Sensorless speed tracking of induction motor with ... - Semantic Scholar
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 49, NO. 4, AUGUST 2002
911
Sensorless Speed Tracking of Induction Motor With Unknown Torque Based on Maximum Power Transfer Hou-Tsan Lee, Member, IEEE, Li-Chen Fu, Senior Member, IEEE, and Hsin-Sain Huang
Stator voltage vector. Rotor voltage vector. Stator flux vector. Rotor flux vector. Stator power. Effective power.
Abstract—In this paper, we first derive the maximum power transfer theorem for an induction motor. Then, a nonlinear indirect adaptive sensorless speed tracking controller for the motor with the maximum power transfer is proposed. In this controller, only the stator currents are assumed to be measurable. The rotor flux and speed observers are designed to relax the need of flux and speed measurement. In addition, the rotor resistance estimator is also designed to cope with the problem of the fluctuation of rotor resistance with temperature. Stability analysis based on Lyapunov theory is also performed to guarantee that the controller design here is stable. Finally, the computer simulations and experiments are conducted to demonstrate the satisfactory tracking performance of our design subject to maximum power transfer.
. . . . . .
Index Terms—Adaptive control, induction machine, maximum power transfer, sensorless.
. . .
NOMENCLATURE
.
Number of pole pairs. Slip angular velocity. Stator self-inductance. Rotor self-inductance. Mutual inductance. Stator resistance. Rotor resistance. Stator currents in – frame (reference fixed frame). Rotor fluxes in – frame (reference fixed frame). Stator voltages in – frame (reference fixed frame). Electric torque. Load torque. Torque constant. Rotor inertia. Damping coefficient. Manuscript received January 10, 2001; revised December 17, 2001. Abstract published on the Internet May 16, 2002. This work was supported by the National Science Council, R.O.C., under Grant NSC 90-2213-E-002-054. H.-T. Lee was with the Department of Electrical Engineering, National Taiwan University, 106 Taipei, Taiwan, R.O.C. He is now with ChungHwa Telecom Company, Keelung, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). L.-C. Fu is with the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, 106 Taipei, Taiwan, R.O.C. (e-mail: [email protected]). H.-S. Huang was with the Department of Electrical Engineering, National Taiwan University, 106 Taipei, Taiwan, R.O.C. He is now with Quanta Display Inc., Taipei, Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier 10.1109/TIE.2002.801245.
I. INTRODUCTION
I
N THE EARLY research on induction motors, all system states are assumed to be measurable and all parameters are considered known. Under these assumptions, techniques such as field orientation [1] and input–output feedback linearization [2], [3] are utilized to design the controller. In particular, the controller in [3] is adaptive with respect to both load torque and rotor resistance variations. In these schemes the flux sensors are required, which makes them impractical for implementation. Therefore, the flux observers are then designed to relax the need of flux measurement [4], [5]. These flux observers are designed under the assumption that the rotor resistance is known. Generally, the value of rotor resistance may drift due to heating of rotor and the observers proposed above are sensitive to such value. Therefore, efforts have been made to design an estimator of the rotor resistance [6], [7]. Following this research, more efforts have been made to design controllers and flux observers which are adaptive with respect to both system parameters and/or load torque [8]–[11]. There are many works concerning the sensorless control problem, in which the vector control technique is utilized, and the research results on sensorless vector control have been proposed [12], [13], of which analyses are mainly based on the steady-state behavior and only rough proofs are supplied. In [14], the speed observer is designed and analyzed based on the Lyapunov stability theory. Both observer and controller apply the direct adaptive control scheme coping with the unknown rotor resistance. Because of the complex dynamics of the
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induction motor, the overall system analysis is also complex. In [15], an indirect adaptive scheme is proposed. The controller design is under the assumption that all states are measurable and all parameters except the load torque are known a priori. However, the controller is not actually realizable because the rotor resistance and speed are not available. Therefore, the observers and estimators are designed to provide estimates of those states and parameters, which then replace those measurable quantities so that the closed-loop controller is realizable and stable. Here, we follow this trend to design a full nonlinear adaptive sensorless controller to achieve both speed tracking and maximum power transfer based on the setup with speed and flux observer. Given the above observation, the developed adaptive controller achieves both objectives of speed tracking and maximum power transfer despite lack of precise information of rotor resistance, payload, and speed. Specifically, to relax the need of speed and flux measurement, we design a set of adaptive rotor speed and flux observer and a rotor resistance estimator in order to replace those unavailable signals in controller design. In this paper, the simulation and experimental results are also given to verify the performance of the controller design. This paper is organized as follows. In Section I, motivation of this research, related research results in the literature, and the research contribution are addressed. The main part of this paper is from Section II to Section IV. The maximum power transfer theory is derived in Section II and will be used in controller design in Section IV. Section III includes the procedure of observer design which is then analyzed in detail. Also, the simulation and experimental results are provided in Section V and Section VI, respectively. Finally, we make some conclusions in Section VII. II. MAXIMUM POWER TRANSFER OF INDUCTION MOTOR
Fig. 1. Spatial vector diagram of induction motor in d –q reference frame.
Taking the relationship (2.2) into (2.1), we have the new dynamic equation
(2.3) is, in Since the stator current exists only along the axis, for the vector fact, equal to a scalar . After substituting and the scalar in (2.3), we can then derive the following equation after splitting the real part and the imaginary part of the original equation:
(2.4) and Rearranging (2.4) with we come up with the new set of equations as follows:
,
If we define a new – frame, called the – frame, rotating to make the stator current vector lie on the axis, in speed a new set of governing dynamic equations can be derived. We rearrange these equations in vector form as (2.5)
(2.1)
and it For simplicity, we define in [16]. has been verified that In the steady state, the stator current is defined as , where is the equivalent impedance at the stator input. Thus, the input power of the induction motor can be defined as
where the symbols are defined in the Nomenclature. The spatial reference frame is shown in diagram of vectors in the Fig. 1. The relationship between the flux and the currents are as follows: which is related to in the steady state. Referring to (2.3) and (2.4), the input power can be briefly noted as
so that
where
(2.2)
Note that only the terms with rotor flux are transferred to the rotor part of the induction motor and the rest of them become
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the loss. Thus, the power transferred to the rotor of induction motor can be rearranged by applying (2.4) and (2.5) as
Now, all the terms in the imaginary parts are obviously summed are the power transferred into up to zero, and the terms with can be expressed as the rotor. Therefore, those terms with
On
the
other
hand, the matrix product can also be expanded into the
following equation:
by applying (2.4), (2.5), and the definition of . Therefore, the power transferred to the rotor of induction motor in the steady bestate is maximal if and only if is a fixed comes maximal. Under the constraint that (or ) is the - (or -) axis constant at any time, where rotor flux, the latter objective can obviously be achieved by realization of the field-oriented control law in the following (using the optimal LaGrange method with a bounded condition [20]): (2.6)
which clearly satisfies the voltage constraint “ ” at any time. In the above discussion, we have made the conclusion that the maximum power transfer can be achieved when (2.6) is satisfied. It implies that the stator voltage vector and the rotor flux vector are perpendicular to each other and such relationship holds in the fixed reference frame. Therefore, we can also derive the maximum power transfer property here in the fixed reference frame under the constraints (2.7)
