On the Robust Control of Buck-Converter DC-Motor Combinations

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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 8, AUGUST 2013

On the Robust Control of Buck-Converter DC-Motor Combinations Hebertt Sira-Ram´ırez and Marco Antonio Oliver-Salazar

Abstract—The concepts of active disturbance rejection control and flatness-based control are used in this paper to regulate the response of a dc-to-dc buck power converter affected by unknown, exogenous, time-varying load current demands. The generalized proportional integral observer is used to estimate and cancel the time-varying disturbance signals. A key element in the proposed control for the buck converter-dc motor combination is that even if the control input gain is imprecisely known, the control strategy still provides proper regulation and tracking. The robustness of this method is further extended to the case of a double buck topology driving two different dc motors affected by different load torque disturbances. Simulation results are provided. Index Terms—Active disturbance rejection control, cascaded buck converters, generalized proportional integral (GPI) observer, time-varying loads.

I. INTRODUCTION INEAR regulators are unattractive in modern electronic applications due to high dissipation, transformer size, and large storage capacitors requirements for nominal power operation values, aside from the fact that they cannot provide the extended hold-up time required for the controlled shutdown in digital storage systems (see Pressman [1]). The buck power converter is an appropriate alternative for the dc-to-dc bus voltage regulation as it has low internal losses and hence high-power conversion efficiency. Moreover, the buck–buck converter (a tandem connection of two buck converters) is a reliable alternative to simultaneously regulate two dc output voltage signals normally required in electronic applications (for example, a 12-V supply for electronic components/sensors and a 5-V supply for TTL circuits or two voltage-regulated dc motors driving a robot arm and forearm with variable torques each). All this by considering the voltage step-down feature of this converter. The buck–buck topology requires an adequate control scheme to achieve the required output voltage regulation specially under the action of load variations. Therefore, the buck topology together with a proper control scheme can provide, among other features, an efficient energy managing and robustness against

L

Manuscript received March 23, 2012; revised May 24, 2012 and August 8, 2012; accepted October 22, 2012. Date of current version January 18, 2013. Recommended for publication by Associate Editor J. Hur. H. Sira-Ram´ırez is with the Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional, Electrical Engineering, Colonia San Pedro Zacatenco, 07300 Mexico City, Mexico (e-mail: [email protected]). M. A. Oliver-Salazar is with the Centro Nacional de Investigacion y Desarrollo Tecnologico, Mechatronics, 62490 Cuernavaca, Morelos, Mexico (e-mail: [email protected]). Digital Object Identifier 10.1109/TPEL.2012.2227806

exogenous or endogenous disturbances in a dc distribution system. Different research works have addressed the problem of disturbance rejection in dc-to-dc electronic converters: In [2], Hwu and Yau propose a PID controller plus a linear-to-nonlinear translator to enhance the performance of a boost converter. In the work of Boora et al. [3], load disturbance rejection is achieved in a positive buck–boost converter by decoupling the capacitor voltage from the inductor current with the disadvantage of an increase in the switching loss. The work presented by Morales et al. in [4] uses current-mode control in a quadratic buck converter to obtain a certain degree of robustness in the transient response due to step changes in the load. The transient response for step changes in the load resistance of a boost converter is analyzed by Zengshi et al. in [5] under the action of a cascade control of the PI type and under a sliding-mode-control scheme. A hybrid digital adaptive control is used by Babazadeh and Maksimovic in [6] to achieve near-time-optimal responses for a wide range of step-load transients in a buck converter. The basis of active disturbance rejection control (ADRC) relies on the on-line estimation of the system disturbance for its proper cancelation. For earlier works on this topic see Johnson [7] and [8]. The concept has been further addressed by Han et al. (see [9]–[13]), where the extended state observer (ESO) is used to estimate internal and/or exogenous disturbances in the interest of their subsequent cancelation. Practical applications of this type of control can be found in the work of Su et al. [14], She et al. [15], and Zheng and Cao [16]. The ESO alternative is a convenient method as it provides an efficient disturbance estimation even without an exact knowledge of the system parameters. However, the ESO approach is based on a nonlinear configuration that is complemented by nonlinear state feedback, making it somewhat difficult for implementation (see Feng et al. [17]). In contrast, the generalized proportional integral observer (GPI), used in this study, also provides an efficient disturbance estimation without the requirement of an exact knowledge of the system parameters, with the additional advantage of been a simpler and easier to implement observer as it is based on a linear configuration. There are applications in areas such as mobile robotics, induction motors, electronic circuits, and robotic manipulators (see Sira-Ramirez et al. [18]–[20] and Ramirez-Neira et al. [21], respectively), where the use of the ADRC based on GPI observers has improved the uncertain system response compared to other control methods. An application of GPI controllers can be found in [22] where the effect of voltage sags in active rectifiers are compensated without the use of a specific voltage sag detection algorithm. These results motivate the authors of this study to rely on the GPI observerbased ADRC methodology to address the complex problem of

