Download Feedback Linearized Model of DC Motor using Differential Geometry

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) –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 nonlinear 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:
𝑑𝑖
Field circuit Equation : 𝜐𝑓 = 𝑅𝑓 𝑖𝑓 + 𝐿𝑓 𝑓
, where 𝜐𝑓 is the voltage of the field circuit,𝑅𝑓 is the resistance of the
dx
field circuit, 𝑖𝑓 is the current of the field circuit, 𝐿𝑓 is the inductance of the field circuit.
Let the time constant of the field circuit, 𝑇𝑓 = 𝐿𝑓 ⁄𝑅𝑓
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International Journal of Enhanced Research in Science, Technology & Engineering
ISSN: 2319-7463, Vol. 5 Issue 1, May-2016
𝑑𝑖
Armature circuit Equation : 𝜐𝑎 = 𝑐1 𝑖𝑓 𝜔 + 𝐿𝑎 𝑎 + 𝑅𝑎 𝑖𝑎
,where 𝜐𝑎 is the voltage of the armature circuit,𝑅𝑎 is
dx
the resistance of the armature circuit,𝑖𝑎 is the current of the armature circuit,𝐿𝑎 is the inductance of the armature
circuit, 𝑐1 𝑖𝑓 𝜔 is the back e.m.f. induced in the armature circuit.
Let the time constant of the armature circuit, 𝑇𝑎 = 𝐿𝑎 ⁄𝑅𝑎
𝑑𝜔
Torque Equation for the shaft: 𝐽
= 𝑐2 𝑖𝑓 𝑖𝑎 − 𝑐3 𝜔 ,where 𝐽 is the rotor inertia, 𝑐3 is the damping inertia and
dx
𝑐2 𝑖𝑓 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 = 𝑉𝑎 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 (𝐼𝑓 , 𝐼𝑎 , 𝛺)as a nominal operating point, thusthe system has a unique equilibrium point at
𝐼𝑓 =
𝑈
𝑅𝑓
, 𝐼𝑎 =
𝑐3 𝑉𝑎
𝑐3 𝑅𝑎 + 𝑐1 𝑐2 𝑈 2 ⁄𝑅𝑓2
, 𝛺 =
𝑐2 𝑉𝑎 𝑈 ⁄𝑅𝑓
𝑐3 𝑅𝑎 + 𝑐1 𝑐2 𝑈 2 ⁄𝑅𝑓2
Now,let the state equations have discontinuous dependence on a small perturbation parameter𝜀and as 𝑇𝑓 ≫ 𝑇𝑎 ,the
above system can be modeled as a singularly perturbed system with slow variables (𝑖𝑓 and𝛺) and fast
variables(𝑖𝑎 ),which can written as
𝑥˙1 = −𝑥1 + 𝑢 ; 𝑥˙2 = 𝑎(𝑥1 𝑧 − 𝑥2 ) ; 𝜀 𝑧˙ = −𝑧 − 𝑏𝑥1 𝑥2 + 𝑐
where, 𝑥1 = 𝑖𝑓 ⁄𝐼𝑓 ; 𝑥2 = 𝜔 ⁄𝛺 ; 𝑧 = 𝑖𝑎 ⁄𝐼𝑎 ; 𝑢 = 𝜐𝑓 ⁄𝑈 ; 𝜀 = 𝑇𝑎 ⁄𝑇𝑓 ; 𝑡 ′ = 𝑡⁄𝑇𝑓 ; 𝑎 = 𝐿𝑓 𝑐3 ⁄𝑅𝑓 𝐽 ;
𝑏 = 𝑐1 𝑐2 𝑈 2 ⁄𝑐3 𝑅𝑎 𝑅𝑓2 ; 𝑐 = 𝑉𝑎 ⁄𝐼𝑎 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 :
𝑥˙1 = −𝜃1 𝑥1 − 𝜃2 𝑥2 𝑢 + 𝜃3 ; 𝑥˙2 = −𝜃4 𝑥2 − 𝜃5 𝑥1 𝑢 ; 𝑦 = 𝑥2
where,𝑥1 is the armature current ; 𝑥2 is the speed; 𝑢 is the field current ;𝜃1, 𝜃2, 𝜃3, 𝜃4, 𝜃5 are positive constants
𝑦𝑅2 < (𝜃32 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 variables 𝑥1 (𝑡), 𝑥2 (𝑡), . ..
