An adaptive fuzzy sliding-mode controller - IEEE Xplore

18

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 1, FEBRUARY 2001

An Adaptive Fuzzy Sliding-Mode Controller Ralph G. Berstecher, Rainer Palm, Member, IEEE, and Heinz D. Unbehauen, Fellow, IEEE

Abstract—This paper deals with a new adaptive fuzzy sliding-mode controller and its application to a robot manipulator arm. The theory for this approach and for the heuristics-based linguistic adaptation is presented, and a mathematical description is derived. Furthermore, an application of this adaptive controller for a two-link robot arm is shown. The obtained results show the high efficiency of the new controller type. Index Terms—Adaptive control, fuzzy control, robotics, sliding mode.

NOMENCLATURE State vector. Switching variable of the fuzzy sliding-mode controller (FSMC). Desired state vector. Error vector. slope of the switching line Manipulated variable. Optimal manipulated variable. Degree of membership. Moment and area of the membership function, respectively. th supporting point. Adaptation vector. Point in state space. Linguistic values for , and in the th rule. Characteristic of the FSMC. switching variable in the adaptation loop. Slope of the adaptation line . Correction terms for , and , respectively. State vector at a distance from . Optimal parameter vector. Model uncertainties of , respectively. Lyapunov-like function. Width of the boundary layer.

I. INTRODUCTION

I

N THIS PAPER, a linguistic heuristics-based adaptation algorithm for an FSMC is presented. The algorithm relies on linguistic knowledge in the form of fuzzy IF–THEN rules which reflect how an experienced operator would adapt the controller in order to obtain desired closed-loop behavior. The major reason for the choice of an adaptive FSMC is to better cope with changing system dynamics, unknown model uncertainties, and disturbances. This combination of an FSMC with an adaptation block is superior to, e.g., a pure sliding-mode controller plus integrator. Since the latter increases the system’s order by one, the closed-loop behavior is more sensitive to fast changes of the system parameters and disturbances. The adaptation parameters are grouped into the parameter vector . The linguistically defined adaptation law uses the switching variable of the FSMC, its time derivative , and the manipulated variable in order to compute a resulting adaptation parameter change . The analytic description of the adaptation law reveals similarities to conventional control theory and allows one to relate uncertainties in the system description with the expected adaptation parameter convergence. Then, the heuristics-based linguistic description of the adaptation law is analyzed with conventional methods and stability and convergence properties are studied via Lyapunov’s second method. In Fig. 1, the structures of the underlying FSM control and FSM adaptation loops are shown. In Section II, the basic principles of FSMCs are briefly reviewed (see [7] and [8]). In Section III, the heuristics-based motivation for the linguistic adaptation law is presented and an equivalent mathematical description is derived. The latter is used in Section IV to analyze stability properties of the closed-loop system, including the adaptation block, and the convergence properties of the adaptation law. In Section V, additional aspects of the adaptation algorithm are discussed. In Section VI, guidelines for the implementation of the adaptive FSMC are presented. In Section VII, the adaptation law is applied to the control of a simulated two-link robot arm. Section VIII presents real-time experiments for a robot arm [2]. Section IX presents conclusions. II. FSMC

Manuscript received June 6, 1999; revised October 1, 2000. Abstract published on the Internet November 15, 2000. R. G. Berstecher was with Corporate Technologies, Siemens AG, 81730 Munich, Germany. He is now with Goldman Sachs International, London, EC4A 2BB, U.K. R. Palm is with Corporate Technologies, Siemens AG, 81730 Munich, Germany. H. Unbehauen is with the Control Engineering Laboratory, Faculty of Electrical Engineering, Ruhr-University Bochum, 44780 Bochum, Germany. Publisher Item Identifier S 0278-0046(01)01115-7.

In the following, we consider a nonlinear th-order system of the form (1) Let

0278–0046/01$10.00 © 2001 IEEE

and the state has to follow the desired trajectory

BERSTECHER et al.: ADAPTIVE FSMC

Fig. 1.

19

Control and adaptation loop.

TABLE I RULE BASE FOR AN FSMC Fig. 2. Nonlinear transfer characteristic of the FSMC.

(2)

the choice of the switching variable and the nonlinear interpolation in the boundary layer. The shape of the nonlinear transfer characteristic of this type of FSMC depends not only on the values of , but also on the membership functions of the rule antecedents (IF parts) and consequents (THEN parts) and the defuzzification method. In represents the transfer characteristic of the what follows, FSMC and describes the dependence of the manipulated variand the adaptation parameable on the switching variable that are to be defined later. ters By choosing the center-of-sums defuzzification method, the output of the FSMC is computed as

(3)

(4)

