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International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 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) โElectrical 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 nonlinear model. In the field controlled, let the control input is the voltage of the field circuit, ๐๐ , then we have the following set of equations: ๐๐ Field circuit Equation : ๐๐ = ๐ ๐ ๐๐ + ๐ฟ๐ ๐ , where ๐๐ is the voltage of the field circuit,๐ ๐ is the resistance of the dx field circuit, ๐๐ is the current of the field circuit, ๐ฟ๐ is the inductance of the field circuit. Let the time constant of the field circuit, ๐๐ = ๐ฟ๐ โ๐ ๐ Page | 1 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, May-2016 ๐๐ Armature circuit Equation : ๐๐ = ๐1 ๐๐ ๐ + ๐ฟ๐ ๐ + ๐ ๐ ๐๐ ,where ๐๐ is the voltage of the armature circuit,๐ ๐ is dx the resistance of the armature circuit,๐๐ is the current of the armature circuit,๐ฟ๐ is the inductance of the armature circuit, ๐1 ๐๐ ๐ is the back e.m.f. induced in the armature circuit. Let the time constant of the armature circuit, ๐๐ = ๐ฟ๐ โ๐ ๐ ๐๐ Torque Equation for the shaft: ๐ฝ = ๐2 ๐๐ ๐๐ โ ๐3 ๐ ,where ๐ฝ is the rotor inertia, ๐3 is the damping inertia and dx ๐2 ๐๐ ๐๐ is the torque produced by the interaction of the armature current with the field circuit flux. Let ๐๐ , ๐๐ be constant such that ๐๐ = ๐๐ and ๐๐ = ๐. A singularly perturbed system can be modeled if we assume the voltages of the armature circuit and the field circuit be constant and choose (๐ผ๐ , ๐ผ๐ , ๐บ)as a nominal operating point, thusthe system has a unique equilibrium point at ๐ผ๐ = ๐ ๐ ๐ , ๐ผ๐ = ๐3 ๐๐ ๐3 ๐ ๐ + ๐1 ๐2 ๐ 2 โ๐ ๐2 , ๐บ = ๐2 ๐๐ ๐ โ๐ ๐ ๐3 ๐ ๐ + ๐1 ๐2 ๐ 2 โ๐ ๐2 Now,let the state equations have discontinuous dependence on a small perturbation parameter๐and as ๐๐ โซ ๐๐ ,the above system can be modeled as a singularly perturbed system with slow variables (๐๐ and๐บ) and fast variables(๐๐ ),which can written as ๐ฅห1 = โ๐ฅ1 + ๐ข ; ๐ฅห2 = ๐(๐ฅ1 ๐ง โ ๐ฅ2 ) ; ๐ ๐งห = โ๐ง โ ๐๐ฅ1 ๐ฅ2 + ๐ where, ๐ฅ1 = ๐๐ โ๐ผ๐ ; ๐ฅ2 = ๐ โ๐บ ; ๐ง = ๐๐ โ๐ผ๐ ; ๐ข = ๐๐ โ๐ ; ๐ = ๐๐ โ๐๐ ; ๐ก โฒ = ๐กโ๐๐ ; ๐ = ๐ฟ๐ ๐3 โ๐ ๐ ๐ฝ ; ๐ = ๐1 ๐2 ๐ 2 โ๐3 ๐ ๐ ๐ ๐2 ; ๐ = ๐๐ โ๐ผ๐ ๐ ๐ 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 ๐ฅ1 > ๐3 