On Some Nonlinear Current Controllers for Three-Phase Boost Rectifiers

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 2, FEBRUARY 2009

On Some Nonlinear Current Controllers for Three-Phase Boost Rectifiers Albrecht Gensior, Hebertt Sira-Ramírez, Joachim Rudolph, and Henry Güldner, Member, IEEE

Abstract—Several flatness-based current controllers for threephase three-wire boost rectifiers are compared. For this purpose, the flatness of a rectifier model is shown, and a trajectory planning algorithm that nominally achieves voltage regulation in finite time is given. The main focus lies on the inner loop current controllers. On one hand, linearization-based controllers using exact feedback linearization, exact feedforward linearization, and input–output linearization are discussed. On the other hand, two passivity-based approaches are compared. The first one is the energy shaping and damping injection method, and the other one uses exact tracking error dynamics passive output feedback. Furthermore, a reduced-order load observer is given, and a method that allows the prevention of invalid switching patterns is presented. The presented control algorithms are tested by simulations on a switched model. Index Terms—Exact feedforward linearization, flatness-based control, input–output linearization, observers, passivity-based control.

I. I NTRODUCTION

T

HE CONTROL of three-phase boost rectifiers is often accomplished by using a cascaded structure. The outer control loop is used for voltage regulation, and the inner loop is used for current control. Often, linear proportional–integral (PI) compensators are implemented for voltage regulation and parameterized for a linearized model of the plant. Usually, the load is modeled as a resistor or a constant current source. Other approaches make use of an alternative modeling of the load as a constant power sink. In [1], a load power estimate is used for feedforward in order to improve transient performance. In [2], the load estimate enables voltage regulation in finite time. Finally, [3] and [4] exploit the flatness of the converter model and suggest flatness-based control algorithms. Flatness is a mathematical property of a system of differential equations. It can be roughly characterized as the possibility of freely parameterizing the trajectories of all system variables by trajectories of a so-called flat output. This property will be used for the controller designs presented here.

Manuscript received December 20, 2007; revised July 24, 2008. First published August 19, 2008; current version published January 30, 2009. This work was supported in part by the Deutsche Forschungsgemeinschaft, Germany, under Grant GU 372/9-1. A. Gensior and H. Güldner are with the Professur Leistungselektronik, Elektrotechnisches Institut, Technische Universität Dresden, 01062 Dresden, Germany (e-mail: [email protected]). H. Sira-Ramírez is with the Sección de Mecatrónica, Departamento de Ingeniera Eléctrica, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, 07300 México City, México. J. Rudolph is with the Institut für Regelungs- und Steuerungstheorie, Technische Universität Dresden, 01062 Dresden, Germany. Digital Object Identifier 10.1109/TIE.2008.2003370

For the inner loop, there exists a variety of control schemes. A survey is given in [5]. The main focus of this paper lies on some nonlinear current controllers for fixed frequency pulsewidth modulation (PWM)-driven converters. Probably, one of the first references using a nonlinear control concept based on input–output linearization for three-phase three-wire boost rectifiers is [6]. In [7], this approach has also been used, and it has been shown that a suitable choice of the output yields stable zero dynamics. A so-called exact feedforward linearization controller has been introduced in [8], where it has been used for dc-motor control. Here, this concept will be adapted to the present application. Another possibility to implement the current controller is the use of passivity-based approaches. Energy shaping and damping injection controllers are presented in [9] for a variety of systems. For the present application, a comparison with a linear controller can be found in [10]. In [11], an alternative concept that is termed here as Exact Tracking Error Dynamics Passive Output Feedback (ETEDPOF) has been suggested and leads the present application to similar results as obtained in [12]. This paper introduces and compares three linearization-based and two passivity-based control concepts. Among the latter is a new passivity-based concept which is introduced formally and compared with the well-known damping injection method. The controllers are considered as flatness-based controllers, i.e., controllers for the stabilization of the motion of the system around trajectories that are planned in advance. Furthermore, a reduced load observer is given. This paper is organized as follows. In Section II, a model is presented and the flatness of the model is shown. A trajectory planning algorithm and a reduced-order load observer are developed. In Sections III and IV, different stabilization approaches are discussed. The linearization-based methods “exact feedback linearization,” “exact feedforward linearization,” and “input–output linearization” are presented in Section III, and two passivity-based methods are presented in Section IV. A passivity-based “energy shaping and damping injection” controller is reviewed, and an ETEDPOF controller is developed. The application of the latter is discussed for the present problem. Section V presents a method to modify invalid switching patterns in order to avoid a distortion of the currents. Simulation results are discussed in Section VI, and the Appendix briefly reviews the concept of flatness. II. M ATHEMATICAL M ODEL , F LATNESS , AND A L OAD O BSERVER A. Mathematical Model Fig. 1 shows the electrical circuit of an ideal three-phase three-wire boost converter with bidirectional switches where

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GENSIOR et al.: ON SOME NONLINEAR CURRENT CONTROLLERS FOR THREE-PHASE BOOST RECTIFIERS

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In the literature, the load is often modeled as a resistor. In this case, one would replace the last term in (2c) by −zc /R, with resistance R. Usually, the transformation is carried out in such a way that Vq —or at least its average—is zero; i.e., the reference frame is adjusted to the positive sequence of the mains voltage. This is done by adding a phase offset to the arguments of the trigonometric functions in (1). This alignment is assumed in the sequel. Fig. 1.

Electrical circuit of an ideal three-phase three-wire boost rectifier.