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SIRA-RAMÍREZ AND OLIVER-SALAZAR: ON THE ROBUST CONTROL OF BUCK-CONVERTER DC-MOTOR COMBINATIONS

independently controlling dc motor driven by cascaded buck power converters. Another important contribution to ADRC has been recently provided by Fliess (see [23] and [24]) who proposed the intelligent PID control scheme for controlling systems with state-dependent perturbation inputs. The theoretical basis of these developments can be found in [25]. On the other hand, recalling that flatness is a system property that extends the notion of linear controllability to nonlinear dynamical systems, a system with the flatness property (called a flat system), has a flat output that can be used to explicitly express all states, and inputs, in terms of the flat output and a finite number of its time derivatives. The basis of flatness-based control can be found in the work by Fliess et al. [27]. Linear GPI observers are used to locally estimate the joint effects of exogenous and endogenous additive disturbances, while the control input gain is assumed to be known and, at most, output dependent. GPI observers are capable of accurate on-line estimations of: 1) the output related phase variables; 2) the, state dependent, additive perturbation input signal itself; and 3) the estimation of a certain number of the perturbation input time derivatives. This last feature facilitates the task of perturbation input prediction as GPI observers are most naturally applicable to the control of perturbed differentially flat nonlinear systems with measurable flat outputs (see the work of Sira-Ramirez et al. [28], [29] and [30]). One of the objectives of this study is to present an essentially linear methodology providing a solution to the problem of controlling combinations of electrical machines and dcto-dc power converters of rather complex analytical nature. This study is organized as follows: Section II presents the problem formulation, the fundamental background results under which the proposed control methodology is established and the generalities about the GPI observers. In Section III, two particular cases are considered in the study of the proposed robust control scheme under the action of load variations: 1). The single buck-converter dc-motor topology; and 2). The cascaded buck converters dc-motors system. Section IV presents the obtained simulation results for each one of the two aforementioned cases. The conclusions of this study are presented in Section V. Section VI contains the appendixes where some complex explicit differential parametrization expressions for the treated systems state variables and control inputs are presented.

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B. Background Results An important difference (and a new type of equivalence) between nonlinear differential equations and the corresponding linear differential equations with injected exogenous timevarying signals is established. Consider the following nonlinear, time-varying, vector differential equation with a linear term y(t) ˙ = Ay + φ(t, y), y(0) = y0 , y ∈ n

(1)

where A is a Hurwitz, n × n, matrix of constant coefficients and φ(t, y) is a vector of nonlinearities including time-varying signals. The solution of the aforementioned nonlinear system of differential equations is denoted by y(t, y0 ). Let z be an n-dimensional vector and consider the following linear system with a time-varying exogenous disturbance injection: z(t) ˙ = Az + φ(t, y(t, y0 )), z(0) = y0 + b

(2)

where b is a constant. The following two remarks take place: Remark 1: The time-varying vector y(t, y0 ) trivially satisfies the identity y(t, ˙ y0 ) = Ay(t, y0 ) + φ(t, y(t, y0 )), y(0, y0 ) = y0 .

(3)

Remark 2: The linear differential equation for z includes a copy of the nonlinear term φ(t, y) particularized for the solution y = y(t, y0 ) of the nonlinear differential equation. As such, for every fixed initial condition y0 the term φ(t, y(t, y0 )) is a function of time and is denoted by ξ(t) = φ(t, y(t, y0 )). If the error vector is defined as e = y − z, then, e satisfies the following linear differential equation: e˙ = Ae, e(0) = −b. Since A is a Hurwitz matrix then limt→∞ e(t) = 0 for all b ∈ n . Moreover, if b is set to zero, and the initial conditions for y and z coincide, then y ≡ z for all t ≥ 0. The natural consequence of this simple fact is that the nonlinear system y(t) ˙ = Ay + φ(t, y), y(0) = y0 , y ∈ n and the linear system with exogenous injection z(t) ˙ = Az + φ(t, y(t, y0 )), z(0) = y0

II. MAIN STATEMENT AND SUPPORTING REMARKS A. Problem Formulation Given a system composed by the cascade connection of a controlled buck converter and a dc motor, it is desired to have the motor shaft motion track a given smooth reference angular velocity profile in the presence of an unknown time varying load torque acting on the motor. The monovariable results are to be extended to the decoupled angular velocity trajectory tracking control of two identical dc motors, also subject to independent time varying unknown load torque variations, fed by two cascaded controlled buck converters.

may be regarded as equivalent. The distinction between z and y becomes irrelevant. The practical implication of this is that any problem (such as an observer design problem) defined on the nonlinear system, may be now posed on the equivalent linear dynamics, viewed without any ambiguity as y˙ = Ay + ξ(t), y(0) = y0 .

(4)

Customarily, in practice, the nonlinear function φ(t, y) is unknown due to unknown parameter values or due to complex nonlinearities, or simply due to the presence of unknown exogenous time signals affecting φ(t, y). Let ξˆ be an estimate of the ˆ unknown time function ξ(t) such that the difference ξ(t) − ξ(t) is guaranteed to be uniformly absolutely bounded, in norm, by

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a small positive constant , i.e., ˆ supt ξ(t) − ξ(t) ≤

(5)

then it can be shown that the error vector e is ultimately uniformly absolutely bounded in norm by a small positive constant δ() that depends on the eigenvalue λ of A with the largest (negative) real part, i.e., that which is closest to the imaginary axis in the complex plane. Since A is Hurwitz, for any given symmetric positive definite matrix Q = QT > 0. There exists a positive definite matrix P , such that AT P + P A = −Q. Consider the Lyapunov function candidate V (e) = 12 eT P e, P = P T > 0. Then, along the solutions of the perturbed error system ˆ e˙ = Ae + ξ(t) − ξ(t) (6) the time derivative of V (e) satisfies 1 ˆ V˙ = eT AT P + P A e + eT P ξ(t) − ξ(t) 2 1 = − eT Qe + e P 2 1 ≤ − e 2 λm in (QQT ) + P e 2

