Sliding Mode Control Synthesis Using Fuzzy Logic - IEEE Xplore

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Center for Manufacturing Research and Technology Utilization. Ali T. Alouani. Department of Electrical Engineering. Tennessee Technological University.
Pmtssdlnpt of lhs Am%dtanConlml Conlenncs Sestllo, Washimpton *June !ODs

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ode Control Synthesis Using

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Ali T. Alouani Mounir Ben Ghalia Department of Electrical Engineering Center for Manufacturing Research and Technology Utilization Tennessee Technological University Cookeville, TN 38505, U.S.A. mbg999iQtntech.edu

[email protected]

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the sliding mode. In other words, state trajectories starting off the slidin surface remain sensitive to parameter perturbations an8 external disturbances. Moreover, the convergence to the sliding surface may only be a s y m p totic resulting in a long reaching time (time needed for the state trajectory to hit the switching surface) during which the advantages of VSC are not achieved.

In variable structure control. algorithms sliding mode plays a crucial role in making the closed ioop system insensitive to modeling uncertainties and external disturbances, and also in transforming the original system dynamics into prescribed reduced order dynamics restricted to a switching manifold on which desired system dynamic behavior is achieved. The control law used to realize the desired slidin mode dynamics is discontinous on the switching manifAd. However, due to imperfections in switching, such as time delays, the system trajector chatters instead of sliding along the switching manifold: This chatterin is undesirable because it may excite unmodeled high-gequency dynamics in the physical system. To overcome this drawback, in this paper, a new sliding mode control algorithm usin concepts from fuzzy set theory and fuzzy logic is dev5oped. The eccentricity of the new sliding mode control a1Forithm resides in its generalized form of switching manifold which allows for a smooth transition between the reaching mode and the sliding mode. To demonstrate its performance, the r p o s e d control algorithm is applied to a one-degree of eedom robot arm.

The second shortcomin is control chattering. In the design and analysis of V S 6 systems, it is assumed that the control can be switched from one structure to another infinitely fast. However, in practice it is impossible to achieve a hi h switching control which is a re uirement for most V f C designs. This is due.to FeverJ reasons. One of them is the existence of smtching time delays resulting from delays in control computation. Another reason is the limitations of physical actuators which cannot handle the switching of control signal at infinite rate. As a result of these imperfections in switching between control structures, the system tra’ector chatters instead of slidin along the switching suriace. *his chattering is undesira%le in practice because it may serve as a source to excite the unmodeled high-frequency dynamics of the system [3].

To overcome the first shortcoming, that is to reduce the

reaching time, the use of high-gain control signal was suggested in [12]. However, this may result in the saturation of the actuator and also in higher chattering henomenon which is undeshable in a physical s stem. 8 n the other hand, a time-varying switching sur%ce was suggested in [lo] in order to eliminate the reaching phase, where the initial tracking control errors were assumed to be zero. However, this assumption rules out many practical situations in which the system initial conditions are located arbitrarily.

1. I n t r o d u c t i o n The problem of designing feedback control for dynamical systems subject to modeling uncertainties and external disturbances has been the focus of an extensive research effort. If statistical characterizations of the uncertainties in the system dynamic and of the external disturbances affecting the system are available, then stochastic control theory [I] is a prime candidate t o this control problem. However, if such statistics are not available, but bounds on the uncertainties and disturbances =re known, then one has to opt for a deterministic control ap roach. Within the deterministic framework, a nonlinear gedback control design method of interest to this paper is based on the theory of variable structure control (VSC) systems [3].

The problem of chattering has been addressed by many researchers. In [lo], the discontinous control is approximated inside a boundary layer located .around the switchin surface. However, although chatterlng can be reduced, ro%ustness and tracking accuracy are compromised.

Variable structure control s stems constitute a class

While research efforts continue in variable structure control theory, another control strategy, called fuzzy logic control (FLC), which uses concepts from fuzzy set theory [13] and fuzzy logic [14], has been attracting the attention of the control research community during the last two decades [4]. Fuzzy logic control has emerged as a paradigm of intelligent control capable of dealing with complex and ill-defined systems [4]. New results have been made recently to identify the connection between fuzzy logic and variable structure control [2,8,9]. It has been shown that fuzzy logic control is a general form of variable structure control. This connection has suggested the integration of the two control approaches in control

of nonlinear feedback control systems whose structure changes depending upon the state of the system. Although, neither structure is necessarily stable, their combination results in a sliding mode, that is, the system trajectory slides along a switching surface (also called sliding surface). A VSC with sliding modes is often called sliding mode control (SMC). An important feature of SMC is that while in sliding mode, the system remains insensitive tQparameter perturbations and disturbances [3]. Despite the benefits of VSC control, the latter suffers from two ma’or shortcomings. First, the insensitivity property of a VdC system i s present only when the system is in