B. Flatness of the Model

zi , i ∈ {1, 2, 3} denote the line currents and zc denotes the capacitor voltage. The switching functions are σi : R → {−1, 1}, i ∈ {1, 2, 3}. In the sequel, an averaged model1 of the system is considered. The averaged switching functions are si : R → [−1, 1], i ∈ {1, 2, 3}. In a practical application with finite switching frequency, the switchings are implemented as duty ratios. The grid voltages Vi , i ∈ {1, 2, 3} are assumed to be sinusoidal. The load is modeled as a constant power sink, i.e., P˙ = 0. The switches are usually implemented as two power electronic devices in series, each having an antiparallel diode, which is clamped to the dc side, while the node between these switches is connected to the inductor. This construction ensures zc > 0 for nonzero input voltages because the diodes start conducting—independently of the gate signals applied to the power electronic switches—as soon as the absolute value of any line to line voltage exceeds zc . The mathematical model of the system can be written in the rotating dq-frame using the transformation (zd , zq , z0 )T = T (z1 , z2 , z3 )T with ⎞ ⎛ cos(ωt) cos ωt − 2π cos ωt + 2π 3 3 2⎝ ⎠. T (t) = − sin ωt + 2π − sin(ωt) − sin ωt − 2π 3 3 3 1/2 1/2 1/2 (1) Thus, the constraint z1 + z2 + z3 = 0 translates to z0 = 0. Transforming the other system variables in a similar way, the differential equations in the dq-frame read L z˙d = Vd − zc sd /2 + Lωzq

(2a)

L z˙q = Vq − zc sq /2 − Lωzd

(2b)

C z˙c = 3(sq zq + sd zd )/4 − P/zc

(2c)

where L and C are constant parameters for the inductance and the capacitance of the passive components, respectively. The voltage ze between the negative electrode of the output and the mains star point is given by ze = V0 − s0 zc /2 − zc /2.

(3)

In the literature, this voltage is usually not modeled. It is mentioned here 1) for the sake of completeness of the model and 2) because it reveals that s0 and V0 influence ze only and none of the line currents. 1 The averaged model is derived by averaging the switched model over one switching period while the switching frequency is assumed to be infinite [13].

The concept of flatness is briefly reviewed in the Appendix. The mathematical model of the rectifier given by (2) and (3) is flat, and y = (y1 , y2 , y3 ) with y1 = 3 Lzd2 + Lzq2 /4 + Czc2 /2 (4a) y 2 = zq

(4b)

y 3 = ze

(4c)

forms a flat output. Its first component is the energy stored in the system. Let ψ be given by ψ(y1 , y2 , y˙ 1 ) = 6C 3Vd2 4y1 − 3Ly22 −L(2y˙ 1 − 3y2 Vq + 2P )2 . (5) The system variables can be calculated from the flat output and its time derivatives zd = ψzd (y2 , y˙ 1 ) = (2y˙ 1 + 2P − 3y2 Vq )/(3Vd )

(6a)

zq = ψzq (y2 ) = y2

(6b)

zc = ψzc (y1 , y2 , y˙ 1 ) =

ψ(y1 , y2 , y˙ 1 )/(6CVd )

ze = ψze (y3 ) = y3 .

(6c) (6d)

The averaged switching functions sd , sq , and s0 can be calculated from y¨1 , y˙ 2 , and y3 using (6a)–(6d) sd = ψsd (y1 , y2 , y˙ 1 , y¨1 )

V˙ q 2¨ y1 2L V˙ d Vq Vd − = zd + y˙ 2 + zq + + ωzq (6e) zc V d Vd Vd L 3Vd 2 (Vq − y˙ 2 L − ωLzd ) zc 2 s0 = ψs0 (y1 , y2 , y˙ 1 , y3 ) = (V0 − y3 ) − 1. zc sq = ψsq (y1 , y2 , y˙ 1 , y˙ 2 ) =

(6f) (6g)

The capacitor voltage zc is positive definite; thus, ψ ≥ 0 and the possible singularity in (6e)–(6g) is taken into account. Remember that, if zc is smaller than the absolute value of any line to line voltage, the switching signals are overridden anyway, as explained in Section II-A. In the sequel, trajectories which lead to si (t) ∈ [−1, 1], i ∈ {1, 2, 3}, t ∈ R are termed invalid. Because y is a flat output of the system, it is possible to plan nominal trajectories for y1 , y2 , and y3 independently from each


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other. From (6a)–(6g), it follows that then, the trajectories of all other system variables, including the input variables, are fixed. Note that the system is static state feedback linearizable with the feedback defined by (6e)–(6g). The required state transformation is given by (6a)–(6c). This implies that the system is flat. Note further that, if a system is flat, there is an infinite number of flat outputs, but the choice made in (4) for the present case simplifies trajectory planning. C. Load Observer In the former considerations, the system parameters were supposed to be known. Often, this is indeed the case for the parameters L and C which are known from the design process. The voltages Vd and Vq are measured. To avoid a measurement of the load current, a load observer can be used instead. In [3], a second-order load observer for a three-phase four-wire boost rectifier has been suggested, and in [14], an algebraic load identification algorithm can be found. Here, a reduced-order observer will be used. Introducing the new variable zobs = Czc + kP

(7)

y˙ 1∗ = 3Vd zd∗ /2 − P.

with k constant, its derivative reads z˙obs = 3(sq zq + sd zd )/4 − P/zc

(8)

with the model P˙ = 0. A reduced-order observer is given by zˆ˙ obs = 3(sq zq + sd zd )/4 − Pˆ /zc ˜ c /k Pˆ = zobs − (C + C)z

(9) (10)

where the perturbation C˜ has been introduced in order to model a parameter uncertainty. This means that the value C + C˜ is used in the observer while C is the “correct” parameter. With P˜ = P − Pˆ , the error system reads P˜ C˜ z˙c P˜˙ = − . + kzc k

The variable to be controlled to a certain reference is not always a component of the flat output used for trajectory planning. For instance, in the present case of the three-phase boost rectifier, it is required to control the output voltage of the converter while the input currents are sinusoidal and a certain amount of reactive power is delivered to the grid. In most cases, a unity power factor is needed, which leads to the desired trajectory y2∗ = 0. In the sequel, a trajectory planning algorithm will be discussed. It parameterizes the energy y1 in terms of the desired current zd∗ by the integration of (6a), as presented in [3] for a four-wire system. A similar algorithm for a single-phase power factor correction application can be found in [15] and [16]. For the sake of simplicity, a symmetrical grid is assumed here, with Vq ≡ 0 and Vd constant. For the general case with asymmetrical grid voltages, see [3]. The algorithm allows for voltage regulation in finite time. First, the desired current trajectory has to be specified. Here, the very simple choice of a piecewise constant current zd (t) = zd∗ for t ∈ [ti , ti + T ), i ∈ Z is made, with T being the length of the planning horizon. Rewriting (6a) for the desired trajectory y1∗ under consideration of Vq = 0 yields