, 𝑥𝑛 (𝑡). Then, we get state-space equations defined as
System 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 inputs 𝑢, 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 time 𝑡, it makes the system an autonomous system, written as
𝑥˙ = 𝑓(𝑥)
(2.4)
where,𝑓 ∶ 𝐷 → 𝑅𝑛 is a locally Lipchitz map from a domain 𝐷 ⊂ 𝑅𝑛 into 𝑅𝑛 .
A function 𝑓 is said to satisfy a Lipchitz condition of order 𝛼 at 𝑐 if there exists a positive number 𝑀and a 1-ball 𝐵(𝑐)
such that | 𝑓(𝑥) − 𝑓(𝑐) | < 𝑀|𝑥 − 𝑐|𝛼 , whenever 𝑥 ∈ 𝐵(𝑐), 𝑥 ≠ 𝑐.
And 𝑓(𝑥) will be 'locally Lipchitz' on open and connected set,if each point of 𝐷 has a neighborhood 𝐷0 such that
𝑓satisfies the Lipchitz condition for all points in 𝐷0 with some Lipchitz constant 𝐿0.
Let 𝑥 = 0 be the equilibrium point for (2.4) and let n-dimensional Euclidean space is denoted by 𝑅𝑛 , and 𝐷 ⊂ 𝑅𝑛 be a
domain containing the origin, i.e.𝑥 = 0. Now using Lyapunov stability theorem, which states that 𝑥 = 0 is stable, if
𝑉 ∶ 𝐷 → 𝑅 is a continuously differentiable function, such that 𝑉(0) = 0 and 𝑉(𝑥) > 0 in 𝐷 − {0}
(2.5)
and 𝑉˙ (𝑥) ≤ 0 in 𝐷
(2.6)
where, 𝑉˙ (𝑥), is the derivative of 𝑉 along the trajectories of (2.4), given by
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑉˙ (𝑥) = ∑𝑛i=1 𝑥˙𝑖 = ∑𝑛i=1 𝑓𝑖 (𝑥) =
𝑓(𝑥)
(2.7)
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕𝑥
Thus, the equilibrium point is stable if, for each 𝜀 > 0,there is 𝛿 = 𝛿(𝜀) > 0such that for all 𝑡 ≥ 0,we
have ‖𝑥(0) ‖ < 𝛿 ⇒ ‖𝑥(0) ‖ < 𝜀
(2.8)
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We know that if we have a map 𝑀 ∶ 𝑈 ⊂ 𝑉 → 𝑊 such that there is bijection 𝑀 between open sets 𝑈𝛼 ⊂ 𝑉 and
𝑈𝛽 ⊂ 𝑊 , then it is called a 𝐶 𝑟 - diffeomorphism if and only if 𝑀and 𝑀−1 are both 𝐶 𝑟 -differentiable.
