12th International Conference on Frontiers of Information Technology
Robust Takagi-Sugeno Fuzzy Speed Regulator for DC Series Motors Umar Farooq1, Jason Gu2, Muhammad Usman Asad3, Ghulam Abbas4 1
Department of Electrical Engineering, University of The Punjab Lahore-54590 Pakistan Department of Electrical and Computer Engineering, Dalhousie University Halifax, N.S., Canada 3,4 Department of Electrical Engineering, The University of Lahore, Lahore Pakistan Email:
[email protected],
[email protected],
[email protected],
[email protected] 2
the current measurements. The proposed approach is verified through experimentation in real time. The use of back stepping method in controlling the speed of DC series motors is reported in [9]. The recursive introduction of virtual control inputs and control Lyapunov functions produces a stabilizing control law. An improved version of this method is reported in [10] for achieving better transient performance.
Abstract—This paper presents the design of fuzzy logic based speed regulator for DC series motor. A Takagi-Sugeno (TS) fuzzy model of DC series motor considering parametric uncertainty is first constructed using sector nonlinearity and set of operating point techniques. An integral state is then introduced in each fuzzy plant sub-model to guarantee exact tracking of speed commands. A parallel distributed compensation (PDC) controller under H-, decay rate and control input constraints, is finally designed for the augmented system to regulate the motor speed in the presence of load torque and parameter variations. MATLAB simulations are performed to prove the effectiveness of the proposed regulator.
The use of knowledge based techniques, which do not require the mathematical description of the plant to be controlled such as fuzzy logic and neural networks, has also been investigated for the control of dc series motors [11-15]. A PID-ANN controller is proposed in [11] which acquires training data from a conventional PID controller and the trained controller drives the dc motor through a chopper. Real time results using an 80C51 microcontroller show the effectiveness of the designed scheme. The design of a rule based fuzzy logic controller is presented in [12] which drives the dc series motor through speed and current controllers. The inputs to both the fuzzy controllers are deviation and rate of deviation from the set point. Each of these inputs are represented by seven fuzzy sets with triangular membership functions. A total of 49 rules for each of these controllers are evaluated and firing signals are generated for thyristors fed dc series motor. The performance of the designed fuzzy scheme shows improvement compared with classical PI control strategy. The power of fuzzy logic controller for speed of dc series motors is presented in [13] where a simple fuzzy logic controller with a single input and single output is shown to perform better than classical PI controller. To better deal with uncertainties, type-2 fuzzy logic controller is designed in [15] to regulate the speed of DC series and shunt motors. Similar to [12], a rule base type-2 fuzzy logic controller, employing seven type-2 membership functions to represent each of inputs and containing 49 rules, is deployed. Simulation results are included to show the superior performance of type-2 fuzzy logic controller over type-1 fuzzy logic controller with respect to the transient performance parameters such as settling time, maximum overshoot and integral absolute error.
Keywords— DC series motor;TS fuzzy model;LMI regulator;Parallel distributed compensation;MATLAB/Simulink
I.
INTRODUCTION
The demand of high starting torque at low speeds by large industrial loads such as cranes, elevators, conveyors, trolleys etc., is often fulfilled by deploying DC series motors [1-4]. Such high torques result from series connection of motor armature and field windings which gives rise to a square law relationship between the torque produced by the DC series motor and the current supplied to it. This constitutes the first source of nonlinearity in the motor model. The second nonlinearity stems from the back EMF which is proportional to the product of current and speed. These nonlinear effects tend to saturate the field magnetic core with the load torque variations which ultimately results in poor speed regulation. A controller is thus needed to regulate the speed of DC series motors when load torque is changing. The design of controller can be carried out based upon the linearized model of the motor which will guarantee the desired system performance over a small range of operating conditions. However, for the wider range operation, nonlinear model will be required for designing the motor controller. A number of nonlinear control methods have been deployed by researchers to address the speed control problem of DC series motors [5-15]. Speed regulation is achieved in [5] using feedback linearization technique. By introducing a nonlinear state transformation, a linear control law is formed to regulate the speed of the motor. Further, a nonlinear observer is also constructed for speed and load torque estimation based on 978-1-4799-7505-1/14 $31.00 © 2014 IEEE DOI 10.1109/FIT.2014.24
This paper follows TS fuzzy model based control scheme to design a robust controller for speed regulation of DC series motor. The nonlinear motor model containing parametric uncertainty is converted into a set of linear models using sector
79
TABLE I.