. Furthermore, define the switching variable

with

represents a switching surface in the state space. is a parameter specifying and therewith the dynamics of the closed-loop behavior according to [9]. to Fuzzy controllers which use the switching variable calculate the manipulated variable belong to the family of FSMCs [7]. Their fuzzy rules have the form given in Table I. denotes negative, positive, zero, big, In Table I, medium, and small linguistic values whose meaning is defined by corresponding membership functions. The manipulated variable is the output of the FSMC while the switching is its input. variable A boundary layer is introduced for conventional sliding-mode controllers in order to reduce chattering. The output value is linearly interpolated between the positive and negative control value in the boundary layer. However, the robust control properties of sliding-mode controllers are maintained [9]. That is, the attractiveness of the boundary layer is guaranteed for given upper bounds on disturbances and modeling uncertainties. Furthermore, starting inside the boundary layer a tracking precision is guaranteed where depends on the width of the boundary layer. In our case, the FSMC is described by

where is the degree to which the antecedent of the th fuzzy and rule is satisfied by a particular crisp value of , and are the moment and area of the membership function defining the linguistic value of in the th rule. The point at which a triangular membership function has points , value 1 is called a supporting point. Here, the , with membership 1 to the membership functions defining the linguistic values of , are the supporting points diagram (see Fig. 2). At the th supporting in the rules have degrees of point the antecedents of the other satisfaction equal to zero and therefore these rules do not fire. , Thus, the output of the FSMC at the th supporting point according to (4) and the th rule R in Table I, is

(5) lie on the nonlinear transfer charThe points diagram. acteristic of the FSMC in the

20


III. LINGUISTIC ADAPTATION LAW A. Motivation for the Linguistic Adaptation Strategy It has to be emphasized that, because of the costly computation of the inverse dynamics of the system, the control law (4) does not include any feedforward compensation for the system dynamics. Instead, an adaptation of the parameters of the FSMC with respect to changing system dynamics and disturbances is developed. The adaptation strategy is defined by heuristics-based linguistic rules describing how an expert would adapt the FSMC in order to achieve a desired closed-loop behavior [1]. Such a description by fuzzy rules allows an obvious and easy representation of the adaptation law. of every fuzzy rule is adapted at its correThe output sponding supporting point. At such a point the output of the FSMC can be calculated directly, i.e., without interpolation. The nonlinear transfer characteristic is represented by the paramat the supporting points following Fig. 2. eter values supporting points are The adaptation parameters for the grouped together in the adaptation vector (6)

Fig. 3.

Change of the location of the output membership functions.

of the FSMC, Depending on the adaptation parameters the resulting parameter adaptation is linear or nonlinear. If the output of the FSMC depends linearly on the vector , then the output can be described by (7) is a nonlinear function of the where the vector function error vector . In order to deal with a linear adaptation problem in , the output value of the FSMC is only adapted at the supporting by changing the moments in (4). A change in points the areas , however, would result in a nonlinear adaptation problem. The vector of adaptation parameters then is (8) By choosing the membership functions of the consequents as singletons, the moments are equal to the location of the respective singletons. If the membership functions of the consequents are described by triangular membership functions such for each in the whole domain, that then the shift of a supporting point would result in a change of the moment as shown in Fig. 3. In turn, the areas of the neighboring membership functions would be altered. Therefore, the output value of the FSMC would depend nonlinearly on the parameters and (7) would then approximate the “correct” output value. If the membership functions of the consequents are realized by triangluar membership functions, then by changing only their location and keeping their area, the output is given by (7). The desired closed-loop behavior is formulated by an expert at the supporting points, where the adaptation rules for every supporting point are put together into a rule base corresponding to this supporting point. of the The adaptation strategy changes the th output in a manner that FSMC by adapting its corresponding depends on the desired closed-loop behavior. For systems of

Fig. 4. Representation of the linguistically defined adaptation algorithm in the phase plane.

second order in the -phase plane, the point in Fig. 4 (see the vector from the origin to the error trajectory) . In Fig. 4, the switching line represents the error , the distance between and the switching line, are shown. and its time derivative The adaptation law uses the following: of from the switching surface in • distance accordance with (2); of to the switching surface • the approach velocity ; • manipulated variable ; in the antecedents in the fuzzy rules describing the linguistic adaptation strategy. Based on these, each fuzzy rule computes a parameter change for the th component of the parameter vector . The rules R , defining the adaptation strategy for at , thus, have the form R

If

is

and

then

is

is

and

is (9)


