Stabilization of a Class of Underactuated Mechanical Systems via Interconnection and Damping Assignment

Transcription

1 Stabilization of a Class of Uneractuate Mechanical Systems via Interconnection an Damping Assignment Romeo Ortega Lab. es Signaux et Systèmes CNRS-SUPELEC Gif sur Yvette 99 FRANCE tel. no:(33) fax. no:(33) Romeo.Ortega@lss.supelec.fr Fabio Gómez-Estern Dept. Ing. e Sistemas y Automática Escuela Superior e Ingenieros C. e los Descubrimientos, 49 Sevilla SPAIN tel. no:(34) fax. no:(34) fgs@cartuja.us.es May 9, Mark W. Spong Coorinate Science Lab University of Illinois Urbana, IL 68 USA tel. no:() fax. no:() m-spong@uiuc.eu Guio Blankenstein EPFL/DMA MA Ecublens CH-5 Lausanne SWITZERLAND tel fax guio.blankenstein@epfl.ch Keywors: Uneractuate mechanical systems, Hamiltonian systems, nonlinear control, energy shaping, passivity. Abstract In this paper we consier the application of a new formulation of Passivity Base Control, known as Interconnection an Damping Assignment, or IDA-PBC, to the problem of stabilization of uneractuate mechanical systems, which requires the moification of both the potential an the kinetic energies. Our main contribution is the characterization of a class of systems for which IDA-PBC yiels a smooth asymptotically stabilizing controller with a guarantee omain of attraction. The class is given in terms of solvability of certain partial ifferential equations. One important feature of IDA PBC, stemming from its Hamiltonian (as oppose to the more classical Lagrangian) formulation, is that it provies new egrees of freeom for the solution of these equations. Using this aitional freeom, we are able to show that the metho of controlle Lagrangians in its original formulation may be viewe as a special case of our approach. As illustrations we esign asymptotically stabilizing IDA-PBC s for the classical ball an beam system an a novel inertia wheel penulum. For the former we prove that for all initial conitions (except a set of zero measure) we rive the beam to the right orientation. Also, we efine a omain of attraction for the zero equilibrium that ensures that the ball remains within the bar. For the inertia wheel we prove that it is possible to swing up an balance the penulum without switching between separately erive swing up an balance controllers an without measurement of velocities.

2 Introuction Passivity base control (PBC) is a esign methoology for control of nonlinear systems which is well known in mechanical applications. In some regulation problems it provies a natural proceure to shape the potential energy preserving in close loop the Euler Lagrange (EL) structure of the system. As thoroughly iscusse in [6, PBC is also applicable to a broa class of systems escribe by the EL equations of motion, incluing electrical an electromechanical systems. In [7 we propose the utilization of ynamic EL controllers, couple with the plant via power preserving interconnections (hence preserving the EL structure), to shape the potential energy of a class of uneractuate EL systems via partial state measurements. It is well known, however, that to stabilize some uneractuate mechanical evices, as well as most electrical an electromechanical systems, it is necessary to moify the total energy function. Unfortunately, total energy shaping with the classical proceure of PBC, where we first select the storage function to be assigne an then esign the controller that enforces the issipation inequality, estroys the EL structure. That is, in these cases, the close loop although still efining a passive operator is no longer an EL system, an the storage function of the passive map (which is typically quaratic in the errors) is not an energy function in any meaningful physical sense. As explaine in Section.3. of [6, this situation stems from the fact that these esigns carry out an inversion of the system along the reference trajectories, inheriting the poor robustness properties of feeback linearizing esigns an imposing an unnatural stable invertibility requirement to the system. To overcome this problem we have evelope in [8 (see also [9, 8) a new PBC esign methoology calle interconnection an amping assignment or IDA PBC. The main istinguishing features of IDA PBC are that: ) it is formulate for systems escribe by so calle port controlle Hamiltonian moels, which is a class that strictly contains EL moels, an ) the close loop energy function is obtaine via the solution of a partial ifferential equation (PDE) as a result of our choice of esire subsystems interconnections an amping. It is well known that solving PDE s is, in general, not easy. However, the particular PDE that appears in IDA PBC is parameterize in terms of the esire interconnection an amping matrices, which can be juiciously chosen invoking physical consierations to solve it. This point has been illustrate in several practical applications incluing mass balance systems, electrical motors, magnetic levitation systems, power systems, power converters, satellite control an unerwater vehicles. See [8 for a list of references. The present paper, which is an expane version of [ an [, is concerne with the application of IDA PBC to the problem of stabilization of uneractuate mechanical systems. Our main contribution is the characterization of a class of systems for which IDA PBC yiels a smooth stabilizing controller. The class is given in terms of solvability of two PDE s that correspon to the potential an kinetic energy shaping stages of the esign. As illustrations we esign asymptotically stabilizing IDA PBCs for the well-known ball an beam system an for a novel inverte penulum. For the ball an beam we prove that for all initial conitions (except a set of zero measure) we rive the beam to the right orientation an efine a omain of attraction that ensures that the ball remains within the bar. For the inertia wheel we present a ynamic nonlinear output feeback which, again for almost all initial conitions, stabilizes its upwar position. That is, we show that it is possible to swing up an balance this penulum without either switching or measurement of velocities. Another concern of our paper is to place the new Hamiltonian approach in perspective with the metho of controlle Lagrangians for stabilization of simple mechanical systems evelope We shoul point out that, as presente here, IDA PBC applies only to smoothly stabilizable systems, ruling out the large class consiere in [3. However, the version of IDA PBC of [9, applies as well to problems involving time varying Hamiltonians, allowing to consier stabilization with time varying controllers.

