Tracking Control of Dynamic Nonlinear Systems via Improved Adaptive Fuzzy Control

Tracking Control of Dynamic Nonlinear Systems via Improved Adaptive Fuzzy Control

Pham Van Thiem, Lai Khac Lai

Electronics Faculty ,

Thai Nguyen University of Technology, Thai Nguyen City,

Viet Nam

Nguyen Thi Thanh Quynh

Centre de Recherche en Sciences et Technologies de IInformation et de la Communication (CRESTIC), Université de Reims Champagne Ardenne,

France

AbstractIn this paper, an Adaptive Fuzzy combined with Model Adaptive Reference System (MARS) is proposed to a single input-single output nonlinear system. The main goals of this paper are estimation and control of second order nonlinear system. Firstly, adaptive fuzzy system is applied to design the estimator and the controller. Next, a performance of system is improved by adjusting parameters of the controller via MARS. Finally, two examples are presented to illustrate of the proposed methods. By applying Lyapunov stability theory the adaptive law that is derived in this study is robust and convergent quickly. The simulation results and analysis show that the proposed method have better than Adaptive Fuzzy in the sense of robustness against disturbance.

KeywordsAdaptive Fuzzy System; Model Reference Adaptive Systems (MARS); Coupled Tank Liquid Level; Robot Arm.

1. INTRODUCTION

   The model of object is usually obtained by identification methods, then we use this model to design controller. If the characteristic of an object is not changing in all working process, the closed loop system with this controller is good for demanded designing. However, the parameters of object are unpredictably time variant in practice. Consequently, we need to have an online identification method based on adaptive algorithm which can be adapted by the changing parameters. To do this, the neural network is used [1, 2] to identify the dynamic systems with the back propagation algorithm and found on adaptive fuzzy control which is proposed by Li-Xin

   Wang [3, 4].

   in order to transform a given nonlinear system with outputs into a controllable and observable linear one.

   With the nonlinear system [5, 6, and 9], K. Khalil use the exact feedback linearization approach to define a control law with assumption that the modeled object is known clearly and states are measured. If we dont know exactly parameters of object, two trends will appear. The first trend consists in introducing a direct adaptive fuzzy control [3, 4] based on Lyapunov theory. The second trend, with which we are concerned, is carried out in controller design based on universal approximation [3, 10], the indirect adaptive fuzzy control. On the other hand, the controller in [3, 4, and 5] guarantees still the stable closed loop system, but it does not concentrate to performance of closed loop system, for example: the settling time, overshoot and error between the output signal and referenceTherefore, to improve performance of system, concretely, the redue error between reference trajectory and output trajectory, and rise the settling time of transient response.. Parameters of controller will be adjusted by adaptive law found on model adaptive reference system (MARS).

   The remainder of this paper is organized as follows. In Section II, we provide the exact linearization of nonlinear. Section III estimation and control based on adaptive fuzzy mode control. Section IV then consider the improved adaptive fuzzy mode control. Simulation results are given in Section V and Section VI concludes this work.

2. EXACT LINEARIZATION OF NONLINEAR WITH OUTPUT

   Consider the SISO nonlinear system given as:

   1 2 n

   1 2 n

   dx T n

   The nonlinear systems has attracted widespread attention in

   f (x) h(x).u

   x (x x … x ) R

   (1)

   the recent decades [5, 6]. There are many controlled methods derived from linear controller as gain scheduling, Jacobi matrix [7]. However, a question is that of when there exists a global change of coordinates, a diffeomorphism, in the state

   space that carries the nonlinear system into linear system.

   dt u,y, g(x) R; f (x),h(x ) Rn

   y g(x)

   We always find the coordinates transformation in order to transform the nonlinear system into linear system.

