Fuzzy Application Library/Technical Applications/Automotive Engineering
Fuzzy Logic in Automotive Engineering
by Constantin von Altrock
Citation Reference: This paper was published at the Embedded Systems Conferences in 1994, 1995, and 1996 in Santa Clara.
Fuzzy Logic is a powerful technology which allows designers to quickly incorporate engineering expertise into products. In this presentation, we focus on automotive applications. We will discuss development methodologies, tools used, and code speed/size requirements of three case studies. Also, we will compare dedicated hardware for fuzzy logic systems to software implementations. A final discussion of stability analysis and verification of fuzzy logic systems concludes the presentation.
The first case study of a fuzzy logic anti-lock braking system (ABS) shows how fuzzy logic and conventional design techniques can complement each other. Fuzzy logic is used to optimize the existing ABS system to achieve a better performance in all braking conditions. An estimation system for the road surface state has been built that parameterizes the conventional ABS. Road tests show that the fuzzy logic estimator is able to improve the performance of the conventional ABS, especially on dry surfaces.
The second case study of an engine control system focuses on the engineering process of building up a fuzzy logic system. We show how engineering expertise and experimental results can be used together with model-based system design to reduce time-to-market. Also, we present design technologies and tools that expedite design and support verification of fuzzy logic systems.
The third case study of an automatic gearbox control gives an outlook into future user adaptive systems. Such a system automatically identifies the user's needs in any situation and adapts the control strategies to his specific requirements and preferences. The gearbox control example shows that fuzzy logic is an enabling technology for these applications.
A fourth case study concerning an anti-skid steering system exemplifies what further innovations can result from the use of fuzzy logic in automotive engineering.
1. Fuzzy Logic in Automotive Engineering
One of the areas where fuzzy logic is a common design technology is the automotive industry. In Japan, Germany and France, cars with fuzzy logic controlled components already sell very successfully.
The reasons are manifold: first, control systems in cars are complex and involve multiple parameters. Second, the optimization of most systems is based on engineering expertise rather than mathematical models. "Good handling", "Fahrvergnügen", and "riding comfort" are optimization goals that cannot be defined mathematically. Third, automotive engineering is very competitive on an international scale. A technology that provides a competitive advantage is soon commonly used.
The following discussion assumes the reader is familiar with basic fuzzy logic design principles. For a comprehensive hands-on course on practical fuzzy logic design, refer to [13].
2. Anti-Lock Braking Systems with Fuzzy Logic
In 1947, Boeing Corporation developed the first anti-lock braking system for airplanes as a mechanical system. Today, anti-lock braking systems (ABS) are standard equipment with most cars. They use electronic sensors that measure the speed of every wheel and microcontrollers to control the fluid pressure for the brake cylinders. Mathematical models for the braking system of a car do exist, but the interaction of the braking system with the car and the road is by far too complex to model adequately. Hence, today's anti-lock braking systems contain the engineering knowledge of years of testing in different roads and climates.
Fuzzy Anti-lock Braking Systems in Production
Fuzzy logic is a very efficient technology for putting engineering knowledge directly into a technical solution. Hence, it is no surprise that already many applications with anti-lock braking systems are on the market. Currently Nissan and Mitsubishi ship cars with fuzzy ABS. Honda, Mazda, Hyundai, BMW, Mercedes-Benz, Bosch and Peugeot are working on solutions.
Another reason why anti-lock braking systems benefit from fuzzy logic is the high computational efficiency of its implementation. In an ABS, the control loop time is in the order of 5 milliseconds. Within this interval, the microcontrollers must fetch all sensor data, preprocess it, compute the ABS algorithm, drive the bypass valves, and conduct the test routines. Any additional functions must be very computationally efficient. Most ABS systems use 16 bit microcontrollers, which can compute a medium size fuzzy logic system in about ½ millisecond using only about 2 KB ROM space.
Adapting the ABS to the Road Surface
The implementations of fuzzy logic ABS are very different. The implementation of Nippondenso [4], which I am now presenting as a case study, also exhibits an intelligent combination of conventional techniques with fuzzy logic. First, some basics of the braking process.
