Robust Back Stepping Control Of Im Drives Using Artificial Neural Networks

Robust Backstepping Control of Induction Motor Drives Using Artificial Neural Networks *

J. Soltani , R. Yazdanpanah** Electrical and Computer Engineering, Isfshan University of Technology, Isfahan, Iran *

**

[email protected] [email protected]

Abstract-- In this paper, using the three-phase Induction Motor(IM) fifth order model in a stationary two axis reference frame whit stator current and rotor flux as state variables, a conventional backstepping controller is designed first for speed and rotor flux control of an IM drive. Then in order to make the control system stable and robust against the parameter uncertainties as well as the unknown load torque, in the next stage the backstepping controller is combined with an Artificial Neural Network (ANN). It will be shown that the proposed composite controller is capable of compensating the parameters variations and rejecting the external load torque disturbance. The overall system stability is proved by Lyapunov theory. It is also shown that the method of ANN training, guarantees the boundedness of errors and ANN weighs. The validity and effectiveness of the controller is verified by computer simulation.

feedback nonlinear systems. This method has been applied to a single arm robot[5] and to a rotor-flux Field Oriented Control (FOC) IM drive[6]. One may note that the FOC methods are in fact a type of partial feedback linearization control technique in which the zero dynamic stability can not be proved. As a result, it is not guaranteed that system model would be robust to parameters variation. In addition in these control methods, the field orientation can be achieved only in the system steady state conditions. To overcome the above problems, in this paper, using the fifth order model of IM in a fixed stator reference frame, based on control theory described in [4], a composite nonlinear controller is designed that makes the IM drive system control robust and stable against the parameter uncertainties and external load torque. In this control approach, a two level SVPWM inverter feeds the IM drive.

Keywords- backstepping; induction motor; ANN; nonlinear systems; robust

II. IM MODEL

I. INTRODUCTION In the recent two decades, nonlinear control methods such as input-output feedback linearization and SlidingMode (SM) control have been applied to the IM drive. Specially in these years, in the field of adaptive and robust control, there has been a tremendous amount of activity on a special control scheme known as “backstepping” [1],[2],[3]. A major problem of backstepping control approach is that certain function must be “linear in the unknown system parameters” and some very tedious analysis is needed to determine a “regression matrix” [3]. It must be noted that in adaptive backstepping control, problem of finding regression matrix is more difficult in comparison with conventional backstepping method. To overcome the above problem, in [4], a combination of the backstepping control with ANN has been proposed. According to this method, in the process of backstepping controller design, two ANN are used to estimate a nonlinear function. Therefore no need to find the regression matrix for on-line estimation of unknown parameters. In [4], using the ANN, the theory of robust backstepping control has been presented for strictly 1-4244-0449-5/06/$20.00 ©2006 IEEE

The IM fifth order model in fixed two axis reference frame with rotor fluxes and stator currents as state variables [7] is given as T dω 3n p M (ψ ra isb −ψ rb isa ) − l = dt 2 JLr J

(1)

dψ ra R = − r ψ ra − n p ωψ rb + dt Lr

Rr Misa Lr

(2)

dψ rb R R = − r ψ rb + n p ωψ ra + r Misb dt Lr Lr

(3)

np M disa MRr = ψ ra + ωψ rb 2 σ Ls Lr dt σ Ls Lr  M 2 R + L 2 R  1 r r s − u  isa + 2 σ Ls sa  σ Ls Lr 

(4)

np M disb MRr = ψ rb − ωψ ra 2 dt σ Ls Lr σ Ls Lr  M 2 R + L 2 R  1 r r s u −  isb + 2 σ Ls sb  σ Ls Lr 

(5)

IPEMC 2006

where isa , isb ,ψ ra ,ψ rb , usa , usb are the stator currents, rotor fluxes and stator voltages, respectively. Subscripts a, b indicate a vector components in the fixed stator reference frame. Subscripts r , s indicate rotor and stator components. ω is rotor angular mechanical speed and

σ = 1 − M 2 /( Ls Lr ) . Ls , Lr are per-phase stator and rotor spatial inductances, respectively. M is per phase magnetizing inductance. n p is number of pole pairs. Rs , Rr are stator and rotor resistances, respectively.

