2019 16th International Conference on 2019 16th International Conference on Electrical Engineering, Computing Science andAutomatic Control (CCE)Mexico City, Mexico. September 11-13, 2019

DC Motor Control based on Robust run-timeOptimization Algorithm

1st Cesar U. SolisDepartment of Automatic Control

CINVESTAV-IPNMexico City, Mexico

csolis@ctrl.cinvestav.mx

2nd Julio B. ClempnerEscuela Superior de Fısica y Matematicas

Instituto Politecnico NacionalMexico City, Mexicojulio@clempner.name

3rd Alexander S. PoznyakDepartment of Automatic Control

CINVESTAV-IPNMexico City, Mexico

apoznyak@ctrl.cinvestav.mx

Abstract—This conference paper suggests an interesting appli-cation of a Robust Extremum Seeking Algorithm for dynamicplants with uncertainties and perturbations. The main idea isdevelop a signal such drives a DC motor to a specific angularvelocity, taking into account a perturbation of the inertia (changeof mass of the load, density, spatial distribution, etc) and viscousfriction and optimizing an specific functional in real-time to reachsome angular velocity.

Index Terms—Extremum Seeking, Real-time optimization, DCmotor application

I. RELATED WORKS

Liu and Krstic [24] proposed a stochastic averaging anddeterministic extremum seeking procedure when perturbationsexist in dynamical model but not in observations and measure-ments. Wang et al. [39] presented an experimental applicationof extremum seeking algorithm. Zinkevich et al. [40] devel-oped a parallelized stochastic gradient descent in discrete time.Iyasere et al. [17] suggested an optimum seeking-based robustnon-linear controller to maximize wind energy captured byvariable speed wind turbines at low-to-medium wind speeds.Solis et al. [30], [31] suggested an interesting fast terminaloptimal control to control a chaotic oscillator and a 3D cranewith perturbations based on sliding mode manifolds. Bensafiaet al. [8] presented a robust method based on fractional controltheory and adaptive PI controller to set-point a DC motor.Zhang et al. [4] suggested an H−∞ control strategy forelectric electromotive. Velagic and Galijasevic [38] designed afuzzy logic control for a DC motor under real constrains anddisturbances. Hai-Ping and Xia [3] presented a global robustsiding mode control for DC motors bye exact linearization.Dobra [13] suggested an interesting robust PI control for servomotors. Brisilla et al. [12] showed an observer based slidingmode control for permanent magnet motors. Tijerina and Meza[36] presented a variable structure control for DC motors basedon sliding mode manifolds.

Although the concept of extremum seeking is interestingfor the solution of real-time optimization problems like motorset-point with uncertainties and perturbations, etc.

II. MAIN RESULTS

This paper presents the following main contributions:

1) the Robust Extremum Seking Algorithm and its generalformulation problem,

2) the application of the Algorithm to a DC motor modelwith uncertainties and with convex functional,

3) a numerical simulation to validate the results.

III. GENERAL FORMULATION OF THE PROBLEM

Let us consider the problem:

argminx1∈Rn

f (·)

s. t.x1 = x2

x2 = g (x1,x2, t) + u

y (·) = f (·)

(1)

with the following assumptions:

1) x1, x2 ∈ Rn the known states of the plant and u ∈ Rnthe input control,

2) g : Rn × Rn × R → Rn defined by g (x1,x2, t) is asmooth function that represents the description of thethe dynamical plant with non-modeled uncertainties andsatisfies that ‖g (x1,x2, t)‖ ≤ c0 + c1 ‖x1‖ + c2 ‖x2‖for some known constants c0, c1, c2 > 0,

3) f : Rn → R is an unknown twice differentiable stronglyconvex function such that h−I < ∇2f (x1) < h+I forsome known constants h+, h− > 0,

4) y (·) ∈ Rn is a measurable output signal about theaforementioned function.

The problem considers a continuous-time strongly convexunknown function f (·) with bounded dynamical constrainsincluding uncertainties. We will propose a robust closed-loop controller that stabilizes the system around an optimalconvergence zone.

IV. ROBUST EXTREMUM SEEKING

Let x be a Rn vector. We define sign(x) := [sign(xm)]mwith m ∈ J1, nK.

978-1-5386-7033-0/18/$31.00 c©2019 IEEE

Fig. 1. Block diagram structure for the algorithm.

The suggested algorithm to solve the problem (1) is asfollows:

D = 1αζy (x1 + αζt)

Jo = β(D − Jo

)J = β (Jo − J)

u = −µJ− Umax [c0 + c1 ‖x1‖+c2 ‖x2‖] sign (µJ+ x2)

(2)

with the following considerations:1) constants β, µ, α > 0 and Umax ≥ 1,2) the auxiliary states D, J, J0 ∈ Rn,3) the dither signal ζt is defined as follows

ζt := [sin (ω0t) , sin (2ω0t) , . . . , sin (nω0t)]ᵀ. (3)

where ω0 > 0 is sufficiently large. The dither signal ζt isdefined as in Eq. (3) in order to satisfy the orthogonalityproperties of the system and to obtain a correct representationfor the time-dependent functions.

