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A Robust Linear Field-Oriented Voltage Control for the Induction Motor: Experimental Results Hebertt Sira-Ramírez, Senior Member, IEEE, Felipe González-Montañez, John Alexander Cortés-Romero, and Alberto Luviano-Juárez
Abstract—A field-oriented armature-input-voltage outputfeedback control approach is proposed for the robust linear controller design on an induction motor. The scheme simultaneously solves an angular-velocity reference-trajectory tracking task and a flux magnitude regulation in the presence of arbitrary timevarying load torques and unknown nonlinearities. The fieldoriented input-voltage scheme is combined with linear high-gain asymptotic observers, of the generalized proportional–integral type, and linear active-disturbance-rejection output-feedback controllers. The linear observers online estimate, in a simultaneous manner, the output phase variables and the lumped effects of the following: 1) unknown time-varying load torques and unmodeled frictions and 2) rather complex state-dependent nonlinearities present in the electric and magnetic circuits. The field-oriented part of the scheme uses the classical flux observer or simulator. The proposed control laws naturally decouple, while linearizing, the extended second-order dynamics for the angular velocity and the squared flux magnitude. The proposed control scheme is here tested on an experimental induction motor setup. Index Terms—Field-oriented control, generalized proportional– integral (GPI) control, induction motors.
I. I NTRODUCTION
A
SURVEY of available methods for the control of induction motors is certainly beyond the scope of this paper. For a detailed background on induction motor control, the readers are referred to the excellent books by Leonhard [1], Chiasson [2], and Marino et al. [3]. The attention is centered on the field-oriented control of these machines (see Trzynadlowski [4] and Ortega et al. [5]), particularly in its combination with a total active disturbance rejection (ADR) scheme. The specific departure of our work from most available control schemes Manuscript received August 24, 2011; revised December 26, 2011 and February 15, 2012; accepted April 16, 2012. Date of publication May 25, 2012; date of current version April 11, 2013. This work was supported in part by the Instituto de Ciencia y Tecnología del Distrito Federal, México, under Research Project PICSO11-23 and in part by the Secretaría de Investigación y Posgrado, Instituto Politécnico Nacional, under Research Project 20120231. H. Sira-Ramírez is with the Sección de Mecatrónica, Departamento de Ingeniería Eléctrica, Centro de Investigación y de Estudios Avanzados, Instituto Politécnico Nacional, México City 07300, México (e-mail: hsira@ cinvestav.mx). F. González-Montañez is with the Universidad Autónoma Metropolitana, Azcapotzalco 02200, México (e-mail:
[email protected]). J. A. Cortés-Romero is with the Departamento de Ingeniería Eléctrica y Electrónica, Facultad de Ingeniería, Universidad Nacional de Colombia, Bogotá, Colombia (e-mail:
[email protected]). A. Luviano-Juárez is with the Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Instituto Politécnico Nacional, Ticomán 07340, México (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2012.2201430
lies in directly using the field-oriented voltage control scheme in combination with ADR. This is achieved via a linear generalized proportional–integral (GPI) observer approximately estimating state-dependent nonlinearities appearing in the extended second-order dynamics of both the angular velocity and the squared flux magnitude. The first-order dynamics of the complex flux argument constitutes the zero dynamics of the multivariable closed-loop system. Such zero dynamics belongs to the particular class of unstable but unharmful nonobservable dynamics. Asymptotic estimation of external unstructured perturbation inputs, with the aim of exactly (or approximately) canceling their influences at the controller stage, has been treated in the existing literature under several headings: Disturbance accommodation control is represented by the work of Professor C. D. Johnson. This line of research dates from the 1970s (see [6]). The approach has been evolving, as evidenced by the survey paper by Johnson [7], including extensions to discrete-time systems [8] and to the decoupling of nonlinear systems [9]. A second avenue of research is the ADR control method, represented by the works of Han [10]. The main idea is that of observerbased disturbance estimation and subsequent cancellation. His emphasis, though, lies on nonlinear-observer-based disturbance estimation. The work of Han has been extended, and applied to various fields, by Gao and his colleagues among other important contributors (see [11]–[17]). A third methodology, known as intelligent proportional–integral–derivative (PID) control (IPIDC) [18], was developed by Fliess and Join. The theoretical and practical support of IPIDC and, to a large extent, that of GPI-observer-based control stem from a radically new viewpoint in nonlinear state estimation based on differential algebra. For details, the reader is referred to Fliess et al. [19]. GPI observers, which are dual counterparts of GPI controllers (see [20]), were introduced in [21] in the context of