Simultaneous Interconnection and Damping Assignment Passivity Based Control: Two Practical Examples

Transcription

1 Simultaneous Interconnection and Damping Assignment Passivity Based Control: Two Practical Examples Carles Batlle, Arnau Dòria-Cerezo Gerardo Espinosa-Pérez MA4, DEE and IOC, UPC DEPFI UNAM EPSEVG, Av. V. Balaguer s/n Apartado Postal Vilanova i la Geltrú, , México D.F. Spain México (carles.batlle@upc.edu,arnau.doria@upc.edu) gerardoe@servidor.unam.mx Romeo Ortega Laboratoire des Signaux et Systèmes CNRS-SUPÉLEC Gif-sur-Yvette, France Romeo.Ortega@lss.supelec.fr January 13, 2006 Abstract We argue in this paper that the standard two stage procedure used in Interconnection and Damping Assignment Passivity Based Control (IDA PBC) consisting of splitting the control action into the sum of energy shaping and damping injection terms is not without loss of generality, and effectively reduces the set of systems that can be stabilized with IDA PBC. To overcome this problem we suggest in this paper to carry out simultaneously both stages and refer to this variation of the method as SIDA PBC. To illustrate the application of SIDA PBC we consider two practically important examples. First, we show that torque and rotor flux regulation of the induction motor cannot be solved with two stage IDA PBC. It is, however, solvable with SIDA PBC. Second, we prove that with SIDA PBC we can shape the total energy of the full (electrical and mechanical) dynamics of a doubly fed induction generator used in power flow regulation tasks, while with two stage IDA PBC only the electrical energy can be shaped. Simulation results of these examples are presented to illustrate the performance improvement obtained with SIDA PBC. Keywords: Nonlinear control, Passivity-based control, Energy-shaping, Induction motors. 1 Introduction Passivity based control (PBC) is a generic name given to a family of controller design techniques that achieves system stabilization via the route of passivation, that is, rendering the closed loop system passive with a desired storage function (that usually qualifies as a Lyapunov function for the stability analysis.) If the passivity property turns out to be output strict, with an output signal with respect to which the system is detectable, then asymptotic stability is ensured. See the monographs 8, 16], and 9] for a recent survey. As is well known, 16], a passive system can be rendered strictly passive simply adding a negative feedback loop around the passive output an action sometimes called L g V control 14]. For this reason, it has been found convenient in some applications, in particular for mechanical systems 15], 11], to split the control action into the sum of two terms, an energy shaping term which, as indicated by its name, is responsible of assigning the desired energy/storage function to the passive map, and a second L g V term that injects damping for asymptotic stability. The purpose of this paper is to bring to the readers attention the fact that splitting the control action

2 in this way is not without loss of generality, and effectively reduces the set of problems that can be solved via PBC. This assertion is, of course, not surprising since it is clear that, to achieve strict passivity, the procedure described above is just one of many other possible ways. Our point is illustrated with the IDA PBC design methodology proposed in 7]. To enlarge the set of systems that can be stabilized via IDA PBC we suggest to carry out simultaneously the energy shaping and the damping injection stages and refer to this variation of the method as SIDA PBC. We illustrate the application of SIDA PBC with two practically important examples. First, we show that the fundamental problem of induction motor torque and rotor flux regulation cannot be solved with two stage IDA PBC. It is, however, solvable with SIDA PBC. Second, we prove that with SIDA PBC we can shape the total energy of the full (electrical and mechanical) dynamics of a doubly fed induction generator used in power flow regulation tasks while, as reported in 1], with two stage IDA PBC only the electrical energy could be shaped. Simulation results of these examples are presented to illustrate the performance improvement obtained with SIDA PBC. 2 Simultaneous Energy Shaping and Damping Injection Control We consider the problem of stabilization of an equilibrium point for nonlinear systems of the form ẋ = f(x, t) + g(x)u (1) where x R n is the state vector, u R m, m < n is the control action and g(x) is assumed full rank. In two stage IDA PBC this objective is achieved as follows, see 7, 11] for further details. First, decompose the control signal in two terms u = u es + u di (2) where u es is responsible of the energy shaping stage and u di injects the damping. Second, solve the key matching equation 1 g (x)f(x, t) = g (x)j d (x, t) H d (3) for some functions J d : R n R R n n, H d : R n R, satisfying the skew symmetry condition for the interconnection matrix and the equilibrium assignment condition for the desired total stored energy J d (x, t) + J d (x, t) = 0, (4) x = arg min H d (x) (5) with x R n the equilibrium to be stabilized 2 and g (x) R (n m) n a full rank left annihilator of g(x), that is, g (x)g(x) = 0 and rank g (x) = n m. As shown in 7] system (1) in closed-loop with the control (2), with u es = g (x)g(x)] 1 g (x){j d (x, t) H d f(x, t)}. (6) yields a port controlled Hamiltonian (PCH) system of the form ẋ = J d (x, t) H d + g(x)u di y = g (x) H d. (7) The system (7) without damping injection term is conservative, i.e., Ḣ d = 0, with x a stable equilibrium (with Lyapunov function H d (x)). To add dissipation we feedback the passive output y, for instance, with u di = K di y, K di = K di > 0, 1 All vectors in the paper are column vectors, even the gradient of a scalar function denoted ( ) = ( ). When clear from the context the subindex will be omitted. 2 That is, x is a member of the set { x R n g ( x)f( x, t) = 0, t R}.

