IDA-PBC under sampling for Port-Controlled Hamiltonian systems

Transcription

1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June -July, WeC7.4 IDA-PBC under sampling for Port-Controlled Hamiltonian systems Fernando Tiefensee, Salvatore Monaco and Dorothée Normand-Cyrot Abstract Taking in mind that the lost of the passivity property under sampling reflects into the degradation of the stabilizing performances of emulated controllers, this paper is a first attempt to the sampled-data version of the interconnection and damping assignment-passivity based controllers IDA-PBC). A sampled-data controller, preserving asymptotically the energetic behavior of a target dynamics and achieving stabilization to a suitable equilibrium is described for hamiltonian dynamics. A mechanical case study illustrates the results in a comparative perspective. I. INTRODUCTION Nonlinear control gradually adopts new techniques conserving and even exploiting the nonlinear structure instead of those based on nonlinearity cancellations. In a continuoustime context, the Passivity Based Control-PBC methodology introduced in [9] has been further developed versus the Interconnection and Damping Assignment - Passivity Based Control -IDA-PBC see [], [] and the references therein) so renewing stabilizing strategies. The basic philosophy is to shape a desired internal structure and then to inject damping to dissipate the total system s energy. Such an approach is very successful in the context of mechanical, electrical and electromechanical systems represented through the Euler-Lagrange formulation. However, it cannot still be considered as a systematic tool. Necessary and sufficient like conditions characterizing the problem solvability are not completely available and constructive solutions are developed for some specific classes of systems. Among them, Hamiltonian systems provide natural meaningful examples and inspire insights for new solutions []. The object of this work is to discuss the effects of sampling on the IDA-PBC approach set on Hamiltonian dynamics. As well known, sampled-data design performances strongly depend on the availability of a reliable sampled-data model. Eventhough some robust algorithms to construct discretetime Hamiltonian dynamics are proposed see []), they are oftently not suitable to control design purposes. Moreover, to set the IDA-PBC problem in a purely discrete-time context is not an easy task since the definition of discrete-time Hamiltonian structures preserving the essential properties of losslessness and Hamiltonian conservativeness is still a challenge for both control and mathematical communities. In [], the preservation of passivity of the interconnection F. Tiefensee and D. Normand-Cyrot are Laboratoire des Signaux et Systèmes, CNRS-Supelec, Plateau de Moulon, 99 Gif-sur-Yvette, France. tiefensee@lss.supelec.fr, cyrot@lss.supelec.fr S. Monaco is Dipartimento di Informatica e Sistemistica Antonio Ruberti, Università La Sapienza, via Ariosto 5, 85, Roma, Italy. salvatore.monaco@dis.uniroma.it of a continuous-time system a discrete-time Port Control Hamiltonian system is studied. In [7], we shown that Hamiltonian conservation can be verified under sampling respect to a suitably defined output mapping. In [], a direct sampled-data controller is built design for a separable Hamiltonian system through Euler sampling procedures and generalized in [4] to the non-separable case in terms of discrete gradient. These results surely contribute to the IDA- PBC sampling-data design and improve the performances of strictly emulated strategies but are strongly limited by the Euler approximation validity which is not guaranteed except for very small sampling periods [5]. Here after, to solve the problem in a quite systematic way, we adopt the approach developed in some recent papers [5], [8], []) to design a sampled-data controller in such a way to match, at the sampling instants, some key behaviors of the continuous-time