Adaptive fuzzy sliding mode control for uncertain nonlinear systems

Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India

Adaptive fuzzy sliding mode control for uncertain nonlinear systems Petr Huˇ sek Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 166 27, Prague 6, Czech Republic (e-mail: [email protected]). Abstract: An application of adaptive fuzzy sliding mode control is presented in this paper. The parameters of sliding mode control (the extended feedback gain and switching gain) are adapted during the control process using a fuzzy self-learning mechanism such that better control performance in comparison with sliding mode control with fixed parameters without chattering is achieved. Moreover, the nonlinear functions describing state space model of the controlled plant do not need to be known. The method is applied on control of a laboratory model of a pendulum on a cart. Keywords: sliding mode control, adaptive fuzzy control, pendulum on cart 1. INTRODUCTION Over past few decades sliding mode control (SMC, Utkin (1992)) has become the most popular technique for control of nonlinear systems, especially because of simplicity of the control law, easy implementation and high robustness. Unfortunately, when used with fixed parameters SMC has several drawbacks. The most important are chattering of control input (leading e.g. to high moving of mechanical parts and heat losses in electrical power circuits), slow convergence and nonzero steady state error. The usual way how to decrease the chattering phenomenon consists in introduction of boundary layer (Slotine and Li (1991), Duan et al. (2002), Chen et al. (2005)). However attenuation of chattering in this case decreases control performance. To avoid this effect different adaptive mechanisms for online tuning of parameters of sliding mode control has been introduced in past years. One of the most often used adaptation mechanisms is based on fuzzy logic approach (Liu et al. (2005), Hung et al. (2007), Chang et al. (2012)). Adaptive fuzzy control has achieved a big number of applications, e.g. Ismail (1998), Huˇsek (2013), Oltean et al. (2007), Xu and Pan (2008), Huˇsek (2011a), Su et al. (2005), Mendes et al. (2013). Different approaches to adaptive fuzzy control can be found in Hung et al. (2007), Huˇsek (2011b), Lian (2012) or Onieva et al. (2013). Great popularity in last few years has gained especially the self-learning mechanisms based on fuzzy logic since only a very rough model of the controlled plant is sufficient for successful control (Layne and Passino (1992), Yuan et al. (2004), Huˇsek and Cerman (2013), Ling (2009)). A survey of different approaches to self-learning and self-adaptive systems can be found in Mac´ıas-Escriv´ a et al. (2013). ? This work has been supported by the projects P103/12/1187 sponsored by the Grant Agency of the Czech Republic and LG13045 sponsored by the Ministry of Education of the Czech Republic.

978-3-902823-60-1 © 2014 IFAC

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In the recent years, AFSMC methods have enjoyed a great popularity that is supported by many successful applications, e.g. in automotive industry (Huang and Lin (2003), Huang and Chen (2006)), aeronautics (Bao et al. (2006), transportation (Lee and Tomizuka (2003), Wai and Su (2006)), Luo et al. (2008)), Bessa et al. (2008)), mobile robotics (Lian (2012), control of underwater vehicles (Javadi-Moghaddam and Bagheri (2010), Yeh et al. (2009)), manipulators (Guo and Woo (2003)), hydraulic units (Ho and Ahn (2011), Bessa et al. (2010)) or servomechanisms (Wai et al. (2002), Wai et al. (2004)). The parameter which is typically adapted is the width of boundary layer (Lee et al. (2001)). Unfortunately, design of the corresponding rules is very difficult and not transparent, especially for high order systems. An interesting method that consists in adaptation of the extended feedback gain and switching gain using selftuning mechanism based on fuzzy logic was presented in Cerman and Huˇsek (2012). The presented method does not require exact knowledge of the nonlinear functions characterizing the state space model of the controlled plant. The main drawback of the approach that seems to overcome the methods presented above is that it was applied only on a simulation example of a low order system. In this paper the adaptive fuzzy sliding mode control using adaptive mechanisms based on fuzzy logic for the control gains is applied on a real laboratory system pendulum on a cart that represents a highly nonlinear 4-th order system. The achieved results prove that such approach achieves better tracking performance than the sliding mode control with fixed boundary layer. 2. CONVENTIONAL SLIDING MODE CONTROL Let us recall the principle of conventional sliding mode control (Utkin (1992)). Consider a general class of SISO n-th order nonlinear systems in the following form: 10.3182/20140313-3-IN-3024.00114