, , , and represent the linguistic values for the variables involved in the rule antecedent and consequent, respectively. Each such linguistic value is defined by a membership function. The switching variable is calculated according is the same for all the rules that constitute the to (2) and (3). th rule base. These rules are responsible for the adaptation of the FSMC at the th supporting point. The aim of the adaptation rules is to achieve an optimal approach velocity to the switching : an approach velocity which is too slow or even surface vanishing must be avoided. On the other hand, at small distances , the approach velocity to the switching surface should not be too high in order to prevent undesired chattering. As already mentioned, fuzzy controllers which calculate by using the switching variable the manipulated variable are FSMCs. Fuzzy rules that calculate the derivative of the switching variable by using the switching variable and the manipulated variable characterize the closed-loop system behavior. In [4] and [10], adaptation rules are derived and . In contrast, we derive a from the assessment of from , parameter change of the adaptation parameters , and , according to (9). Such fuzzy rules represent the desired closed-loop behavior in a heuristics-based linguistic manner since they heuristically relate linguistically expressed system states, , to linguistic values of the control effort and the system output [3]. Thus, the FSMC performance is enhanced by a linguistic adaptation law. At the same time, the transparency of the controller design is maintained. Observe here that, in order to avoid local minima, the adaptation of the . supporting points is restricted to B. Description of the Linguistic Adaptation Algorithm with Conventional Control Methods A deliberate change of the controller parameters forms the basis of the adaptation algorithm. The values themselves have immediate impact on the manipulated variable of the FSMC. The desired closed-loop behavior depends on the set of fuzzy rules from (9). The fuzzy rules calculate, depending on the desired behavior at the supporting points, a change of . For the adaptation of each parameter , a separate fuzzy rule base and, therewith, adaptation strategy is defined. only at the supporting By influencing the parameters , coupling effects between the different parameters points does are avoided. Fig. 5 illustrates that a change in by , not alter the output of the FSMC at the supporting point . The switching variable characterizes one supporting point of the transfer characteristic and, therefore, is identical for all adaptation rules. By representing , and as in (9) and omitin the lookup table representation from Fig. 6, the simiting larity of the adaptation algorithm with the structure of an FSMC becomes apparent. This fuzzy rule base has the structure of an FSMC because, as shown in Fig. 6, in the cells parallel to the main diagonal (cells with values Z), the same output value is always chosen. This FSMC in the adaptation loop is used here for the adaptation of the FSMC in the underlying control loop. The FSMC in the adaptation loop is used for the adaptation of the param. The index indicates the switching variable chareter

21

Fig. 5. Dependence of the transfer characteristic of the FSMC on the parameter value r at s .

Fig. 6. Linguistically defined adaptation algorithm for f_ where the antecedent “s is PB ” from (9) is omitted.

acterizing the adaptation at the th supporting point whereas the index characterizes the switching variable of the FSMC in the comes into play control loop. Since the switching line with respect to the adaptation of the parameter , it is called an adaptation line in the following. The adaptation line can—in analogy to —analytically be described by (10) indicates the slope of the In the above expression, , and defines a point on the adaptation line . The point on the adaptation adaptation line represents a pair of reference values of and at line which the parameter is not adapted. Taking into account the , we obtain the set of all points at which no slope adaptation is performed. Because for points on the adaptation holds, we have line (11) Examined analytically, the adaptation rules define a and (local) switching line that is given by the parameter . The adaptation rules following reference points and at the supporting from (9) shows the combination of points at which is to be adapted. The choice of the parameters , and then is driven by the desired closed-loop behavior at the th supporting point as well as the characteristics of the underlying system. The linguistically expressed heuristic knowledge about the process is incorporated here by choosing and . The choice of the the membership functions for

22

Fig. 7. Adaptation line of the ith fuzzy rule base in the (s_


; u)

plane.

membership degrees at the th supporting point corresponds to , , and the slope of the adaptation line the parameters . The variable is calculated by differentiating (2) and a consequent substitution in (1). That is, (12) and are estiThe unknown nonlinear functions and . Equation (12) describes a straight line mated by -plane which is also called the system for and in the for line. Fig. 7 represents a moment at which and is supposed to hold. At the supporting point, represents the optimal parameter with re. Conspect to the adaptation criterion, which is given by sider now the intersection point of the adaptation line from (10) with the system line from (12). This intersection , the optimal compoint then represents, at supporting point and . For the intersection point, the control pabination of is not adapted because is equal to rameter (follows from (10)). Then the maa parameter change . The nipulated variable at time has its optimal value equation

to (13), is equivalent to the optimal parameter value . and , in (12) , , , For variable and therewith also change. and the desired Starting from a variable state vector , in (10) means that the closed-loop behavior parameter value is optimal and does not need to be adapted, . When and in (10) change over time, because then even for fixed parameters the values of the switching also change. A changing then leads variables even if parameter is to variable parameter values with . These changes of the paoptimal at rameters, however, are not the result of an intentional parameter and . In the neighborhood of change but are due to and do not vary too much. does not vary too much For this reason, the parameter differs from , the more difficult it is either. The more and . In order to estimate the effects from changes in to reduce the undesired parameter changes, the membership ” should only have a small support functions of “ is . The effect of the around the respective supporting point parameter fluctuations then is not completely cancelled but is instead significantly reduced. on and , In order to eliminate the dependence of and are subsequently replaced by state-dependent and, therewith, time-dependent correction terms and . Formally speaking, the adaptation line for the is described by variables and (15) The correction term

(16) Here,

(17)