โ 2๐1 ,as : ๐ฅห1 = โ๐1 ๐ฅ1 โ ๐2 ๐ฅ2 ๐ข + ๐3 ; ๐ฅห2 = โ๐4 ๐ฅ2 โ ๐5 ๐ฅ1 ๐ข ; ๐ฆ = ๐ฅ2 where,๐ฅ1 is the armature current ; ๐ฅ2 is the speed; ๐ข is the field current ;๐1, ๐2, ๐3, ๐4, ๐5 are positive constants ๐ฆ๐ 2 < (๐32 ๐5 ) โ (4 ๐1, ๐2, ๐4 ) 2. FEEDBACK LINEARIZATION CONCEPT USING DIFFERENTIAL GEOMETRY APPROACH Consider a MIMO (multiple-input,-output) system involving๐integrators having ๐ inputs ๐ข1 (๐ก), ๐ข2 (๐ก), ๐ข3 (๐ก), . . . , ๐ข๐ (๐ก); ๐ outputs ๐ฆ1 (๐ก), ๐ฆ2 (๐ก), . . . , ๐ฆ๐ (๐ก) and the outputs of these integrators are defined as state variables ๐ฅ1 (๐ก), ๐ฅ2 (๐ก), . .. , ๐ฅ๐ (๐ก). Then, we get state-space equations defined as System State Equation : ๐ฅห = ๐(๐ฅ, ๐ข, ๐ก) (2.1) and Output Equation : ๐ฆ = ๐(๐ฅ, ๐ข, ๐ก). (2.2) Now if the state equation (2.1) is represented without inputs ๐ข, it is called unforced state equation ,written as ๐ฅห = ๐(๐ฅ, ๐ก); (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 time ๐ก, it makes the system an autonomous system, written as ๐ฅห = ๐(๐ฅ) (2.4) where,๐ โถ ๐ท โ ๐ ๐ is a locally Lipchitz map from a domain ๐ท โ ๐ ๐ into ๐ ๐ . A function ๐ is said to satisfy a Lipchitz condition of order ๐ผ at ๐ if there exists a positive number ๐and a 1-ball ๐ต(๐) such that | ๐(๐ฅ) โ ๐(๐) | < ๐|๐ฅ โ ๐|๐ผ , whenever ๐ฅ โ ๐ต(๐), ๐ฅ โ ๐. And ๐(๐ฅ) will be 'locally Lipchitz' on open and connected set,if each point of ๐ท has a neighborhood ๐ท0 such that ๐satisfies the Lipchitz condition for all points in ๐ท0 with some Lipchitz constant ๐ฟ0. Let ๐ฅ = 0 be the equilibrium point for (2.4) and let n-dimensional Euclidean space is denoted by ๐ ๐ , and ๐ท โ ๐ ๐ be a domain containing the origin, i.e.๐ฅ = 0. Now using Lyapunov stability theorem, which states that ๐ฅ = 0 is stable, if ๐ โถ ๐ท โ ๐ is a continuously differentiable function, such that ๐(0) = 0 and ๐(๐ฅ) > 0 in ๐ท โ {0} (2.5) and ๐ห (๐ฅ) โค 0 in ๐ท (2.6) where, ๐ห (๐ฅ), is the derivative of ๐ along the trajectories of (2.4), given by ๐๐ ๐๐ ๐๐ ๐ห (๐ฅ) = โ๐i=1 ๐ฅห๐ = โ๐i=1 ๐๐ (๐ฅ) = ๐(๐ฅ) (2.7) ๐๐ฅ๐ ๐๐ฅ๐ ๐๐ฅ Thus, the equilibrium point is stable if, for each ๐ > 0,there is ๐ฟ = ๐ฟ(๐) > 0such that for all ๐ก โฅ 0,we have โ๐ฅ(0) โ < ๐ฟ โ โ๐ฅ(0) โ < ๐ (2.8) Page | 2 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, May-2016 We know that if we have a