λm in AAT P e 2 + P e . ≤− V˙ (e) is negative outside the ball B defined by e ∈ n | e ≤ δ() =

B=

ˆ = z1 ξ(t) y˙ 1 = y2 + λp+n −1 (y − y1 ) y˙ 2 = y3 + λp+n −2 (y − y1 ) (7)

(8)

and, hence, all solution trajectories e(t) approach the ball B from the outside. Otherwise, they evolve uniformly bounded inside B. Clearly, the more negative the eigenvalues of A, the smaller the radius of the ball B. The implications of the previous result on the observation, or control, of the nonlinear scalar system y

(n )

= ϕ(t, y)u + φ(t, y, y, ˙ ...,y

(n −1)

)

(9)

where the control input gain ϕ(t, y) is assumed to be known and φ(t, y, y, ˙ . . . , y (n −1) ) being unknown are immediate. Indeed, an active disturbance rejection control designed on the basis of the equivalent system y (n ) = v + ξ(t) which is of the form ˆ − ϕ(t, y)u = v = [y ∗ (t)](n ) − ξ(t)

n −1

κn −i e(i) y

(10)

i=0

for suitably selected constant parameters κ yields a tracking error system governed by the perturbed vector equation ˆ χ˙ = F χ + g(ξ − ξ(t))

.. . y˙ n = φ(t, y)u + z1 + λp (y − y1 )

λm in (AAT )

time-varying quantity that is easily shown to be observable in the sense of Diop and Fliess [26]. The linear observer design strategy consists of estimating this time-varying quantity ξ(t) using an, instantaneous, internal time polynomial model, realized in the form of a chain of integrators of length p − 1 at the observer stage for a fixed, sufficiently large integer p. When forcing the dominantly linear, perturbed, output estimation error dynamics to exhibit an asymptotically convergent behavior, the internal model for ξ(t) is automatically and continuously self-updated. It is now assumed that ξ(t) and a finite number of its time derivatives ξ (k ) (t) are uniformly absolutely bounded, for k = 1, 2, . . . , p, for some sufficiently large integer p. The following general result takes place: Theorem 1: The GPI observer-based dynamical feedback controller −1 ˆ κj [yj − (y ∗ (t))(j ) ] − ξ(t) [y ∗ (t)](n ) − nj =0 u= ϕ(t, y)

z˙1 = z2 + λp−1 (y − y1 ) .. . z˙p−1 = zp + λ1 (y − y1 ) z˙p = λ0 (y − y1 ) (k )

asymptotically drives the tracking error phase variables, ey = y (k ) − [y ∗ (t)](k ) , k = 0, 1, .., n − 1 to an arbitrary small neighborhood of the origin, of the tracking error phase space, which can be made as small as desired from the appropriate choice of the controller gain parameters {κ0 , . . . , κn −1 }. Moreover, the estimation errors: e˜(i) = y (i) − yi+1 , i = 0, . . . , n − 1 and the perturbation estimation errors: zj − ξ (j −1) (t), j = 1, . . . , p, asymptotically exponentially converge toward a small as desired neighborhood of the origin of the reconstruction error space, with the appropriate choice of the observer gain parameters {λ0 , . . . , λp+n −1 }. The proof of this theorem can be found in [29].

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(n −1)

with χ = (ey , e˙ y , . . . , ey )T and F in companion form, with, g, being a column vector of zeros with a 1 in the last entry. C. GPI Observers The function ξ(t) = φ(t, y, y, ˙ . . . , y (n −1) ) considered as a perturbation input, needs to be online estimated for a subsequent cancelation at the controller stage. This function, viewed from the observer design perspective, represents an exogenous

III. CASE STUDIES Two cases are considered in the study of robust control under the effect of load variations: the buck-converter dc-motor combination and cascaded buck-converter dc-motor combination. A. Buck-Converter DC-Motor Combination Consider the following model (12) for the composite system consisting of a buck dc-to-dc power converter in cascade with a


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differential functions of x4 , which is now renamed as F, yielding x4 = F 1 ˙ β F + QB F + TL x3 = k αQB + Qa β ˙ Qa QB αβ ¨ x2 = F+ F+ +k F k k k Qa α TL + T˙L k k αQQB + QQa β + αβ ¨ αβ (3) F + F x1 = k kQ QQa QB + k 2 Q + αQB + βQa β ˙ + + F kQ k Qa QB + k 2 + QQB Qa + Q + F+ TL kQ kQ α QQa + α ˙ TL + T¨L + kQ k αβ (4) (3) (14) F + θ F (3) , F¨ , F˙ , F, TL , T˙L , T¨L , TL u= k +

Fig. 1.