(13) of the parameter vector at implicitely expresses at time . At this supporting point, the supporting point , according to (4), do not have to be conparameters , sidered. This is illustrated in Fig. 5. components , As mentioned before, each of the , of the parameter vector of the FSMC is adapted by its own fuzzy rule base. For the adaptation of the parameter vector , its velocity vector, given as (14) is used. In the preceding, it was shown how the parameter values are adapted for at the supporting points , . The intersection of system and adaptation lines which, according represents the optimal control value

is defined as

has the time derivative

(18) is a state vector at a distance from The state vector and thus corresponds to the th the switching surface is constant. supporting point. For constant , , represents a traFor variable from the switching jectory of the error vector at a distance . In the sequel, time dependencies are not given surface in order to enhance representational clarity. and are estimated by a model of the plant. Therefore, for the calculation of the correction terms an estimated


model of the plant is included in the form of and . That is,

23

,

In order to enhance the representational clarity of (24) and to be able to find upper bounds for the parameter uncertainty, the term

,

(19) (25)

(20) Thus, is obtained from (18) by taking into account (19) and (20). Namely,

is added to the right-hand side of (24). The terms in (24), characterizing the behavior of the controlled plant at the supporting , can now be grouped together as points

(21) Substituting (21) in (16), we obtain

(26) where

(22) The correction term

is the estimated value of at the th supporting point is the optimal manipulated control value at the same and supporting point

is calculated by (23)

(27) Equations (22) and (23) represent the coordinates of the point , through which the adaptation line with slope is drawn (see Fig. 7). The membership values are also shown for , and along the axis. The rule conse-plane for “ is .” The quents of (9) are given in the and the system line [see (10) and (12)] adaptation line plane for . The arrows show are drawn in the and are adapted through and the direction in which for fixed and . The new correction terms are dependent on: 1) the state vector ; 2) the parameter vector ; and 3) the estimates and . Then, is calcuthe switching variable of the adaptation line lated as

At supporting point parameter vector of (26)

holds. Therefore, the optimal corresponding to (10) is a solution (28)

is reached, then When the optimal parameter vector holds and, in particular, . Therefore, for the parameter adaptation is done. From (24) and (25) and taking (26) into account, we obtain

(29) The model uncertainties with respect to by

and

are described

(30) (31) (24) From (29) it then follows that For the adaptation, the closed-loop behavior is important. Again, the model of the plant is incorporated in the adaptation process by using and which can be identified through an ordinary identification algorithm.

(32)

24


In (32), the model uncertainties in the form of and are related to the deviations of the manipulated variable from the optimal manipulated variable value . Inexact estimates in the form of (30) and (31) . The adaptalead to an adaptation switching variable . tion algorithm adapts the parameter value , such that This parameter value then is optimal for the given system description. If a system identification block is used in order to estimate and , then the adaptation law is an indirect one. If no such block is used, then the adaptation scheme is said to be direct. IV. STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM AND PARAMETER CONVERGENCE Parameter is adapted via by using the switching variable according to (32). All parameters, to , of the parameter vector are adapted at their corresponding supporting points. The adaptation implicitly refers to the plant model at the . supporting points The convergence of the adaptation parameters and the stability of the closed loop including the adaptation block are treated with a special Lyapunov-like function [11], [5]. The positive-semidefinite function is chosen as

(33) is always stable refering to (2). The switching surface ensures that the parameter conAccording to (32), . For the latter, verges to its optimal value holds at the th supporting point. In order to deal with a Lyapunov-like function, the time derivative must obey

Fig. 8.

Direction of the error in the phase space.

or sgn

(35)

then the boundary layer of the FSMC is a region of attraction. If for the manipulated variable of the FSMC we have that the inequality from the following holds:

sgn sgn

then the boundary layer is a region of attraction. For the coordinates of point in the plane, the two following differential equations hold:

(36)

-phase

(34) If each term in (34) is negative, then the above is guaranteed. . The control law is realized by an FSMC with A. Closed-Loop Stability Properties of the FSMC For the derivative of the switching variable cording to (12)) that

it follows (ac-

If for the switching variable we have that either one of the following two inequalities holds:

d d

(37)

drives the vector The manipulated variable to the switching surface and makes it then move to the origin -phase plane. From it of the . follows that The first equation of (37) stands for the movement of point toward the switching surface . The second equation of (37) describes the movement of parallel to the switching . surface The resulting movement of is then given by the superposi-phase plane as tion of the two velocity components in the shown in Fig. 8. Outside the boundary layer of the FSMC, the component usually dominates the approach velocity to the switching surface. In the boundary layer, the manipulated variable is reduced thus slowing the approach velocity . Hence, the imporincreases. tance of the component parallel to


25

B. Convergence of the Adaptation Parameters

C. Choice of the Adaptation Law for Exact Modeling

The transfer characteristic of the FSMC is adapted at the supporting points. The convergence of the adaptation parameis negative such that ters is guaranteed if each term is negative [see (34)]. Each term is negative if the following inequality holds:

In order to simplify the representation, first the case of exact modeling of the plant (1) is considered. According to (30) and (31), holds. The switching variables are obtained in this case from (32) as (42)

(38) For the time derivative of condition follows:

By differentiating (32), we obtain

, by the use of (38) the following

sgn d d d

sgn d (43) (39)

in The term (39) represents the influence of the system uncertainties on the switching variable of the adaptation. These uncertainties are the result of inexact or insufficient process knowledge in the form and . The term of

For , an adaptation law has to be chosen such that (43) holds has a negative value in (34). and the term , only the th component of At the th supporting point for is of interest because all other parameters , do not have any influence on . First, we obtain from (4)

in (39) represents the effect which the uncertainty has d d on the parameter adaptation. The term in (39) describes the result of the time-dependent change of the estimated input coefficient . Finally, the term

The dependence of the FSMC on the parameter vector

(40) (44) in (39) describes the dependence of the FSMC on the chosen parameters and the adaptation law at the th supporting point . From (38) and taking into account (39), the following convergence condition follows:

at the th supporting point

is given by

(45) For (40), we then obtain d d sgn (41)

(46)

26


Then, at , only the th component of the velocity vector in the adaptation law is of interest. Equation (43) then becomes

The transfer characteristic of the FSMC is strictly monotonously increasing between the supporting points. If at the same time the manipulated variable at the th supporting point obeys

sgn d d sgn

(51)

(47) for

changes For slowly time-varying reference trajectories, d can be neglected for such reference slowly, i.e., d trajectories. is then chosen with the signumThe adaptation law for as function

(48)

are such that has a negative The positive constants value in (34) and the adaptation parameters converge. Using the vector functions , sgn sgn , and with = and = diag the diagonal matrices , , the adaptation laws

then we have (52)

Then, the part of the boundary layer described by is a region of attraction. The same holds for all the other supporting points. With this approach, the effective width of the boundary layer can be reduced. If the plant from (1) is known, then by (38) the convergence of the adaptation parameters to their linguistically defined optimal values is guaranteed. If (52) holds, then the manipulated variable is such that it also guarantees the stability of the boundary layer. E. Stability and Convergence Analysis for Approximate Modeling According to (32), the adaptation algorithm for parameter refers to the switching variable : the parameter values are is reached. Referring to (32) and for changed such that we obtain

(49) (53)

can be grouped together. D. Closed-Loop Stability Behavior in the Boundary Layer of the FSMC The adaptation law at the switching variable that

is defined in such a way so that for at the th supporting point we have

(50) The parameter value is adapted by (48) so that the term of (47) is negative at the supporting points. Sliding-mode controllers can be used for the control of a stationary operating point as well as for trajectory following. The does not enter the stability considdesired control value erations explicitely because sliding-mode controllers are supposed to switch instantaneously. However, the upper bound of the manipulated variable of the sliding-mode controller deas can be seen from (36). pends on the th derivative of changes, the bigger the manipulated value has The faster can to be in order to guarantee stability. If a quasi-static be assumed, then a smaller value of the manipulated variable is sufficient for plant stabilization.

Equation (53) can be interpreted as a balance condition. In this equation, model uncertainties are related to deviations of from the optimal value . means that for a model is matching the real plant, the optimal parameter vector reached. According to (53), the model uncertainties and will necessarily lead for to a deviation of the parameter value from the optimal value . describes the effect of the model The design parameter uncertainties in the balance condition. From (53), we obtain

(54) to states the importance The relation of of the model uncertainties for the deviation of the adaptation pa. The bigger is, the rameters from their optimal values stronger the undesired effect of the model uncertainties feeds. In case the plant is completely known, the parameter value converges to its optimal value. Equation (53) identifies a relation between the model-plant match quality on the one side and convergence behavior on the other side. The better the match, the


27

VI. ADAPTIVE CONTROLLER IMPLEMENTATION

smaller the maximally expected parameter uncertainties are. The term

A. Design Steps for the Controller and the Adaptation Algorithm (55)

via a local, linguistically deresults in the adaptation of at fined adaptation criterion. It enables the adaptation throughout is the whole definition space of . Also, convergence of guaranteed. For a different initialization of the state vector and the control parameter vector , the correction term in (24) ensures that the always same parameter value is obtained after convergence is reached. V. REMARKS ON THE ADAPTATION ALGORITHM A. Parameter Chattering and Implementation of an FSMC in the Adaptation Loop During adaptation, a chattering of the adaptation parameters occurs according to (48) and(49), respectively. The parameter chattering is the result of a too-high approach velocity of the parameters and the inability of the FSMC to realize infinitely fast switching. Therefore, the sliding-mode controller in the adaptation loop [see (48) or (49)] is substituted by is defined by an FSMC. The maximal parameter velocity the above adaptation laws. Between these, the parameter adaptation rate in the boundary layer of the FSMC is interpolated. This structure was already defined through the linguistic adaptation algorithm. During the analytical examination of the adaptation law, however, first a condition was derived to guarantee convergence of the adaptation parameters, starting from the adap. The resulting sliding-mode controller, tation line however, produces undesired parameter chattering. Therefore, as with classic sliding-mode controllers, the sliding-mode controller in the adaptation loop is replaced with a linguistically defined FSMC. B. Quasi-Continuous Adaptation Rate For the purpose of implementation, time-discrete controllers are used, and the continuous adaptation rate [see (49)] is approximated by (56) where represents the sampling period. If the controller parameters are only adapted after every th sampling interval, is approximated by (57) If the adaptation law from (49) is given in a time-discrete form, then by taking into account (56) and (57), we obtain (58)