(11)

As zc > 0, stability is obvious for k > 0 and C˜ = 0. For finite C˜ = 0, the perturbation is finite and vanishes in steady state. Thus, in steady state, the estimate of P is not affected by this kind of parameter uncertainty. Note that, for the following discussions on the stability of the controllers, a correct load estimate is assumed. D. Trajectory Planning The property of flatness allows for the parameterization of all trajectories of the system in terms of a flat output trajectory. Trajectory planning has the following objectives: • transition between constant operating points; • consideration of state and input constraints; • planning with avoidance of possible singularities. Concerning the third point, it is obvious that any suitable trajectory for y implies zc > 0. Thus, the singularities in (6e)–(6g) are avoided also for the nominal case.

(12)

Integrating (12) gives the corresponding trajectory t → y1∗ (t) = (3Vd zd∗ /2 − P ) (t − ti ) + K

(13)

in which the constant parameters zd∗ and K are still unknown. The planned trajectory should continuously extend the measured one at ti , which means K = y1 (ti ) = y1∗ (ti ).

(14)

Furthermore, at t = ti + Tp , a final value y1,ref = y1∗ (ti + Tp ) is to be reached. According to (13), this means zd∗ = 2 (y1,ref − y1 (ti ) + P Tp ) /(3Vd Tp ).

(15)

However, y1,ref is not directly prescribed. Instead, a final value zc,ref of the desired capacitor voltage is given. This is used in (4a) together with (6a) in which y˙ 1∗ = 0, which yields the value

2 2P 3L C 2 2 + y2∗ + zc,ref . (16) y1,ref = 4 3Vd 2 Due to (14), the nominal trajectory may become discontinuous because the planned trajectory continuously extends the measured one but not necessarily the planned trajectory from the former planning horizon. This resetting does not impose problems because the derivative of the nominal trajectory y1∗ is calculated via (12). Furthermore, the current trajectory zd∗ is possibly not continuous at the time instants where a trajectory update is made because its nominal trajectory is stepwise constant. Thus, y ∗ is an invalid trajectory. Nevertheless, this trajectory can be used, because the length of the planning horizon is chosen according to the dynamics of the voltage which is much “slower” than the dynamics of the current.



For reasons of performance and security, the aforementioned planning algorithm is modified in the following three points. 1) Due to the fact that P is unknown, the estimate Pˆ is used. 2) A periodically executed planning algorithm may be improved by doing additional planning if significant load changes are detected or if the desired output voltage zc,ref changes. This is why the planning takes place with a frequency of at least 1/Tp and, furthermore, if |Pˆ − Pˆ−1 | > Ptol or zc,ref = zc,ref,−1 , where Pˆ−1 and zc,ref,−1 denote the values of Pˆ and zc,ref , respectively, which have been used for the last trajectory update. 3) In particular, for large transitions between equilibrium points, the calculated current zd∗ via (15) may become very large. One may thus implement a bound by replacing (15) with zd∗ = max {2 (y1,ref − y1 (ti ) + P Tp ) /(3Vd Tp ), zd,max } . The trajectory of y2∗ can be chosen independently of y1∗ according to the required amount of reactive power. Usually, there are no requirements with respect to y3 , i.e., one may choose an arbitrary trajectory that does not violate the constraints. In Section V, this freedom is used in order to guarantee si (t) ∈ [−1, 1], i ∈ {1, 2, 3}, t ∈ R without distorting the currents. III. L INEARIZATION -B ASED M ETHODS

Exploiting the flatness of the model for trajectory planning, it is straightforward to employ a static state feedback tracking controller, as it has been shown for a four-wire system in [3]. Introducing the (fictitious) inputs v1 = y¨1 and v2 = y˙ 2 and substituting these into (6e) and (6f) leads to an exact linearization of the model. Now, choosing any stable dynamics for (yi∗ − yi ), i ∈ {1, 2}, e.g.,

0

= y˙ 2∗

− v2 +

kp2 (y2∗

− y2 )

Introducing the inputs v1 = y¨1 and v2 = y˙ 2 in (6e) and (6f) and using the planned trajectory y ∗ —instead of y—the controller can be chosen as sd = ψsd (y1∗ , y2∗ , y˙ 1∗ , v1 )

(19a)

sq = ψsq (y1∗ , y2∗ , y˙ 1∗ , v2 )

(19b)

v1 = y¨1∗ + kd1 e˙ 1 + kp1 e1 + kcp2 e2

(20a)

v2 = y˙ 2∗ + kp2 e2 + kcd1 e˙ 1 + kcp1 e1

(20b)

with

where e1 = y1∗ − y1 and e2 = y2∗ − y2 denote tracking errors and kd1 , kp1 , kcd1 , kcp1 , kp2 , kcp2 ∈ R are controller parameters. Showing the stability of the tracking error dynamics can be facilitated by considering the linearization of the tracking error around the nominal trajectory e˙ = (A + B)e

(21)

with e = (e1 , e˙ 1 , e2 )T . The (time-varying) matrices read ⎞ ⎛ ⎞ ⎛ 0 0 0 0 1 0 6C B = ∗ ⎝ 21 22 23 ⎠ A = ⎝ γ21 γ22 γ23 ⎠ ψ γ31 γ32 γ33 31 32 33 with

A. Exact Feedback Linearization

0 = y¨1∗ − v1 + kd1 (y˙ 1∗ − y˙ 1 ) + kp1 (y1∗ − y1 )

363

(17a) (17b)

γ21 = −kp1 −

54ωCVd3 y2∗ ψ∗

36ωVd LCP ψ∗

2 54Vd2 LCy2∗ 3ωVd 1+ = −kcp2 + 2 ψ∗

γ22 = −kd1 + γ23

γ31 = −kcp1 +

24ωVd CP ψ∗ 3Ly2∗ − 4y1∗ ψ∗ 2

with kp,1 , kp,2 , kd,1 > 0 enforces tracking convergence. Thus, solving (17) for v1 and v2 and using (6e) and (6f), the controller is given by sd = ψsd (y1 , y2 , y˙ 1 , v1 )

(18a)

sq = ψsq (y1 , y2 , y˙ 1 , v2 ).

