Intell Ind Syst (2016) 2:371–380DOI 10.1007/s40903-016-0061-x

ORIGINAL PAPER

Control of DC–DC Converter and DC Motor Dynamics UsingDifferential Flatness Theory

G. Rigatos1 · P. Siano2 · P. Wira3 · M. Sayed-Mouchaweh4

Received: 10 July 2016 / Revised: 31 August 2016 / Accepted: 19 September 2016 / Published online: 1 October 2016© Springer Science+Business Media Singapore 2016

Abstract The article proposes a method for nonlinear con-trol of the dynamical system that is formed by a DC–DCconverter and a DC motor, making use of differential flat-ness theory. First it is proven that the aforementioned systemis differentially flat which means that all its state vector ele-ments and its control inputs can be expressed as differentialfunctions of primary state variables which are defined to bethe system’s flat outputs. By exploiting the differential flat-ness properties of themodel its transformation to a linearizedcanonical (Brunovsky) form becomes possible. For the lat-ter description of the system one can design a stabilizingfeedback controller. Moreover, estimation of the nonmea-surable state vector elements of the system is achieved byapplying a new nonlinear Filtering method which is knownas Derivative-free nonlinear Kalman Filter. This filter con-sists of the Kalman Filter recursion applied on the linearizedequivalent model of the system and of an inverse transfor-mation that is based on differential flatness theory and which

B G. Rigatosgrigat@ieee.org

P. Sianopsiano@unisa.it

P. Wirapwira@uha.fr

M. Sayed-Mouchawehmoamar.sayed-mouchaweh@mines-douai.fr

1 Unit of Industrial Automation, Industrial Systems Institute,26504 Rion Patras, Greece

2 Department of Industrial Engineering, University of Salerno,84084 Fisciano, Italy

3 Laboratoire MIPS, Université de Haute Alsace, 68093Mulhouse Cedex, France

4 Ecole des Mines de Douai, Laboratoire d’ Automatique,59508 Douai, France

enables to obtain estimates of the initial nonlinear state-spacemodel.Moreover, to compensate for parametric uncertaintiesand external perturbations the filter is redesigned as a distur-bance observer. By estimating the perturbation inputs thataffect the joint model of the DC–DC converter and of the DCmotor their compensation becomes possible. The efficiencyof the proposed control scheme is further confirmed throughsimulation experiments.

Keywords DC–DC converter · DC motor · Nonlinearcontrol · Differential flatness theory · Global linearization ·Asymptotic stability · Kalman Filtering · Disturbanceobserver

Introduction

A nonlinear control method, based on differential flatnesstheory, is proposed for the model that is formed after con-necting a DC motor to a DC–DC converter. This scheme canbe used for the exploitation of the power produced by photo-voltaic units, since the DC–DC converter controls the levelof the produced output voltage. In turn the DC output volt-age can be fed into DC motors, as for instance in the caseof actuators, mechatronic devices and pumps [1–3]. Nonlin-ear control approaches for DC–DC converters connected toDCmotors have been presented in [4–8]. In particular globallinearization approaches and flatness-based control of suchsystems have been presented in [9–15]. In this article, a state-space description is obtained first for the system consistingof the DC–DC converter and the DC motor. Next,the differ-ential flatness properties of this model are proven using asflat output the rotation angle of the motor. Differential flat-ness means that all state variables and the control input of thesystem can be expressed as functions of the flat output andits derivatives [16–21].

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372 Intell Ind Syst (2016) 2:371–380

By proving that the system is a differentially flat one, it isconfirmed that it can be transformed to the linear canonical(Brunovsky) form [22–28]. After such a transformation thecomputation of a stabilizing feedback control input becomespossible. To estimate the non-measurable state variables ofthe system theDerivative-free nonlinearKalmanFilter is pro-posed [19–21]. This consists of the Kalman Filter recursionapplied on the equivalent linearized model of the system[29–31]. Moreover, it makes use of an inverse transforma-tion which is based on differential flatness theory and whichallows to obtain estimates of the state variables of the initialnonlinearmodel. Next, theDerivative-free nonlinear KalmanFilter is designed as a disturbance observer. This enablessimultaneous estimation of the state variables of the systemand of disturbance terms affecting it [32,33]. By identifyingsuch disturbance inputs, their compensation becomes possi-ble with the inclusion of an additional control term in theaggregate feedback control signal.

