Interconnection and Damping Assignment PassivityâBased. Control: Towards a Constructive ProcedureâPart II. Eloısa GarcıaâCanseco, Alessandro Astolfi and ...
ThB07.2
43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas
Interconnection and Damping Assignment Passivity–Based Control: Towards a Constructive Procedure—Part II Elo´ısa Garc´ıa–Canseco, Alessandro Astolfi and Romeo Ortega
Abstract— In Part I of this paper, we presented the fundamentals of the theory of IDA–PBC. In this part, we discuss the main new results and the practical applications of this control design approach as well as the current open problems and future directions.
to collect and present in a unified way some of the new theoretical results and to discuss the current research and future directions.
Keywords: Passivity, nonlinear systems, stabilization, interconnection, passivity–based control, Hamiltonian systems.
We have argued in Subsection II-B in Part I [1] that it is sometimes desirable to restrict the desired energy function to a certain class. In this section we discuss how this variation of the method simplifies its application to mechanical systems.
I. INTRODUCTION It was shown in Part I [1] that the IDA–PBC, introduced in [2], [3] is a procedure that allows to design a static state feedback that stabilizes the equilibria of nonlinear systems of the form x˙ = f (x) + g(x)u, where x ∈ Rn is the state vector and u ∈ Rm , m < n is the control action. By assigning to the closed–loop the PCH form1 x˙ = [Jd (x) − Rd (x)]∇Hd
(1)
where the matrices Jd (x) = −Jd (x) and Rd (x) = R d (x) ≥ 0, which represent the desired interconnection structure and dissipation, respectively, are fixed by the designer—hence the name IDA—and Hd (x) is the desired total energy, that should satisfy x = arg min Hd (x), with x ∈ Rn the equilibrium to be stabilized. As stated in Proposition 1 in [1], the admissible energy functions are characterized by the matching equation g ⊥ (x)f (x) = g ⊥ (x)[Jd (x) − Rd (x)]∇Hd
(2)
where g ⊥ (x) is a full–rank left annihilator of g(x), i.e., g ⊥ (x)g(x) = 0. Since the introduction of IDA–PBC many theoretical extensions and practical applications of this control design technique have been reported in the literature. Among the practical applications one has: mass–balance systems [4], electrical motors [5], [6], magnetic levitation systems [7], [8], power systems [9], [10], power converters [11], [12], underwater vehicles [13], spacecrafts [14] and mechanical systems [15], [16], [17]. The purposes of this paper are This work was partially supported by CONACYT, Mexico, and the European sponsored GeoPlex project with reference code IST–2001– 34166. Further information is available at http://www.geoplex.cc Elo´ısa Garc´ıa–Canseco and Romeo Ortega are with the Laboratoire des Signaux et Syst`emes, Supelec, Plateau du Moulon, 91192, Gif-sur-Yvette, cedex, France. (e–mail: {garcia,ortega}@lss.supelec.fr) Alessandro Astolfi is with the Electrical Engineering Department, Imperial College, Exhibition Road, London, SW7 2BT, UK. (e–mail: a.astolfi@ic.ac.uk) 1 All vectors defined in the paper are column vectors, even the gradient of a scalar function that we denote with the operator ∇x = ∂∂x . When clear from the context the subindex will be omitted. Also we use (·) to denote differentiation for functions of scalar arguments.