sliding-mode observers for flexible robotic systems. The nonsliding version appears in [22], as applied to chaotic-system synchronization. The linear GPI observer naturally incorporates iterated output error integral injections for attenuating the effects, on the estimation error dynamics, of exogenous and statedependent perturbation input signals affecting the input–output model of the plant. GPI observers are capable of obtaining accurate online estimations of the following: 1) the output related phase variables of the underlying pure integration input–output system; 2) the nonlinear state-dependent additive perturbation input signal itself; and 3) the estimation of a certain number of the perturbation input time derivatives. Reported results for other applications, such as controlling power converters [23],
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encourage the use of GPI control schemes as an alternative of improvement in relation to classic control schemes. Traditional controllers for the induction motor are based on the so-called two-stage controller design method. In the first stage, the armature current signals are taken as auxiliary control inputs acting in an “outer loop” angular-velocity controller. In the second stage, the designed currents are taken as reference signals, to be tracked, on the basis of the armature voltages acting as an “inner loop” controller. In this paper, a different approach is proposed by which the armature input voltage is directly used to robustly control, in a decoupled fashion, both the angular velocity and the magnitude of the flux. This is accomplished via extended expressions of the angular velocity and of the flux magnitude. This technique implies a more complex system dynamics. However, this complexity is overcome by means of an observer-based ADR controller approach. In this case, a linear-GPI-observer-based controller is shown to efficiently handle the complexity of the extended system. The result is the achieving of an accurate tracking of the given angular-velocity reference trajectory for the induction motor while simultaneously efficiently regulating the flux magnitude with a linear observer and a field-oriented plus linear controller. This paper is organized as follows. Section II presents some generalities about the induction motor model and formulates the robust field-oriented armature-voltage output-feedback control problem. Section III presents the main results. Section IV describes the experimental implementation of the proposed feedback control scheme. Section V is devoted to the conclusions and suggestions for further work. The Appendix contains background on GPI observers. II. F ORMULATION OF THE P ROBLEM AND BACKGROUND R ESULTS
dω np Lsr = Im(iλ) − Bω − τL (t) dt Lr dλ = − (Rr − jnp Lr ω)λ + Rr Lsr i Lr dt di Lsr Ls σ = 2 (Rr − jnp Lr ω)λ − γLs σi + u dt Lr
Lr
(1) (2) (3)
where ω is the shaft’s angular velocity, i is the complex armature current, λ is the complex flux, u is the complex input voltage, and the variable τL (t) is the unknown time-varying load torque perturbation input. Also, the following auxiliary parameters are defined in terms of the machine parameters: σ := 1 − L2sr /(Ls Lr ) and γ := Rs L2r + Rr L2sr /(Ls σL2r ). The following complex variable notation is used: √ i = ia + jib , λ = λa + jλb , and u = ua + jub , where j = −1 is the imaginary unit and z is the conjugate of z. A. Flux Simulator Of all variables in the induction motor model, the flux variable λ cannot be easily measured. For this reason, an observer is
ˆ dλ ˆ + Rr Lsr i = −(Rr − jnp Lr ω)λ dt
(4)
ˆ satisfies the estimation error, defined as eλ := λ − λ, Lr
deλ = −(Rr − jnp Lr ω)eλ . dt
(5)
Consider the Lyapunov candidate function V (eλ ) = (1/2)Lr |eλ |2 . Then, along solutions of (5), V˙ (eλ ) = −Rr |eλ |2 = −2(Rr /Lr )V (eλ ). Hence, the origin of the complex simulation error space eλ = 0 is a globally asymptotic exponential equilibrium point for (5). B. Indirect- and Direct-Control Decoupling Properties of the Induction Motor System The induction motor model (1)–(3), written in complex variables, may be replaced by the following model using polar coordinates for the complex flux, i.e., λ = |λ|ejφ , and writing the differential equations for |λ|2 and φ with the help of (2) and (3), one obtains: J
dω np Lsr = Im(iλ) − Bω − τL dt Lr
d 2 Rr Rr Lsr |λ| = − 2 |λ|2 + 2 Re(iλ) dt Lr Lr Lr
Consider the following model of the induction motor in a fixed stator frame (see Martin and Rouchon [25]): J
usually devised for the flux dynamics given in (2). In this case, the observer is simply given by a replica of the system itself (see [25]). If the observer is proposed as
d 1 φ = np Lr ω + 2 Rr Lsr Im(iλ) dt |λ|
Ls σ
di Lsr = 2 (Rr − jnp Lr ω)λ − γLs σi + u. dt Lr
(6)
The first two equations in (6) reveal an interesting indirectcontrol decoupling property: The mechanical part of the system, represented by the angular-velocity equation, is ruled by Im(iλ), while the electromagnetic part, represented by the squared flux magnitude equation, is governed by Re(iλ). Therefore, viewing the currents i, as auxiliary control inputs, both constitutive parts of the system can, in principle, be controlled independently of each other. This classical indirectcontrol decoupling property is inherited by the field-oriented voltage control approach here explored, and it has been used in feedback linearization control schemes for the induction motor by Kim et al. [26] and Bodson et al. [27]. Indeed, it is not difficult to see that ω and |λ|2 both have a relative degree of two and that, moreover, they satisfy the following set of real nonlinear controlled differential equations: 2 d2 ω np Lsr 2 d|λ| Im(uλ) + ξω ω, ω, , τL , τ˙L = ˙ |λ| , dt2 JLr Ls σ dt 2 d2 2 2Rr Lsr 2 d|λ| Re(uλ) + ξλ ω, ω, , τL |λ| = ˙ |λ| , (7) dt2 Lr Ls σ dt