3 to finally obtain the PCH system with dissipation ẋ = J d (x, t) R d (x)] H d + g(x)v where the damping matrix R d (x) = Rd (x) 0 is defined by y = g (x) H d. (8) R d (x) = g(x)k di g (x), and we have added a signal v to (2) to define the port variables. Since the new closed loop system (with v = 0) satisfies Ḣ d = y K di y, it is easy to prove (e.g., Lemma of 16]) that the equilibrium x will now be asymptotically stable if it is detectable from y, i.e., if the implication (y(t) 0 lim t x(t) = x ) is true. Obviously, the key for the success of IDA PBC is the solution of the matching equation (3). With the motivation of enlarging the class of systems for which this equation is solvable we propose in this paper to avoid the decomposition of the control into energy shaping and damping injection terms. Instead, we suggest to carry out simultaneously both stages and replace (3), with the SIDA PBC matching equations to replace the constraint (4) by the strictly weaker condition and define the control as g (x)f(x, t) = g (x)f d (x, t) H d, (9) F d (x, t) + F d (x, t) 0, (10) u = g (x)g(x)] 1 g (x){f d (x, t) H d f(x, t)}. Since the set of skew symmetric matrices is strictly contained in the set of matrices with negative semi definite symmetric part, it is clear that the set of functions {f(x, t), g(x)} for which (3) subject to the constraint (4) is solvable is strictly smaller than the set for which (9), subject to (10), is solvable. Remark 1 There exists several techniques to solve the matching equations (3) (resp., (9)), with two extremes being the purely algebraic approach of 3] and the PDE approach of 7]. In the former H d (x) is a priori fixed, which makes (3) (resp., (9)) an algebraic equation that is solved for J d (x, t) (resp., F d (x, t)) subject to the constraint (4) (resp., (10)). On the other hand, in the latter J d (x, t) (resp., F d (x, t)) is fixed making (3) (resp., (9)) a PDE that is solved for H d (x). We refer the interested reader to 9] for a detailed discussion on these, as well as other, methods of solution of the matching equations. In this paper we will adopt the algebraic approach. Remark 2 Similarly to IDA PBC, application of SIDA PBC also yields a closed loop PCH system of the form (8) with J d (x, t) = 1 2 F d(x, t) F d (x, t)], R d (x, t) = 1 2 F d(x, t) + F d (x, t)]. Remark 3 To make IDA PBC applicable to non autonomous systems, which will be required in the induction motor application, we have presented above a slight variation of the method. Notice that the matrices J d and R d may depend explicitly on time. Clearly, their skew symmetry and non negativity properties must now hold uniformly in time as well. 3 Induction Motor Control via SIDA PBC In this section we will show that the problem of output feedback torque control of induction motors with quadratic in the increments desired energy function is not solvable via two stage IDA PBC. That is, it is not possible to solve (3) subject to (4). On the other hand, we will find a solution of (9) subject to (10), hence the problem is solvable with SIDA PBC. An interesting feature of our SIDA PBC is that, in contrast with the large majority of controllers proposed for the induction motor where signal convergence is inferred from the stability analysis of some kind of error dynamics 2, 8,?], we establish here (Lyapunov) stability of a given equilibrium that generates the desired torque and rotor flux amplitude.