where the meanings of all the variables are listed in the Nomenclature. As mentioned in Section I, the control objective is to fulfill the requirement of speed tracking but under the adverse conditions that no speed and flux measurements are allowed and the rotor resistance is unknown a priori. Furthermore, the whole developed scheme will satisfy the so-called maximal power transfer property. Given such circumstances, the speed tracking controller will have to be developed with ingenious design of various observers which meet the needs of estimating unknown parameters as well as unmeasurable states. Before we proceed to derive the observers and the controller in the following section, some assumptions will be introduced below to manifest the problem and to make it more tractable. Assumptions: (A1) All parameters of the induction motor are known, except the rotor resistance . (A2) Among all states, only the stator currents are measurable. (A3) The desired rotor speed should be a bounded smooth function with known first- and second-order time derivatives. (A4) The load torque is an unknown constant. Assumption (A1) comes from the uncertainty of rotor resistance. Its value may vary up to 150% during the operation due to variation of temperature or working condition. Assumption (A2) is a realistic consideration in the actual application of induction motors as has been mentioned in Section I. Finally, assumptions (A3) and (A4) are mainly for technical soundness and are also quite realistic in practical applications. Later in our experiment, step-type desired speed trajectory in fact can be shown to be permissive to our scheme, i.e., assumption (A3) can, in reality, be relaxed. is asAs has been mentioned earlier, the rotor resistance sumed unknown due to its fluctuation with temperature and the rotor speed is not measurable since no speed sensor is and as mounted. To cope with this, we first rewrite
where is the nominal value of rotor resistance and is the desired rotor speed. Apparently, and stand for the disand . For subsequent crepancies from what we know about simplicity of the problem, we will assume is a constant and , although it is not a constant, will be slowly varying, i.e., is small enough. According to the structure of the dynamics in (3.1), the observers are proposed as follows:
III. OBSERVER DESIGN In general, the dynamics of an induction motor can be expressed as [18]
(3.2)
(3.1)
, and and are where the auxiliary control signals to be designed later. Note that the symbol above indicates an observed value whereas the symbol
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denotes the associated observation error. Now, we can derive the error dynamics by subtracting (3.2) from (3.1) to yield
After investigating the error dynamics of the stator current observer, we now turn to the study of the flux observer. The flux observer dynamics will not be analyzed directly, but instead will be through the analysis of ( ). Our intention is to define ) as follows: control signals (
(3.8) so that
(3.3)
(3.9)
By careful observation, the right-hand side (RHS) of the first two equations in (3.3) has many terms identical to those on the RHS of the last two equations also in (3.3), but they are with different signs. In order to utilize this property to cancel the unmeasurable terms, we introduce two auxiliary observation errors as
So far, we have derived the error dynamics of the observation errors of the stator currents, rotor flux, and those of the auxiliary ). signals and have specified the controller input ( ) and ( ) are not yet defined, However, the variables ( but will be specified in the later Lyapunov stability analysis so that stability condition will hold in both observer and estimator design. Consider the following Lyapunov function candidate:
(3.4) ) to approach zero asymptotically, we To force ( ) and ( ) introduce another set of auxiliary signals ( into the system, which are related as (3.5) Note that the control signals ( coupled terms as
(3.10) and are all strictly positive. where the control gains , and the means By carefully evaluating the time derivative of , can be made more concise as of definitions of follows:
) are devised to cancel the
(3.6) and are design variables, but and then Obviously, will not be available from (3.5). After substituting them into the first two equations of (3.3) and defining new symbols
Here, we introduced four additional functions which in order to simplify the above exare similar to and pression of , where the difference between lies in that ( ) are replaced by ( ) since and are not obtainable as has been mentioned earlier. Thus, considerable simplification of will be contributed by the following assignment:
then, we get
(3.11) (3.7) where
and
.
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(3.12)
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so that
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readily becomes
(3.13) and . We now adopt variablewhere structure design (VSD) in our observer design in order to assure the upper bound of the RHS in (3.13) to be a definite negative ) in sign. To proceed, we first obtain the upper bound of ( the following lemma. ) can be explicitly Lemma 3.1: The upper bound of ( ) as some positive constants derived by proper design of ( and . Proof: The proof of the lemma can be referred to in [15]. Q.E.D. ) is acquired from Lemma Since the upper bound ( , 3.1, we can design the additional functions , and the control signals as follows:
and devise
Lemma 3.3: The rotor speed is bounded if ( ) is bounded. Proof: The proof of the lemma can be referred to in [15]. Q.E.D. is bounded from the previous argument, it follows Since is also bounded. That will lead to the following Lemma that 3.4 to guarantee the boundedness of the estimated rotor flux. Lemma 3.4: The error dynamics of the rotor flux in (3.2) are rearranged in vector form as
(3.16) , is kept posiThus, if the estimate of the rotor resistance, ) will be tive and bounded away from the origin, then ( ) are bounded. bounded provided ( Proof: The proof of the lemma can be referred to in [15]. Q.E.D. ) are bounded, which together From Lemma 3.4, ( with the boundedness of and from (3.15) readily implies ) are also bounded from (3.4). Then, it follows that ( ) are thus bounded. Now, from (3.3) from (2.5) that ( are proved bounded as well since signals on the RHS are all bounded, which together with the property that is revealed in (3.15) implies and
where if if Seemingly, and may not be realizable due to lack of knowland . However, in reality, one can always choose edge of , , initially to establish close observahigher gains and tend to small residual values and then tions first, i.e., gradually reduce the gains. As the result of the above designs, becomes
via Barbalat’s Lemma [19]. Now, we have proved the convergence of the observation errors of the stator currents under the premises of Lemma 3.4. The next problem is to prove if the estimation errors of the rotor resistance and the rotor speed have the same convergence property. To solve this problem, we first make the following definitions:
(3.14) and rearrange the error dynamics in (3.7) and those equations in (3.11) to yield
From (3.10) and (3.14), it immediately follows is bounded and
(3.15) (3.17)
In order to confirm the above claim on the proper design of and , we soon show that and will converge to zero under appropriate conditions besides the high gain condition in the following. Before that, we first present the following working lemmas. ) are bounded provided Lemma 3.2: The states ( ) are bounded in finite time. ( Proof: The proof of the lemma can be referred to in [15]. Q.E.D. Now, we need to establish another result, which guarantees presuming boundedness of ( ) in boundedness of Lemma 3.3 as follows.
where
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Now, to prove tends to zero asymptotically, we have to use the conclusion stated in Lemma 3.5 given below. Lemma 3.5: The system with the form
(3.18)
is persistently exciting is exponentially stable if and only if (PE), i.e., there exist two positive constant and such that
Proof: The proof of the lemma can be referred to in [15]. Q.E.D. By Lemma 3.5, we conclude that the system (3.18) is expois PE. However, from (3.17) the nentially stable provided and may no longer asymptotic convergence property of hold due to the forcing term added to the homogeneous system (3.18). To cope with this, we first note that the forcing term in (3.17) with respect to the homogeneous part in (3.18) will be under the ultimately in the order of the magnitude of ) will then tend to zero. premises of Lemma 3.4, since ( Thus, in our observer design, we gradually decrease the values ) to zero as tends to zero. Under of the control signals ( ) of (3.17) resuch design, the equilibrium point ( ) tends to zero mains asymptotically stable so that ( asymptotically. In turn, this will confirm the hypothesis where is kept positive and bounded the rotor resistance estimate away from the origin in Lemma 3.4. Finally, from (3.3) the fact and tend to zero asymptotically and that the control that and will also signals and converge to zero by (3.8), tend to zero asymptotically. Up to now, we can conclude the satisfactory convergence property provided PE condition holds ) are bounded. After the analysis above, the design of and ( observers can be detailed as in Theorem 1 shown below, whose proof is self-evident from the above arguments. Theorem 1: Consider the induction motor whose dynamics are governed by (3.1) under the assumptions (A1)–(A4). If the stator current observers and the rotor flux observers are designed as in (3.2), where
where the auxiliary signals
are designed as follows:
for some constants
.
