Abstract
Fuzzy Inf. Eng. (2010) 2: 157-186 DOI 10.1007/s12543-010-0043-8 ORIGINAL ARTICLE Fuzzy Control Strategies in Human Operator and Sport Modeling Tijana T. Ivancevic· Bojan Jovanovic· Sasa Markovic Received: 27 September 2009/ Revised: 8 April 2010/ Accepted: 20 May 2010/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China 2010 Abstract The motivation behind mathematically modeling the human operator is to help explain the response characteristics of the complex dynamical system includ- ing the human manual controller. In this paper, we present two diﬀerent fuzzy logic strategies for human operator and sport modeling: fixed fuzzy-logic inference con- trol and adaptive fuzzy-logic control, including neuro-fuzzy-fractal control. As an application of the presented fuzzy strategies, we present a fuzzy-control based tennis simulator. Keywords Human operator modeling· Fuzzy control strategies· Fuzzy sport mod- eling 1. Introduction Despite the increasing trend toward automation, robotics and artificial intelligence (AI) in many environments, the human operator will probably continue for some time to be integrally involved in the control and regulation of various machines (e.g., missile-launchers, ground vehicles, watercrafts, submarines, spacecrafts, helicopters, jet fighters, etc.). A typical manual control task is the task in which control of these machines is accomplished by manipulation of the hands or f ingers [1]. As human- computer interfaces evolve, interaction techniques increasingly involve a much more continuous form of interaction with the user, over both human-to-computer (input) Tijana T. Ivancevic () Society for Nonlinear Dynamics in Human Factors, Adelaide, Australia CITECH Research IP Pty Ltd, Adelaide, Australia email: tijana.ivancevic@alumni.adelaide.edu.au Bojan Jovanovic () Sports Academy, Belgrade, Serbia email: jovanboj@yahoo.com Sasa Markovic () Faulty of Sport Sciences, University of Nis, Serbia email: markovic@fsfv.ni.ac.rs 158 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) and computer-to-human (output) channels. Such interaction could involve gestures, speech and animation in addition to more ‘conventional’ interaction via mouse, joy- stick and keyboard. This poses a problem for the design of interactive systems as it becomes increasingly necessary to consider interactions occurring over an interval, in continuous time. The so-called manual control theory developed out of the eﬀorts of feedback con- trol engineers during and after the World War II, who required models of human per- formance for continuous military tasks, such as tracking with anti-aircraft guns [2]. This seems to be an area worth exploring, firstly since it is generally concerned with systems which are controlled in continuous time by the user, although discrete time analogues of the various models exist. Secondly, it is an approach which models both system and user and hence is compatible with research eﬀorts on ‘synthetic’ models, in which aspects of both system and user are specified within the same framework. Thirdly, it is an approach where continuous mathematics is used to describe functions of time. Finally, it is a theory which has been validated with respect to experimental data and applied extensively within the military domains such as avionics. The premise of manual control theory is that for certain tasks, the performance of the human operator can be well approximated by a describing function, much as an inanimate controller would be. Hence, in the literature frequency domain representa- tions of behavior in continuous time are applied. Two of the main classes of system modelled by the theory are compensatory and pursuit systems. A system where only the error signal is available to the human operator is a compensatory system. A sys- tem where both the target and current output are available is called a pursuit system. In many pursuit systems the user can also see a portion of the input in advance; such tasks are called preview tasks [3]. A simple and widely used model is the ‘crossover model’ [8], which has two main parameters, a gain K and a time delayτ, given by the transfer function in the Laplace transform s domain −τs H = K . Even with this simple model we can investigate some quite interesting phenomena. For example consider a compensatory system with a certain delay, if we have a low gain, then the system will move only slowly towards the target, and hence will seem sluggish. An expanded version of the crossover model is given by the transfer func- tion [1] −(τs+α/s) (T s+ 1) e H = K , (T s+ 1)(T s+ 1) I N where T and T are the lead and lag constants (which describe the equalization of the L I human operator), while the first-order lag (T S + 1) approximates the neuromuscular lag of the hand and arm. The expanded term α/s in the time delay accounts for the ‘phase drop’, i.e., increased lags observed at very low frequency [4]. Alternatively if the gain K is very high, then the system is very likely to over- shoot the target, requiring an adjustment in the opposite direction, which may in turn overshoot, and so on. This is known as ‘oscillatory behavior’. Many more detailed models have also been developed; there are ‘anthropomorphic models’, which have Fuzzy Inf. Eng. (2010) 2: 157-186 159 a cognitive or physiological basis. For example the ‘structural model’ attempts to reflect the structure of the human, with central nervous system, neuromuscular and vestibular components [3]. Alternatively there is the ‘optimal control modeling’ ap- proach, where algorithmic models which very closely match empirical data are used, but which do not have any direct relationship or explanation in terms of human neu- ral and cognitive architecture [9]. In this model, an operator is assumed to perceive a vector of displayed quantities and must exercise control to minimize a cost functional given by [1] 2 2 2 J = E{ lim [q y (t)+ (r u (t)+ g u ˙ (t))]dt}, i i i T→∞ which means that the operator will attempt to minimize the expected value E of some weighted combination of squared display error y, squared control displacement u and squared control velocity u ˙ . The relative values of the weighting constants q, r, g i i i will depend upon the relative importance of control precision, control eﬀort and fuel expenditure. In the case of manual control of a vehicle, this modeling yields the ‘closed-loop’ or ‘operator-vehicle’ dynamics. A quantitative explanation of this closed-loop behavior is necessary to summarize operator behavioral data, to understand operator control actions, and to predict the operator-vehicle dynamic characteristics. For these rea- sons, control engineering methodologies are applied to modeling human operators. These ‘control theoretic’ models primarily attempt to represent the operator’s con- trol behavior, not the physiological and psychological structure of the operator [5, 6]. These models ‘gain in acceptability’ if they can identify features of these structures, ‘although they cannot be rejected’ for failing to do so [7]. One broad division of human operator models is whether they simulated a continu- ous or discontinuous operator control strategy. Significant success has been achieved in modeling human operators performing compensatory and pursuit tracking tasks by employing continuous, quasi-linear operator models. Examples of these include the crossover optimal control models mentioned above. Discontinuous input behavior is often observed during manual control of large amplitude and acquisition tasks [8, 10, 11, 12]. These discontinuous human operator responses are usually associated with precognitive human control behavior [8, 13]. Discontinuous control strategies have been previously described by ‘bang-bang’ or relay control techniques. In [14], the authors highlighted operator’s preference for this type of relay control strategy in a study that compared controlling high-order system plants with a proportional verses a relay control stick. By allowing the oper- ator to generate a sharper step input, the relay control stick improved the operators’ performance by up to 50 percent. These authors hypothesized that when a human controls a high-order plant, the operator must consider the error of the system to be dependent upon the integral of the control input. Pulse and step inputs would re- duce the integration requirements on the operator and should make the system error response more predictable to the