4 3.1 Motor Model, Control Objective and Equilibria The standard three-phase induction motor represented with a two-phase model defined in an arbitrary reference frame, which rotates at an arbitrary speed ω s R, is given by 5] in which ẋ 12 = γi 2 + (n p ω + u 3 )J ] x 12 + α 1 (I 2 n p ωj ) x 34 + α 2 u 12 (11) ẋ 34 = ( 1 I 2 + J u 3 )x 34 + L sr x 12 (12) ω = α 3 x 12J x 34 τ L J m (13) I 2 = ] 0 1, J = 1 0 ] = J, x 12 R 2 are the stator currents, x 34 R 2 the rotor fluxes, ω R the rotor speed, u 12 R 2 are the stator voltages, τ L R is the load torque and u 3 := ω s n p ω. The parameters, all positive, are defined as γ := Rs L s σ + L2 sr σl s L r, σ := 1 L2 sr L s L r, α 1 := Lsr σl s L r, α 2 := 1 σl s, α 3 := nplsr J m L r, := Lr R r with L s, L r the windings inductances, R s, R r the windings resistances and L sr the mutual inductance, n p the number of pole pairs and J m the rotor moment of inertia. As first pointed out in the control literature in 10], the signal u 3 (known in the drives literature as slip frequency) effectively acts as an additional control input. Below, we will select u 3 to transform the periodic orbits of the system into constant equilibria. We are interested in this paper in the problem of regulation of the motor torque and the rotor flux amplitude ] ] y1 Jm α y = = 3 x 12J x 34, (14) x 34 y 2 to some constant desired values y = col(y 1, y 2 ), where is the Euclidean norm. The following important practical restriction of the induction motor control problem has to be imposed: (OF) The only signals available for measurement are x 12 and ω. To solve this problem using (S)IDA PBC it is necessary to express the control objective in terms of a desired equilibrium. As is well known 6], the zero dynamics of the induction motor is periodic, a fact that is clearly shown computing the angular speed of the rotor flux ρ = R r y 1 n p y2 2 u 3, where ρ := arctan x 4 x 3 is the rotor flux angle. From this relation it is clear that, if u 3 is fixed to a constant, say ū 3, and y = y, x 34 is a vector of constant amplitude rotating at speed ρ(t) = ( R r y 1 n p ū y2 2 3 )t + ρ(0). Therefore, the only choice of u 3 for which a (constant) equilibrium compatible with the desired outputs y exists is u 3 = u 3 := R r y 1. (15) n p From this observation and (12) we have the following simple lemma whose proof is obtained via direct substitution. Lemma 1 Consider the induction motor model (11) (13) with output functions (14) and u 3 = u 3. Then, the set of assignable equilibrium points, denoted col( x 12, x 34, ω) R 5, which are compatible with the desired outputs y is defined by with ω arbitrary. x 12 = 1 L sr y Lr y 1 n p y2 2 1 L r n p y 1 y 2 2 ] x 34 x 34 = y 2 (16)

5 Among the set of assignable equilibria defined above we select, for the electrical coordinates, the one that ensures field orientation 5] and, defining x := col(x 12, x 34 ), denote it x := col( y 2 L r y 1,, y 2, 0). (17) L sr n p L sr y 2 Remark 4 From (13) and (14) we see that to operate the system in equilibrium, y 1 = τ L hence, to define the desired equilibrium the load torque needs to be known. In practical applications, an outer loop PI control around the velocity error is usually added. The output of the integrator, on one hand, provides an estimate of τ L while, on the other hand, ensures that speed also converges to the desired value as shown via simulations in Section Algebraic (S)IDA PBC Problem Formulations As indicated in Remark 1, in this paper we will adopt the algebraic approach to solve the matching equations. Fixing f(x, t), g (x) and H d (x), the two algebraic equations, (3) and (9), are the same and the difference between IDA PBC and SIDA PBC resides only on the constraints imposed on the matrices J d (x, t) and F d (x, t). To simplify the problem formulation we will use the generic symbol F (x, t) to denote either J d (x, t) or F d (x, t), the distinction will be made imposing either F (x, t) + F (x, t) = 0 or F (x, t) + F (x, t) 0, respectively. Since we are interested here in torque control, and this is only defined by the stator currents and the rotor fluxes, its regulation can be achieved applying IDA PBC to the electrical subsystem only. Boundedness of ω will be established in a subsequent analysis. Although with IDA PBC it is possible, in principle, to assign an arbitrary energy function to the electrical subsystem, we will consider for simplicity a quadratic in errors form H d (x) = 1 2 (x x ) P (x x ), (18) with P = P > 0 a matrix to be determined. As indicated in Remark 1, fixing H d (x) transforms the matching equation (3) into a set of algebraic equations see also 13] for application of this, so called Algebraic IDA PBC, to general electro mechanical systems. The electrical subsystem (11), (12) with u 3 = u 3 can be written in the form (1) with u = u 12, and γi2 + (n p ω(t) + u 3 )J ] x 12 + α 1 (I 2 n p ω(t)j ) x 34 f(x, t) = ( 1 I 2 + J u 3 )x 34 + Lsr x 12 ], g = α 2 I ]. Notice that we are treating the dependence of the dynamics on angular speed ω as a time dependence. Since g = I 2 ], the matching equations (3) and (9) concern only the third and fourth rows of f(x, t) and they take the form ( 1 I 2 + J u 3 )x 34 + L sr x 12 = F 3 (x, t) F 4 (x, t)]p (x x ), (19) where, to simplify the notation, we partition F (x, t) into 2 2 sub-matrices as ] F1 (x, t) F F (x, t) = 2 (x, t). (20) F 3 (x, t) F 4 (x, t) The output feedback condition (OF) imposes an additional constraint that involves now the first and second rows of f(x, t). Indeed, from (6) we see that the control can be written as α 2 u 12 = γi 2 + (n p ω(t) + u 3 )J ] x 12 α 1 (I 2 n p ω(t)j ) x 34 + F 1 (x, t) F 2 (x, t)]p (x x ). Factoring the components that depend on the unmeasurable quantity x 34 we can express u 12 as u 12 = û 12 (x 12, ω) + 1 α 2 S(x, t)x 34 where we have defined S(x, t) := α 1 n p ω(t)j I 2 ] + F 1 (x, t)p 2 + F 2 (x, t)p 3,