If 𝑟 = ∞ then 𝑀 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
𝑥˙ = 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 over 𝑅𝑛 ; 𝑥 ∈ 𝑅𝑛 is the state vector ;
𝑢 ∈ 𝑅𝑟 is the vector of inputs ;
𝑦 ∈ 𝑅𝑚 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
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 space 𝑋𝑛 and an infinitesimal point transformation such that, 𝑇 ∶
′𝜉 𝑥 = 𝜉 𝑥 + 𝜈 𝑥 𝑑𝑡 and let the dragging along (𝜒) → (𝜒′)of the coordinate system (𝜒) by the infinitesimal point
transformation 𝑇 −1 ∶ ′𝜉 → 𝜉 inverse to 𝑇 is given by
𝜉 𝑥′ = 𝜉 𝑥 + 𝜈 𝑥 𝑑𝑡
(2.13)
Suppose we have a covariant vector field 𝑤𝜆 (𝜉) in 𝑋𝑛 and a new covariant vector field ′𝑤𝜆 (𝜉) at 𝜉 as a field whose
components ′𝑤𝜆′ (𝜉) w.r.t.(𝜒′) at 𝜉 such that ′𝑤𝜆′ (𝜉) ≝ 𝑤𝜆 (′𝜉)
Then, the Lie derivative of 𝑤𝜆 is given by 𝐿𝜈 𝑤𝜆 = 𝜈 𝜇 𝜕𝜇 𝑤𝜆 + 𝑤𝜇 𝜕𝜆 𝜈 𝜇
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 ℎ along 𝑓 as
𝜕ℎ
𝐿𝑓 ℎ(𝑥) =
𝑓(𝑥)
(2.14)
𝜕𝑥
Similarly, we can define other Lie derivatives as
𝐿𝑔 𝐿𝑓 ℎ(𝑥) =
𝐿𝑘𝑓 ℎ(𝑥) =
𝜕(𝐿𝑓 ℎ)
𝜕𝑥
𝜕𝐿𝑘−1
𝑓 ℎ
𝜕𝑥
𝑔(𝑥)
(2.15)
𝑓(𝑥)
(2.16)
𝐿0𝑓 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 𝐿𝑖𝑓 ℎ(𝑥) = 0 , 0 ≤ 𝑖 ≤ 𝑛 − 2
(2.18)
and 𝐿𝑔 𝐿𝑛−𝑖
ℎ(𝑥)
≠
0
,
∀
𝑥
∈
𝐷
(2.19)
𝑥
𝑓
Thus, the system in (2.9) and (2.10) has a relative degree 𝑛 in 𝐷𝑥 and the change of variables (2.12) can be applied as
𝑧 = 𝑇(𝑥) = [ ℎ(𝑥), 𝐿𝑓 ℎ(𝑥), . . . , 𝐿𝑛−1
𝑓 ℎ(𝑥) ]
𝑇
(2.20)
Now, since for all𝑥 ∈ 𝐷,the row vectors and column vectors are linearly independent, thus the Jacobian matrix
𝜕𝑇
[ ] (𝑥) is non-singular ∀ 𝑥 ∈ 𝐷𝑥 .Hence, for each 𝑥0 ∈ 𝐷𝑥 ,there is neighborhood 𝑁 of 𝑥0 such that 𝑇(𝑥), restricted
𝜕𝑥
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 ∃ domain
𝐷𝑥 ⊂ 𝐷, such that matrix 𝐺(𝑥) has a rank 𝑛 ∀ 𝑥 ∈ 𝐷𝑥
(2.21)
and distribution, Δ is involutive in 𝐷𝑥
(2.22)
𝑛−1
,where
matrix𝐺(𝑥) = [𝑔(𝑥), 𝑎𝑑𝑓 𝑔(𝑥), . . . , 𝑎𝑑𝑓 𝑔(𝑥)] and distribution, Δ= span{𝑔, 𝑎𝑑𝑓 𝑔, . . . , 𝑎𝑑𝑓𝑛−2 𝑔}
A non-linear system (2.9),where 𝐷𝑥 ∈ 𝑅𝑛 , 𝑓: 𝐷𝑥 → 𝑅𝑛 , 𝐺 ∶ 𝐷𝑥 → 𝑅𝑛×𝑝 , is said to be input-state linearizable, if there
exists a diffeomorphism 𝑇: 𝐷𝑥 → 𝑅𝑛 , such that 𝐷𝑧 = 𝑇(𝐷𝑥 ),contains the origin and (2.12) transforms (2.9) into
𝑧˙ = 𝐴𝑧 + 𝐵𝛽 −1 (𝑥)[ 𝑢 − 𝛼(𝑥) ]
(2.23)
with(𝐴, 𝐵)controllable and 𝛽(𝑥)non-singular for all,𝑥 ∈ 𝐷𝑥 .