map ๐ โถ ๐ โ ๐ โ ๐ such that there is bijection ๐ between open sets ๐๐ผ โ ๐ and ๐๐ฝ โ ๐ , then it is called a ๐ถ ๐ - diffeomorphism if and only if ๐and ๐โ1 are both ๐ถ ๐ -differentiable. If ๐ = โ then ๐ 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 form ๐ฅห = ๐(๐ฅ) + ๐(๐ฅ)๐ข (2.9) ๐ฆ = โ(๐ฅ) (2.10) where,๐(๐ฅ) is a locally Lipchitz ; ๐(๐ฅ), โ(๐ฅ)are continuous over ๐ ๐ ; ๐ฅ โ ๐ ๐ is the state vector ; ๐ข โ ๐ ๐ is the vector of inputs ; ๐ฆ โ ๐ ๐ is 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 ๐ข = ๐ผ(๐ฅ) + ๐ฝ(๐ฅ)๐ (2.11) and the change of variables by ๐ง = ๐(๐ฅ) (2.12) From the theory of โLie derivativesโ, we know that Let there be a scalar field ๐(๐) in an n-dimensional space ๐๐ and an infinitesimal point transformation such that, ๐ โถ โฒ๐ ๐ฅ = ๐ ๐ฅ + ๐ ๐ฅ ๐๐ก and let the dragging along (๐) โ (๐โฒ)of the coordinate system (๐) by the infinitesimal point transformation ๐ โ1 โถ โฒ๐ โ ๐ inverse to ๐ is given by ๐ ๐ฅโฒ = ๐ ๐ฅ + ๐ ๐ฅ ๐๐ก (2.13) Suppose we have a covariant vector field ๐ค๐ (๐) in ๐๐ and a new covariant vector field โฒ๐ค๐ (๐) at ๐ as a field whose components โฒ๐ค๐โฒ (๐) w.r.t.(๐โฒ) at ๐ such that โฒ๐ค๐โฒ (๐) โ ๐ค๐ (โฒ๐) Then, the Lie derivative of ๐ค๐ is given by ๐ฟ๐ ๐ค๐ = ๐ ๐ ๐๐ ๐ค๐ + ๐ค๐ ๐๐ ๐ ๐ 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 โ along ๐ as ๐โ ๐ฟ๐ โ(๐ฅ) = ๐(๐ฅ) (2.14) ๐๐ฅ Similarly, we can define other Lie derivatives as ๐ฟ๐ ๐ฟ๐ โ(๐ฅ) = ๐ฟ๐๐ โ(๐ฅ) = ๐(๐ฟ๐ โ) ๐๐ฅ ๐๐ฟ๐โ1 ๐ โ ๐๐ฅ ๐(๐ฅ) (2.15) ๐(๐ฅ) (2.16) ๐ฟ0๐ โ(๐ฅ) = โ(๐ฅ) (2.17) The system in (2.9) is input-state linearizable if and only if โ โ(๐ฅ) satisfies (2.18) and (2.19) ๐ฟ๐ ๐ฟ๐๐ โ(๐ฅ) = 0 , 0 โค ๐ โค ๐ โ 2 (2.18) and ๐ฟ๐ ๐ฟ๐โ๐ โ(๐ฅ) โ 0 , โ ๐ฅ โ ๐ท (2.19) ๐ฅ ๐ Thus, the system in (2.9) and (2.10) has a relative degree ๐ in ๐ท๐ฅ and the change of variables (2.12) can be applied as ๐ง = ๐(๐ฅ) = [ โ(๐ฅ), ๐ฟ๐ โ(๐ฅ), . . . , ๐ฟ๐โ1 ๐ โ(๐ฅ) ] ๐ (2.20) Now, since for all๐ฅ โ ๐ท,the row vectors and column vectors are linearly independent, thus the Jacobian matrix ๐๐ [ ] (๐ฅ) is non-singular โ ๐ฅ โ ๐ท๐ฅ .Hence, for each ๐ฅ0 โ ๐ท๐ฅ ,there is neighborhood ๐ of ๐ฅ0 such that ๐(๐ฅ), restricted ๐๐ฅ to ๐,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 โ domain ๐ท๐ฅ โ ๐ท, such that matrix ๐บ(๐ฅ) has a rank ๐ โ ๐ฅ โ ๐ท๐ฅ (2.21) and