A. ANN Basics Define W as the collection of ANN weighs, then the net output is [4] (6) y = W T φ ( x) Let S be a compact simply connected set of n

, with

m

map f : S → , define C ( s ) the functional space such that f is continuous. A general nonlinear function m

f ( x) ∈ C (S ) , x(t ) ∈ S can be approximated by a neural network as (7) f ( x) = W T φ ( x ) + ε ( x)

with ε ( x ) a ANN functional reconstruction error vector and φ ( x ) is sigmoid activation function. B. Robust Backstepping Control of IM Using ANN Using the well known fifth order IM model in a stator two axis reference frame where the rotor fluxes and stator currents are assumed as state variables [7], the robust nonlinear controller is designed in the following way. Dividing the above IM model into two nonlinear subsystems, where isa , isb are the outputs for the first subsystem which are simultaneously assumed the fictitious inputs of the second sub-system. Assume that: Assumption 1: The reference trajectories ω r and ψ rr are differentiable and bounded. Assumption 2: The load torque is an unknown constant and resistances, inductances and moment of inertia are unknown and bounded. In the first step of the controller design, isa , isb are assumed as fictitious controls for the second sub-system. The main objective is to obtain these controls so that the desired rotor speed and rotor flux amplitude signals are perfectly tracked in spite of machine parameters and external load torque uncertainties. Considering ω r and

ψ rr

as references for ω and ψ r , tracking error equations

are

e1 = ω − ω r e2 = ψ ra 2 + ψ rb2 −ψ rr 2 = ψ r 2 −ψ rr 2

i  −T L  L  l r − J r ωr  i M  M  D1 e = F1 + G1i , F1 =  i   − 2 (ψ 2 ) − 2 Lr ψ r ψ r  r r  M r Rr M    Lr  3n p 0  J  −3n p   ψ rb ψ ra  M  ,D =  G1 =  2 2   1  Lr  0 2ψ rb   2ψ ra  Rr M  

(9)

It is clear that G1 is known and invertible. By treating

III. ROBUST BACKSTEPPING CONTROL

n

Then

(8)

i as a fictitious input, a controller for the ideal i is designed as ∧

i = G1−1[− F 1 − K1e]

(10)

, K1 > 0 ∧

where K1 a design parameter and F1 the estimate of F1 which will be estimated in the next section with a two layer ANN. Substituting (10) into (9) gives ∧

i

−

(11)

D1 e = F1 − F1 − K1e + G1η , η = i − i

In the second step, the control u (usa , usb ) are obtained in such a way that η in equation (11), becomes as small as possible. Differentiating η with respect to time, yields i

D2 η = F2 + G2 u where usa  1 0 1 0  u =   , G2 =   , D2 = σ Ls   0 1 0 1  usb  i −1

i

∧

−1 1

∧

F2 = ... + D2{G1 ( F1 + K1e) + G F1

(12)

(13)

∧

+ G1−1K1 D1−1 ( F1 − F1 − K1e + G1η )}

To make η as small as possible, the following control is chosen ∧

u = G2−1[− F2 − K 2η − G1T e]

(14)

∧

In (14), F2 is an estimate of F2 that like the first step, a two layer ANN is used to estimate it. In addition a term −G1T e is added in (14) which is necessary to cancel the effect of G1η in (11). Combining (12) and (14), gives i

∧

D2 η = F2 − F2 − K 2η − G1T e

(15)

C. F1 , F2 Approximation Using ANN In this section, functions F1 , F2 are approximated by two two-layer ANN. In adaptive backstepping control, it is assumed that functions F1 , F2 are linear in term of known regression matrices, however in ANN method, there is no limitation for these functions. Using ANNs

approximation property, F1 , F2 as outputs of two twolayer ANN with constant weights Wi , is assumed to be as follows F1 = W1T φ1 + ε1

, ε1 < ε1N = cte

F2 = W2T φ2 + ε 2

, ε 2 < ε 2 N = cte

(16)

where φ1 , φ2 provide suitable basis functions. From (16), one can find that net reconstruction error ε i ( x) is bounded by a known constant ε iN . Assumption 3: The ideal weighs are bounded by known positive values so that (17) W1 ≤ W1M , W2 ≤ W2 M F

F

Or equivalently: Z F ≤ Z M , Z = diag {W1 , W2 }

(18) i

i

The actual inputs to ANN1 are ψ r , ωr ,ψ rr ,ψ rr and i

actual inputs to ANN2 are ω , ω r , ω r ,ψ r ,ψ rr ,ψ ra ,ψ rb i

,ψ rr , isa , isb , e1 , e2 .