SIRA-RAMÍREZ et al.: ROBUST LINEAR FIELD-ORIENTED VOLTAGE CONTROL FOR THE INDUCTION MOTOR
where np Lsr Lr Rr np ω|λ|2 × −σ Im(iλ)−np ωRe(iλ)− Lr Ls Lr σ −B ω− ˙ τ˙L 2 4Rr 2Rr2 L2sr ξλ (·) = + |λ|2 L2r Ls σL3r −6Rr2 Lsr 2Rr γLsr + − Re(iλ) L2r Lr 2 2 2Rr Lsr np 2Rr Lsr 2 + (8) ωIm(iλ)+ i . Lr L2r
ξω (·) =
As it can be easily verified, the nature of the nonlinear terms ξω (·) and ξλ (·) is quite involved. The key observation of the linear-GPI-observer-based control approach is that such timevarying perturbation inputs can be approximately estimated (and then canceled at the controller stage) using linear observers equipped with sufficient output-estimation-error iterated integral injections. This procedure, which is equivalent to postulating a self-updating internal model for the unknown perturbation, is incidentally in the very same spirit of total ADR and nonlinear algebraic estimation theory (see [12], [19], and [28]). C. Problem Formulation It is desired to have the rotor angular velocity ω track, even if in an arbitrarily closed manner, a given desired angularvelocity reference trajectory ω ∗ (t), while the squared magnitude of the complex flux is independently controlled, as closely as desired, toward a given constant reference value α2 . Moreover, the tracking process must be carried out regardless of the unknown time-varying load input torque τ (t), the presence of both viscous and Coulomb friction terms in the rotor dynamics, and the rather complex nonlinearities repre˙ |λ|2 , (d|λ|2 /dt)τL (t), τ˙L (t)) sented by the functions ξω (ω, ω, 2 2 ˙ |λ| , (d|λ| /dt), τL (t)), respectively, affecting the and ξλ (ω, ω, second-order dynamics of the angular velocity and the secondorder dynamics of the squared magnitude of the flux. D. Field-Oriented Armature-Voltage Control The armature input voltage u can be directly used to robustly control, in a naturally decoupled fashion, both the angular velocity of the motor shaft and the squared magnitude of the flux via extended second-order controlled equations. For this, we propose to consider the observer construction problem in the context of the following simplified expressions of the angular velocity and the flux magnitude extended ˙ |λ(t)|, dynamics. Denote by ξω (t) the quantity ξω (ω(t), ω(t), (d|λ(t)|2 /dt), τL (t), τ˙L (t)), and by ξλ (t) the function ˙ |λ(t)|, (d|λ(t)|2 /dt), τL (t)) ξλ (ω(t), ω(t), d2 ω np Lsr Im(uλ) + ξω (t) = dt2 JLr Ls σ d2 2 2Rr Lsr Re(uλ) + ξλ (t). |λ| = dt2 Lr Ls σ
(9)
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The aforementioned fourth-order dynamics is complemented with the following perturbed dynamics for the complex flux argument φ: Rr d φ = np ω + (J ω˙ + Bω + τL (t)) . dt np |λ|2
(10)
This last equation will play the role of the zero dynamics corresponding to the controlled fourth-order system described before. Note that ω˙ is just a state of the extended angularvelocity dynamics written in (9). Suppose, for a moment, that the complex flux λ is perfectly known. Then, it is possible to set, with the help of the auxiliary complex control input variable v = va + jvb , the following input-voltage field-oriented controller: λ u= v (11) |λ|2 yielding the following set of control-decoupled linear disturbed systems: np Lsr d2 ω = vb + ξω (t) dt2 JLr Ls σ 2Rr Lsr d2 2 |λ| = (12) va + ξλ (t). dt2 Lr Ls σ Naturally, the lack of measurability of λ prompts us to rely on ˆ For a discussion on the asymptotic complex flux estimate λ. how the “nonlinear separation principle” has been traditionally used in induction motor control [5, pp. 359–360] and for its justification in feedback linearization [29]. III. GPI-O BSERVER -BASED L INEAR C ONTROL OF THE I NDUCTION M OTOR ω Theorem: Let the set of parameters {κλ1 , κλ0 } and {κω 1 , κ0 } be such that the polynomials in the complex variable s, given by
pλ,c (s) = s2 +κλ1 s+κλ0
ω pω,c (s) = s2 +κω 1 s+κ0
(13)
are Hurwitz polynomials. Similarly, let the set of parameλ λ ters, for some given integers p and m,1 {γm+1 , γm , . . . , γ0λ }, ω ω ω {πp+1 , πp , . . . , π0 }, be such that the polynomials in the complex variable s, given by λ λ m sm+1 + γm s + · · · + γ0λ pλ,o (s) = sm+2 + γm+1 ω pω,o (s) = sp+2 + πp+1 sp+1 + πpω sp + · · · + π0ω
(14)
are also Hurwitz polynomials, with roots located sufficiently far into the left half of the complex plane. Let, furthermore, (m) (p) ξλ (t) and ξω (t) be functions of time uniformly absolutely 1 The integers p and m are, in principle, sufficiently large, indicating the number of iterated output-estimation-error integral injections needed to attenuate the effect of unmodeled plant nonlinearities in the GPI observation error dynamics. In practice, however, they are small and chosen within the range of three to five. We recall here a quote by J. von Neumann: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk!”