The structure of the article is as follows: in Section“Dynamics of the Model Formed by the DC–DC Converterand the DC Motor” an analysis is given about the dynamicsof the model which consists of the DC–DC converter con-nected to the DC motor. In Section “Differential Flatnessof the Model of the DC–DC Converter and the DC Motor”the differential flatness properties of the aforementioned sys-tem are proven. In Section “Transformation of the DynamicModel into the Canonical Form” the state-space model ofthe system is transformed into the linear canonical form. InSection “Disturbances Compensation with the Derivative-Free Nonlinear Kalman Filter” the Derivative-free nonlinearKalman is used for estimating the non-measurable state vari-ables of the system. Moreover, this filter is redesigned as adisturbance observer so as to estimate additive perturbationterms affecting this model. In Section “Simulation Tests”the performance of the control scheme is evaluated throughsimulation experiments. Finally, in Section “Conclusions”concluding remarks are stated.

Dynamics of theModel Formed by theDC–DCCon-verter and the DC Motor

DC motors controlled through DC–DC converters can befound in several applications, as for instance in photo-voltaic power pumps or in desalination units (Fig. 1). Controlis implemented through a pulse-width-modulation (PWM)approach. The equivalent circuit of the system that is formedafter connecting a DC motor to a DC–DC (buck) converteris depicted in Fig. 2.

PWM is applied for the converter’s control. The amplitudeof the output voltage Vo is determined by the duty cycle ofthe PWM. The on/off state of the switch Q sets voltage u to Eor to 0 for specific time intervals within the sampling period.

DC motor / pump

Water resources

Ba�ery

Converter

PhotovoltaicCells

To irriga�on

Fig. 1 Photovoltaic powered pump

E

u

Q L

D C R0V

mL

mJ

mLI

cI RI

aI

Fig. 2 Circuit of the DC–DC converter connected to a DC motor

The ratio between the time interval in which u = E and thesampling period defines the duty cycle. By varying the dutycycle one can control the voltage output Vo, as if a variableinput voltage E was applied to the circuit.

The dynamics of the electrical part of the circuit comesfrom the application of Kirchhoff’s laws. It holds:

Ld I

dt+ Vc = u

Lmd Iadt

+ Rm Ia + Keω = Vc (1)

I = Ic + IR + Ia or I = CdVcdt

+ VcR

+ Ia

The dynamics of the mechanical part of the circuit comesfrom the laws of rotational motion. It holds that:

θ = ω

J ω = −Bω + Ka Ia + τL (2)

where τL is the load’s torque. By defining the state variablesx1 = θ , x2 = ω, x3 = I , x4 = Vc and x5 = Ia one obtainsthe following state-space model:

x1 = x2

x2 = − B

Jx2 + Ka

Jx5 + 1

Jτd

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Intell Ind Syst (2016) 2:371–380 373

x3 = − 1

Lx4 + 1

Lu

x4 = 1

Cx3 − 1

RCx4 + 1

Cx5

x5 = 1

Lmx4 − Rm

Lmx5 + Ke

Lmx2 (3)

Without loss of generality it is considered that the load’storque is τL = mglsin(x1) (the motor is considered to berotating a rod of length l having a mass m at its end), theprevious state-space description is written in the followingmatrix form

⎛⎜⎜⎜⎜⎝

x1x2x3x4x5

⎞⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎝

x2− B

J x2 + KaJ x5 + 1

J mglsin(x1)− 1

L x41C x3 − 1

RC x4 + 1C x5

KeLm

x2 + 1Lm

x4 − RmLm

x5

⎞⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎝

001L00

⎞⎟⎟⎟⎟⎠u (4)

Differential Flatness of the Model of the DC–DCConverter and the DC Motor

Next, the differential flatness properties of the model areproven. The flat output of the system is taken to be y = x1,that is the turn angle of the rotor. From the first row of themodel one gets

x2 = x1 = y (5)