Figure 1 shows the block structure of the proposed algo-rithm given in (2).

A. Gradient estimationFor simplicity, in this section, we consider the time-

dependent vector xt := x1 and we introduce the followingdefinition and notation.

Definition 1. For a time-dependent integrable real valuefunction h (t), the Fourier transform, denoted by h (t) (ω) inthe frequency domain is given by:

h (t) (ω) :=

∞∫−∞

h (t) exp (−iωt) dt (4)

Lemma 1. For an integrable real value function h (t) and forsome constant ω0 > 0 we have that

sin (nω0t)h (t) (ω) =12i

(h (t) (ω + nω0)− h (t) (ω − nω0)

) (5)

sin (nω0t) sin (mω0t)h (t) (ω) =14

(h (t) (ω − (m− n)ω0) + h (t) (ω − (n−m)ω0)

−h (t) (ω − (n+m)ω0)− h (t) (ω + (m+ n)ω0)) (6)

In particular we have that

sin2 (nω0t)h (t) (ω) =12h (t) (ω)−

14

(h (t) (ω − 2nω0) + h (t) (ω + 2nω0)

)(7)

Proof. See Hsu [15].

Notice that if in Eq. (5) if ω0 > 0 the spectrum of h (t)is displaced around ±ω0 central frequency and in the Eq. (7)there is exists a spectral component around the origin. Thisproperty will be used to separate the spectrum of the gradientestimation signal of another components.

Definition 2. Heaviside function Let us define the complexvalue Heaviside function as follows:

Heaviside (z) :=

{1 < (z) > 0

0 < (z) ≤ 0

where < (z) is the real part of z ∈ C. This definition can beextended to a vector form as follows, let z be a complex vectorin Cn then Heaviside (z) := [Heaviside (zm)]m for every entrym ∈ J1, nK.

Proposition 1. Let xt be a time-dependent vector in Rn andthe function f (·) given in (1) such that admits a Fouriertransform given by f (·) (ω). With the accepted assumptionsabove, let ωd be the bandwidth of the gradient ∇f (·) then forsome small α > 0 and sufficiently large dither frequency ω0

of the dither signal ζt, we have that

1

αζtf (xt + αζt) (ω) ·H (ω) =

1

2∇f (εt) (ω) (8)

where εt = atxt + (1− at) (xt + αζt) for some at ∈ [0, 1]and

H (ω) := Heaviside (ω + ωd)− Heaviside (ω − ωd) (9)

with 0 < ωd << ω0.

Proof. See Solis et al. [33].

Remark 1. Notice that formula (8) represents the frequencyspectrum of the gradient ∇f (·) with some error introducedby εt. Notice that H (ω) is a transfer function for an idealfilter implying that it is not realizable. Then, for practicalconsiderations it is important to design a low-pass filter withflat response and minimum phase change of the output.

In order to attend the previous remark, let us calculate theTaylor expansion of f (xt + αζt) and taking into account Eq.(8) with some algebraic manipulation we have that

∇f (εt) = ∇f (xt) + o(α2)

Now, let us consider the realizable second order filter givenby: {

Jo = β(D − Jo

)J = β (Jo − J)

(10)

It is possible to analyze every entry as a transfer functionas follows [

J]m[

D]m

=1(

1β iω + 1

)2then we have that

[J]m

=

[1αζf (xt + αζt)

]m(ω)(

1β iω + 1

)2

Clearly, it is not possible to obtain an analytic inverse of theFourier transform. However, we can consider the limit pointsto approximate the expression. Then, remembering that ∇f (·)is a limited bandwidth ωd it is possible to choose β > 0 suchthat for ω ≥ 0

[J]m≈

1αζtf (xt + αζt) (ω) ·H (ω) ω ≤ ωd << β << ω0

0 ω >> β

Lemma 2. With the assumptions and notation given in (1)and (2) the dynamics x1 = −µJ is stable in a zone aroundthe optimal point x∗1.

Proof. See Solis et al. [33].

Remark 2. Notice two facts, 1) substituting the value ε in ρwe get ρ = µh− this controls the velocity of convergence andgiven it is possible to solve as follows

V (t) ≤ ε0µh−

+ V (0) exp(−µh−t)

and, 2) if we choose a very small parameter α > 0 it ispossible to guarantee a small convergence to a zone, but theterm D can present an undesirable numerical singularity.

