )Kk mau (qu )m− uu (qu ), is positive definite and the function 1 (33) Vd (q) := ke ku Vu (qu ) + ∥ka qa + ku VN (qu )∥2KI , 2 satisfies condition (4), and the minimum is isolated. The following proposition is the main stabilization result of [7]. Proposition 3. Consider the underactuated mechanical system (29) with mau , muu and Vu satisfying Assumptions A4 and A5. The control v : Rn × Rn → Rm given by
1 v(q, p) = −K −1 ku Kk mau m− uu ∇qu Vu ku 2
1 Kk mau m− uu
1 −1 −1 × ∇q⊤u [m− uu pu ]pu + ku Kk ∇q [mau muu pu ]muu pu 1 − KP K ⊤ (ka pa − ku mau m− uu pu ),
(34)
2 The authors thank Dr A. Satici for the fruitful discussions on this equivalence.
1 2
˜ −1 p ] Md − Md−1 ∇q⊤ [M
˜ −1 ∇q⊤ [Md−1 p] − Md−1 GK ˜ −1 +M .. −1 ⊤ −1 × 0 . ku Kk mau muu ∇qu [muu pu ] Md 1 −1 − 2ku Kk ∇q [mau m− uu pu ]muu
A5. There exist constants ke , ka , ku ∈ R, Kk , KI ∈ Rm×m , Kk , KI ≥ 0 such that the following hold. (a) det[K (qu )] ̸= 0, ∀qu ∈ Rs , where K : Rs → Rm×m is defined as 1 K (qu ) := ke Im + ka Kk + ku Kk mau (qu )m− uu
+ KI (ka qa + ku VN ) −
(35)
2
and
˜ (qu ) := G
1 ⊤ −1 p Md p + Vd (q), 2
˜ −1 p) − ∇q V + G˜ v − ∇q (p⊤ M
˜ (q, p) = 1 p⊤ M ˜ −1 (qu )p + Vu (qu ) H 2 where
Hd (q, p) =
− G˜ (q)KP G˜ ⊤ (q).
(37)
Application to the inverted pendulum on a cart To illustrate Proposition 4 we consider here the controller for classical cart-pendulum example reported in [7]. This is a 2-dof system with potential energy given by V (qu ) = mg ℓ cos(qu ), mass matrix M (qu ) =
Mc + m mℓ cos(qu )
mℓ cos(qu ) , mℓ2
and the input matrix is G = col(1, 0), where qa is the position of the car and qu denotes the angle of the pendulum with respect to the up-right vertical position. The parameter Mc is the mass of the car, m is the mass of the pendulum and ℓ its length. The control objective is to stabilize the up-right vertical position of the pendulum. The system satisfies assumptions A1–A4, thus, after using a partial-feedback linearizing control (28), the dynamics can be written as in (29), with momentum vector p = col(pa , pu ) = col(˙qa , m1ℓ2 q˙ u ), muu = mℓ2 and mau = mℓ cos(qu ). In [7] Proposition 3 was used to derive the (locally stabilizing) controller
1 2 v = −ku Kk m sin(qu ) pu − g cos(qu ) K (qu ) m 2 ℓ3 ku − Kp K (qu ) pa − cos(qu )pu ℓ 1
(38)
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
where KI = 0, ka = 1 and
Now, define the vector
K (qu ) = ke + Kk + ku Kk m cos (qu ) 2
ke + Kk Md (qu ) = ku Kk − cos(qu ) ℓ Vd (qu ) = ke ku mg ℓ cos(qu ).
−
C (q, p) := g (q) + f (q, p) + Gu +
ku Kk
ke ku mℓ
2
cos(qu )
ℓ
−1
+
k2u Kk
ℓ
2
cos2 (qu )
ku < 0,
(39)
ke + Kk + ku Kk m cos (qu ) < 0,
2
Md
−
−
0 ka ku Kk ml3
sin(qu )pu
1 K 1 −mℓ cos(qu ) P
2ka ku Kk ml3
ka ku Kk ml3
sin(qu )pu
sin(qu )pa
−mℓ cos(qu ) .