operator. Although operators may employ a bang-bang control strategy, they often impose an internal limit on the magnitude of control inputs. This internal limit is typically 160 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) less than the full control authority available [8]. Some authors [15] hypothesized that this behavior is due to the operator’s recognition of their own reaction time delay. The operator must tradeoﬀ the cost of a switching time error with the cost of limiting the velocity of the output to a value less than the maximum. A significant amount of research during the 1960’s and 1970’s examined discon- tinuous input behavior by human operators and developed models to emulate it [13, 16, 17, 18, 19, 20, 21, 22, 23]. Good summaries of these eﬀorts can be found in [24], [10], [8] and [5, 6]. All of these eﬀorts employed some type of relay element to model the discontinuous input behavior. During the 1980’s and 1990’s, pilot models were developed that included switching or discrete changes in pilot behavior [25, 26, 27, 28, 11, 12]. Recently, the so-called ‘variable structure control’ techniques were applied to model human operator behavior during acquisition tasks [5, 6]. The result was a coupled, multi-input model replicating the discontinuous control strategy. In this formulation, a switching surface was the mathematical representation of the human operator’s con- trol strategy. The performance of the variable strategy model was evaluated by con- sidering the longitudinal control of an aircraft during the visual landing task. For a review of classical feedback control theory in the context of human operator modelling see [29, 4, 30] and contrast it with nonlinear and stochastic dynamics (see [31, 32, 33]). For similar approaches to sport modelling, see [34]. In this paper, we present two diﬀerent fuzzy logic strategies for human operator and sport modeling: fixed fuzzy-logic inference control and adaptive fuzzy-logic con- trol, including neuro-fuzzy-fractal control. As an application of the presented fuzzy strategies, we present a fuzzy-control based tennis simulator. 2. Fixed Fuzzy Control in Human Operator Modeling Modeling is the name of the game in any intelligence, be it human or machine. With the model and its exercising we can look forward in time with predictions and pre- scriptions and backward in time with diagnostics and explanations. With these time binding information structures we can make decisions and estimations in the here and now for purposes of eﬀiciency, eﬀicacy and control into the future. We and our machines hope to look into the future and the past so we may act intelligently now. Recall that fuzzy logic is a departure from classical two-valued sets and logic, that uses ‘soft’ linguistic (e.g., large, hot, tall) system variables and a continuous range of truth values in the interval [0,1], rather than strict binary (true or false) decisions and assignments. Formally, fuzzy logic is a structured, model-free estimator that approximates a function through linguistic input/output associations. Fuzzy rule-based systems apply these methods to solve many types of ‘real-world’ problems, especially where a system is diﬀicult to model, is controlled by a human operator or expert, or where ambiguity or vagueness is common. A typical fuzzy system consists of a rule base, membership functions, and an inference procedure. The fuzzy cognitive map, Fuzzy systems engineering. Fuzzy Inf. Eng. (2010) 2: 157-186 161 The key benef its of fuzzy logic design are: 1. Simplified & reduced development cycle; 2. Ease of implementation; 3. Can provide more ‘user-friendly’ and eﬀicient performance. Some fuzzy logic applications include: 1. Control (Robotics, Automation, Tracking, Consumer Electronics); 2. Information Systems (DBMS, Info. Retrieval); 3. Pattern Recognition (Image Processing, Machine Vision); 4. Decision Support (Adaptive HMI, Sensor Fusion). Recall that conventional controllers are derived from control theory techniques based on mathematical models of the open-loop process, called system, to be con- trolled. On the other hand, in a fuzzy logic controller, the dynamic behavior of a fuzzy system is characterized by a set of linguistic description rules based on expert knowledge. The expert knowledge is usually of the form: IF (a set of conditions are satisfied) THEN (a set of consequences can be inferred). Since the antecedents and the consequents of these IF-THEN rules are associated with fuzzy concepts (linguistic terms), they are often called fuzzy conditional state- ments. In this terminology, a fuzzy control rule is a fuzzy conditional statement in which the antecedent is a condition in its application domain and the consequent is a control action for the system under control. Basically, fuzzy control rules provide a convenient way for expressing control policy and domain knowledge. Furthermore, several linguistic variables might be involved in the antecedents and the conclusions of these rules. Furthermore, several linguistic variables might be involved in the antecedents and the conclusions of these rules. When this is the case, the system will be referred to as a multi-input multi-output fuzzy system. The most famous fuzzy control application is the subway car controller used in Sendai (Japan), which has outperformed both human operators and conventional au- tomated controllers. Conventional controllers start or stop a train by reacting to posi- tion markers that show how far the vehicle is from a station. Because the controllers are rigidly programmed, the ride may be jerky: the automated controller will apply the same brake pressure when a train is, say, 100 meters from a station, even if the train is going uphill or downhill. In the mid-1980s engineers from Hitachi used fuzzy rules to accelerate, slow and brake the subway trains more smoothly than could a deft human operator. The rules encompassed a broad range of variables about the ongoing performance of the train, such as how frequently and by how much its speed changed and how close the actual speed was to the maximum speed. In simulated tests the fuzzy controller beat an automated version on measures of riders’ comfort, shortened riding times and even achieved a 10 percent reduction in the train’s energy consumption [57]. 162 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) 2.1. Fuzzy Inference Engine Recall that a crisp (i.e., ordinary mathematical) set X is defined by a binary charac- teristic function μ (x) of its elements x 1, if x ∈ X, μ (x) = X ⎪ 0, if x X, while a fuzzy set is defined by a continuous characteristic function μ (x) = [0, 1], including all (possible) real values between the two crisp extremes 1 and 0, and in- cluding them as special cases. A fuzzy set X is a collection of ordered pairs X = {(x,μ(x))}, (1) where μ(x)isthe membership function representing the grade of membership of the element x in the set X. A single pair is called a fuzzy singleton. For technical details on fuzzy sets and fuzzy state machines, see [42, 43]. Fig. 1: Basic structure of the fuzzy inference engine Like neural networks, the fuzzy logic systems are generic nonlinear function ap- proximators [44]. In the realm of fuzzy logic this generic nonlinear function approxi- mation is performed by means of fuzzy inference engine. The fuzzy inference engine is an input-output dynamical system which maps a set of input linguistic variables (IF−part) into a set of output linguistic variables (THEN−part). It has three sequen- tial modules (see Fig. 1): 1. Fuzzif ication; in this module numerical crisp input variables are fuzzified; this is performed as an overlapping partition of their universes of discourse by means of fuzzy membership functionsμ(x) (1), which can have various shapes, like triangular, trapezoidal, Gaussian-bell, −(x− m) μ(x) = exp , 2σ Fuzzy Inf. Eng. (2010) 2: 157-186 163 (with mean m and standard deviationσ), sigmoid −1 x− m μ(x) = 1+ , or some other shapes (see Fig. 2). Fig. 2: Fuzzification example: Set of triangular-trapezoidal membership functions partitioning the universe of discourse for the angle of the hypothetical steering wheel; notice the white overlapping triangles B. Kosko and his students have done extensive computer simulations looking for the best shape of fuzzy sets to model a known test system as closely as possible. They let fuzzy sets of all shapes and sizes compete against each other. They also let neural systems tune the fuzzy-set curves to improve how well they model the test system. The main conclusion from these experiments is that ‘triangles never do well’ in such contests. Suppose we want an adaptive fuzzy n n system F : R → R to approximate a test function or approximand f : R → R as closely as possible in the sense of minimizing the mean-squared error between them, f − F . Then the ith scalar ‘sinc’ function (as commonly used in signal processing), x−m sin μ (x) = , i = 1,..., n, (2) x−m with center m and dispersion (width) d = σ > 0, often gives the best per- i i formance for IF−part mean-squared function approximation, even though this generalized function can take on negative values (see [50]). 