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system design applications [6,8,9]. However, there has not been a unified integration strategy of these two control a proaches. In this paper a new sliding mode control Jgorithm using concepts h o m fuzzy set theory and fuzzy logic is developed. Unlike the studies in [6,8,9],the control approach fjroposed in this paper is based on the mathematical mo el of the system.

take values of either 0 or 1, which means that an object either belongs to or does not belong to a given set, the characteristic function (called membership function in fuzzy set theory) of a fuzzy set can take on vallues in the interval [O,1][13].

This paper is organized

tant concepts of fuzzy lo ic. represents inference rules whose premises contain k z z y propositions. Unlike inference in calssical logic, in its computation inference, approximate reasoning uses fuzzy sets which represents the meanin of a collection of fuzzy propositions. For inand p~ represtance,qet the membership function sent the meanin of a fuzzy proposition "x is A." and the meaninF of an i f t h e n fuzzy rule "if z is A thein y is B", respectively, where A and B are fuzzy sets. Then, one can compute the membership function representing the meaning of the conclusion "y is B" [4]. Now consider a collection of L fuzzy if-then rules:

2.2.2 Fuzzy Logic and Approximate Reasoning: Approximate reasonin is one of the most impor-

It

as follows. Section 2 gives some background on variable structure control with sliding modes and on fuzz logic. Section 3 presents the new slidin mode contro! algorithm. In Section 4, an application o? the above results to a one-de ree of freedom robot arm is illustrated. Finally, Section presents some conclusions and directions for future research.

2. Background 2.1. Basic Concepts of VSC Let a nonlinear dynamic system be represented by the

Rule(') : if

state equation:

2i

= f ( z , t ) -tg ( x , t ) u

input to the system. The VSC design consists in achieving the following steps:: 1) Design a switching manifold S in the state space to represent a desired system d namics, which is of lower order than the dimension of t i e given plant; S is defined by s := {x E R."ls(z) = O}, (2) where s(z> E R is called the switching function. 2) Design a variable structure control ut(x) *U-(.)

when s(z) > 0 when s(x) < 0

P E ( Y ) = maxi min ( ~ A w ( z * ) ,P E ( ' )b)) V 3.

(3)

where y1, represents a crisp value for which the membership function pg(q reaches its maximum, 1 = 1,.. .,L .

3. Sliding Mode Control Using Fuzzy ][logic 3.11. Problem Formulation Consider a system whose dynamics are described by the vector differential equation (1). As discussed in the previous section, the transient dynamics of a VSC s stem consist of two modes: a reaching mode and a sli& mode. The desired sliding mode dynamics are specifief by designing an appropriate switching function s(~). For the reaching mode, the desired system response path is the one that allows to reach the switching manifold S given in (2).

Once the state is can the sliding surface,.the system remains insensitive to arameter perturbations and external disturbances. d e r e f o r e , it is important to determine the condition under which the state will move toward and reach the sliding surface. This condition is called the reaching condition. The s stem trajectory under the reaching condition is calledr the reuchin phase or the reaching mode. To specify the reaching consition a Lyapunov function is used [3]. Let the Lyapunov function candidate be defined as: V(x, I) = +),

(7)

To determine the correspondin crisp value of y, a defuezification procedure is applief to the inferred fuzzy set B [4]. One of the most used defuzzification scheme is the center-of-area (COA) method [4]. Applying COA to (7) yields:

such that any state x outside the switching surface is driven to reach this surface in finite time, that is, the condition s(x) = 0 .issatisfied in finite time. Once on the switching surface, the sliding mode takes lace, following the desired system dynamics. This procesure makes the VSC system globally asymptotically stable [3].

1

A(') then y is B('), 1 = 1 , . . . , L (6)

where A(') and B('),1 = 1,..., L, are fuzzy sets of the variables x and y, respectively. Given a crisp value z* of the variable x, then the fuzzy set B representing y resulting from the firing of the fuzzy rules is given by [4]:

(1)

where z E R" is the state vector of the system, f , g : '.R x R" -+ IZ" are vector fields, U E R is the control

u(t) =

2 is

An important part in the sliding mode control desi n is to specify a control law for the reaching mode such tfat the reaching condition (5) is met. An important characteristic of VSC system is its invariance to system perturbations and external disturbances once in the sliding mode [3]. However, during the reaching phase the system (does not possess the invariance property. Most of current d e s i p techniques of VSC do not specif the reaching dynamics andl they are concerned only wit{ the specification of the reaching condition to make sure that the state reaches the switching surface in finite time. This fact does not allow for the specification of the speed with which the sliding manifold is reached and the control chattering around the switching surface.