IV. CONTROLLER DESIGN In this section, the controller that achieves speed tracking with maximal power transfer is proposed. The controller can also overcome the variation of the unknown load torque under the indirect adaptive control scheme, i.e., the observer provides all the necessary information to the controller, such as unknown parameter and unmeasurable states. Therefore, throughout the rest of sections we will proceed with our discussion pretending all parameters are known and all state signals are measurable. Thus, all notations used in the sequel will be without “” attached. Before we apply the input-output linearization scheme, we will introduce an assumption on model so that such a scheme can hereby actually applicable. Recall the mathematical model of the induction motor is derived in Section III and is rewritten here in (4.1), where all notations introduced are explained in the Nomenclature. Assumption (A5): The damping term of the mechanical subsystem will have negligible effect relative to our control objective so that it will be omitted from our controller design. The new system dynamics of the induction motor are as follows:
then the observed speed and flux of the rotor will be driven to the actual speed and flux and the estimate of rotor resistance will also converge to the actual one subject to the control signals designed as follows: (4.1) where the symbols are defined in the Nomenclature. First, we define the load torque variation as
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where is the nominal value of the load torque and is the actual value. Now, we introduce a set of coordinates as follows:
Rewrite the dynamical (4.1) in terms of the new coordinate variables with the maximal power transfer property (see [21]) as
TABLE I SPECIFICATIONS AND PARAMETERS OF THE CONTROLLED MOTOR
and , where . Up to this point, the adaptive term has not yet been determined, but it will be defined later. Note that the decoupled dynamics can be expressed concisely as
(4.2) (4.6) To claim our control objective, it is natural to build a reference model a priori and generate the relevant reference signal. The reference model can be defined as follows:
where
To make the representation more concise, we introduce another coordinate transformation (4.3) (4.7) where is the estimate of the load torque variation . Define the estimation error of the load torque variation as follows:
where
(4.4) Substituting the new coordinates in (4.3) and the new definition in (4.4) into the system dynamics (4.2), we then obtain the following set of differential equations as:
(4.5)
Before we start to design and analyze the controller, Lemma 4.1 is given to ensure the feasibility of the controller. Lemma 4.1: The stator currents ( , ) will be bounded prois bounded in a realistic operating envided the input power vironment, i.e., the stator voltages are kept bounded. Proof: The proof of the lemma can be referred to in [21]. Q.E.D. Theorem 2: If the dynamic system of an induction motor can be described as in (4.5), then the rotor speed can be driven to the desired speed with unknown load torque by the controller designed as (2.7), where
Now we introduce some modification into the decoupling control in (2.7) as shown below
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Fig. 2. Simulation result of !
= 300(1 0 e
) r/min with unknown
R
and T .
subject to the payload adaptation law as
Now, letting block diag , the positive-definite symmetric solution to the Lyapunov equation
Proof: To analyze the convergence property of the previously developed controller, we first define the error signals
where block diag and , are positive-definite symmetric matrices. To proceed with the Lyapunov analysis, we define a Lyapunov function candidate as
For convenience of analysis, we rearrange the error dynamics as whose time derivative can be then derived as
If we let ately follows that
(4.9) , then from (4.4), it immedi-
(4.8)
where
which reveals our adaptation law of the load torque. It is designed to overcome variation of load torque, which then helps to simplify the RHS of (4.9) to yield (4.10)
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Fig. 3. Simulation result of !
919
= 500 sin(1:5t) r/min with unknown R
and T .
The fact that is negative implies that , are bounded and the negative quadratic form of the RHS of (4.10) implies that . is bounded, from (4.6) it can be proved that the Since states ( , ) are bounded, which immediately implies that is bounded. In addition, according to the fact that both and are bounded from (4.4), the boundedness of can also be proved. Then, from (4.3) we know that is bounded and it immedi) is bounded. Due to the fact that ately follows that ( , then it can be directly shown that is bounded. If we recall the power formula
which immediately deduces that is also bounded. With the results from Lemma 4.1 and from the definition of , we can also prove that is a bounded function. Then, is a bounded function, it folfrom (4.6) and the fact that lows that are bounded
(4.11)
TABLE II VARIOUS GAINS AND INITIAL VALUES FOR SPEED CONTROL IN CASE 1
Considering (4.8), we can prove that is bounded. Therefore, from Barbalat’s Lemma, we have the final result: . In other words, the controller goal
Q.E.D. Up to this point, we thus conclude that through the adaptive input-output feedback linearization, a speed controller with
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Fig. 4. Simulation result of speed tracking benchmark result.
TABLE III VARIOUS GAINS
AND INITIAL VALUES FOR BENCHMARK TEST
SPEED CONTROL
IN
Fig. 5. Block diagram of experiment system.
maximum power transfer property is designed as in Theorem 2 and it can be further proved the controller can operate in the environment whose load torque is an unknown constant. V. SIMULATION RESULTS The simulation and experiment are done with a four-pole three-phase squrrel-cage induction motor with rated power 3 hp with a 1000-pulse/rev encoder which is manufactured by TACO
Fig. 6. Control block diagram of the induction motor system.
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Fig. 7.
Experimental result of !
= 300(1 0 e
921
) r/min with no load.
Company Ltd., Taiwan, R.O.C., with a delta-connected stator. Detailed parameters and specifications will be found Table I. The software we adopt in simulation is Simulink 3.0 and Matlab 5.2, an excellent product of The MathWorks Inc., Natick, MA. In addition, we use Simu-Drive to combine the motor control card with the Simulink/Real Time Workshop. Then, we can directly apply the simulation program to proceed with our experiment. In the speed tracking problem, we design the desired speed as a smooth function which satisfies our assumption (A3). Then, the desired speed is a smooth function with known first- and second-order time derivatives. The simulation results will show the performance of the proposed control scheme. In addition, we will show the properties of the state observers and the parameter estimator. Case 1—Speed Tracking With Unknown Rotor Resistance and Load Torque: We design two kinds of speed trajectories and the results are shown in Figs. 2 and 3. About the variation of the rotor resistance, its nominal value is 0.53 and it is assumed to vary 50% due to the temperature excursion, i.e., it can reach the maximum value 0.795 . Furthermore, the load torque is assumed to be 5.0 N m. There are several parts in each figure. The results of speed tracking are shown in Fig. 2(a) and (b) and Fig. 3(a) and (b). Figs. 2(c) and 3(c) show the tracking of rotor
resistance, and Figs. 2(d) and 3(d) are the observed rotor flux on the axis in the stationary reference frame. Fig. 2(e) and (f) and Fig. 3(e) and (f) are stator voltage and current on the axis in the stationary reference frame, respectively. Finally, the controller gains and observers gains are shown in Table II. From these figures, we can find that the tracking error will tend to zero asymptotically when time goes to infinity. Furthermore, the stator current response, stator voltage, and the performance of flux tracking are satisfactorily shown. From the results, we can conclude that our controller can overcome the uncertainty of rotor resistance and load torque. Case 2—Speed Tracking Benchmark Results: We perform computer simulations for the benchmark example [17]. The , rotor speed is required to change between these values: , 0.25 , and 1.5 during the time interval 0.1 is 500 r/min. The load torque is [0,10] s, where the at s and changes to 0.25 at s, 0.5 is 10 N m, and the rotor resistance variation where the is 30%. The simulation result for the speed tracking benchmark is shown in Fig. 4. In order to apply the theory developed in this paper, especially the assumption on the smoothness of desired speeds, we adopt a smooth function to approximate the step function so that the desired speed is continuously differentiable.
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Experimental result of !
= 500 sin(1:5t) r/min with no load.
One can find that the speed tracking is insensitive to the load torque variation. Furthermore, the speed tracking error will converge despite the variation of the desired rotor resistance. However, when the rotor resistance variation is considered, the error convergence for speed tracking will be slightly slower. Finally, the various gains and initial values used for simulation are listed in Table III.
VI. EXPERIMENTAL RESULTS In this section, we first show the overall system in Figs. 5 and 6. Then, in order to check the performance, we have done two experiments with exponential and sinusoidal speed command as in Figs. 7 and 8. For all of the cases, they are subjected to the load free condition. And the nominal rotor resistance is set as 0.53 (different to the maximal 0.795 in simulation). To make the experiment the realization of the simulation, the desired speed trajectories of the experiments are assigned as the same as those in simulation. Therefore, it is easy to validate the tracking performance of the proposed controller, and we conduct an experiment to compare our results with those from another controller based on the same scheme of I/O linearization [3], [21]. Fig. 9 shows the performances of both control schemes
under the same conditions (with speed feedback) with the speed command of 120 rad/s. The control performance can be summarized as follows. • Due to the effect of load torque and damping effect, the rotor hardly rotates very smoothly, especially in the case of extremely low speed. Therefore, there will be a relatively large error when the rotor is operated near the zero-crossing point. • For the exponential trajectories, the speed error is nearly limited within 10 r/min. In addition, the speed error is larger for the sinusoidal trajectories because of the phase shift, but are always limited within 50 r/min. • The estimated rotor resistance will approach the actual value. • Both the stator current and the rotor speed error are smaller than the compared control scheme [3]. VII. CONCLUSION In this paper, we have proposed an indirect adaptive sensorless controller with maximum power transfer based on the I/O feedback linearization scheme. In Section II, we derived the maximum power transfer theory via the concept of coupled field and Lagrange multiplier. We can achieve not only better energy
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Fig. 9. Comparison of proposed control scheme and [3] at speed 120 rad/s (see [21]).
efficiency through this theory, but can also add one constraint to our control signals [21]. To solve the speed tracking problem, we use the I/O feedback linearization scheme to design our controller. In addition, to cope with the situation where some states, such as flux and rotor speed, are not available and the parameter (rotor resistance) has a shift problem, we first construct the state observers and the rotor resistance estimator to provide the asymptotic accurate values of the states and the parameter. Then, we proposed an adaptive I/O feedback linearization controller subject to the unknown load torque. Finally, we substitute the observed states and estimated rotor resistance into the controller for realization. Finally, both the simulation and experimental results confirm the effect of our control design and the power-saving effect.