2. Inference; this module has two submodules: (i) The expert-knowledge base consisting of a set of IF− THEN rules relating input and output variables, and 164 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) (ii) The inference method, or implication operator, that actually combines the rules to give the fuzzy output; the most common is Mamdani Min-Max infer- ence, in which the membership functions for input variables are first combined inside the IF − THEN rules using AND (∩,or Min) operator, and then the output fuzzy sets from diﬀerent IF − THEN rules are combined using OR (∪, or Max) operator to get the common fuzzy output (see Fig. 3). Fig. 3: Mamdani’s Min-Max inference method and Center of Gravity defuzzification 3. Defuzzif ication; in this module fuzzy outputs from the inference module are converted to numerical crisp values; this is achieved by one of the several de- fuzzification algorithms; the most common is the Center of Gravity method, in which the crisp output value is calculated as the abscissa under the center of gravity of the output fuzzy set (see Fig. 3). In more complex technical applications of general function approximation (like in complex control systems, signal and image processing, etc.), two optional blocks are usually added to the fuzzy inference engine [44, 51, 52]: 0. Preprocessor, preceding the fuzzification module, performing various kinds of normalization, scaling, filtering, averaging, diﬀerentiation or integration of input data; and 4. Postprocessor, succeeding the defuzzification module, performing the analog operations on output data. Common fuzzy systems have a simple feedforward mathematical structure, the so-called Standard Additive Model (SAM, for short), which aids the spread of appli- cations. Almost all applied fuzzy systems use some form of SAM, and some SAMs in turn resemble the ANN models (see [50]). Fuzzy Inf. Eng. (2010) 2: 157-186 165 n p In particular, an additive fuzzy system F : R → R stores m rules of the patch n p form A × B ⊂ R × R , or of the word form ‘If X = A Then Y = B ’ and adds the i i i i ‘fired’ Then-parts B (x) to give the output set B(x), calculated as n n B(x) = w B (x) = wμ (x)B (x), i = 1,..., n, (3) i i i i i=1 i=1 for a scalar rule weight w > 0. The factored form B (x) = μ (x)B (x) makes the i i i additive System (3) a SAM system. The fuzzy system F computes its output F(x)by taking the centroid of the output set B(x): F(x) = Centroid(B(x)). The SAM Theorem then gives the centroid as a simple ratio, F(x) = p (x)c, i = 1,..., n, i i i=1 where the convex coeﬀicients or discrete probability weights p (x) depend on the input x through the ratios wμ (x)V i i i p (x) = , i = 1,..., n. (4) w μ (x)V k k k k=1 V is the finite positive volume (or area if p = 1 in the codomain space R ) [50], V = b (y ,..., y )dy ...dy > 0, i i 1 p 1 p and c is the centroid of the Then-part set B (x), i i yb (y ,..., y )dy ...dy i 1 p 1 p c = . b (y ,..., y )dy ...dy p i 1 p 1 p 2.2. Fuzzy Decision Making Recall that f inite state machines (FSMs) are simple ‘machines’ that have a finite number of states (or conditions) and transition functions that determine how input to the system changes it from one state to another [38]. Fuzzy State Machines (FuSMs) are a modification of FSMs. In FuSMs, the inputs to the system (that cause the transitions between states) are not discrete. The real value of FuSMs comes from the interaction of the system inputs. For example, a character in a video game may of a simple combat scenario decides how aggressive he will be depending on his health, the enemy’s health, and his distance from the enemy. The combination of these inputs cause the state transitions to happen. This can result in very complex behaviors from a small set of rules. For example, The health variables have three sets: Near death, Good, and Excellent. The distance variable has three sets: Close, Medium, and Far. Finally, the output (aggressiveness) has three sets: Run away, Fight defensively, and All out attack!. Fuzzy Control Language (FCL) is a standard for Fuzzy Control Programming pub- lished by the International Electrotechnical Commission (IEC). 166 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) Fuzzy-logic decision maker (FLDM) breaks the decision scenario down into small parts that we can focus on and input easily. It then uses theoretically optimal methods of combining the scenario pieces into a global interrelated whole with an indication as to which alternative is the best within the constraints and goals of the decision scenario. The assumption in FLDM is that a judgment consists of a known here and now (the constraints), a hoped for future there and then (the goals), and various paths (the alternatives) for getting from the present here and now to the future there and then. The problem is then the selection of the path (alternative) that optimally supports the present constraints and the future goals. Decision making when faced with several alternatives, which initially appear equally good or desirable, can be a time consuming and often painful process. The FLDM overcomes the (human) memory and processor limitations by allowing the decision maker to selectively evaluate small amounts of the necessary information at any one time (i.e., the fuzzy values of goal and constraint satisfaction and simple, one-at- a-time paired comparisons). Then, when it becomes necessary to evaluate all the pertinent data, the computer can be utilized to perform the decision task in a straight forward manner. 2.3. Fuzzy Logic Control The most common and straightforward applications of fuzzy logic are in the domain of control [44, 51, 52, 53]. Fuzzy control is a nonlinear control method based on fuzzy logic. Just as fuzzy logic can be described simply as computing with words rather than numbers, fuzzy control can be described simply as control with sentences rather than diﬀerential equations. A fuzzy controller is based on the fuzzy inference engine, which acts either in the feedforward or in the feedback path, or as a supervisor for the conventional PID controller. A fuzzy controller can work either directly with fuzzified dynamical variables, like direction, angle, speed, or with their fuzzified errors and rates of change of errors. In the second case we have rules of the form: 1. If error is Neg and change in error is Neg then output is NB. 2. If error is Neg and change in error is Zero then output is NM. The collection of rules is called a rule base. The rules are in IF − THEN format, and formally the IF−side is called the condition and the THEN−side is called the conclusion (more often, perhaps, the pair is called antecedent-consequent). The input value Neg is a linguistic term short for the word Negative, the output value NB stands for Negative Big and NM for Negative Medium. The computer is able to execute the rules and compute a control signal depending on the measured inputs error and change in error. The rulebase can be also presented in a convenient form of one or several rule matrices, the so-called FAM−matrices, where FAM is a shortcut for Kosko’s fuzzy associative memory [44, 51]. For example, a 9× 9 graded FAM matrix can be defined in a symmetrical weighted form: Fuzzy Inf. Eng. (2010) 2: 157-186 167 ⎛ ⎞ ⎜ 0.6S4 0.6S4 0.7S3 ... CE ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0.6S4 0.7S3 0.7S3 ... 0.9B1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ FAM = 0.7S3 0.7S3 0.8S2 ... 0.9B1 , ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ... ... ... ... 0.6B4 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ CE 0.9B1 0.9B1 ... 