design, usually some suitable inputoutput link. Arguing so in the present context, we assume the existence of a continuous-time IDA-PBC controller and look for a sampled-data controller matching the energy behavior of the closed loop system, fixed by the designer. The so designed sampled-data controller is described by its series expansion around the continuous-time one. In practice, the first terms of the series only are computed and implemented to show efficiency in terms of maintaining the desired energy losslessness and Hamiltonian conservativeness). We illustrate robustness to larger sampling time and improved intersample behaviors. The example studied in [] and [] is chosen to verify the performances of the proposed digital controller in a comparative perspective the solutions therein proposed. The paper is organized as follows. After some preliminaries and the problem settlement in section II, section III describes the sampled-data IDA-PBC strategy. The example of an inertia wheel pendulum is illustrated simulated tests in section IV. Some notations - In the paper, x X = R n,t R,u U, the set of admissible inputs - all valued piecewise continuous functions are defined on R; vectors, when non differently specified, are assumed column vectors; functions, operators and vector fields are smooth i.e. C ) on their domains of definition and vector fields are complete. Standard notations are used when no ambiguity is possible. In particular e f = L + i f i i!, indicates the operator Lie series associated a smooth vector field f regarded as the Lie derivative L f = n i= f ix) x i, indicates the identity operator and I n the identity function on R n while I d indicates the identity matrix of appropriate dimension. Manipulations over the series //$6. AACC 8

2 described by their asymptotic expansions are formal and no convergence study is performed. H defines the gradient of H : R n R. gx) is a full-rank left annihilator of gx), i. e. gx) gx) =. The matrix g + x) is the Moore-Penrose inverse of gx), i.e. gg + := g[g T x)gx)] g T x) = I d. The evaluation of a function at time t = kδ indicated by t=kδ is omitted, when obvious from the context. II. PRELIMINARIES AND PROBLEM SETTLEMENT A. The continuous-time IDA-PBC strategy Consider a Port Controlled Hamiltonian-PCH dynamics [4] ẋt) = fx)+gx)u = [J x) Rx)] H + gx)u ) where x R n is the state vector, u R m the control vector m < n, H : R n R is the total energy, J x) = J T x),rx) = R T x) are the usual interconnection and damping matrices respectively. Let us recall the IDA- PBC design procedure proposed in [9] and [] for such a system s class. Proposition.: Let ) and assume the existence of matrices g T x),j d x) = Jd Tx),R dx) = Rd T x) and a function H d : R n R that verify the Partial Derivative Equation-PDE g x) fx) = g x)[j d x) R d x)] H d ) where H d x) is the target energy such that x = argminh d x) x the equilibrium to be stabilized. Setting u = u es +u da u es = g + x)j d x) H d fx)) ) u da = g + x)r d x) H d 4) the closed-loop dynamics has a locally stable equilibrium in x. If in addition x is an isolated minimum of H d x) and if the largest invariant set under the closed loop dynamics contained in {x R n s.t.[ H d ] T R d x) H d = } equals {x }, then x is asymptotically stable. An estimate of its domain of attraction is given by the largest bounded level set {x R n s.t.h d x) c}. The following comments are verified by construction : provided H qualifies as a storage function, then the dynamics ) output y = g T x) H 5) is passive; the first part u es of the controller shapes the energy so achieving in closed loop ẋt) = f es x)+gx)u = J d x) H d + gx)u; 6) such a dynamics output y d = g T x) H d is lossless and conservative in the absence of control, u = ; i.e. provided H d qualifies as a storage function, one verifies i) L fes H d = ii) Ḣ d x) = y T d u or equivalently, in its integral version for all u,x,t ii) H d xt)) H d x ) = t y T d τ)uτ)dτ; 7) the damping part u da of