2014 ACODS March 13-15, 2014. Kanpur, India

x˙ n = f (x, t) + g(x, t)u(t) + d(t) y =x

(1) T

(n−1)

n

where x = [x1 , x2 , . . . , xn ] = [x, x, ˙ ...,x ] ∈ < is the state vector of the system which is assumed to be available for measurement, f (x, t) and g(x, t) are unknown but bounded nonlinear functions, u ∈ < and y ∈ < is the input and output of the system, respectively, d(t) is unmeasurable bounded external disturbance, |d(t)| ≤ D < ∞. The nonlinear system (1) is supposed to be controllable and the input gain g(x, t) 6= 0. Without loss of generality, we assume that g(x, t) > 0. The control objective is to design a control law for the state x to track a desired (n−1) ) in the reference state trajectory xd = (xd , x˙d , . . . , xd presence of model uncertainties and external disturbances. The tracking error is defined as e = x − xd = [e, e, ˙ . . . , e(n−1) ]T ∈
(2)

Then the sliding surface in the state space of the error state is defined as

Now let us suppose that the nonlinear functions f (x, t) and g(x, t) are known in advance. We will try to approximate them by a fuzzy mapping. 2.1 Fuzzy system The fuzzy system of Mamdani type is a collection of the IF-THEN rules where the l-th rule, j = 1, . . . , m, is in the following form: (l)

where x = [x1 , · · · , xn ]T ∈
T

s((e), t) = k1 e + k2 e˙ + · · · + kn−1 e(n−2) + e(n−1) = k e T

(j)

R(j) : IF x1 is A1 and · · · and xn is An THEN y is B (l)

y(x) =

n

Qn l l=1 y ( i=1 µAli (xi )) Pm Qn l=1 i=1 µAli (xi )

(7)

where k = [k1 , k2 , . . . , kn−1 , 1] ∈ < is the vector of the coefficients of a Hurwitz polynomial h(λ) = λn−1 + kn−1 λn−2 + . . . + k1 where λ is the Laplace operator. For the zero initial condition e(0) = 0, the tracking problem x = xd can be considered as keeping the error state vector on the sliding surface s((e), t) = 0 for all t ≥ 0.

where µAli (xi ) is the value of the membership function of the linguistic variable xi , y l represents a crisp value at which the output membership function µB l reaches its maximum value, µB l (y l ) = 1.

The sliding process can be divided into two phases: the first one is the approaching phase with s 6= 0 and the second one is the sliding phase s = 0. A sufficient condition to guarantee that the trajectory of the error vector e will translate from the approaching phase to the sliding surface is to choose the control strategy such that

y(x) = θ ζ(x) = ζ(x)T θ,

s · s˙ ≤ −η|s|,

η > 0.

(3)

Let us assume that f (x, t) and g(x, t) are known. The system is controlled in such a way that it always moves towards the sliding surface and hits it. In the sliding phase, the control law can be easily derived from the condition s˙ = 0: # " n−1 X 1 (n) (4) ueq = ki e(i) − f (x, t) + xd − g(x, t) i=1 The control term ueq is called equivalent control law. In the approaching phase, where s 6= 0, the switching control usw term has to be added to the control law in order to hit the sliding surface, 1 η∆ sgn(s) (5) usw = g(x, t) where η∆ > D + η and the signum function is defined as ( −1 s < 0, s = 0, sgn(s) = 0 +1 s > 0. The total control law is then given by u∗ = ueq − usw . (6) Switching parameter η∆ has to be determined in advance. 541