(18b)

If subordinate current controllers, e.g., hysteresis controllers are used, it is possible to take the currents zd and zq as inputs. In this case, one would choose y˙ 1 as a fictitious input, which also leads to an exact linearization of the model. B. Exact Feedforward Linearization Another approach to ensure tracking is the use of the nominal control law with an additional stabilizing controller around the nominal trajectory. This method was introduced in [8] as “exact feedforward linearization.”

γ32 = −kcd1 + 12ωCVd γ33 = −kp2 −

36ωVd LCP y2∗ ψ∗

21 = 6Vd2 y¨1∗ 22 = 2L (−2¨ y1∗ y˙ 1∗ + 3ω y˙ 1∗ y2∗ Vd − 2¨ y1∗ P ) 23 = −9LVd2 y2∗ y¨1∗ 31 = 2Vd (2ω y˙ 1∗ + 3Vd y˙ 2∗ ) 32 = −4Ly˙ 2∗ (P + y˙ 1∗ ) 33 = −3LVd y2∗ (2ω y˙ 1∗ + 3Vd y˙ 2∗ ) and ψ ∗ = ψ(y1∗ , y2∗ , y˙ 1∗ ) according to (5). Note that ψ ∗ > 0 is guaranteed by an appropriate choice of the nominal trajectory y ∗ . Note further that the matrix B vanishes if y ∗ is a stationary trajectory, i.e., if y ∗ is an equilibrium point.


364


The matrix A can be chosen arbitrarily. A first choice would be ⎞ ⎛ 0 1 0 (22) 0 ⎠ A = ⎝ −ωc2 −2δc ωc 0 0 −1/τc with ωc , δc , τc ∈ R positive, which makes A time invariant. Note that comparing to (17), the cross-coupling parameters kcd1 , kcp1 , and kcp2 allow for decoupling in the linearized model. With this choice, A becomes time invariant and the stability of the system e˙ = Ae is obvious. For stationary trajectories y ∗ , the term Be becomes zero which implies the stability of the error sytem in steady state—see [17, Ex. 9.6].

which reduces to z˙c = 0 if input–output power balance is considered for trajectory planning. IV. P ASSIVITY -B ASED M ETHODS Many power electronic systems can be represented in the form ([9] and [11], for example) Az˙ = J(s)z + RD z + Bs + E

(27)

with the state vector z, the vector of input variables s = (s1 , s2 , . . . , sn )T , a positive diagonal matrix A, a skew symmetric matrix J(s) = J0 + ni=1 Ji si , a vector of external sources E, and a diagonal and negative semidefinite matrix RD . The latter corresponds to the inherent or “natural” (resistive) damping.

C. Input–Output Linearization For the concept of input–output linearization, it is necessary to specify an output y ◦ = h(z) of the system which is to be controlled. The control law is chosen accordingly, such that the input–output behavior is linearized. Note that, if choosing y ◦ = (y1 , y2 ), this is equivalent to exact feedback linearization. It has been shown in [7] that y ◦ = (zc , zq ) is not a suitable output of the system [(2)], because the remaining dynamics on zd is unstable. However, the same output has been proposed in [6], [18], and [19] for a modified model, where (2c) has been replaced by C z˙c = 3(Vd zd + Vq zq − 2P )/(2zc ).

(23)

Here, the current delivered to the capacitor and the load has been calculated by the power balance between the ac input toward the dc output (right before capacitor and load) neglecting the energy stored in the inductors. Note that, for the models (2a), (2b), and (23), the output y ◦ = (zc , zq ) is a flat output which means that input–output linearization is equivalent to exact feedback linearization. Instead, using the output y ◦ = (zd , zq ) for model (2) and its desired trajectories zd∗ = ψzd (y2∗ , y˙ 1∗ ) and zq∗ = ψzq (y2∗ ), the control law yielding v1 = z˙d and v2 = z˙q reads 2 (Vd − L v1 + Lωzq ) zc 2 sq = (Vq − L v2 − Lωzd ) zc

sd =

(24a) (24b)

where z˙d and z˙q have been substituted by the inputs v1 and v2 . As a stabilizing feedback controller, one may use v1 = z˙d∗ + kp,d (zd∗ − zd ) v2 = z˙q∗ + kp,q zq∗ − zq

(25a) (25b)

which yields exponential dynamics for zd∗ − zd and zq∗ − zq , respectively. The zero dynamics reads C z˙c = 3(Vd zd + Vq zq − 2P )/(2zc )

(26)

A. Damping Injection Derivation of passivity-based energy shaping and damping injection controllers for a lot of mechanical and electrical systems can be found in [9]. For the sake of completeness, the main idea is briefly sketched here. A typical way to derive such a controller starts with the establishment of an “auxiliary system” [9]. This is made of a copy of the system, thus sharing all important properties of the “original system,” but having additional damping. An auxiliary system for (27) is Azˆ˙ = J zˆ + RD zˆ + Bs + E − RA (z − zˆ).

(28)

The matrix RA enables the injection of additional damping as one can verify easily after calculation of the system describing the dynamics of the error z˜ = z − zˆ Az˜˙ = J z˜ + (RD + RA )˜ z.

(29)

Stability can be shown using the function V = (˜ z T A˜ z )/2

(30)

V˙ = z˜T J z˜ + z˜T (RD + RA )˜ z.