1) Design of the FSMC in the Control Loop: The membership functions of the input of the FSMC are chosen as triangular membership functions, with overlap of 1. The membership functions of the output are also represented by triangular membership functions. The rule base itself is given in diagonal form. The inference method is realized by the Larsen operator. The rule consequents of different rules are merged through addition. From this, the crisp output of the FSMC is calculated through defuzzification by the method of weighted sums. The implementation steps are as follows. 1) Define ranges for the state variables. 2) Calculate the error vector from the state vector and according to (3). the reference vector for (2). 3) Choose parameter according to (2). 4) Define the switching hyperplane 5) Characterize the range of the switching variable and, thus, of the operating range of the FSMC; 6) Determine the minimal and maximal manipulated variand for values of the manipulated variable able that are outside the boundary layer. Mostly, . has to be chosen greater than the sum of the absolute values of the model uncertainties and disturbances in order to ensure an attracting boundary layer. 7) Without boundary layer, the closed-loop behavior is . However, chattering is undesired. given by Therefore, a small boundary layer is introduced by providing a balance between the desired tracking accuracy and the control activity. It has to be taken into for time-discrete account that the switching variable control varies around the switching hyperplane. This is due to the fact that the manipulated variable cannot be switched at any instant. The bigger the variation of the manipulated variable, the bigger value for it is chosen. 8) Choose the number and position of the supporting points of the FSMC; sometimes symmetry around operating points can be used in order to halve the number of adaptation parameters. 9) Choose the membership functions of the inputs and outputs of the FSMC. 10) Define the parameter values of the FSMC in (8) determining the nonlinear transfer characteristic. 11) Design the FSMC from the rules of Table I. 2) Implementation of the FSMC in the Adaptation Loop: The number and the position of the supporting points are defined heuristically. The bigger the boundary layer of the FSMC is, the more supporting points are chosen. As a rule of thumb, the supporting points are chosen closer around the origin of the transfer characteristic. With increasing , the distance between the supporting points increases as well. For the definition of the correction terms, a model of the plant is used. According to the balance condition in (53), a more exact knowledge about the plant leads to a better convergence behavior of the controller parameters. The design can be done in the following steps.

28


1) Heuristically choose the optimal approach velocity and the corresponding optimal manipulated variable of and at the th supporting the FSMC through and point by choosing the membership functions for in (10). of the adaptation line 2) Determine the slope in (10). According to Fig. 7, means is reached even for large that the approach velocity , the changes in the manipulated variable . For manipulated variable changes more slowly relative to the approach velocity . 3) Include the description of the model in the form of the and for the correction terms estimated model and in (15). of the adaptation law according to 4) Choose the factors (48), such that convergence of the adaptation parameters is guaranteed. 5) Introduce a boundary layer in the adaptation loop in order to prevent an exaggerated varying of the parameters. 6) Design the linguistic adaptation block with the structure of the FSMC. VII. TWO-LINK ROBOT ARM The adaptation algorithm is tested on an example of a two-link robot arm whose dynamics are described by the following nonlinear equations [8]: (59) where

and

tion of the inverse problem. According to the kinematic properties of the robot, i.e., (60) the Cartesian coordinates have to be transformed into angular coordinates according to (61) The function describes the kinematic model of the robot and the geometric relations between coordinates in the work space and internal coordinates. For a planar two-link robot, this function is bijective. A. Determination of the Reference Trajectories An efficient approach to solve the inverse problem is the differential method [6] where by differentiation of (60) with respect to time we obtain d

d

(62)

d d is a Jacobian matrix where is the number of work space coordinates and the number of internal link coordinates. Then, the incremental angular change reads d

(63)

Because of discretization effects, an angular correction has to be introduced during the incremental approach in order to guarantee exact following of the robot trajectory according to (60). If a smooth differential velocity profile is given in terms of work-space coordinates, then it is transformed through the matrix into a smooth, respectively differentiable, angular velocity profile as long as the matrix is not singular. The reference trajectory chosen is a square. In the case where a trapezoidal velocity profile is given, then this results in a nonsmooth, noncontinuous acceleration profile. As the required acceleration cannot be supplied immediately by the motor, control oscillations will occur for the velocity profile. In order to avoid such osciallations, a continuously differentiable velocity profile is chosen. The acceleration then is chosen as (64)

and are damping coefficients, is the is the control input vector of joint angles, and vector. The mechanical parameters of the system are