REFERENCES [1] W. Leonhard, “Microcomputer control of high dynamic performance AC-drives—A survey,” Automatica, vol. 22, pp. 1–19, 1986. [2] A. D. Luca and G. Ulivi, “Design of an exact nonlinear controller for induction motors,” IEEE Trans. Automat. Contr., vol. 34, pp. 1303–1307, Dec. 1989. [3] R. Marino, S. Peresada, and P. Valigi, “Adaptive input-output linearizing control of induction motors,” IEEE Trans. Automat. Contr., vol. 38, pp. 208–221, Feb. 1992. [4] A. Bellini, G. Figalli, and G. Ulivi, “Adaptive and design of a microcomputer-based observer for an induction machine,” Automatica, vol. 24, pp. 549–555, 1988.
[5] G. C. Verghese and S. R. Sanders, “Observers for flux estimator in induction mahines,” IEEE Trans. Ind. Electron., vol. 35, pp. 85–94, Feb. 1988. [6] J. Stephan, M. Bodson, and J. Chiasson, “Real-Time estimation of the parameters and fluxes of induction motors,” IEEE Trans. Ind. Applicat., vol. 30, pp. 746–748, May/June 1993. [7] R. Marino, S. Peresada, and P. Tomei, “Exponentially convergent rotor resistance estimation for induction motors,” IEEE Trans. Ind. Electron., vol. 42, pp. 508–515, Oct. 1996. [8] J. Hu and D. M. Dawson, “Adaptive control of induction motor systems despite rotor resistance uncertainty,” in Proc. American Control Conf., June 1996, pp. 1397–1402. [9] J. H. Yang, W. H. Yu, and L. C. Fu, “Nonlinear observer-based adaptive tracking control for induction motors with unknown load,” IEEE Trans. Ind. Electron., vol. 42, pp. 579–586, Dec. 1995. [10] R. Marino, S. Peresada, and P. Tomei, “Adaptive observer-based control of induction motors with unknown rotor resistance,” Int. J. Adapt. Control Signal Process., vol. 10, pp. 345–362, 1996. [11] , “Global adaptive output feedback control of induction motors with unknown rotor resistance,” in Proc. 35th Conf. Decision and Control, 1996, pp. 4701–4706. [12] H. Kubota and K. Matsuse, “Speed sensorless field-oriented control of induction motor with rotor resistance adaptation,” IEEE Trans. Ind. Applicat., vol. 30, pp. 2158–2162, Sept./Oct. 1994. [13] M. Feemster, P. Aquino, D. M. Dawson, and D. Haste, “Sensorless rotor velocity tracking control for induction motors,” in Proc. American Control Conf., June 1999, pp. 1397–1402. [14] R. J. Chen and L. C. Fu, “Nonlinear adaptive speed and torque servo control of induction motors with unknown rotor resistance,” Masters thesis, Dept. Elect. Eng. Comput. Sci., National Taiwan Univ., Taipei, Taiwan, R.O.C, 1996. [15] Y. C. Lin and L. C. Fu, “Nonlinear sensorless indirect adaptive speed control of induction motors with unknown rotor resistance and load,” Int. J. Adapt. Control Signal Process., vol. 14, 2000.
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[16] E. Bassi, F. P. Benzi, S. Bolobnani, and G. S. Buja, “A field orientation scheme for current-fed induction motor drives based on the torque angle closed-loop control,” IEEE Trans. Ind. Applicat., vol. 28, pp. 1038–1044, Sept./Oct. 1992. [17] H. T. Lee, J. S. Chang, and L. C. Fu, “Exponentially stable nonlinear control for speed regulation of induction motor with field-oriented PI-controller,” Int. J. Adapt. Control Signal Process., vol. 14, pp. 297–312, 2000. [18] P. C. Krause, Analysis of Electric Machinery. New York: McGrawHill, 1987. [19] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989. [20] P. Y. Papalambros and D. J. Wilde, Principles of Optimal Design—Modeling and Computation. Cambridge, U.K.: Cambridge Univ. Press, 1988. [21] H. T. Lee, L. C. Fu, and H. S. Huang, “Speed tracking control of induction motor with maximal power transfer,” in Proc. 39th Conf. Decision and Control, 2000, pp. 925–930.
Hou-Tsan Lee (S’01–M’02) was born in Keelung, Taiwan, R.O.C., in 1963. He received the B.S. degree from National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., in 1989, the M.S. degree from National Taiwan Ocean University, Keelung, Taiwan, R.O.C., in 1992, and the Ph.D. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2001. He served in the Army of Taiwan from 1984 to 1986. He was an Electronic Engineer with Delta Electronics Company, Taiwan, R.O.C., from 1986 to 1988. Since 1988, he has been with ChungHwa Telecom Company, Keelung, Taiwan, R.O.C. His areas of research interest include induction machines, positioning and tracking, power electronics, and control theory and applications. Dr. Lee is a member of the Chinese Automatic Control Society. He received the Best Student Paper Award from the Chinese Automatic Control Society in 2001.
Li-Chen Fu (M’88–SM’02) was born in Taipei, Taiwan, R.O.C., in 1959. He received the B.S. degree from National Taiwan University, Taipei, R.O.C., in 1981 and the M.S. and Ph.D. degrees from the University of California, Berkeley, in 1985 and 1987, respectively. Since 1987, he has been a member of the faculty, and is currently a Professor in both the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, where he also served as the Deputy Director of the Tjing Ling Industrial Research Institute from 1999 to 2001. His research interests include robotics, FMS scheduling, shop floor control, home automation, visual detection and tracking, E-commerce, and control theory and applications. He has been the Editor of the Journal of Control and Systems Technology and an Associate Editor of the prestigious control journal, Automatica. In 1999, he became the Editor-in-Chief of a new control journal, Asian Journal of Control. Prof. Fu is a member of the IEEE Robotics and Automation and IEEE Automatic Control Societies. He is also a Member of the Boards of the Chinese Automatic Control Society and Chinese Institute of Automation Engineers. During 1996–1998 and 2000, he was appointed as a member of the IEEE Robotics and Automation Society AdCom and will serve as the Program Chair of the 2003 IEEE International Conference on Robotics and Automation. He received the Excellent Research Award for the period 1990–1993 and Outstanding Research Awards in 1995, 1998, and 2000 from the National Science Council, R.O.C. He also received the Outstanding Youth Medal in 1991, the Outstanding Engineering Professor Award in 1995, and the Best Teaching Award in 1994 from the Ministry of Education, the Ten Outstanding Young Persons Award in 1999 of the R.O.C., the Outstanding Control Engineering Award from the Chinese Automatic Control Society in 2000, and the Lee Kuo-Ding Medal from the Chinese Institute of Information and Computing Machinery in 2000.
Hsin-Sain Huang received the B.S. degree in power mechanical engineering from National Tsing-Hua University, Hsin Chu, Taiwan, R.O.C., in 1995 and the M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in and 2000. He is currently a Computer Integrated Manufacture Engineer with Quanta Display Inc., Taipei, Taiwan, R.O.C. His research interests include nonlinear control system analysis, adaptive control, and real-time applications.