0.6B4 in which the vector of nine linguistic variables L partitioning the universes of dis- course of all three variables (with trapezoidal or Gaussian bell-shaped membership functions) has the form 9 T L = {S 4, S 3, S 2, S 1, CE, B1, B2, B3, B4} , to be interpreted as: ‘small 4’, ... , ‘small 1’, ‘center’, ‘big 1’, ... , ‘big 4’. For example, the left upper entry (1, 1) of the FAM matrix means: IF red is S4 and blue is S4, THEN result is 0.6S4; or, entry (3, 7) means: IF red is S2 and blue is B2, THEN result is center, etc. Here we give design examples for three fuzzy controllers. Temperature Control System. In this simple example, the input linguistic variable is temperature error = desired temperature− current temperature. The two output linguistic variables are: hot fan speed, and cool fan speed. The universes of discourse, consisting of membership functions, i.e., overlapping triangular- trapezoidal shaped intervals, for all three variables are: invar: temperature error = {Negative Big, Negative Medium, Negative S mall, Zero, Positive S mall, Positive Medium, Positive Big}, with the range [−110, 110] degrees; outvars: hot fan speed and cool fan speed={zero, low, medium, high, very high}, with the range [0, 100] rounds-per-meter. Car Anti-Lock Braking System. The fuzzy–logic controller for the car anti-lock braking system consists of the following input variables: slip r (rear wheels slip), slip fr (front right wheel slip), slip fl (front left wheel slip), with their membership functions: NZ= Near Zero, OP= Optimal, AO = Above Optimal, and the following output variables: bp r (rear wheels brake pressure), bp fr (front right brake pressure), bp fl (front left brake pressure), 168 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) with their membership functions: LW= Low, MD= Medium, HG = High. The inference rule-base for this example consists of the following fuzzy implications: IF slip fl is NZ THEN bp fl is MD; IF slip fr is NZ THEN bp fr is MD; IF slip r is NZ THEN bp r is MD; IF slip fl is OP THEN bp fl is HG; IF slip fr is OP THEN bp fr is HG; IF slip r is OP THEN bp r is HG; IF slip fl is AO THEN bp fl is LW; IF slip fr is AO THEN bp fr is LW; IF slip r is AO THEN bp risLW. Truck Backer-Upper Steering Control System. In this example there are two in- put linguistic variables: position and direction of the truck, and one output linguistic variable: steering angle (see Fig. 4). The universes of discourse, partitioned by over- lapping triangular-trapezoidal shaped intervals, are defined as: Fig. 4: Truck backer-upper steering control system invars: position = {NL, NS, ZR, PS, PL}, and direction = {NL, NM, NS, ZR, PS, PM, PL}, where NL denotes Negative Large, NM is Negative Medium, NS is Negative Small, etc. Fuzzy Inf. Eng. (2010) 2: 157-186 169 outvar: steering angle = {NL, NM, NS, ZR, PS, PM, PL}. The rule-base is given as: IF direction is NL and position is NL, THEN steering angle is NL; IF direction is NL and position is NS , THEN steering angle is NL; IF direction is NL and position is ZR, THEN steering angle is PL; IF direction is NL and position is PS , THEN steering angle is PL; IF direction is NL and position is PL, THEN steering angle is PL; IF direction is NM and position is NL, THEN steering angle is ZR; ............. IF direction is PL and position is PL, THEN steering angle is PL, which can also be rewritten in the above FAM form. The so-called control surface for the truck backer-upper steering control system is depicted in Fig. 5. Fig. 5: Control surface for the truck backer-upper steering control system 2.3.1. Characteristics of Fixed Fuzzy Control Fuzzy logic oﬀers several unique features that make it a particularly good choice for many control problems, among them [52, 53]: 1. It is inherently robust since it does not require precise, noise-free inputs and can be programmed to fail safely if a feedback sensor quits or is destroyed. The output control is a smooth control function despite a wide range of input variations. 2. Since the fuzzy logic controller processes user-defined rules governing the tar- get control system, it can be modified and tweaked easily to improve or dras- tically alter system performance. New sensors can easily be incorporated into the system simply by generating appropriate governing rules. 170 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) 3. Fuzzy logic is not limited to a few feedback inputs and one or two control outputs, nor is it necessary to measure or compute rate-of-change parameters in order for it to be implemented. Any sensor data that provides some indication of a systems actions and reactions is suﬀicient. This allows the sensors to be inexpensive and imprecise thus keeping the overall system cost and complexity low. 4. Because of the rule-based operation, any reasonable number of inputs can be processed (1-8 or more) and numerous outputs (1-4 or more) generated, al- though defining the rulebase quickly becomes complex if too many inputs and outputs are chosen for a single implementation since rules defining their inter- relations must also be defined. It would be better to break the control system into smaller chunks and use several smaller fuzzy logic controllers distributed on the system, each with more limited responsibilities. 5. Fuzzy logic can control nonlinear systems that would be diﬀicult or impossible to model mathematically. This opens doors for control systems that would normally be deemed unfeasible for automation. A fuzzy logic controller is usually designed using the following steps: 1. Define the control objectives and criteria: What am I trying to control? What do I have to do to control the system? What kind of response do I need? What are the possible (probable) system failure modes? 2. Determine the input and output relationships and choose a minimum number of variables for input to the fuzzy logic engine (typically error and rate-of-change of error). 3. Using the rule-based structure of fuzzy logic, break the control problem down into a series of IF X AND Y THEN Z rules that define the desired system output response for given system input conditions. The number and complexity of rules depends on the number of input parameters that are to be processed and the number fuzzy variables associated with each parameter. If possible, use at least one variable and its time derivative. Although it is possible to use a single, instantaneous error parameter without knowing its rate of change, this cripples the systems ability to minimize overshoot for a step inputs. 4. Create fuzzy logic membership functions that define the meaning (values) of Input/Output terms used in the rules. 5. Test the system, evaluate the results, tune the rules and membership functions, and re-test until satisfactory results are obtained. Therefore, fuzzy logic does not require precise inputs, is inherently robust, and can process any reasonable number of inputs but system complexity increases rapidly with more inputs and outputs. Distributed processors would probably be easier to im- plement. Simple, plain-language rules of the form IF X AND Y THEN Z are used to describe the desired system response in terms of linguistic variables rather than Fuzzy Inf. Eng. (2010) 2: 157-186 171 mathematical formulas. The number of these is dependent on the number of inputs, outputs, and the designers control response goals. Obviously, for very complex sys- tems, the rule-base can be enormous and this is actually the only drawback in applying fuzzy logic. 2.3.2. Pro and Contra Fuzzy Logic Control According to [54] there are the following pro and contra arguments regarding fuzzy logic control: 1. Fuzzy logic control is more useful than its detractors claim. 2. Fuzzy logic control is less useful than its proponents claim. 3. Fuzzy logic does not generate a control law. It maps an existing control law from one set of rules into a logic set. 4. Fuzzy logic control is most useful in ‘common sense’ control situations, i.e., ones where it might be diﬀicult to write down the equations of motion, but a human would know how to control it. Examples of this are the ‘truck backer upper’, car parking, train control, and helicopter control problems. 5. Fuzzy logic sets eﬀectively quantize their input and output space. However, the quantization intervals are rarely uniform. 