On line ANN approximation of F1 is ∧

∧

F1 = W1T φ1 Then error dynamic equation of (11) becomes i

T

D1 e = W1 φ1 − K1e + G1η + ε1

(19) (20)

∧

where W1 = W1 − W1 . Similarly, approximation of F2 is assumes as ∧ ∧ F2 = W2T

φ2 Then error dynamic (15) will be i

T

− G1T e + ε 2

(22)

−G1T e

in (22). This means there are couplings between the error dynamics (20) and (22). D. Updating ANNs Weights In this part, the stability of proposed controller, is proved based on Lyapunov stability theory. This analysis shows that tracking errors and updated weighs are Uniformly Ultimately Bounded (UUB). Theory : Let the desired trajectories ω r ,ψ rr be bounded. Take the control input (14) with weigh updates be provided by i

∧

W 1 = Γ1φ1eT − kω Γ1 ζ W 1 i

∧

magnitudes of ζ

(23)

∧

W 2 = Γ 2φ2 eT − kω Γ 2 ζ W 2

with any constant matrices Γ1 = Γ1T > 0, Γ2 = ΓT2 > 0 and scalar positive constant kω . Then the errors

and Z

smaller ζ and a larger Z

F

F

, a smaller kω yields a

, and vice versa.

∧

Note 2 : If W i (0) are taken as zeroes the linear proportional control term − K ζ stabilizes the system on an interim basis.

IV. SYSTEM SIMULATION Based on proposed control strategy described in previous section, the block diagram of IM drive control is shown in Fig. 1. A C ++ computer program was developed for system simulation. In this program, the nonlinear equations are solved based on static forth order Range-Kutta method. The proposed control method, is tested for a three-phase IM with parameters shown in Table (1). In this simulation, the controller gains are obtained by trial and error method which are given as K1 = diag{1525,1550} , K2 = diag{5000,1550} kω = 1 , Γi = 10I

(21)

D2 η = W2 φ2 − K 2η Note that there is a term G1η in (20) and a term

∧

η (t ), e(t ) are UUB. ANN updated weights are bounded. The errors η (t ), e(t ) can be kept as small as desired by increasing gains Ki . Proof of this theory can be find in [4]. Note 1: Small tracking error bounds may be achieved by selecting large control gain K . The parameter kω offers a design tradeoff between the relative eventual

Table 1

: IM PARAMETERS

Stator resistance

Rs = 0.18Ω

Rotor resistance

Rr = 0.15Ω

Rotor nominal flux linkage Number of pole pairs

ψ rr = 1.3Wb.turns np = 1

Stator inductance

Ls = 0.0699H

Rotor inductance

Lr = 0.0699H

Mutual inductance

M = 0.068H

Nominal rotor speed Moment of inertia

ω r = 220rad / s J = .0586kgm2

Simulation results shown in Fig. 2 , are obtained in the case of an exponential reference flux rising up from zero to 1.3W .T at t = 0s , down to 0.8W .T at t = 3s with a time constant of τ = 0.05s , an exponential reference speed from zero to 220rad / s at t = 0.3s , rising up to 350rad / s at t = 3s with a time constant of τ = 0.1s , a step load torque disturbance from zero to 40 N .m. at t = 2s and motor electromechanical parameters assumed to be twice their nominal values at t = 1s . In addition, the