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II. MECHANICAL SYSTEMS
A. A Parameterized Energy Function In [15] we have applied Parameterized IDA–PBC for mechanical systems described by the Hamiltonian equations q˙ 0 In ∇q H 0 = + u (3) p˙ −In 0 ∇p H G(q) with total energy H(q, p) = 12 p M −1 (q)p + V (q), where q ∈ Rn , p ∈ Rn are the generalized position and momenta, respectively, M (q) = M (q) > 0 is the inertia matrix, V (q) is the potential energy and rank(G) = m < n. It has been proposed to fix the desired energy function 1 −1 p Md (q)p + Vd (q), 2 where Md (q) and Vd (q) represent the (to be defined) closed–loop inertia matrix and potential energy function, respectively, and we require that Md (q) = Md (q) > 0 and q = arg min Vd (q). Fixing the desired energy function also fixes the desired interconnection matrix as 0 M −1 (q)Md (q) Jd (q, p) = J2 (q, p) −Md (q)M −1 (q) Hd (q, p) =
where Jd (q, p) = −Jd (q, p), with the skew–symmetric matrix J2 (q, p) a free parameter. On the other hand, it is also proposed in [15] to split the control into two parts, namely, u = ues (q, p)+udi (q, p). This restricts the desired damping matrix to be of the form 0 0 Rd (q) = ≥0 0 G(q)Kv G (q) where Kv > 0. As shown in [15] the PDEs of the IDA– PBC can be naturally separated into the terms that depend on p and terms which are independent of p, i.e., those corresponding to the kinetic and the potential energies, respectively. This leads to G⊥ ∇q (pM −1p)−Md M −1 ∇q (pMd−1 p)+2J2 Md−1 p = 0
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G⊥ {∇q V − Md M −1 ∇q Vd } = 0
where G⊥ (q)G(q) = 0. The first equation is a nonlinear PDE that using J2 (which has to be linear in p) has to be solved for the unknown elements of the closed–loop inertia matrix Md (q). There are two “extreme” particular cases of our procedure. First, if we do not modify the interconnection matrix then we recover the well–known potential energy shaping procedure of PBC that has its roots in [18]. Indeed, if Md (q) = M (q) and J2 (q, p) = 0, then the energy– shaping term of the controller reduces to ues (q) = (G (q)G(q))−1 G (q)(∇q V − ∇q Vd ) which is the familiar potential energy shaping control. On the other extreme, if we do not change the potential energy, but only modify the kinetic energy then, it is shown in [15], [19], that for a particular choice of J2 (q, p) we recover the controlled–Lagrangian method of [20]. Now, if we shape both kinetic and potential energies, but still fix J2 (q, p), then IDA–PBC coincides with the method proposed in [21], [22]. Actually, the approach of [20] is more modest than IDA–PBC and [21], [22] in that a physically inspired parameterization for the modified kinetic energy is assumed a priori. As indicated in [23] the Lagrangian and Hamiltonian settings can be made equivalent adding to the former gyroscopic forces. This fact was exploited in [24] to prove that, considering these enlarged set of desired Lagrangians, both methods are equivalent. The benefits of adding gyroscopic forces (or equivalently, of adopting the PCH formalism of IDA–PBC) are clearly shown in [25] where, thanks to the addition of these terms, the calculations for a classical example can be worked out. B. Underactuation Degree One In [26] it has been shown that, for a class of mechanical systems with underactuation degree one, it is possible to completely obviate the PDEs. The class consists of systems of the form q˙ = M −1 (qr )p (4) p˙ = s(qr ) + G(qr )u, where qr , with r an integer taking values in the set {1, . . . , n}, is a distinguished element of q ∈ Rn , p ∈ Rn , and u ∈ Rn−1 . Structures of the form (4) result from the reduction, via singular perturbations or a preliminary feedback action, of certain classes of mechanical systems. A complete characterization of the class of mechanical systems feedback equivalent to (4) is also given in the paper. Two key properties of (4) are exploited in [26]: (i) Since M (qr ) and Md (qr ) depend only on the coordinate qr the kinetic energy PDE (4) becomes an ODE; (ii) restricting Md (qr ) to a particular structure, the latter can be explicitly solved with a suitable choice of J2 (qr ) and G⊥ (qr )—the latter being possible to define because of the underactuation degree one. See Fact 1 in [1]. The main result of the paper is the following:
Proposition 1 [26] Consider system (4). Fix qr G(µ)Ψ(µ)G (µ)dµ + Md0 , Md (qr ) = qr
and assume there are Ψ(qr ) = Ψ (qr ) ∈ R(n−1)×(n−1) and Md0 = (Md0 ) ∈ Rn×n , Md0 > 0 such that for some ˜ ⊥ (qr ) of G(qr )2 left annihilator G ˜ ⊥ (qr )Md (qr )M −1 (qr )er | ≥ > 0. |G Then, there exists a matrix J2 (qr ) and a function η(qr ) such that the kinetic energy PDE (4) with G⊥ (qr ) = ˜ ⊥ (qr ) is solved. Furthermore, the solution of the η(qr )G potential energy PDE (4) is given by 1 qr ⊥ G (µ)s(µ)dµ + Φ(z(q)), (5) Vd (q) = − ρ 0 where z(q) is an n − 1 dimensional vector whose elements are of the form 1 qr ⊥ zi (q) := qi − G (µ)Md (µ)M −1 (µ)ei dµ, (6) ρ 0 i = 1, . . . , n, i = r, Φ(z) is an arbitrary differentiable function, and ρ is an arbitrary constant. Using this construction we automatically verify that Md (qr ) > 0, hence the only condition needed to ensure stability is q = arg min Vd (q). From (5), and taking into account that Φ(z) is free, we see that this translates into the “Hessian” assumption for the qr coordinate −
1 d {G⊥ s}(qr ) > 0 ρ dqr
It is interesting to compare these results with those of [27], particularly Theorem 3.1 of that paper. Auckly and Kapitanski do not consider structure modification (equivalently, gyroscopic forces), but they find similar simplifications in the matching conditions and they derive an interesting integrability result for systems which are underactuated by one degree. C. Friction Effects Let us now study the effect of (linear) friction on the system, which adds a matrix R2 = diag{ri } ≥ 0 to the (2, 2)–block of the interconnection matrix of (3) as 0 In J −R= . −In R2 On one hand, we see that for potential energy shaping controllers the damping condition R(x)(∇Hd − ∇H) = 0 showed in [1] for energy–balancing stabilization becomes ∇q (Vd − V ) 0 0 =0 0 0 R2 which is clearly satisfied for all V (q), Vd (q), R2 . Hence, if kinetic energy is not modified, IDA–PBC is energy– balancing. Unfortunately, in some underactuated system
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2e i
∈ Rn are the vectors of the standard Euclidian basis.
applications we have to modify the kinetic energy to shape the potential energy function. On the other hand, in the recent interesting paper [28] it has been shown that when the kinetic energy is also modified, and friction is not taken into account in the design, then it may have a destabilizing effect. Indeed, if friction is neglected the closed–loop system contains a perturbation term 0 q˙ = [Jd (q, p) − Rd (q, p)]∇Hd + R2 M −1 (q)p p˙
where u ∈ Rne is the vector of voltages applied to the windings, R = R > 0 is the matrix of electrical resistance of the windings. The coupling between the electrical and the mechanical subsystems is established through the torque of electrical origin 1 (9) τ (i, θ) = i L i + i µ 2 The model is completed replacing the latter in the mechanical dynamics J θ¨ = −rm θ˙ + τ (i, θ) − V (10)
and the derivative of the energy function becomes
where J > 0 is the rotational (translational) inertia of the mechanical subsystem, rm ≥ 0 is the viscous friction coefficient, and the scalar function V (θ) is the potential energy. As shown in [33] the model contains, as particular cases, magnetic levitated systems, as well as permanent magnet synchronous and stepping motors. The system may be written in PCH form ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ λ˙ −R 0 0 I ∇λ H ⎣ θ˙ ⎦=⎣ 0 0 1 ⎦⎣ ∇θ H ⎦+⎣ 0 ⎦u (11) 0 −1 −rm 0 ∇p H p˙
H˙ d = −p Md−1 [GKv G + R2 M −1 Md ]Md−1 p.