The corresponding velocity profile is, therewith, (65)

kg m

kg

m kg m s

k gm s

Since the arm acts horizontally as a planar robot, the gravitation , which simplifies the model constant can be set to zero, significantly. As the robot arm is planar, the internal link coordinates can be directly calculated from the coordinates of the work space, possibly Cartesian coordinates. This is also known as the solu-

if the motor starts from zero initial condition. After reaching the , the arm is no longer accelerated and maximal velocity the angle inreases proportionally with time. The deceleration is analogous to the acceleration procedure. B. Control of the Robot Arm In the following, the quality of the control design is examined for the links of a simulated two-link planar robot with the angles


Fig. 9.

29

00) and actual (–) square trajectory.

Reference (

Fig. 10.

State space and manipulated variable for the arm.

and . Each link controller consists of an FSMC similar to that in Table I and a corresponding adaptation block according to Fig. 6. According to the first two paragraphs in Section VI, the following parameters have been chosen. FSMC , , 4 supporting points for both the positive and the negative range of . Fuzzy rules IF is NVB THEN is PVB IF is NB THEN is PB IF is NM THEN is PM IF is NS THEN is PS IF is NVS THEN is PVS IF is Z THEN is Z IF is PVS THEN is NVS IF is PS THEN is NS IF is PM THEN is NM IF is PB THEN is NB IF is PVB THEN is NVB where N—Negative, P—Positive, S—Small, B—Big, Z—Zero, V—Very Adaptation block , , 25 adaptation rules according to Fig. 6. The relationship between the control inputs at the supporting is i points and the adaptation parameters j . In Fig. 9, the reference trajectory is given in . The task of the controller is to make the form of a square the system (–) follow the desired system trajectory starting from an initial state far from the reference trajectory. In Fig. 10, the ) and output values of the two angles of the robot arm ( (–) and their respective velocities are presented and the manip( ) and (–) are shown. Fig. 11 shows ulated variables the adaptation rates for the four parameters of the first FSMC. Fig. 12 shows the evolution of the parameters during adaptation. In Fig. 11, one can see that the initial adaptation rate is bounded,

Fig. 11. Parameter adaptation rate for the first controller.

Fig. 12.

Parameter adaptation for the first controller.

leading nevertheless to quickly convergent parameter values in Fig. 12. By using the FSMC in the adaptation block, a robust

30


Fig. 13.

Fig. 14.

Parameter adaptation rate for the second controller. Fig. 15.

View of the robot arm.

Fig. 16.

Reference trajectory (

Fig. 17.

Desired values (

Parameter adaptation for the second controller.

adaptation behavior can be obtained in spite of bounded adaptation rates. At the same time, the parameter convergence is fast enough. In Figs. 13 and 14, the respective values for the adaptation rate and parameters of the second FSMC are presented. VIII. REAL-TIME EXPERIMENTS WITH ROBOT ARM

A

00) and measured trajectory (–).

THREE-LINK

The described adaptive controller was also applied to a three-link robot arm where only the second and third links are considered (see Fig. 15). The robot arm is installed on a mobile platform and is used for environmental exploration, as well as for shifting of obstacles. Thus, the position control system for the end-effector at the third link has to cope with widely varying operation conditions like dry friction between the end-effector and the ground surface. Each link is driven by a special dc motor plus a planetary drive with a high inertia. Therefore, the two links can be regarded as nearly dynamically decoupled. Thus, the whole system can be considered as being composed of two single-input–single-output (SISO) systems that are only weakly disturbed by each other. The main disturbance comes from the

00) and actual values of the second link (–).

position/velocity-dependent friction between end-effector and ground. Nonlinear effects in the form of position-dependent


31

[8] R. Palm, D. Driankov, and H. Hellendoorn, Model Based Fuzzy Control. Berlin, Germany: Springer-Verlag, 1997. [9] J.-J. Slotine and W. Li, Applied Nonlinear Control. Englewood Cliffs, NJ: Prentice-Hall, 1991. [10] Z.-M. Yeh, “A performance approach to fuzzy control design for nonlinear systems,” Fuzzy Sets Syst., vol. 64, pp. 339–352, 1994. [11] T. Yoshizawa, “Stability and boundedness of solutions,” Arch. Ration. Mech. Anal., vol. 6, pp. 409–421, 1960.

Ralph G. Berstecher received the B. Eng. degree from the University of Essex, Colchester, U.K., and the Dr.-Ing. degree from Ruhr-University Bochum, Bochum, Germany, in 1994 and 1997, respectively. From 1995 to 1997, he was with the Neuro-Fuzzy Research Laboratory, Siemens AG, Munich, Germany. He is currently with Goldman Sachs, London, U.K. His research interests are adaptive control, sliding-mode control, and fuzzy logic control. Fig. 18.