1.3W .t at t = 0s , down to 0.8W .t at t = 2 s and rising up to 1.3W .t at t = 3.5s , a step up load torque from zero to 40 N .m at t = 1s is shown in Fig. 4. In addition the IM speed control is obtained for an exponential reference flux rising up from zero to 1.3W .t at t = 0s and an exponential reference speed from zero to 220rad / s at t = 0.3s , down to −220rad / s at t = 2 s , rising up to 220rad / s at t = 3.5s , a step load torque from zero to 40 N .m. at t = 1s is shown in Fig. 5. In flux and speed control performance, motor electromechanical parameters assumed to be twice their nominal values at t = 0s .

steady state tracking errors e2 , e1 are also shown in this figure. Fig. 3 shows the simulation results obtained for an exponential reference flux rising up from zero to 1.3W .t at t = 0s and an exponential reference speed rising up from zero to 220rad / s at t = 0.3s , a load torque profile which is also shown in Fig.3 and motor electromechanical parameters assumed to be twice their nominal values at t = 0s . The IM rotor flux control is obtained for an exponential reference speed rising up from zero to 220rad / s at t = 0.3s and an exponential flux reference from zero to

F1

i

i

η

− K1

F2

−K 2

ω ,ψ r

e1 , e2

r r

system states

u

−G1T

Fig. 1 : The overall block diagram of IM drive control

1

1.3

ψ r (W .t )

220

0.5

0.8

0

-100

0

1

2

sec. 3

4

0

5

0

1

2

sec.

3

4

5

0

-0.5 -1

50

-1.5

0

-100

-50

0

1

2

sec.

3

4

5

600

-100

1

2

sec.

3

4

5

400

usb (V )

0 -200 -400 -600 1.9

1.95

2

sec.

2.05

2.1

1

2

3

sec.

4

5

0

1

2

3

4

5

0.01 0

500

200

0

0

0

-0.01

0

-500 1.9

e2 ((W .t )2 )

isb ( A)

100

50

isa ( A)

100

-50

usa (V )

e1 ( rad / s)

ω ( rad / s )

350

1.95

2

sec.

2.05

2.1

Fig. 2 : IM performance using robust backstepping controller

-0.02

sec.

∧

ψ r (W .t )

Tl ( N .m)

ω ( rad / s )

sec.

sec.

sec.

∧

ψ r (W .t )

Tl ( N .m)

ω (rad / s )

Fig. 3 : IM torque control

sec.

sec.

sec.

Tl ( N .m)

∧

ψ r (W .t )

ω (rad / s)

Fig. 4 : IM flux control

sec.

sec.

sec.

Fig. 5 : IM speed control

CONCLUSIONS

In this paper, a composite nonlinear controller has been proposed for the IM rotor flux and speed tracking control. The nonlinear controller is designed based on the IM fifth order model in a fixed two axis reference frame, combining the backstepping control and ANN. The overall stability of this controller is proved by Lyapunov theory. Computer simulation results obtained, confirm the effectiveness and validity of the proposed controller. These results also confirm that the drive system control is robust and stable against the parameters uncertainties and unknown load torque disturbance. REFERENCES [1] I. Kanellakopoulos, P. V. Kokotovic and A. S. Morse, “Systematic Design of Adaptive Controllers for Feedback Linearizable

Systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1241–1253, 1991. [2] P. V. Kokotovic, “Bode lecture: The joy of feedback,” IEEE Contr. Syst. Mag., No. 3, pp. 7–17, June 1992. [3] M. Krstic, I. Kannellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design, Wiley and Sons Inc., New York, 1995. [4] C. M. Kwan, F. L. Lewis, “Robust Backstepping Control of Nonlinear Systems Using Neural Networks,” IEEE Trans. Systems, Man and Cybernetics, vol. 30, No. 6, Nov. 2000. [5] O. Kuljaca, N. Swamy, F. L. Lewis and Ch. M. Kwan “Design and Implementation of Industrial Neural Network Controller Using Backstepping,” IEEE Trans. Industrial Electronics, vol. 50, No. 1, Feb. 2003. [6] C. M. Kwan, F. L. Lewis, “Robust Backstepping Control of Induction Motors Using Neural Networks,” IEEE Trans. Neural Networks, vol. 11, No. 5, Sep. 2000. [7] P. C. Krause, Analysis of Electric Machinery, McGraw-Hill Inc., 1986.