Sensorless PBC of induction motors: A separation principle from ISS properties

Transcription

1 Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec , 27 FrC9.2 Sensorless PBC of induction motors: A separation principle from ISS properties Jaime A. Moreno and Gerardo Espinosa Pérez Abstract In this paper the design of Sensorless Controllers for Induction Motors is approached from the perspective of establishing structural properties of the control law that allow for dealing with the observer design in a systematic way. In particular it is shown that sensorless Passivity based Control (PBC) of this kind of machines leads to some Input to State Stability properties of the control error dynamics that establish a Separation principle for the closed loop composed by the motor, the PBC and any sensorless observer. The importance of the result concerns the fact that it is proved that no matter how the observer is designed, boundedness (convergence to zero) of the observation error guarantees that the control error also will be bounded (tend to zero). Evidently, the well-known lack of global observability properties of the motor model is still present in the system operation. The usefulness of the contribution is illustrated via some numerical simulations. I. INTRODUCTION Sensorless Control of Induction Motors (SCIM) is a topic that has attracted the attention due to its implications in the use of these kind of devices, namely, cost dropping and reliability improvement 1. The main feature of this technique is the elimination of mechanical sensors for both position and speed of the machine. This, in addition to the well known limitation for measuring rotor (flux and/or current) variables, imposes a highly complex control problem since the aim is to achieve similar performances as those obtained with sensor based schemes but using less information for the controller design. As a natural way for approaching the SCIM problem the observability properties of the machine have been analyzed both in a local 2 and in a global setting 3. As a result, the limitations for obtaining a solution are completely understood. In particular it is currently recognized the impossibility for providing a global solution due to several unavoidable obstacles like the existence of indistinguishable trajectories, i.e. pairs of different state trajectories (including motor speed) with the same input/output behavior. In spite of the complexity for designing sensorless controllers, several solutions have been provided to the problem. As expected, all of them take as a fundamental step the design of state observers for estimating the unmeasurable (rotor and mechanical) variables. These efforts can be, roughly speaking, divided into three main approaches: the Part of the work of the first author was supported by DGAPA UNAM (IN11195) while part of the work of the second author was supported by DGAPA UNAM (IN1336) and CONACYT (24761). J. A. Moreno is with Instituto de Ingeniera UNAM, A.P , 451 México D.F., MEXICO. jmorenop@ii.unam.mx G. Espinosa Pérez is with DEPFI UNAM, A.P , 451 México D.F., MEXICO. gerardoe@servidor.unam.mx first is given by the proposition of sensorless observers with the objective of replace the unmeasurable variables for their estimates in a given sensor based controller, see for example 4, 5. The second is characterized by the fact that the observer is obtained in an embedded way into the controller design. This is done in some cases by enhancing the observability properties of the motor assuming that the rotor variables can be measured, which allows for the design of globally defined controllers 6, to later on substitute them by some estimates obtained from open loop observers 7, while in other cases the simultaneous design is developed without appealing to this assumption leading to local schemes 8. The third approach aims to generate sensorless versions of control strategies currently used in industrial applications like Field Oriented Control 9. Nevertheless the currently achieved advances in SCIM are remarkable, the aim of this paper is to go one step behind by showing that this control problem can be solved in a systematic way if the sensorless version of the passivity based control previously reported in 1 is considered. To this end, the objective is the establishment of structural properties of the closed loop system that simplify the observer based controller design. The rational behind this approach is to consider that exploiting generic stability properties of the system composed by the plant and the control law would lead to observer designs that can be carried out in a simpler way. Regarding the aforementioned structural properties, the main contribution of this paper is to prove