(7)
The matrix in brackets may be sign indefinite disqualifying Hd (q, p) as a Lyapunov function. Notice, however, that this does not mean that the IDA–PBC design is unstable. Actually, some subsequent analysis has revealed that, for the examples reported in [15], it is always possible to increase the damping injection to ensure a margin of stability vis– a` –vis this unmodelled effect. A detailed analysis of this phenomenon has been reported in [29] where it has been established that a necessary and sufficient condition to ensure passivity of the closed–loop, is that the restriction of R2 M −1 (q)Md (q) to the kernel of G(q) is positive semidefinite. That is, the effect of the damping on the non–actuated coordinates is not detrimental.
where H=
1 1 p2 [λ − µ(θ)] L(θ)−1 [λ − µ(θ)] + V (θ) + 2 2J
p = J θ˙ is the mechanical momentum. The control objective is asymptotic regulation of θ to a constant position θ , hence the desired equilibria are of the III. ELECTROMECHANICAL SYSTEMS form [λ , θ , 0] , where λ = L(θ )i + µ(θ ) and i is IDA–PBC has been successful in various types of electhe solution of (9), (10) for θ = θ , that is tromechanical systems applications including magnetic lev1 itation [30], [31], motor control [5], [6] and transient i L (θ ) i + i (12) µ (θ ) − V (θ ) = 0 2 stabilization of synchronous generators [9], [10], [32]. We It is interesting at this point to compare the problem dereview briefly some of these results providing, in some scribed above with the one of mechanical systems discussed cases, new insights and alternative solutions. in Section II. The equilibria (of the open–loop system) are A. Position Control not only determined by the mechanical potential energy, and IDA–PBC was used in [31] to solve a, quite general, po- they (generally) correspond to nonzero electric energy. To sition control problem. Similarly to the mechanical systems be able to assign the equilibria of the closed–loop system of Subsection II-B, the structure of the energy function was via a selection of the potential energy only we will choose fixed, and the PDE for the new parameters was explicitly a desired energy function of the form 1 solved using the elements of Jd . [λ − µd (θ, p)] L(θ)−1 [λ − µd (θ, p)] Hd (λ, θ, p) = We consider an electromechanical system consisting of 2 1 2 ne windings, with possible permanent magnets or a salient p +Vd (θ) + (13) rotor where the relationship between the flux linkage vector 2J λ ∈ Rne and the current vector i ∈ Rne is given by where µd (θ, p) is a function to be defined, on which we λ = L(θ)i + µ(θ), with θ ∈ R the mechanical angular impose the constraint λ = µd (θ , 0). position, and L(θ) = L (θ) > 0 the ne × ne multiport In this way, the equilibria will coincide with the extrema inductance matrix of the windings. The vector µ(θ) rep- of Vd (θ), and we simply have to select a function with a resents the flux linkages due to the possible existence of unique isolated minimum at θ . permanent magnets. Assuming all the electrical coordinates Now, to assign the proposed energy function preserving are actuated,3 the voltage balance equation yields the PCH structure we propose to modify the original interconnection and damping structures to take the form λ˙ + Ri = u (8) ⎡ ⎤ ⎡ ⎤ ⎤⎡ −R α γ λ˙ ∇λ Hd 3 This assumption is essential to avoid fixing some elements of the matrix ⎣ θ˙ ⎦ = ⎣ −α 0 ⎦ ⎣ ∇θ Hd ⎦ (14) 1 Jd , as in the case of mechanical systems. ∇p Hd −γ −1 −ra (p) p˙ 3420