Control of buck-converter dc-motor combination.

dc motor, as Fig. 1 shows dv v di = −v + uE, C = i − − ia dt dt R dia dω = −Ra ia − Km ω + v, J La dt dt = −Bω + Km ia − τL L

(12)

where the first line of (12) corresponds to the controlled buck power converter and the second line corresponds to the armature controlled dc motor. The two equations in the first line are taken in an average sense with a continuous input u ∈ [0, 1]. The resistance R (or its normalized version Q defined later) represents the traditional resistive load at the output of the dc-to-dc converter. Although R demands additional power consumption, the regulation or tracking features of the dc motor under the proposed control scheme are not affected by the presence of R. Moreover, R does not intervene in the control input gain expression, which is the only system parameter required in the design of the proposed controller. Conveniently, the parameter R contributes to conform an output low-pass filter that substantially improves the converter output current transmitted to the dc motors armature circuit ia , since the nonaveraged converter current i is “noisy” due to the high-frequency (control input) switchings. To simplify the analysis without affecting its dynamics, the system is now normalized by defining the following state and time coor L/C, x2 = v/E, x3 = dinates transformation; √ x1 = (i/E) √ (ia /E) L/C, x4 = ω LC, τ = t/ LC, the following normalized average system model (13) is obtained: dx1 = −x2 + u, dτ

dx2 x2 = x1 − − x3 dτ Q

dx3 = −Qa x3 − kx4 + x2 dτ dx4 = −QB x4 + kx3 − TL β dτ

α=

F˙0 = F1 + λm +3 (F − F0 ) F˙1 = F2 + λm +2 (F − F0 ) F˙2 = F3 + λm +1 (F − F0 ) k F˙3 = u + z1 + λm (F − F0 ) αβ

z˙1 = z2 + λm −1 (F − F0 ) (13)

.. . z˙m −1 = zm + λ1 (F − F0 )

where C/L, Qa = Ra C/L, QB =

(3) where ξ(τ ) = −( αkβ )θ(F (3) , F¨ , F˙ , F, TL , T˙L , T¨L , TL ). By denoting Fj , j = 0, 1, 2, 3 as the estimates of F (j ) , the following observer is proposed:

.. .

α

Q=R

where θ(.) is a function of F , TL and the corresponding time derivatives. See Appendix A for the expression u. Observer design: For observer construction purposes, the complete system can be seen as the following linear perturbed system: k F (4) = u + ξ (τ ) αβ

E2 C

B

z˙m = λ0 (F − F0 ). C/L

La J Km τL , β= , k= √ , TL = 2 . 2 2 L LC E E C E LC

The normalized average system is flat, with flat output given by x4 , and therefore, all system variables can be written as

The estimation error defined as e = F − F0 satisfies the following linear perturbed differential equation: e(m +4) + λm +3 e(m +3) + · · · + λ1 e˙ + λ0 e = ξ (m ) (τ ). If ξ (m ) (τ ) is uniformly absolutely bounded, then a suitable choice of the set of coefficients {λ0 , . . . , λm +3 } yields

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ultimately uniformly absolutely bounded estimation error phase variables: e(j ) , j = 0, . . . , m + 3, which evolve inside a small as desired neighborhood of the origin of the estimation error phase space, characterized by the vector: χ = (e, e, ˙ . . . , e(m +3) )T . (m ) If ξ (τ ) is not uniformly absolutely bounded, then solutions of the plant equations k (3) ¨ ˙ k (3) u− φ F , F , F , F, TL , T˙L , T¨L , TL F (4) = αβ αβ do not even exist for any finite u. On the other hand, convergence of e, and its time derivatives, to a vicinity of the origin of the χ-space, along with the equation

Fig. 2.

Control of cascaded buck-converter dc-motor combination.

e(4) = ξ(τ ) − z1 − λm e demonstrate that z1 constitutes an arbitrarily close estimate of the unknown function ξ(τ ). An active disturbance rejection feedback controller, (ADRC), with ξˆ = z1 is proposed as 3

αβ (4) (i) ∗ ∗ − ξˆ . [F (τ )] − κi Fi − [F (τ )] u= k i=0 Given that F (i) = Fi + e(i) , i = 1, 2, 3 the closed-loop tracking error system, with eF = F − F ∗ (τ ), (i) eF = F (i) − [F ∗ (τ )](i) , i = 1, 2, 3, satisfies (4)

(3)

eF + κ3 eF + · · · + κ0 eF = ξ(τ ) − ξˆ +

3

κi e(i) .

i=1

Since the e(i) ’s converge to a small as desired vicinity of zero and the difference ξ(τ ) − ξˆ is as small as desired, the right-hand side of the previous linear tracking error equation is arbitrarily small. It follows that eF and its time derivatives converge to a small vicinity of the origin of the tracking error phase space (3) θ = (eF , e˙ F , . . . , eF )T provided the set of gains, {κ0 , . . . , κ3 }, are chosen moderately but sufficiently high so as to have the poles of the corresponding dominating characteristic polynomial: p(s) = s4 + κ3 s3 + · · · + κ0 , far into the left half of the complex plane.