that the sensorless version, i.e. without measuring neither rotor variables nor motor speed and assuming unknown load torque, of the passivity based control (PBC) reported in 1 defines a mapping from the observation error (given as the difference between the actual and the estimated values of the unmeasurable variables) to the control error (defined as the difference between the actual and the desired behavior for the motor state) that is Input to State Stable (ISS). The importance of establishing this ISS property comes from the well known fact 11 that for systems that enjoy it is possible to state (in a relatively simple way) a separation principle to prove the stability of a closed loop system composed by the plant and an observer based controller. Hence, the contribution of this paper is the establishment of a Separation Principle for passivity based sensorless controlled induction motors. It is important to notice that, as expected, the limitations imposed by the lack of global observability of the sensorless induction motor model are not removed by the establishment /7/$ IEEE. 694

2 46th IEEE CDC, New Orleans, USA, Dec , 27 FrC9.2 of the ISS property. Thus, the usefulness of the control scheme still depends on the observer performance, since the control error will tend to zero only if the observation error tends to zero, condition that depends in its turn on the operation regime considered for the machine. However, it must also noticed, on the other hand, that the ISS property indeed states that no matter how the sensorless observer is designed, stability for the closed loop system is guaranteed. The paper is organized as follows: The induction motor model considered for the analysis is presented in Section II while the sensorless PBC is developed in Section III by introducing the ISS properties of the error dynamic. With the aim to illustrate the usefulness of the presented result, in Section IV the results obtained from the numerical evaluation of the sensorless PBC operating with a recently developed sensorless observer 12 are presented. The paper ends with some concluding remarks included in Section V. II. INDUCTION MOTOR MODEL In order to develop the proposed sensorless controller, in this paper it is considered the standard 2φ equivalent model of the unsaturated IM given by 13 Dẋ + C(x)x + Rx = Q (1) with x T = i T,ψ T,ω, D = diag { σi 2, I 2, J} n p MJ ψ C(x) = n p ωj (2) n p Mψ T J T R = L rσγi 2 β 3 I 2 β 3 I 2 β 1 I 2 ; Q = B u T L (3) where i T = i a,i b R 2 are the stator currents, ψ T = ψ a,ψ b R 2 are the rotor fluxes, ω R is the rotor speed, u T = u a,u b R 2 are the stator (control) voltages and T L is the (externally applied) load torque. The (all positive) motor parameters are β 1 = R r ; β 3 = R rm σ = L s M 2 ; γ = M2 R r + L 2 rr s σl 2 r with L s, the stator and rotor inductances, M the mutual inductance, R s,r r the stator and rotor resistances, J the rotor inertia, B the rotor friction coefficient and n p the number of pole pairs. The matrices J R 2 2, I R 2 2 are given by J = 1 1 = J T, I = 1 1 Under sensorless operation, the only available for measurement variables are the stator currents i and the control voltages u, i.e. the controller design must be carried out by assuming that the rotor fluxes ψ and the motor speed ω can not be measured. In addition, the external disturbance given by the load torque T L is unknown. III. SENSORLESS PBC DESIGN In this section the sensorless version of the PBC reported in 1 is developed to later on establish the ISS properties of the error dynamics obtained as a result of this design. Although the followed methodology is the same as the presented in 1, for the sake of clarity the complete design is included. Define the control error as the difference between the actual and the desired state e = x x d = e s e r e ω ; x d = i d ψ d ω d with x d the desired behavior for the motor state. In terms of these variables the IM model (1) can be written as with Dė + C(x)e + Re = Φ (4) Φ = Q {Dẋ d + C(x)x d + Rx d } (5) For classical PBC, the purpose of the design is to exploit the passivity properties of the error dynamics (4) by defining the control law in such a way that Φ = Ke. This implies that the error dynamics takes the form Dė + C(x)e + R + Ke = (6) Thus, by applying standard stability arguments, it can be shown that lim t e(t) = whenever the symmetric part of R + K became a positive definite matrix. For sensorless design, the classical PBC can not be longer applied since equality (6) is not satisfied. In the next proposition it is presented the structure of the error dynamics that is obtained when rotor and mechanical variables are not measurable. Proposition 1: Consider the induction motor model (1). Assume that A.1 The only available variables are stator currents i and stator voltages u. A.2 The load torque T L is an unknown disturbance. A.3 The motor parameters are known. Define the control law as u = σ di d dt + n pm J L ˆψω d + σγi d β 3 ψ d k 1 e s (7) r where ˆψ is the estimate of the rotor fluxes ψ while the desired dynamic behavior for the rotor fluxes satisfies ψ d = n pˆω + R r β n p β 2 T d J ψ d ; ψ d () = (8) with β = ψ d the norm of the desired rotor flux vector and ˆω the estimated speed. The desired stator currents are such that i d = β 1 3 Rr n p β 2 T dj ψ d + β 1 ψ d (9) 695