Md
(40)
Several works have proposed an approach using direct Lyapunov method for control design of underactuated mechanical systems (see e.g. [18–20]). In the following, we summarize the main idea proposed in these works. Consider a mechanical system with dynamics as follows q˙ = M −1 (q)p
p˙ = g (q) + f (q, p) + Gu.
(41)
This dynamics could result from a change of coordinate or a preliminary feedback (or change of coordinates) on the mechanical system (1) that may not preserve either Lagrangian or Hamiltonian structure. Notice that the system (41) coincides with the standard mechanical system (1) if g (q) + f (q, p) ≡ −∇q H (q, p),
(42)
and M is the inertia matrix. To proceed with the design, the Lyapunov function candidate 1 ⊤ −1 p Md (q)p + Vd (q), 2
(43)
p⊤ Md−1 Λ(q, p)Md−1 p ≤ 0.
(45)
Proposition 5. The control law obtained via the so-called direct Lyapunov approach is a SIDA-PBC with dissipative forces. Proof. The proof follows noting that, from the derivations above, the control law should verify g (q) + f (q, p) + Gu 1 −1 ⊤
= − Md M ∇q [Md−1 p]p − Md M−1 ∇ Vd + ΛMd−1 p 2
(46)
which coincides with the matching equation (25), if we consider a more general class of open-loop dynamics for the momenta. This matching equation together with the stability condition (45) shows that the controller is a SIDA-PBC with dissipative forces, and the closed-loop takes the form (21). Application to the ball and beam system We present here the 2-dof example of the ball and beam solved in [18] using the direct Lyapunov method, and show that the resulting controller is a SIDA-PBC with dissipative forces. The design in [18] first applies a partial-feedback linearizing control and a change of coordinate that allows us to write the dynamics of the system as follows
1 q˙ a 2(ϵ + q2u ) = q˙ u
0
pa
0
p˙ a = p˙ u
qu p2a 2(ϵ +
q2u
)
− δ pu
(47)
pu
1
0
+
0 + G u, − sin(qa )
(48)
where qa is the angle of the beam and qu is the position of the ball on the beam. The momentum vector is defined as p = M (qu )˙q, with M (qu ) = diag( 2(ϵ + q2u ), 1), and G = col(1, 0). The control objective is to stabilize the equilibrium q⋆ = (0, 0). The Lyapunov function candidate has the form (43) with
with Md > 0 and q⋆ = arg min Vd (q) is proposed. The control law is computed to ensure that the time derivative of (43) along the dynamics (41) is negative semidefinite. That is,
2ϵ + q2u
Md−1 (qu ) =
− ϵ + q2u
− ϵ + q2u , 2ϵ + q2u
and
H˙ d = p⊤ Md−1 [g (q) + f (q, p) + Gu]
√
Vd (q) = ϵ 2[1 − cos(qa )] +
1
+ ∇q⊤ [p⊤ Md−1 p]M−1 p + ∇ ⊤ Vd M−1 p 2 1 ⊤ −1 = p Md g (q) + f (q, p) + Gu + Md M−1 2 × ∇q⊤ [Md−1 p]p + Md M−1 ∇ Vd ≤ 0.
Replacing the equations above in (44) yields
= −Md M−1 ∇q Hd + Λ(q, p)Md−1 p,
4.2. Lyapunov approach for control of underactuated mechanical systems
Hd (q, p) =
C (q, p) = Λ(q, p)Md−1 p.
2
Λ(q, p) =
Md M−1 ∇q⊤ [Md−1 p]p
and, recalling observation O1, rewrite it (without loss of generality) as
for all qu ∈ (− π2 , π2 ). Some lengthy, but straightforward, calculations show that the control law (38) satisfies the matching condition (36) with Λ, derived from (37), given by 1
1 2
+ Md M−1 ∇ Vd
The conditions of Proposition 3 are satisfied if the controller gains verify ke > 0,
123
K 2
1 qa − √ sinh−1 2
qu
√
2ϵ
.