where α, γ, ra (p) are the free parameters (possibly functions of the state space variables) that we will use to assign the desired energy function, and we select ra (p) > 0. Proposition 2 [31] For any scalar function ra (p) > 0 and any function Vd (θ), there exists a function µd (θ, p) satisfying (III-A), vector functions α, γ, and a static state– feedback control β(λ, θ, p) such that the original dynamics (11) in closed–loop with u = β(λ, θ, p) matches the desired dynamics (14) in some neighborhood of the equilibrium.4 In the proposition above no assumptions are made about the form of the parameters that define the system dynamics. In particular, no assumption of Blondel–Parks transformability—essential for the developments in [33]— is required. The price paid for achieving this level of generality, is complexity, requiring full state measurement and knowledge of system parameters. Some results on partial state feedback control are reported on [34]. On the other hand, it is our contention that the solid theoretical foundation of the proposed controller structure provides a suitable starting point for the derivation of (more practical) control schemes under some simplifying assumptions on the model. B. An Interlaced Algebraic–Parameterized IDA–PBC To solve a power systems problem, yet another variation of IDA–PBC was introduced in [32]. The main idea is to interlace the steps of solution of the PDE and determination of Jd as follows: (i) first “evaluate” the PDE in some subspace of the state–space where the “solution” can be easily computed, and then (ii) find the matrices Jd , Rd , g that will ensure the solution is valid in the whole state–space. To explain the procedure we use here a permanent magnet synchronous motor (PMSM) example. Application of IDA– PBC to speed control of PMSMs has been reported in [5], see also Subsection III–C in [1]. The controllers were tested experimentally and shown to be downward compatible with existing engineering practice, in the sense that the proposed controllers reduced to the standard ones for some particular tuning parameters and under some simplifying assumptions on the models. The model in the so–called dq coordinates takes the form did = −Rs id + ωLq iq + vd Ld dt diq = −Rs iq − ωLd id − ωΦ + vq Lq dt dω = np ((Ld − Lq )id iq + Φiq ) − τl J dt where ω is angular velocity, vd , vq , id , iq are voltages and currents, np is the number of pole pairs, Ld and Lq are stator inductances, Rs is stator winding resistance, τl is a 4 In [31] it is shown that for typical applications, including rotating machines, matching (and subsequent stabilization) is global.
constant unknown load torque, Φ and J are the dq back emf constant and the moment of inertia respectively, and all parameters are positive. The desired equilibrium—applying a “maximum torque L τ per ampere” principle— is (0, nqp Φl , nJp ω ). We then fix the structure of the desired energy function Hd (id , iq , ω) =
1 2 1 γ1 i + ψ(iq ) + γ2 (ω − ω )2 2 d 2
where ψ(iq ) is a function to be defined, γi > 0 are free parameters, and select ⎤ ⎡ −r1 J12 (id , iq ) J13 (id , iq ) Jd (id , iq )−Rd=⎣ −J12 (id , iq ) −r2 J23 (id , iq ) ⎦ 0 −J13 (id , iq ) −J23 (id , iq ) with ri > 0, and the functions Jij (id , iq ) to be defined. The ODE to be solved, corresponding to the unactuated coordinate, is −J13 (id , iq )γ1 id − J23 (id , iq )ψ = 1 (np (Ld − Lq )id iq + np Φiq − τl ) J
(15)
The key step here is to “evaluate” the ODE on the plane {id = 0}. This yields ψ =
1 (τl − np Φiq ) J23 (0, iq )J
that can be easily integrated if we take J23 to be constant. We note that the (Hessian) minimum condition, ψ > 0 imposes J23 < 0. Now, we plug back the proposed ψ into the ODE (15) and compute the function J13 (iq ) as J13 (iq ) =
np (Lq − Ld )iq γ1 J
The design is completed computing the control law vd
vq
Rs 1 J12 (id , iq )(τl − np Φiq ) id + Ld J23 J Lq np γ2 (Lq − Ld )iq (ω − ω ) − iq ω − r1 γ1 id + Ld γ1 J Rs Ld Φ = iq + id ω + ω − J12 (id , iq )γ1 id Lq Lq Lq r2 (τl − np Φiq ) + J23 γ2 (ω − ω ) − J23 J =
These expressions can be considerably simplified selecting the free parameters r1 , r2 , γ1 , γ2 > 0, J12 ∈ R, J23 < 0. Also, as in [5] an observer for τl may be added. The procedure explained above to solve (15) admits the following alternative interpretation. Fixing J23 to be constant and J13 independent of id , we can split (15) into a part that is linear on id and another one, containing ψ , that does not depend on id .