Figure 1: Plot of brake effect over the wheel slack s for dry, wet and
snowy road surfaces (µ: friction coefficient, measure of brake effect) (large |
If a wheel rotates exactly as fast as it corresponds to the speed of the car, the wheel is experiencing no braking effect at all. If a wheel does not rotate at all, it is blocked. This situation has two disadvantages. First, a car with blocked wheels is impossible to steer. Second, the brake effect is not optimal. The point of optimum brake effect is between these two extremes.
Road condition | Optimum slack |
Dry road | 0.2 |
Slippery or wet asphalt | 0.12 |
Ice or Snow | 0.05 |
Figure 2: The Slack Value for Maximum Brake Effect Depends on the Type of Road
The speed difference between the car and the wheel during braking is called "slack". Its definition is:
s = ( V_{Car} - V_{Wheel} ) / V_{Car}where:
s : slack, always between 0 (no
braking) and 1 (blocking)
V_{Car} : velocity of the car
V_{Wheel} : velocity of the wheel
Figure 1 plots the relation between brake effect and slack for different road surfaces. For s = 0, the speed of the wheel equals the speed of the car. In the case of s = 1, the wheel blocks completely. The curves show that the optimum brake effect lies between these two extremes. However, the point of maximum brake effect depends on the type of road. Figure 2 lists typical values.
Estimation of Road Surface using Fuzzy Logic
A conventional anti-lock braking system controls the bypass valves of the brake fluid so that the slack equals a set value. Most manufacturers program this set value to a slack of 0.1, because this is a good compromise value for all road conditions. As figures 1 and 2 show, this set value is not optimal for every road type. By knowing the road type, the braking effect could further be enhanced.
The problem is how to know what the road type is. Asking the driver to push a button on the dash board before an emergency brake is not feasible. An alternative is the use of sensors. Many companies have evaluated different types of sensors. The result is that sensors that deliver good road surface identification are not robust enough and are too expensive.
The idea of the cited fuzzy logic application is simple. Consider sitting in your own car equipped with a standard ABS. After driving at a known speed, you would jam the brake pedal such that the ABS starts to work. Even if you would not know what the road surface is like, you could now have a good guess just from the reaction of the car. Now, if you can estimate the road surface just from the car's reaction, why not implement this in the ABS using fuzzy logic?
This is just what Nippondenso did. When the ABS first detects blocking of a wheel, it starts to control the brake fluid valves so that each wheel rotates with a slack of 0.1. The fuzzy logic system then evaluates the reaction of the car to the braking and estimates the current road surface. Considering this estimation, the ABS then corrects the set value for the slack so that it achieves the best braking effect.
The fuzzy logic system only uses input data that stems from the existing sensors of the ABS. Such input variables are deceleration or speed of the car, deceleration or speed of the wheels and hydraulic pressure of the brake fluid. These input variables are an indirect indicator of the current operation point of the braking (Figure 1) and its behavior over time. Experiments showed that a first prototype with just six (6) fuzzy logic rules already improved performance significantly. One test track alternates from snowy to wet road. Here, the fuzzy logic ABS detected the change even during braking.
Other Applications of Fuzzy Logic in ABS
Due to the high competition in this area, most manufacturers are reluctant to publish any details about the technologies they use. The cited application only shows results from an experimental fuzzy logic system. The details about the final product are not published.
Also, some companies worry about the negative connotation of the word "fuzzy". As it implies "imprecision" and "inexact", manufacturers are afraid drivers could think of a "fuzzy ABS" as something inferior. Others feel threatened by the scenario in which a clever lawyer sues them by suggesting to a laymen's jury that fuzzy logic ABS is something hazardous. In Japan, where an appreciation for ambiguity lies in the culture, "fuzzy" does not have a negative connotation. In contrast, it is considered an advantage because it enables intelligent systems such as the cited application. Hence, companies are proud of its use and even use it in advertising
In Germany for example, the situation is different. Here, the concepts of fuzziness and engineering masterpiece do not fit in the public perception. Hence, most manufacturers that use fuzzy logic in ABS actually hide the fact. After all, a fuzzy logic system is only a segment of assembly code in a microcontroller once implemented. Who could actually tell that this code implements fuzzy logic? Some engineers already talk about "hidden fuzzy logic"
Stability of a Fuzzy ABS
Some engineers, especially those who went to a well-reputed university, argue the stability of fuzzy logic systems. For a discussion on stability analysis of fuzzy logic systems, refer to [13]. In the case of the Fuzzy ABS shown, this is not an issue. The conventional ABS was considered to be stable for any slack value in the interval from 0.05 to 0.25. Hence, a fuzzy logic road surface estimator that only tunes this value to the optimum can principally not make this system unstable.