(4)

where s(x) is the switching surface and z E R" is the state vector of the s stem. Then the reachin condition for the existence of t i e sliding mode motion ofgthe system under consideration is given by [3]:

2.2. Fuzzy Sets and Fuzzy Lo ic 2.2.1 The Concept of A%zzy Set: A fuzzy set is a generalization of the classical notion of a set.

Since the reaching mode resents an important part of the transient dynamics of a $SC system, special attention has

Whilst the characteristic function of a classical set can

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value of the switching function s. Then, these rules are:

to be given to the specification of the control law which a desired system d namics in the reaching mode. n addition, the control raw has to satisfy the reaching condition, which guarantees the existence of sliding mode on the switching manifold.

Rule Rule Rule Rule

The control of the dynamics of a VSC system in the reaching mode may be made possible by specifying the dynamics of a switching function s(x) [7]. More specifically, the dynamics of the switching function s(x) are described by a differential equation of the form [7]:

i(x) = -Ksgn(s),

K > 0.

1: 2: 3: 4:

if ] s ( x ( t ) ) l i s S L if I s ( z ( t ) ) / i s S M if /s(z(t))l i s SS i f ( s ( z ( t ) ) / i s SZ

then then then then

K ( t )= K L K ( t )= K M K ( t )= K S K ( t ) = KZ

where SL (Large Is!), SM (Medium Isl), SS (Small !SI), and SZ (Zero are fuzzy sets of the variable Is(z(t))l. K L , K M , K S KZ, are different values of the control gain corresponding to a large, medium, small, and zero gain value, respectively.

IS~),

(9)

Note that we no longer need to verif the reaching condition because it is inherent in the Jfferential equation of the function ~(x). By specifying the dynamics of the function s ( x ) , we can predetermine the speed with which the system state approaches the switching manifold; the rate of convergence is given by K.

Given the value of / s ( z ( t ) ) la t the instant time t , then the value of the control gain K a t time t is inferred using the above four fuzzy if-then rules following the same procedure explained in subsection 2.2.2.

After choosing the dynamics of the reachin mode, we now determine the associated control law. Dikerentiating s(x) with respect to time along the trajectory of (1) gives This strategy of selecting the variable control gain K ( t ) has the following advantages over choosing a fixed control gain K : a) A large control gain is applied only when the system state is far away from the sliding manifold. b) When the system state is close to the sliding manifold, a small control gain is used.

Selving(l0) for the control law gives

In the traditional sliding mode control design, the system motion that takes place strictly on the constrained manifold S is called ideal sliding [5]. If the control input can be switched at an infinitely fast rate then an ideal switching mode may be obtained. The condition i ( s )= 0 is necessary for the state trajectory t o stay on the switching manifold S. The control input under the ideal sliding mode is called the equivalent control. The latter is the computed control input such that the state trajectory stays on the switching manifold S. To compute the equivalent control U E we differentiate s(z) with respect to time along the trajectory (1) and we let i ( x ) = 0,

where the existence of the inverse of the matrix [ is a necessary condition.

gg(x)]-’

3.2. Design Considerations The above formulation gives our main framework for the synthesis of sliding mode control. However, there are some design considerations which represent specific guidelines for our control design. These considerations are as follows: a ) The invariance property of VSC systems is present on the sliding manifold (switching manifold) and not during the reaching phase in which the system is affected by the system perturbations and the external disturbances. Therefore it is important to minimize the time required to reach the sliding manifold. b) One way of reducing the reaching time is to apply a very large gain. However, this may not be possible due to saturation problems and also due to the chattering effects which may result. c ) Selecting smaller values of the control gain K results in less chattering while on the sliding mode. Therefore, once the s stem tra‘ectory is about to reach the sliding manifold, should i e kept as small as possible in order to reduce the chattering effects. d) The assumption of saying that the control input can be switched from one value t o another infinitely fast cannot be achieved in practice. This is due to the resence of finite time delays for control computation a n 8 also to the limitations of ph sical actuators. The fact that the control input cannot ge switched a t an infinitely fast rate is responsible for the occurrence of the chattering phenomenon in the sliding and the steady state modes of a VSC system. Consequently, a ”more realistic” sliding mode has to be defined.

as

i ( x ) = -f(z)

ax

Solving (13) for

U

as + -ax g(z)u

= 0.

