Feedback Linearized Model of DC Motor using · Web viewIn this case, for any controllable pair...

International Journal of Enhanced Research in Science, Technology & EngineeringISSN: 2319-7463, Vol. 5 Issue 1, May-2016

Feedback Linearized Model of DC Motor using Differential Geometry

Abhishek Bansal

M.S- Mathematics, M.Tech.(P) –Electrical Engg.(Power System & Control Engg.), Delhi, India

ABSTRACT

This research paper discusses the existing technique of obtaining an exact feedback linearization mathematical model of non linear model of a DC motor using differential geometry approach. The singularly perturbed system, input-state linearization and the input-output linearization has also been developed along with these tools and the results of geometric approach are compared and found similar. In this paper, all equations are developed for DC motor which is field controlled rather than armature controlled.

Keywords: Lyapunov stability, Lie derivative, DC motor, feedback, differential geometry, linearization, perturbation,field control.

1. INTRODUCTION

A DC motor is a DC machine which converts DC (Direct Current) electrical power into mechanical power.

Figure 1. A DC machine

A DC motor contains a current carrying armature which is connected to the supply end through commutator segments and brushes and placed within the poles of electromagnet. Fig. 1 shows the basic part of a DC machine, and the field windings which are the windings (i.e. many turns of a conductor) wound round the pole core and the current passing through this conductor creates electromagnet. DC motors can be either shunt wound or series wound or a combination of both known as compound DC motor. The speed of these DC motors can be controlled by either armature control method or field control method.

The field controlled method affects the flux per pole. In the field control of DC series motor, either field diverter or tapper field method is used ,whereas in shunt and compound DC motors, generally field rheostat control method is employed. This paper is not of the speed controller designing but of the linearized mathematical modeling of a non-linear model.

In the field controlled, let the control input is the voltage of the field circuit, υf , then we have the following set of equations:

Field circuit Equation : υf =Rf if +L f

d if

dx , where υf is the voltage of the field circuit,R f is the resistance of the

field circuit, if is the current of the field circuit, Lf is the inductance of the field circuit.

Page | 1


Let the time constant of the field circuit, T f =Lf / Rf

Armature circuit Equation : υa=c1 if ω+La

d ia

dx+Ra ia ,where υais the voltage of the armature circuit,Rais the

resistance of the armature circuit,ia is the current of the armature circuit,Lais the inductance of the armature circuit, c1i f ω is the back e.m.f. induced in the armature circuit.Let the time constant of the armature circuit, T a=La / Ra

Torque Equation for the shaft: J d ωdx

=c2if ia−c3 ω ,where J is the rotor inertia,c3 is the damping inertia and

c2i f ia is the torque produced by the interaction of the armature current with the field circuit flux.

Let υa ,υf be constant such that υa=V a and υf =U .

A singularly perturbed system can be modeled if we assume the voltages of the armature circuit and the field circuit be constant and choose ( I f , I a, Ω )as a nominal operating point, thusthe system has a unique equilibrium point at

I f =UR f

, I a=c3V a

c3 Ra+c1c2U 2/R f2 , Ω=

c2V a U / Rf

c3 Ra+c1 c2U 2/ Rf2

Now,let the state equations have discontinuous dependence on a small perturbation parameterεand as T f ≫T a ,the above system can be modeled as a singularly perturbed system with slow variables (if andΩ) and fast variables(ia ) ,which can written asx1=−x1+u ; x2=a ( x1 z−x2 ) ; ε z=−z−b x1 x2+cwhere, x1=if /I f ; x2=ω /Ω ;z=ia/I a ;u=υf /U ; ε=T a/T f ; t '=t /T f ;a=Lf c3/ Rf J ;b=c1 c2 U2 /c3 Ra R f

2 ; c=V a/I a Ra

Now, suppose the field circuit would have been driven by a current source ,then the control input would have been the field current instead field voltage of the field circuit. In this case, this system can be modeled, if the domain of operation is restricted to x1>θ3 / 2θ1 ,as :x1=−θ1 x1−θ2 x2 u+θ3 ; x2=−θ4 x2−θ5 x1u; y=x2

where,x1is the armature current ; x2is the speed; uis the field current ;θ1, θ2, θ3,θ4, θ5are positive constants

y R2 <(θ3

2 θ5 ) / (4 θ1, θ2, θ4 )