By introducing the basis function the output of the fuzzy system (7) can be expressed as T

(8)

where θ = [y 1 , . . . , y m ]T is the parameter vector and ζ(x) = [ζ 1 (x), . . . , ζ m (x)]T is a regressive vector with the regressor defined as Qn

i=1 µAl (xi ) . ζ (x) = Pm Qn i l=1 i=1 µAli (xi ) l

(9)

3. ADAPTIVE FUZZY SLIDING MODE CONTROL To achieve faster convergence of the error vector to zero a high value of the switching gain η has to be chosen. Unfortunately, such an arrangement leads to higher chattering of the control signal that is undesirable. To have a possibility to decrease chattering extending proportional term to the switching term is proposed in Cerman and Huˇsek (2012). The switching term then becomes uswp = η∆ sgn(s) + KP ds

(10)

with some strictly positive constant KP > 0 and η∆ > D+ η. ds is the signed distance between the point in the error state space e and the sliding surface calculated as s ds = q . 2 2 k1 + · · · + kn−1 The control law (4) then becomes

(11)

2014 ACODS March 13-15, 2014. Kanpur, India

" n−1 X 1 (n) ki e(i) − f (x, t) + xd − η∆ sgn(s) − u = g(x, t) i=1 #

can be proved. Another drawback is that the rules cannot be easily implemented for higher order systems.

− KP ds .

(12)

As it was mentioned above, since the functions f (x, t) and g(x, t) are usually unknown and are tough to be identified we will approximate them by the estimates fˆ(x|θf ) and gˆ(x|θg ) that are implemented by a fuzzy system (8). After substitution of (8) into both functions the proposed control law is obtained as " n−1 X 1 (n) ∗ u = ki e(i) − fˆ(x|θf ) + xd − η∆ sgn(s) − gˆ(x|θg ) i=1 # − KP ds

(13)

where T fˆ(x|θf ) = θf ζ(x), T

gˆ(x|θg ) = θg ζ(x).

(14)

The key point is to establish an adaptation law for the parameter vectors θf and θg such that all signals and parameters in the closed loop are bounded and the output error converges to zero. The following theorem suggests such an adaptation law. Theorem (Cerman and Huˇsek (2012)) Consider the nonlinear system (1) controlled by the control law (13). Then if the parameter vectors θf , θg are adjusted by the adaptation law θ˙ f = r1 ds ζ(x), θ˙ g = r2 ds ζ(x)u (15) with r1 > 0 and r2 > 0 being learning rates the closed-loop system signals will be bounded and the tracking error will converge to zero asymptotically. The proof can be found in Cerman and Huˇsek (2012). A typical way how to eliminate the chattering phenomenon is to introduce boundary layer (Slotine and Li (1991)), i.e. to replace the signum function by the saturation function

In this paper we propose to eliminate chattering while preserving fast convergence by adaptation of the sliding mode control gains – the extended proportional gain KP and the switching gain η. In Cerman and Huˇsek (2012) fuzzy rules for such an adaptation were proposed. Unfortunately, the fuzzy rule base is too complicated and cannot be easily modified for systems with different dynamics since the gains are changed by absolute values. Here we substantially simplify the rules and apply them to a real system to prove improved control performance of sliding mode control. At first let us establish the rules for adaptation of KP . The corresponding fuzzy system has two inputs: one is the distance between the point in the error state space and sliding surface |ds | and its time derivative, d˙s . The output of the EG fuzzy system is the change of KP , ∆KP , applied in the next simulation step. The larger distance from the sliding surface and lower rate of approaching to the sliding surface, the larger increment ∆KP should be applied to quickly get the system state closer to the sliding surface. If the distance |ds | is small as well as the approaching rate the value of KP should be decreased to eliminate chattering. If the approaching rate is high the KP should be decreased more quickly to avoid crossing of the sliding surface. As soon as the trajectory hits the sliding surface and thus |ds | = 0 the adaptation of KP has to be stopped. The rules are summarized in the Table 1. We use 4 triangular membership functions for the input |ds | and 5 for the rate d˙s , see Fig. 1. For the output of the fuzzy system ∆KP seven equidistantly placed singletons normalized to the interval [−1, 1] are used, see Fig. 2. The inputs are normalized to the interval [0, 1] or [−1, 1], respectively. The membership functions are labelled by linguistic values Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZR), Positive Small (PS), Positive Medium (PM) and Positive Big (PB).