   Theorem 1: Consider the SISO nonlinear system (1) with

   Krener [8] showed the importance of the Lie algebra of a vector fields associated with the system in studying such a question and gave an answer to this problem. D. Cheng [9] illustrated that we always find the coordinates transformation

   r is the characteristic number at x . We assumern thespace, under the diffeomorphism:

   f f

   f f

   z m(x) (g(x), L g(x),…, Ln1g(x))T

   in (2)

   the nonlinear system will be transformed into linear system:

   z Az bw

   (3)

   where bl is the point at which output membership function

   A

   A

   achieves its maximum value and are input membership

   where

   Bl

   functions. If we fix the

   l j

   (x ) and view thebl as adjustable

   0 1 … 0

   0

   Al j

   j

   j

   A = ; b ; w F(x) G(x).u

   parameters, then (5) can be written as:

   0

   0

   1

   1

   0

   0

   0

   0

   b(x) T x

   (10)

   T 1 M

   0 0 0 0

   1

   where

   (b

   ,…,b

   ) is a parameter vector of the output

   Proof (see Appendix)

   membership functions, and

   x (1(x ),…, M (x ))T is a

3. ESTIMATION AND CONTROL VIA ADAPTIVE FUZZY MODE regressive vector with the regression l (x ) defined as:

   CONTROL (AFC)

   Without loss of generality, we consider the dynamic

   (x)

   2

   j 1

   j

   Aij

   (x j )

   (11)

   nonlinear has two state variables linearized by Theorem 1 and

   M 2

   j (x )

   given a structure as:

   l 1

   j 1

   Aij j

   x x

   Assume that the functions F(x) andG(x) describing the

   1 2

   2

   2

   x

   F(x) G(x)u

   (4)

   system dynamics are unknown, we can be estimated the function F(x) andG(x) by fuzzy logic system with the

   y x1

   adjusted parameters of output membership functions based on

   1 2

   1 2

   G

   G

   where x (x x )T R2; u,y R; F(x) R2;G(x) R2

   and F(x);G(x) C are smooth functions. The control target

   adaptive law.

   F(x) T

   x; (x) T

   x

   (12)

   of system (4) is stable tracking a prior trajectory y

   (t) , called

   F F G G

   where the parameter vector T ;T are updated online so that

   m F G

   reference trajectory, as means:

   the approximate error between F(x);G(x) and F(x);G(x) is

   lime(t) 0 ,

   t

   e(t)

   is finite (5)

   minimal. Define the optimal parameter vector as:

   where e(t) y

   (t) y(t) is tracking error.

   * arg min(sup | T (x) F(x))

   F F F

   F F F

   m F x

   (13)

   From (4), we express: * arg min(sup | T (x) G(x))

   x

   x

   y x1 2

   If

   G

   F

   * ;

   x

   * then

   G G

   F(x) F(x), G(x) G(x) that

   y x2 F(x) G(x)u

   y x

   (6)

   F F G G

   means the fuzzy system (9) can approximate smooth nonlinar

   1

   We assume that the parameters of system (6) are known exactly. Then, the state feedback controller shown as:

   functions with the arbitrary small error if the number of fuzzy rules is large enough [10]. The state feedback controller (7)

   can be expressed:

   u 1 (y F(x) k e k e)

   (7) 1

   G(x) m p d

   u* F(x) y

   k e k e

   (14)

   where kp and kd are coefficients of Hurwitz polynomial:

   d p

   d p

   h(s) s2 k s k (8)

   This controller will be made stable closed loop system as means (4).

   In formula (7), the controller depends on functions F(x),G(x) . If they are unknown, the controller cannot be implemented. Consequently, we need to estimate an online

   G(x) m p d

   From (10), we see that if the parameters of systems have change by the time, the functions F(x),G(x) will change and

   the control signal u will be suitable adjusted for alterative parameters of object.