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IV. CONCLUDING REMARKS AND FUTURE RESEARCH We have reviewed a series of applications and extensions of the basic IDA–PBC design methodology laid out in [2]. The wealth of material contained here reveals the wide diversity of this research line, which is being enthusiastically pursued by many different control groups. Some extensions to the basic methodology of IDA–PBC presented in Part I [1] that incorporate additional features and allow to handle control scenarios other than just stabilization are given in [35]. Among the manifold of research directions that are currently being investigated or remain open we can include: • Tracking The basic IDA–PBC is restricted to stabilization of fixed points and, with the notable exceptions of [8], [36], [37], tracking remains an essentially open issue. As pointed out in [37] the question of tracking exosystem–generated references may be cleanly recasted as a damping injection problem, but unfortunately with “unmatched” signals. In some cases it is possible to adapt the procedure to treat the stabilization of periodic orbits, see [17] for an example, where a “Mexican sombrero” Hd (x) is assigned to the closed–loop. See also [38] where similar objectives are pursued. • Solving the matching equation The need to have an explicit solution of the matching equation (2) remains an important stumbling block to make IDA–PBC a systematic design procedure. For mechanical systems, it has been shown in [19] that the λ–method of [21], developed for the Controlled Lagrangian method, can be adapted for IDA–PBC, yielding a bilinear PDE. See also the ν– method described in [27]. Using all the available degrees of freedom, (i.e., Jd (x), Rd (x), g ⊥ (x), change of coordinates, perturbed target dynamics) and combining the two extreme approaches described in the companion paper [1], this obstacle has been overcome for a large collection of practical applications. A particularly promising approach, that allowed the solution of the longstanding problem of transient stabilization of multimachine power systems with transfer conductances [32], is the interlaced technique described in Subsection III-B. • Robustness and adaptation The current framework to assess robustness of controller designs is based on contrived (but mathematically convenient) uncertainty structures that are difficult to justify from physical considerations.5 Developing a robustness and adaptation theory that would accommodate interconnection of (partially uncertain) parameterized PCH systems seems to be a reasonable alternative to reverse this trend. • Asymptotic matching The final aim of IDA–PBC is, in essence, model matching which in many applications might be a too strong requirement. Another possibility, that would clearly relax the stringent conditions imposed
by the method, is to aim at asymptotic model matching. This perspective was adopted in [14] to develop a working design in a satellite application. This research led to the development of a new, immersion and invariance, technique for stabilization of general nonlinear systems [39], that could be instrumental to derive systematic tools to attain our objective of asymptotically matching a PCH system. • Power shaping A practical drawback of energy–shaping control is the limited ability to “speed up” the transient response leading to somehow sluggish transients and below par overall performance levels. In a recent paper [40] we prove that for a class of RLC circuits with convex energy function and weak electromagnetic coupling it is possible to “add a differentiation” to the port terminals preserving passivity—with a new storage function that is directly related to the circuit power. A complete characterization of the linear RLC circuits that enjoy these new property is given in [41]. A corollary of this result is that it is possible to formulate the stabilization problem in terms of power (as opposed to energy) shaping, adding “derivative” actions in the control that naturally yield faster responses. Current research is under way in two directions: extending the technique to other physical systems and assessing the effective advantages of the alternative framework with respect to energy–shaping based schemes like IDA–PBC. (See [42] for the geometric formalization of the ideas in [40] and [43].) • Infinite dimensional systems The PCH modelling framework for these systems has been already laid out in [44] and some preliminary results are available on control by interconnection [45], [46]. See also [47]. From a control engineering viewpoint the introduction of a “new” controller design technique is justified only if it can outperform the existing schemes for a practical problem or provide solutions to a challenging task that remained open. As witnessed by the list of references IDA–PBC has amply and satisfactorily fulfilled these expectations. The wide attention that PBC in general, and IDA–PBC in particular, have attracted makes us confident that the technique will prove instrumental to solve other practical problems in the future.
5 The prevalent configuration assumes some kind of matching condition that can be “dominated” with high–gain.
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