3 46th IEEE CDC, New Orleans, USA, Dec , 27 FrC9.2 and the desired generated torque is given by T d = J ω d + Bω d + ˆT L k 2 (ˆω ω d ) (1) with ˆT L the estimated load torque. Under these conditions the error dynamics (4) can be written as Dė + C(x)e + Re = M x (11) with R = and M = ( σγ + k 1 )I 2 β 3 I 2 β 3 I 2 β 1 I 2 n p Mi T d J B + k 2 n pmj ω d n p J ψ d k 2 ; x = (12) ψ ω T L Proof: The first two rows of (5) are given by { Φ 1 = u σ di } d dt + n pmj ψω d + σγi d β 3 ψ d (13) which under the definition of the control law (7) takes de form Φ 1 = n p MJ ψω d k 1 e s (14) Regarding the second two rows of (5), they are given by Φ 2 = { ψ d n p ωj ψ d β 3 i d + β 1 ψ d } (15) which in terms of the observation error for the rotor speed ω = ˆω ω can be written as Φ 2 = { ψ d n p (ˆω ω)j ψ d β 3 i d + β 1 ψ d } Thus, if the dynamic desired behavior for the rotor fluxes is defined as (8) and considering T d = n p M ψ T d J i d then equation (15) takes the form Φ 2 = { R r n p β 2 T dj ψ d + n p ωj ψ d β 3 i d + β 1 ψ d } Hence, substitution of the desired stator currents expression (9) leads to Φ 2 = n p e ω J ψ d (16) Finally, the fifth row of equation (5) is given by Φ 3 = T L { J ω d n p Mψ T J T i d + Bω d } (17) which, by considering that e r = ψ ψ d, can be written as { } M Φ 3 = T L + J ω d T d n p e T r J T i d + Bω d If (1) is considered, then equation (17) becomes { } Φ 3 = T M L n p e T r J T i d + k 2 (ˆω ω d ) (18) Noting that ω = ˆω ω equation (18) can be finally written as Φ 3 = TL + n p Mi T d J T e r k 2 ω k 2 e ω (19) The proof is completed by noting that substitution of (14), (16) and (19) into equation (4) gives as a result expression (11). The main result of the paper, i.e. the establishment of ISS properties for the error equation (11), is presented in the next Proposition 2: Consider the induction motor model (1) in closed loop with the output feedback sensorless controller (7), (8), (9), (1). If the controller gains satisfy k 2 > n2 p M2 4 β k 1 > β2 3 β 1 β 2 3 β 1 ( σγ+k 1 ) i d 2 then, the error dynamics (11) defines an input to state stable mapping considering x as input and e as state. Proof: Consider the positive definite function V = 1 2 et De whose time derivative along the trajectories of (11) is given by V = e T Rs e + e T M x (2) where R s is the symmetric part of matrix R given in (12). Noting that if the conditions imposed over the controller gains k 1 and k 2 are satisfied R s becomes, by a straightforward application of the Schur s complement, positive definite, then it can be proved that x = implies that e = is globally exponentially stable since V = e T Rs e < On the other hand, if x then equation (2) can be equivalently written as V = (1 θ)e T Rs e θe T Rs e + e T M x with θ a positive constant which belongs to the set (, 1). Thus V (1 θ)e T Rs e for all e M θλ min ( R s ) x Then by applying well known arguments (see Theorem 5.2 in 14) it can be proved that the map is input to state stable. Σ : x e The following remarks about the presented result are in order: Remark 1: The importance of the obtained ISS property of the error dynamics (11) can hardly be overestimated. This result states that no matter how the observer for obtaining x 696

4 46th IEEE CDC, New Orleans, USA, Dec , 27 FrC9.2 is designed, convergence of this error to zero guarantees that the control error e will also tend to zero, i.e. a separation principle for the proposed PBC has been established. Remark 2: Evidently, although strong, the presented ISS property does not remove the limitations imposed by the lack of globally defined observability properties of the induction motor model (1). However an interesting feature of ISS is related with the fact that if the estimation error x is bounded then the control error e will be also bounded. Remark 3: It is clear that (perhaps) the main drawback of the presented result lies on Assumption A.3, i.e. the knowledge on the model parameters. Although current research is carried out with the aim of relaxing this requirement, it must be noticed that the constraints imposed to the controller gains can be satisfied by allowing some degree of