Theorem 1. Under the above assumptions and considerationsthe Optimal Seeking Algorithm given in (2) applied to theproblem (1) satisfies:

(Th1). the dynamics σ := µJ+ x2 = 0 is stable,(Th2). the global system reaches the sliding surface σ = 0

in finite time,(Th3). the global system is stable and converges to a zone

around the optimal point.

Proof. See Solis et al. [33].

V. MATHEMATICAL MODEL OF THE DC MOTOR

The standard mathematical model for a DC motor (seePoznyak [27]) is given by

P (s) =ω(s)

V (s)=

K

(Js+ b)(Ls+R) +K2(11)

where J is the total inertia of the motor and the load, b the totalviscous friction, L the equivalent inductor, R the equivalentresistor and K the electromotive force constant that is equal tothe motor torque constant. In figure 2 we can see al diagramof the system under discussion.

VI. NUMERICAL EXAMPLE

For numerical example we consider a system with variabletotal inertia and total viscous friction, then the followingparameters: L = R = K = 1, J and b change at time 30sfrom J = b = 1 to J = 1.5 and b = 2.

The functional to optimize is given by

1

2ω2 + (ω − ωref )2

Fig. 2. Motor mathematical model.

Fig. 3. Angular velocity.

where ωref is some reference for the angular velocity, in ourcase is equal to 10. This functional was selected in orderto optimize the desired angular velocity and at same timeminimice the angular acceleration to reduce the consumptionof energy and the input signal was bounded.

In order to employ the algorithm we choose the followingparameters: α = 1, β = 2, µ = 10, c0, c1, c2 = 2, Umax = 1.5.

In figure 3 we can see the angular velocity behavior underthe algorithm with reference 10, notice that at 30s time thechange in the parameters induce a perturbation but the controlsystem compensates it. the figure 4 shows the optimal behaviorfor the selected functional, finally the input control signal isshowed in the figure 5.

VII. CONCLUSIONS

This conference paper presented an interesting applicationof the algorithm Robust Extremum Seeking for dynamic plantswith perturbations applied to a DC motor mathematical modeland the result was satisfactory in order to hold the sameangular velocity under changes in the inertia and frictioncoefficients.

The optimization was reached in the energy and minimizingthe error with the angular velocity reference.

REFERENCES

[1] P-A Absil and K Kurdyka. On the stable equilibrium points of gradientsystems. Systems & control letters, 55(7):573–577, 2006.

Fig. 4. Optimization of the functional.

Fig. 5. Input control.

[2] S. V. Ambesange, S. Y. Kamble, and D. S. More. Application of slidingmode control for the speed control of DC motor drives. In Proc. IEEEInt. Conf. Control Applications (CCA), pages 832–836, August 2013.

[3] and Hai-Ping Pang and Xia Chen. Global robust sliding mode tracldngcontrol for DC motors by exact linearization. In Proc. Chinese Controland Decision Conf, pages 5130–5133, July 2008.

[4] and Zhang Chuanwei. H? drive control strategy for mining electriclocomotive. In Proc. Asia-Pacific Conf. Computational Intelligence andIndustrial Applications (PACIIA), volume 1, pages 433–435, November2009.

[5] M. T. Angulo. Nonlinear extremum seeking inspired on second ordersliding modes. Automatica, 57:51–55, 2015.

[6] K. B. Ariyur and M. Krstic. Real-time optimization by extremum-seekingcontrol. John Wiley & Sons, 2003.

[7] A. Benamor, L. Chrifi-Alaui, H. Messaoud, and M. Chaabane. Slidingmode control, with integrator, for a class of mimo nonlinear systems.Engineering, 3(05):435, 2011.

[8] Y. Bensafia, K. Khettab, and S. Ladaci. DC-motor velocity controlusing a robust fractionalized adaptive pi controller. In Proc. 16th Int.Conf. Sciences and Techniques of Automatic Control and ComputerEngineering (STA), pages 622–627, December 2015.

[9] D. P. Bertsekas. Dynamic programming and optimal control, volume 1.Athena Scientific Belmont, MA, 1995.

[10] S. Bhatnagar, H.L. Prasad, and L.A. Prashanth. Lecture notes incontrol and information sciences. In Stochastic Recursive Algorithms forOptimization, volume 434 of 0170-8643, pages XVIII, 302. Springer-Verlag London, 2013.

[11] B. F. Blackman. An Exposition of Adaptive Control, chapter Extremum-seeking regulators, pages 36–50. Macmillan, New York, 1962.

[12] R. M. Brisilla, V. Sankaranarayanan, and Joseph Godfrey A. . Extendedstate observer based sliding mode control of permanent magnet DCmotor. In Proc. Annual IEEE India Conf. (INDICON), pages 1–6,December 2015.

[13] P. Dobra. Robust pi control for servo DC motor. In Proc. Int. Conf.Control Applications, volume 1, pages 100–101 vol.1, September 2002.