3 in a series of papers by Bloch et al. (e.g., [5, 6), an followe up by [, 3. IDA PBC an the controlle Lagrangians metho are proceures to generate state feebacks that transform a given Hamiltonian (respectively, EL) system into another Hamiltonian (respectively, EL) system. A central ifference between the methos is that while the target EL ynamics in the controlle Lagrangian metho is obtaine moifying only the generalize inertia matrix an the potential energy function, in IDA PBC we have also the possibility of changing the interconnection matrix, i.e., the Poisson structure of the system. In this respect, the following questions naturally arise:. What is the specific choice of this free parameter that yiels the same controllers for both methos? Although the answer to this question has alreay been reporte in our previous papers, e.g., [9, we use here the more recent results of [4 to sharpen the statement.. Can we use this aitional egree of freeom to simplify the task of solving the aforementione PDE s? To answer this question we write the PDE s in a form where the free parameters appear explicitly as esigner chosen control inputs, an work out two classical examples to illustrate how to select these inputs. 3. Is the set of EL close loop moels achievable via IDA PBC larger than the one achievable with Lagrangian methos? An if so, can we characterize the gap? We provie here an answer to both questions, showing that with IDA PBC we can, still preserving the EL structure, a gyroscopic terms to the close loop system, a feature that is not inclue in the reporte literature with the Lagrangian approach. 3 To further clarify the connections between IDA PBC an controlle Lagrangians we again invoking [4 prove that the matching conitions of [5 characterize a class of mechanical systems such that its kinetic energy functions can be shape (to a take a particular form) without solving the aforementione PDE s. Obviating the nee of solving the PDE s, which is the main stumbling block in both approaches, clearly enlarges the applicability of these esign methos. The remaining of the paper is organize as follows. In Section we present the application of IDA PBC to uneractuate mechanical systems. Section 3 is evote to the comparison between IDA PBC an the metho of controlle Lagrangians. Section 4 summarizes some results available on solvability of the PDE s. Sections 5 an 6 contain the erivations of IDA PBCs for the inertia wheel penulum an the ball an beam system, respectively. We wrap up the paper with some concluing remarks in Section 7. Stabilization of Uneractuate Mechanical Systems In this section we apply the IDA-PBC approach to regulate the position of uneractuate mechanical systems with total energy H(q, p) = p M (q)p + V (q) (.) where q IR n, p IR n are the generalize position an momenta, respectively, M(q) =M (q) > is the inertia matrix, an V (q) is the potential energy. If we assume that the system has no natural See [ for an elegant extension to general Lagrangian systems. 3 Recently, Bloch et al. [7 have extene the controlle Lagrangian approach to obtain a metho equivalent to the IDA-PBC. See Section 3 for more information. 3

4 amping, then the equations of motion can be written as 4 [ q ṗ [ = I n I n [ q H p H [ + G(q) u (.) The matrix G IR n m is etermine by the manner in which the control u IR m enters into the system an is invertible in case the system is fully actuate, i.e. m = n. We consier here the more ifficult case where the system is uneractuate, an assume rank(g) = m<n. In the IDA-PBC metho we follow the two basic steps of PBC [: ) energy shaping, where we moify the total energy function of the system to assign the esire equilibrium (q, ); an ) amping injection to achieve asymptotic stability. As explaine below, to preserve the energy interpretation of the stabilization mechanism we also require the close loop system to be in port controlle Hamiltonian form [8.. Target Dynamics Motivate by (.) we propose the following form for the esire (close loop) energy function H (q, p) = p M (q)p + V (q) (.3) where M = M > anv represent the (to be efine) close loop inertia matrix an potential energy function, respectively. We will require that V have an isolate minimum at q, that is q =argminv (q) (.4) In PBC the control input is naturally ecompose into two terms u = u es (q, p)+u i (q, p) (.5) where the first term is esigne to achieve the energy shaping an the secon one injects the amping. The esire port controlle Hamiltonian ynamics are taken of the form 5 [ q ṗ [ q H =[J (q, p) R (q, p) p H (.6) where the terms J = J = [ M M M M J (q, p) ; R = R = [ GK v G represent the esire interconnection an amping structures. The following observations are in orer: From (.), (.) we have that q = M p. Since this is a non actuate coorinate this relationship shoul hol also in close loop. Fixing (.3) an (.6) etermines the (, ) block of J. 4 Throughout the paper we view all vectors, incluing the graient, as column vectors; except in the Hessian matrix where they appear as rows. 5 See [8, 8 for the physical an analytical justification of this choice. 4

5 The matrix R is inclue to a amping into the system. As is well known, this is achieve via negative feeback of the (new) passive output (also calle L g V control), which in this case is G p H. That is, we will select the secon term of (.5) as u i = K v G p H (.7) where we take K v = K v >. This explains the (, ) block of R. We will show below that the skew symmetric matrix J (an some of the elements of M ) can be use as free parameters in orer to achieve the kinetic energy shaping. Proviing these egrees of freeom is the essence of IDA PBC. 6. Stability For the esire close loop ynamics we have the following proposition, which reveals the stabilization properties of our approach: Proposition The system (.6), with (.3) an (.4), has a stable equilibrium point at (q, ). This equilibrium is asymptotically stable if it is locally etectable from the output 7 G (q) p H (q, p). An estimate of the omain of attraction is given by Ω c where Ω c = {(q, p) IR n H (q, p) <c} an c =sup{c >H (q, ) Ω c is boune} (.8) Proof. From (.3) an (.4) we have that H is a positive efinite function in a neighborhoo of (q, ). A straightforwar calculation shows that, along trajectories of (.6), Ḣ satisfies Ḣ = ( q H ) q +( p H ) ṗ = ( p H ) GK v G p H λ min {K v } ( p H ) G since J is skew symmetric an K v is positive efinite. Hence, (q, ) is a stable equilibrium. Furthermore, since (by efinition) H is proper on its sub level set Ω c, all trajectories starting in Ω c are boune. Asymptotic stability, uner the etectability assumption, is establishe invoking Barbashin Krasovskii s theorem an the arguments use in the proof of Theorem 3. of [8. Finally, the estimate of the omain of attraction follows from the fact that Ω c is the largest boune sub level set of H..3 Energy Shaping To obtain the energy shaping term, u es, of the controller we replace (.5) an (.7) in (.) an equate it with (.6) 8 [ [ [ [ I n q H M + u I n p H G es = [ M q H M M J (q, p) p H 6 In Section 3 we will show that, for a particular choice of skew symmetric J, IDA PBC reuces to the controlle Lagrangian schemes of [,, 6. It is reasonable to expect that the possibility of choosing among the whole class of skew symmetric matrices enlarges the class of stabilizable systems. 7 That is, for any solution (q(t),p(t)) of the close loop system which belongs to some open neighborhoo of the equilibrium for all t, the following implication is true: G (q(t)) ph (q(t),p(t)), t lim (q(t),p(t)) = (q t, ). 8 Notice that the amping matrix cancels with u i. 5