The controller proposed in [18] is as follows 2ϵ + q2u 1 u = − sin(qa ) + ∇qu Vd − ca pa − cu pu 2 ϵ + qu ϵ + q2u
(44)
−
δ + KP 2ϵ +
q2u
pa + KP
ϵ + q2u pu ,
(49)
124
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
−
0
1 Λ(q, p) := − Md 2
qu pu
−
qu pu
ϵ + q2u
2ϵ + q2u qu pa
ϵ + q2u
+
2ϵ + q2u qu pa
(ϵ + q2u ) 0
(ϵ + q2u ) δ ϵ + q2u 1 δ 2ϵ + q2u + ϵ KP . − ϵ δ ϵ + q2u δ 2ϵ + q2u
Md
(50)
Box I.
with the functions qu pu qu pa cu (q, p) := − + 2(ϵ + q2a ) 2 2ϵ + q2u ϵ + q2u qu pa ca (q, p) := − , 2 2ϵ + q2u ϵ + q2u and parameters K , Kp and ϵ positive constants to be chosen. We show in Appendix B that the controller (49) satisfies the matching equation (46) with Eq. (50) given in Box I. We now verify that the matrix Λ defined in (50) satisfies the stability condition (45). For, we notice that the first matrix in (50) is skew symmetric. Now, factoring the term ϵδ , the second matrix can be partitioned as
2ϵ + q2u + ϵ KP
ϵ + q2u 2ϵ + q2u = ϵ + q2u
ϵ + q2u
2ϵ + q2u
ϵ+
q2u
2ϵ +
q2u
+
˜= Md−1 G
ka Im
1 ⊤ −ku m− uu mau
K.
(A.1)
The proof of Proposition 4 is divided in two parts. First, we verify that (34) satisfies the matching equation (25). Second, we prove that Λ given in (37) satisfies the stability condition (22). The matching equation (25) for the system (29) is equivalent to 1 ˜ −1 p] − Md−1 ∇q V + Md−1 G˜ v − Md−1 ∇q [p⊤ M 2
1 −1 ˜ −1 ∇q Vd + Md−1 ΛMd−1 p. ˜ ∇q [p⊤ Md−1 p] − M =− M
2 Then, the control law should satisfy
˜ as in (32) and (30) , Lemma 2. Given the matrices Md−1 and G respectively. The following relation holds
ϵ KP 0
˜v = Md−1 G
0 . 0
This matrix is positive definite because δ , KP and ϵ are positive constants and the determinant of the first right hand matrix equals ϵ . Therefore, the control law (49) is a SIDA-PBC, and the closed-loop dynamics can be written in the form (21). 5. Conclusions An extension to the well known IDA-PBC method for mechanical systems has been reported. It essentially consists of two parts: (i) allowing the presence in the target dynamics of forces, which are more general than the usual gyroscopic ones, and (ii) the proposition of simultaneously carrying out the energy shaping and damping injection steps—instead of doing them as separate steps. These two modifications have been previously reported in [11,12], respectively. It has been shown that several recent controller designs that do not fit in the standard IDA-PBC paradigm, actually belong to this new class of SIDA-PBC with dissipative forces. In this