L

di2 dv2 v2 = −v2 + u2 v1 , C = i2 − − ia2 dt dt R2

dia2 = −Ra ia2 − Km ω2 + v2 dt dω2 = −Bω2 + Km ia2 − τL 2 J dt

La

where, for simplification purposes and without loss of generality, R1 = R2 = R. The model (15) represents the first and second buck power converters and the armature-controlled dc motors. The equations are taken in an average sense with continuous inputs u1 , u2 ∈ [0, 1]. Similar arguments regarding the presence of the resistance R in the buck-converter dc-motor combination of Fig. 1 apply here for the presence of R1 and R2 in the cascaded buck-converter dc-motor combination of Fig. 2. To simplify the analysis while preserving its dynamical properties, the system is now normalized by defining the following x1 = state and time coordinates transformation: /E) L/C, x2 √ = v1 /E, x3 = (ia1 /E) L/C, x4 = (i1√ ω1 LC, τ = t/ LC, z1 = (i2 /E) L/C, z2 = √ v2 /E, z3 = (ia2 /E) L/C, z4 = ω2 LC, the following normalized average system model (16) is obtained: x˙ 1 = −x2 + u1 , x˙ 2 = x1 −

β x˙ 4 = −Qb x4 + kx3 − TL 1 z˙1 = −z2 + u2 x2 , z˙2 = z1 − αz˙3 = −Qa z3 − kz4 + z2

Consider the cascaded dc-to-dc buck power converters with two identical dc motors connected to the output capacitor branches of the converters, as shown in Fig. 2. The system model becomes

β z˙4 = −Qb z4 + kz3 − TL 2

di1 dv1 v1 = −v1 + u1 E, C = i1 − − ia1 − i2 u2 dt dt R1

dia1 = −Ra ia1 − Km ω1 + v1 dt dω1 = −Bω1 + Km ia1 − τL 1 J dt

La

x2 − x3 − z1 u2 Q

αx˙ 3 = −Qa x3 − kx4 + x2

B. Cascaded Buck-Converter DC-Motor Combination

L

(15)

z2 − z3 Q (16)

where “ ˙ ” stands for d/dτ , and Q=R

La J Km , β= , k= √ L LC 2 E 2 E LC τL 1 τL 2 = 2 , TL 2 = 2 . E C E C

α= TL 1

C/L, Qa = Ra C/L, QB =

E2 C

B

C/L


¨4 , x˙ 4 , x4 , z¨4 , z˙4 , z4 , u2 , T¨L 1 , T˙L 1 , TL 1 , + θ2 x T¨L 2 , T˙L 2 , TL 2

For desired, constant, values of the normalized angular velocities x∗4 and z4∗ , the equilibrium point becomes x ¯3 = x ¯2 =

x ¯1 u ¯1 z¯3 z¯2 z¯1 u ¯2

Qb k

x∗4 u1 =

Qa Qb + k x∗4 k

Qa Qb k Qb (z4∗ )2 ∗ + + = x4 + ∗ kQ Q k x4 Qa Qb + k x∗4 = k Qb = z4∗ k Qa Qb + k z4∗ = k Qa Qb k Qb + + = z4∗ kQ Q k 1 Qa Qb z¯2 z∗ + k z4∗ = = = 4∗ . x ¯2 k x ¯2 x4

(17)

β Qb TL 2 z˙4 + z4 + k k k Qa β Qa Qb αQb αβ z2 = z¨4 + + + k z4 z˙4 + k k k k z3 =

αT˙L 2 Qa TL 2 + + k k Qa β Qa Qb αQb αβ αβ (3) z4 + + + +k z1 = z¨4 + k k k kQ k Qa Qb αQb β k Qb Qa β + + + + + z˙4 + z4 kQ kQ k kQ Q k α 1 Qa ˙ Qa αT¨L 2 + + TL 2 + + + TL 2 k kQ k k kQ 1 αβ (4) (3) (3) ¨ ˙ z + θ1 z4 , z¨4 , z˙4 , z4 , TL 2 , TL 2 , TL 2 , TL 2 u2 = x2 k 4 β Qb TL 1 x˙ 4 + x4 + k k k Qa β Qa Qb αQb αβ x2 = x ¨4 + + + k x4 x˙ 4 + k k k k αT˙L 1 Qa TL 1 + k k αβ (3) (3) x1 = x4 + z4 u2 k +

αβ (4) (4) (3) x4 + z4 u2 + z4 u˙ 2 k (3) (3) + θ3 x4 , x ¨4 , x˙ 4 , x4 , z4 , z¨4 , z˙4 , z4 , u˙ 2 , u2 , (3) (3) TL 1 , T¨L 1 , T˙L 1 , TL 1 , TL 2 , T¨L 2 , T˙L 2 , TL 2 .

See Appendix B for the exact form of u2 , x1 , and u1 . From the previous expressions, the relation between the flat output highest derivatives and the control inputs requires of a dynamic extension of order one for u2 . By replacing some of the long expressions by state variables the control inputs are denoted as αβ (4) αβ z1 (5) x4 + u1 = z4 + · · · k k x2

The system (16) is flat, with flat outputs, x4 and z4 and therefore, all the variables are differentially parameterizable in terms of these two variables. Moreover, the perturbed differential parametrization includes the influence of the unknown load torques TL 1 and TL 2 , and therefore

x3 =

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u˙ 2 =

αβ (5) z + ··· kx2 4

which in matrix form becomes ⎤ ⎡ k k (4) − z1 ⎢ αβ ⎥ u1 x4 ξ1 (τ ) αβ ⎢ ⎥ =⎣ ⎦ u˙ + ξ (τ ) . (5) k 2 2 z4 0 x2 αβ Observer design: The following GPI observers are proposed with the measured outputs being f1 = x4 and g1 = z4 f˙1h = f2h + n6 (f1 − f1h ) f˙2h = f3h + n5 (f1 − f1h ) f˙3h = f4h + n4 (f1 − f1h ) k (u1 − z1 u˙ 2c ) + ζ1 + n3 (f1 − f1h ) f˙4h = αβ ζ˙1 = ζ2 + n2 (f1 − f1h ) ζ˙2 = ζ3 + n1 (f1 − f1h ) ζ˙3 = n0 (f1 − f1h ) g˙ 1h = g2h + m7 (g1 − g1h ) g˙ 2h = g3h + m6 (g1 − g1h ) g˙ 3h = g4h + m5 (g1 − g1h ) g˙ 4h = g5h + m4 (g1 − g1h ) g˙ 5h =

kx2 u˙ 2 + ς1 + m3 (g1 − g1h ) αβ

ς˙1 = ς2 + m2 (g1 − g1h ) ς˙2 = ς3 + m1 (g1 − g1h ) ς˙3 = m0 (g1 − g1h ).