6. In most fuzzy logic control success stories the sample rates are incredibly high relative to the dynamics of the system. Much of their success is because of this. Most of the examples of fuzzy logic control being successfully applied fall into the category of things that humans do well [54, 55, 56]. Recall that in Japan, there is a train (Sendai subway), which is controlled by fuzzy logic. The train pulls into the station within a few inches of its target. More accurate, but nevertheless replacing human control [55]. Also in Japan, there are experiments in controlling a small model helicopter (Spec- trum, July 1992) via radio control. The helicopter can respond to commands such as take oﬀ and land, hover, forward, backwards, left and right [54, 55, 56]. Proponents assert that a conventional control scheme would be incredibly hard to design because it would be really tough to model the helicopter dynamics. The ‘model free’ nature of fuzzy logic control makes the problem trivial. This might be true, at least from a practical application point of view, but it obscures some key facts [54]: 1. The model helicopter was designed so that a human operator with a joystick could control it, i.e., it was designed to respond well to intuitive control rules. Because of this, the helicopter has been designed to be very robust to impreci- sion. (Robustness to imprecision is one of the features that many proponents claim fuzzy logic brings to the problem. It is possible that this feature is more a feature of the dynamic system than of fuzzy logic itself. In fact, L. Zadeh, the creator of fuzzy logic, points out that fuzzy logic takes advantage of a sys- tem’s inherent robustness to imprecision rather than creating a robustness to imprecision). 172 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) 2. The human operator has an implicit model in his mind of the input-output be- havior of the helicopter. This is how he generates his control law for using the joystick. 3. Fuzzy logic maps the human’s control law and therefore is based on the hu- man’s implicit model of the helicopter. This in turn works because the heli- copter was designed to be robust to human control actions. 4. The human being’s ‘bandwidth’ is quite low, certainly less than 100 Hz. Fur- thermore, it is unlikely that a toy helicopter, a train, or a truck would respond to anything about 1 Hz and certainly not 10 Hz. (Since it must be an issue in every digital control problem and since any implementation of fuzzy logic control in- volves using some digital processor, the natural conclusion is that the sample rates are chosen so high above the system time constants that they seemingly stop being important.) The train control problem, as well as the car parking and truck backer upper prob- lem are all described by (1-4) above. So we can conclude that high sample rates are an inherent part of using fuzzy logic. The seemingly unimportant high sample rate may be precisely why the simple control rules work well. Fast sampling does lead to a greater computational burden. However, the computational cost many be oﬀset by being able to use a simpler control law. If we look in any fuzzy logic article we will see a picture of membership functions for fuzzy sets (see, e.g., [55]). These sets eﬀectively quantize the interval that they are on: they span the space so that any value on the line must fall into at least one of the sets. However, they do not behave quite like what we think of as quantizers since a particular value can be a member of more than one set. The sets are typically fairly coarse in terms of what we would consider eﬀective quantization. Combinations of these coarse quantizers provide various fuzzy conditions. The coarse quantizations and simple rules may oﬀset the higher sample rate requirement. In summary, fuzzy logic does not generate a control law, it merely maps a law from one form to another. The simple rules for train control or truck backing up are not generated by fuzzy logic control. These are already present in the mind of the human operator. Fuzzy logic merely maps the intuitive rules into a computer program. What seems to be the newest feature of fuzzy logic control is that because the borders are fuzzy, more than one logic state can be true to some degree. This allows for a smooth transition between one control action and another, since they can both go on but at diﬀerent activation levels, or gain. Quite often control systems have diﬀerent operating regimes. Handling the transitions between these tends to be ad hoc. Things, which are already ad hoc, are perfect candidates for using fuzzy logic. Thus, fuzzy logic might be a good solution for smoothly switching a control system from one operating regime to another. In the transition, both control laws would be active, but their outputs would be scaled by the how much the system is in one regime or another. Clearly, this means that both control laws would have to be run in parallel during the transition. On the other hand, quite a lot has been said about the model-free nature of a fuzzy Fuzzy Inf. Eng. (2010) 2: 157-186 173 logic control system. The notion is that rather than trying to construct these com- plicated dynamic models for a system, the ‘simple fuzzy rules’ allow the designer to design a control system. Clearly, this hides the notion that buried in those ‘sim- ple fuzzy rules’ is an implicit model of the system. Following [54], we believe that no intelligent action is possible without a model. Any general behavior trend con- stitutes a model, whether explicit (e.g., dynamic systems model) or implicit (i.e., as encompassed in the fuzzy logic rules). Another general idea that seems to permeate the fuzzy logic control hype is the notion that someone with very little skill can design a controller using fuzzy logic, while using classical control takes years of training. In fact, the advantages and disadvantage of fuzzy systems result of the fact that fuzzy logic represents a decision making process. In control field, this provides a wide range of viable ways to solve naturally control problems while the basic control knowledge is not needed. Another thing to point out is that usually the fuzzy logic rules use more external sensors, including acceleration, velocity, and the position information. So they natu- rally perform better than conventional controllers (position feedback loop) based only on position sensors. Many proponents of fuzzy logic control argue that fuzzy logic works much better than conventional control when the system is nonlinear. However, the conventional controller they are comparing it to is a PID controller based on a linear system model. In the sense that the fuzzy logic rules encompass a better model (implicit but there) of the system than an inappropriately applied linear model, the fuzzy logic rules will work better. Recall that the linear model has its faults as well. If a control system is designed using a linear model that doesn’t characterize the system behavior well, then the control system will probably fail to work well. However, a fair comparison would be one made between a fuzzy logic controller and a nonlinear state feedback controller that measures all the same variables at the same sampling rate as the fuzzy logic controller. If such a comparison is made there is no guarantee that the fuzzy logic controller will work better. 3. Adaptive Fuzzy Control in Human Operator Modeling 3.1. Neuro-Fuzzy Hybrid Systems In many applications, desired system behavior is partially represented by data sets. In control systems, these data sets may represent operational states. In decision support systems and data analysis applications, these data sets may represent sample cases. Discussing the respective strengths and weaknesses of fuzzy logic and neural net technology, a simple comparison indicates that the strongest benefit of a neural net is that it can automatically learn from sample data. However, a neural net remains a black box, thus manual modification and verification of a trained net is not possible in a direct way. This is where fuzzy logic excels: In a fuzzy logic system, any component is defined as close as possible to human intuition, making it very easy to manually 174 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) modify and verify a designed system. However, fuzzy logic systems can not auto- matically learn from sample data. This is where neuro-fuzzy system provides ‘the best of both worlds’. Take the explicit representation of knowledge in linguistic variables and rules from fuzzy logic and add the learning approach used with neural nets. In the neuro-fuzzy system, both fuzzy rules and membership functions are adjusted by some form of backpropagation