6 While the first row of equations above is clearly satisfie, the secon set of equations can be expresse as Gu es = q H M M q H + J M p Now, it is clear that if G is invertible, i.e., if the system is fully actuate, then we may uniquely solve for the control input u es given any H an J. In the uneractuate case, G is not invertible but only full column rank, an u es can only influence the terms in the range space of G. This leas to the following set of constraint equations, which must be satisfie for any choice of u es, G { q H M M q H + J M p} = (.9) where G is a full rank left annihilator of G, i.e., G G =. Equation (.9), with H given by (.3), is a set of nonlinear PDE s with unknowns M an V,withJ a free parameter, an p an inepenent coorinate. If a solution for this PDE is obtaine, the resulting control law u es is given as u es =(G G) G ( q H M M q H + J M p) (.) The PDE s (.9) can be naturally separate into the terms that epen on p an terms which are inepenent of p, i.e., those corresponing to the kinetic an the potential energies, respectively. Thus, (.9) can be equivalently written as { } G q (p M p) M M q (p M p)+j M p = (.) G { q V M M q V } = (.) The first equation is a nonlinear PDE that has to be solve for the unknown elements of the close loop inertia matrix M.GivenM, equation (.) is a simple linear PDE, hence the main ifficulty is in the solution of (.). Equation (.) can be expresse in a more explicit form with the following erivations. First, we use the fact that q (z P (q)z) =[ q (P (q)z) z, for all z IR n an all symmetric P IR n n, to write (.) as Then, we apply the ientity G {[ [ q (M p) M M [ q (M q (P (q)z) = n q (P (,k) )z k k= } p) +J M p = which hols for all z IR n an all P IR n n, where P (,k) enotes the k th column of the matrix P, reparametrize J in terms of the matrices U k (q) = U k (q) IRn n as J = n U k p k (.3) k= an equate terms in p k to obtain n PDE s, expresse only in terms of the inepenent variable q, of the form { [ G q (M (,k) ) M M [ } q (M ) (,k) + Uk M = (.4) where U k are esigner chosen matrices. The key iea is then to chose the free parameters U k in such a way that (.4) amits a solution with M symmetric an positive efinite. Then, we replace it into (.), which is a linear PDE, 6

7 an look for a solution V which satisfies (.4). The aitional egree of freeom provie by U k is the main feature that istinguishes IDA PBC from the Lagrangian methos that we review in the next section. The following remarks are in orer The erivations above characterize a class of uneractuate mechanical systems for which the newly evelope IDA PBC esign methoology yiels smooth stabilization. The class is given in terms of solvability of the nonlinear PDE (.), (or the more explicit (.4)), an the linear PDE (.). Although it is well known that solving PDE s is in general har, it is our contention that the ae egree of freeom the close loop interconnection J (equivalently, U k ) simplifies this task. We will elaborate further on this point in Section 3 an in the examples below. There are two extreme particular cases of our proceure. First, if we o not moify the interconnection matrix then we recover the well known potential energy shaping proceure of PBC. Inee, if M = M an J =, then the controller equation (.) reuces to u es =(G G) G ( q V q V ) which is the familiar potential energy shaping control. On the other extreme, if we o not change the potential energy, but only moify the kinetic energy then, as we will show in the next section, for a particular choice of J, i.e., (3.6), we recover the controlle Lagrangian metho of [5. Now, if we shape both kinetic an potential energies, but fix J to (3.6), then IDA PBC coincies with the metho propose in [, 3. 3 Comparison with the Lagrangian Approach The IDA PBC metho may be interprete as a proceure to generate state feeback controllers that transform a given port controlle Hamiltonian system into another port controlle Hamiltonian system with some esire stability properties, e.g., in the case of mechanical systems to transform (.), (.) into (.3), (.6) for some positive efinite inertia matrix M,apotential function V satisfying (.4), an an arbitrary skew symmetric matrix J. Viewe from this perspective, the PDE s (.) an (.) constitute some matching conitions (that ensure that the solutions (q(t),p(t)) of both systems are the same.) A similar approach can be taken proceeing from a Lagrangian perspective. That is, starting from an EL system t ql(q, q) q L(q, q) =G(q)u where L is the Lagrangian, efine as the ifference between the kinetic an the potential energies, we want to fin a static state feeback such that the behavior of the close loop is escribe by the controlle Lagrangian system t ql c (q, q) q L c (q, q) = (3.) where L c is the esire Lagrangian. This problem has been stuie in great generality in [, see also [4. For the case of interest here, that is, restricte to Lagrangians of the form L = q M(q) q V (q), L c = q M c (q) q V c (q), (3.) 7