(13)

yields the equivalent control

Here, again the matrix e g ( x ) is assumed t o be nonsingular.

2

Using the notion of fuzzy sets, we introduce the following defirutions: Definition 1: Fuzzy sliding mode: A Fuzzy sliding mode IS a mode where the system motion takes place on the constrained manifold S with degree of possibility 1 and on the neighborhood [ - e , €1 of the switching manifold with degree of ossibility less than 1. A fuzzy sliding mode is represented gy the fuzzy set Sf,which is characterized by its membership function given by: 1)

3.3. Proposed Sliding Mode C o n t r o l A l g o r i t h m In this section we present a solution to the shding mode control design usin concepts from the fuzzy set theory. In the light of the Jesign considerations given above, the control gain K has to be selected according to the following rules. At each instant time, t , s ( z ( t ) ) is the algebraic

if s(z) = 0 if ~ ( 2 E) [-f, elsewhere

€1

(15)

Deflnition 2: Fuzzy reaching mode: A fuzzy reaching mode, re resented by the fuzzy set S,, is the complementary moae of the fuzzy sliding mode. This means that

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(iii) Compute the reaching control law U R given in (11), where the control gain K is computed as in (12); (iv) Compute the equivalent control law UE given in (14); (v) The overall control gain is given in (118).

fuzzy reaching mode is characterized by its membership function given by: PS,(3(.))

=1 -PS,,W)).

(16)

Figure 1 shows the membership functions of the fuzzy sliding mode and the -fuzzy reachin mode in the case where the sliding manifold is a line. 8 o t e that an ideal sliding mode may be characterized by its membership function given by:

4. Design Example

To illustrate the above design approach, a single-link robot arm is considered. The dynamics of the robot arm can be derived using the Euler-Lagrange equations of motion [Ill,

Characterizing the sliding mode as in (17) is ideal and cannot be achieved in practice. Instead, a realistic characterization of the sliding mode, as given in (15), should be considered.

+

m128”(t) mglsin(@(t)) = u(t), where B ( t ) is the angular position of the robot arm, u ( t ) is the control torque applied to the robot arm, m is the mass of the robot arm, 1 is the length of the arm, g = 9.8(m/secz) is the gravitational constant. If we choose m = 1kg and 1 = I m , and we let z l ( t ) = B ( t ) and z z ( t ) = j ( t ) , then Equation (19) can be put in the following form:

To guarantee a smooth transition of the control law when the system mode changes from the reaching mode to the sliding mode, two i‘uzzy rules are used:

Rule(’) :

i.f s ( z ( t ) ) is S,

then

U

= UR

Rule(’) :

ijms ( z ( t ) ) is Sfs then

ZL

= UE

= zz(t) i Z ( t ) = gsin(m(t)) u(t). The control objective is to maintain 2 1 ( t ) = 0 and z z ( t )= 0. In this design example we choose the switching surface given by s(2) = c21 2 2 = 0. (20) The reaching mode control (11) is given by il(1)

The above fuzzy rules mean that if the system motion corresponds to the fuzzy reachin mode then the control applied to the system is the reackng control UR given in (11). On the other hand, if the system motion corresponds to the fuzzy sliding mode then the control applied to the system is the equivalent control UE given in (14).

+

The degree of correspondence of the actual system motion to one of these modes is iven by the degree of membership obtained by matching &e computed switching function, s ( z ( t ) ) , with the membership functions psv and ps,, depicted in Figure 1.

UR = - ( - g s i n ( Z l )

+ c22 + K s g n ( s ) ) .

(21)

The equivalent control is given by

Using the results of subsection 2.2.2, the proposed overall control law is given by

In simulations, the initial conditions are chosen to be 21(0) = 0 . 2 ( r a d ) and 22(0) = O(rad/sec). The coefficient c was chosen to be equal to 10. The parameter t defined in (15) was chosen to be 0.02. The sampling time was lms.

where U E is given by (14) and U R is given by (11). Note that from Equation (18), the control input gradually switches from UR to U E when the system trajectory is moving towards the sliding manifold. In fact, by referring to Figure 1, the closer the system trajectory gets to the sliding manifold the smaller p ~ , ( s and ) the larger ps,, (s) becomes. When the system trajectory is far from the sliding manifold (see for example the reference point A in Figure 1 ) then ,us,(s) = 1 and ~ s , , ( s ) = 0, only the reaching control U, is active (see Equation (18)). At the reference point B in Figure 1, both control laws, UR and UE,are active. However, the more the state trajectory ets closer to the sliding manifold, the smaller is the contrbution of the reaching control UR to the overall control, until the system reaches the sliding manifold (point C in Figure 1) where the reaching control is turned off completely and the system is totally under the equivalent control UE.