learning to adapt the system behavior according to the sample data. The neuro-fuzzy system can also be used to optimize existing fuzzy logic systems. Starting with an existing fuzzy logic system, the neuro-fuzzy system interactively tunes rule weights and membership function definitions so that the system converges to the behavior represented by the data sets. To distinguish between more and less important rules in the knowledge base, we can put weights on them. Such weighted knowledge base can be then trained by means of artificial neural networks. In this way we get hybrid neuro-fuzzy trainable expert systems. Another way of the hybrid neuro-fuzzy design is the fuzzy inference engine such that each module is performed by a layer of hidden artificial neurons, and ANN- learning capability is provided to enhance the system knowledge (see Fig. 6). Fig. 6: Neuro-fuzzy inference engine Again, the fuzzy control of the backpropagation learning can be implemented as a set of heuristics in the form of fuzzy IF − THEN rules, for the purpose of achiev- ing a faster rate of convergence. The heuristics are driven by the behavior of the instantaneous sum of squared errors. As another alternative, we can consider the well-known fuzzy ARTMAP system, which is essentially a clustering algorithm (vector quantizer), with supervision that redirects training inputs which would be grouped in an incorrect category to a diﬀer- ent cluster. A fuzzy ARTMAP system consists of two fuzzy ART modules, each of which clusters vectors in an unsupervised fashion, linked by a map field. Fuzzy ART clusters vectors based on two separate distance criteria, match and choice. For more details, see [58]. Finally, most feedback fuzzy systems are either discrete or continuous generalized Fuzzy Inf. Eng. (2010) 2: 157-186 175 SAMs [50], given respectively by n n x(k+ 1) = p (x(k))B (x(k)), or x ˙(t) = p (x(t))B (x(t)), i i i i i=1 i=1 with coeﬀicients p given by (4) above. 3.2. Neuro-Fuzzy-Fractal Operator Control Although the general concept of learning, according to the schematic recursion NEW VALUE = OLD V ALUE + INNOVAT ION t t n+1 n can be implemented in the framework of nonlinear control theory (as seen in the previous subsection), its natural framework is artificial intelligence. For the purpose of neuro-fuzzy-fractal control [38, 39], the general model for a nonlinear plant can be modified as [35, 36] x ˙= f (x, D,α)−β f (x, D,α), (5) 1 2 y ˙=β f (x, D,α), n m where x ∈ R is a vector of state variables, y ∈ R is a vector of the system outputs, β ∈ R is a constant measuring the eﬀiciency of the conversion process, D ∈ (0, 3) is the fractal dimension of the process, and α ∈ R is a fuzzy-inference selection parameter. For a complex dynamical system it may be necessary to consider a set of math- ematical models to represent adequately all of possible dynamic behaviors of the system. In this case, we need a decision scheme to select the appropriate model to use according to the linguistic value of a selection parameter. We use a fuzzy infer- ence system for diﬀerential equations to achieve the model selection. We have fuzzy rules of the form : For programming purposes, recall that basic logical control structures in the pseudocode include IF-THEN and SWITCH statements, respectively defined as: IF-THEN if ((condition1) (condition2)) {action1; } else if ((condition3) && (condition4)) { action2; } else { default action; SWITCH switch (condition) { case 1: action1; break; case 2: action2; break; default: default action; break; } 176 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) IF α is A AND D is B THEN M , 1 1 1 ... ... ... IF α is A AND D is B THEN M , n n n where A ,..., A are linguistic values forα, B ,..., B are linguistic values for the frac- 1 n 1 n tal dimension D, and M ,..., M are mathematical models of the form given by 5. The 1 n selection parameter α represents the environment variable, like temperature, humid- ity, etc. Following [35, 36], we combine adaptive model-based control using neural net- works with the method for model selection using fuzzy logic and fractal theory, to obtain a hybrid neuro-fuzzy-fractal method for control of nonlinear plants. This gen- eral method combines the advantages of neural networks (ability for identification and control) with the advantages of fuzzy logic (ability for decision and use of ex- pert knowledge) to achieve the goal of robust adaptive control of nonlinear dynamic plants. We also use the fractal dimension to characterize the plant-output processes in modeling these dynamical systems. 3.2.1. Fractal Dimension for Machine Output Identification The experimental identification of a nonlinear biologic transducer is often approached via consideration of its response to a stochastic test ensemble, such as Gaussian white noise [46]. In this approach, the input-output relationship a deterministic transducer is described by an orthogonal series of functionals. Laboratory implementation of such procedures requires the use of a particular test signal drawn from the idealized stochastic ensemble; the statistics of the particular test signal necessarily deviate from the statistics of the ensemble. The notion of a fractal dimension (specifically the ca- pacity dimension) is a means to characterize a complex time series. It characterizes one aspect of the diﬀerence between a specific example of a test signal and the test ensemble from which it is drawn: the fractal dimension of ideal Gaussian white noise is infinite, while the fractal dimension of a particular test signal is finite. The fractal dimension of a test signal is a key descriptor of its departure from ideality: the fractal dimension of the test signal bounds the number of terms that can reliably be identified in the orthogonal functional series of an unknown transducer [47]. Definition of the fractal dimension. Recall that for a smooth (i.e., nonfractal) line, an approximate length L(r) is given by the minimum number N of segments of length r needed to cover the line, L(r) = Nr.As r goes to zero, L(r) approaches a finite limit, the length L of the curve. Similarly one can define the area A or the volume V of nonfractal objects as the limit of an integer power law of r, A = lim Nr, V = lim Nr , r→0 r→0 where the integer exponent is the Euclidean dimension E of the object. This definition can not be used for fractal objects: as r tends to 0, we enter finer and finer details of the fractal and the product Nr may diverge to infinity. However, a real number D exists so that the limit of Nr stays finite. This exponent is called Fuzzy Inf. Eng. (2010) 2: 157-186 177 Hausdorﬀ dimension D , defined by log N D = lim . r→0 log(1/r) Another popular definition of dimension proposed for fractal objects is the correla- tion dimension D , given by log C(r) D = lim , r→0 log(r) where C(r) is the number of points which have a smaller (Euclidean) distance than a given distance r. This measure is widely used because it is easy to evaluate for exper- imental data, when the fractal comes from a ‘dust’ of isolated points. A method for measuring D of strange attractors can be found in [49]. D may also be used to de- 2 2 termine whether a time-series derives from a random process or from a deterministic chaotic system. m−dimensional data vectors are constructed from m measurements spaced equidistant in time, and D is evaluated for this m−dimensional set of points. If the time-series is a random process, D increases with m; if the time-series is a deterministic signal, D does not increase further when the embedding dimension m exceeds D . Thus a plot of the correlation dimension as a function of the embedding dimension may easily show whether a signal is random noise of deterministic chaos. Note that D ≤ D . 