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The observer coefficients are chosen in accordance with their underlying characteristic polynomials of the predominant linear part governing the error behavior. In the previous cases, a seventh and an eighth degree polynomials are chosen, re2 3 ) (s + p0 ) and spectively, of the form (s2 + 2z01 ω01 s + ω01 2 2 4 (s + 2z02 ω02 s + ω02 ) , where n6 = p0 + 6z01 ω01 2 2 2 n5 = 3ω01 + 12z01 ω01 + 6z01 ω01 p0 3 2 2 3 3 2 n4 = 12z01 ω01 + 12z01 ω01 p0 + 8z01 ω01 + 3ω01 p0 3 3 2 4 3 4 n3 = 8z01 ω01 p0 + 12z01 ω01 + 12z01 ω01 p0 + 3ω01 5 2 4 4 n2 = 6z01 ω01 + 12z01 ω01 p0 + 3ω01 p0 5 6 n1 = 6z01 ω01 p0 + ω01 6 n0 = ω01 p0

m7 = 8z02 ω02 2 2 2 m6 = 24z02 ω02 + 4ω02 3 3 3 m5 = 24z02 ω02 + 32z02 ω02 4 4 4 2 4 m4 = 6ω02 + 16z02 ω02 + 48z02 ω02 3 5 5 m3 = 32z02 ω02 + 24z02 ω02 2 6 6 m2 = 24z02 ω02 + 4ω02 7 m1 = 8z02 ω02 8 m0 = ω02 .

The linear controllers are synthesized as disturbance canceling controllers with x∗4 (τ ) = f ∗ (τ ) and z4∗ (τ ) = g ∗ (τ ), so that

Fig. 3.

Simulation results of the buck-converter dc-motor combination.

2 2 (s2 + 2zc2 ωc2 s + ωc2 ) (s + pc2 ), with

k3 = 4zc1 ωc1 2 2 2 k2 = 2ωc1 + 4zc1 ωc1 3 k1 = 4zc1 ωc1 4 k0 = ωc1

s4 = pc2 + 4zc2 ωc2 2 2 2 s3 = 2ωc2 + 4zc2 ωc2 + 4zc2 ωc2 pc2 3 2 2 2 s2 = 4zc2 ωc2 + 2ωc2 pc2 + 4zc2 ωc2 pc2 3 4 s1 = 4zc2 ωc2 pc2 + ωc2 4 s0 = ωc2 pc2 .

u1 = μ11 v1 + μ12 v2

v1 = [f ∗ (τ )](4) − k3 f4h − [f ∗ (τ )](3) − k2 f3h − f¨∗ (τ ) − k1 f2h − f˙∗ (τ ) − k0 (f1 − f ∗ (τ )) − ζ1

u˙ 2 = μ22 v2 − c1

v2 = [g ∗ (τ )](5) − s4 g5h − [g ∗ (τ )](4) − s3 g4h − [g ∗ (τ )](3) − s2 (g3h − g¨∗ (τ )) − s1 (g2h − g˙ ∗ (τ )) − s0 (g − g ∗ (τ ))

IV. SIMULATION RESULTS The following simulation results have been obtained for the buck-converter dc-motor system and the cascaded buckconverter dc-motor combination system. A. Simulation Results for the Buck-Converter DC-Motor System Consider the buck-converter dc-motor system of Fig. 1 with the following physical parameter values: E = 40 V, C = 57.6e−6 F, L = 0.0089 H R = 492.6 Ω, Ra = 6.14 Ω, La = 0.01591 H

where μ11

αβ αβ αβ , μ12 = (ζ1 /x2 ), μ22 = = . k k kx2

The controller coefficients are chosen according to the desired stability features impressed on the characteristic polynomial dominating the linear evolution of the trajectory tracking errors. In this particular case, such coefficients are chosen, respectively, as the coefficients of the following fourth- and 2 2 ) and fifth-order polynomial expressions (s2 + 2zc1 ωc1 s + ωc1

B = 40.92e−6 N·m, J = 7.95e−6 Kg·m2 , Km = 0.04913. Tracking of a reference angular velocity ω ∗ (t) is required including the influence of a variable load torque τL (t). The obtained (nonnormalized) simulation results are presented in Fig. 3. ˆ allows a As it can be seen, the estimated disturbance ξ(t) proper cancelation of the torque variations so that the system angular velocity ω(t) conveniently tracks the reference velocity ω ∗ (t). The disturbance effects at the load are absorbed by


Fig. 4. Simulation results: robustness with respect to the control input gain, f > 1.