the controller is a negative gain output feedback u da = K d y d K d > and the complete controller u c = u es + u da achieves the closed loop PCH form ẋt) = f d x) = [J d x) R d x)] H d 8) R d x) = K d gx)g T x). It is worthy to note that the key point of the result in Proposition. is the solvability of the PDE ). B. Problem settlement Hereafter, assuming the conditions of Proposition. satisfied, we set the problem of the digital implementation of the controller -4). More precisely, does it exist a piecewise constant controller achieving asymptotic stabilization at x of the PCH dynamics )? To solve the problem in a systematic way, we adopt the approach developed in some recent papers [5], [] to design a sampled-data controller in such a way to match, at the sampling instants, some key behaviors of the continuous-time design, usually some suitably defined inputoutput link. In the present context, assuming the existence of a constructive IDA-PBC controller achieving stabilization to a suitable equilibrium and damping, we look for a sampleddata controller matching the energy behavior fixed by the designer. III. THE SAMPLED-DATA IDA-PBC A. The sampled-data equivalent model Assuming the control in ) constant over time intervals of length δ ],T ] a sufficiently small time interval), and denoting by u k its value over [kδ, k + )δ[ and by x k the value of xt) at time t = kδ for k, the sampled equivalent model is defined by the δ-parameterized map x k x k+ = F δ x k,u k ) = e δ f+u kg) x k. 9) The state evolutions of F δ.,u k ) coincide that of ) at the sampling instants: xt = kδ) = x k for k whenever x = xt = ). To get a closed form solution requires the integrability of ) which does not hold in general. The sampled dynamics 9) is thus described by its asymptotic series expansion in powers of δ; i.e. e δ f+ug) = +δl f + ul g )+ δ! δ p p! Lp f+ug +... L f + ul f L g + L g L f )+u L g Truncations in δ yield to approximate sampled models of order p in δ error in Oδ p+ )). Remark. It is worthy to note that the specific PCH state-space structure in ) as well as losslessness and conservativeness of the continuous-time system -5) are lost under sampling as soon as terms of order in the equivalent sampleddata dynamics 9) are taken into account. In [] and [4], the design of sampled-data IDA-PBC controllers is restricted to Euler approximate models truncations at the st order in δ). ) 8

3 B. The sampled-data IDA-PBC matching solution Let us reformulate the problem in terms of step-by-step matching at the sampling instants of the energetic behavior of the continuous-time target Hamiltonian function H d. Denoting by u c resp. uk δ ) the continuous-time resp. sampled-data) controller, by x c resp. x d ) the continuous-time state trajectory under u c resp. sampled-data under uk δ ), one computes at the sampling instants t = kδ, k and respectively H d x c t)) t=k+)δ = e δ f+u cg) H d t=kδ. ) H d x dk+ ) = e δ f+uδ k g) H d x dk ). ) The design works out assuming the piecewise constant control uk δ of the form uk δ = u + i+)! u i ) i and solving the following equality of series δ i H d x c t)) t=k+)δ = H d x dk+ ) or equivalently, setting x c k) = x dk in ) and ) k+)δ L f+uc gh d x c τ))dτ = e δ f+uδ k g) H d x dk ) H d x dk ). ) kδ Theorem.: Given ) an assuming the existence of a continuous-time IDA-PBC as in Proposition. H d a C -function describing the target Hamiltonian, then : there exists a piecewise constant controller of the form ) ensuring matching at the sampling instants t = kδ, k of the desired Hamiltonian behavior H d ; asymptotic stabilization at the target equilibrium x is achieved; the successive terms in the expansion ) can be computed iteratively so obtaining for the first ones u δ k = u k + δ u k + δ u k = u es t)+u da t)) t=kδ! u k 4) u k = u es t)+ u da t)) t=kδ ) ad [ f,g] H d u k = ü es t)+ü da t)+u k L g H d. t=kδ Proof: Rewriting ) as an equality of series after formal integration of its left hand side, the existence of a