(31)

and its time derivative

With a suitable choice of RA , it can be shown that the point z˜ = 0 is (asymptotically) stable in the sense of Lyapunov. Note that the term z˜T J z˜ is zero due to the skew symmetry of J. Note further that the stability of the solutions of (29) is not related to the stability of those of (27) and (28). Following [9], system (28) is simulated as part of the controller. Furthermore, a switching mode controller is implemented in order to achieve tracking between some variables of zˆ and their steady state values. One has to check that the zero dynamics is stable. This approach does not make use of the fact that system (27) is possibly flat. However, one may modify this concept in two aspects. 1) The control law can be obtained by solving (28) for s, which means that a suitable output y ◦ of (28) is chosen in order to perform an input–output linearization [10]. The remaining dynamics can be calculated as controller states, and



one has to check that it is stable. 2) The components used for the output y ◦ are set to the corresponding nominal trajectories generated by a trajectory planning algorithm [20]. This has the following advantages: Modification 1) reduces the number of equations that have to be integrated, and if the system is flat, modification 2) allows the parameterization of the system in terms of a flat output. Both modifications are assumed in the sequel. An auxiliary system for (2) is zq + ld (zd − zˆd ) L zˆ˙ d = Vd − zˆc sd /2 + Lωˆ

(32a)

zd + lq (zq − zˆq ) L zˆ˙ q = Vq − zˆc sq /2 − Lωˆ

(32b)

C zˆ˙ c = 3(sq zˆq + sd zˆd )/4 − zˆc /R + lc (zc − zˆc ) (32c) where the load is modeled as a resistor for compatibility with the passivity-based flavor of the method. With (2) and (32), the error system reads

of this paper. The derivation of such a controller will be given here in a general setting for a better understanding and also to point out the difference with respect to the damping injection controller from Section IV-A. Planning valid trajectories for system (27) may cause invalid control signals, because the control is derived using (28). This is why, now, the auxiliary system is an exact copy of the “original system” s+E Azˆ˙ = Jˆzˆ + RD zˆ + Bˆ

(37)

with Jˆ = J(ˆ s). The error system reads Az˜˙ = J0 z˜ +

n

Ji si z −

i=1

= J z˜ +

n

n

Ji (si − s˜i )ˆ z + RD z˜ + B˜ s

i=1

Ji s˜i zˆ + RD z˜ + B˜ s

i=1

zq − ld z˜d L z˜˙ d = − z˜c sd /2 + Lω˜

(33a)

L z˜˙ q = −˜ zc sq /2 − Lω˜ zd − lq z˜q

(33b)

C z˜˙ c = 3(sq z˜q + sd z˜d )/4 − z˜c /R − lc z˜c

(33c)

with z˜d = zd − zˆd , z˜q = zq − zˆq , and z˜c = zc − zˆc . The stability of (˜ zd , z˜q , z˜c ) = (0, 0, 0) can be shown using the following Lyapunov function candidate: (34) zq2 /2 + C z˜c2 /2. V = 3L˜ zd2 /2 + 3L˜ With (33), the derivative reads 3 z˜2 3 V˙ = − ld z˜d2 − lq z˜q2 − c − lc z˜c2 . 2 2 R

365

(35)

The derivative V˙ is negative definite with ld , lq , lc > 0. Thus, with this choice of the design parameters, the origin is asymptotically stable in the sense of Lyapunov. zd , zˆq ), analogously to Choosing the output y ◦ = (ˆ Section III-C, the controller is given by 2 Vd − L zˆ˙ d + Lωˆ zq + ld (zd − zˆd ) (36a) sd = zˆc 2 zd + lq (zq − zˆq ) (36b) sq = Vq − L zˆ˙ q − Lωˆ zˆc 3 C zˆ˙ c = (sq zˆq + sd zˆd ) − zˆc /R + lc (zc − zˆc ). (36c) 4 With lc = 0, the zero dynamics of (36c) is stable [10]. Remember that, due to modification 2), zˆd = ψzd (y2∗ , y˙ 1∗ ) and zˆq = ψzq (y2∗ ). Notice that (36a) and (36b) are similar to (24) and (25), except that the measured voltage zc is used for input–output linearization, while the estimated voltage zˆc from (36c) is used for the passivity-based approach. B. Exact Tracking Error Dynamics Passive Output Feedback In [11], an alternative passivity-based approach has been suggested which will be termed ETEDPOF in the remainder

= J z˜ + RD z˜ + B ∗ s˜

(38)

with s˜ = s − sˆ, B ∗ = [b1 + J1 zˆ, . . . , bn + Jn zˆ], and bi ’s as the columns of B. In order to show the stability of (38), the function V from (30) can be used again. Its derivative reads V˙ = z˜T J z˜ + z˜T RD z˜ + z˜T B ∗ s˜.

(39)

Recall that z˜T J z˜ is zero because J is skew symmetric. Choosing the controller as s˜ = ΓB ∗ z˜ T

(40)

with a negative definite symmetric matrix Γ such that RD + T B ∗ ΓB ∗ is negative definite, V˙ becomes negative definite, and thus, the origin is, in principle, globally asymptotically stable. However, due to possible saturations of the limited control actions, one may conclude on semiglobal asymptotical stablility of the origin. T Note that the condition that RD + B ∗ ΓB ∗ must be negative definite means that the natural dissipative structure of the system must possibly be completed by the dissipation achieved by feedback. There might be cases where this becomes a difficult task particularly because B ∗ depends on zˆ. This approach does not require the computation of controller states, which is an important advantage with respect to the controller presented in Section IV-A. Using this alternative approach, an auxiliary system for (2) is given by zq L zˆ˙ d = Vd − zˆc sˆd /2 + Lωˆ

(41a)

zd L zˆ˙ q = Vq − zˆc sˆq /2 − Lωˆ

(41b)

sq zˆq + sˆd zˆd )/4 − zˆc /R. C zˆ˙ c = 3(ˆ

(41c)

Again, the load is modeled as a resistor for the same reasons as in Section IV-A.