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bounded by finite constants. Then, the armature-voltage fieldoriented controller ˆ λ v, v = va + jvb u = ua + jub = ˆ2 |λ|
2 Lr L s σ λ d|λ| λ 2 2 ˆ ξλ + κ1 va = − + κ0 |λ| − α 2Rr Lsr dt
JLr Ls σ ˙ − ω˙ ∗ (t) vb = − ¨ ∗ (t) + κω ξω − ω 1 ω np Lsr ∗ (ω − ω (t)) (15) + κω 0 2 /dt) given by the variables ϑλ and ζ λ , with ξ λ and i(d|λ| 1 2 respectively, generated by the following linear high-gain GPI observer:
λ ˆ 2 − ζλ |λ| ζ˙1λ = ζ2λ + γm+1 1
ˆ 2 − ζλ ˙ζ λ = ϑλ + 2Rr Lsr va + γ λ |λ| 2 1 m 1 Lr Ls σ
λ ˆ 2 − ζλ |λ| ϑ˙ λ1 = ϑλ2 + γm−1 1
λ ˆ 2 − ζλ ϑ˙ λ2 = ϑλ3 + γm−2 |λ| 1
.. .
ˆ 2 − ζλ ϑ˙ λm−1 = ϑλm + γ1λ |λ| 1
ˆ 2 − ζλ ϑ˙ λm = γ0λ |λ| 1
(16)
ω (ω − ζ1ω ) ζ˙1ω = ζ2ω + πp+1 np Lsr ˙ζ ω = ϑω + vb + πpω (ω − ζ1ω ) 2 1 JLr Ls σ ϑ˙ ω = ϑω + π ω (ω − ζ ω ) 2
p−1
ϑ˙ ω 2
= ϑω 3
ω πp−2
+
1
(ω − ζ1ω )
respectively. The estimation errors are easily seen to satisfy the following linear disturbed equations: ω e˜(p+2) + γp+1 e˜(p+1) + · · · + γ0ω e˜ω = ξω(p) (t) ω ω (m+2)
e˜λ
(m+1)
λ + πm+1 e˜λ
(m)
+ · · · + π0λ e˜λ = ξλ (t).
(18)
In the Appendix, it is proven in general terms that, under all the assumptions specified before, the GPI observer estimation error trajectories, and those of its various time derivatives, converge toward arbitrarily small neighborhoods of the estimation error phase space where they remain ultimately bounded. As for the control part, the closed-loop dynamics of the secondorder tracking error dynamics for the angular velocity ω and the closed-loop dynamics of the squared flux magnitude |λ|2 ˆ → λ, exponentially) are given by (given that λ ω ω˙
e¨ω + κω ˜ω 1 e˙ ω + κ0 eω = ξω (t) − ξω (t) + κ1 e
(19)
The convergence of ξ ω (t) toward an arbitrarily small vicinity of ξω (t) and that of e˜˙ ω toward a small vicinity of zero establish that the tracking errors eω and e˙ ω ultimately absolutely converge toward a small as-desired vicinity of the origin for ω gains κω 1 and κ0 appropriately chosen so that the roots of the dominant characteristic polynomial in the complex variable s, ω pc,ω (s) = s2 + κω 1 s + κ0 are located sufficiently far into the left half of the complex plane. A corresponding statement can be made for ξ λ (t) and e˜˙ λ with the same implications on eλ and e˙ λ . The “steady state” of the load torque perturbed complex flux argument dynamics, corresponding to the previously defined controller, is described by d Rr φ = np ω ∗ (t) + (J ω˙ ∗ (t) + Bω ∗ (t) + τL (t)) . dt np α2
.. . ω ω ω ϑ˙ ω p−1 = ϑp + π1 (ω − ζ1 ) ω ω ϑ˙ ω p = π0 (ω − ζ1 )
Experimental setup.
e¨λ + κλ1 e˙ λ + κλ0 eλ = ξλ (t) − ξ λ (t) + κλ1 e˜˙ λ .