From the second row of the model one gets

x5 = x2 + BJ x2 − 1

J mglsin(x1)KaJ

⇒x5 = fa(y, y, y) (6)

From the fifth row of the model one gets

x4 = x5 + KeLm

x2 + RmLm

x51Lm

⇒x4 = fb(y, y, y, y(3)) (7)

From the fourth row of the model one gets

x3 = x4 + 1RC x4 + 1

C x51C

⇒x3 = fc(y, y, y, y(3), y(4)) (8)

From the third row of the model one gets

u = L(x3 + 1

Lx4

)⇒ u = fd

(y, y, y, y(3), y(4), y(5)) (9)

Consequently, all state variables and control inputs of themodel are differential functions of the flat output and themodel is a differentially flat one.

Transformation of the Dynamic Model into theCanonical Form

Knowing that the dynamicmodel of the system that is formedby the DC–DC converter and the DCmotor is a differentiallyflat, its transformation to the linear canonical (Brunovsky)form is assured. Indeed by differentiating the first row of thestate-space model with respect to time one obtains:

x1 = x2⇒x1 = x2⇒x1 = − B

Jx2 + Km

Jx5 + mglsin(x1)

(10)

By differentiating oncemorewith respect to time one gets:

x (3)1 = − B

Jx2 + Km

Jx5 + mglcos(x1)x1⇒

x (3)1 = − B

J

[− B

Jx2 + Ka

Jx5 + mglsin(x1)

]

+ Km

J

(Ke

Lmx2 + 1

Lmx4 − Rm

Lmx5

)

+mglcos(x1)x1 (11)

After intermediate operations one obtains:

x (3)1 =

[(B

J

)2

+(KaKe

J Km

)]x2

+[− Bmgl

Jsin(x1) + mglcos(x1)x2

]

+ Ka

J Lmx4 +

[− BKa

J 2− KaRm

J Lm

]x5 (12)

By differentiating once more with respect to time onearrives at:

x (4)1 =

[(B

J

)2

+(KaKe

J Lm

)]x2

+[− Bmgl

Jcos(x1)x1 − mglsin(x1)x1x2

+mglcos(x1)x2]

+ Ka

J Lmx4 +

[− BKa

J 2− KaRm

J Lm

]x5 (13)

which after intermediate operations gives

x (4)1 =

[(B

J

)2 (KaKe

J Lm

)+ mglcos(x1)

]

×[− B

Jx2 + Ka

Jx5 + mglcos(x1)

]

+[− Bmgl

Jcos(x1)x2 − mglsin(x1)x

22

]

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374 Intell Ind Syst (2016) 2:371–380

+ Ka

J Lm

[1

cx3 − 1

RCx4 + 1

Cx5

]

+[− BKa

J 2− KaRm

J Lm

]·[1

cx3 − 1

RCx4 + 1

Cx5

]

(14)

By differentiating oncemorewith respect to time one gets:

x (5)1 = − [mglsin(x1)x1] ·

[− B

Jx2 + Ka

Jx5 + mglcos(x1)

]

+[(

B

J

)2

+ KaKe

J Lm

]+ mglcos(x1)

·[− B

Jx2 + Ka

Jx5 + mglcos(x1)x1

]

+[Bmgl

Jsin(x1)x1x2 − Bmgl

Jcos(x1)x2

−mglcos(x1)x1x22 − mglsin(x1)2x2 x2]

+ Ka

J LmC

(− 1

L

)x4 + Ka

J LmC

(1

L

)u (15)

The input–output linearized system is also written in theconcise form:

y(5) = f (y, . . . , y(5)) + g(y, . . . , y(5))u (16)

and by defining v = f (y, . . . , y(5)) + g(y, . . . , y(5))u onegets the equivalent description

y(5) = v (17)

For the previous description, the design of a stabilizingfeedback controller gives:

u = y(5)d − k1

(y(4) − y(4)

d

) − k2(y(3) − y(3)

d

)