366


The error system reads zc sd /2 − zˆc s˜d /2 + Lω˜ zq L z˜˙ d = −˜ ˙ L z˜q = −˜ zc sq /2 − zˆc s˜q /2 − Lω˜ zd C z˜˙ c = 3(sq z˜q + s˜d zˆd + sd z˜d + s˜q zˆq )/4 − z˜c /R

(42a) (42b) (42c)

with z˜d = zd − zˆd , z˜q = zq − zˆq , z˜c = zc − zˆc , s˜d = sd − sˆd , and s˜q = sq − sˆq . The stability of its origin can be shown using the Lyapunov function V as defined in (34). Choosing zc z˜d + z˜c zˆd )/4 s˜d = 3ld (−ˆ

(43a)

s˜q = 3lq (−ˆ zc z˜q + z˜c zˆq )/4

(43b)

which corresponds to a diagonal matrix Γ with entries ld and lq in (40) V˙ = 9 ld (−ˆ zc z˜d + z˜c zˆd )2 + lq (−ˆ zc z˜q + z˜c zˆq )2 /16 − z˜c2 /R. (44) Choosing ld , lq < 0, V˙ is negative definite, which implies the asymptotic stability of the origin in the sense of Lyapunov. Finally, the control input applied to the system reads sd = ψsd (y1∗ , y2∗ , y˙ 1∗ , y¨1∗ ) + s˜d

(45a)

sq = ψsq (y1∗ , y2∗ , y˙ 1∗ , y˙ 2∗ )

(45b)

+ s˜q

where the nominal values in (43) are calculated as zˆd = ψzd (y2∗ , y˙ 1∗ ), zˆq = ψzq (y2∗ ), and zˆc = ψzc (y1∗ , y2∗ , y˙ 1∗ ) by (6a)–(6c). Note that, for y˙ 1∗ ≡ 0 and y2∗ ≡ 0, this leads to the same controller as proposed in [12]. Here, the difference with respect to [12] is that the stability of the solution of (42) is shown also for nonstationary trajectories. V. D EALING W ITH I NVALID S WITCHING F UNCTIONS If the value of one of the switching functions si , i = 1, 2, 3 is larger than 1 or lower than −1, a natural limitation applies. This may cause distortions in the line currents. However, the trajectory planning algorithm given in Section II-D did not impose any requirements on y3 . In the sequel, this freedom will be exploited in order to deal with invalid switching patterns. Introducing the converter voltages vci = Vi − Lz˙i , i ∈ {1, 2, 3} as the voltages at the switches with respect to the midpoint of the grid leads to vc = V0 (1, 1, 1)T +

zc Ms 6

(46)

with vc = (vc1 , vc2 , vc3 )T , s = (s1 , s2 , s3 )T , and ⎛ ⎞ 2 −1 −1 M = ⎝ −1 2 −1 ⎠ . −1 −1 2 The triple of the switching functions s may be interpreted as a vector in a cartesian coordinate system. Then, all combinations of images of valid switching functions can be visualized as a cube with an edge length of two.

The matrix M is of rank 2 and has the kernel ker M = λ(1, 1, 1)T , λ ∈ R. It shows the characteristics of an orthogonal projection. The points of the “cube” are projected along ker M on a plane, constituting the well-known hexagonal shape. It is straightforward that using s∗ = s + λ(1, 1, 1)T instead of s leads to the same converter voltages. This implies that s∗0 = s0 + λ. If some points of the calculated trajectory of the switching functions are located “outside the cube,” which means that they take values that are not in the interval [−1, 1], it is possible to make the trajectory valid by an appropriate choice of λ if |si (t) − sj (t)| ≤ 2

∀i, j ∈ {1, 2, 3}.

(47)

Inequality (47) ensures that vc does not lie outside the wellknown hexagon, marking possible converter voltages. Assuming that condition (47) is satisfied, λ can be chosen according to ⎧ − max ({si (t)}) + 1, if max ({si (t)}) > 1 ⎪ ⎨ i=1,2,3 i=1,2,3 λ(t) = − min ({si (t)}) − 1, if min ({si (t)}) < −1 ⎪ i=1,2,3 ⎩ i=1,2,3 0, else. (48) Thus, the invalid points of s are projected along the kernel of M on the “surface of the cube.” In contrast to the use of three independent limiters, this method does not affect the converter voltages. An application of this method on a Vienna rectifier can be found in [21]. Note that this concept is closely related to any method increasing the modulation index by adding a zero sequence s0 , such as third harmonic injection (see, e.g., [22]). Moreover, it is possible to influence the distribution of the freewheeling states2 which itself influences the power losses due to switching and current ripple [23]. The latter is not in the focus of this paper. If condition (47) is not satisfied, one may find a similar algorithm that yields switching functions corresponding to converter voltages which are “as close as possible” to the desired, but invalid, ones. VI. S IMULATION R ESULTS Simulations with MATLAB/Simulink are performed using a switched model of the plant equipped with additional parasitic components. The parameters of the converter are C = 680 μF and L = 1 mH, the parasitic resistance of the inductors is 1 mΩ, the forward voltage of the power electronic switches is 4 V, and for the diodes, it is 1 V. The load used for simulations is modeled as a constant power load, also for the passivity-based approaches. The rated dc-load power is Pr = 30 kW. √ The line voltage is parameterized by ω = 2π 50 s−1 , Vd = 2 230 V, and Vq = 0. The model operates at fixed switching frequency fs = 10 kHz, where the switching functions were transformed using the inverse of (1) and subsequently implemented as duty ratios. Measurements, observer, and controller output are 2 Freewheeling states mean σ = 1, respectively, σ = −1 for all i ∈ i i {1, 2, 3}.