3. Engine Control with Fuzzy Logic
The control of car and truck engines becomes increasingly more complex due to higher emission standards and the constant push for higher fuel efficiency. Twenty years ago, control systems were mechanical (carburetor, distributor and breaker contact). Now, microcontroller based systems control fuel injection and ignition point. Since the control strategy for an engine depends strongly on the current operating point (revolutions, momentum,...), linear control models, such as PID, are not suitable. On the other hand, no mathematical model that describes the complete behavior of an engine exists. Because of this, most engine controllers use a look-up-table to represent the control strategy. The look-up-table is generated from the results of extensive testing and the engineer's experience.
The generation of such a look-up-table, however, is only suitable for three dimensions (2 inputs, 1 output). Also, the generation and interpretation of these look-up-tables is difficult and considered a "black art". Replacing these look-up-tables hence is a potential for fuzzy logic. Alas, most manufacturers are unwilling to publish any details on their fuzzy logic engine control solution. This is due to the fact that the rules of the fuzzy logic system contain the entire engine control knowledge of the company and are completely transparent. Hence, manufacturers are afraid that their competitors could learn too much about the solution by disassembling the fuzzy logic rules.
Identification of Driving Condition
The case study of an engine control system by NOK and Nissan [3] illustrates the benefits of using fuzzy logic. Figure 3 sketches the components of the engine controller that contains three fuzzy logic modules. The basic idea of the system is that it first identifies the operational condition of the engine by the linguistic variable "Situation" that has the following linguistic terms:
linguistic variable Situation {
Term 1: Start
Control strategy is that the cold engine runs smooth -
ignition is timed early and the mix is fat;
Term 2: Idle
Control ignition timing and fuel injection depending on engine
temperature to ensure that the engine runs smooth;
Term 3: Normal drive, low or medium load
Maximize fuel efficiency by meager mix, watch knocking;
Term 4: Normal drive, high load
Fat mix and early ignition to maximize performance - the only
constraint is the permitted emission maximum;
Term 5: Coasting
fuel cut-off, depending on situation;
Term 6: Acceleration
Depending on the load, fattening of the mix
}
The determination of the linguistic variable "Situation" is a state estimation of the operation point. As "Situation" is a linguistic variable, more than one term can be valid at the same time. This allows for expressing combinations of the operational points defined by the terms. Hence, a possible value of "Situation" could be: {0.8; 0; 1; 0; 0; 0.3}. Linguistically, this value represents the driving condition engine started just a short while ago, normal drive condition at medium or low load, slightly accelerating. From this operation point identification, the individual fuzzy logic modules control injection, fuel cut-off, and ignition.
Similar to ABS systems described in the previous section, engine control requires a very short loop time. Some systems are as fast as 1 millisecond for an entire control loop. For this, some manufacturers design the system using fuzzy logic, but then translate it into a look-up-table for faster processing. Even though, a look-up-table is very fast to compute, the memory requirements may be prohibitive. For a look-up-table with 2 inputs and 1 output, all 8 bit resolution already requires 64 Kbytes of ROM. Restricting the resolution of the input variables to 6 bit each, the look-up-table still requires 4 Kbytes. A look-up-table with 3 inputs and 1 output, all inputs of 6 bit resolution would require ¼ Mbyte. Some engineers tried to implement a look-up-table with a very limited resolution and to use an interpolation algorithm. Comparison, however, shows that the interpolation requires about as much computing time as the fuzzy logic system itself [5].
Figure 3: Modules of the Fuzzy Logic Engine Controller of NOK (large) |
Another published application of fuzzy logic in engine control is an idle control unit by Ford Motor Corporation [1].
4. Adaptive 5-Speed Automatic Transmission
When the first 3-speed automatic transmissions appeared on the market about 30 years ago, the engine power of most cars was just sufficient enough to keep the car in pace with traffic. The necessity of getting maximum momentum from the engine determined the shift points for the gears. Now that most car engines can deliver much more power than necessary to keep the car in pace with traffic, automatic transmission systems have up to five (5) speeds and fuel efficiency is an important issue, plus the control is much more complex.