uncertainty in motor parameters, giving then some robustness to the ISS property. Remark 4: Notice that even in the case that B =, i.e. when the mechanical damping is disregarded, the positivity of matrix R s is still preserved, giving as a result the existence of the ISS property under these conditions. Remark 5: From a methodological point of view, the importance of the stated separation principle lies in the fact that the controller designer must concentrate all the attention only on the observer design without any concern about the stability of the whole system, besides the required to assure the convergence and boundedness of the estimation scheme. IV. ILLUSTRATIVE EXAMPLE With the aim of illustrating the usefulness of the proposed controller, in this section some numerical results are presented which were obtained by considering a closed loop composed by the motor model (1), the sensorless controller (7) (1) and an observer recently introduced in 12. Although the intention of including this numerical evaluation is only to illustrate how the stability of the control system is preserved if the observer guarantees bounded trajectories, it is interesting to notice the obtention of an important byproduct given by the performance achieved by the scheme. This last feature, exhibited under some experiments that included zero crossing speed trajectories, comes as a consequence of the properties of both the controller and the observer. On the other hand, since the topic of study of this paper does not concern on the sensorless observer design and, in addition, the complete proof of its convergence properties is excessively long, only the main characteristics of the observation scheme used in the simulations will be included. In this sense, its structure appears as an important advantage since, in contrast with complex schemes reported in the literature, it is given by a copy of the plant corrected with an output injection to stabilize the error dynamics. Interestingly enough, the convergence properties are obtaining by exploiting (again) the passivity properties of the motor model. The second feature is the usual one in sensorless observers, i.e. the scheme is able to deal only with trajectories that are distinguishable while the convergence properties depend on the distance to unobservability of the trajectory. The structure of the observer is presented in the next Proposition 3: Consider the IM model (1). Assume O.1 The only measurable signals are (i,u). O.2 The load torque is constant but unknown. O.3 All parameters are known. Under these conditions, the sensorless observer defined by dî dt = β (ai+n pˆωj) ˆψ (Ma + b)i + cu K i (î i), ˆω = f ˆω + α ˆψ T Ji ˆT L J K ω (î i), (21) ˆψ = (ai+n pˆωj) ˆψ + Mai K ψ (î i), ˆT L = K T (î i), with output injection gains given as K i =k i I; k i K ω = α β it J + k n p β ( 1 + g T (t)g (t) ) ˆψT J T + g1 T (t) ( ai + n pˆωj T), ( K z = k n p βg 1 (t) ˆψ T J T + aj + n pˆωj T), K T = kn p βg 2 (t) ˆψ T J T. where g1 (t) g (t) = g 2 (t) = exp ( BJ ) t t and a = β 1 ; β = β 3 σr r ; α = n pβ 3 ( ) B α exp J τ β J T i(τ) 1 J JR r ; b = R rr s β 3 dτ ; c = R r β 3 satisfies: 1) For every pair (r,ǫ), such that r > ǫ >, and any trajectory of the plant with bounded state, if { the initial estimation error belongs to the set C r = ω r, ψ } βĩ r, bounded in ω and ψ βĩ, it will converge to the compact set C ǫ = { ĩ ǫ } C r, and will stay there for all future times. 2) If the trajectory is distinguishable the observation error is finally and uniformly bounded. 3) Moreover, if the distinguishable trajectory is far enough from an indistinguishable one, the observation error converges uniformly and asymptotically. In order to carry out the numerical simulations, there were considered the following motor parameters: =.76 H, L s =.142 H, R r =.93 Ω, R s = Ω, M =.99 H, n p = 2, J =.29 Kg m 2, f =.13 s 1 while the applied unknown load torque was constant of value T L = 5 Nm. Regarding the controller structure, the speed reference was set to ω d (t) = 3sin(.25t) with a desired value for the rotor flux norm equal to β =.8Wb. This kind 697