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Sensorless Speed Tracking of Induction Motor With Unknown Torque Based on Maximum Power Transfer Hou-Tsan Lee, Member, IEEE, Li-Chen Fu, Senior Member, IEEE, and Hsin-Sain Huang
Stator voltage vector. Rotor voltage vector. Stator flux vector. Rotor flux vector. Stator power. Effective power.
Abstract—In this paper, we first derive the maximum power transfer theorem for an induction motor. Then, a nonlinear indirect adaptive sensorless speed tracking controller for the motor with the maximum power transfer is proposed. In this controller, only the stator currents are assumed to be measurable. The rotor flux and speed observers are designed to relax the need of flux and speed measurement. In addition, the rotor resistance estimator is also designed to cope with the problem of the fluctuation of rotor resistance with temperature. Stability analysis based on Lyapunov theory is also performed to guarantee that the controller design here is stable. Finally, the computer simulations and experiments are conducted to demonstrate the satisfactory tracking performance of our design subject to maximum power transfer.
. . . . . .
Index Terms—Adaptive control, induction machine, maximum power transfer, sensorless.
. . .
NOMENCLATURE
.
Number of pole pairs. Slip angular velocity. Stator self-inductance. Rotor self-inductance. Mutual inductance. Stator resistance. Rotor resistance. Stator currents in – frame (reference fixed frame). Rotor fluxes in – frame (reference fixed frame). Stator voltages in – frame (reference fixed frame). Electric torque. Load torque. Torque constant. Rotor inertia. Damping coefficient. Manuscript received January 10, 2001; revised December 17, 2001. Abstract published on the Internet May 16, 2002. This work was supported by the National Science Council, R.O.C., under Grant NSC 90-2213-E-002-054. H.-T. Lee was with the Department of Electrical Engineering, National Taiwan University, 106 Taipei, Taiwan, R.O.C. He is now with ChungHwa Telecom Company, Keelung, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). L.-C. Fu is with the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, 106 Taipei, Taiwan, R.O.C. (e-mail: [email protected]). H.-S. Huang was with the Department of Electrical Engineering, National Taiwan University, 106 Taipei, Taiwan, R.O.C. He is now with Quanta Display Inc., Taipei, Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier 10.1109/TIE.2002.801245.
I. INTRODUCTION
I
N THE EARLY research on induction motors, all system states are assumed to be measurable and all parameters are considered known. Under these assumptions, techniques such as field orientation [1] and input–output feedback linearization [2], [3] are utilized to design the controller. In particular, the controller in [3] is adaptive with respect to both load torque and rotor resistance variations. In these schemes the flux sensors are required, which makes them impractical for implementation. Therefore, the flux observers are then designed to relax the need of flux measurement [4], [5]. These flux observers are designed under the assumption that the rotor resistance is known. Generally, the value of rotor resistance may drift due to heating of rotor and the observers proposed above are sensitive to such value. Therefore, efforts have been made to design an estimator of the rotor resistance [6], [7]. Following this research, more efforts have been made to design controllers and flux observers which are adaptive with respect to both system parameters and/or load torque [8]–[11]. There are many works concerning the sensorless control problem, in which the vector control technique is utilized, and the research results on sensorless vector control have been proposed [12], [13], of which analyses are mainly based on the steady-state behavior and only rough proofs are supplied. In [14], the speed observer is designed and analyzed based on the Lyapunov stability theory. Both observer and controller apply the direct adaptive control scheme coping with the unknown rotor resistance. Because of the complex dynamics of the
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induction motor, the overall system analysis is also complex. In [15], an indirect adaptive scheme is proposed. The controller design is under the assumption that all states are measurable and all parameters except the load torque are known a priori. However, the controller is not actually realizable because the rotor resistance and speed are not available. Therefore, the observers and estimators are designed to provide estimates of those states and parameters, which then replace those measurable quantities so that the closed-loop controller is realizable and stable. Here, we follow this trend to design a full nonlinear adaptive sensorless controller to achieve both speed tracking and maximum power transfer based on the setup with speed and flux observer. Given the above observation, the developed adaptive controller achieves both objectives of speed tracking and maximum power transfer despite lack of precise information of rotor resistance, payload, and speed. Specifically, to relax the need of speed and flux measurement, we design a set of adaptive rotor speed and flux observer and a rotor resistance estimator in order to replace those unavailable signals in controller design. In this paper, the simulation and experimental results are also given to verify the performance of the controller design. This paper is organized as follows. In Section I, motivation of this research, related research results in the literature, and the research contribution are addressed. The main part of this paper is from Section II to Section IV. The maximum power transfer theory is derived in Section II and will be used in controller design in Section IV. Section III includes the procedure of observer design which is then analyzed in detail. Also, the simulation and experimental results are provided in Section V and Section VI, respectively. Finally, we make some conclusions in Section VII. II. MAXIMUM POWER TRANSFER OF INDUCTION MOTOR
Fig. 1. Spatial vector diagram of induction motor in d –q reference frame.
Taking the relationship (2.2) into (2.1), we have the new dynamic equation
(2.3) is, in Since the stator current exists only along the axis, for the vector fact, equal to a scalar . After substituting and the scalar in (2.3), we can then derive the following equation after splitting the real part and the imaginary part of the original equation:
(2.4) and Rearranging (2.4) with we come up with the new set of equations as follows:
,
If we define a new – frame, called the – frame, rotating to make the stator current vector lie on the axis, in speed a new set of governing dynamic equations can be derived. We rearrange these equations in vector form as (2.5)
(2.1)
and it For simplicity, we define in [16]. has been verified that In the steady state, the stator current is defined as , where is the equivalent impedance at the stator input. Thus, the input power of the induction motor can be defined as
where the symbols are defined in the Nomenclature. The spatial reference frame is shown in diagram of vectors in the Fig. 1. The relationship between the flux and the currents are as follows: which is related to in the steady state. Referring to (2.3) and (2.4), the input power can be briefly noted as
so that
where
(2.2)
Note that only the terms with rotor flux are transferred to the rotor part of the induction motor and the rest of them become
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the loss. Thus, the power transferred to the rotor of induction motor can be rearranged by applying (2.4) and (2.5) as
Now, all the terms in the imaginary parts are obviously summed are the power transferred into up to zero, and the terms with can be expressed as the rotor. Therefore, those terms with
On
the
other
hand, the matrix product can also be expanded into the
following equation:
by applying (2.4), (2.5), and the definition of . Therefore, the power transferred to the rotor of induction motor in the steady bestate is maximal if and only if is a fixed comes maximal. Under the constraint that (or ) is the - (or -) axis constant at any time, where rotor flux, the latter objective can obviously be achieved by realization of the field-oriented control law in the following (using the optimal LaGrange method with a bounded condition [20]): (2.6)
which clearly satisfies the voltage constraint “ ” at any time. In the above discussion, we have made the conclusion that the maximum power transfer can be achieved when (2.6) is satisfied. It implies that the stator voltage vector and the rotor flux vector are perpendicular to each other and such relationship holds in the fixed reference frame. Therefore, we can also derive the maximum power transfer property here in the fixed reference frame under the constraints (2.7)