nonlinearity [16-17] and set of operating point technique [18]. The convex combination of these local models according to weighted average defuzzification method will exactly reproduce the system dynamics in the compact region. To minimize the tracking error, an integral state is added to each local model. The regulator gains for these augmented local models are obtained after solving a set of LMIs under H-, decay rate and control input constraints. The fuzzy blending of these gains according to PDC scheme yields the final control law. MATLAB simulations are performed to validate the proposed speed regulator under load torque and parameter variations.
Parameters R L M J B
B ª ª•º « − x J « 1» = « «•» « M ¬ x2 ¼ « − x2 ¬ L
d ( La i + φ f ( i ) ) + ( R f + Ra ) i + K mφ f ( i ) ω dt
dω J + Bω + TL = K mφ f ( i ) i dt
•
x = Ax + Bu + Ed (1)
y = Cx
B ª « −J A=« «− M x «¬ L 2
the motor constant, ‘ TL ’is the load torque, and φ f ( i ) is the field flux which is a function of field current. If the field circuit is not in magnetic saturation, then field flux is linearly related to the field current through the field inductance i.e., φ f ( i ) = L f i f . Under this assumption, the dynamics in (1) can be simplified as:
J
dω + Bω + TL = Mi 2 dt
M º x2 ª0º ª 1º − J » » ,B = « 1 » ,E = « J » « » « » R − » «¬ 0 »¼ ¬L¼ » L ¼
(6)
It can observed that system matrix in (4) corresponds to a family of matrices due to its dependence on state and uncertain parameter. We can define a compact region to cover the state ( x2 ∈ [ 20 200] ) and parameter ( B ∈ [ 0.1 0.5] ) variations as: D = {( x2 , B ) ∈ 2 : x2 L ≤ x2 ≤ x2U , BL ≤ B ≤ BU }
(2)
(7)
Let M 1 and M 2 be the fuzzy sets covering the universe of discourse for the state variable x2 as defined in compact region:
where, L = L f + La , R = R f + Ra and M = K m L f .
M1 =
The motor parameters are adopted from [9] and are listed in Table 1. In addition, friction coefficient is assumed to be uncertain. By declaring speed and current as the state variables, ( x1 = ω , x2 = i, y = x1 ), we can write (2) as:
dx1 B M 2 TL x2 − = − x1 + dt J J J dx2 M R u = − x2 x1 − x2 + dt L L L
(5)
Where ‘ A ’is the system matrix, ‘ B ’is the input vector, ‘ E ’is the disturbance vector, ‘ x ’ is the system state vector, ‘ u ’is the control input which is the motor voltage, ‘ d ’is the disturbance input which in this case is taken to be the load torque and ‘ C ’is the output vector.
circuit, ‘ Ra ’is the resistance of armature circuit, ‘ R f ’is the resistance of field circuit, ‘ J ’is the inertia associated with both the motor and the load, ‘ B ’ is the friction coefficient, ‘ K m ’is
di + Ri + Miω dt
(4)
The general form of (4) is given as:
where, ‘ i ’is the armature (or field) current, ‘ V ’is the terminal voltage, ‘ ω ’is the rotation speed of the motor, ‘ La ’is inductance of armature circuit, ‘ L f ’is inductance of field
V =L
M º x2 ª 1º ª0º J » ª x1 º + « » u + « − » T »« » 1 J L « » « » x R − » ¬ 2¼ ¬L¼ «¬ 0 ¼» L ¼»
ªx º y = [1 0] « 1 » ¬ x2 ¼
TS FUZZY MODEL OF DC SERIES MOTOR
The dynamical model of DC series motor can be expressed as [5]:
V=
Values 1Ω 0.05 H 0.027 H 0.5 Kgm2 [0.1,0.5]Nms/rad
where, ‘ u ’is the terminal voltage to be generated by the controller for regulating the speed. We can write (3) in matrix form as:
We start by constructing the TS fuzzy model of uncertain DC series motor in section II. Controller design is presented in section III followed by simulation results in section IV. Conclusions are drawn in section V. II.