The higher engine power and five (5) speeds to chose from give the automatic transmission system a much higher degree of freedom. Driving at 35 mph (56 km/h), a 3-speed automatic transmission has no choice but selecting the second gear. A 5-speed automatic transmission with a powerful engine could select the second gear if maximum acceleration is required, the third gear in a normal driving condition, and the fourth gear if only a little acceleration is needed.
Acceleration vs. Fuel Efficiency
Unfortunately, the goal for the control strategy is in a dilemma. For maximum fuel efficiency, you must select the next higher gear as early as possible. For maximum performance, you switch to the next higher gear later. If you have a shift gear box, you can choose the strategy depending on the traffic conditions. An automatic gearbox has no understanding of the traffic conditions and the driver's wishes.
In this section, I will show how intelligent control techniques can enhance automatic transmissions. As this intelligent control technique is based on experience and engineering knowledge rather than on mathematical models, fuzzy logic proves to be an efficient technology for implementation. Existing applications are used as case studies. In 1991, Nissan introduced fuzzy logic controlled automatic 5-speed transmission systems [9, 2]. Honda followed one year later [8] and GM/Saturn first introduced its solution in a car in 1993.
The job for the fuzzy logic system in these applications is similar:
Figure 4: A 5-speed automatic transmission with fixed shift points always switches between 4th and 5th gear on a winding road. A driver with a shift gear box would stay in the 4th gear. (large) |
Figure 4 shows a typical situation on a winding road. While a driver with a shift gear box would stay in the 4th gear, a 5-speed automatic transmission switches between the 4th and the 5th gear.
The fuzzy logic controller in the automatic transmission evaluates more than just the current speed of the car. It also analyzes how the driver accelerates and brakes. To detect the condition of a winding road for instance, the fuzzy logic controller looks at the number of accelerator pedal changes within a certain period. Figure 5 shows the definition of the linguistic variable "Accelerator pedal changes". Also, the variance of the accelerator pedal changes is an input to the fuzzy logic controller.
Figure 5: Classification of driving condition using a linguistic variable. The variable linguistically interprets the amplitude of accelerator pedal changes within a certain period (large) |
Some of the rules that estimate the driving conditions from these input variables are:
The interesting part of this application is that it uses the driver as the actual sensor for the driving condition. The fuzzy logic controller only interprets the driver's reaction to the driving condition and adapts the car's performance accordingly. This could be used as the definition of an intelligent control system. The technical system tries to understand whether the human is satisfied with its performance. If not, the technical system adapts itself to suit the needs of the human that uses it.
"Intelligent" Automatic Transmissions
Another example of an automatic transmission system, currently under development in Germany, shows this even better. The following situation is typical, if the driver is not satisfied with the acceleration of a car. The driver pushes down the accelerator pedal and within 1 to 1.5 seconds, he pushes the pedal down even a little bit more. This is the subconscious reaction to a non-satisfying acceleration. Most drivers do not even consciously realize that they like the car to accelerate faster. If the automatic transmission system is capable of detecting this, it could move the shift points higher to achieve more acceleration. The opposite case is similar. If the automatic transmission detects that the driver accelerates very carefully and takes the foot off from the accelerator long before red lights, chances are that the driver wants high fuel efficiency.
Why Fuzzy Logic?
The question remains, why do you need fuzzy logic to implement these intelligent functions? The answer is, certainly you can use other techniques to implement these control strategies. But fuzzy logic may be the most efficient one:
Active stability control systems in cars have a long history. First, anti-locking brakes (ABS) improved braking performance by reducing the amount of brake force to what the road conditions can handle. This avoids sliding and results in shorter braking distances. Second, traction control systems that do essentially the same thing improved acceleration. Reducing engine power applied to the wheels to what the road can handle, it maximizes acceleration and minimizes tire wear. The next logical step, after skid-controlled braking and skid-controlled acceleration, is skid-controlled steering. Such an anti-skid steering system (ASS) reduces the steering angle to the amount the road can take. As such, it optimizes the steering action and avoids sliding. A sliding car is very hard to stabilize, especially for drivers not used to it.