where the meanings of all the variables are listed in the Nomenclature. As mentioned in Section I, the control objective is to fulfill the requirement of speed tracking but under the adverse conditions that no speed and flux measurements are allowed and the rotor resistance is unknown a priori. Furthermore, the whole developed scheme will satisfy the so-called maximal power transfer property. Given such circumstances, the speed tracking controller will have to be developed with ingenious design of various observers which meet the needs of estimating unknown parameters as well as unmeasurable states. Before we proceed to derive the observers and the controller in the following section, some assumptions will be introduced below to manifest the problem and to make it more tractable. Assumptions: (A1) All parameters of the induction motor are known, except the rotor resistance . (A2) Among all states, only the stator currents are measurable. (A3) The desired rotor speed should be a bounded smooth function with known first- and second-order time derivatives. (A4) The load torque is an unknown constant. Assumption (A1) comes from the uncertainty of rotor resistance. Its value may vary up to 150% during the operation due to variation of temperature or working condition. Assumption (A2) is a realistic consideration in the actual application of induction motors as has been mentioned in Section I. Finally, assumptions (A3) and (A4) are mainly for technical soundness and are also quite realistic in practical applications. Later in our experiment, step-type desired speed trajectory in fact can be shown to be permissive to our scheme, i.e., assumption (A3) can, in reality, be relaxed. is asAs has been mentioned earlier, the rotor resistance sumed unknown due to its fluctuation with temperature and the rotor speed is not measurable since no speed sensor is and as mounted. To cope with this, we first rewrite
where is the nominal value of rotor resistance and is the desired rotor speed. Apparently, and stand for the disand . For subsequent crepancies from what we know about simplicity of the problem, we will assume is a constant and , although it is not a constant, will be slowly varying, i.e., is small enough. According to the structure of the dynamics in (3.1), the observers are proposed as follows:
III. OBSERVER DESIGN In general, the dynamics of an induction motor can be expressed as [18]
(3.2)
(3.1)
, and and are where the auxiliary control signals to be designed later. Note that the symbol above indicates an observed value whereas the symbol
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denotes the associated observation error. Now, we can derive the error dynamics by subtracting (3.2) from (3.1) to yield
After investigating the error dynamics of the stator current observer, we now turn to the study of the flux observer. The flux observer dynamics will not be analyzed directly, but instead will be through the analysis of ( ). Our intention is to define ) as follows: control signals (
(3.8) so that
(3.3)
(3.9)
By careful observation, the right-hand side (RHS) of the first two equations in (3.3) has many terms identical to those on the RHS of the last two equations also in (3.3), but they are with different signs. In order to utilize this property to cancel the unmeasurable terms, we introduce two auxiliary observation errors as
So far, we have derived the error dynamics of the observation errors of the stator currents, rotor flux, and those of the auxiliary ). signals and have specified the controller input ( ) and ( ) are not yet defined, However, the variables ( but will be specified in the later Lyapunov stability analysis so that stability condition will hold in both observer and estimator design. Consider the following Lyapunov function candidate:
(3.4) ) to approach zero asymptotically, we To force ( ) and ( ) introduce another set of auxiliary signals ( into the system, which are related as (3.5) Note that the control signals ( coupled terms as
(3.10) and are all strictly positive. where the control gains , and the means By carefully evaluating the time derivative of , can be made more concise as of definitions of follows:
) are devised to cancel the
(3.6) and are design variables, but and then Obviously, will not be available from (3.5). After substituting them into the first two equations of (3.3) and defining new symbols
Here, we introduced four additional functions which in order to simplify the above exare similar to and pression of , where the difference between lies in that ( ) are replaced by ( ) since and are not obtainable as has been mentioned earlier. Thus, considerable simplification of will be contributed by the following assignment:
then, we get
(3.11) (3.7) where
and
.
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(3.12)
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so that
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readily becomes
(3.13) and . We now adopt variablewhere structure design (VSD) in our observer design in order to assure the upper bound of the RHS in (3.13) to be a definite negative ) in sign. To proceed, we first obtain the upper bound of ( the following lemma. ) can be explicitly Lemma 3.1: The upper bound of ( ) as some positive constants derived by proper design of ( and . Proof: The proof of the lemma can be referred to in [15]. Q.E.D. ) is acquired from Lemma Since the upper bound ( , 3.1, we can design the additional functions , and the control signals as follows:
and devise
Lemma 3.3: The rotor speed is bounded if ( ) is bounded. Proof: The proof of the lemma can be referred to in [15]. Q.E.D. is bounded from the previous argument, it follows Since is also bounded. That will lead to the following Lemma that 3.4 to guarantee the boundedness of the estimated rotor flux. Lemma 3.4: The error dynamics of the rotor flux in (3.2) are rearranged in vector form as
(3.16) , is kept posiThus, if the estimate of the rotor resistance, ) will be tive and bounded away from the origin, then ( ) are bounded. bounded provided ( Proof: The proof of the lemma can be referred to in [15]. Q.E.D. ) are bounded, which together From Lemma 3.4, ( with the boundedness of and from (3.15) readily implies ) are also bounded from (3.4). Then, it follows that ( ) are thus bounded. Now, from (3.3) from (2.5) that ( are proved bounded as well since signals on the RHS are all bounded, which together with the property that is revealed in (3.15) implies and
where if if Seemingly, and may not be realizable due to lack of knowland . However, in reality, one can always choose edge of , , initially to establish close observahigher gains and tend to small residual values and then tions first, i.e., gradually reduce the gains. As the result of the above designs, becomes
via Barbalat’s Lemma [19]. Now, we have proved the convergence of the observation errors of the stator currents under the premises of Lemma 3.4. The next problem is to prove if the estimation errors of the rotor resistance and the rotor speed have the same convergence property. To solve this problem, we first make the following definitions:
(3.14) and rearrange the error dynamics in (3.7) and those equations in (3.11) to yield
From (3.10) and (3.14), it immediately follows is bounded and
(3.15) (3.17)
In order to confirm the above claim on the proper design of and , we soon show that and will converge to zero under appropriate conditions besides the high gain condition in the following. Before that, we first present the following working lemmas. ) are bounded provided Lemma 3.2: The states ( ) are bounded in finite time. ( Proof: The proof of the lemma can be referred to in [15]. Q.E.D. Now, we need to establish another result, which guarantees presuming boundedness of ( ) in boundedness of Lemma 3.3 as follows.
where
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Now, to prove tends to zero asymptotically, we have to use the conclusion stated in Lemma 3.5 given below. Lemma 3.5: The system with the form
(3.18)
is persistently exciting is exponentially stable if and only if (PE), i.e., there exist two positive constant and such that
Proof: The proof of the lemma can be referred to in [15]. Q.E.D. By Lemma 3.5, we conclude that the system (3.18) is expois PE. However, from (3.17) the nentially stable provided and may no longer asymptotic convergence property of hold due to the forcing term added to the homogeneous system (3.18). To cope with this, we first note that the forcing term in (3.17) with respect to the homogeneous part in (3.18) will be under the ultimately in the order of the magnitude of ) will then tend to zero. premises of Lemma 3.4, since ( Thus, in our observer design, we gradually decrease the values ) to zero as tends to zero. Under of the control signals ( ) of (3.17) resuch design, the equilibrium point ( ) tends to zero mains asymptotically stable so that ( asymptotically. In turn, this will confirm the hypothesis where is kept positive and bounded the rotor resistance estimate away from the origin in Lemma 3.4. Finally, from (3.3) the fact and tend to zero asymptotically and that the control that and will also signals and converge to zero by (3.8), tend to zero asymptotically. Up to now, we can conclude the satisfactory convergence property provided PE condition holds ) are bounded. After the analysis above, the design of and ( observers can be detailed as in Theorem 1 shown below, whose proof is self-evident from the above arguments. Theorem 1: Consider the induction motor whose dynamics are governed by (3.1) under the assumptions (A1)–(A4). If the stator current observers and the rotor flux observers are designed as in (3.2), where
where the auxiliary signals
are designed as follows:
for some constants
.