[14] W. H Fleming and R. W Rishel. Deterministic and stochastic optimalcontrol, volume 1. Springer Science & Business Media, 2012.

[15] Hwei P. Hsu. Fourier analysis. Revised edition. Simon and Schuster,New York, 1970.

[16] M. Ibnkahla. Signal processing for mobile communications handbook.CRC press, 2004.

[17] E. Iyasere, M. Salah, D. Dawson, J. Wagner, and E. Tatlicioglu. Op-timum seeking-based non-linear controller to maximise energy capturein a variable speed wind turbine. IET Control Theory and Applications,6(4):526–532, 2012.

[18] R. Johnson and T. Zhang. Accelerating stochastic gradient descentusing predictive variance reduction. In Advances in Neural InformationProcessing Systems, pages 315–323, 2013.

[19] F. C. Klebaner. Introduction to stochastic calculus with applications,volume 57. World Scientific, 2005.

[20] M. Krstic. Performance improvement and limitations in extremumseeking control. Systems & Control Letters, 39(5):313–326, 2000.

[21] M. Leblanc. Sur l’electrification des chemins de fer au moyen decourants alternatifs de frequence elevee. Revue Generale de l’Electricite,1922.

[22] H. Li, H. Gao, P. Shi, and X. Zhao. Fault-tolerant control of markovianjump stochastic systems via the augmented sliding mode observerapproach. Automatica, 50(7):1825–1834, 2014.

[23] S.-J. Liu and M. Krstic. Stochastic averaging in continuous time andits applications to extremum seeking. IEEE Transactions on AutomaticControl, 55(10):2235–2250, 2010.

[24] S.-J. Liu and M. Krstic. Stochastic averaging and stochastic extremumseeking. Springer Science & Business Media, 2012.

[25] K. R. Parthasarathy. An introduction to quantum stochastic calculus.Springer Science & Business Media, 2012.

[26] A. S. Poznyak. Advanced Mathematical tools for Automatic ControlEngineers. Deterministic technique, volume 1. Elsevier, Amsterdam,Oxford, 2008.

[27] A. S. Poznyak. Advanced Mathematical Tools for Control Engineers:Stochastic Techniques, volume 2. Elsevier, 2010.

[28] L. A. Rastrigin. Systems of extremal control. Mir, Moscow, 1974.[29] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K Levitt. Spread

spectrum communications handbook, volume 2. Citeseer, 1994.[30] C. U Solis, J. B Clempner, and A. S. Poznyak. Designing a terminal

optimal control with an integral sliding mode component using a saddlepoint method approach: a cartesian 3d-crane application. NonlinearDynamics, 86(2):911–926, 2016.

[31] C. U. Solis, J. B. Clempner, and A. S. Poznyak. Fast terminal sliding-mode control with an integral filter applied to a van der pol oscillator.IEEE Transactions on Industrial Electronics, 64(7):5622–5628, July2017.

[32] C. U. Solis, J. B. Clempner, and A. S. Poznyak. Constrained extremumseeking with function measurements disturbed by stochastic noise. InProc. Computing Science and Automatic Control (CCE) 2018 15th Int.Conf. Electrical Engineering, pages 1–4, September 2018.

[33] C. U. Solis, J. B. Clempner, and A. S. Poznyak. Continuous-timeextremum seeking with function measurements disturbed by stochasticnoise: A synchronous detection approach. In Proc. Computing Scienceand Automatic Control (CCE) 2018 15th Int. Conf. Electrical Engineer-ing, pages 1–5, September 2018.

[34] M. S. Stankovic and D. M. Stipanovic. Discrete time extremum seekingby autonomous vehicles in a stochastic environment. In CDC, pages4541–4546, 2009.

[35] M. S. Stankovic and D. M. Stipanovic. Extremum seeking under stochas-tic noise and applications to mobile sensors. Automatica, 46(8):1243–1251, 2010.

[36] A. Tijerina Araiza and J. L. Meza Medina. Variable structure with slidingmode controls for DC motors. In Proc. IV IEEE Int. Power ElectronicsCongress. Technical CIEP 95, pages 26–28, October 1995.

[37] H. S. Tsien. Engineering Cybernetics. McGraw-Hill, New York., 1954.[38] J. Velagic and A. Galijasevic. Design of fuzzy logic control of permanent

magnet DC motor under real constraints and disturbances. In Proc.(ISIC) 2009 IEEE Control Applications, (CCA) Intelligent Control, pages461–466, July 2009.

[39] H.-H. Wang, S. Yeung, and M. Krstic. Experimental application ofextremum seeking on an axial-flow compressor. IEEE Transactions onControl Systems Technology, 8(2):300–309, 2000.

[40] M. Zinkevich, M. Weimer, L. Li, and A. J Smola. Parallelized stochasticgradient descent. In Advances in neural information processing systems,pages 2595–2603, 2010.