2. FEEDBACK LINEARIZATION CONCEPT USING DIFFERENTIAL GEOMETRY APPROACH

Consider a MIMO (multiple-input,-output) system involvingnintegrators having r inputs u1 ( t ) , u2 ( t ) , u3 (t ) ,... , ur ( t );m outputs y1 ( t ) , y2 (t ) ,... , ym ( t ) and the outputs of these integrators are defined as state variablesx1 ( t ) , x2 (t ) , ... , xn (t ) . Then, we get state-space equations defined asSystem State Equation : x=f (x ,u ,t ) (2.1)and Output Equation : y=g ( x ,u , t ) . (2.2)Now if the state equation (2.1) is represented without inputsu ,it is called unforced state equation ,written as

x= f (x , t ) ; (2.3)where now input can either be zero or input could be specified as a function of time or a given feedback function of the state.And if the equation (2.3) is invariant of timet , it makes the system an autonomous system, written as

x= f (x ) (2.4)where,f : D →Rnis a locally Lipchitz map from a domainD⊂RnintoRn .

A functionf is said to satisfy a Lipchitz condition of orderα atcif there exists a positive numberMand a 1-ballB (c )

Page | 2


such that|f (x )−f (c )|<M|x−c|α ,wheneverx∈B ( c ) , x≠ c .And f ( x ) will be 'locally Lipchitz' on open and connected set,if each point ofDhas a neighborhoodD0such that fsatisfies the Lipchitz condition for all points inD0with some Lipchitz constantL0.

Let x=0be the equilibrium point for (2.4) and let n-dimensional Euclidean space is denoted by Rn ,and D⊂Rnbe a domain containing the origin, i.e.x=0 . Now using Lyapunov stability theorem, which states that x=0is stable, if V : D→ Ris a continuously differentiable function, such that V (0 )=0and V ( x )>0in D− {0 } (2.5) and V ( x )≤ 0in D (2.6) where, V ( x ) , is the derivative of V along the trajectories of (2.4), given by

V ( x )=∑i=1

n ∂V∂ xi

xi=∑i=1

n ∂V∂ xi

f i ( x )= ∂V∂ x

f ( x ) (2.7)

Thus, the equilibrium point is stable if, for eachε>0 ,there isδ=δ (ε )>0such that for allt ≥ 0 ,we have

‖x (0 )‖<δ⇒‖x (0 )‖<ε (2.8)

We know that if we have a mapM :U⊂V →W such that there is bijectionMbetween open setsUα ⊂V andU β⊂W ,then it is called aC r- diffeomorphism if and only if Mand M−1are both C r-differentiable. Ifr=∞ thenM is called diffeomorphism.

In general, to obtain a linear mathematical model for a nonlinear system, it is assumed that the variables deviate only slightly from some operating point. To get feedback linearization model, nonlinear system should be of the formx= f (x )+g ( x )u (2.9)y=h (x ) (2.10)where,f ( x )is a locally Lipchitz ; g ( x ) , h ( x )are continuous overRn ; x∈ Rnis the state vector ; u∈Rris the vector of inputs ; y∈Rmis the vector of outputs

Then, the equivalent linearized system is obtained using the change of variables and a control input, which then presents a linear input-output map between the new input and the output.Thus, we a state feedback control, given as u=α ( x )+β ( x ) υ (2.11)and the change of variables by z=T (x ) (2.12)

From the theory of “Lie derivatives”, we know that Let there be a scalar field p (ξ )in an n-dimensional spaceX nand an infinitesimal point transformation such that,

T : ' ξx=ξx+νx dt and let the dragging along( χ ) → ( χ ' )of the coordinate system( χ )by the infinitesimal point transformationT−1: ' ξ → ξinverse toT is given by ξ x'=ξ x+νx dt (2.13)Suppose we have a covariant vector field w λ (ξ )inX nand a new covariant vector field' w λ (ξ ) atξ as a field whose components' w λ ' (ξ ) w.r.t.( χ ' )atξ such that' w λ ' (ξ )≝w λ ( ' ξ ) Then, the Lie derivative of w λis given by Lν w λ=ν μ ∂μ w λ+wμ∂λ νμ

Thus, Lie derivative evaluates the change of a field along the flow of another vector field. Using this concept in (2.9) and (2.10), we can write the Lie derivative of halongf as

Lf h ( x )=∂ h∂ x

f ( x ) (2.14)