IV. CONTROLLER DESIGN In this section, the controller that achieves speed tracking with maximal power transfer is proposed. The controller can also overcome the variation of the unknown load torque under the indirect adaptive control scheme, i.e., the observer provides all the necessary information to the controller, such as unknown parameter and unmeasurable states. Therefore, throughout the rest of sections we will proceed with our discussion pretending all parameters are known and all state signals are measurable. Thus, all notations used in the sequel will be without “” attached. Before we apply the input-output linearization scheme, we will introduce an assumption on model so that such a scheme can hereby actually applicable. Recall the mathematical model of the induction motor is derived in Section III and is rewritten here in (4.1), where all notations introduced are explained in the Nomenclature. Assumption (A5): The damping term of the mechanical subsystem will have negligible effect relative to our control objective so that it will be omitted from our controller design. The new system dynamics of the induction motor are as follows:
then the observed speed and flux of the rotor will be driven to the actual speed and flux and the estimate of rotor resistance will also converge to the actual one subject to the control signals designed as follows: (4.1) where the symbols are defined in the Nomenclature. First, we define the load torque variation as
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where is the nominal value of the load torque and is the actual value. Now, we introduce a set of coordinates as follows:
Rewrite the dynamical (4.1) in terms of the new coordinate variables with the maximal power transfer property (see [21]) as
TABLE I SPECIFICATIONS AND PARAMETERS OF THE CONTROLLED MOTOR
and , where . Up to this point, the adaptive term has not yet been determined, but it will be defined later. Note that the decoupled dynamics can be expressed concisely as
(4.2) (4.6) To claim our control objective, it is natural to build a reference model a priori and generate the relevant reference signal. The reference model can be defined as follows:
where
To make the representation more concise, we introduce another coordinate transformation (4.3) (4.7) where is the estimate of the load torque variation . Define the estimation error of the load torque variation as follows:
where
(4.4) Substituting the new coordinates in (4.3) and the new definition in (4.4) into the system dynamics (4.2), we then obtain the following set of differential equations as:
(4.5)
Before we start to design and analyze the controller, Lemma 4.1 is given to ensure the feasibility of the controller. Lemma 4.1: The stator currents ( , ) will be bounded prois bounded in a realistic operating envided the input power vironment, i.e., the stator voltages are kept bounded. Proof: The proof of the lemma can be referred to in [21]. Q.E.D. Theorem 2: If the dynamic system of an induction motor can be described as in (4.5), then the rotor speed can be driven to the desired speed with unknown load torque by the controller designed as (2.7), where
Now we introduce some modification into the decoupling control in (2.7) as shown below
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Fig. 2. Simulation result of !
= 300(1 0 e
) r/min with unknown
R
and T .
subject to the payload adaptation law as
Now, letting block diag , the positive-definite symmetric solution to the Lyapunov equation
Proof: To analyze the convergence property of the previously developed controller, we first define the error signals
where block diag and , are positive-definite symmetric matrices. To proceed with the Lyapunov analysis, we define a Lyapunov function candidate as
For convenience of analysis, we rearrange the error dynamics as whose time derivative can be then derived as
If we let ately follows that
(4.9) , then from (4.4), it immedi-
(4.8)
where
which reveals our adaptation law of the load torque. It is designed to overcome variation of load torque, which then helps to simplify the RHS of (4.9) to yield (4.10)
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Fig. 3. Simulation result of !
919
= 500 sin(1:5t) r/min with unknown R
and T .
The fact that is negative implies that , are bounded and the negative quadratic form of the RHS of (4.10) implies that . is bounded, from (4.6) it can be proved that the Since states ( , ) are bounded, which immediately implies that is bounded. In addition, according to the fact that both and are bounded from (4.4), the boundedness of can also be proved. Then, from (4.3) we know that is bounded and it immedi) is bounded. Due to the fact that ately follows that ( , then it can be directly shown that is bounded. If we recall the power formula
which immediately deduces that is also bounded. With the results from Lemma 4.1 and from the definition of , we can also prove that is a bounded function. Then, is a bounded function, it folfrom (4.6) and the fact that lows that are bounded
(4.11)
TABLE II VARIOUS GAINS AND INITIAL VALUES FOR SPEED CONTROL IN CASE 1
Considering (4.8), we can prove that is bounded. Therefore, from Barbalat’s Lemma, we have the final result: . In other words, the controller goal
Q.E.D. Up to this point, we thus conclude that through the adaptive input-output feedback linearization, a speed controller with
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Fig. 4. Simulation result of speed tracking benchmark result.
TABLE III VARIOUS GAINS
AND INITIAL VALUES FOR BENCHMARK TEST
SPEED CONTROL
IN
Fig. 5. Block diagram of experiment system.
maximum power transfer property is designed as in Theorem 2 and it can be further proved the controller can operate in the environment whose load torque is an unknown constant. V. SIMULATION RESULTS The simulation and experiment are done with a four-pole three-phase squrrel-cage induction motor with rated power 3 hp with a 1000-pulse/rev encoder which is manufactured by TACO
Fig. 6. Control block diagram of the induction motor system.
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LEE et al.: SENSORLESS SPEED TRACKING OF INDUCTION MOTOR
Fig. 7.
Experimental result of !
= 300(1 0 e
921
) r/min with no load.
Company Ltd., Taiwan, R.O.C., with a delta-connected stator. Detailed parameters and specifications will be found Table I. The software we adopt in simulation is Simulink 3.0 and Matlab 5.2, an excellent product of The MathWorks Inc., Natick, MA. In addition, we use Simu-Drive to combine the motor control card with the Simulink/Real Time Workshop. Then, we can directly apply the simulation program to proceed with our experiment. In the speed tracking problem, we design the desired speed as a smooth function which satisfies our assumption (A3). Then, the desired speed is a smooth function with known first- and second-order time derivatives. The simulation results will show the performance of the proposed control scheme. In addition, we will show the properties of the state observers and the parameter estimator. Case 1—Speed Tracking With Unknown Rotor Resistance and Load Torque: We design two kinds of speed trajectories and the results are shown in Figs. 2 and 3. About the variation of the rotor resistance, its nominal value is 0.53 and it is assumed to vary 50% due to the temperature excursion, i.e., it can reach the maximum value 0.795 . Furthermore, the load torque is assumed to be 5.0 N m. There are several parts in each figure. The results of speed tracking are shown in Fig. 2(a) and (b) and Fig. 3(a) and (b). Figs. 2(c) and 3(c) show the tracking of rotor
resistance, and Figs. 2(d) and 3(d) are the observed rotor flux on the axis in the stationary reference frame. Fig. 2(e) and (f) and Fig. 3(e) and (f) are stator voltage and current on the axis in the stationary reference frame, respectively. Finally, the controller gains and observers gains are shown in Table II. From these figures, we can find that the tracking error will tend to zero asymptotically when time goes to infinity. Furthermore, the stator current response, stator voltage, and the performance of flux tracking are satisfactorily shown. From the results, we can conclude that our controller can overcome the uncertainty of rotor resistance and load torque. Case 2—Speed Tracking Benchmark Results: We perform computer simulations for the benchmark example [17]. The , rotor speed is required to change between these values: , 0.25 , and 1.5 during the time interval 0.1 is 500 r/min. The load torque is [0,10] s, where the at s and changes to 0.25 at s, 0.5 is 10 N m, and the rotor resistance variation where the is 30%. The simulation result for the speed tracking benchmark is shown in Fig. 4. In order to apply the theory developed in this paper, especially the assumption on the smoothness of desired speeds, we adopt a smooth function to approximate the step function so that the desired speed is continuously differentiable.
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Fig. 8.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 49, NO. 4, AUGUST 2002
Experimental result of !
= 500 sin(1:5t) r/min with no load.
One can find that the speed tracking is insensitive to the load torque variation. Furthermore, the speed tracking error will converge despite the variation of the desired rotor resistance. However, when the rotor resistance variation is considered, the error convergence for speed tracking will be slightly slower. Finally, the various gains and initial values used for simulation are listed in Table III.
VI. EXPERIMENTAL RESULTS In this section, we first show the overall system in Figs. 5 and 6. Then, in order to check the performance, we have done two experiments with exponential and sinusoidal speed command as in Figs. 7 and 8. For all of the cases, they are subjected to the load free condition. And the nominal rotor resistance is set as 0.53 (different to the maximal 0.795 in simulation). To make the experiment the realization of the simulation, the desired speed trajectories of the experiments are assigned as the same as those in simulation. Therefore, it is easy to validate the tracking performance of the proposed controller, and we conduct an experiment to compare our results with those from another controller based on the same scheme of I/O linearization [3], [21]. Fig. 9 shows the performances of both control schemes
under the same conditions (with speed feedback) with the speed command of 120 rad/s. The control performance can be summarized as follows. • Due to the effect of load torque and damping effect, the rotor hardly rotates very smoothly, especially in the case of extremely low speed. Therefore, there will be a relatively large error when the rotor is operated near the zero-crossing point. • For the exponential trajectories, the speed error is nearly limited within 10 r/min. In addition, the speed error is larger for the sinusoidal trajectories because of the phase shift, but are always limited within 50 r/min. • The estimated rotor resistance will approach the actual value. • Both the stator current and the rotor speed error are smaller than the compared control scheme [3]. VII. CONCLUSION In this paper, we have proposed an indirect adaptive sensorless controller with maximum power transfer based on the I/O feedback linearization scheme. In Section II, we derived the maximum power transfer theory via the concept of coupled field and Lagrange multiplier. We can achieve not only better energy
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LEE et al.: SENSORLESS SPEED TRACKING OF INDUCTION MOTOR
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Fig. 9. Comparison of proposed control scheme and [3] at speed 120 rad/s (see [21]).
efficiency through this theory, but can also add one constraint to our control signals [21]. To solve the speed tracking problem, we use the I/O feedback linearization scheme to design our controller. In addition, to cope with the situation where some states, such as flux and rotor speed, are not available and the parameter (rotor resistance) has a shift problem, we first construct the state observers and the rotor resistance estimator to provide the asymptotic accurate values of the states and the parameter. Then, we proposed an adaptive I/O feedback linearization controller subject to the unknown load torque. Finally, we substitute the observed states and estimated rotor resistance into the controller for realization. Finally, both the simulation and experimental results confirm the effect of our control design and the power-saving effect.