6 partitioned P as P1 P P := 2 P2 P 3 ], P i R 2 2, i = 1, 2, 3, and defined a function, û 12 (x 12, ω), whose expression is given in (29). It is clear that, to verify the output feedback condition (OF), S(x, t) has to be set to zero, that is α 1 n p ω(t)j I 2 ] + F 1 (x, t)p 2 + F 2 (x, t)p 3 = 0. (21) (S)IDA PBC Problems. Find matrices F (x, t) and P = P constraint that > 0 satisfying (19) (21) with the additional (Energy shaping) F (x, t) + F (x, t) = 0, (22) or the strictly weaker (Simultaneous energy shaping and damping injection) F (x, t) + F (x, t) 0. (23) 3.3 Solvability of the (S)IDA PBC Problems Proposition 1 The energy shaping problem is not solvable. damping injection one is solvable. However, the simultaneous energy shaping and Proof. First, we write the matching equation (19) in terms of the errors ( ) L sr 1 (x 12 x 12 ) I 2 + u 3 J (x 34 x 34 ) = F 3 (x, t) F 4 (x, t)]p (x x ), which will be satisfied if and only if F 3 (x, t) F 4 (x, t)]p = Lsr I 2 ( )] 1 I 2 + u 3 J, (24) Since P is full rank and the right hand side of the equation is constant, we conclude that F 3 and F 4 should also be constant. (To underscore the nature of the F i matrices we will explicitly write their arguments in the sequel.) Let us consider first the energy shaping problem. From (20) and (22) we have that F 2 = F 3. Replacing the latter in (21) it is obtained that F 1 (x, t)p 2 F 3 P 3 = α 1 I 2 n p ω(t)j ] (25) On the other hand, from the first two columns of (24) it follows that ( ) Lsr F 3 = I 2 F 4 P2 P1 1 (26) Substitution of (26) into (25) leads to F 1 (x, t)p 2 P 1 1 ( ) Lsr I 2 P 2 F4 P 3 = α 1 I 2 α 1 n p ω(t)j. (27) Invoking again (22) we have that F 1 (x, t) must be skew-symmetric. Since, furthermore, it is only a function of t we can, without loss of generality, express it in the form F 1 (t) = β 1 (t)j + β 2 J, where β 2 R. Looking at the t dependent terms we get β 1 (t)j P 2 + α 1 n p ω(t)j = 0,

7 which can be achieved only if P 2 = λi 2, with λ R, λ 0, and β 1 (t) = λ 1 α 1 n p ω(t). The constant part of (27), considering that P 2 = λi 2, reduces to ( ) λβ 2 J P1 1 Lsr I 2 λf4 P 3 = α 1 I 2 which using the fact that P 3 is full rank can be expressed as F4 = GP3 1, where we have defined the constant matrix G := 1 ] Lsr P 3 + P 1 (α 1 I 2 λβ 2 J ) λ Finally, since F 4 must also be skew symmetric, we have that G = P 1 3 ( G )P 3. Consequently, G must be similar to G, and both necessarily have the same eigenvalues. A necessary condition for the latter is that trace(g) = 0, that is not satisfied because trace(g) = 1 L sr trace(p 3 ) +α 1 trace(p 1 ), λ }{{}}{{} >0 >0 which is different from zero. This completes the proof of the first claim. We will now prove that if we consider the larger class of matrices (23) the problem is indeed solvable, and actually give a very simple explicit expression for F (x) and P. For, we set P 2 = 0, thus it is easy to see that F 2 (t) = α 1 I 2 n p ω(t)j ] P 1 3 F 3 = L sr P1 1 F 4 = ( 1 I 2 + u 3 J ) P 1 3 with F 1 (x, t), P 1, P 3 free, provide a solution to (24) and (21). It only remains to establish (23). For, we fix P 1 = Lsr I 2, P 3 = α 1 I 2 and F 1 (t) = K(ω(t)), with K(ω(t)) = K (ω(t)) > 0, then ] K(ω(t)) I2 ( n p ω(t)j ) F (t) = I 2 α I 2 + u 3 J A simple Schur complement analysis shows that F (t) + F (t) < 0 if and only if L sr K(ω(t)) > 1 L s L r L ] (n p ω(t)) 2 I 2. (28) sr 3.4 Proposed Controller and Stability Analysis Once the solvability of the SIDA PBC problem has been established, the final part of the design is the explicit definition of the resulting controller and the assessment of its stability properties. This is summarized in the proposition below whose proof follows immediately from analysis of the closed loop electrical sub system ẋ = F (t) H d and the mechanical dynamics (13), where H d is given by (18) and F (t) + F (t) < 0. Proposition 2 Consider the induction motor model (11) (13) with outputs to be regulated given by (14). Assume that A.1 The measurable states are the stator currents x 12 and the rotor speed ω. A.2 All the motor parameters are known.