Figure 6: Model car for high-speed driving experiments (large) |
Though an anti-skid steering system (ASS) makes a lot of sense from a technical point of view, such a system is hard to market. For an ABS, one can prove that it never performs worse than a traditional brake system. For an ASS, this is very hard to prove. Also, it may be difficult to sell cars that "take over the steering" in emergency situations. Even ABS faced a long period of rejection by customers, because they felt uneasy about a system inhibiting their brake action. For these reasons, it may take a long time before an ASS system is implemented in a production car. All results shown in this section stem from a research project of a German car manufacturer [10]. As this system is one of the most complex fuzzy logic embedded systems, it shows well the potential of the technology.
The Test Vehicle
Real experiments were made on a modified Audi sedan and on a 20" model car (Figure 6). In the following, I will only present the results derived from the model car experiments. A one-horsepower electric motor powers the car, rendering the power-to-weight-ratio of a race car. This allows it to perform skidding and sliding experiments in extreme situations at high speeds. On a dry surface, the car reaches a velocity of 20 mph in 3.5 seconds with top speed up to 50 mph. The speed for most experiments ranges from 20 to 30 mph. Each wheel features individual suspension and has a separate shock absorber. The car has disk brakes and a lockable differential [10].
Figure 7: Three ultrasound sensors guide the car on the track (large) |
The controller of the car uses the motherboard of a 12 MHz notebook 286-PC connected to an interface board that drives actors and sensors. Actors are power steering servo, disk brake servo, and pulse width modulated motor control. Sensors are three ultrasound distance sensors for track guidance and infrared reflex sensors in every wheel for speed. The control loop time, from reading in sensor signals to setting the values for the actors, is 10 milliseconds.
To measure the dynamic state of the car, such as skidding and sliding, infrared sensors measure the individual speed of all four wheels. Evaluating wheel speed differences, the fuzzy logic system interprets the current situation. Three fixed mounted ultrasound sensors measure the distance to the next obstacle to the front, the left and the right. This allows for an autonomous operation of the car. Intentionally, low-cost sensors have been used in this study rather than CCD cameras and picture recognition techniques, to prove that expensive sensors can be replaced by a fuzzy logic control strategy.
Figure 8 shows the example of an experiment involving the model car. The obstacle (block) is placed right after the curve, so that the ultrasound sensors of the car will "see" the obstacle too late. To not hit the obstacle, the car has to decide for a very rapid turn. To optimize the steering effect, the anti-skid controller has to reduce the desired steering angle to the maximum the road can take, avoiding both sliding and hitting the obstacle.
Figure 8: Example of an experiment. The ultrasound sensors of the car detect the obstacle placed right after the curve very late, making a quick turn necessary (large) |
Model Based Solution vs. Fuzzy Logic Control
In theory, you can build a mechanical model for the car and derive a mathematical model with differential equations to implement a model based controller. In reality, the complexity of this approach would be overwhelming and the resulting controller would be very hard to tune. Here is the point for fuzzy logic: race car drivers can perform this control task very well without solving differential equations. Hence, there must be an alternative way for anti-skid steering control. This alternative way is to represent the driving strategy in engineering heuristics.
Though there are multiple ways of expressing engineering heuristics, fuzzy logic has proven to be very effective, due to the following reasons:
Design and Implementation of a Fuzzy Logic Controller
Figure 9 shows the first version of a fuzzy logic controller for the car. The objective for this controller was autonomous guidance of the car in the track at slow speed, where no skidding and sliding occurs. In Figure 9, the lower rule block uses the distances measured by the three ultrasound sensors to determine the steering angle. The upper rule block implements a simple speed control by using the distance to the next obstacle measured by the front ultrasound sensor and the speed of one front wheel only. Due to the slow speeds, no skidding or sliding occurs. Hence, all wheel speeds are the same. The first version contains about 200 rules and took only a few hours to implement.
The second version of a fuzzy logic controller implements a more complex fuzzy logic system for dynamic stability control. It includes anti-locking brakes, traction and anti-skid steering control (Figure 10). This 600 rule fuzzy logic controller has two stages of fuzzy inference. The first stage, represented by the three left rule blocks, estimates the state variables of the dynamic situation of the car out of sensor data. The two lower rule blocks estimate skidding and sliding states out of speed sensor signals, while the upper rule block estimates position and orientation of the car in the test track. Note, the output of the left three rule blocks is linguistic rather than numerical. An estimated state of the car therefore could be: "the position is rather left, while the orientation is strongly to the right, and the car skids over the left front wheel".