00) and actual values of the third link (–).

Desired values (

friction and remaining coupling effects are neglected and will be rejected during control. The sampling interval is chosen as 0.008 s and the width of the boundary layer (dashed line) . Fig. 16 shows a desired was chosen to be reference square together with actual tracking results. Figs. 17 and 18 show the corresponding desired and actual values of and during tracking. Obviously, a good tracking behavior is achieved. It must be observed, however, that despite the adaptation a small chattering of the last link could not be avoided. This is a result of the relatively high friction between end-effector and ground. IX. CONCLUSIONS This paper has presented a new type of linguistically defined adaptive FSMC. The mathematical description and the heuristics-based linguistic adaptation law as well as the practical application in simulation and real-time experiments have been presented. The adaptation strategy was applied to a two-link planar robot. Through the adaptation of the nonlinear control characteristics, chattering effects have been suppressed and a good performance of the tracking behavior has been achieved. REFERENCES [1] K. J. Åström and B. Wittenmark, “A survey of adaptive control applications,” in Proc. 34th IEEE Conf. Decision and Control, New Orleans, LA, 1995, pp. 649–654. [2] R. G. Berstecher, R. Palm, and H. Unbehauen, “Adaptive fuzzy sliding-mode controller for a robot arm,” in Proc. 14th IFAC World Congr., vol. B, Beijing, China, 1999, pp. 13–18. [3] P. Eykhoff, “Every good regulator of a system must be a model of that system,” Model., Identif. Control, vol. 15, no. 3, pp. 135–139, 1994. [4] G.-C. Hwang and S. C. Lin, “A stability approach to fuzzy control design for nonlinear systems,” Fuzzy Sets Syst., vol. 48, pp. 279–287, 1992. [5] J. P. Lasalle and S. Lefschetz, Die Stabilitätstheorie von Ljapunow-Die direkte Methode mit Anwendungen. Mannheim, Germany: Bibliographisches Institut, 1967. [6] R. Palm, “Control of a redundant manipulator using fuzzy rules,” Fuzzy Sets Syst., vol. 45, pp. 279–298, 1992. [7] , “Robust Control by Fuzzy Sliding Mode,” Automatica, vol. 30, no. 9, pp. 1429–1437, 1994.

Rainer Palm (M’94) received the Dipl.-Ing. degree in 1975 from the Technical University of Dresden, Dresden, Germany, the Dr.-Ing. degree from Humboldt University, Berlin, Germany, and the Dr. sc.techn. degree from the Academy of Sciences, Berlin, Germany. From 1972 to 1981, he was with the Institute of Automatic Control, Berlin, Germany. From 1982 to 1990, he was a Project Leader and Head of a robotic research group at the Academy of Sciences, Berlin, Germany. From 1990 to 1991, he was a Guest Researcher at the Fraunhofer Institute for Production Systems and Design Technology, Berlin, Germany. In 1991, he joined the Neuro-Fuzzy Research Laboratory, Siemens AG, Munich, Germany, where he is currently a Research Scientist. He has authored, edited, or coedited three books. He has authored or coauthored numerous papers and articles published in technical editions, journals, and conference proceedings. He is the coauthor and Associate Editor of the Handbook of Fuzzy Computations (Bristol, U.K.: Institute of Physics Publishing, 1998) and coauthor of the Wiley Encyclopedia of Electrical and Electronics Engineering (New York: Wiley, 1999). His research interests include sliding-mode control, fuzzy control, multiple-model approaches, hybrid systems, decentralized control, and robotics. Dr. Palm is an Associate Editor of the IEEE TRANSACTIONS ON FUZZY SYSTEMS and a member of the German Society for Computer Science.

Heinz D. Unbehauen (SM’92–F’00) received the Dipl.-Ing., Dr.-Ing., and Dr.-Ing. Habil. degrees from the University of Stuttgart, Stuttgart, Germany, in 1961, 1964, and 1969, respectively. In 1969, he was awarded the title of Docent, and in 1972 he was appointed Professor of Control Engineering, in the Department of Energy Systems, University of Stuttgart. Since 1975, has been a Full Professor in the Faculty of Electrical Engineering, RuhrUniversity Bochum, Bochum, Germany, where he is the Head of the Control Engineering Laboratory and was the Dean of the Faculty during 1978–1979. He has been a Visiting Professor at universities in Japan, India, China, and the U.S. He is also an Honorary Professor of Tongji University, Shanghai, China. He has authored or coauthored more than 400 journal articles and conference papers and seven books. He has delivered many invited lectures and special courses at universities and companíes around the world and has served as a Consultant to many companies, as well as to public organizations such as UNIDO and UNESCO. His main research interests are system identification, adaptive control, robust control, and process control of multivariable systems.