solution is guaranteed by the condition L g H d. Such a solution is expressed through formal series inversion so taking the form ). Setting y d = L g H d, the required relative degree like condition is ensured by the zero state detectability like assumptions set in Proposition. which expresses that no solution of the uncontrolled dynamics ẋ = f d x) can stay in the set {x R n s.t h d x) = } other than solutions xt) converging asymptotically to x. Remark. The approximate sampled-data controller 4) guarantees Hamiltonian matching up to the third order error in Oδ 4 )) assuming H d to be a C function only while u δ a = u + δ u guarantees Hamiltonian matching up to the second order error in Oδ )) assuming H d to be a C function only. The following comments specify the result. The so constructed digital controller achieves at the sampling instants the equality H d F δ x k,u δ k )) = eδ f+uδ k g) H d x k ) = e δ f d H d x k ) the desired closed loop dynamics f d described in 8). This means that, even though the closed loop sampled state dynamics x k F δ x k,uk δ ) does not match at the sampling instants the dynamics, x k e δ f d x k, they have the same energetic behavior in terms of H d. As far as energy shaping matching is concerned, we note that setting u da = or K d =, the resulting digital controller ukes δ satisfies the equality H d F δ x k,u δ kes )) = eδ f es H d x k ) = H d x k ) i.e. the sampled-data energy shaping controller ukes δ proposed in Theorem. ensures Hamiltonian conservativeness at the sampling instants. As far as the preservation of the Hamiltonian state space structure is concerned, one easily verifies that under the proposed sampled-data controller uk δ, the target Hamiltonian structure is preserved into δ error in Oδ )). As far as the losslessness property 7) is concerned, setting uk δ = uδ kes + v k and following the lines proposed in [6], we can easily verify the lossless criteria H d x k+ ) H d x k ) = δy δ d k)t v k the so defined sampled-data output yd δ k) := k+)δ δ kδ L f+u δ k g L gh d xτ))dτ deduced from y d in 7). Then, a direct digital damping controller can be designed over v k as proposed in [] regarding the Single Machine Infinite Bus-SMIB system. Work is progressing in this direction. C. Hamiltonian and separable Hamiltonian dynamics Setting x = [q, p] T, where q are the generalized configuration coordinates and p the generalized momenta of the system n/ degrees of freedom, let Hq, p) = pt M q)p+pq) 5) be the sum of the kinetic and potential energies Mq), a symmetric and definite positive inertia matrix. Let the PCH representation [ qt) ṗt) ] = [ Id Id ][ q H p H ] [ + Gq) ] 6) y = Gq) T M q)p = Gq) q. 7) 8

4 Setting the target Hamiltonian dynamics in the form [ M J d = ] q)m d q) M d q)m q) J where J q, p) = J q, p) T is a design parameter and M d q) = M d q) T a desired inertia matrix so that H d q, p) = pt Md q)p+p dq) has an isolated minimum at the desired equilibrium. Then, setting ) u c = G + q) q H M d M q H d + J Md p K d Md q)p the dissipative matrix [ ] R d = Gq)K d Gq) T K d > is assigned. 6)-7) is a separable Hamiltonian system, if the inertia matrix M.) is constant so that the kinetic and potential energies are decoupled, Hp,q) = pt M p + Pq). The target system can also be set in the form of a separable Hamiltonian system and the designer parameter J can be chosen equal to zero []. In such a case, the sampled-data controller simplifies as u k = G + q) q Pq) P d q)) }{{} u es u k = u es + u da u k = u es + u da u k K d G T q)m p } {{ } u da G T q)m d p) G T q)m q P d q) + u k G+ q) q Gq)M p being zero the last term in u k when G is a constant matrix. IV. THE INERTIA WHEEL PENDULUM Fig.. I qq I q u The Inertia Wheel Pendulum The inertia wheel pendulum example is an interesting example of separable Hamiltonian system of the form 6-7) for which a continuous-time IDA-PBC has been developed in [] and its digital implementation in []. As depicted in Fig., it consists in a pendulum a