the motor armature current ia since the angular velocity ω(t) is forced to track a smooth prescribed trajectory ω ∗ (t) that ultimately becomes constant. The disturbance effects are canceled from the input–output dynamics (relating the angular velocity to the average control input). Hence, the first equation in (12) is free from the influence of such disturbances. This equation predicts that the output voltage v will become proportional to the average control input u through the factor E/L in steady state, indicating that there should be no lags between the capacitor voltage and the control input in stationary conditions. The disturbance cancelation attenuates other possible effects on this relation. The converter dynamics is rather fast compared to the motor dynamics and the inertial effects are thus negligible. 1) Simulation of Robustness With Respect to the Control Input Gain. Buck-Converter DC Motor: In the previous simulation, the control input gain μ = αβ/k of the linear system was assumed to be known. If the control input gain μ is not precisely known, but instead an estimate, μ ˆ, is available. For simulation purposes, the following estimated gain μ ˆ = (αβ/k) f can be used in both, the GPI observer and the linear ADRC. The constant factor f is a real number modeling the percentage of error committed in the gain estimation with respect to its actual value. Naturally, f = 1 represents exact knowledge of the control input gain. When the simulations were carried out, it was found that for a factor f approximately ranging in the set f ∈ [0.7, 11] the closed-loop system exhibits an accurate trajectory tracking of the desired angular velocity profile ω ∗ (t). This can be verified in Figs. 4 and 5 for f > 1 and f < 1, respectively. B. Simulation Results for the Cascaded Buck Converters Dc-Motor System Consider the cascaded buck-converter dc-motor system (see Fig. 2) with the following physical parameter values: E = 40 V, C = 57.6e−6 F, L = 0.01591 H R = 492.6 Ω, Ra = 6.14 Ω, La = 0.0089 H B = 40.92e−6 N·m, J = 7.95e−6 Kg·m2 , Km = 0.04913.

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Fig. 5. Simulation results: robustness with respect to the control input gain, f < 1.

Fig. 6. Simulation results: performance of the cascaded buck converters dcmotor system first stage variables.

In this system, the controllers were set to have the first motor follow a constant angular velocity reference ω1∗ (t) while the second motor smoothly increases its angular velocity from an initial constant reference value toward a prespecified final constant reference value ω2∗ (t). The transition is accomplished with the help of a Bézier interpolating polynomial. The controller parameters for the normalized system were chosen to be zc1 = 1, wc1 = 0.3, zc2 = 1, wc2 = 0.6, pc2 = 0.6 while the observer parameters were set to be z01 = 1, w01 = 5, p0 = 5, z02 = 1, w02 = 5. Figs. 6 and 7 correspondingly present, the (nonnormalized) simulation results of the first stage and the second stage of the cascaded buck-converter dc-motor system.

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Fig. 8. Simulation results: robustness with respect to the control input gain, f > 1.

Fig. 7. Simulation results: performance of cascaded buck converters dc-motor system second stage variables.

As it can be noticed, the first motor follows the constant angular velocity reference ω1∗ (t), and simultaneously, the second motor tracks the required trajectory reference velocity ω2∗ (t). This is achieved under the action of variable load torque disturbances in each motor. 1) Robustness With Respect to the Control Input Gain. Cascaded Buck-Converter dc-Motor System: The key parameter expression appearing in the controllers and the observers is represented by the control input gain μ = αβ/k, or its inverse value. Therefore, knowledge of the true value of μ becomes relevant for a proper tracking of the angular velocity in both motors. Unfortunately, this is not always the case as limited knowledge about the system parameters is common to occur or even μ may experience small variations in time regarding the nonlinear behavior of the electrical components. To highlight the convenience of this ADRC method in tracking the angular velocities let μ be modulated by an uncertainty knowledge factor f representing the lack of knowledge (in percentage) of the true value of μ. A precise knowledge of the gain parameter μ occurs for f = 1. Then, by considering, μ ˆ = (αβ/k)f , instead of μ in the expressions for the controllers and the observers, the corresponding performance graphs in terms of the angular velocity trajectory tracking for different values of f were obtained. To quantify this effect an integral square error (ISE) of the trajectory tracking errors was set for assessing the effect of various degrees of the control input gain parameter uncertainty represented by the factor f . The ISE is defined as t (x4 − x∗4 (τ ))2 + (z4 − z4∗ (τ ))2 dτ (18) ISEf (t) = nf 0

with nf being a normalization factor that sets the steady-state value of the ISE error to 1 (i.e., it sets ISEf (∞) = 1) for f = 1. The value of nf = 1/0.7202 was found. Figs. 8 and 9 present

Fig. 9. Simulation results: robustness with respect to the control input gain, f < 1.