8 A.3 The load torque is constant and known. Fix the desired equilibrium to be stabilized as (17), with y = col (τ L, y 2 ), y 2 > 0, and set u 3 = R r y 1 n p y2 2 u 12 = û 12 (x 12, ω) with and û 12 (x 12, ω) = 1 α 2 γi 2 + (n p ω + u 3 )J ] x 12 α 1 α 2 (I 2 n p ωj ) x 34 L sr α 2 K(ω)(x 12 x 12 )] (29) and K(ω) satisfying (28). Then, x is a globally exponentially stable equilibrium of the x subsystem with Lyapunov function H d (x) = L sr 2 x 12 x α 1 2 x 34 x 34 2, that satisfies Ḣd κh d, for some κ > 0. Consequently, for all initial conditions, lim x(t) = x, t lim y(t) = y, t exponentially fast. Furthermore, ω remains bounded and converges to some constant value, that is, lim t ω(t) = ω. Remark 5 The resulting SIDA PBC is a simple output feedback scheme that ensures global exponential stability of the desired equilibrium, hence exponential convergence of the generated torque and the rotor flux norm to their desired (constant) values. To the best of our knowledge, this is the first output feedback scheme that ensures such strong stability properties for this system. Although, for the establishment of the theoretical result it is assumed that τ L is known, as indicated in Remark 4 and further validated in the simulations below, in practical applications a PI speed loop provides an adequate estimate of τ L. A scheme that removes this assumption has recently been proposed in 4]. 3.5 Simulation Results The performance of the proposed SIDA PBC was investigated by simulations considering two experiments described below. The considered motor parameters, taken from 10], were n p = 1, L s = 84mH, L r = 85.2mH, L sr = 81.3mH, R s = 0.687Ω and R r = 0.842Ω, with an unitary rotor moment of inertia. Regarding the controller parameters, following field oriented ideas, the rotor flux equilibrium value was set to x 34 = col (β, 0) with β = 2, while x 12 where computed according to (16). In order to satisfy condition (28), it was defined K(ω) = k(ω)i 2 with L sr k(ω) = T 2 (L s L r L 2 r ω ] sr) In a first experiment the motor was initially at standstill with a zero load torque. At startup, the load torque was set to τ L = 20Nm and this value was maintained until t = 40sec when a new step in this variable was introduced changing to τ L = 40Nm. Figure 1 shows the behavior of the stator currents where it can be noticed how, according to the field oriented approach, one of the stator currents remains (almost) constant while the second one is dedicated to produce the required generated torque. In this sense, in Figure 2 it can be observed how the rotor flux is aligned with the reference frame since one of the components equals β while the other is zero. The internal stability of the closed loop system is illustrated in Figure 3 where the rotor speed is presented. As expected, besides its boundedness, it can be noticed that when the load torque is increased, this variable decreases. In Figure 4 the main objective of the proposed controller is depicted. Here it is shown how the generated torque regulation objective, both before and after the load torque change, is achieved. Figure 5 shows the boundedness of the control (stator voltages) inputs. The second experiment was aimed to illustrate the claim stated in Remark 4. In this sense, the control input u 3 was set to u 3 = û 3 = R r ŷ 1 y 2 2 where the estimate of the load torque is obtained as the output of a PI controller, defined over the speed error between the actual and the desired velocities, of the form t ŷ 1 (t) = k p (ω(t) ω ) + k i (ω(s) ω ) ds 0

9 Figure 6 shows the rotor speed behavior when the desired velocity is (initially) ω = 100rpm and at t = 50sec it is changed to ω = 150rpm. In this simulation it was considered τ L = 10Nm, k i =.1 and k p = 1. All the other parameters were the same than in the first experiment stator currents amp] time sec] Figure 1: Stator currents 4 Total Energy Shaping of a Doubly Fed Induction Generator Doubly-fed induction machines (DFIM) have been proposed in the literature, among other applications, for high performance energy storage systems, wind-turbine generators or hybrid engines see 12] for an extended discussion. In 1] we considered a DFIM controlled through the rotor voltage and connected to a variable local load that acts as an energy switching device between a local energy storing element (a flywheel) and the electrical power network. The control objective is to change the direction of the power flow (towards or from the flywheel) depending on the load demand. This is achieved commuting between two different controllers that drive the system towards two given steady state regimes. The equilibria associated to these regimes is stabilized with IDA PBCs that, similarly to the induction motor controller proposed in the previous section, shape the electrical energy, treating the mechanical dynamics as a cascaded subsystem. The nested loop architecture, with an inner loop to control the electrical subsystem and an outer loop (usually a simple PI) to control the mechanical variables, is prevalent in classical electromechanical systems applications, where it is justified invoking time scale separation arguments. Intrinsic to the nested loop configuration is the fact that the time response of the mechanical subsystem is subordinated to the electrical transient. This may lead to below par performances in small stand alone DFIM based generating units where a fast response of the mechanical speed is needed to ensure efficient control of the power flow and, furthermore, the mechanical and electrical time constants may be close. The purpose of this section is to show that using SIDA PBC it is possible to shape the energy function of the complete system dynamics, resulting in a controller with improved power flow regulation performance. To the best of our knowledge, this is the first control algorithm for this class of systems that provides for this additional degree of freedom.