8 it has been shown in [, 3 that the set of achievable Lagrangians L c is characterize by the solvability of the PDE s 9 { G [ q (M q) MMc q (M c q) q [ q ( q M q) MMc q ( q M c q) } = (3.3) G { q V MMc q V c } = (3.4) which, similarly to (.), (.) of IDA PBC, match the kinetic an the potential energy terms. Recalling that p = M q an efining a matrix M c (q) = MM M (3.5) which is clearly symmetric an positive efinite, we see that (3.4) exactly coincies with (.) setting V c = V. Furthermore, equation (.) reuces to (3.3) if an only if { } J (q, p) =M M [ q (MM p) q (MM p) M M (3.6) Although the expression above can be verifie via irect substitution, a more elegant proof is given in [4 checking that the (energy conserving part of the) port controlle Hamiltonian system (.3), (.6) is equivalent to the EL (mechanical) system [ q ṗ c [ = I n I n [ q H c pc H c where p c = M c q an H c (q, p c )= p c Mc (q)p c + V c (q). An alternative form for J can be obtaine as (J ) (i,j) = p M M [ (M M ) (,i), (M M ) (,j) with [, the stanar Lie bracket an ( ) (i,j) enotes the (i, j) term of a matrix. (This expression was first reporte in [9, although with swappe sub-inices ue to an unfortunate typo.) It has also been shown in [4 that we can a to (3.6) a (skew-symmetric) matrix of the form M M {[ q Q(q) q Q(q)}M M,withQan arbitrary function of q, still preserving the close loop EL structure (3.). This correspons to the close loop Lagrangian L c = q M c q + q Q V c that inclues the gyroscopic terms q Q, which are proven to be intrinsic roughly speaking, this means that they cannot be remove with a change of coorinates. In other wors, IDA-PBC naturally generates a richer class of matchable EL systems of the form (3.), which have not been consiere in the literature of the Lagrangian approach. Recently Bloch et al. [7 have shown that a small ajustment in the controlle Lagrangian approach yiels a metho which is fully equivalent with IDA-PBC as escribe in this paper. Essentially, instea of restricting to systems of the form (3.), they also allow to inclue some external forces into the close-loop Euler-Lagrange system (i.e., the right han sie of (3.) is not necessarily equal to zero, but can be any external force). In this way, it is possible to write any mechanical Hamiltonian system in Euler-Lagrange format by incluing the non-integrable part of the Hamiltonian system as an external (gyroscopic) force into the Euler-Lagrange system. Notice that this metho only works for the class of simple mechanical systems, as presente in this paper. Consiering this larger class of close-loop EL systems [7 establishes that the controlle Lagrangian metho is equivalent to the IDA-PBC metho. 9 In the cite references the PDE (3.3) is expresse using the Christoffel symbols of the secon kin, which leas to a more elegant an compact notation. To avoi introucing aitional notation we prefer to use the form given here. In [7, see also [8, we erive matching conitions expresse in terms of algebraic constraints for potential energy shaping of EL systems in close loop with ynamic EL controllers. 8

9 4 Methos for Solving the Matching PDE s Our erivations above showe that in both approaches, Lagrangian or Hamiltonian, it is necessary to solve some PDE s a task that is, in general, ifficult. We recall now some results reporte in the literature that allows us to simplify this problem. First, it has been shown in [ that if m = n (i.e., the system is uneractuate only by one egree of freeom), an the kinetic energy matrix M epens only on the unactuate coorinate, then the nonlinear PDE (.) can be transforme, with a suitable choice of M an J,intoaset of orinary ifferential equations (ODE s), hence easier to solve. More precisely, if we let k be the inex of the non actuate coorinate, an assume that G = e k q M (i,j) = M (i,j) e k q k Then, restricting M to be only a function of q k, it is possible to show that (.) reuces to the ODE s ( ) (M ) q (,k) = k (M M M (M )M (4.) ) (k,k) q k (,k) The class of systems that satisfies these assumptions is quite common in the control literature incluing the two examples consiere here, namely, the inertia wheel penulum an the classical ball an beam, as well as the cart an penulum which is stuie in [. For the first example the inertia matrix M is constant an we can take M also inepenent of q an J =, obviating the solution of (.). On the other han, we will show later that for the secon example the reuction to orinary ifferential equations is instrumental to solve the problems. Secon, we have shown in Section 3 that if we fix J as (3.6), which ensures that the close loop system amits an EL representation of the form (3.), (3.), then the nonlinear PDE (.) reuces to (3.3). An important contribution of [, see also [3, [, is the proof that all of the solutions of these PDE s may be obtaine by sequentially solving a set of three first orer linear PDE s. It has been show in [4 that the techniques use in [ can be extene to the stuy of the PDE (.), which incorporate the free parameter U k. In fact, this leas to a set of one quaratic an two linear first orer PDEs. The first PDE is quaratic in the sense that it contains terms quaratic in the to be solve-variables, the erivatives however still appear linearly in the equation. Thir, in a series of interesting papers, e.g., [5, 6, Bloch et al. have proven that, in some cases which inclues some well known examples, the solution of these PDE s can actually be obviate. In [4 we have interprete these conitions using the notation employe in this paper. We now briefly summarize these results. Towars this en, we fin it convenient to partition the generalize coorinates as q =[x,θ,withx IR r, θ IR n r. This inuces a natural partition of the open loop inertia matrix as [ M xx M M = xθ M θx As in [5, we now introuce the following assumptions: (i) The Lagrangian is inepenent of the θ coorinates, i.e., they are cyclic variables. In this case the Lagrangian takes the form L(x, q) = q M(x) q V (x). M θθ (ii) The θ coorinates are fully actuate, that is G =[ I. 9