REFERENCES [1] W. Leonhard, “Microcomputer control of high dynamic performance AC-drives—A survey,” Automatica, vol. 22, pp. 1–19, 1986. [2] A. D. Luca and G. Ulivi, “Design of an exact nonlinear controller for induction motors,” IEEE Trans. Automat. Contr., vol. 34, pp. 1303–1307, Dec. 1989. [3] R. Marino, S. Peresada, and P. Valigi, “Adaptive input-output linearizing control of induction motors,” IEEE Trans. Automat. Contr., vol. 38, pp. 208–221, Feb. 1992. [4] A. Bellini, G. Figalli, and G. Ulivi, “Adaptive and design of a microcomputer-based observer for an induction machine,” Automatica, vol. 24, pp. 549–555, 1988.
[5] G. C. Verghese and S. R. Sanders, “Observers for flux estimator in induction mahines,” IEEE Trans. Ind. Electron., vol. 35, pp. 85–94, Feb. 1988. [6] J. Stephan, M. Bodson, and J. Chiasson, “Real-Time estimation of the parameters and fluxes of induction motors,” IEEE Trans. Ind. Applicat., vol. 30, pp. 746–748, May/June 1993. [7] R. Marino, S. Peresada, and P. Tomei, “Exponentially convergent rotor resistance estimation for induction motors,” IEEE Trans. Ind. Electron., vol. 42, pp. 508–515, Oct. 1996. [8] J. Hu and D. M. Dawson, “Adaptive control of induction motor systems despite rotor resistance uncertainty,” in Proc. American Control Conf., June 1996, pp. 1397–1402. [9] J. H. Yang, W. H. Yu, and L. C. Fu, “Nonlinear observer-based adaptive tracking control for induction motors with unknown load,” IEEE Trans. Ind. Electron., vol. 42, pp. 579–586, Dec. 1995. [10] R. Marino, S. Peresada, and P. Tomei, “Adaptive observer-based control of induction motors with unknown rotor resistance,” Int. J. Adapt. Control Signal Process., vol. 10, pp. 345–362, 1996. [11] , “Global adaptive output feedback control of induction motors with unknown rotor resistance,” in Proc. 35th Conf. Decision and Control, 1996, pp. 4701–4706. [12] H. Kubota and K. Matsuse, “Speed sensorless field-oriented control of induction motor with rotor resistance adaptation,” IEEE Trans. Ind. Applicat., vol. 30, pp. 2158–2162, Sept./Oct. 1994. [13] M. Feemster, P. Aquino, D. M. Dawson, and D. Haste, “Sensorless rotor velocity tracking control for induction motors,” in Proc. American Control Conf., June 1999, pp. 1397–1402. [14] R. J. Chen and L. C. Fu, “Nonlinear adaptive speed and torque servo control of induction motors with unknown rotor resistance,” Masters thesis, Dept. Elect. Eng. Comput. Sci., National Taiwan Univ., Taipei, Taiwan, R.O.C, 1996. [15] Y. C. Lin and L. C. Fu, “Nonlinear sensorless indirect adaptive speed control of induction motors with unknown rotor resistance and load,” Int. J. Adapt. Control Signal Process., vol. 14, 2000.
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[16] E. Bassi, F. P. Benzi, S. Bolobnani, and G. S. Buja, “A field orientation scheme for current-fed induction motor drives based on the torque angle closed-loop control,” IEEE Trans. Ind. Applicat., vol. 28, pp. 1038–1044, Sept./Oct. 1992. [17] H. T. Lee, J. S. Chang, and L. C. Fu, “Exponentially stable nonlinear control for speed regulation of induction motor with field-oriented PI-controller,” Int. J. Adapt. Control Signal Process., vol. 14, pp. 297–312, 2000. [18] P. C. Krause, Analysis of Electric Machinery. New York: McGrawHill, 1987. [19] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence and Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989. [20] P. Y. Papalambros and D. J. Wilde, Principles of Optimal Design—Modeling and Computation. Cambridge, U.K.: Cambridge Univ. Press, 1988. [21] H. T. Lee, L. C. Fu, and H. S. Huang, “Speed tracking control of induction motor with maximal power transfer,” in Proc. 39th Conf. Decision and Control, 2000, pp. 925–930.
Hou-Tsan Lee (S’01–M’02) was born in Keelung, Taiwan, R.O.C., in 1963. He received the B.S. degree from National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., in 1989, the M.S. degree from National Taiwan Ocean University, Keelung, Taiwan, R.O.C., in 1992, and the Ph.D. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2001. He served in the Army of Taiwan from 1984 to 1986. He was an Electronic Engineer with Delta Electronics Company, Taiwan, R.O.C., from 1986 to 1988. Since 1988, he has been with ChungHwa Telecom Company, Keelung, Taiwan, R.O.C. His areas of research interest include induction machines, positioning and tracking, power electronics, and control theory and applications. Dr. Lee is a member of the Chinese Automatic Control Society. He received the Best Student Paper Award from the Chinese Automatic Control Society in 2001.
Li-Chen Fu (M’88–SM’02) was born in Taipei, Taiwan, R.O.C., in 1959. He received the B.S. degree from National Taiwan University, Taipei, R.O.C., in 1981 and the M.S. and Ph.D. degrees from the University of California, Berkeley, in 1985 and 1987, respectively. Since 1987, he has been a member of the faculty, and is currently a Professor in both the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, where he also served as the Deputy Director of the Tjing Ling Industrial Research Institute from 1999 to 2001. His research interests include robotics, FMS scheduling, shop floor control, home automation, visual detection and tracking, E-commerce, and control theory and applications. He has been the Editor of the Journal of Control and Systems Technology and an Associate Editor of the prestigious control journal, Automatica. In 1999, he became the Editor-in-Chief of a new control journal, Asian Journal of Control. Prof. Fu is a member of the IEEE Robotics and Automation and IEEE Automatic Control Societies. He is also a Member of the Boards of the Chinese Automatic Control Society and Chinese Institute of Automation Engineers. During 1996–1998 and 2000, he was appointed as a member of the IEEE Robotics and Automation Society AdCom and will serve as the Program Chair of the 2003 IEEE International Conference on Robotics and Automation. He received the Excellent Research Award for the period 1990–1993 and Outstanding Research Awards in 1995, 1998, and 2000 from the National Science Council, R.O.C. He also received the Outstanding Youth Medal in 1991, the Outstanding Engineering Professor Award in 1995, and the Best Teaching Award in 1994 from the Ministry of Education, the Ten Outstanding Young Persons Award in 1999 of the R.O.C., the Outstanding Control Engineering Award from the Chinese Automatic Control Society in 2000, and the Lee Kuo-Ding Medal from the Chinese Institute of Information and Computing Machinery in 2000.
Hsin-Sain Huang received the B.S. degree in power mechanical engineering from National Tsing-Hua University, Hsin Chu, Taiwan, R.O.C., in 1995 and the M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in and 2000. He is currently a Computer Integrated Manufacture Engineer with Quanta Display Inc., Taipei, Taiwan, R.O.C. His research interests include nonlinear control system analysis, adaptive control, and real-time applications.
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