10 rotor fluxes wb] time sec] Figure 2: Rotor fluxes 4.1 Overall System Model and Equilibria We consider the configuration for the DFIM studied in 1] to which we refer the reader for additional details. Here, we will concentrate on the problem of stabilization of equilibria of the overall system. Before presenting the systems model a word on notation is in order. We chose a representation in the dq framework rotating at the (constant) angular speed of the AC source (ω s ), and state vector z = col(λ s, λ r, J m ω) R 5, where λ s, λ r R 2 are the stator and rotor fluxes, respectively, ω R is the mechanical speed, and J m is the machine inertia. To simplify the derivations we will find convenient to work with a mixed notation, writing some of the equations using also the coordinates χ = col(i s, i r, ω) R 5 5, where i s, i r R 2 are the stator and rotor currents, respectively. The relation between z and χ is given by with L = L si 2 L sr I 2 O 2 1 L sr I 2 L r I 2 O 2 1 O 1 2 O 1 2 J m z = Lχ, (30) = L > 0 the generalized inductance matrix, where L s, L r, L sr are the DFIM inductances. The energy function of the overall system is H(z) = 1 2 z L 1 z, and the model given by ż = J(i s, ω) R] H + v s O 2 1 τ L + O 2 2 I 2 O 2 2 u (31) where v s R 2, τ L R are, respectively, the stator voltage and the external mechanical torque, which are constant,

11 rotor speed rpm] time sec] Figure 3: Rotor speed and u R 2 are the rotor voltages that act as control inputs. 3 The structure and damping matrices are ω s L s J ω s L sr J O 2 1 J(i s, ω) := ω s L sr J (ω s ω)l r J L sr J i s = J (i s, ω), R := R si 2 O 2 2 O 2 1 O 2 2 R r I 2 O 2 1 > 0, O 1 2 L sr i s J 0 O 1 2 O 1 2 B r where R s, R s are DFIM resistances and B r is the friction coefficient. All parameters, including B r, are strictly positive. Clearly, the fixed point equations for (31) are given by z = Lχ, with χ := col(i s, i r, ω ) the solutions of (ω s L s J + R s I 2 )i s ω s L sr J i r + v s = 0 L sr i s J i r B r ω + τ L = 0, (32) which constitute a set of three nonlinear algebraic equations in the variables i s, ω. As thoroughly discussed in 1], the direction of the power flow towards or from the inertia wheel can be regulated commuting between two controllers that stabilize two different equilibrium points. We will assume in the sequel that the desired equilibrium z (or, correspondingly, χ ) is given and concentrate on the task of designing a SIDA PBC that stabilizes this equilibrium point. 4.2 SIDA PBC Design As done in Section 3 we will design our SIDA PBC adopting the algebraic approach. For, we fix the desired energy function as H d (z) = 1 2 (z z ) P (z z ), P = P > 0. (33) 3 We underscore the difference of the DFIM with respect to the classical (squirrel cage) induction motor see, e.g., previous section where the control signals are the stator voltages and the rotor is short circuited. Also, in DFIM both, rotor and stator, currents are available for measurement.

12 generated and load torque Nm] time sec] Figure 4: Generated and load torque The SIDA PBC design reduces to finding a matrix F d (z) solution of ] v J(i s, ω) R] H + s + O 2 2 I O 2 u = F d (z)p (z z ), (34) 4 1 O 2 2 and verifying F d (z) + Fd (z) 0. To simplify the solution we restrict P to be diagonal and for reasons that will become clear below impose the {13} element of F d (z) to be zero, that is, P = p si 2 O 2 2 O 2 1 F 11 (z) F 12 (z) O 2 1 O 2 2 p r I 2 O 2 1 > 0, F d (z) = F 21 (z) F 22 (z) F 23 (z), O 1 2 O 1 2 p ω F31(z) F32(z) F 33 (z) where the partition of F d (z) is conformal with the partition of P. From the first two rows of (34), after using (30) and the first equation in (32), one obtains (ω s L s J + R s I 2 )ĩ s ω s L sr J ĩ r = (L s F 11 p s + L sr F 12 p r )ĩ s + (L sr F 11 p s + L r F 12 p r )ĩ r, which, for all ĩ s, ĩ r, has a unique solution given by F 11 = 1 ( ω s J + L ) r p s µ R si 2 (35) F 12 = L sr p r µ R si 2 (36) where µ := L s L r L 2 sr > 0. We remark that this simple calculation was possible because the element F 13 of F d (z) was set to zero. The price paid for having this term equal to zero is that it makes the selection of F 31 (z) critical. Indeed, this term will appear in the corners of F d (z) + F d (z), that we recall should be negative semi definite. The term F 32(z), on