   Theorem 2: Consider the nonlinear system (4), the state

   feedback controller (10) and F(x),G(x) in (12) are applied

   with the T and T updated by adaptive law:

   function F(x),G(x) based on adaptive fuzzy mode algorithm. F G

   Q1 (x)s (e); Q1 (x)s (e)u

   1 2

   1 2

   In the [3], consider the MISO fuzzy logic control system has two inputs x (x x )T R2 and an outputb . This

   j

   j

   fuzzy logic system with the center-average defuzzifier, product inference, and singleton fuzzifier is of the following form:

   F F F f G G G f

   The closed loop system will be asymptotic stable.

   Proof:

   The second derivative of the output error is expressed:

   y F(x) G(x)u

   M bl

   l 1

   2

   j 1 Al

   (x )

   F(x) F(x)G(x) G(x)u F(x) G(x)u

   m p d

   e k e k e T (x) *T (x) T (x) *T (x)u

   m p d

   e k e k e T (x) *T (x) T (x) *T (x)u

   M 2

   M 2

   l

   (x )

   l

   (x )

   b(x) j

   (9)

   y y

   k e k e F(x) F(x)G(x) G(x)u

   l 1

   j 1 Aj j

   p d F F F F G G G G

   s T *T (x) T *T (x)u

   f F F F G G G

   F F G G

   F F G G

   T (x) T (x)u

   T T *T , T T *T are the parameter error * 1

   1. F F

   2. G G

   u F(x) y

   k e k e

   Consider the Lyapunov candidate function:

   G(x) m p d

   V 1 s 2 1 TQ

   T 1 TQ

   T

   ; Q 0; Q 0

4. IMPROVED ADAPTIVE FUZZY MODE CONTROL(IAFC)

   2 f 2

   F F F

   2 G G G F G

   i i

   i i

   whereQ Rdd (d dim ) is a positive definite matrix. Take

   i i

   i i

   the derivative ofV with respect to time and notice that , we have:

   In formula (14), two parameters kp ;kd of the state feedback controller are chosen in (8) that they still guarantee the stable closed loop system. However, it does not concentrate to

   performance of closed loop system, for example: the settling

   V s s TQ TQ

   time, overshoot and error between the output signal and

   f f F F F G G G

   V s

   T (x ) T (x )u TQ TQ

   referenceTherefore, to improve performance of system,

   f F F G G F F F G G G

   concretely, the decrease error between reference trajectory and

   V T s (x ) Q T s (x )u Q 0

   F f F F F G f G G G

   Choose the parameters update law as follow:

   output trajectory, and rise the settling time, two parameters

   kp ;kd will be adjusted by adaptive law based on model

   Q1s (x); Q1s (x)u

   F F f F G G f G

   with the error surface is defined:

   adaptive reference system

   The controller (14) can be rewritten as:

   s (e) k e k de

   (15)

   u*

   1 (x) y

   T

   (18)

   f 1 2 dt

   F

   G(x)

   m k e

   where T (k k ); (e e)T . The T will be updated by

   k p d e k

   the adaptive law based on reference system in Theorem 3.

   Theorem 3: Consider the nonlinear system (1), the state

   feedback controller (18) and F(x),G(x) in (12) are applied

   with the T ; T and T updated by adaptive law:

   a b k

   Q1 (x); Q1 (x)u; Q1

   F F F G G G k k e

   The closed loop system will be asymptotic stable.

   Proof:

   The second derivative of the output error between output and reference is expressed:

   y F(x) G(x)u

   F(x) F(x)G(x) G(x)u F(x) G(x)u

   Fig. 1. The overall scheme of identification and control via adaptive

   y y

   T

   x) F(x)

   x) G(x) u

   fuzzy system (AFC)

   m k e

   F(

   G(

   2m

   2m

   2

   2

   F F F F G G G G

   F F F F G G G G

   k e

   k e

   x x T (x) *T (x) T (x) *T (x)u T

   where SVF is state variable filter. The derivative of the error can be created by using SVF. The parameters of this state variable filter are chosen in such as the parameters of the reference model.