DC SERIES MOTOR PARAMETERS
− x2 + x2U x2U − x2 L
(8)
M 2 = 1 − M1 Let N1 and N 2 be the fuzzy sets covering the universe of discourse for B as defined in compact region: N1 =
(3)
− B + BU BU − BL
N 2 = 1 − N1
80
(9)
The fuzzy sets in (8)-(9) are shown in Fig. 1. We can now define the following fuzzy plant rules whose averaged weighted combination will reproduce the nonlinear model (4) at least in the compact region (7):
M2/N2
M1/N1 1.0
Rule 1: IF x2 is M 1 AND B is N1
THEN x = A1x + B1u + E1d
x2/B
Rule 2: IF x2 is M 1 AND B is N 2 x2L/BL
THEN x = A 2 x + B 2 u + E 2 d
Fig. 1. Fuzzy sets for the premise variable
Rule 3: IF x2 is M 2 AND B is N1 hi ( z ( t ) ) =
THEN x = A3 x + B3u + E3 d
wi ( z ( t ) )
(12)
4
¦ wi ( z (t )) i =1
Rule 4: IF x2 is M 2 AND B is N 2
where, wi ( z ( t ) ) is the firing strength of the ith rule and is
THEN x = A 4 x + B 4 u + E 4 d
given by the product of the fuzzy sets associated with the ith rule.
where,
wi ( z ( t ) ) = M i ( x2 ( t ) ) × N i ( B )
BL ª « − J A1 = « «− M x «¬ L 2 L BL ª « − J A3 = « «− M x «¬ L 2U
M M º ª BU º x2 L » x2 L » « − J J J » , A2 = « » R » M R » « − − x2 L − L »¼ L ¼» ¬« L BU M M º ª º x2U » x2U » « − J J J » , A4 = « » R » R » «− M x − − 2 U L »¼ L ¼» ¬« L ª0º B1 = B 2 = B3 = B 4 = « 1 » « » ¬L¼ ª 1º − E1 = E 2 = E3 = E 4 = « J » « » ¬« 0 ¼» The net TS fuzzy model can now be constructed as: •
x2U/BU
III.
(13)
TS FUZZY CONTROLLER
This section describes the design of a TS fuzzy PDC regulator for tracking constant reference speed commands under jumping load torque disturbances and parameter variations. An integral control structure is adopted for regulation task and is depicted in Fig. 2. From the integral controller in Fig. 2, we can write the error dynamics as:
(10)
•
ξ = ωref − y = −Cx + ωref
(13)
Using (11) and (13), the ith subsystem of augmented model can be given as:
ª• º « xi » = ª A i « • » «¬ −Ci ¬«ξi ¼» yi = [Ci
4
x = ¦ hi ( z ( t ) ) ( A i x + B i u + Ei d ) i =1
(11)
4
y = ¦ hi ( z ( t ) ) ( Ci x )
0 º ª xi º ª Bi º ªE º ª0 º u + « i » d + « » ωref + 0 »¼ «¬ξi »¼ «¬ 0 »¼ 0 ¬ ¼ ¬1 ¼
(14)
ªx º 0] « i » ¬ξ i ¼
The general formulation of (14) can be expressed as: •
i =1
xi = A i x + Bi u + Ei d + F ωref
where, hi ( z ( t ) ) is the normalized firing strength of i rule th
(15)
We can now define the following control rules for the augmented model in (15):
which varies according to the scheduling vector z ( t ) = ª¬ x2 ( t ) B º¼ . Here ‘ B ’is assumed as constant which can take any value from compact region. However, the model will still hold even if ‘ B ’is time varying but bounded by the compact region.