ω ˙ given by the variables ϑω and with ξ ω and ω 1 and ζ2 , respectively, generated by the following linear high-gain GPI observer:
1
Fig. 1.
(17)
globally ultimately drives the angular velocity ω(t) and the squared magnitude of the flux |λ|2 toward small asdesired vicinities of the reference trajectories: ω ∗ (t) and α2 , regardless of the nonlinearities present in the func˙ λ, i, τL , τ˙L ) defined tions ξλ (ω, λ, i, d|λ|2 /dt) and ξω (ω, ω, previously. Proof: Define the tracking error eω := ω − ω ∗ (t) and the stabilization error eλ := |λ|2 − α2 . Likewise, let e˜ω := ω − ζ1ω and e˜λ := |λ|2 − ζ1λ denote the estimation errors associated with the angular velocity and the squared flux magnitude,
The instability of the complex flux argument φ is of no concern whatsoever, much as it is of no concern the “instability” of the unbounded uniformly growing angular position of the rotor, for any ultimately constant angular-velocity reference trajectory. Moreover, it has been seen that φ has absolutely no influence on the closed-loop dynamics of ω and |λ|2 . IV. E XPERIMENTAL R ESULTS An induction motor coupled with a dc motor generating a time-varying load torque input was used for the experimental tests (see Fig. 1). The induction motor, manufactured by WEG, has the following parameters: a rated power of 0.75 hp,
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Fig. 3. Angular-velocity reference-trajectory tracking and tracking error.
Fig. 2.
Block diagram of the control system.
J = 2 × 10−3 kg · m2 , np = 1, M = 0.2374 H, LR = 0.2505 H, Ls = 0.2505 H, Rs = 4.32 Ω, and Rr = 2.8807 Ω. The position measurement was carried out by a 10 000-PPR optical encoder, directly coupled to the rotor shaft, and the induction motor current signals and the armature current of the load dc motor were measured by Hall-effect current sensors (LEM LA-55P). The torque load was measured indirectly through the dc motor current, where τL (t) = km iL , with km = 1.4285 N · m/A. The reference value for the flux magnitude was chosen so as to maximize the induced torque when subject to nominal √ currents. It was set that |λ∗ | = α = M inom / 2 = 0.5036 Wb, where inom = 3 A. The GPI-observer-based controller was devised in a MATLAB–xPC Target environment using a sampling period of 0.125 ms. The communication between the plant and the controller was performed by two data acquisition devices. Analog data were channeled through a National Instruments PCI-6025E data acquisition card. The digital outputs and the position encoder readings were gathered in a National Instruments PCI-6602 data acquisition card. The voltage and current signals were low-pass filtered with a cutoff frequency of 1.0 kHz. A schematic diagram of the control system is shown in Fig. 2. The angular-velocity output reference trajectory ω ∗ (t) was defined as a series of ramps, which takes values of 0 to 35, 35 to −5, −5 to 15, 15 to −5, and −5 to 20 rad · s−1 , during time intervals of 2.0 s. The characteristic polynomial associated with the velocity control loop was set to be of the ω ω ω form (s2 + κω 1 s + κ0 ), with κ1 = 208 and κ0 = 6400. The characteristic polynomial for the flux control loop was set to be of a similar form, with κλ1 = 20 and κλ0 = 100. The characteristic polynomial associated with the angular-velocity control loop disturbance observer was chosen to be of the form 2 3 ) (s + poω ), with ζoω = 11, ωoω = 60, (s2 + 2ζoω ωoω s + ωoω and poω = 60, while the characteristic polynomial associated with the flux control loop disturbance observer was set to be 2 3 ) (s + pλo ), with ζλo = 1, ωλo = 200 (s2 + 2ζλo ωλo s + ωλo and pλo = 200.
Fig. 4. Flux magnitude regulation and its stabilization error.
Fig. 5. Armature voltages and currents in the a, b reference frame.
Fig. 6. DC-motor-generated chaotic disturbance torque input τL (t).
Fig. 3 shows a rather accurate angular-velocity tracking of the desired reference trajectory. Fig. 4 shows the remarkable flux magnitude stabilization. In Fig. 5, the armature control input voltage and armature currents are shown in the reference frame a, b. The chaotic nature of the dc-motor-generated mechanical load torque is shown in Fig. 6. Fig. 7 shows the disturbance estimations, associated with the flux and velocity control loops. The chaotic load torque was synthesized using
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Fig. 10.
Performance comparison with PD and PID control schemes.
Fig. 11.
Performance comparison; power consumption.
Fig. 7. Estimated disturbance inputs ξω (t) and ξλ (t).