𝜕𝑇
𝜕𝑇
[𝑓(𝑥) + 𝐺(𝑥)𝑢]
Thus, we get 𝑧˙ =
𝑥˙ =
𝜕𝑥
𝜕𝑥
and
˙𝑧 = 𝐴𝑇(𝑥) + 𝐵𝛽 −1 (𝑥)[ 𝑢 − 𝛼(𝑥) ]
(2.24)
(2.25)
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International Journal of Enhanced Research in Science, Technology & Engineering
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From (2.24) and (2.25), we get
𝜕𝑇
𝜕𝑥
[𝑓(𝑥) + 𝐺(𝑥)𝑢] = 𝐴𝑇(𝑥) + 𝐵𝛽 −1 (𝑥)[ 𝑢 − 𝛼(𝑥) ]
Now if we put 𝑢 = 0 in (2.26), we get
𝜕𝑇
𝑓(𝑥) = 𝐴𝑇(𝑥) − 𝐵𝛽 −1 (𝑥) 𝛼(𝑥)
(2.27)
−1 (𝑥)
(2.28)
𝜕𝑥
𝜕𝑇
𝜕𝑥
𝐺(𝑥) = 𝐵𝛽
(2.26)
Thus, if there is a map 𝑇( ⋅ ) ,which satisfies (2.27) and (2.28) for some 𝛼, 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 singular 𝑀, then the state equation in the 𝜍 −
coordinates are given by
𝜍˙ = 𝑀 𝐴 𝑀−1 𝜍 + 𝑀 𝐵𝛽 −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 matrix 𝑀 that transforms (𝐴, 𝐵) into canonical form
𝑀 𝐴 𝑀−1 = 𝐴𝑐 + 𝐵𝑐 𝜆𝑇
(2.30)
and 𝑀 𝐵 = 𝐵𝑐
(2.31)
where
0 1 0 … 0
0 0 1 … 0
𝐴𝑐 = ⋮ ⋮ ⋱ ⋱ ⋮
(2.32)
⋮ ⋮ ⋮ 0 1
[0 … … 0 0]𝑛 × 𝑛
0
0
and 𝐵𝑐 = ⋮
(2.33)
0
[1]
𝑇1 (𝑥)
𝑇2 (𝑥)
⋮
Let 𝑇(𝑥) =
(2.34)
𝑇𝑛−1 (𝑥)
[ 𝑇𝑛 (𝑥) ]
Thus,
𝑇2 (𝑥)
𝑇3 (𝑥)
⋮
𝐴𝑐 𝑇(𝑥) − 𝐵𝑐 𝛽 −1 (𝑥) 𝛼(𝑥) =
(2.35)
𝑇𝑛 (𝑥)
[−𝛼 (𝑥)⁄𝛽 (𝑥)]
0
0
⋮
and
𝐵𝑐 𝛽 −1 (𝑥) =
(2.36)
0
[1⁄𝛽 (𝑥)]
Using these expressions in (2.27),we get simplified partial differential equations as
𝜕𝑇1
𝑓(𝑥) = 𝑇2 (𝑥) ;
(2.37)
𝜕𝑥
𝜕𝑇2
𝑓(𝑥) = 𝑇3 (𝑥) ;
⋮
𝜕𝑇𝑛−1
𝑓(𝑥) = 𝑇𝑛 (𝑥) ;
(2.38)
𝜕𝑥
(2.39)
𝜕𝑥
𝜕𝑇𝑛
𝜕𝑥
𝑓(𝑥) = −𝛼 (𝑥)⁄𝛽 (𝑥)
(2.40)
Using these expressions in (2.28), we get simplified partial differential equations as
𝜕𝑇1
𝑔(𝑥) = 0 ;
(2.41)
𝜕𝑥
𝜕𝑇2
𝜕𝑥
𝑔(𝑥) = 0 ;
⋮
(2.42)
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𝜕𝑇𝑛−1
𝜕𝑥