10 (iii) The matrix M has the block M θθ constant, an satisfies xj M x iθ k = xi M x jθ k, i,j =,...,r; k =,...,n r Now, let us take the controlle Lagrangian as L c (x, q) = q M c (x) q V (x). (Notice that, as in [5, we only aim at kinetic energy shaping.) A first immeiate observation is that in this case, uner the assumptions (i), (ii), the matching conition (3.4) reuces to an algebraic equation [I (I MM c ) [ I x V = (4.) A central contribution of [5 is the proof that, for the following particular class of M c, satisfying the so calle, simplifie matching assumptions, [ κ(κ +)M M c = M + xθ (M θθ ) M θx κm xθ κm θx, with κ IR, the PDE s (3.4), (3.3) are automatically satisfie. In [4 we have shown that, for the class of M c consiere in [5, their first matching conition M- is equivalent to the algebraic conition [ [I (I MMc I ) =, which in the light of (4.), clearly obviates the potential energy PDE (3.4). Furthermore, it is also establishe that the matching conitions M- an M-3 of [5 exactly coincie with the PDE (3.3). We shoul recall that solving the PDE s (3.3), (3.4) is just the first step in the esign proceure. Inee, to establish stability using Proposition the matrix M shoul be positive efinite (at least in a neighborhoo of the equilibrium q ), further, V shoul have an isolate local minimum in q. In Theorem 3.4 of [5 it has been shown that relative equilibria, i.e., equilibria of the form (x = x, ẋ =, θ =),ofconservative systems are stable if the Hessian of the total energy is efinite (either positive or negative). Although har to justify from a physical viewpoint, we can in this way use these methos to locally stabilize (relative equilibria of) conservative mechanical systems with a negative efinite close loop inertia matrix. This feature is essential in [5 where the potential energy is not moifie by the control, an will typically have a maximum at the esire equilibrium, hence the kinetic energy shoul also have a maximum at this point. The qualifier conservative is also important because it is not clear to these authors how to hanle the presence of physical amping in this framework. 5 The Inertia Wheel Penulum In this section we apply the preceing esign methoology to the problem of stabilizing the inverte position of the inertia wheel penulum shown in Fig., which consists of a physical penulum with a balance rotor at the en. The motor torque prouces an angular acceleration of the en-mass which generates a coupling torque at the penulum axis. We show that the IDA-PBC provies an affirmative answer to the question of existence of a continuous control law that, for all initial conitions except a set of zero measure, swings up an balances in the upwar position the penulum. We also show that, as usual in PBC [8, the passivity property allows us to replace the state-feeback control by a ynamic output feeback that oes not require the measurements of velocities. These are the simplifie matching assumptions an 4 of [5, respectively. This means that the basin of attraction of our controller is an open ense set in the state space, which is the best one can hope for using a continuous feeback [6.

11 q u q Figure : Inertia Wheel Penulum. 5. Moel The ynamic equations of the evice can be written in stanar form using the EL formulation [7 as [ [ [ [ I + I I θ mgl sin(θ ) + = u I I θ where θ, θ an I, I are the respective angles an moments of inertia of the penulum an isk, m is the penulum mass, L its length, g the gravity constant, an u is the control input torque acting between the isk an penulum. The change of coorinates [ q q [ = θ θ + θ leas to the simplifie escription [ I [ q I q [ m3 sin(q + ) = [ u (5.) where m 3 = mgl. The system can be written in Hamiltonian form (.) with p =[I q,i q, the total energy function (.) an [ I M = I [, G =, V(q )=m 3 (cos q ) The equilibrium to be stabilize is the upwar position with the inertia isk aligne, which correspons to q = q =.3 Caveat The moel of the inertia wheel penulum can be seen either as a system efine over IR 4 or, taking q moulo π, overs IR 3,withS the unit circle. To allow for feeback laws that are not necessarily π perioic on q we aopt the former one for the controller esign. However, to etermine the omain of attraction we look at the system in the cyliner. 3 Since the mass istribution of the inertia isk is symmetric, it may be argue that there is no particular reason for aligning it. In spite of that, we impose this objective to illustrate the generality of the approach.

12 5. Controller Design We will esign our IDA-PBC in three steps. First, we will obtain a state-feeback that shapes the energy to globally stabilize the upwar position, then we a the amping for asymptotic stability by feeback of the passive output. Finally, we show that it is possible to replace the velocity feeback by its irty erivative preserving asymptotic stability. A. Energy Shaping First, notice that the inertia matrix M is inepenent of q, hence, we can take J = an M to be a constant matrix too, which we enote by [ a a M = a a 3 a >, a a 3 >a (5.) The inequalities for the coefficients are impose to ensure M is positive efinite. The only PDE to be solve is then the potential energy PDE given by Equation (.), that is ( ) ( ) a + a V a + a 3 V + = m 3 sin(q ) I q I q This is a trivial linear PDE whose general solution is of the form V (q) = I m 3 cos(q )+Φ(z(q)) (5.3) a + a z(q) = q + γ q (5.4) where Φ is an arbitrary ifferentiable function that we must choose to satisfy the conition (.4) for q =. Wehavealsoefineγ = I (a + a 3 )/(I (a + a )). Some simple calculations show that the necessary conition q V () = is satisfie if an only if Φ(z()) =, while the sufficient conition qv () > will hol if the Hessian of Φ at the origin is positive, an a < a (5.5) These conitions on Φ are satisfie with the choice 4 Φ(z) = k z, where k > represents an ajustable gain. The energy shaping term of the control input (.) is given as u es = (G G) G ( q V M M q V ) = [( ) ( ) a a V a a 3 V + + m 3 sin(q ) I q I q = γ sin(q )+k p (q + γ q ) (5.6) where we have efine γ = a a + a m 3 k p = k [ a a 3 a I (a + a ) We recall that a,a,a 3 shoul satisfy the inequalities (5.) an (5.5), some simple calculations show that these conitions translate into γ >m 3 γ > I γ (5.7) I γ m 3 which, together with k p >, efine the amissible region for the tuning gains. 4 For simplicity we have taken here a quaratic function Φ. Clearly other options, that may be selecte to improve the transient performance, are possible; e.g., a saturate function.