5 46th IEEE CDC, New Orleans, USA, Dec , 27 FrC9.2 of reference establishes a challenge for the control scheme since it starts in a well-known unobservable trajectory, i.e zero motor speed. The experiment is further complicated by noting the subsequent zero crossing behavior of the reference signal. Regarding the observer information, the considered initial values were: ˆω() = 5rpm for the estimated motor speed, ˆψ() =.9Wb for the estimated rotor fluxes and î() = 1A for the estimated stator currents. The observer gains were set to k i = 1 and k = 2. Under the conditions described above, in Figure 1 the motor speed response is presented. In this picture three important elements must be noticed. First, how the transient response due to uncertainty in the motor variables is quickly compensated. Second, that the control error is very small, and, third, that even that the reference trajectory crosses the zero level, the control objective is still achieved. stator currents A time s Fig. 3. Stator currents response stator voltages V speed rpm time s 2 Fig. 4. Stator voltages response time t Fig. 1. Motor speed response. In order to illustrate the internal stability properties and, at the same time, the fact that the controller does not demand stringent requirements for its operation, in Figure 2 the rotor fluxes are presented while the stator currents are shown in Figure 3. The control signals are depicted in Figure 4. Notice that besides their boundedness, the maximum values that they reach correspond to a physical achievable operation..8.6 rotor fluxes, motor speed and load torque are, respectively, presented. From these figures, it is clear how the the actual value for the motor variables is achieved very quickly. Rotor fluxes observation error rotor fluxes Wb time t Fig. 5. Rotor fluxes observation error response time s Fig. 2. Rotor fluxes response. Finally, to illustrate the observer performance, in Figure 5, Figure 6 and Figure 7 the observation errors for the V. CONCLUDING REMARKS In this paper a separation principle for a passivity based sensorless controller for induction motors has been established. This contribution was obtained by finding some Input to State Stability properties of the error dynamic that result from the closed loop composed by the induction motor and the controller. The importance of this result lies in the 698

6 46th IEEE CDC, New Orleans, USA, Dec , 27 FrC9.2 Load torque observation error Nm Speed observation error rpm time t Fig. 6. Speed observation error response. 6 Marino, R., P. Tomei, and C. Verrelli: A global tracking control for speed-sensorless induction motors, AUTOMATICA, Vol. 4, No. 6, pp , Feemster, M., P. Aquino, D.M. Dawson and A. Behal: Sensorless Rotor Velocity Tracking Control of Induction Motors, IEEE Transactions on Control Systems Technology, Vol. 9, No. 4, pp , R. Marino, P. Tomei and C.M. Verrelli: A nonlinear tracking control for sensorless induction motors, AUTOMATICA, Vol. 41, No. 6, pp , M. Montanari, S. Peresada and A. Tilli: Sensorless indirect field oriented control of induction motor via adaptive speed observer, Proceedings of the 23 American Control Conference, Vol. 6, pp , Colorado, USA, Espinosa Perez, G., and R. Ortega: An Output Feedback Globally Stable Controller for Induction Motors, IEEE Transactions on Automatic Control, Vol. 4, No. 1, pp , Angeli, D., B. Ingalls, E.D. Sontag and Y. Wang: Separation Principles for Input Output and Integral Input to State Stability, SIAM Journal Control Optim., Vol. 43, No. 1, pp , Moreno, J.A., and G. Espinosa Perez: A Low Order Semiglobal Sensorless Observer for Induction Motors, Int. Report National University of Mexico, J. Meisel: Principles of Electromechanical Energy Conversion, Mc Graw Hill, H. Khalil: Nonlinear Systems, 2nd. edition, Prentice Hall, time t Fig. 7. Load torque observation error response. fact that the separation principle is valid for every sensorless observer, although convergence of the control error still depends on the observer performance. The main drawback of the proposed controller lies on the assumption that all the motor parameters are known. Relaxing this disadvantage is the topic of current research. The usefulness of the contribution was illustrated via the numerical evaluation of the reported sensorless controller operating in conjunction with a sensorless observer whose structure is a copy of the plant corrected with an output injection. Remarkable performances were obtained even in the presence of zero crossing speed references. REFERENCES 1 Rajashekara, K., A. Kawamura and K. Matsuse: Sensorless Control of AC Motor Drives, IEEE Press, Canudas, C., A. Youssef, J.P.Barbot, P. Martin and F. Maltrait: Observability Conditions of Induction Motors at Low Frecuencies, Proceedings of the 39th IEEE Conference on Decision and Control, CD ROM, Sydney, Australia, 2. 3 Ibarra Rojas, S., J. Moreno and G. Espinosa Perez: Global Observability Analysis of Sensorless Induction Motors, AUTOMATICA, Vol. 4, No. 6, pp , Besancon, G., and T. Alexandru: Simultaneous State and Parameter Estimation in Asynchronous Motors under Sensorless Speed Control, Proceedings of European Control Conference 23, pp , Cambridge, UK, Y. Kim, S. Sul and M. Park: Speed sensorless vector control of induction motor using extended Kalman filter, IEEE Trans. on Industry Applications, Vol. 3, No. 5, pp ,