− k3(y − yd) − k4(y − yd) − k5(y − yd) (18)

while the control input that is actually exerted on the sys-tem is

u = 1

g(y, . . . , y(5))

[v − f (y, . . . , y(5))

](19)

with the previous control law the closed-loop dynamicsbecomes

e(5) + k1e(4) + k2e

(3) + k3e + k4e + k5e = 0 (20)

Thus, through suitable selection of the feedback controlgains ki i = 1, . . . , 5 so as the associated characteristic poly-nomial to be a Hurwitz stable one, it is assured that

limt→∞ e(t) = 0 (21)

Disturbances Compensation with the Derivative-Free Nonlinear Kalman Filter

Next, the problem of disturbances compensation is treated.Without loss of generality it is assumed that additiveinput disturbances d affect the linearized model of theconverter

y(5) = v + d (22)

and that these disturbances are described by their time deriv-atives up to order 3, that is d(3) = fd . Actually, every signald can be equivalently described by its time derivatives upto order n and the associated initial conditions. However,since estimation of the signal d and of its derivatives isgoing to be performed with the use of Kalman Filtering,there is no prior constraint about knowing these initial con-ditions.

The state vector of the DC–DC converter’s model isextended by introducing as additional state variables the dis-turbance inputs and their derivatives. Thus, z1 = y, z2 = y,

z3 = y, z4 = y(3), z5 = y(4), z6 = d , z7 = ˙d and z8 = ¨d.In this manner, and by defining the extended state vectorZ = [z1, z2, . . . , z8]T one arrives at a state-space descrip-tion into the linear canonical (Brunovsky) form:

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 10 0 0 0 0 0 0 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

z1z2z3z4z5z6z7z8

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 00 00 00 01 00 00 00 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(v

fd

)(23)

zm = (1 0 0 0 0 0 0 0

)Z (24)

The previous description of the system is written in thelinear state-space form:

z = Az + Bv

zm = Cz (25)

Toperformsimultaneous estimationof thenon-measurablestate vector elements and of the disturbance inputs the fol-lowing disturbance observer is defined

˙z = Aoz + Bov + K f (zm − zm)zm = Coz (26)

where Ao = A, Co = C and

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Intell Ind Syst (2016) 2:371–380 375

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

ω

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

I

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

Vc

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

I a

(a) (b)

Fig. 3 Tracking of setpoint 1 for the system consisting of the DC–DC converter connected to the DC motor. a State variables x2 = ω and x3 = I ,b state variables x4 = Vc and x5 = Ia

Bo = (0 0 0 0 1 0 0 0

)T(27)

The observer’s gain is computed through the Kalman Fil-ter’s recursion [29–31]. To apply Kalman Filtering on thelinearized equivalent model of the system, which is alsoknown as Derivative-free nonlinear Kalman Filter, matricesAo, Bo,Co are subjected to discretization with the use ofcommon discretization methods. The filter also comprisesan inverse transformation based on differential flatness the-ory and on Eqs. (5), (6), (7) and (8), which enables toobtain state estimates of the initial nonlinear model. Thediscrete-time equivalents of the aforementioned matricesare denoted as Ad , Bd and Cd . The Kalman Filter’s algo-rithm consists of a measurement update and a time-updatestage. By denoting as Q, R the process and measure-ment noise covariance matrices, as P− the state vectorerror covariance matrix prior to measurement taking andby P the state vector error after measurement taking onehas

Measurement update

K f (k) = P−(k)CTd [Cd P

−(k)CTd + R]−1

x(k) = x−(k) + K f (k)[zm − zm]P(k) = P−(k) − K f (k)Cd P

−(k) (28)

Time update

P−(k + 1) = Ad P(k)ATd + Q

x−(k + 1) = Ad x(k) + Bdv(k) (29)

Simulation Tests

The efficiency of the proposed control scheme was testedthrough simulation experiments. Flatness-based control wasapplied to the system consisting of the DC–DC converterand of the DC motor. The only measurable state variable ofthe system was taken to be the rotor’s turn angle x1 = θ .The Derivative-free nonlinear Kalman Filter was used asa disturbance observer, for estimating simultaneously thenon-measurable state variables of the model and the addi-tive disturbance inputs that were affecting it. Indicativevalues for the parameters of the electric circuit formed bythe DC–DC converter and the DC motor, that is shownin Fig. 2 are as follows [13]: E = 56V, R = 61.7�,L = 118.6 mH, C = 114.4µF, Ra = 0.965� andLa = 2.22 mH while the moment of inertia of the rotoris J = 118.2× 10−3 kgm2.

