Sliding mode fuzzy control

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lmax. Nu = K ~ u t t N lmax '. The FC with partial compensation has the form. U = -Ad - KFUzE(e, d, .... Furthermore] assume that sP = eNT . nN is the projection of ...
Sliding Mode Fuzzy Control Rainer Palm Siemens AG Corporate Research and Development Dept. ZFE ST SN 4 Otto-Hahn-Ring 6 8000 Munich 83 Germany Abstract Most Fuzzy Controlers (FCs) for nonlinear 2nd order systems are designed with a two-dimensional phase plane in mind. We show that the performance and the robustness of this kind of FCs stems from their property of driving the system into the so-called sliding mode (SM) in which the controlled system is invariant to parameter fluctuations and disturbances. Additionally, the continuous distribution of the control values in the phase plane causes a behavior similar to that of a sliding mode controler (SMC) with a boundary layer (BL) near the switching line. This gives assured tracking quality even in the presence of high model uncertainties. Further improvement can be obtained by the introduction of a boundary layer at the FC. Tracing the FC back to the principle of an SMC one obtains evidence about the stability of the closed-loop system. The choice of the scaling factors for the crisp inputs and outputs can be guided by the comparison of the FC with the SMC and with the modified SMC, respectively. At the end of the paper, an FC for a higher-order system is proposed.

1

Introduction

Fuzzy controlers are usually applied to highly non-linear systems with large parameter fluctuations. Commonly, for a large class of nonlinear systems FCs are designed with respect to the phase plane constituted by error and change of error of the respective state [Ray 84, Tang 87, Wakileh 881. A fuzzy value for the manipulated variable (control output) is attached to error and change of error depending on the location of the states in the phase plane. The general approach is the division of the phase plane into two semi-planes by means of a switching line. Within the semi-planes positive and negative control outputs are produced, respectively. The magnitude of the control outputs depends on the distance of the state vector from the switching line. Considering the large field of FC applications one is tempted to ask why this control method is so successful. In this paper, this is answered for a certain class of systems and FCs as follows: For a definite class of nonlinear systems there is an appropriate robust control method called sliding mode control [Utkin 77, Slotine 851. This control method can be applied very well in the presence of model uncertainties, parameter fluctuations and disturbances provided that the upper bounds of their absolute values are known. The sliding mode control (SMC) is especially appropriate for the tracking control of robot manipulators and also for motors whose mechanical loads change over a wide range. The disadvantage of this method is the drastic changes of the manipulated variable. However, this can be avoided by a small modification: a boundary layer is introduced near the switching line which smoothes out the control behavior and ensures the states remaining within the layer. Given that the upper bounds of the model uncertainties etc. are known, stability and high performance of the controlled system are guaranteed. In principle, Fuzzy controlers work like modified SMCs. Compared to ordinary SMCs, however, FCs have the advantage of still higher robustness. In the following, the structure of an FC is derived from a non-linear state equation representing a large class of physical systems. Furthermore, the following aspects are discussed: stability conditions, scaling (normalization) of the state vector, choice of the switching line and determination of the break frequencies of the controler. By the choice of an additional boundary layer in the phase plane the FC is modified so that drastic changes of the manipulated variable

0-7803-0236-2 /92 $3.00 0 1992IEEE

519

can be avoided especially at the boundary of the normalized phase plane. In this context, the higher robustness of the modified FC over the modified SMC is discussed. Finally, the method is extended to the n-dimensional case.

2

Sliding Mode

Let

d n ) ( t )= f (x,t ) -tU

+ d(t)

with

x ( t ) = ( I ,k,...)2("-1))T

x ( t ) - state vector;

d ( t ) - disturbances;

U

- manipulated variable.

Furthermore, let

f(x,t ) = f(x7t ) + Af(x, t ) be a nonlinear function of the state vector x and of time t where A f - model uncertainties;

f - estimate of f .

Let furthermore A f , d and zl;"'(t) have upper bounds with known values F , D and

lAfl I F(x,t);

Id1 I D(x,t);

1$'(2)1

I v(t).

The control problem is to obtain the state x for tracking a desired state uncertainties and disturbances. With the tracking error

xd

e = x ( t ) - x d ( t )= ( e , e, ..., e("-l)IT

U:

(3) in the presence of model

(4)

a sliding surface (switching line for 2nd order systems)

s(x,t ) = 0 s(x,t) = (d/dt

(5)

+ X)("-')e;

X20

is defined. Starting from the initial conditions

e(0) = 0

(7)

the tracking problem x = xd can be considered as the state vector e remaining on the sliding surface s ( x , t ) = 0 for all t 2 0. A sufficient condition for this behavior is to choose the control value so that 1 d 2 . -dt( s 2 ( x 7 t ) ) 5 -77

-

*

IsI;

77 2 0

Considering s2(x,t ) a Lyapunov function, it follows from eq.(8) that the system controlled is stable. Looking a t the phase plane we obtain: The system is controlled in such a way that the state always moves towards the sliding surface. The sign of the control value must change at the intersection of state trajectory e ( t ) and sliding surface. In this way, the trajectory is forced to move always towards the sliding surface (see fig. 1). A sliding mode along the sliding surface is thus obtained. By remaining in the sliding mode of eq.(8) the system is invariant despite model uncertainties, parameter fluctuations and disturbances. However, sliding mode causes high control activities which is an evident drawback for technical systems. Returning to eq.(8) one obtains the conventional notation for sliding mode s .s

5 -7. Is1

or alternatively

S. sgn(s)

5 -7.

(9)

In the following, without loss of universality, we focus on 2nd order systems. Hence, from eq.(6) follows s

= Xe + e

and

S = Xe + e = Xe + x - x d .

520

From this and eq.( 1) follows

Rewriting this equation leads to [f(X,t )

+d +

- z d ] sgn(s) + U sgn(s) 5 -7.

(11)

To achieve the sliding mode of eq.(9) we choose U so that U

=

(-i- Xe) - K ( x ,t ) . sgn(s)with K ( x ,t ) > 0

(12)

where (-f - Xe) is a compensation term and the 2nd term is the controler. With this, eq.(ll) can be written as (f(X, t ) - f(& t ) d ( t ) - z d ) sgn(s) - K(& t ) 5 -v. (13)

+

Installing the upper bounds of eq.(3) in eq.(13) one obtains finally

K ( x ,t),,, L F

+ D + + v. 21

(14)

To avoid drastic changes of the manipulated variable mentioned above, we substitute function sgn(s) by s a t ( s / @ ) in eq.(12), where if 1x1 < 1 sat(x)

=

This substitution corresponds to the introduction of a boundary layer (BL) Is1 5 0 (see fig.1). Thus, we have U = -f - X i - K ( x ,t),,, . s ~ ( s / @ ) where @ > 0; K ( z ,t)ma, > 0. (15) From this follows with eq.( 10) and (13) the filter function

for unmodelled disturbances, model fluctuations and the acceleration of the desired values Ed for the input. The output s of the filter is the distance to the switching line. With the slope X of the switching line s = 0 one obtains the guaranteed tracking precision

e=

@ x.

The break frequency of filter (16) yields

Figure 1: Sliding mode principle with boundary layer

521

On the other hand, even eq.(6) is a filter with s as input and e as output. Hence, the break frequency X of this filter should be, like U@, small compared to unmodelled frequencies v~,,: A