13 B. Damping Injection an Stability Analysis The control law u = u es + u i results in the system [ [ q M = [ M q H ṗ M M p H [ + G u i for which we have Ḣ = p H Gu i. Clearly, without amping injection the origin is a stable equilibrium. To make this equilibrium asymptotically stable we propose to a amping feeing back the new passive output p H G, which can be compute as G p H = a a 3 a [ (a + a 3 )p +(a + a )p = k ( q + γ q ) (5.8) where we efine k = I (a +a ) >, with positivity following from (5.) an (5.5). a a 3 a We are in position to present our first stabilization result, which establishes that for all initial conitions except a set of zero measure the IDA PBC rives the penulum to its upwar position with the isk aligne. Proposition The inertia wheel penulum (5.) in close-loop with the static state feeback IDA PBC u = γ sin(q )+k p (q + γ q ) k v ( q + γ q ) (5.9) where γ,γ satisfy (5.7), k p > is a proportional gain, an k v > is a amping injection gain, has an asymptotically stable equilibrium at zero, with omain of attraction the whole state space minus a set of Lebesgue measure zero. Proof. We first observe that the close loop system has equilibria ( q, ), where q =(jπ,kπ),j,k IN, are the solutions of q V ( q) =. We recall from Fig. that the points q with k even correspon to the esire upwar position, while the ones with o k are with the penulum hanging From Proposition an the erivations above we know that the equilibria corresponing to the upwar position of the penulum are stable. Further, with some basic signal chasing, it is possible to show that in a neighborhoo of zero the trajectories of the close loop system satisfy the (stronger) observability conition: Ḣ (q(t),p(t)). Therefore, Proposition insures that the esire equilibrium is asymptotically stable. Although Proposition provies also an estimate of its omain of attraction, we will show below that asymptotic stability is almost global in the sense of Proposition. Towars this en, we make the important observation that the introuction in the control (5.9) of a non perioic function of q forces us to consier the system in IR 4 instea of S IR 3, an then H is not a proper function of (q, p) an we cannot insure bouneness of trajectories (starting outsie Ω c.) Nevertheless, in the coorinates (q, q,p,p ), with q = q + γ q, the close loop system is inee efine over S IR 3, an the energy function H (q, q,p,p )= p M p + I m 3 a + a [cos(q ) + k q is positive efinite an proper throughout S IR 3. Then, since H = k v k q, we have that all solutions are boune in S IR 3. From the analysis above we know that the zero equilibrium is asymptotically stable. We will now show that the other equilibria are unstable. Inee, the 3

14 linearization of the close loop system at these equilibria has eigenvalues with strictly positive real part an at least one eigenvalue with strictly negative real part. Associate to the latter there is a stable manifol, an trajectories starting in this manifol will converge to the ownwar position. However, it is well known that an s imensional invariant manifol of an n imensional system has Lebesgue measure zero if s<n; see, e.g., [9. Consequently, the set of initial conitions that converges to the ba equilibrium has zero measure. To complete the proof we establish now that trajectories are also boune in IR 4. 5 For, we see that our erivations above have proven that, in S IR 3, all trajectories are boune an ten to one of the equilibria (,,, ) or (π,,, ). The first equilibrium is asymptotically stable an the secon one is hyperbolic. We have furthermore shown that almost any solution converges to the stable equilibrium, being trappe in finite time in a sufficiently small neighborhoo of it, say N. Let us now immerse the cyliner in the Eucliean space IR 4, an consier the sets of IR 4 which correspon to N on the cyliner. If N is chosen sufficiently small, then these sets in IR 4 o not intersect. Since the solutions on the cyliner o not leave N the immersion of the trajectories will not leave the corresponing neighborhoo in IR 4. This completes the proof. C. Output Feeback It is well known that in PBC esigns it is sometimes possible to obviate velocity measurement feeing back instea the irty erivative of positions [5. This feature stems from the fact that for feeback interconnection of passive maps we can replace a constant feeback by a feeback through any positive real transfer function preserving stability. In particular, to implement the amping injection we can use the feeback u i = k vd τd + (q + γ q ) with D = t,ank v,τ > some filter parameters. The important point is that u i is implementable without velocity feeback. This consieration leas us to our final result containe in the proposition below. The proof follows along the lines of the previous proposition, an is only outline for brevity. Proposition 3 Consier the inertia wheel penulum (5.) in close-loop with the ynamic output feeback IDA PBC u = γ sin(q )+k p (q + γ q )+u i where γ,γ satisfy (5.7), k p > is a proportional gain; u i is a amping injection term generate from the irty erivative of the positions as ϑ = τ ϑ + k v τ (q + γ q ) u i = ϑ k v τ (q + γ q ) (5.) with τ,k v >. Then, for all initial conitions except a set of zero measure the penulum converges to its upwar position with all internal signals uniformly boune. Proof. First, notice that from (5.8) an (5.) we get u i = τ u i k v k τ G p H 5 We thank anonymous Reviewer No. 5 for this proof of global bouneness. 4

15 Consequently, if we efine the new energy function W (q, p, u i ) = H + k τ close loop system in port controlle Hamiltonian form as [ q M [ M ṗ = M M k v [ k τ G q W p W u i ui W kv k τ G kv k τ k v u i, we can write the We have Ẇ = k k v (u i ). The proof is complete proceeing as in Proposition. 5.3 Simulation Results We simulate the response of the inertia wheel penulum using the system parameters I =.,I =.,m 3 = an the full state feeback controller with tuning gains γ =3,γ =.. The following plots show the response of the system starting at rest with initial configuration q() = (3, ). Thus the penulum is hanging nearly vertically ownwar. In orer to illustrate the influence of the choice of k p an k v on the transient behavior, we have put together several plots of q (t) anq (t), first only changing k p with k v kept constant at a value of as seen in Fig.. A simple observation shows that larger values of k p slow own the convergence towars the equilibrium point. Leaving k p constant an changing k v from to, the plots in Fig. 3 are obtaine. These plots show that, counter intuitively but not totally unexpecte, the oscillations vanish faster for lower values of k v! Finally, Fig. 4 shows the applie control torque when k p =. ank v =. 3.5 Kp= Kp=. Kp= Kp=. q (ra).5 q (ra) time (secons) time (secons) Figure : Evolution of q (t) (left) an q (t) for ifferent values of k p, letting k v = Kv= q (ra).5 Kv= Kv= q (ra) 3 Kv=.5 Kv= time (secons) Kv= time (secons) Figure 3: Evolution of q (t) (left) an q (t) for ifferent values of k v,ank p =. 5