For traditional sliding mode control the control gain K was kept constant and equal to 5 ( N m ) . For the ;proposed control al orithm, the membership functions of the fuzzy SS, S M , and SL are shown in Figures 2. subsets The levels of control gains for the proposed algorithm are K L = 5.O(NM), K M = 2.O(Nm), KS = 1 . 5 ( N ” ) , and KZ = 0.5(Nm). Note that for comparison reasons, we have chosen the maximum control gain to be same for the traditional and the proposed sliding mode control. Fi ure 3 shows the simulation results using traditional sliting mode control. The simulation results of the application of the pro osed control design ap roach are depicted in Figure 4. is clear from the simufations results that the proposed control approach reduces the chattering while maintaining a very small tracking error.

5’5,

fi

5. Conclusions

In Summary, the proposed slidin mode control algorithm using fuzzy logic consists of the following steps: m

+

In this paper, sliding mode control has been develsoped using a fuzz logic approach. T h e proposed control 1s characterized its sim licity and its capability in alleviating the chattering usu&y resent in traditional sliding mode control. The notion offuzzy sliding mode is introduced since ideal sliding mode cannot be realized in practice. Future work will address issues of quantificatioln of the chatterin reduction and the ap lication of the proposed control &orithm to more c o m p f k systems.

&

(i) Design a switching manifold S in the state space to represent desired system dynamics as in classical

vsc;

(ii) Construct the sliding mode fuzzy set Sf,and the reaching mode fuzzy set S, (see Figure 1);

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References [ll - - Astrcm, J. K., Introduction to Stochastic Control Theory, Academic Press: New York, 1970. [2] Ben Ghalia, M. and A. T. Alouani, "Analysis of the mechanism of fuzzy controllers," Technical Report, MCTR, Tennessee Tech. University, 1994. [3] DeCarlo, R. A., et al, "Variable structure control of nonlinear multivariable systems: a tutorial," Proceedings of the I E E E , vol. 76, no. 3, pp.212-232, 1988. [4] Driankov, D. et al., A n Introduction to Fuzzy Control, Springer-Verlag: New York, 1993. [5] Itkis, U., Control Systems of Variable Structure, John Wiley & Sons: New York, 1976. [6] Lin, S.-C. and C.-C. Kung, "A linguistic fuzzysliding mode controller " em Proceedings of the American Control Conference, Chicago, ILL, pp. 1904-1905, 1992. 171 Morgan, G. R., and 0.ezgiiner, "A decentralized variable structure control algorithm for robotic manipulators," I E E E Journal of Robotics and Automation, vol. 1, no. 1,pp. 57-65, 1985. [8] Palm, R., "Sliding mode fuzzy control," Proceedings of the I E E E International Conference on Fuzzy Systems, San Diego, CA, pp. 519-526, 1992. Palm, R., "Robust control by fuzzy sliding mode," [9] Automatica, vol. 30, no. 9, pp. 1429-1437, 1994. [IO] Slotine, J. J., and S. S. Sastry, "Tracking control of non-linear systems using slidin surfaces with ap lications to robot manipulators, Int. fournal of Controy vol. 38, pp. 465-492, 1983. [Ill Spong, M. W. and M. Vidyasagar, Robot Dynamics and Control, John Wiley 8z Sons: New York, 1989. [12] Young, K.-K. D., "Controller design of a m a n i p ulator using theory of variable structure systems," I E E E Trans. Syst. Man, and Cybern. vol. 8 , no. 12, pp. 101-109, 1977. [13] Zadeh, L A . , "Fuzzy sets," Information and Control, vol. 8, pp. 338-353, 1965. [14] Zadeh, L. A., "Fuzzy algorithms," Information and Control, vol. 12, pp. 94-102, 1968.

,

Tlme

Time (seconds)

(seconds)

Time (seconds)

Angular Poiition (Radian)

Figure 3: Simulations results using traditional sliding mode control. Applied Control Torque

......:................... W

..... a

i .

.

0

1 Tlme

C B

A

1

2 3 (stconds)

4

Angular Vdocity of the Robot km I

I

I

Figure 1: Fuzzy sliding and reaching modes.

I

P

o

1 2 3 Time (sdconds)

1> f

k!

Phase Portrait

..................

Q

U

h c

..... .... ....

z8 -OS 7

Time (second$)

P

0

O,l

Angular Position (Radian)

4.)

Figure 2 : Membership functions for the switching function ~(z).

Figure 4: Simulations results using the proposed sliding mode control.

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