V. C ONCLUSION
Fig. 8. Performance comparison with PD and PID control schemes.
Fig. 9. Performance comparison with PD and PID control schemes.
Chua’s circuit output acting as a reference trajectory for the armature current in the dc motor. The performance of our proposed control scheme was compared with those of other two armature-voltage field-oriented control strategies, differing only in the linear part of the design while excluding the GPI observer. A proportional–derivative (PD) controller and a PID control scheme with well-tuned gains were chosen for the comparison. The experimental comparison results are shown in Figs. 8–11, where the GPI observer significantly improves the performance of the single PD control loop (the control gains were even larger than the ones used in the GPI-observer-based control). The PD control scheme exhibits poor tracking results. On the other hand, the PID controller shows a quite satisfactory performance in both the velocity and flux control. However, its behavior in the presence of the applied torque input is slightly affected. The proposed GPIobserver-based controller achieved the most accurate results in steady state. In Fig. 11, the power consumption analysis is shown for a mechanical torque generation Pm . In this case, even though all strategies had a similar power consumption, the GPI control exhibited the lowest results.
An armature-voltage field-oriented approach has been combined with a robust linear-GPI-observer-based output-feedback controller for the induction motor. The control objectives consisted of the following: 1) an angular-velocity referencetrajectory tracking task and 2) the regulation to a prespecified constant value of the flux magnitude. The scheme was shown, with the aid of a laboratory experimental platform, to be remarkably robust with respect to arbitrary external sudden load torque inputs, as provided by an attached dc motor generating a chaotic load torque profile. The scheme was found to be robust with respect to unmodeled torques of mechanical nature while efficiently circumventing the need for knowledge about significant state-dependent nonlinearities, and parameters, present in the flux dynamics and the armature current dynamics. The proposed approach effectively represents an alternative to the classical two-stage feedback controller design procedure. The scheme, however, still relies, for the field-oriented part of the strategy, in the classical open-loop flux observer or flux simulator. A. Future Works A problem that needs attention, in the ADR approach for the direct field-oriented voltage control of induction motors, resides in avoiding the use of mechanical sensors [30]. Within the control approach here introduced, the sensorless control alternative still remains to be explored. Also, the problem of active estimation, for a fault-tolerant control scheme [31], is suitable for the proposed ADR scheme. For a better performance of the observer, some antiwindup techniques may be applied [32]. Finally, a natural step forward in the GPI ADR scheme is the case including nonlinearities in the actuator, such as hysteresis and dead zones [24].
SIRA-RAMÍREZ et al.: ROBUST LINEAR FIELD-ORIENTED VOLTAGE CONTROL FOR THE INDUCTION MOTOR
so that the characteristic polynomial in the complex variable s
A PPENDIX Consider the following problem: It is desired to estimate the phase variables of the scalar nth-order nonlinear system
(20) y (n) = φ t, y, y, ˙ . . . , y (n−1) for which it is assumed that a solution y(t) exists, uniformly in (n−1) t, for every given set of initial conditions: y0 , y˙ 0 , . . . , y0 , specified at time t = 0. Assume, however, that the function φ(·) is completely unknown. Any solution y(t) of the above differential equation trivially satisfies y (n) (t) = φ(t, y(t), y(t), ˙ . . . , y (n−1) (t)). It is assumed based on this fact that, as a time function, the nth derivative of the solution y(t), given by y (n) (t), admits m further time derivatives which are all uniformly absolutely bounded. In other words, there exists a constant K such that2
˙ . . . , y (n−1) (t) ≤ K. sup φ(m) t, y(t), y(t), (21) t
˙ . . . , yn = y (n−1) , a state space model Setting y1 = y, y2 = y, for such an uncertain system is given by y˙ j = yj+1 ,
j = 1, . . . , n − 1
y˙ n = φ(t, y1 , y2 , . . . , yn ).
(22)
Let us propose the following observer for the phase variables {y1 , y2 , . . . , yn } associated with y, characterized by the states yˆ1 , . . . , yˆn , and complemented by m output-estimation-error iterated integral injections, characterized by the variable z1 . We have yˆ˙ j yˆ˙ n z˙i z˙m
= yˆj+1 + λn+m−j (y1 − yˆ1 ), j = 1, . . . , n − 1 = z1 + λm (y1 − yˆ1 ) = zi+1 + λm−i (y1 − yˆ1 ), i = 1, . . . , m − 1 = λ0 (y1 − yˆ1 ) (23)
Let the estimation error ey be defined as ey = e1 := y1 − yˆ1 = y − yˆ1 with e2 = y2 − yˆ2 , etc., e˙ j e˙ n z˙i z˙m
= ej+1 − λn+m−j e1 , j = 1, . . . , n − 1 = φ (t, y1 (t), y2 (t), . . . , yn (t)) − z1 − λm e1 = zi+1 + λm−i e1 , i = 1, . . . , m − 1 = λ 0 e1 .