Similarly, we can define other Lie derivatives as

Lg Lf h ( x )=∂ ( Lf h )

∂ xg (x ) (2.15)

Lfk h ( x )=

∂ Lfk−1 h

∂ xf (x ) (2.16)

Page | 3


Lf0 h (x )=h ( x ) (2.17)

The system in (2.9) is input-state linearizable if and only if ∃h ( x ) satisfies (2.18) and (2.19)

Lg Lfi h ( x )=0 , 0≤ i≤ n−2 (2.18)

andLg Lfn−i h ( x ) ≠ 0 ,∀ x∈Dx (2.19)

Thus, the system in (2.9) and (2.10) has a relative degreen inD xand the change of variables (2.12) can be applied as

z=T (x )=[h ( x ) , Lf h ( x ) ,... , Lfn−1h ( x ) ]T (2.20)

Now, since for allx∈D ,the row vectors and column vectors are linearly independent, thus the Jacobian matrix

[ ∂ T∂ x ] ( x )is non-singular ∀ x∈D x .Hence, for eachx0∈Dx ,there is neighborhoodNofx0such that T ( x ) ,restricted to

N ,is a diffeomorphism.

Thus, it can be said that the system is input-state linearizable if and only if it satisfies (2.21) and (2.22) and ∃ domainD x⊂D ,such that matrixG ( x )has a rankn ∀ x∈D x (2.21)and distribution, Δ is involutive in D x (2.22)

,where matrixG ( x )=[ g ( x ) , a d f g ( x ) ,... , ad fn−1 g ( x ) ] and distribution, Δ= span{g , ad f g , ... , ad f

n−2 g}

A non-linear system (2.9),whereD x∈Rn , f : D x → Rn ,G : D x → Rn× p , is said to be input-state linearizable, if there

exists a diffeomorphismT : D x → Rn , such thatD z=T ( D x ) ,contains the origin and (2.12) transforms (2.9) into

z=Az+B β−1 (x ) [u−α ( x ) ] (2.23)with( A , B )controllable andβ (x )non-singular for all,x∈Dx .

Thus, we get z= ∂ T∂ x

x= ∂T∂ x [ f ( x )+G ( x ) u ] (2.24)

and z=AT ( x )+B β−1 ( x ) [u−α ( x ) ] (2.25)

From (2.24) and (2.25), we get ∂T∂ x [ f ( x )+G ( x )u ]=AT ( x )+B β−1 ( x ) [u−α ( x ) ] (2.26)

Now if we putu=0in (2.26), we get∂T∂ x

f ( x )=AT ( x )−B β−1 ( x ) α ( x ) (2.27)

∂T∂ x

G ( x )=B β−1 ( x ) (2.28)

Thus, if there is a mapT (⋅ ) ,which satisfies (2.27) and (2.28) for someα , β , A , B , then (2.12) transforms (2.9) into (2.23).

Now, if apply linear state transformation in (2.23) with ς=M z with a singularM ,then the state equation in theς−¿coordinates are given byς=M A M−1 ς+ M B β−1 ( x ) [u−α ( x ) ] (2.29)

Now if we have single input system instead of multiple-input, thus, now p=1.In this case, for any controllable pair( A , B ) ,we can find a nonsingular matrixM that transforms( A , B )into canonical form M A M−1=A c+B c λT (2.30)and M B=Bc (2.31)where

Page | 4


Ac=[0 1 0 … 00 0 1 … 0⋮ ⋮ ⋱ ⋱ ⋮⋮ ⋮ ⋮ 0 10 … … 0 0

]n× n

(2.32)

and Bc=[00⋮01] (2.33)

Let T ( x )=[T 1 ( x )T 2 ( x )⋮

Tn−1 ( x )T n ( x )

] (2.34)

Thus,

Ac T ( x )−Bc β−1 ( x ) α ( x )=[T2 ( x )T3 ( x )⋮

Tn ( x )−α ( x ) /β ( x )

] (2.35)

and Bc β−1 ( x )=[00⋮0

1/ β (x )] (2.36)

Using these expressions in (2.27),we get simplified partial differential equations as∂T 1

∂ xf (x )=T2 ( x ); (2.37)

∂T 2

∂ xf (x )=T3 ( x ); (2.38)

⋮∂T n−1

∂ xf (x )=Tn ( x ); (2.39)

∂T n

∂ xf ( x )=−α ( x ) / β (x ) (2.40)