Rule 1: IF x2 is M 1 AND B is N1
THEN u ( t ) = − K1 x ( t )
81
TL
Ȧref
y
ȟ
u
DC Series Motor
KI
_
I2x2
_ K
TSFLC
Fig. 2. Fuzzy LMI integral controller • § · § · V ¨ x ( t ) ¸ < −2α V ¨ x ( t ) ¸ © ¹ © ¹
Rule 2: IF x2 is M 1 AND B is N 2
THEN u ( t ) = − K 2 x ( t )
Third, we place a limit on the control input since the motor cannot be driven by a control signal which is beyond the rated voltage of the motor, i.e.:
Rule 3: IF x2 is M 2 AND B is N1
THEN u ( t ) = − K 3 x ( t )
u (t ) 2 ≤ μ
Rule 4: IF x2 is M 2 AND B is N 2
T ª 1§ T º · « − ¨ A c ,ij P + P A c ,ij + A c , ji P + P A c, ji ¸ ∗ ∗ » 2 ¹ « © » « » T 1 § · « γ 2I 0» ≥ 0 − ¨ Ei + E j ¸ P « » 2© ¹ « » 1§ · « 0 I» ¨ Ci + C j ¸ 2© « » ¹ ¬ ¼
The TS fuzzy PDC control law is obtained by blending the local control gains according to average defuzzification methodology:
u ( t ) = −¦ hi ( z ( t ) ) K i x ( t )
(16)
i =1
where, K i = ª¬ K i − K I i º¼ is the control gain for i th rule. The control gains for the fuzzy sub-systems are computed under certain performance constraints. First, we impose H- constraint which will help to minimize the effect of load torque variations while motor is running at desired speed. Mathematically, ‘ γ ’in the following inequality needs to be minimized:
sup
d (t ) 2 ≠ 0
y (t )
d (t )
2
≤γ
T
§A · §A · ¨ c ,ij + A c , ji ¸ ¨ c ,ij + A c , ji ¸ © ¹ P+P© ¹ + 2α P ≤ 0, i < j 2 2 T
ª º T ª 1 P P K iT » x ( 0) º « ≥0 « » ≥ 0, « » P »¼ 2 «¬ x ( 0 ) «¬K i P μ I »¼
(17)
2
T
(20)
A c ,ii P + P A c ,ii + 2α P < 0, ∀i
Secondly, to improve the response time of the regulator, we add the decay rate constraint. The system states will decay at a • rate ‘ α ’, if the derivative, V §¨ x ( t ) ¸· of quadratic Lyapunov © ¹
(19)
The satisfaction of the design constraints in (17)-(19) is subject to the existence of a symmetric positive definite matrix P > 0 and the following inequalities:
THEN u ( t ) = − K 4 x ( t )
4
(18)
(21)
(22)
where, ‘ ∗ ’denotes the transposed entry, and
A c ,ij = Ai − Bi K j
function, V §¨ x ( t ) ·¸ = x P x satisfies the following inequality: © ¹
82
(23)
ª 0.0123 -0.1185 0.0008 º P = «« -0.1185 2.3259 -0.0002 »» × 103 ¬« 0.0008 -0.0002 0.0001 »¼
The matrix inequalities in (20)-(22) are not linear. However, by using congruence transformation and change of variables method, we can convert them into LMIs which can then be solved efficiently using powerful MATLAB LMI toolbox to determine the control gains. By pre- and postmultiplying (20) with matrices X = diag {P −1 , I, I} and by
IV.