(24)
It is not difficult to see that the estimation error ey = e1 satisfies, after elimination of all variables z, the following n + mthorder perturbed linear differential equation e(n+m) + λn+m−1 e(n+m−1) + · · · + λ1 e˙ y + λ0 ey y y (m) =φ (t, y1 (t), y2 (t), . . . , yn (t)) .
(25)
Clearly, if φ(m) (·) is uniformly absolutely bounded, then choosing the gain coefficients λj , j = 0, 1, . . . , n + m − 1, 2 This
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assumption cannot be verified a priori when φ(·) is completely unknown. However, in cases where the nonlinearity is known except for some of its parameters, as it is the case of our motor system, its validity can be assessed with some work.
po (s) = sn+m + λn+m−1 sn+m−1 + · · · + λ1 s + λ0
(26)
exhibits all its roots sufficiently far from the imaginary axis, in the left half of the complex plane, then the trajectories for ey and for its time derivatives ultimately absolutely converge, in an exponentially dominated manner, toward a small as-desired vicinity of the origin of the estimation (n+m−1) }, where they remain error phase space {ey , e˙ y , . . . , ey ultimately bounded. The farther away the roots are located in the left half of the complex plane, the smaller the vicinity of ultimate boundedness around the origin of the estimation error phase space. To prove this result, we proceed as follows: Let x = (e1 , . . . , en+m )T denote the phase variables of (25). The perturbed linear system (25) is of the form x˙ = Ax + bφ(m) (t), with A being a Hurwitz matrix written in companion form and b is a vector of zeroes except for the last component being equal to one. The Hurwitz character of A implies that, given a positive definite matrix Q, there exists a positive definite matrix P , such that −Q = AT P + P A. The largest (real negative) eigenvalue of −Q denoted by σmax (−Q) < 0 satisfies |σmax (−Q)| ≤ 2P |ρmax (A)|, with |ρmax (A)| being the absolute value of the largest negative real part of the eigenvalues of A. The Lyapunov function candidate V (x) = (1/2)xT P x exhibits, along the solutions of the linear perturbed system, a time derivative of the form V˙ (x, t) = (1/2)xT (AT P + P A)x + bT P xφ(m) (t). It follows, using the form of b, the uniform bound on φ(m) (t), and the just established matrix inequalities, that the time derivative V˙ (x, t) is strictly negative everywhere outside the sphere S = {x ∈ Rn+m |x2 ≤ K 2 /(|ρmax (A)|)2 } while its sign is undefined inside the sphere S. Hence, all trajectories x(t) starting outside this sphere, defined in the estimation error phase space, converge toward its interior, and all those trajectories starting inside S will never abandon it. The more negative the real parts of all the eigenvalues of A, the larger the (ρmax (A))2 and the smaller the radius of the ultimate bounding sphere S in the x space. From (24), it follows that z1 = φ (t, y1 (t), y2 (t), . . . , yn (t)) − λm e1 − e˙ n .
(27)
Hence, as e1 and en evolve toward the small bounding sphere in the estimation error phase space, the trajectory of z1 tracks arbitrarily close to the unknown function φ(t, y1 (t), y2 (t), . . . , yn (t)). Clearly, zi converges toward a vicinity of φ(i−1) (t), i = 1, . . . , m. From the definition of the estimation errors for y, and its time derivatives, it follows that yˆj , j = 1, . . . , n, reconstructs, in an arbitrarily close fashion, the time derivatives of y. Let ψ(t, y, u) be a perfectly known smooth scalar function. The aforementioned result extends immediately to nonlinear controlled systems of the form y (n) = φ(t, y, y, ˙ . . . , y (n−1) ) + ψ(t, y, u) by considering the observer yˆ˙ j = yˆj+1 + λn+m−j (y1 − yˆ1 ),
j = 1, 2, . . . , n − 1
yˆ˙ n = ψ(t, y1 , u) + z1 + λm (y1 − yˆ1 ) z˙i = zi+1 + λm−i (y1 − yˆ1 ), z˙m = λ0 (y1 − yˆ1 ).
i = 1, 2, . . . , m − 1 (28)
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 8, AUGUST 2013
As a final remark, it should be pointed out that low-pass filtering is required for the implementation of the GPI-observerbased algorithm. Real life noises do not preclude the application of high-gain observers, as they can be inferred from the experimental results here presented.