Using these expressions in (2.28), we get simplified partial differential equations as∂T 1

∂ xg ( x )=0 ; (2.41)

∂T 2

∂ xg ( x )=0 ; (2.42)

⋮ ∂T n−1

∂ xg ( x )=0 ; (2.43)

Page | 5


∂T n

∂ xg ( x )=1/ β (x ) ≠ 0 (2.44)

Now, the functionT 1 ( x )has to be searched, which satisfies∂T i

∂ xg ( x )=0 ; (n−1 )≤ i≤ 1 (2.45)

where,T i+1 (x )=∂ T i

∂ xf ( x ) , ( n−1 ) ≤i ≤1

∂T n

∂ xg ( x )≠ 0 (2.46)

If there is a function which satisfies (2.45) and (2.46), then

β (x )= 1( ∂T n/∂ x) g ( x ) (2.47)

α (x )=−( ∂T n/∂ x ) f ( x )

(∂ Tn/∂ x ) g ( x ) (2.48)

From differential geometry approach, we also noticed that for all x∈D , k≥ 0 , 0≤ j≤ r−1, we haveLad f

i g Lfk h ( x )=0 , 0≤ j+k<r−1 (2.49)

Lad fi g Lf

k h ( x )=(−1 ) j Lg L fr−1 h (x ) ≠ 0 , j+k=r−1 (2.50)

and the row vectors [dh ( x ) , …, d Lfr−1 h ( x ) ]and column vectors[ g ( x ) … a d f

r−1 g ( x ) ] are linearly independent. From Jacobian identity, we know that for any real-valuedλand any vector fieldsf and β ,L[ f , β ] λ ( x )=Lf Lβ λ ( x )−Lβ L f λ ( x ) (2.51)

Also the distribution, Δ generated byf 1 , …, f ris said to be completely integrable if for eachx0∈D ,∃ a neighborhoodNofx0andn−rreal-valued smooth functionsh1 ( x ) , …, hn−r ( x )satisfy the partial differential equations∂ hi

∂ xf i ( x )=0 , ∀1≤ i≤ r ,1≤ j ≤ n−r (2.52)

and the covector fieldsd h j ( x )are linearly independent for allx∈D .Thus, a nonsingular distribution is completely integrable if and only if it is involutive, and is called as Frobenius theorem.

3. DC MOTOR LINEARIZATION

Let us now apply all these concepts in obtaining the exact linearization model of the DC motor.Let the field controlled motor is represented by the system as

x= f (x )+gu , (3.1)

where f ( x )=[ −a x1

−b x2+ ρ−c x1 x3

θ x1 x2];g=[100 ] (3.2)

¿a , b , c ,θ , ρare positive constants

Letx1=0 , x2=ρ/b ,be the equilibrium point for the open loop system and the desired operating point be~xgiven as~x=( 0 , ρ/b ,ω0 )T ,whereω0is a desired set point for the angular velocityx3.

Let T 2 ( x )=∂ T1

∂ xf ( x );T 3 ( x )=

∂ T 2

∂ xf ( x ) (3.3)

Page | 6


Then, withT 1 (~x )=0 ,we can findT 1=T1 ( x ) ,which satisfies the conditions (2.41),(2.42) and∂T 3

∂ xg ( x )≠ 0 (3.4)

thus, ∂T 1

∂ xg=

∂T 1

∂ x1=0 (3.5)

and therefore,

T 2 ( x )=∂ T1

∂ x2[−b x2+ρ−c x1 x3 ]+

∂ T1

∂ x3θ x1 x2 (3.6)

Using (2.42) in (2.46),we get c x3

∂T 1

∂ x2=θ x2

∂T 1

∂ x3, (3.7)

Equation (3.7) will be satisfied ifT 1is of the form T 1=c1 [θ x22+c x3

2 ]+c2 (3.8)

Let c1=1and c2=−θ (~x )22−c (~x )3

2=−θ ( ρ /b )2−c ω02 (3.9)

using (3.9), we get T 2 ( x )=2θ x2 ( ρ−b x2) (3.10)

and T 3 ( x )=2θ ( ρ−2b x2 ) (b x2+ ρ−c x1 x3 ) (3.11)hence,

∂T 3

∂ xg=

∂T 3

∂ x1=2cθ ( ρ−2b x2 ) ( x3) (3.12)

and the (3.4) is satisfied, wheneverx2≠ ρ /b , x3≠ 0.