T ª § T º · T « 1 ¨ P A i + A i P − Q j Bi − B i Q j ¸ » − ∗ ∗ « 2¨ » ¸ T T « ¨ +P A j + A j P − Q T B j − B j Q ¸ » i i ¹ « © » T « » 1§ · − ¨ Ei + E j ¸ γ 2 I 0 » ≥ 0 (24) « 2© ¹ « » « » 1§ · 0 I» « ¨ Ci + C j ¸ P 2© ¹ « » « » «¬ »¼
Similarly, by pre- and post-multiplying (21) with P −1 and
T
T
T
T
T
the following LMIs are
The designed regulator is also compared with pole placement controller. A pole placement controller is designed to exhibit the same settling time and overshoot as offered by TS fuzzy regulator. Using Ts = 0.68 and ξ = 1 desired characteristic equation is found to be:
P A i + A i P − Qi T B i − Bi Qi + 2α P < 0, ∀i P A i + A i P − Q j T Bi − B i Q j + T
(25)
P A j + A j P − Qi T B j − B j Qi + 4α P ≤ 0, ∀i < j
s 3 + 129.5s 2 + 1420 s + 4074.7 = 0
The inequality (22) is converted to LMI by the change of
Q iT º »≥0 μ 2I ¼
(26)
The solution of LMIs (24)-(26) will give the matrix P and vectors Qi . The control gains can be determined as:
K i = Q i P −1
(27)
By assuming x2 L = 20 , x2U = 200 , γ = 0.2 , α = 5 , μ = 230 ; the following control gains and symmetric positive definite matrix are found using LMI toolbox of MATLAB:
K1 = [149.7905 8.6727 -826.0353]
K 2 = [148.2703 8.7282 -823.6647 ]
K 3 = [115.8027 9.3937 -669.8904]
K 4 = [115.6837 9.3929 -671.9141]
(33)
Note that the third pole is placed twenty times farther than the dominant poles to form (33) and averaged model is used to design the control gains of the pole placement controller. The simulation results showing the comparison of TS FLC and PPC are depicted in Fig. 6. It can be observed that rise time, undershoot and overshoot are higher in speed response of PPC as compared to TS FLC. Also the deviation from set speed in the presence of load torque disturbances is found to be 1.7rad/sec which is eight times greater than what is achieved with TS FLC. The disturbance rejection time of PPC is observed to be 1.5sec which is also greater as compared to TS FLC.
variable as defined for (24)-(25) i.e. Qi = K i P : T ª 1 ªP x (0) º « » ≥ 0, « P ¼» ¬Q i ¬« x ( 0 )
RESULTS
The proposed regulator is simulated in MATLAB/Simulink environment for tracking reference speed commands under load torque and parameter variations. Figure 3 shows the response of the motor for tracking a set speed of 50rad/sec when the applied load torque is 50Nm. From the graph, the settling time is found to be 0.68sec while no overshoot is observed in speed response. The control input and current drawn by the motor are also shown. It can be seen that control input is bounded as per the constraint used to design the regulator. With the same load torque applied, DC series motor is made to track a set of reference speeds ({50,60,80} rad/sec). The result is depicted in Fig. 4. A sequence of load torque disturbances ({60,70,80}Nm) are now introduced while the motor is running at a set speed of 80rad/sec. The disturbance rejection result is shown in Fig. 5. The deviation in reference speed is found to be 0.22 rad/sec when load torque is suddenly changed and reference speed is regained in a time interval of 0.5sec which constitutes the disturbance rejection time of the regulator. The value of friction coefficient B is assumed to be 0.3 in all the above simulations.
defining P = P −1 , Qi = K i P we have the following LMI for H constraint which will hold for all i ≤ j :
by defining P = P −1 , Qi = K i P obtained:
(32)
The comparison of TS FLC and PPC under parameter variations is shown in Fig. 7. The value of friction coefficient is varied between the limits and response at the extreme values is recorded. It can be observed that TS FLC exhibits the same transient performance with the variation of uncertain friction coefficient. However, speed response is varying in case of PPC with the change of friction coefficient.
(28)
(29) (30) (31)
83
(a)
(a)
(b)
(b)
(c)
(c) Fig. 3. Simulation results for
ωref
Fig. 4. Simulation results for
= 50 rad/s and TL=50 Nm
84
ωref
= {50,60,80} rad/s and TL=50 Nm
(a) (d) Fig. 5. Simulation results for
ωref
= 80 rad/s and TL= {60,70,80} Nm
(b)
(a)
(c)
(b) Fig. 6. Comparison of Pole Placement and TS FLC under load torque variations
85
[4] [5]
[6]
[7]
[8]
[9]
[10] Fig. 7. Comparison of Pole Placement and TS FLC under parameter variations
V.
[11]
CONCLUSIONS
[12]
The design of TS fuzzy regulator for DC series motor is presented. TS fuzzy model of uncertain DC series motor is formed using sector nonlinearity and set of operating point techniques. Based upon this model, a TS fuzzy PDC regulator is designed under certain performance constraints. The proposed regulator is then validated through simulations in MATLAB environment where it is also compared with pole placement controller. Simulations reveals that TS fuzzy regulator has exhibited better transient performance as compared to PPC under load torque and parameter variations.
[13]
[14]
[15]
[16]
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