balanced rotor at the end. The rotor torque produces an angular acceleration of the end-mass generating a coupling torque at the pendulum axis. q denotes the pendulum angular deviation from the horizontal axis while q represents the angular deviation of the disk respect to the pendulum. The control objective is to stabilize the pendulum in its inverted position. The model takes the form 6), Hamiltonian 5) [ ] [ ] I M =, G = I Pq ) = mglcosq ) ) where I and I are the moments of inertia of the pendulum and disk respectively, L and m the pendulum length and mass. Gravity g is assumed constant and the system is controlled by a torque u acting between the disk and the pendulum. The equilibrium to be stabilized corresponds to the upward position the inertia disk aligned, q = q =. ) Continuous-time design, []: Since we are dealing a separable Hamiltonian system, M is constant so that J can be set equal to and the desired inertia matrix M d can be set equal to [ ] a a M d = a a a > and a a > a, such that M d = Md T >. The PDE to be solved is ) ) a + a Vd a + a Vd + = mglsinq ). I q I q Setting P d = I mgl cosq + b a + a γ q + q ), to get an isolated minimum at q = q =, the energy shaping and damping terms are given by u es t) = γ sinq )+γ p γ q + q ) K d u da t) = a a a a + a )p +a + a )p ) = K α γ q + q ) γ := a a +a mgl, γ := I a +a ) I a +a ), γ a a p = b a I a +a ) I and K α = K a +a ) d. γ a a a > m, γ > I γ I γ mgl, γ p > and K α > define the admissible region for the tuning gains. A. Sampled-data design The digital controller 4) specifies as follows u k = γ sinq )+γ γ q + q ) K α γ q + q ) u k = q γ γ + γ cosq ))+γ q K α γ q + q ) u k = q γ cosq )+γ γ )+γ q γ q sinq... K α γ q + q... )+ ξq, p) ξq, p) = α q + γ q )α q + γ q )+α sinq )) α = I K α I K, α d = K I I γ ) and α = I I mgl The solution proposed in [] results in u al = u k + δ u T u T = K al p + p ), K al >. a a. It is used in the next section in a comparative perspective. ) 84

5 4 5 x f ree xuc) x k u a δ ) x k u a δ ) x k u al ).5.5 H d u c ) H d u ) H d u δ a ) H d u δ a ) H d u al ) x.5.5 a) q p evolution x b) Desired Hamiltonian H d evolution Fig.. Performances of Inertia Wheel Pendulum, δ = 5ms..5 x f ree xuc) x k u a δ ) x k u a δ ).5.5 H d u c ) H d ua δ ) H d ua δ ) H d u al ) x x a) q p evolution b) Desired Hamiltonian H d evolution Fig.. Performances of Inertia Wheel Pendulum, δ = 75ms. B. Simulations Setting the parameters values equal to I =., I =. and mgl = ; a =, a = and a = 5; b = 5 and K d = or K α = and K al = 7, the simulated tests compare the inertia wheel pendulum free evolution x f ree the closed loop behaviors under continuous-time IDA-PBC controller u c t) = u es t)+u da t), emulated strategy u k, approximate sampleddata controllers at order one or two, denoted respectively by ua δ = u k + δ u k and ua δ = uδ a + δ! u k and u al proposed in []. Fig.a) depicts the state trajectories q and p regarding the pendulum. Since the model is lossless, R = ), the uncontrolled system ẋ = Fx) = x f ree does not converge to the desired equilibrium point q, p ) =,). The continuoustime IDA-PBC controller shapes the total system s energy so that H d has a minimum at this point Fig. b)) and asymptotically stabilizes the system Fig.a)). We note that state trajectories q and p regarding the disk also converge to the equilibrium point q, p ) =,) more details about the continuous-time responses are in []). The performances of the sampled-data controllers are illustrated through the trajectory q p while q p is omitted because it reflects into the convergence of H d. The length of the sampling period δ directly affects the performances of the sampled-data controllers. This is clear from Fig. to Fig. 4 illustrating the successive cases for δ = 5ms,75ms,ms. In Fig., the emulated strategy does not work anymore while ua δ and uδ a better match the target Hamiltonian evolution H d than u al, keeping the q p trajectory very close to the target trajectory. In Fig., the