2 H Fractal behavior and singularities in time series. The functions y(t) typically studied in mathematical analysis are continuous and have continuous derivatives. Hence, they can be approximated in the vicinity of some time t by Taylor series (or power series) 2 3 y(t) = a + a (t− t )+ a (t− t ) + a (t− t ) +··· (6) 0 1 i 2 i 3 i For small regions around t , just a few terms of the expansion (6) are necessary to approximate the function y(t). In contrast, most time series y(t) found in ‘real-life’ applications appear quite noisy). Therefore, at almost every point in time, they can- not be approximated either by Taylor series (or by Fourier series) of just a few terms. Moreover, many experimental or empirical time series have fractal features, i.e., for some times t , the series y(t) displays singular behavior [48, 48]. By this, we mean that at those times ti, the signal has components with non-integer powers of time which appear as step-like or cusp-like features, the so-called singularities, in the sig- nal. Formally, one can write 2 3 h y(t) = a + a (t− t )+ a (t− t ) + a (t− t ) +···+ a (t− t ) , (7) 0 1 i 2 i 3 i h i where t is inside a small vicinity of t , and h is a non-integer number quantifying the i i local singularity of y(t)at t = t . The next problem is to quantify the ‘frequency’ in the signal of a particular value h of the singularity exponents h .Diﬀerent possibilities can be considered. For ex- ample, the set of times with singular behavior {t} may be a finite fraction of the i 178 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) time series and homogeneously distributed over the signal. But {t} may also be an asymptotically infinitesimal fraction of the entire signal and have a very heteroge- neous structure. That is, the set {t} may be a fractal itself. In either case, it is useful to quantify the properties of the sets of singularities in the signal by calculating their fractal dimensions D or D . 2 H Fractal dimension of a machine output signal. This method uses the fractal di- mension to make a unique classification of the diﬀerent types of machine behavior, because diﬀerent types of signals have diﬀerent geometrical forms. The problem is then of finding a one-to-one map between the diﬀerent types of machine behaviors and their corresponding fractal dimension. The first step in obtaining this map is to find experimentally the diﬀerent geometrical forms for machine output signals. The second step is to calculate the corresponding fractal dimensions for these signals. This fractal dimension can be calculated for a selected type of signals with several samples, to obtain as a result a statistical estimation of the fractal dimension and the corresponding error of the estimation. In order to make an eﬀicient use of this map between the diﬀerent types of machine behaviors and their corresponding estimated dimensions, we need to implement it as a module in the computer program. 3.2.2. Fuzzy Logic Model Selection for Dynamical Systems For a real-world dynamical system it may be necessary to consider a set of mathe- matical models to represent adequately all of the possible dynamic behaviors of the system. In this case, we need a fuzzy decision procedure to select the appropriate model to use according to the value of a selection parameter vector α. To implement this decision procedure, we need a fuzzy inference system that can use diﬀerential equations as consequents. For this purpose, we can follow the fuzzy decision sys- tem developed in [35, 36], that can be considered as a generalization of the classical Sugeno’s inference system [40, 41, 44], in which diﬀerential equations as conse- quents of the fuzzy rules are used instead of simple polynomials like in the original Sugeno’s method. Using this method, a fuzzy model for a general dynamical system can be expressed as follows [38]: IFα is A ANDα is A ... ANDα is A THEN y ˙ = f (y,α), 1 11 2 12 m 1m 1 IFα is A ANDα is A ... ANDα is A THEN y ˙ = f (y,α), 1 21 2 22 m 2m 2 ... ... ... IFα is A ANDα is A ... ANDα is A THEN y ˙ = f (y,α), 1 n1 2 n2 m nm n where A is the linguistic value ofα for the ith rule,α = [α ,··· ,α ] ∈ R , and y ∈ ij j 1 m R is the output obtained by the numerical solution of the corresponding diﬀerential equation (it is assumed that each diﬀerential equation in this fuzzy model locally approximates the real dynamical system over a neighborhood (or region) of R ). For example, if a complex dynamical system is modelled by using four diﬀerent mathematical models (M , M , M , M ) of the form (5), the decision scheme can be 1 2 3 4 expressed as a single-input fuzzy model [35, 36] IFα is small THEN y ˙ = f (y,α), 1 Fuzzy Inf. Eng. (2010) 2: 157-186 179 IF α is regular THEN y ˙ = f (y,α), IFα is medium THEN y ˙ = f (y,α), IF α is large THEN y ˙ = f (y,α), where the output y is obtained by the numerical solution of the corresponding diﬀer- ential equation. 3.2.3. Parametric Adaptive Control Using Neural Networks A feedforward neural network model takes an input vector X and produces an output vector Y . The input-output map NN : X → Y is determined by the network architec- ture (see, e.g., [44, 45]). The feedforward network generally consists of at least three layers: one input layer, one output layer, and one or more hidden layers. In our case, the input layer with y + 1 processing elements, i.e., one for each predictor variable plus a processing element for the bias. The bias element always has an input of one, X = 1. Each processing element in the input layer sends signals X (i = 1,··· , y+1) y+1 i to each of the q processing elements in the hidden layer. The q processing elements in the hidden layer (indexed by j = 1,··· , q) produce an ‘activation’ a = F w X j ij i where w are the weights associated with the connections between the y+ 1 process- ij ing elements of the input layer and the jth processing element of the hidden layer. Once again, processing element q+ 1 of the hidden layer is a bias element and always has an activation of one, i.e., a = 1. Assuming that the processing element in the q+1 output layer is linear, the network model will be ⎛ ⎞ p+1 p+1 p+1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Y = π x + θ F ⎜ w X ⎟ . (8) t l it j ⎜ ij it⎟ ⎝ ⎠ l=1 j=1 j=1 Here π are the weights for the connections between the input layer and the output layer, and θ are the weights for the connections between the hidden layer and the output layer. The main requirement to be satisfied by the activation function F(·)is that it be nonlinear and diﬀerentiable. Typical functions used are the sigmoid, F(x) = 1/(1+exp(−x)) and hyperbolic tangent, F(x) = (exp(x)−exp(−x))/(exp(x)+exp(−x)). Feedforward neural nets are trained by supervised training, the most popular being some form of the backpropagation algorithm. As the name suggests, the error com- puted from the output layer is backpropagated through the network, and the weights are modified according to their contribution to the error function. Essentially, back- propagation performs a local gradient search, and hence its implementation does not guarantee reaching a global minimum. A number of heuristics are available to partly address this problem, for practical purpose the best one is the Levenberg-Marquardt algorithm. Instead of distinguishing between the weights of the diﬀerent layers as in (8), we refer to them generically as w in the following. After some mathematical ij simplification the weight changeΔw equation suggested by backpropagation can be ij expressed as (see, e.g., [45, 44]) Δw = −η(∂E /∂w )+θΔw , (9) ij 1 ij ij where η is the learning coeﬀicient and θ is the momentum term. One heuristic that is used to prevent the neural network from getting stuck at a local minimum is the 180 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) random presentation of the training data. Another heuristic that can speed up con- vergence is the cumulative update of weights, i.e., weights are not updated after the presentation of each input-output pair, but are accumulated until a certain number of presentations are made, this number referred to as an ‘epoch’. In the absence of the second term in (9), setting a low learning coeﬀicient results in slow learning, whereas a high learning coeﬀicient can produce divergent behavior. The second term in (9) reinforces general trends, whereas oscillatory behavior is cancelled out, thus allow- ing a low learning coeﬀicient but faster learning. Last, it is suggested that starting the training with a large learning coeﬀicient and letting its value decay as training progresses speeds up convergence. Now, parametric adaptive control is the problem of controlling the output of a system with a known structure but unknown parameters. These parameters can be considered as the elements of a vector y.If y is known, the parameter vector of a controller can be chosen as θ so that the plant together with the fixed controller behaves like a reference model described by a diﬀerence (or diﬀerential) equation with constant coeﬀicients [37]. If y is unknown, the vector θ(t) has to be adjusted on-line using all the available information concerning the system. Two distinct approaches to the adaptive control of an unknown plant are (i) direct control and (ii) indirect control. In direct control, the parameters of the controller are directly adjusted to reduce some norm of the output error. In indirect control, the parameters of the plant are estimated as y(t) at any time instant and the parameter vector θ(t) of the controller is chosen assuming that y(t) represents the true value of the plant parameter vector. Even when the plant is assumed to be linear and time- invariant, both direct and indirect adaptive control results in non-linear systems. When indirect control is used to control a nonlinear system, the plant is param- eterized using a mathematical model of the general form (5) and the parameters of the model are updated using the identification error. The controller’s parameters in turn are adjusted by backpropagating the error (between the identified model and the reference model outputs) through the identified model. 