13 stator voltages v] time sec] Figure 5: Stator voltages the other hand, is not critical because its contribution to F d (z) + Fd (z) can be countered by a suitable selection of F 23 (z) that, in view of the presence of the control, is totally free. These issues will become clearer as we go through the calculations below. From the fifth row of (34) and (30) we get ] F 31 (z)p s F32(z)p r F 33 (z)p ω L x = Lsr i s J i r B r ω = L sr i r J i s J ] ĩ B r ω with the second identity obtained adding and substracting the equilibrium equation (32). From the ω term we get while the remaining equations can be arranged as ĩ (L ps F 31 (z) p r F 32 (z) where we have defined L := F 33 = B r p ω J m, (37) ] L sr J ir J i s ]) = 0, (38) ] Ls I 2 L sr I 2. L sr I 2 L r I 2 Although a solution for F 31 (z) and F 32 (z) of this equation can be easily obtained inverting L to set the term inside the parenthesis equal to zero it turns out that we don t have enough flexibility in the control to generate a matrix F d (z) that satisfies the skew symmetry constraint (4), and we have to look for an alternative solution. Towards this end, notice that we can add to (38) any vector G(z) R 4 ĩ (L ps F 31 (z) p r F 32 (z) ] ] ) J ir L sr G(z) = 0 (39) J i s

14 Speed rpm] time sec] Figure 6: Speed behavior as long as ĩ G(z) = 0. Setting the term inside the parenthesis of (39) equal to zero we get ] ps F 31 (z) = L sr Lr J i r + L sr J i s p r F 32 (z) µ L r J i s L sr J i r ] GC (z) + G D (z) ], (40) where, for convenience, we have introduced the partition ] GC (z) = L 1 G(z) G D (z) with G C (z), G D (z) R 2. As indicated above, to satisfy the skew symmetry condition it is necessary to generate a solution with F 31 (z) constant. This is easily achieved selecting With this selection G(z) results in and, in order to ensure ĩ G(z) = 0, we fix G(z) = Finally, replacing (41) and (42) in (40) we get G C (z) = L2 sr µ J ĩ s. (41) L 2 sr Lr µ J ĩ s + L sr G D L3 sr µ J ĩ s + L r G D ] G D (z) = L2 sr µ J ĩ r (42)

15 F 31 = L sr p s µ J (L sri s + L r i r ) = L sr p s µ J λ r (43) F 32 (z) = L sr p r µ J (L si s + L sr i r ) = L sr p r µ J λ s. The next step is to select the remaining terms of F d (z) that are directly affected by the control action to satisfy the skew symmetry constraint (4). To simplify the condition, we select F 21 = F 12, F 23 (z) = F 32 (z), F 22 = k r 2p r I 2 < 0 which yields F d (z) + F d (z) = 2L rr s p s µ I 2 O 2 2 L sr p s µ J λ r O 2 2 kr p r I 2 O 2 1 Lsr p s µ λ r J O 1 2 2Br p ω J m. (44) A simple Schur s complement analysis establishes that F d (z) + Fd (z) < 0 if and only if the free parameters p s and p ω satisfy ( Jm L 2 ) sr p s > 4B r L r R s µ λ r 2 p ω. (45) Remark 6 The inequality (45) clearly reveals the critical role played by B r. If this parameter is small p s p ω has to be large. In the next section we compute the control law and we see that this can be achieved injecting a high gain in the current loop or reducing the gain on the speed error feedback both options inducing obvious detrimental effects. 4.3 Proposed Controller and Stability Analysis Once we have solved the SIDA PBC matching equations, the design is completed computing the controller and assessing its stability properties. This is summarized in the proposition below. Proposition 3 Consider the DFIM system (31) in closed loop with the static feedback control where k r > 0, k ω > 0 and u = R r i r + (ω s ω)j (L sr i s + L r i r ) k s (L s ĩ s + L sr ĩ r ) k r (L sr ĩ s + L r ĩ r ) + k ω J λ s ω (46) k s > L2 sr 4B r L r µ λ r 2 k ω. The closed loop system is then described by ż = F d (z) H d, with H d (z) defined in (33), and F d (z) + Fd (z) < 0. Consequently, the equilibrium z is globally exponentially stable. Proof. From the second row of (34) we obtain u = R r i r + (ω s ω)j (L sr i s + L r i r ) + F 21 p s λs + F 22 p r λr + F 23 (z)p ω J m ω. The control law (46) follows from this expression after substitution of the values of F ij, using (30) and defining k s := p sl sr R s p r µ, k ω := p ωj m L sr. p r µ Finally, the lower bound on k s k ω is obtained from the inequality (45).