sampled-data controller u al does not work anymore while both ud δ and uδ d still match H d but the q p trajectory degrades under ud δ. In fact, the interest of the second order term is to fit well the continuoustime process so ensuring better performances limited control amplitude the amplitude of ua δ tends to be smaller as depicted in Fig.5a)). With δ = ms, the first order does not ensure the system stability, while ua δ achieves stabilization in few steps, a good intersample behaviour of the H d response and control amplitude close to that of u c t), Fig.5b)). V. CONCLUSION A sampled-data controller, imposing asymptotically the energetic behavior of a target dynamics and preserving, up to order, a desired Hamiltonian structure has been described. Its efficiency has been verified by comparing its performances that of a continuous-time design implemented through emulation and a sampled-data solution proposed in []. While the continuous-time design is composed two parts to ensure suitable shaping and to improve damping, the sampled-data controller satisfies both requirements the same accuracy provided differentiability of the Hamiltonian. A sampled-data strategy satisfying energy shaping and direct damping in a purely discrete-time context could be developed following recent results aiming to preserve passivity like conditions under sampling [6]. 85

6 4 5.5 x f ree xuc) H d u c ) x k u δ a ).5 H d u δ a ) H d u δ a ).5 x a) q p evolution x b) Desired Hamiltonian H d evolution Fig. 4. Performances of Inertia Wheel Pendulum, δ = ms. 4 4 u c u c u δ a u δ a u δ a a) Controllers for δ = 75ms b) Controllers for δ = ms Fig. 5. Controllers for different δ VI. ACKNOWLEDGMENTS Work partially supported by a research mobility grant VINCI- of the UFI/UFI - French/Italian - Italian/French University. REFERENCES [] A. Astolfi, R. Ortega and R. Sepulchre ) Stabilization and disturbance attenuation of nonlinear systems using dissipativity theory: A survey, European J of Control, 8, [] O. Gonzalez 996) The stability of nonlinear dissipative systems, Journal of Nonlinear Science, 6, [] Laila, D. S. and A. Astolfi 5) Discrete-time IDA-PBC design for separable Hamiltonian systems, Proc. 6th IFAC World Congress, Prague. [4] Laila, D. S. and A. Astolfi 7) Direct Discrete-Time Design for Sampled-Data Hamiltonian Control Systems, Lecture Notes in Control and Information Sciences, Springer Berlin/Heidelberg, 66, [5] S. Monaco and D. Normand-Cyrot 7) Advanced Tools for Nonlinear Sampled-Data Systems Analysis and Control, Mini-Tutorial, ECC-7, Kos, Invited Paper), Special Issue Fundamental Issues in Control, European Journal of Control, Hermès Sciences, Paris,-,, 4. [6] S. Monaco, D. Normand-Cyrot and F. Tiefensee 8) From passivity under sampling to a new discrete-time passivity concept, Proc. 47th IEEE-CDC, Cancun, [7] S. Monaco, D. Normand-Cyrot and F. Tiefensee 9) Nonlinear port controlled Hamiltonian systems under sampling, Proc. 48th IEEE- CDC, Shanghai, [8] D. Nesic, L. Grune 5) Lyapunov based continuoustime nonlinear controller redesign for sampleddata implementation. Automatica, 4, 456. [9] R.Ortega and M. Spong 989) Adaptive motion control of rigid robots; a tutorial, Automatica, 5-6, [] R.Ortega, M. Spong, F. Gomez-Estern and G. Blankenstein ) Stabilization of a class of underactuated mechanical systems via interconnection and damping assignement, IEEE Trans on AC, AC47-8, 8. [] R. Ortega and E. Garcia-Canseco 4) Interconnection and damping assignment passivity based control: A survey. European J of Control, [] S. Stramigioli, C. Secchi, A. Van der Schaft, and C. Fantuzzi ) Sampled-data systems passivity and discrete port-hamiltonian systems, Proc. IEEE Trans on AC,, [] F. Tiefensee, S. Monaco and D. Normand-Cyrot 9), Lyapunov design under sampling for a Synchronous Machine Infinity Bus System, Proc. ECC, Budapest, [4] A. Van der Schaft, and C. Fantuzzi ) L Gain and Passivity Techniques in Nonlinear Control. Berlin, Germany, Springer-Verlag. 86