Fuzzified value [−]

∗

1

ZR

PM

PS

PB

0.8 0.6 0.4 0.2 0

if if

s | | < 1, φ s | |≥1 φ

0

0.2

0.4

0.6 |d |; |s|

0.8

1

1.2

s

where φ is the thickness of the boundary layer. Unfortunately, a fixed thickness of boundary layer usually does to provide a satisfactory bahaviour. For narrow boundary layer the chattering persists whereas high values of φ result in slow convergence rate. Solution of this problem consists in using variable boundary layer (Slotine and Li (1991), Lee et al. (2001)), typically tuned by linguistic rules. The drawbacks of the variable boundary layer are very complicated and not transparent composition of the rules such that the convergence of the overall control scheme 542

Fuzzified value [−]

 µ ¶ s s sat = φ s  sgn( ) φ φ

1

NB

NM

ZR

PM

PB

0.8 0.6 0.4 0.2 0 −1

−0.8 −0.6 −0.4 −0.2 0 0.2 dds/dt; ds/dt

0.4

0.6

0.8

1

Fig. 1. Input fuzzy sets: a) on input |ds | and |s| b) on input d˙s and s˙

2014 ACODS March 13-15, 2014. Kanpur, India

1

NB

NS

NM

ZR

PS

PM

rod is loosely fastened in the middle of the cart. The input to the system is a voltage u [V] applied on the DC motor driving the cart, u ∈ [−10, 10]. The output of the system is the angle of the rod φ [◦ ] measured from the downward position. The position of the cart x [m] is measured as well. The goal is to control the pendulum around its unstable upper position.

PB

Fuzzified value [−]

0.8

0.6

0.4

0.2

0

−1

−0.5

0 ∆K ,∆η

0.5

1

p

Fig. 2. Fuzzy sets: a) on input do of SG fuzzy system b) on output of both fuzzy systems Since the switching gain η influences the error convergence rate to the sliding surface the inputs to its adaptation fuzzy rule base were chosen as |s| and its time derivative s. ˙ Otherwise the strategy remains the same as in previous case. The output of the fuzzy system is the change of the switching gain, ∆η. Its value equals to zero if the system is on the sliding surface or is approaching withe rate that is proportional to the value of |s|. If the trajectory is moving too slowly to the sliding surface the value of η should be increased and it it should be decreased whenever the approaching rate is too high in the near vicinity of the sliding surface. The same number, shape and range for the support of membership functions as in the previous fuzzy system were used. The fuzzy rule base is depicted in Table 2. |ds |\d˙s ZR PS PM PB

NB PM PB PS ZR

NS PS PS ZR NS

ZR ZR ZR NS NM

PS NS NS NM NM

NB PM PS ZR ZR

NS PS ZR NS NS

ZR ZR NS NS NM

PS NS NM NM NB

Using the Euler-Lagrange equation the following nonlinear state space equations describing the 4-th order model can be derived: ml K b x˙ + (φ¨ cos φ − φ˙ sin φ) + u M +m 2(M + m) M +m 3g 3 φ¨ = −2δ φ˙ + sin φ − x ¨ cos φ (16) 2l 2l x ¨=−

PB NM NB NB NB

where b [kg. s−1 ] is friction coefficient of the cart, δ [s−1 ] is viscose damping coefficient of the rotational movement of the pendulum, K [NV−1 ] is transfer constant between the voltage applied on the motor and its inferred force and g [ms−2 ] is gravity acceleration.