16 Control Torque (Nm) time (secons) 6 The Ball an Beam System Figure 4: Control signal for k v =, an k p =. In this section we will esign an IDA PBC for the well known ball an beam system epicte in Fig. 5. First, we will prove that for all initial conitions (except a set of zero measure) we rive the beam to the right orientation. Then, we efine a omain of attraction for the zero equilibrium that ensures that the ball remains within the bar. u q q Figure 5: Ball an Beam System. 6. Moel The ynamic behavior of the ball an beam system, uner time scaling an some assumptions on mass constants for simplicity, is escribe by the EL equations q + g sin(q ) q q = (L + q) q +q q q + gq cos(q ) = u (6.) where q,q are the ball position an the bar angle, respectively, 6 an L is the length of the bar. Since we are intereste in ensuring that the ball remains in the bar, we have explicitly inclue L in the moel. We refer the reaer to [4 for further etails on the moel. The Hamiltonian moel (.), (.) is obtaine efining the matrices [ [ M(q )= L + q,g= an the potential energy function V (q) =gq sin(q ). The control objective is to stabilize the ball an beam in its rest position with q = q =. 6 The caveat of Subsection 5. applies here as well. Inee, the moel can be seen either as a system efine over IR 4 or over IR S IR. 6

17 6. Controller Design For peagogical reasons, we will split now the esign of the IDA-PBC into kinetic energy shaping, potential energy shaping, asymptotic stability an transient performance analysis. A. Kinetic Energy Shaping First, notice that M is a function of q only, hence it is reasonable to propose M of the form (5.), but with the coefficients a,a,a 3 functions (to be efine) of q also. We will enote [ J (q, p) = j(q, p) j(q, p) with the function j also to be efine. Since this is the scenario of [ iscusse in Section 4 we apply irectly the formula (4.), with G = [, to erive the ODE s for M. This leas to the system of ODE s a (q ) q = a (q ) q = q a (L + q ) a q a a 3 (L + q ) a that has to be solve for a,a, an we view a 3 as a free parameter. To obtain two equations with two unknowns, we fix a 3 = a a, hence we can easily get explicit solutions of the ODE s as a = (κ + q ), a = L + q (6.) with κ a free integration constant that must be chosen to ensure that M is positive efinite. It is clear that a > for all κ>, furthermore the eterminant of M results a 3 a a = ( L + q )( q +κ L ) which is positive, as esire, for all κ> L. For simplicity, we will take κ = L in the sequel. The resulting M is then [ (L + q M =(L + q) ) / (L + q ) (6.3) The kinetic energy shaping is complete evaluating j from (.) an the M calculate above [ j = q p (L + q) / p (6.4) B. Potential Energy Shaping Once we have etermine the esire inertia an interconnection matrices we procee now to efine the close loop potential energy from the solution of (.), which in this case is expresse as a (q ) V (q)+ a (q ) V q L + q (q) =g sin(q ) (6.5) q 7

18 Substituting the solutions obtaine for a an a in (6.) yiels (L + q ) V + V = g sin(q ) q q We will solve this PDE using the symbolic language Maple. To help Maple fin a suitable solution we introuce the change of coorinates x = L q,x = q, the PDE can then be rewritten as a +x V (x)+b V (x) =sin(x ) x x g where we have efine a = an b = g. The require Maple commans are >equ:=a*iff(v(x,x),x)*sqrt(+x\^)+b*iff(v(x,x),x)=sin(x); >sol:=psolve(equ); which prouces (after transforming back the coorinates) V (q) = g cos(q )+Φ(z(q)) z(q) = q ( q ) arcsinh L with Φ an arbitrary ifferentiable function of z. This function must be chosen to ensure the equilibrium assignment, i.e., to satisfy (.4) with q =. Towars this en, let us evaluate the graient [ q V (q) = (L +q ) zφ(z(q)) g sin(q )+ z Φ(z(q)) Clearly, a necessary an sufficient conition to assign the zero equilibrium is z Φ(z()) =. Remark that, inepenently of the choice of Φ, (or the integration constant κ), the close loop has other equilibrium points. Inee, since z() = an sinh( ) is an increasing first thir quarant function, q V ( q) = has a countable number of roots given by q =(Lsinh( iπ),iπ), with i IN. On the other han, we have that the only equilibrium with q L is the zero equilibrium. Of course, this property is practically meaningful only if we can show that the trajectories q (t) L for all t, which will be one later. It is important to recall that, the equilibria q = iπ with i even correspon to the bar in its original orientation, while for i o the bar is rotate 8. We will prove now that, with a suitable choice of Φ, we can make the former equilibria stable an the latter ones unstable. To stuy the stability of the equilibria we check positivity of the Hessian of V,evaluatea q, which yiels qv ( q) = (L + q ) zφ(z( q)) (L + q zφ(z( q)) ) (L + q zφ(z( q)) g cos( q )+ zφ(z( q)) ) Taking into account that cos( q )=±, epening on whether the bar is at its original position or rotate 8, the eterminant of this matrix is ± g (L + q ) Φ(z( q)), an stability (instability) of the equilibrium is etermine by the sign of Φ(z( q)). We then choose Φ(z) = kp z,withk p >. In this way, the equilibria corresponing to the bar in its original orientation will be stable, while the ones with the rotate bar are unstable. The new potential energy takes the final form [ q arcsinh (6.6) V = g[ cos(q ) + k p ( q ) L where we have ae a constant to shift the minimum to zero. See Fig. 6. 8