way, it is shown that these controllers, that were derived invoking less systematic procedures, are obtained following the wellestablished SIDA-PBC methodology. In future research we will focus on the development of systematic procedures to choose the free parameters of SIDA-PBC to avoid the task of solving the PDEs that result from matching the open and target closed-loop dynamics. Appendix A. Proof of Proposition 4
2
˜ −1 p]p + Md−1 ∇q V − Md−1 ∇q⊤ [M
1 2
˜ −1 ∇q⊤ [Md−1 p]p M
˜ −1 ∇q Vd + Md−1 ΛMd−1 p −M =
1 2
˜ −1 p]p + Md−1 ∇q V − Md−1 ∇q⊤ [M
1 2
˜ −1 ∇q⊤ [Md−1 p]p M
1 ⊤ −1 ˜ −1 ku Kk mau m− ˜ −1 ∇q Vd − 1 Md−1 GK −M uu ∇qu [muu pu ] 2 1 −1 − 2ku Kk ∇q [mau m− uu pu ]muu pu 1 ˜ −1 ∇q⊤ [Md−1 p]p ˜ −1 p]p + 1 M − Md−1 ∇q⊤ [M 2
2
˜ P G˜ ⊤ Md−1 p − Md−1 GK ˜ −1 ∇q Vd − Md−1 GK ˜ P G˜ ⊤ Md−1 p = Md−1 ∇q V − M 1 1 ⊤ −1 ˜ −1 ku Kk mau m− − Md−1 GK uu ∇qu [muu pu ] 2 1 −1 − 2ku Kk ∇q [mau m− uu pu ]muu pu 1 −ka ku Kk mau m− uu = ∇qu V 1 2 −1 ⊤ −1 ke ku m− uu + ku muu mau Kk mau muu k a Im 0m×s − −1 ∇qu V − −1 ⊤ ke ku muu −ku muu mau ˜ −1 × KI (ka q1 + ku VN ) − Md−1 GK
ku 2
1 ⊤ −1 Kk mau m− uu ∇qu [muu pu ]pu
−1 ˜ 1 −1 ˜ −1 ku Kk ∇q [mau m− ˜ ⊤ −1 + Md−1 GK uu pu ]muu pu − Md GKP G Md p k a Im −1 =− 1 ⊤ ku Kk mau muu ∇qu V −k u m − uu mau ka Im −1 ˜ −1 − 1 ⊤ KI (ka q1 + ku VN ) − Md GK −ku m− uu mau
× We state first the following lemma, whose proof is established using straightforward calculations, that will be used below.
1
ku 2
−1 ˜ −1 1 ⊤ −1 Kk mau m− ku Kk ∇q uu ∇qu [muu pu ]pu + Md GK
−1 ˜ 1 −1 ˜ ⊤ −1 × [mau m− uu pu ]muu pu − Md GKP G Md p
A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
1 ⊤ −1 −1 −1 −ka ku Kk mau m− uu ∇q (muu pu ) − 2∇q (mau muu pu )muu 1 ⊤ −1 ⊤ −1 −1 −1 k2u m− uu mau Kk mau muu ∇q (muu pu ) − 2∇q (mau muu pu )muu 1 ⊤ −1 ka ku Kk mau m− uu ∇q (muu pu ) 1 ⊤ −1 2 −1 ⊤ −1 ⊤ −1 −ke ku m− uu ∇q (muu pu ) − ku muu mau Kk mau muu ∇q (muu pu )
0m×m
∆1 = ∆2 =
125
0s×m
0m×m 0s×m
(A.3)
(A.4)
0m×m
0m×s 1 ⊤ −1 ⊤ −1 ⊤ −1 −ka ku m− uu ∇q (muu mau Kk pa ) + ke ku muu ∇q (muu pu ) . 2 −1 ⊤ −1 ⊤ −1 + ku muu ∇q (muu mau Kk mau muu pu )
∆3 = 1 ⊤ −1 −ka ku m− uu ∇ (Kk mau muu pu )
(A.5)
Box II.
1 ⊤ −1 − ∇q⊤ (m− uu mau Kk mau muu pu )
1 ˜ −1 ku Kk mau m− = −Md−1 GK uu ∇qu V + KI (ka q1 + ku VN )
−
ku 2
1 ⊤ −1 −1 −1 Kk mau m− uu ∇qu [muu pu ]pu + ku Kk ∇q [mau muu pu ]muu pu
˜ P G˜ ⊤ Md−1 p − p⊤ Md−1 GK ˜ P G˜ ⊤ Md−1 p, = −p⊤ Md−1 GK which shows that the condition (22) is satisfied since KP > 0.
1 ˜ P K ⊤ (ka pa − ku mau m− − Md−1 GK uu pu ).