Table 1. Rule base for adaptation of the extended gain KP |s|\s˙ ZR PS PM PB

Fig. 3. Inverted pendulum on a cart

PB NM NB NB NB

Table 2. Rule base for adaptation of the switching gain η The main drawback of adaptive sliding mode control with adaptation governed by fuzzy logic is that until now only simulations results are available. In this paper the presented method will be applied on control of laboratory model composed from pendulum on a cart. 4. LABORATORY MODEL The laboratory model is composed from a cart that is moving along the x-axis with the mass M [kg] and a rod of the length l [m] and the mass m [kg], see Fig. 3. The 543

The method presented above is compared with sliding mode control with fixed boundary layer method that is known to have a good control performance. Two thickness of the boundary layer, φ1 = 0.5 corresponding to a narrow one and φ2 = 50 corresponding to a broad one were chosen. The initial values of the control gains were chosen as η = 0.33 and KP = 7. The coefficients of the sliding surface were set k1 = 1, k2 = 100 and k3 = 10. Let us remain that the equations (16) were not used for derivation of the controlled law. The responses of the pendulum angle on a step reference signal for the proposed method and classical sliding mode control with narrow and broad boundary layers are depicted in Fig. 4. The corresponding control action is shown in Fig. 5. The adaptation process of both the extended feedback gain and switching gain is depicted in Fig. 6. The results confirm that the tracking performance of the proposed method is similar with the one achieved by narrow boundary layer but with significantly less chattering in control action that corresponds to broad boundary layer.

2014 ACODS March 13-15, 2014. Kanpur, India

a)

a)

190

0.4 0.3

reference signal r SMC with narrow boundary SMC with large boundary layer the proposed method

170

160

η [−]

φ [°]

180

2.2

2.4

2.6

2.8

3 3.2 Time [s] b)

3.4

3.6

3.8

0.1 0 2

4

20

2.5

3

3.5 Time [s] b)

4

4.5

5

2.5

3

3.5 Time [s]

4

4.5

5

10 SMC with narrow boundary SMC with large boundary layer the proposed method

10

5 Kp [−]

error [°]

0.2

0

0 −5 −10

2.2

2.4

2.6

2.8

3 3.2 Time [s]

3.4

3.6

3.8

−10 2

4

Fig. 4. Reference signal tracking performance

Fig. 6. Progress of adaptation

a) control action u [V]

10 SMC with narrow boundary 5 0 −5 −10 2

2.5

3

3.5 Time [s] b)

4

4.5

5

control action u [V]

10 SMC with large boundary layer the proposed method

5 0 −5 −10 2

2.5

3

3.5 Time [s]

4

4.5

5

Fig. 5. Comparison of control actions 5. CONCLUSION The paper presents an application of adaptive fuzzy sliding mode control with adaptation of both extended feedback and switching gain using self-learning mechanism based on fuzzy logic. The results achieved on a real laboratory model of pendulum on a cart confirm better reference tracking performance and decreased chattering of control action of the proposed method in comparison with commonly used sliding mode control with fixed boundary layer. REFERENCES Bao, Y., Du, W., Tang, D., Yang, X., and Yu, J. (2006). Adaptive fuzzy sliding-mode control for non-minimum phase overload system of missile. Intelligent Control and Automation, 344(1), 219–228. Bessa, W., Dutra, M., and Kreuzer, E. (2008). Depth control of remotely operated underwater vehicles using an adaptive fuzzy sliding mode controller. Robotics and Autonomous Systems, 56(8), 670–677. 544

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2014 ACODS March 13-15, 2014. Kanpur, India

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