   T *T (x) T *T (x)u T

   F F G G

   F F G G

   k e

   k e

   F F F G G G k e

   F F F G G G k e

   T (x) T (x)u T

   T T *T , T T *T are the parameter error

   The following steps to design the estimation and control via

   1. F F

   2. G G

   adaptive fuzzy system:

   Step1: The regressive vector l (x ) as:

   Consider the Lyapunov candidate function:

   V 1 2 1 TQ T 1 TQ T 1 TQ

   2 2 F F F 2 G G G 2 k k k

   2

   2

   j (x )

   Q 0; Q 0; Q 0

   M

   M

   2

   2

   (x )

   (x )

   whereQ

   whereQ

   Rdd (d dim ) is a positive definite matrix. Take

   Rdd (d dim ) is a positive definite matrix. Take

   (x)

   j 1 Aij j

   (16)

   F G k

   l 1 j 1 Aj j i i

   i i

   i i

   ij

   Step 2: The adaptive law:

   the derivative ofV with respect to time and notice that ,

   Q1s (x); Q1s

   (x)u

   (17)

   we have:

   F F f F G G f G

   V TQ TQ

   TQ

   Step 3: Estimation the function:

   F F F G G G

   k k k

   V T (x) T (x)u T TQ

   TQ

   F(x) T x ; G(x) T

   x

   F F G G

   k e F F F G G G

   F F G G

   TQ

   Step 4: Define the error surface:

   k k k

   T

   T

   V F F (x) QF F G G (x )u QG G

   s (e) k e k de

   T (Q ) 0

   f 1 2 dt

   k k k e

   Step 5: The state feedback controller:

   Choose the parameters update law as follow

   (x) Q 0 Q1

   (x)

5. ILLUSTRATIVE EXAMPLE

   1. F F F F F

      (x)u Q 0 Q1

      (x)u

      In this section, we present two examples of tracking

   2. G G

   G G G

   reference set point with a 1 DOF robot arm and a coupled tank

   Q 0

   Q1

   k k e

   k k e

   liquid level system. The simulation illustrates the convergence

   where a weighting matrixQk given as:

   of error under our proposed AFC and IAFC approach.

   q

   0

   k

   1 q

   0 e

   1. 1 DOF Robot Arm

      Q k 11

      ;

      p> Q1 p

      k 11

      Consider the 1 DOF robot arm [1] has the dynamic

      k 0

      qk 22

      1

      k k e

      k

      0 q

      k 22 e

      equation shown as:

      x1 x2

      d

      d

      kd (t) kd (0)

      (ym y)dt

      g d 1

      qk 22

      1

      x

      2

      2

      l

      sin(x1 )

      ml2

      x

      2

      u

      ml 2

      (22)

      k (t) k (0)

      (y y)dt;

      y

      y (19)

      ' x

      p p q

      m m

      k 11

      y 1

      and kp (0) ; kd (0) are chosen in(8) to guarantee the stability of the closed loop system.

      where x (x x )T ( )T ; is the angle of robot arm and

      1 2

      1 2

      is the velocity of robot arm,l 0.85m is length of arm.

      d 0.85kg / m2 is the coefficient of friction, m 0.9kg is

      weight of robot arm, and g 9.8m / s2 is the gravity acceleration .

      x

      x

      g d 1

      we have

      F (x )

      sin(x )

      2

      1

      2

      1

      2

      and G(x) .

      2

      l ml ml

      We will identify and design the controller following steps as:

      Choose the inputs membership function of F(x); G(x) have

      shaped in Gaussian over the interval [-1 1]:

      x exp x

      A

      A

      cl 2 / 2 l 2

      (23)

      j

      j

      l j

      i j j

      and F(x); G(x) are identified in Theorm 2 for AFC and

      Theorm 3 for IAFC:

      Fig. 2. The overall scheme of identification and control via improved adaptive fuzzy system (IAFC)

      Reference Model: Explicit output reference, derivative output, and acceleration profile set point signals are created using the reference model, which is described by the transfer function:

      H

      H

      w 2

      Fig. 3. Identification nonlinear system via adaptive fuzzy system

      ref

      n

      n n

      n n

      s 2 2zw s w 2

      (20)

      We choose the fix coefficients kp 1; kd 5 of AFC and

      The parameters of the reference model are chosen such as the higher order dynamics of the system will not be excited

      initial parameters kp (0) 1; kd (0) 5 of IAFC. The results of simulation with the two algorithms AFC and IAFC contain

      [11].The following steps to design the identification and

      control based on improved adaptive fuzzy system:

      f

      f

      From Step1 to Step 3 is same the following steps in Section 3 noticed that we replaces with .

      Step 4: The parameters of controller are updated by adaptive law as:

      two parts: the estimation the smooth function F(x); G(x) and error between output and reference trajectory:

      k k e m

      k k e m

      Q1 ; y

      y

      (21)

      Step 5: The state feedback controller:

      u *

      1 (x ) y

      T

      F

      G (x )

      m k e

      Fig. 4. Estimation two nonlinear functions of systems

      b2 the cross sectional area of outlet pipe in tank 2	0.5cm2
      a12 the cross-sectional area interaction pipe between tank 1and tank 2	0.5cm2
      b12 the value ratio of interaction pipe between tank1 and tank 2	1.5
      g acceleration of gravity	981cm2/sec
      a2 the value ratio of outlet pipe of tank 2	1.5

      We use the Theorem 1 in order to convert (24) into (4) where:

      Fig. 5. Tracking error with IAFC (1st line) and with AFC (2nd line)

      F(x) L2 g(x);G(x) L L g(x)

      f h f

      f h f

      Assuming that the function f (x) and

      (25)

      h(x) describing the

      Comparison of angle control simulation between AFC and

      system dynamics are unknow, so that we employ adapative fuzzy system to online identify F(x) andG(x) . The control

      IAFC is present. With F(x); G(x)both AFC and IAFC are able

      to identify (see Fig.4). Response of both AFC and IAFC are almost same. However, it is clearly that the tracking errors for the AFC due to reference trajectory are larger than those by IAFC (see Fig.5), concretely, the tracking errors for IAFC is

      about 100 times as small as those for AFC. The adaptive

      target guarantee for liquid level of Tank 2 at the setpoint. Opened valve ratio of pump 1 is adjusted by control law, and

      pump 2 open arbitrary.

      Fistly, we will estiamte the F(x); G(x) by the adaptive

      fuzzy system. The inputs membership function are shaped in

      k p d

      k p d

      gains T (k k ) of the I

      AFC

      automatically reach stationary

      Gaussian over the interval [0 100] as (19) and T is parameter vector of output membership function which is updated by

      values (see Fig.6).

      Fig. 6. Adaptive gains of the IAFC

   2. Coupled Tank Liquid Level System

   In [12], the modeled equation of coupled tank liquid level system and parameters given as:

   1 2 1 2

   1 2 1 2

   x (x x )T (h h )T

   adaptive law in Thoerm 2 for AFC and Theorm 3 for IAFC as Fig. 3. We choose the fix coefficients kp 1; kd 5 of AFC

   and initial parameters kp (0) 1; kd (0) 5 of IAFC given as:

   Fig. 7. Simulation control structure with MATLAB

   b a 2g x x

   12 12 1 2

   k

   d x1 A

   1 A u(t)

   dt x 1

   1

   2 b a

   2g x x

   b a

   2gx

   0

   (24)

   2

   2

   y x

   dx

   A2

   12 12 1 2 2 2 2

   dt

   f (x) h(x).u

   Fig. 8. Estimation two nonlinear functions of systems

   y g(x)

   Parameter	Value
   A1, A2 the cross-sectional area of tank 1 and tank 2	100cm2
   Qin the flow rate of liquid into tank 1	u(cm3/sec)
   Qout the flow rate of liquid into tank 2	0.8cm3/sec
   p, p the height of liquid in tank1 and tank 2	cm