The obtained results are depicted in Figs. 3, 4, 5, 6, 7,and 8. The real value of the state variable is depicted withthe blue line, the estimated value of it is depicted with thegreen linewhile the reference setpoint is depictedwith the redline. It can be noticed that, despite implementation of a stateestimation-based control scheme and despite the presence ofexternal disturbances, the flatness-based control succeededfast and accurate tracking to the reference setpoints for allstate variables of the system.

The control inputs exerted on the systemwhich consists ofthe DC–DC converter, are shown in the bottom rows of Figs.9, 10, and 11. These diagrams depict the ratio between thecontrol input that is applied to the DC–DC converter and theconstant voltage E (Fig. 2). This ratio provides information

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376 Intell Ind Syst (2016) 2:371–380

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

ω

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

I

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

Vc

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

I a

(a) (b)

Fig. 4 Tracking of setpoint 2 for the system consisting of the DC–DC converter connected to the DC motor. a State variables x2 = ω and x3 = I ,b state variables x4 = Vc and x5 = Ia

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

ω

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

I

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

Vc

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

I a

(a) (b)

Fig. 5 Tracking of setpoint 3 for the system consisting of the DC–DC converter connected to the DC motor. a state variables x2 = ω and x3 = I ,b state variables x4 = Vc and x5 = Ia

about the implementation of PWM control of the converterand the associated duty cycle. Moreover, in the top rowsof Figs. 9, 10, and 11 one can see that the Kalman Filter-based disturbance observer provides accurate estimates ofthe additive disturbance inputs that affected the converter’sand motor’s model. By identifying in real-time these per-turbation terms their compensation became possible, afterincluding an additional input to the feedback control sig-nal.

Remark 1 Differential flatness theory leads to a system-atic approach for achieving global linearization of nonlineardynamical systems. Differential flatness theory allows todefine the class of nonlinear systems which admits a globallinearization transformation. Actually, all differentially flatsystems can be expressed in the linear canonical state-spaceform through transformations of their state variables. It holdsthat: (i) All single input models are differentially flat and canbe transformed into the linear canonical form (this is the case

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Intell Ind Syst (2016) 2:371–380 377

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

ω

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

I

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

Vc

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

I a

(a) (b)

Fig. 6 Tracking of setpoint 4 for the system consisting of the DC–DC converter connected to the DC motor. a state variables x2 = ω and x3 = I ,b state variables x4 = Vc and x5 = Ia

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

ω

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

I

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

Vc

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

I a

(a) (b)

Fig. 7 Tracking of setpoint 5 for the system consisting of the DC–DC converter connected to the DC motor. a State variables x2 = ω and x3 = I ,b state variables x4 = Vc and x5 = Ia

of the DC–DC converter). (ii) Differential flatness holds forMIMO models that admit static feedback linearization.andwhich can be transformed into the linear canonical formthrough a change of variables (diffeomorphism) and feed-back of the state vector, (iii) Differential flatness holds forMIMO models that admit dynamic feedback linearization,This is the case of some underactuated systems. In the lattercase the state vector of the system is extended by consideringas additional state variables some of the control inputs and

their derivatives, (iv) Finally, a more rare case is the so-calledLiouvillian systems. These are systems for which differentialflatness properties hold for part of their state vector (consti-tuting a flat subsystem)while the non-flat state variables canbe obtained by integration of the elements of the flat subsys-tem. In conclusion, a necessary and sufficient condition for asystem to admit a global linearizing transformation into thecanonical form is to be differentially flat. Differentially flatsystems form the widest class to which global linearization-

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378 Intell Ind Syst (2016) 2:371–380