Appendix B. Matching equation (46) for the Ball and Beam
Finally,
From the matching equation (46), we obtain that the control law should satisfy the following equation
1 ˜ v = Md−1 G˜ −K −1 ku Kk mau m− Md−1 G uu ∇qu V
+KI (ka q1 + ku VN ) −
ku 2
Gu = −g (q) − Md M −1 ∇ Vd − f (q, p) −
1 Kk mau m− uu
1 −1 −1 × ∇q⊤u [m− uu pu ]pu + ku Kk ∇q [mau muu pu ]muu pu
=
−1
∆1 = −Md GK −1 ˜
−1
0m×m
.. .
−1
⊤
−1
ku Kk mau muu ∇qu [muu pu ]
+
=
=
pa pu
0m×m 1 ⊤ 1 −ka ku m− ∇ (Kk mau m− uu uu pu )
δ
0 sin(qa )
−
˜ −1 ∇q⊤ [Md−1 p] ∆3 = M
˜ P G˜ ⊤ Md−1 ]p (37) = p⊤ [∆1 + ∆2 + ∆3 − Md−1 GK
0
p−
1 2
×
pa ˜ P G˜ ⊤ Md−1 p − p⊤ Md−1 GK pu
= 2ka ku pa Kk ∇q (mau muu pu )muu pu −1 −1 − p⊤ a ∇q (Kk mau muu pu )muu pu ⊤ −1 −1 −1 − k2u p⊤ u muu 2∇q (mau muu pu )Kk mau muu pu
⊤
−1
2(ϵ + q2u )
0
p
2ϵ + q2u 1 − ∇ V + sin ( q ) qu d a − ϵ + q2u ϵ + q2u sin(qa )
0 qu pa
0
p + 0
ϵ+
2
2ϵ + qu p a
q2u
0
δ
0
0 2(ϵ + q2u ) qu p a
1
(ϵ + q2u ) δ + KP 2ϵ + q2u
q2u
−
p qu pa
(ϵ +
q2u
)
+
qu pu
ϵ+
q2u
2ϵ +
q2u
0
p
−KP ϵ + q2u p 0 δ 2ϵ + q2u 1 ∇qu Vd − sin(qa ) = ϵ + q2u ϵ + q2u −
0
−
0
Md M−1 ∇q⊤ [Md−1 p]p + Λ(q, p)Md−1 p
δ + KP 2ϵ + q2u
−KP ϵ + q2u
0 1 −1 2ka ku Kk ∇ ⊤ (mau m− uu pu )muu 2 −1 ⊤ ⊤ −1 1 −2ku muu mau Kk ∇q (mau muu pu )m− uu −1 ⊤ −1 ⊤ −ka ku muu ∇q (muu mau Kk pa ) 1 ⊤ −1 ⊤ −1 +k2u m− uu ∇q (muu mau Kk mau muu pu )
0 qu pa
+
from which we obtain Eqs. (A.3)–(A.5) given in Box II. Now, we compute (22) using (37) and (A.3)–(A.5)
0 0
1 −1 −2ku Kk ∇q [mau m− uu pu ]muu
˜ −1 p] ∆2 = −Md−1 ∇q⊤ [M
⊤
0 − Md M−1 ∇ Vd − sin(qa )
(A.2)
where we used the definition of Λ in the second equality, and the relation (A.1) in the sixth equality. The control law (34) exactly coincides with the term in curly brackets hence it satisfies (A.2) and, therefore, the matching equation (25). Now, to prove that Λ given in (37) satisfies Eq. (22), we compute first some terms of Λ as follows
Md M−1 ∇q⊤ [Md−1 p]p
+ Λ(q, p)Md−1 p
+ KP K (ka pa − ku mau muu pu ) , ⊤
1 2
qu pa
+ 2 ϵ +
p
0
q2u
2ϵ +
q2u
−
qu pa 2(ϵ +
q2u
)
0
+
qu pu
2 ϵ+
q2u
0
2ϵ + q2u 1 ∇ V − sin ( q ) ca a = ϵ + q2u qu d − ϵ + q2u 0 0
−1
−
δ + KP 2ϵ + q2u 0
−KP ϵ + q2u 0
p,
which is satisfied by the control law (49).
cu p 0
2ϵ +
q2u
p
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A. Donaire et al. / Systems & Control Letters 94 (2016) 118–126
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