distribution, ฮ is involutive in ๐ท๐ฅ (2.22) ๐โ1 ,where matrix๐บ(๐ฅ) = [๐(๐ฅ), ๐๐๐ ๐(๐ฅ), . . . , ๐๐๐ ๐(๐ฅ)] and distribution, ฮ= span{๐, ๐๐๐ ๐, . . . , ๐๐๐๐โ2 ๐} A non-linear system (2.9),where ๐ท๐ฅ โ ๐ ๐ , ๐: ๐ท๐ฅ โ ๐ ๐ , ๐บ โถ ๐ท๐ฅ โ ๐ ๐×๐ , is said to be input-state linearizable, if there exists a diffeomorphism ๐: ๐ท๐ฅ โ ๐ ๐ , such that ๐ท๐ง = ๐(๐ท๐ฅ ),contains the origin and (2.12) transforms (2.9) into ๐งห = ๐ด๐ง + ๐ต๐ฝ โ1 (๐ฅ)[ ๐ข โ ๐ผ(๐ฅ) ] (2.23) with(๐ด, ๐ต)controllable and ๐ฝ(๐ฅ)non-singular for all,๐ฅ โ ๐ท๐ฅ . ๐๐ ๐๐ [๐(๐ฅ) + ๐บ(๐ฅ)๐ข] Thus, we get ๐งห = ๐ฅห = ๐๐ฅ ๐๐ฅ and ห๐ง = ๐ด๐(๐ฅ) + ๐ต๐ฝ โ1 (๐ฅ)[ ๐ข โ ๐ผ(๐ฅ) ] (2.24) (2.25) Page | 3 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, May-2016 From (2.24) and (2.25), we get ๐๐ ๐๐ฅ [๐(๐ฅ) + ๐บ(๐ฅ)๐ข] = ๐ด๐(๐ฅ) + ๐ต๐ฝ โ1 (๐ฅ)[ ๐ข โ ๐ผ(๐ฅ) ] Now if we put ๐ข = 0 in (2.26), we get ๐๐ ๐(๐ฅ) = ๐ด๐(๐ฅ) โ ๐ต๐ฝ โ1 (๐ฅ) ๐ผ(๐ฅ) (2.27) โ1 (๐ฅ) (2.28) ๐๐ฅ ๐๐ ๐๐ฅ ๐บ(๐ฅ) = ๐ต๐ฝ (2.26) Thus, if there is a map ๐( โ ) ,which satisfies (2.27) and (2.28) for some ๐ผ, ๐ฝ, ๐ด, ๐ต, then (2.12) transforms (2.9) into (2.23). Now, if apply linear state transformation in (2.23) with ๐ = ๐ ๐ง with a singular ๐, then the state equation in the ๐ โ coordinates are given by ๐ห = ๐ ๐ด ๐โ1 ๐ + ๐ ๐ต๐ฝ โ1 (๐ฅ)[ ๐ข โ ๐ผ(๐ฅ) ] (2.29) Now if we have single input system instead of multiple-input, thus, now ๐ = 1. In this case, for any controllable pair (๐ด, ๐ต), we can find a nonsingular matrix ๐ that transforms (๐ด, ๐ต) into canonical form ๐ ๐ด ๐โ1 = ๐ด๐ + ๐ต๐ ๐๐ (2.30) and ๐ ๐ต = ๐ต๐ (2.31) where 0 1 0 โฆ 0 0 0 1 โฆ 0 ๐ด๐ = โฎ โฎ โฑ โฑ โฎ (2.32) โฎ โฎ โฎ 0 1 [0 โฆ โฆ 0 0]๐ × ๐ 0 0 and ๐ต๐ = โฎ (2.33) 0 [1] ๐1 (๐ฅ) ๐2 (๐ฅ) โฎ Let ๐(๐ฅ) = (2.34) ๐๐โ1 (๐ฅ) [ ๐๐ (๐ฅ) ] Thus, ๐2 (๐ฅ) ๐3 (๐ฅ) โฎ ๐ด๐ ๐(๐ฅ) โ ๐ต๐ ๐ฝ โ1 (๐ฅ) ๐ผ(๐ฅ) = (2.35) ๐๐ (๐ฅ) [โ๐ผ (๐ฅ)โ๐ฝ (๐ฅ)] 0 0 โฎ and ๐ต๐ ๐ฝ โ1 (๐ฅ) = (2.36) 0 [1โ๐ฝ (๐ฅ)] Using these expressions in (2.27),we get simplified partial differential equations as ๐๐1 ๐(๐ฅ) = ๐2 (๐ฅ) ; (2.37) ๐๐ฅ ๐๐2 ๐(๐ฅ) = ๐3 (๐ฅ) ; โฎ ๐๐๐โ1 ๐(๐ฅ) = ๐๐ (๐ฅ) ; (2.38) ๐๐ฅ (2.39) ๐๐ฅ ๐๐๐ ๐๐ฅ ๐(๐ฅ) = โ๐ผ (๐ฅ)โ๐ฝ (๐ฅ) (2.40) Using these expressions in (2.28), we get simplified partial differential equations as ๐๐1 ๐(๐ฅ) = 0 ; (2.41) ๐๐ฅ ๐๐2 ๐๐ฅ ๐(๐ฅ) = 0 ; โฎ (2.42) Page | 4 