16 4.4 Simulations In this subsection we present some simulations using the DFIM parameters of 1]. That is: (in SI units) L s = L r = 0.011, L sr = 0.01, R r = R s = 0.01, J m = 0.001, B r = 0.005, v s = col(380, 0) and ω s = 2π 50. The controller parameters are selected as k s = 1000, k r = 100 and k ω = The first numerical experiment is performed using the DFIM as a motor (for 0 < t 0.5). In this case τ L = 5 and the desired speed is ω = 320 for 0 < t 0.25 and ω = 305 for 0.25 < t 0.5. The second simulation is done using the DFIM as a generator (for 0.5 < t 1). In this case τ L = 5 and the desired current is i s = col( 2.7, 0) for 0.5 < t 0.75 and i s = col( 2.85, 0) for 0.75 < t 1. Notice that, to improve the power factor, we have set the second (q) component of i s to zero. The behavior of the mechanical speed and the d and q components of the stator current for both simulations are depicted in Figs. 7, 8 and 9, respectively. 325 Mechanical speed (w) Motor mode 320 w rad/s] w rad/s] Mechanical speed (w) Generator mode time s] Figure 7: Mechanical speed, ω. In Figure 10 we compare the speed behavior of the new SIDA-PBC with the IDA PBC reported in 1] that shapes only the electrical energy. The simulation conditions are the same as before but, in view of the slower response of the controller of 1], we had to scale up the time. As expected, in spite of the large damping coefficient used in the IDA PBC (r = 1000), SIDA PBC achieves a much faster speed response. 5 Conclusions We have presented an extension of the highly successful IDA-PBC methodology, called SIDA PBC, where the energy shaping and damping injection tasks are not performed sequentially, but simultaneously. In this way we enlarge the class of systems that can be stabilized using PBC and, furthermore, through the consideration of a broader set of desired damping matrices, we provide the designer with more tuning knobs to improve performance. This new idea has been applied to solve the long standing problem of IDA PBC of induction motors, that turns out to be unsolvable with a two stage design. Also, by avoiding the classical nested loop control configuration prevalent in electromechanical systems, we have been able to improve the mechanical response of a DFIM, working both as a motor and a generator.

17 d Stator Current (isd) Motor mode isd A] d Stator Current (isd) Generator mode 2.5 isd A] time s] Figure 8: Stator current d-component. Experimental validation of the two control algorithms is currently being terminated and will be reported in the near future. Acknowledgements This work has been done in the context of the European sponsored project Geoplex with reference code IST Further information is available at The work of Carles Batlle, Arnau Dòria- Cerezo and Gerardo Espinosa has been (partially) supported by the spanish project Mocoshev, DPI , the European Community Marie Curie Fellowship (in the framework of the European Control Training Site) and CONACyT (51050Y) and DGAPA-UNAM (IN103306), respectively. References 1] Batlle, C., A. Dòria-Cerezo and R. Ortega (2005). Power flow control of a doubly-fed induction machine coupled to a flywheel, European Journal of Control, 11(3), pp ] Dawson, D., J. Hu, and T. Burg (1998). Nonlinear Control of Electric Machinery, Marcel Dekker, New York. 3] Fujimoto, K. and T. Sugie (2001). Canonical transformations and stabilization of generalized hamiltonian systems, Systems and Control Letters, 42(3), pp ] Karagiannis, D., A. Astolfi, R. Ortega and M. Hilairet (2005). A nonlinear tracking controller for voltage-fed induction motors with uncertain load torque, LSS Supelec Int. Report. 5] Krause, P.C., O. Wasynczuk and S.D. Sudhoff (1995). Analysis of Electric Machinery, IEEE Press, USA. 6] Marino, R., S. Peresada and P. Valigi (1993). Adaptive Input-output Linearizing Control of Induction Motors, IEEE Trans. Automatic Control, 38(2), pp

18 q Stator Current (isq) Motor mode iqd A] q Stator Current (isq) Generator mode 0.4 isq A] time s] Figure 9: Stator current q-component. 7] Ortega, R., A. van der Schaft, B. Maschke and G. Escobar (2002). Interconnection and Damping Assignment Passivity based Control of Port controlled Hamiltonian Systems, AUTOMATICA, 38(4), pp ] Ortega, R., A. Loria, P.J. Nicklasson and H. Sira-Ramírez (1998). Passivity based Control of Euler Lagrange Systems, Springer-Verlag, Berlin, Communications and Control Engineering. 9] Ortega, R. and E. Garcia-Canseco (2004). Interconnection and Damping Assignment Passivity Based Control: A Survey, European Journal of Control, 10, pp ] Ortega, R. and G. Espinosa (1993). Torque regulation of induction motors, AUTOMATICA, 29(3), pp ] Ortega, R., M. Spong, F. Gomez and G. Blankenstein (2002). Stabilization of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans. Automatic Control, 47(8), ] Peresada, S., A. Tilli and A. Tonelli (2004). Power control of a doubly fed induction machine via output feedback, Control Engineering Practice, 12, pp ] Rodriguez, H. and R. Ortega (2003). Interconnection and Damping Assignment Control of Electromechanical Systems, Int. J. of Robust and Nonlinear Control 13(12), pp ] Sepulchre, R., M. Janković and P. Kokotović (1997). Constructive Nonlinear Control, Springer-Verlag, London. 15] Takegaki, M. and S. Arimoto (1981). A new feedback for dynamic control of manipulators. Trans. of the ASME: Journal of Dynamic Systems, Measurement and Control, 102, pp ] van der Schaft, A. (2002). L 2 Gain and Passivity Techniques in Nonlinear Control, 2nd ed., Springer Verlag, London.

19 325 Mechanical speed (w) Motor mode 320 w rad/s] w rad/s] Mechanical speed (w) Generator mode time s] Figure 10: Mechanical speed, ω, for SIDA PBC (continuous line) and IDA PBC (dashed line).