LetD x contains the point ~xand~x3>0 ,then

D x={x∈R3 ; x2>ρ

2b, x3>0} (3.13)

Thus, the map (2.12) is diffeomorphism on D xand the state equation in the z−¿coordinates are defined in

D z=T ( D x )= {z∈R3 } (3.14)

such that {z1>θ ϕ2 ( z )2−θ ( ρ /b )2−c ω02 }, {z2<

θρ2 b }andϕ (⋅) is the inverse of the map

2 θ x2 ( ρ−b x2 ) .Thus, domainD zcontains the originz=0.

Now, we can verify the same using differential geometry approach as explained in the previous section. As we know

a d f g=[ f , g ]=[ ac x3

−θ x2];ad f

2 g=[ f , a d f g ]=[ a2

(a+b )c x3

(b−a ) θ x2−θρ];Considering G=[ g ,a d f g , ad f

2 g ] ¿ [1 a a2

0 c x3 (a+b )c x3

0 −θ x2 (b−a ) θ x2−θ ρ ]; thus, the determinant of G , detG=cθ (−ρ+2 b x2 ) x3 . Hence,Ghas a rank 3 forx2≠ ρ /2 b , x3 ≠ 0.Now,

Page | 7


[ g , ad f g ]=∂ (ad f g )

∂ xg=[0 0 0

0 0 c0 −θ 0 ][100]=[000 ]

¿¿

.

Hence the distribution ∆ = span{g ,ad f g }is involutive as [ g ,ad f g ]∈DandD x={x∈R3 ; x2>ρ

2 b, x3>0} ,

which is same as (3.14) and this completes the verification and the development of exact feedback linearized model of the DC motor.

CONCLUSION The results of geometric approach are compared and found similar, if other tools are used e.g.. input-state linearization and the input-output linearization, to obtain the exact feedback linearization mathematical model of non linear model of a DC motor.

REFERENCES

1. Krause,Oleg,Sudhoff,” Analysis of electric machinery and Drives systems,” IEEE Press, 2002, ch. 2.2. Ned Mohan “First Course on Power Electronics and Drives”, MNPERE Minneapolis, 2003, ch. 11.3. Serge Lang,”Differential and Riemannian Manifolds”, Springer-Verlag, 1995, ch.1.4. Tom M. Apostol,”Mathematical Analysis”,Addison-Wesley Publishing Company, World Student Series Edition, 1981, ch.

4, ch. 5.5. Sharipov R. A. ,”Course of Differential Geometry: the textbook”, Russian Federal Committee for Higher Education, 2004,

ch. 2.6. Jeffrey M. Lee,” Manifolds and Differential Geometry”, American Mathematical Society, 2009, ch. 1, ch. 8.7. Michael Müger,” An Introduction to Differential Topology,de Rham Theory and Morse Theory”, 2005.8. M. Vidyasagar,”Non linear system analysis”, Prentice-Hall, 199, ch. 6 , ch.7.9. Horacio J. Marquez ,”Nonlinear Control Systems Analysis and Design”, John Wiley & Sons Inc., 2003, ch. 6, ch. 10.10. Lui Lam,”Nonlinear Physics for Beginners Fractals, Chaos, Solitons, Pattern Formation, Cellular Automata and

Complex”, World Scientific Publishing Co. Pte. Ltd., 199, ch. 1.11. Gildas Besancon (Ed.), ”Nonlinear Observers and Applications”, Springer, 2007, ch.1 ,ch.3, ch. 5.12. Wassim M. Haddad, Vijay Sekhar Chellaboina, “Nonlinear Dynamical Systems and Control: A Lyapunov-Based

Approach”, Princeton University Press, 2008, ch. 2, ch. 4,ch.7,ch.8 ,ch.9.13. Walter Rudin,”Real And Complex Analysis”, McGrawHill, 1970, ch. 17.14. V.S. Varadarajan,”Lie Groups, Lie Algebras, and Their Representations”,Springer-Verlag,New York, 1984, ch.1, ch.2.15. Francesco Iachello,”Lie Algebras and Applications”, Springer-Verlag Berlin Heidelberg, 2006, ch.1 ch.3., ch. 4.

Page | 8

Feedback Linearized Model of DC Motor using · Web viewIn this case, for any controllable pair...

Documents

Transcript of Feedback Linearized Model of DC Motor using · Web viewIn this case, for any controllable pair...