   Parameter	Value
   A1, A2 the cross-sectional area of tank 1 and tank 2	100cm2
   Qin the flow rate of liquid into tank 1	u(cm3/sec)
   Qout the flow rate of liquid into tank 2	0.8cm3/sec
   p, p the height of liquid in tank1 and tank 2	cm

   TABLE I. THE PARAMETERS OF COUPLED LIQUID LEVEL SYSTEM

   Fig. 9. Tracking error with AFC (- – -) and with IAFC ( )

   the influence of load disturbances and measurement noise. Strong properties achieved via the proposed method confirm that improved adaptive fuzzy system is an attractive approach for controlling single input-single output nonlinear systems.

   VII. ACKNOWLEDGMENTS

   This research was supported by the MOET (The Ministry of Education and Training), Viet Nam, under grant no. B2016- TN01-01.

   APPENDIX

   The relative order of single input-single output (SISO) system is defined:

   k

   k

   0, 0 k r 2

   L L g(x)

   Fig. 10. Tracking output with both of AFC and IAFC

   h f

   0,k r 2

   In this coupled tank liquid level system, we consider that

   dy g(x) x g(x) f (x) h(x).u L g(x) L L0 g(x)u

   this system is affected by disturbaced(t) which has changing dt x x

   f h f

   2

   L g(x)

   0

   L g(x)

   frequency by the time as Fig.10. With F(x); G(x)both AFC and

   dy

   dt

   f x

   x

   f f (x) h(x).u

   x

   IAFC coul be evaluated when there is disturbance effect on

   L2 g(x) L L1 g(x)u

   the system (see Fig.8). Comparison of liquid level control simulation between AFC and IAFC is present. Response of both

   AFC and IAFC are almost same (see Fig.10). However, it is obviously that the tracking errors for the IAFC due to reference

   dyr 1

   Lr 2g(x)

   f h f

   0

   Lr2g(x)

   liquid level are less than those by AFC (see Fig.9). The

   f

   dt x

   x

   f f (x) h(x).u

   x

   adaptive T (k k ) of the IAFC automatically alter when

   Lr1g(x) L Lr 2g(x)u

   k p d

   f h f

   disturbance inluence on system (see Fig.11).

   r r 1

   L

   L

   0

   r 1

   L

   L

   dy

   g(x)

   f x

   f g(x) f (x) h(x).u

   dt x

   x

   Lr g(x) L Lr 1g(x)u

   f h f

   0

   with n r , we chose the coordinate transformation as:

   1 2

   1 2

   n k

   n k

   f

   f

   z m(x) m (x), m (x),…,m (x)T ; m (x) Lk 1g(x)

   dz1 m1 x g(x) dx L g(x) z

   dt x x dt f 2

   dz2 m2 x Lf g(x) dx L2 g(x) z

   dt x

   x dt f 3

   dz m

   Lr 2g(x) dx

   Fig. 11. Adaptive gians of the IAFC

   n1 n1 x f

   Lr 1g(x) z

   dt x

   dz m

   x dt f n

   Lr 1g(x) dx

6. CONCLUSION

n n x f

dt x x

Lr g(x) L Lr 1g(x)u

f h f

f h f

dt

Lr g(m1(z)) L Lr1g(m1(z))u

In the paper we have presented adaptive fuzzy system combined with MARS, improved adaptive fuzzy system,

f

F (x )

hf

G (x )

offers a potential on deliver more accurate and high overall performance in the presence of all the preceding issues. We investigate the effect of the identifier and controller from the simulation results. Compare to the case with AFC and IAFC in two illustrated examples (1DOF robot arm and coupled tank liquid level), for instance, can do the following (see Fig.4, Fig.5, Fig.8, and Fig.9): (a) Improve the transient behavior of the system; (b) Decrease the sensitivity to plant parameter changes; (c) Eliminate steady-state errors; and (d) Decrease

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