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[23] E. W. Zurita-Bustamante, J. Linares-Flores, E. Guzman-Ramirez, and H. Sira-Ramirez, “A comparison between the GPI and PID controllers for the stabilization of a DC–DC ‘buck’ converter: A field programmable gate array implementation,” IEEE Trans. Ind. Electron., vol. 58, no. 11, pp. 5251–5262, Nov. 2011. [24] C. Hu, B. Yao, and Q. Wang, “Adaptive robust precision motion control of systems with unknown input dead-zones: A case study with comparative experiments,” IEEE Trans. Ind. Electron., vol. 58, no. 6, pp. 2454–2464, Jun. 2011. [25] P. Martin and P. Rouchon, “Two simple flux observers for induction motors,” Int. J. Adapt. Control Signal Process., vol. 14, no. 2/3, pp. 171– 175, Mar.–May 2000. [26] D. Kim, I. Ha, and M. Ko, “Control of induction motors via feedback linearization with input–output decoupling,” Int. J. Control, vol. 51, no. 4, pp. 863–883, 1990. [27] M. Bodson, J. Chiasson, and R. Novotnak, “High-performance induction motor control via input–output linearization,” IEEE Control Syst. Mag., vol. 14, no. 4, pp. 25–33, Aug. 1994. [28] G. Tian and Z. Gao, “From Poncelet’s invariance principle to active disturbance rejection,” in Proc. ACC, St. Louis, MO, Jun. 2009, pp. 2451–2457. [29] M. Maggiore and K. Passino, “Output feedback tracking: A separation principle approach,” IEEE Trans. Autom. Control, vol. 50, no. 1, pp. 111– 117, Jan. 2005. [30] M. S. Zaky, “Stability analysis of speed and stator resistance estimators for sensorless induction motor drives,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 858–870, Feb. 2012. [31] D. Campos-Delgado and D. Espinoza-Trejo, “An observer-based diagnosis scheme for single and simultaneous open-switch faults in induction motor drives,” IEEE Trans. Ind. Electron., vol. 58, no. 2, pp. 671–679, Feb. 2011. [32] H. B. Shin and J. G. Park, “Anti-windup PID controller with integral state predictor for variable-speed motor drives,” IEEE Trans. Ind. Electron., vol. 59, no. 3, pp. 1509–1516, Mar. 2012.
Hebertt Sira-Ramírez (M’75–SM’85) received the Electrical Engineering degree from the Universidad de los Andes, Mérida, Venezuela, in 1970 and the Electrical Engineering degree and M.Sc. degree in electrical engineering in 1974 and the Ph.D. degree in electrical engineering in 1977 from the Massachusetts Institute of Technology, Cambridge. He is currently with the Sección de Mecatrónica, Departamento de Ingeniería Eléctrica, Centro de Investigación y de Estudios Avanzados, Instituto Politécnico Nacional, México City, México. He is the coauthor of several books on automatic control and the author of over 400 technical papers in journals and international conference proceedings. His research interests include the theoretical and practical aspects of feedback regulation of nonlinear dynamic systems with special emphasis on variable-structure feedback control techniques and their applications in power electronics. Dr. Sira-Ramírez is a Distinguished Lecturer of the IEEE Control Systems Society and a member of the IEEE International Committee.
Felipe González-Montañez was born in México City, México, in 1985. He received the B.Sc. degree in electrical engineering from the Universidad Autónoma Metropolitana (UAM), México City, Mexico, in 2009 and the M.Sc. degree in electrical engineering from the Centro de Investigación y de Estudios Avanzados, Instituto Politécnico Nacional, México City, Mexico, in 2011. He is currently with the Área de Ingeniería Energética Electromagnética, Departamento de Energía, UAM, Azcapotzalco, México. His research interests include the modeling and control of electrical machines.
SIRA-RAMÍREZ et al.: ROBUST LINEAR FIELD-ORIENTED VOLTAGE CONTROL FOR THE INDUCTION MOTOR
John Alexander Cortés-Romero received the Electrical Engineering degree, the M.Sc. degree in industrial automation, and the M.Sc. degree in mathematics from the Universidad Nacional de Colombia, Bogotá, Colombia, in 1995, 1999, and 2007, respectively, and the Ph.D. degree in electrical engineering from the Centro de Investigación y de Estudios Avanzados, Instituto Politécnico Nacional, México City, México, in 2011. He is currently an Associate Professor with the Departamento de Ingeniería Eléctrica y Electrónica, Facultad de Ingeniería, Universidad Nacional de Colombia. His research interests include algebraic identification and estimation methods in feedback control systems.
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Alberto Luviano-Juárez was born in México City, México, in 1981. He received the B.S. degree in mechatronics engineering from the Instituto Politécnico Nacional, México City, Mexico, in 2003, the M.Sc. degree in automatic control from the Department of Automatic Control, Centro de Investigación y de Estudios Avanzados (CINVESTAV), Instituto Politécnico Nacional (IPN), in 2006, and the Ph.D. degree in electrical engineering from the Departamento de Ingeniería Eléctrica, CINVESTAV, IPN, in 2011. Since 2011, he has been with the Postgraduate and Research Section, Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, IPN. His research interests include nonlinear observers in control systems, flatness-based control, and algebraic methods in the estimation and control of mechatronic systems.