updated with twice the switching frequency. Thus, asymmetrical PWM signals are applied to the system. All integrals are implemented using the trapezoidal rule. The parameters of the flatness-based trajectory planning algorithm are Tp = 1 ms, Ptol = 1 kW, and zd,max = 100 A. A white noise with zero mean and a variance of one percent of the nominal value is used to simulate measurement noise. A first-order low-pass filter with cutoff frequency ωf = 2πfs is used as an antialiasing filter. The proposed algorithm for preventing from invalid switching patterns has been used. The parameter of the observer (9), (10) is k = 3 · 10−3 A−1 . The controller parameters are chosen as follows: • exact feedback linearization: kp1 = 2 · 107 s−1 , kd1 = 2 · 104 s−2 , and kp2 = 2 · 104 V · s−1 ; • exact feedforward linearization: δc = 2.2, ωc = 4.5 · 103 s−1 , and τc = 10−4 s according to (22); • input–output linearization: kp,d = kp,q = 4 · 104 s−1 ; • passivity-based damping injection: ld = lq = 40 VA−1 ; (VA)−1 and lq = • ETEDPOF: ld = 1 · 10−4 −4 −1 1 · 10 (VA) . For ease of comparison, the following linear controller (with nonlinear feedforward) is implemented: sd = ψsd (y1∗ , y2∗ , y˙ 1∗ , y¨1∗ ) + kp,d (zd − zd∗ ) sq = ψsq (y1∗ , y2∗ , y˙ 1∗ , y¨1∗ ) + kp,q zq − zq∗

(49) (50)

with zd∗ = ψzd (y2∗ , y˙ 1∗ ), zq = ψzq (y2∗ ), and parameters kp,d = kp,q = 0.1 A−1 . Due to the fact that, in practice, model (2) does not perfectly describe the behavior of the system, the tracking may be affected (particularly for stationary trajectories y ∗ ). This is why one may modify the control laws sd and sq by adding

367

Fig. 2. (Solid) Load adaptation after a load step from 50% to 100% rated power at t = 0 and (dashed) reference. The different responses are due to the variations of the parameter C used in the observer: (Light gray) 0.8 C, (dark gray) C, and (black) 1.2C.

deviation from the nominal value in steady state is due to the nonideal model used for the simulations. In Fig. 3, the flatness-based trajectory planning algorithm is compared with the following linear controller: zd,ref (t) = kp,c (zc (t) − zc,ref (t)) t

∗ s+ d (t) = kc (zC (t) − zC (t)) + ki,c

∗ zC (τ ) − zC (τ )dτ (51)

t1

t s+ q (t)

= ki,q

y2 (τ ) − y2∗ (τ )dτ

(52)

t2

in order to get rid of steady state errors. The times t1 and t2 denote the time instants when y1∗ and y2∗ , respectively, become stationary. The parameters are kc = 0.08 V−1 , ki,c = 4 (Vs)−1 , and ki,q = 50 (As)−1 for all control approaches to get a better comparability of the results. Simulations were carried out in order to compare the responses to a load step from 50% to 100% rated power and a transfer of the output voltage from 650 to 700 V. The load estimation process can be observed in Fig. 2 for stabilization via exact feedback linearization. For other stabilization methods, the behavior is similar. The parameter C used in the observer has been varied in order to test the robustness of the observer with respect to variations of that parameter. The simulations reveal that, corresponding to the dynamics (11) of the observation error, the dynamic response is affected by this kind of perturbation, but the steady-state value is not. The slight

(53)

0

sd (t) = kp (zd (t) − zd,ref (t)) t zd (τ ) − zd,ref (τ )dτ

+ ki

(54)

0

sd (t) = kp (zq (t) − zq,ref (t)) t zq (τ ) − zq,ref (τ )dτ

+ ki t

zc (τ ) − zc,ref (τ )dτ

+ ki,c

(55)

0

with kp,c = −1 AV−1 , ki,c = −0.06 A(Vs)−1 , kp = 0.1 A−1 , and ki = 0.005 (As)−1 . The voltage zc,ref is the desired output voltage which is 650 V until at t = 15 ms when it changes abruptly to 700 V. Between t = 10 ms and t = 20 ms, the desired value zq,ref of the quadrature component of the current linearly increases from 0 to 20 A. In order to prevent from too large currents which may cause damages, zd,ref is limited to zd,max = 100 A. In order to give a “fair” comparison between the approaches, the parameters of the linear controller have been tuned in such a way that the voltage restoration after the load step and the transition time after the equilibrium to equilibrium transfer (without overshoot) are similar to the results obtained with the flatness-based controller. For the linear controller, it can be seen that, directly after the transition, a voltage overshoot occurs, while for the flatness-based algorithm, a finite time transfer is almost achieved. The tracking behavior with respect to the reactive current is similar for both approaches. In Fig. 4, the response to a load step and the transition of the output voltage are compared for the proposed controllers. The graphs show the nominal behavior, as specified by the trajectory planning algorithm (gray) and the simulated behavior (black).


368


Fig. 3. Comparison of the flatness-based trajectory planning algorithm with a linear controller: Response to a load step from 50% to 100% rated power at t = 0 and a transition of the output voltage from 650 to 700 V beginning at t = 15 ms. Furthermore, the reactive current increases from t = 10 to t = 20 ms from 0 to 20 A. In the four upper diagrams, the curves corresponding to zd and zq are drawn in black and gray, respectively. In the two lower diagrams, the grid √ voltages have been plotted (scaled by 1/(2.3 2)) in gray in order to illustrate the generation of reactive power. (a) Flatness-based trajectory planning algorithm. (b) Cascaded linear controller.