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

ω

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

I

0 5 10 15 20 25 30 35 40−50

0

50

time (sec)

Vc

0 5 10 15 20 25 30 35 40−10

−5

0

5

10

time (sec)

I a

(a) (b)

Fig. 8 Tracking of setpoint 6 for the system consisting of the DC–DC converter connected to the DC motor. a State variables x2 = ω and x3 = I ,b state variables x4 = Vc and x5 = Ia

0 5 10 15 20 25 30 35 40−2

0

2

4

time (sec)

d

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

u/E

0 5 10 15 20 25 30 35 40−2

0

2

4

time (sec)

d

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

u/E

(a) (b)

Fig. 9 Top row Kalman Filter-based estimation of disturbance inputsaffecting the model of the DC–DC motor and the DC motor, bottomrow ratio u/E between the time-varying control input u applied to the

DC–DC converter and the constant DC input E , in case of a trackingsetpoint 1 b tracking setpoint 2

based controlmethods can be applied (this is amore extendedclass of systems than the one related with Lie algebra-basedcontrol). For the reasons analyzed above, out of all nonlin-ear control theories, differential flatness theory is the one thatprovides themost efficient tools for solving nonlinear controlproblems.

Remark 2 The purpose of proving that a system is differ-entially flat is that this is a confirmation that the nonlinear

system can be finally transformed into an equivalent linearform through a change of state variables (diffeomorphisms).Therefore, by showing that (i) all state variables of themodel of the DC–DC converter can be expressed as dif-ferential functions of its flat output, (ii) the flat output andits derivatives are differentially independent which meansthat they are not connected through a relation in the formof a differential equation, then it is assured that the con-verter’s model is differentially flat. Next, by keeping the

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Intell Ind Syst (2016) 2:371–380 379

0 5 10 15 20 25 30 35 400

2

4

6

time (sec)

d

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

u/E

0 5 10 15 20 25 30 35 40−5

0

5

10

time (sec)

d

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

u/E

(a) (b)

Fig. 10 Top row Kalman Filter-based estimation of disturbance inputsaffecting the model of the DC–DC motor and the DC motor, bottomrow ratio u/E between the time-varying control input u applied to the

DC–DC converter and the constant DC input E , in case of a trackingsetpoint 3 b tracking setpoint 4

0 5 10 15 20 25 30 35 40−5

0

5

10

time (sec)

d

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

u/E

0 5 10 15 20 25 30 35 40−5

0

5

10

15

time (sec)

d

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

time (sec)

u/E

(a) (b)

Fig. 11 Top row Kalman Filter-based estimation of disturbance inputsaffecting the model of the DC–DC motor and the DC motor, bottomrow ratio u/E between the time-varying control input u applied to the

DC–DC converter and the constant DC input E , in case of a trackingsetpoint 5 b tracking setpoint 6

relations which express the state-variables of the converteras functions of its flat output, and by deriving succes-sively the flat output with respect to time one arrives at theequivalent input–output linearized form of the system, aswell as to an equivalent canonical (Brunovsky) state-spaceform.

Conclusions

Flatness-based control has been proposed for the model con-sisting of a DC–DC converter connected to a DC motor.First,it was proven that this system is differentially flat andthat the turn angle of themotor stood for the flat output. It wasdemonstrated that all state variables and the control input of

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the model could be expressed as differential functions of theflat output. By exploiting its differential flatness propertiesthe state-space model of the system was transformed intothe linear canonical (Brunovsky) form. For the equivalentlinearized description of the system a stabilizing feedbackcontroller was designed.

Next, state estimation-based control was implementedwith the use of the Derivative-free nonlinear Kalman Filter.This filtering method consisted of the Kalman Filter recur-sion applied on the equivalent linearized model of the systemand of an inverse transformation that was based on differ-ential flatness theory and which provided estimates of thestate variables of the initial nonlinear model. Moreover, byredesigning this filter as a disturbance observer it becamepossible to estimate simultaneously the state vector of themodel and additive perturbation terms affecting it. The effi-ciency of the proposed control schemewas further confirmedthrough simulation experiments.

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