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, May-2016 ๐๐๐โ1 ๐๐ฅ ๐๐๐ ๐๐ฅ ๐(๐ฅ) = 0 ; (2.43) ๐(๐ฅ) = 1โ๐ฝ (๐ฅ) โ 0 (2.44) Now, the function ๐1 (๐ฅ) has to be searched, which satisfies ๐๐๐ ๐(๐ฅ) = 0 ; (๐ โ 1) โค ๐ โค 1 (2.45) ๐๐ฅ where, ๐๐ +1 (๐ฅ) = ๐๐๐ ๐๐ฅ ๐๐๐ ๐๐ฅ (๐ โ 1) โค ๐ โค 1 ๐(๐ฅ), ๐(๐ฅ) โ 0 (2.46) If there is a function which satisfies (2.45) and (2.46), then 1 ๐ฝ(๐ฅ) = (๐๐ โ (2.47) ๐ผ(๐ฅ) = โ ๐ ๐ ๐ฅ)๐(๐ฅ) (๐๐๐โ๐ ๐ฅ)๐(๐ฅ) (2.48) (๐๐๐โ๐ ๐ฅ)๐(๐ฅ) From differential geometry approach, we also noticed that for all ๐ฅ โ ๐ท, ๐ฟ๐๐ ๐๐ ๐ฟ๐ ๐ โ(๐ฅ) = 0 ,0 โค ๐ + ๐ < ๐ โ 1 ๐ ๐ฟ๐๐ ๐ โฅ 0, 0 โค ๐ โค ๐ โ 1, we have (2.49) ๐ฟ ๐ โ(๐ฅ) = (โ1)๐ ๐ฟ๐ ๐ฟ๐ ๐โ1 โ(๐ฅ) โ 0 , ๐ + ๐ = ๐ โ 1 (2.50) ๐ ๐ ๐ ๐ and the row vectors [๐โ(๐ฅ), โฆ , ๐๐ฟ๐ ๐โ1 โ(๐ฅ)] and column vectors[๐(๐ฅ) โฆ ๐๐๐ ๐โ1 ๐(๐ฅ)] are linearly independent. From Jacobian identity, we know that for any real-valued ๐ and any vector fields ๐and ๐ฝ, ๐ฟ[๐,๐ฝ] ๐(๐ฅ) = ๐ฟ๐ ๐ฟ๐ฝ ๐(๐ฅ) โ ๐ฟ๐ฝ ๐ฟ๐ ๐(๐ฅ) (2.51) Also the distribution, ฮ generated by ๐1, โฆ , ๐๐ is said to be completely integrable if for each ๐ฅ0 โ ๐ท, โ a neighborhood ๐ of ๐ฅ0 and ๐ โ ๐ real-valued smooth functions โ1 (๐ฅ), โฆ , โ๐โ๐ (๐ฅ) satisfy the partial differential equations ๐โ๐ ๐ (๐ฅ) = 0, โ 1 โค ๐ โค ๐, 1 โค ๐ โค ๐ โ ๐ (2.52) ๐๐ฅ ๐ and the covector fields ๐โ๐ (๐ฅ)are linearly independent for all ๐ฅ โ ๐ท.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 ๐ฅห = ๐(๐ฅ) + ๐ ๐ข, (3.1) โ๐๐ฅ1 1 where ๐(๐ฅ) = [โ๐๐ฅ2 + ๐ โ ๐๐ฅ1 ๐ฅ3 ] ; ๐ = [0] (3.2) ๐๐ฅ1 ๐ฅ2 0 and ๐, ๐, ๐, ๐, ๐ are positive constants Let๐ฅ1 = 0, ๐ฅ2 = ๐โ๐,be the equilibrium point for the open loop system and the desired operating point be ~ ๐ฅ given as ~๐ฅ = (0, ๐โ๐ , ๐0 )๐ , where ๐0 is a desired set point for the angular velocity ๐ฅ3. ๐๐ ๐๐ Let ๐2 (๐ฅ) = 1 ๐(๐ฅ) ; ๐3 (๐ฅ) = 2 ๐(๐ฅ) (3.3) ๐๐ฅ ๐๐ฅ Then, with ๐1 (๐ฅ ~) = 0,we can find ๐1 = ๐1 (๐ฅ),which satisfies the conditions (2.41),(2.42) and ๐๐3 ๐(๐ฅ) โ 0 (3.4) ๐๐ฅ ๐๐1 thus, ๐๐ฅ ๐ = ๐๐1 ๐๐ฅ1 = 0 (3.5) and therefore, ๐2 (๐ฅ) = Using (2.42) in (2.46),we get ๐๐ฅ3 ๐๐1 ๐๐ฅ2 ๐๐1 ๐๐ฅ2 [โ๐๐ฅ2 + ๐ โ ๐๐ฅ1 ๐ฅ3 ] + = ๐๐ฅ2 ๐๐1 ๐๐ฅ3 Equation (3.7) will be satisfied if ๐1 is of the form ๐๐1 ๐๐ฅ3 ๐๐ฅ1 ๐ฅ2 , (3.6) (3.7) ๐1 = ๐1 [๐๐ฅ22 Let ๐1 = 1 and ๐2 = โ๐(๐ฅ ~)22 โ ๐(๐ฅ ~)23 = โ๐(๐โ๐)2 โ ๐๐02 + ๐๐ฅ32 ] + ๐2 (3.8) (3.9) Page | 5 International