𝜕𝑇𝑛
𝜕𝑥
𝑔(𝑥) = 0 ;
(2.43)
𝑔(𝑥) = 1⁄𝛽 (𝑥) ≠ 0
(2.44)
Now, the function 𝑇1 (𝑥) has to be searched, which satisfies
𝜕𝑇𝑖
𝑔(𝑥) = 0 ; (𝑛 − 1) ≤ 𝑖 ≤ 1
(2.45)
𝜕𝑥
where, 𝑇𝑖 +1 (𝑥) =
𝜕𝑇𝑛
𝜕𝑥
𝜕𝑇𝑖
𝜕𝑥
(𝑛 − 1) ≤ 𝑖 ≤ 1
𝑓(𝑥),
𝑔(𝑥) ≠ 0
(2.46)
If there is a function which satisfies (2.45) and (2.46), then
1
𝛽(𝑥) = (𝜕𝑇 ⁄
(2.47)
𝛼(𝑥) = −
𝑛 𝜕 𝑥)𝑔(𝑥)
(𝜕𝑇𝑛⁄𝜕 𝑥)𝑓(𝑥)
(2.48)
(𝜕𝑇𝑛⁄𝜕 𝑥)𝑔(𝑥)
From differential geometry approach, we also noticed that for all 𝑥 ∈ 𝐷,
𝐿𝑎𝑑 𝑖𝑔 𝐿𝑓 𝑘 ℎ(𝑥) = 0 ,0 ≤ 𝑗 + 𝑘 < 𝑟 − 1
𝑓
𝐿𝑎𝑑
𝑘 ≥ 0, 0 ≤ 𝑗 ≤ 𝑟 − 1, we have
(2.49)
𝐿 𝑘 ℎ(𝑥) = (−1)𝑗 𝐿𝑔 𝐿𝑓 𝑟−1 ℎ(𝑥) ≠ 0 , 𝑗 + 𝑘 = 𝑟 − 1
(2.50)
𝑖
𝑓
𝑓 𝑔
and the row vectors [𝑑ℎ(𝑥), … , 𝑑𝐿𝑓
𝑟−1
ℎ(𝑥)] and column vectors[𝑔(𝑥)
…
𝑎𝑑𝑓 𝑟−1 𝑔(𝑥)] are linearly independent.
From Jacobian identity, we know that for any real-valued 𝜆 and any vector fields 𝑓and 𝛽,
𝐿[𝑓,𝛽] 𝜆(𝑥) = 𝐿𝑓 𝐿𝛽 𝜆(𝑥) − 𝐿𝛽 𝐿𝑓 𝜆(𝑥)
(2.51)
Also the distribution, Δ generated by 𝑓1, … , 𝑓𝑟 is said to be completely integrable if for each 𝑥0 ∈ 𝐷, ∃ a neighborhood
𝑁 of 𝑥0 and 𝑛 − 𝑟 real-valued smooth functions ℎ1 (𝑥), … , ℎ𝑛−𝑟 (𝑥) satisfy the partial differential equations
𝜕ℎ𝑖
𝑓 (𝑥) = 0, ∀ 1 ≤ 𝑖 ≤ 𝑟, 1 ≤ 𝑗 ≤ 𝑛 − 𝑟
(2.52)
𝜕𝑥 𝑖
and the covector fields 𝑑ℎ𝑗 (𝑥)are linearly independent for all 𝑥 ∈ 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) + 𝑔 𝑢,
(3.1)
−𝑎𝑥1
1
where 𝑓(𝑥) = [−𝑏𝑥2 + 𝜌 − 𝑐𝑥1 𝑥3 ] ; 𝑔 = [0]
(3.2)
𝜃𝑥1 𝑥2
0
and
𝑎, 𝑏, 𝑐, 𝜃, 𝜌 are positive constants
Let𝑥1 = 0, 𝑥2 = 𝜌⁄𝑏,be the equilibrium point for the open loop system and the desired operating point be ~
𝑥 given
as ~𝑥 = (0, 𝜌⁄𝑏 , 𝜔0 )𝑇 , where 𝜔0 is a desired set point for the angular velocity 𝑥3.