the trajectories of the integral square error index ISEf (t) for different values of the uncertainty factor f . Fig. 8 presents the ISEf (t) for f ∈ [1, 10]. As it can be seen, for values of f > 8 the ISEf (t) monotonically grows producing an increase of the output tracking error signal. On the other hand, Fig. 9 presents the ISEf (t) for f ∈ [0.6, 1] where it can be noticed that for values of f < 0.8 the ISEf (t) grows producing a consequent increase of the output tracking error signal. This information indicated that a convenient angular velocity trajectory tracking is obtained for the system with the uncertainty factor f ranging in the set f ∈ [0.8, 8.0] where the ISEf (t) presents a constant steady state indicating that the tracking error eventually becomes zero or exhibiting small oscillations around the desired reference. Outside this interval, the ISEf (t) increases without bound indicating the growth of


the output tracking error signal and the failure of the controller in achieving a good tracking response. V. CONCLUSION A robust control technique based on ADRC and differential flatness has been presented in this study. The presence of unmodeled exogenous disturbances is estimated by GPI observers followed by a proper cancelation in the control strategy. The control input gain, which is imprecisely known, is efficiently accounted for by the proposed technique. Two special configurations have been addressed: the buck-converter dc-motor combination and two cascaded buck converters each feeding a dc motor. Both cases considered the presence of unknown time-varying load torques (disturbances). It has been shown via realistic digital computer simulations that the proposed linear observer-based linear output feedback control scheme effectively provides robustness to the tracking performance in the presence of unknown disturbances and significant gain variations. Experimental work using this methodology for this application is currently being developed. It is the authors intention to present in the future the obtained experimental results. APPENDIX A The expression for the differential parametrization of u in the buck-converter dc-motor combination is given by αQQB + QQa β + αβ αβ (4) F + u= F (3) k kQ αβ β QQa QB + k 2 Q + αQB + Qa β ¨ + + F + k k kQ Qa QB + k 2 + QB Q αQB + Qa β ˙ + + F kQ k α Qa QB Qa Q + α ¨ (3) +k F + TL + TL + k k kQ Qa + Q α ˙ Qa + TL + + TL . kQ k k APPENDIX B The expressions for differential parameterizations of u2 , x1 , and u1 in the extended cascaded buck-converter dc-motor combination model are given by Qa β + αQb αβ 1 αβ (4) (3) z4 + + u2 = z x2 k k kQ 4 Qa Qb + β(1 + α) Qa β + αQb +k+ + z¨4 k kQ Qa Qb k Qb (1 + α) + Qa β + + + z˙4 kQ Q k (3) αTL 2 α Qa Qb Qa ¨ + k z4 + + + TL 2 + k k kQ k 1 Qa α ˙ Qa + + TL 2 + TL 2 + k kQ k k

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αβ (3) (3) x4 + z4 u2 k Qa β αQb αβ + + + (¨ x4 + z¨4 u2 ) k k kQ Qa Qb Qa β αQb β +k+ + + + (x˙ 4 + z˙4 u2 ) k kQ kQ k Qa Qb k Qb + + + (x4 + z4 u2 ) kQ Q k α Qa ˙ α ¨ ¨ TL 1 + TL 2 u 2 + + TL 1 + T˙L 2 u2 + k kQ k 1 Qa + + (TL 1 + TL 2 u2 ) k kQ αβ (4) (4) (3) x4 + z4 u2 + z4 u˙ 2 u1 = k Qa β + αQb αβ (3) (3) + + x4 + z4 u2 + z¨4 u˙ 2 k kQ Qa Qb Qa β αQb β + +k+ + + (¨ x4 + z¨4 u2 + z˙4 u˙ 2 ) k kQ kQ k Qa Qb k Qb + + + (x˙ 4 + z˙4 u2 + z4 u˙ 2 ) kQ Q k Qa Qb αQb + Qa β αβ x ¨4 + x˙ 4 + + k x4 + k k k Qa α (3) (3) TL 1 + TL 2 u2 + T¨L 2 u˙ 2 + TL 1 + k k α Qa + T¨L 1 + T¨L 2 u2 + T˙L 2 u˙ 2 + kQ k α 1 Qa ˙ + + TL 1 + T˙L 2 u2 + TL 2 u˙ 2 + T˙L 1. k kQ k x1 =

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Hebertt Sira-Ram´ırez received the Electrical Engineer’s degree from the Universidad de Los Andes in Mérida, Venezuela, in 1970, the M.Sc. degree in EE and the Electrical Engineer degree, in 1974, and the Ph.D. degree, also in EE, in 1977, all from the Massachusetts Institute of Technology, Cambridge. He was with the Universidad de Los Andes for 28 years where he held the positions of: Head of the Control Systems Department, Head of the Graduate Studies in Control Engineering, and Vice President of the University. He is currently a Titular Researcher in the Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional, Meéxico City, Meéxico. He is a coauthor of five books on Automatic Control and the author of more than 400 technical papers in credited journals (150) and international conferences (250). His research interests include theoretical and practical aspects of feedback regulation of nonlinear dynamic systems with special emphasis in variable structure feedback control techniques, algebraic methods in control and estimation, and their applications in power electronics and other technological systems.

Marco Antonio Oliver-Salazar received the Electromechanical Engineer´s degree from the Universidad Anáhuac, Naucalpan, México, in 1985, the M.Sc. degree in control and information technology, in 1989, from the University of Manchester Institute of Science and Technology, Manchester, U.K., and the Ph.D. degree in control engineering, in 1994, from the University of Sheffield, Sheffield, U.K. From 1985 to 1996, he was as a Researcher at the Instituto de Investigaciones Eléctricas, Cuernavaca, México. In 1996, he joined the Electronics Department as a Lecturer-Researcher at the Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, México, and since 2005, he has been with the Mechatronics Department in the same research center. He is the author of several technical papers in international conferences and journals and he has headed different research projects. He is the author of several technical papers in international conferences and journals and he has headed different research projects. He is interested in topics related to digital signal processing and nonlinear control applied to power electronics, robotics and mechatronic systems, specially in the design and control of robotic hands, human rehabilitation systems, and power electronic converters.