Journal of Enhanced Research in Science, Technology & Engineering ISSN: 2319-7463, Vol. 5 Issue 1, May-2016 using (3.9), we get and hence, ๐2 (๐ฅ) = 2๐๐ฅ2 (๐ โ ๐๐ฅ2 ) ๐3 (๐ฅ) = 2๐(๐ โ 2๐๐ฅ2 )(๐๐ฅ2 + ๐ โ ๐๐ฅ1 ๐ฅ3 ) ๐๐3 ๐๐ฅ ๐ = ๐๐3 ๐๐ฅ1 (3.10) (3.11) = 2๐๐(๐ โ 2๐๐ฅ2 )(๐ฅ3 ) (3.12) and the (3.4) is satisfied, whenever ๐ฅ2 โ ๐โ๐ , ๐ฅ3 โ 0. Let ๐ท๐ฅ contains the point ~๐ฅ and ~ ๐ฅ3 > 0,then ๐ท๐ฅ = {๐ฅ โ ๐ 3 ; ๐ฅ2 > ๐ 2๐ , ๐ฅ3 > 0} (3.13) Thus, the map (2.12) is diffeomorphism on ๐ท๐ฅ and the state equation in the ๐ง โ coordinates are defined in ๐ท๐ง = ๐(๐ท๐ฅ ) = {๐ง โ ๐ 3 } (3.14) ๐๐ 2 2 2 {๐ง1 > ๐๐ (๐ง)2 โ ๐(๐โ๐) โ ๐๐0 }, {๐ง2 < } and ๐( โ ) is the inverse of the map such that 2๐ 2๐๐ฅ2 (๐ โ ๐๐ฅ2 ). Thus, domain ๐ท๐ง contains the origin ๐ง = 0. Now, we can verify the same using differential geometry approach as explained in the previous section. As we know ๐ ๐2 2 (๐ + ๐)๐๐ฅ3 ] ; ๐๐๐ ๐ = [๐, ๐] = [ ๐๐ฅ3 ] ; ๐๐๐ ๐ = [๐, ๐๐๐ ๐] = [ โ๐๐ฅ2 (๐ โ ๐)๐๐ฅ2 โ ๐๐ Considering ๐บ = [๐, ๐๐๐ ๐, ๐๐๐2 ๐] thus, the determinant of ๐บ, 1 = [0 0 ๐ ๐๐ฅ3 โ๐ ๐ฅ2 ๐2 (๐ + ๐)๐๐ฅ3 ]; (๐ โ ๐)๐ ๐ฅ2 โ ๐ ๐ det๐บ = ๐๐(โ๐ + 2๐๐ฅ2 )๐ฅ3 . Hence,๐บ has a rank 3 for ๐ฅ2 โ ๐โ2 ๐, ๐ฅ3 โ 0. Now, 0 0 0 1 0 ๐(๐๐๐ ๐) [๐, ๐๐๐ ๐] = ๐ = [0 0 ๐ ] [0] = [0]. ๐๐ฅ 0 โ๐ 0 0 0 ๐ Hence the distribution โ = span{๐, ๐๐๐ ๐}is involutive as [๐, ๐๐๐ ๐] โ ๐ทand ๐ท๐ฅ = {๐ฅ โ ๐ 3 ; ๐ฅ2 > , ๐ฅ3 > 0} , 2๐ 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. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Krause,Oleg,Sudhoff,โ Analysis of electric machinery and Drives systems,โ IEEE Press, 2002, ch. 2. Ned Mohan โFirst Course on Power Electronics and Drivesโ, MNPERE Minneapolis, 2003, ch. 11. Serge Lang,โDifferential and Riemannian Manifoldsโ, Springer-Verlag, 1995, ch.1. Tom M. 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Haddad, Vijay Sekhar Chellaboina, โNonlinear Dynamical Systems and Control: A Lyapunov-Based Approachโ, Princeton University Press, 2008, ch. 2, ch. 4,ch.7,ch.8 ,ch.9. Walter Rudin,โReal And Complex Analysisโ, McGrawHill, 1970, ch. 17. V.S. Varadarajan,โLie Groups, Lie Algebras, and Their Representationsโ,Springer-Verlag,New York, 1984, ch.1, ch.2. Francesco Iachello,โLie Algebras and Applicationsโ, Springer-Verlag Berlin Heidelberg, 2006, ch.1 ch.3., ch. 4. Page | 6