𝜕𝑇
𝜕𝑇
Let 𝑇2 (𝑥) = 1 𝑓(𝑥) ; 𝑇3 (𝑥) = 2 𝑓(𝑥)
(3.3)
𝜕𝑥
𝜕𝑥
Then, with 𝑇1 (𝑥
~) = 0,we can find 𝑇1 = 𝑇1 (𝑥),which satisfies the conditions (2.41),(2.42) and
𝜕𝑇3
𝑔(𝑥) ≠ 0
(3.4)
𝜕𝑥
𝜕𝑇1
thus,
𝜕𝑥
𝑔 =
𝜕𝑇1
𝜕𝑥1
= 0
(3.5)
and therefore,
𝑇2 (𝑥) =
Using (2.42) in (2.46),we get
𝑐𝑥3
𝜕𝑇1
𝜕𝑥2
𝜕𝑇1
𝜕𝑥2
[−𝑏𝑥2 + 𝜌 − 𝑐𝑥1 𝑥3 ] +
= 𝜃𝑥2
𝜕𝑇1
𝜕𝑥3
Equation (3.7) will be satisfied if 𝑇1 is of the form
𝜕𝑇1
𝜕𝑥3
𝜃𝑥1 𝑥2
,
(3.6)
(3.7)
𝑇1 =
𝑐1 [𝜃𝑥22
Let 𝑐1 = 1 and 𝑐2 = −𝜃(𝑥
~)22 − 𝑐(𝑥
~)23 = −𝜃(𝜌⁄𝑏)2 − 𝑐𝜔02
+
𝑐𝑥32 ]
+ 𝑐2
(3.8)
(3.9)
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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,
𝑇2 (𝑥) = 2𝜃𝑥2 (𝜌 − 𝑏𝑥2 )
𝑇3 (𝑥) = 2𝜃(𝜌 − 2𝑏𝑥2 )(𝑏𝑥2 + 𝜌 − 𝑐𝑥1 𝑥3 )
𝜕𝑇3
𝜕𝑥
𝑔 =
𝜕𝑇3
𝜕𝑥1
(3.10)
(3.11)
= 2𝑐𝜃(𝜌 − 2𝑏𝑥2 )(𝑥3 )
(3.12)
and the (3.4) is satisfied, whenever 𝑥2 ≠ 𝜌⁄𝑏 , 𝑥3 ≠ 0.
Let 𝐷𝑥 contains the point ~𝑥 and ~
𝑥3 > 0,then
𝐷𝑥 = {𝑥 ∈ 𝑅3 ; 𝑥2 >
𝜌
2𝑏
, 𝑥3 > 0}
(3.13)
Thus, the map (2.12) is diffeomorphism on 𝐷𝑥 and the state equation in the 𝑧 − coordinates are defined in
𝐷𝑧 = 𝑇(𝐷𝑥 ) = {𝑧 ∈ 𝑅3 }
(3.14)
𝜃𝜌
2
2
2
{𝑧1 > 𝜃𝜙 (𝑧)2 − 𝜃(𝜌⁄𝑏) − 𝑐𝜔0 }, {𝑧2 < } and 𝜙( ⋅ ) is the inverse of the map
such that
2𝑏
2𝜃𝑥2 (𝜌 − 𝑏𝑥2 ). Thus, domain 𝐷𝑧 contains the origin 𝑧 = 0.
Now, we can verify the same using differential geometry approach as explained in the previous section. As we know
𝑎
𝑎2
2
(𝑎 + 𝑏)𝑐𝑥3 ] ;
𝑎𝑑𝑓 𝑔 = [𝑓, 𝑔] = [ 𝑐𝑥3 ] ; 𝑎𝑑𝑓 𝑔 = [𝑓, 𝑎𝑑𝑓 𝑔] = [
−𝜃𝑥2
(𝑏 − 𝑎)𝜃𝑥2 − 𝜃𝜌
Considering 𝐺 =
[𝑔, 𝑎𝑑𝑓 𝑔, 𝑎𝑑𝑓2 𝑔]
thus, the determinant of 𝐺,
1
= [0
0
𝑎
𝑐𝑥3
−𝜃 𝑥2
𝑎2
(𝑎 + 𝑏)𝑐𝑥3
];
(𝑏 − 𝑎)𝜃 𝑥2 − 𝜃 𝜌
det𝐺 = 𝑐𝜃(−𝜌 + 2𝑏𝑥2 )𝑥3 . Hence,𝐺 has a rank 3 for 𝑥2 ≠ 𝜌⁄2 𝑏, 𝑥3 ≠ 0. Now,
0 0 0 1
0
𝜕(𝑎𝑑𝑓 𝑔)
[𝑔, 𝑎𝑑𝑓 𝑔] =
𝑔 = [0 0 𝑐 ] [0] = [0].
𝜕𝑥
0 −𝜃 0 0
0
𝜌
Hence the distribution ∆ = span{𝑔, 𝑎𝑑𝑓 𝑔}is involutive as [𝑔, 𝑎𝑑𝑓 𝑔] ∈ 𝐷and 𝐷𝑥 = {𝑥 ∈ 𝑅3 ; 𝑥2 >
, 𝑥3 > 0} ,
2𝑏
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.
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