19 Figure 6: Level curves of V (q) aroun the origin for k p =.5 (left) an k p =. (right). C. Asymptotic Stability Analysis To compute the final control law we first etermine the energy shaping term u es from (.), which in this case takes the form u es = q H (M M ) (,) q H (M M ) (,) q H +(J M ) (,)p +(J M ) (,)p Replacing the functions erive above for M an j, an after some straightforwar calculations, we obtain the expression q u es = [ (L + q ) L + q p + p p + p L + q + ξ(q) (6.7) where ξ(q) = L gq cos q g (L + q )sinq k + q ( p q ( q ) ) arcsinh L The controller esign is complete with the amping injection term (.7), which yiels ( ) u i = k v L + q p L + q p (6.9) It is important to unerscore that, in spite of its apparent complexity, the controller is globally efine an its highest egree is quaratic. This is an important property of the control, since saturation shoul be avoie in all practical applications. Also, the role of the tuning parameters has a clear interpretation, namely: k p is a bona fie proportional gain in position, as it multiplies terms that grow linearly in q, an k v injects amping along a specifie irection of velocities. Commissioning of the controller is simplifie by this feature, as will be illustrate later in the simulations. We give now a first result on asymptotic stability, similar to the one obtaine for the inertia wheel penulum, but with the funamental ifference that convergence to zero of the ball is only local. A more practical result, that takes into account the finite length of the bar, will be reporte in the next subsection. Proposition 4 Consier the ball an beam moel (6.) in close-loop with the static state feeback IDA PBC u = u es + u i, with (6.7) (6.9), an k p,k v >. Then, the origin is an asymptotically stable equilibrium with omain of attraction Ω c where Ω c = {(q, p) IR 4 H (q, p) c}, withh 9

20 of the form (.3), M given by (6.3), an V an c efine by (6.6), (.8), respectively. Further, for all initial conitions, except a set of zero measure, the beam will asymptotically converge to its zero orientation position, i.e., for almost all trajectories lim t q (t) =. Proof. Stability of the zero equilibrium follows verifying the conitions of Proposition, however, to establish asymptotic stability, instea of checking etectability, we invoke Matrosov s theorem see, e.g., Theorem 5.5 of [5. For which we pick an auxiliary function W (q) =q + q whose erivative along the trajectories of the close loop is Ẇ = p + L + q p Now, Ḣ = if an only if p = L + q p see (6.9), which (as q ranges in (, )) efines a triangular sector insie the first thir quarant of the plane p p. On the other han, the sector where Ẇ = lives in the secon fourth quarant, consequently, we have that Ẇ is a non vanishing efinite function at the set {(q, p) Ḣ =}, an the conitions of the theorem are satisfie (in a neighborhoo of zero) with H the require Lyapunov function. Similarly to the proof of Proposition, to prove the almost convergence property state above we look at the system in IR S IR to establish bouneness of all trajectories. To immerse the system in this cyliner we change the first coorinate to q = q ( q ) arcsinh (6.) L an efine accoringly the energy function H ( q,q,p) Ṽ ( q,q ) = p M ( q,q )p + Ṽ( q,q ) (6.) = g( cos q )+ k p q (6.) which is proper an positive efinite in IR S IR, an with negative semiefinite erivative. This proves bouneness of all trajectories in IR S IR. The remaining of the proof mimics the one one for the inertia wheel penulum, noting that for all stable equilibria we have the esire asymptotic behavior for q, while the other equilibria are unstable. Before closing this section we note that in this example it is not possible to replace the measurements of velocities by its irty erivative approximation as one for the inertia wheel. This stems from the fact that in the ball an beam controller, besies the amping injection term, the energy shaping term epens explicitly on p. D. Transient Performance It has been state in Proposition 4 that in close loop the origin is an asymptotically stable equilibrium with Lyapunov function the esire total energy. In this subsection we will refine this analysis, stuying the effect of the tuning parameter k p on the size of the omain of attraction, an explicitly quantifying a set of initial conitions such that the ball remains all the time in the bar, that is, q (t) L for all t.

21 First, we note that as H ecreases an the kinetic energy is non negative, we have that V (q(t)) H (q(),p()), hence the sub level sets of V are invariant sets for q(t). Further, if we can show that the kinetic energy is boune, then the boune sets provie an estimate of the omain of attraction. To stuy these sets we fin convenient to work in the coorinates ( q,q,p,p ), where q was introuce in (6.). 7 In these coorinates the potential energy function becomes (6.) which has the same analytical expression as the total energy of the simple penulum an the associate sub level sets, i.e., {( q,q ) Ṽ( q,q ) c}, are of the form shown in Fig 7. We are intereste here in the boune connecte components that contain the origin, that we will enote Ξ c. The following basic lemma will be instrumental in the sequel. Lemma The set Ξ c is boune if an only if c<g. Proof. The fact that all elements q in Ξ c are boune for all finite c is obvious, as kp q c g( cos q ). Hence, we can concentrate only on bouneness of q. ( ) We have the following implication Ṽ ( q,q ) <c cos(q ) > c g > where we have use the positivity of k p in the first right han sie inequality an c<g to obtain the secon one. Note that the strict inequality exclues an interval aroun q = π an q = π from Ξ c. This proves that Ξ c {( q,q ) q ( π, π)}, an consequently q is boune. ( ) Necessity will be prove by contraiction. For, we suppose that c g. Then, it is clear that Ξ c {( q,q ) q =}, which is an unboune set, an consequently q is unboune. Figure 7: Level curves of Ṽ ( q,q ). We are in position to present the main result of this section. To simplify the notation we will use ( ) o to enote the value of the functions at t =. Proposition 5 Consier the ball an beam moel (6.) in close-loop with the static state feeback IDA PBC u = u es + u i, with (6.7) (6.9), an k v >. (i) We can compute a constant kp M all k p kp M,theset >, function of the initial conitions (q o,p o ), such that for {(q, p) p M (q)p + g( cos q ) < g} (6.3) 7 Note that the coorinate mapping (q,q ) ( q,q ) efines a global iffeomorphism, an recall that bouneness of sub level sets is invariant uner the action of iffeomorphisms.