4. Application: Fuzzy-Control Based Tennis Simulator In this section we present a fuzzy-logic model for the tennis game, consisting of two stages: attack (AT) and counter-attack (CA). For technical details, see [38]. 4.1 Attack Model: Tennis Serve A. Simple Attack: Serve Only. The simple AT-dynamics is represented by a single fuzzy associative memory (FAM) map AT TARGET −−−→ AT T ACK. FAM CAT CAT In the case of simple tennis serve, this AT-scenario reads AT O o −−−→ SR sr , m n OPPONENT−IN S ERV E−OUT Fuzzy Inf. Eng. (2010) 2: 157-186 181 where the two n−categories, O o and SR sr , contain the tempo- dim=2 m dim=3 n ral fuzzy variables {o = o (t)} and {sr = sr (t)}, respectively opponent-related m m n n (target information) and serve-related, partitioned by overlapping Gaussians, μ(z) = −(z−m) exp , and defined as: 2σ o = Opp.Posit.Le f t.Right :(center, medium, wide), O : OPPONENT−IN o = Opp.Antcp.Lf t.Rght :(runCenter, stay, runWide), sr = 1.Serve.S peed :(low, medium, high), SR : sr = 2.Serve.S pin :(low, medium, high), S ERV E−OUT sr = 3.Serve.Placement :(center, medium, wide). In the fuzzy-matrix form this simple serve reads SR: S ERV E−OUT ⎡ ⎤ O: OPPONENT−IN ⎡ ⎤ ⎢ ⎥ sr = 1.Serve.S peed ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ AT ⎢ ⎥ ⎢ o = Opp.Posit.Le f t.Right ⎥ F ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −−−→ ⎢ sr = 2.Serve.S pin ⎥ . ⎣ ⎦ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ o = Opp.Anticip.Le f t.Right ⎣ ⎦ sr = 3.Serve.Place B. Attack-Maneuver: Serve-Volley. The generic advanced AT-dynamics is given by a composition of FAM functors AT AT TARGET −−−→ AT T ACK −−−→ MANEUVER. CAT FAM CAT FAM CAT In the case of advanced tennis serve, this AT-scenario reads AT AT F G O o −−−→ SR sr −−−→ RV rv , m n p OPPONENT−IN S ERV E−OUT RUN−VOLEY where the new n−category, RV rv , contains the opponent-anticipation driven dim=2 p volley-maneuver, expressed by fuzzy variables{rv = rv (t)}, partitioned by overlap- p p ping Gaussians and given by: rv = RV.For :(baseLine, center, netClose), RV : RUN−VOLEY rv = RV.L.R. :(le f t, center, right). In the fuzzy-matrix form this advanced serve reads SR: S ERV E−OUT ⎡ ⎤ O: OPPONENT−IN RV : RUN−VOLEY ⎡ ⎤ ⎢ ⎥ ⎡ ⎤ ⎢ sr = 1.Serve.S peed ⎥ ⎢ ⎥ ⎢ ⎥ AT ⎢ ⎥ AT ⎢ ⎥ o = Opp.Posit.L.R. F G rv = RV.For ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ −−−→ −−−→ . ⎢ ⎥ ⎢ sr = 2.Serve.S pin ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ o = Opp.Anticip.L.R. ⎢ ⎥ rv = RV.L.R 2 ⎣ ⎦ 2 sr = 3.Serve.Place 4.2 Counter-Attack Model: Tennis Return A. Simple Return. The simple CA-dynamics reads: CA CA F G AT T ACK −−−→ MANEUVER −−−→ RESPONSE. CAT FAM CAT FAM CAT 182 Tijana T. Ivancevic · Bojan Jovanovic· Sasa Markovic (2010) In the case of simple tennis return, this CA-scenario consists purely of conditioned- reflex reaction, no decision process is involved, so it reads: CA CA F G B b −−−→ R r −−−→ S s , K J k BALL−IN SHOT−OUT RUNNING where the n−categories B b , R r , S s , contain the fuzzy dim=5 K dim=3 J dim=4 k variables {b = b (t)}, {r = r (t)} and {s = s (t)}, respectively defining the ball K K J J k k inputs, our player’s running maneuver and his shot-response, K.e., B: BALL−IN ⎡ ⎤ S : SHOT−OUT ⎢ b = Dist.L.R. ⎥ ⎡ ⎤ ⎢ 1 ⎥ R: RUNNING ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ s = Backhand ⎥ ⎢ ⎥ 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ b = Dist.F.B. r = Run.L.R. ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ CA ⎢ ⎥ CA ⎢ ⎥ ⎢ ⎥ F ⎢ ⎥ G s = Forehand ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b = Dist.Vert ⎥ −−−→ ⎢ r = Run.F.B.⎥ −−−→ . ⎢ ⎥ ⎢ 3 ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ s = Voley ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b = S peed ⎥ r = Run.Vert ⎣ ⎦ ⎢ 4 ⎥ 3 ⎢ ⎥ ⎢ ⎥ s = Smash ⎣ ⎦ 4 b = S pin Here, the existence of eﬀicient weapons within the S arsenal-space, namely SHOT−OUT s (t): s = Backhand, s = Forehand, s = Voley and s = Smash, is assumed. k 1 2 3 4 The universes of discourse for the fuzzy variables {b (t)}, {r (t)} and {s (t)}, par- K J k titioned by overlapping Gaussians, are defined respectively as: b = Dist.L.R. :(veryLe f t, le f t, center, right, veryRight), b = Dist.F.B. :(baseLine, center, netClose), b = Dist.Vert :(low, medium, high), B : BALL−IN b = S peed :(low, medium, high), b = S pin :(highT opS pin, lowT opS pin, flat, lowBackS pin, highBackS pin). R : RUNNING r = Run.L.R. :(veryLe f t, le f t, center, right, veryRight), r = Run.F.B. :(closeFront, f ront, center, back, f arBack), r = Run.Vert :(squat, normal, jump). s = Backhand :(low, medium, high), s = Forehand :(low, medium, high), S : SHOT−OUT s = Voley :(backhand, block, f orehand), s = Smash :(low, medium, high). B. Advanced Return. The advanced CA-dynamics includes both the information about the opponent and (either conscious or subconscious) decision making. This generic CA-scenario is formulated as the following composition + fusion of FAM functors: CA CA CA F G H AT T ACK −−−→ MANEUV −−−→ DECIS ION −−−→ RES P, CAT FAM CAT FAM CAT FAM CAT CA K ↑ FAM TARGET CAT Fuzzy Inf. Eng. (2010) 2: 157-186 183 where we have added two new n−categories, TARGET and DECIS ION, respec- CAT CAT tively containing information about the opponent as a target, as well as our own aim- ing decision processes. In the case of advanced tennis return, this reads: CA CA CA F G H B b −−−→ R r −−−→ D d −−−→ S s , K J l k BALL−IN DECIS ION SHOT−OUT RUNNING CA K ↑ O o OPPONENT−IN where the two additional n−categories, O o and D d , contain the dim=4 m dim=5 l fuzzy variables {o = o (t)} and {d = d (t)}, respectively defining the opponent- m m l l related target information and the aim-related decision processes, both partitioned by overlapping Gaussians and defined as: o = Opp.Posit.L.R. :(le f t, center, right), o = Opp.Posit.F.B. :(netClose, center, baseLine), O : OPPONENT−IN o = Opp.Anticip.L.R. :(runLe f t, stay, runRight), o = Opp.Anticip.F.B. :(runNet, stay, runBase). d = Aim.L.R. :(le f t, center, right), d = Aim.F.B. :(netClose, center, baseLine), d = Aim.Vert :(low, medium, high), D : DECIS ION d = Aim.S peed :(low, medium, high), d = Aim.S pin :(highT opS pin, lowT opS pin, noS pin, lowBackS pin, highBackS pin). The corresponding fuzzy-matrices read: B: BALL−IN D: DECIS ION ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ b = Dist.L.R. d = Aim.L.R. ⎢ 1 ⎥ ⎢ 1 ⎥ R: RUNNING ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ b = Dist.F.B.⎥ ⎢ r = Run.L.R. ⎥ ⎢ d = Aim.F.B. ⎥ 2 1 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ , , , ⎢ b = Dist.Vert ⎥ ⎢ r = Run.F.B.⎥ ⎢ d = Aim.Vert ⎥ 3 2 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ b = S peed r = Run.Vert d = Aim.S peed ⎢ 4 ⎥ 3 ⎢ 4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ b = S pin d = Aim.S pin 5 5 ⎡ ⎤ O: OPPONENT−IN S : SHOT−OUT ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ o = Opp.Posit.L.R. ⎢ s = Backhand ⎥ ⎢ ⎥ 1 ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ o = Opp.Posit.F.B. ⎢ s = Forehand ⎥ ⎢ ⎥ 2 ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ , . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ o = Opp.Anticip.L.R. ⎥ s = Voley ⎢ ⎥ ⎢ 3 ⎥ 3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ o = Opp.Anticip.F.B. s = Smash ⎣ ⎦ 4 4 5. Conclusion In this paper we have presented several control strategies for human operator and sport modelling. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2010
Keywords: Human operator modeling; Fuzzy control strategies; Fuzzy sport modeling