For all stabilization methods, the responses to the load step look very similar. For the transition between equilibrium points, on one hand, exact feedback linearization, exact feedforward linearization, and the ETEDPOF approach show similar results, while on the other hand, input–output linearization, damping injection, and the linear controller with feedforward give a similar overshoot after the transition. The robustness with respect to parameter uncertainty has been tested by simulation. As a result, changes of up to 20% of the value of the parameter L used in the controllers do not significantly change the response of any stabilization scheme proposed. Variations in the parameter C are taken into account by the integral parts in steady state. Moreover, the robustness to line disturbances has been tested by adding a third harmonic to the line voltages. With this kind of perturbation, no differences between the stabilization strategies have been observed. This is also the case for increased additive measurement noise of up to 5% of the rated values.

systems having a specific structure, and its application for the present system has been shown. The flatness-based trajectory planning algorithm showed a better performance than a cascaded linear controller for transitions between equilibrium points. The response to a load step seemed to be similar for both approaches. Simulations revealed that, in general, the current controllers presented here offer a very similar behavior. There have been slight differences for the transition of the output voltage between the exact feedback and exact feedforward linearization methods as well as the ETEDPOF controller on one side and the remaining controllers on the other side, but they are not significant. It turns out that the use of a flatness-based trajectory planning algorithm is more important for the performance of the system than the use of a special current controller because the subsystem describing the current dynamics is the “faster” subsystem anyway. A PPENDIX

VII. C ONCLUSION The model of the three-phase three-wire boost converter is flat. A trajectory planning algorithm that allows for voltage regulation in finite time has been presented. A reduced-order load observer has been suggested. Five stabilization concepts that make use of the flatness of the model have been presented. The linearization-based methods exact feedback linearization, exact feedforward linearization, and input–output linearization as well as two passivity-based methods have been discussed. The energy shaping and damping injection method has been reviewed, and an alternative approach based on ETEDPOF has been suggested. The latter has been introduced in general for

This appendix gives a brief introduction to the concept of flatness. For more information, the reader is referred to [25]–[27]. Let the system model be given by differential equations of the form ˙ z¨, . . . , z (σi ) = 0, i = 1, . . . , p (56) Si z, z, where z = (z1 , . . . , zn ) are the system variables, including also the input variables. Such a system is called (differentially) flat, if there exists an m-tuple y of functions ˙ z¨, . . . , z (ηi ) , i = 1, . . . , m (57) yi = ψi z, z,



369

Fig. 4. Response to a load step from 50% to 100% rated power at t = 0 and a transition of the output voltage from 650 to 700 V beginning at t = 15 ms with stabilization via (from top to bottom) (a) exact feedback linearization, (b) exact feedforward linearization, (c) input–output linearization, (d) a passivity-based controller with damping injection, (e) an ETEDPOF controller, and (f) a linear PI controller. The gray curves correspond to the nominal behavior, while the curves in black belong to the simulated behavior.

fulfilling the following conditions. 1) The components of y are differentially independent, i.e., there does not exist a (nontrivial3 ) differential equation of the form (58) R y, y, ˙ y¨, . . . , y (χ) = 0 which means such a relation in y only cannot be derived from (56). 2) The system variables in z can be expressed by functions of y and its time derivatives zi = φi y, y, ˙ y¨, . . . , y (γi ) , i = 1, . . . , s. (59) 3) This implies that the trajectories of the flat output y provide a parameterization for the trajectories of the system variables (including the input variables). Such an m-tuple y is called a flat output of the system. 3 Nontrivial

means it cannot be reduced to 0 = 0.

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Albrecht Gensior received the Dipl.-Ing. degree in electrical engineering from the Technische Universität Dresden (TU Dresden), Dresden, Germany, in 2003, where is currently working toward the Dr.-Ing. degree in electrical engineering. He is currently with the Professur Leistungselektronik, Elektrotechnisches Institut, TU Dresden, where he has been involved in a research project on flatness-based control of power electronic systems funded by the Deutsche Forschungsgemeinschaft (DFG). His current research project is related to sliding-mode control for electrical drives, which is also funded by the DFG. His research interests are mainly in nonlinear controller design and observers for power electronic systems.

Hebertt Sira-Ramírez received the degree in electrical engineering from the Universidad de Los Andes (ULA), Mérida, Venezuela, in 1970, and the M.Sc. and Ph.D. degrees in electrical engineering from the Massachusetts Institute of Technology, Cambridge, in 1974 and 1977, respectively. For 28 years, he was with the ULA from where he is a Retired Professor. Since 1998, he has been a Titular Researcher with the Sección de Mecatrónica, Departamento de Ingeniera Eléctrica, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, México City, Mexico. He is the author of 127 technical articles in refereed journals and 23 book chapters and is the coauthor of five books. He has participated in 230 international conferences. He is interested in the switched control of nonlinear systems, particularly in the control of power electronics systems. He has been involved in the development of algebraic approaches to state and parameter estimation for the control of uncertain systems.

Joachim Rudolph received a Doctorat from the Université Paris XI, Orsay, France, in 1991, and the Dr.-Ing. habil. degree from the Technische Universität Dresden (TU Dresden), Dresden, Germany, in 2003. He is a Privatdozent with TU Dresden, where he is with the Institut für Regelungs- und Steuerungstheorie. His main research interests are in controller and observer design for nonlinear and infinite-dimensional systems, algebraic systems theory, and the solution of demanding practical control problems.

Henry Güldner (M’03) received the Dipl.-Ing., Dr.Ing., and Dr.-Ing. habil. degrees in electrical engineering from the Technische Universität Dresden (TU Dresden), Dresden, Germany, in 1967, 1971, and 1979, respectively. He was a Research Assistant (1967–1971) and an Assistant Professor (1971–1976) with TU Dresden. From 1976 to 1982, he was with the Institut für Mikroelektronik, Dresden. From 1982 to 1989, he was an Associate Professor of Electrical Engineering with TU Dresden. He was a Full Professor of Power Electronics with the Hochschule für Verkehrswesen, Dresden, from 1989 to 1993. From 1993 to 2007, he was a University Professor of Power Electronics with TU Dresden. He is currently with the Professur Leistungselektronik, Elektrotechnisches Institut, TU Dresden, as Professor Emeritus. His main research interests are in power electronics, modeling of power-electronic devices and systems, ac drives, and electronic ballast. Prof. Güldner is a member of the Verein Deutscher Ingenieure and the European Power Electronics and Drives Association.