The second stage, represented by the three right rule blocks, uses these estimations as inputs to determine the best control action for this situation. The upper rule block determines the steering angle, the middle one the engine power to be applied, and the lower one controls the brake. Such a two-stage control strategy is similar to the human behavior, which first analyzes the situation and then determines the action. Also, this allows for efficient optimization, since the total of 600 rule structures into 6 rule blocks that can be designed and optimized independently.
Figure 10: Second version of the Fuzzy Logic Controller. The controller uses advanced fuzzy logic design technologies and contains a total of 600 rules (large) |
The first version of the controller was only able to guide the car for autonomous cruising (Figure 9). The second version also succeeded to dynamically stabilize the car's cruising with anti-locking brakes, traction control, and anti-skid steering. However, this version required a lot longer design time before the results were completely satisfactory. The second version also uses advanced fuzzy logic technologies, such as FAM inference [7] and the Gamma aggregational operator [14].
Figure 11: The fuzzyTECH Online Edition features both visualization of the running system and modifications "on-the-fly" (large) |
Online Development
The development of the fuzzy logic system involved the fuzzyTECH Online Edition [6]. After the graphical definition of the system structure (Figure 9 and 10), the linguistic variables and the rule bases, then the compiler of fuzzyTECH generated the system as C code. This code was compiled and implemented on the PC mounted in the car. Figure 11 shows how the running fuzzy logic system was modified "on-the-fly" for optimization. The fuzzy logic code is separated into two segments. One contains all "static" code, which is code that does not need to be modified for system modifications. The other one contains all "dynamic" code, which is the code containing the membership functions of the linguistic variables, the inference structure, and the rules. The "dynamic" segment is doubled with only one of the segments active at a time. As such, the parser, linked to the development PC via a communications manager, can modify the inactive code segment. This allows for modifications of the running system without halting or compiling. At the same time, the entire inference flow inside the fuzzy logic controller is graphically visualized on the PC, since the communications manager also transfers all real-time data.
On the example of a anti-skid steering system, we have shown the applicability of fuzzy logic technologies for a complex control problem found in the automotive industry. The design of the fuzzy logic system has been done straightforward and without a mathematical model of the process. Existing engineering heuristics were implemented in fuzzy logic rules and linguistic variables. During optimization, we found the control strategy easy to optimize due to the linguistic representation in the fuzzy logic system. Tests and verification were expedited due to the transparency of the controller. The poor computational performance of early fuzzy logic software solutions has been overcome with the new generation of software implementation tools.
6. Literature
[1] | Feldkamp, L. and Puskorius, G., "Trainable fuzzy and neural-fuzzy systems for idle-speed control", Second IEEE International Conference on Fuzzy Systems, ISBN 0-7803-0615-5, p. 45 - 51. |
[2] | Ikeda, H. et. al., "An intelligent automatic transmission control using a one-chip fuzzy inference engine", Proceedings of the International Fuzzy Systems and Intelligent Control Conference in Louisville (1992), p. 44 - 50. |
[3] | Kawai, H. et al., "Engine control system", Proc. of the Int'l Conf. on Fuzzy Logic & Neural Networks, IIZUKA, Japan (1990), p. 929 - 937. |
[4] | Matsumoto, N. et. at., "Expert antiskid system", IEEE IECON'87 (1987), p. 810 - 816. |
[5] | N.N., "Benchmark Suites for Fuzzy Logic", Working Group Protocol VDI/VDE GMA (German Association of Mechanical and Electrical Engineers) UA 4.8.1 (1994). |
[6] | N.N., "fuzzyTECH 3.2 Online Edition Manual", INFORM GmbH Aachen / Inform Software Corp., Chicago (1994). |
[7] | N.N., "fuzzyTECH 3.2 NeuroFuzzy Module Manual", INFORM GmbH Aachen / Inform Software Corp., Chicago (1994). |
[8] | Sakaguchi, p. et. al., "Application of fuzzy logic to shift scheduling method for automatic transmission", Second IEEE International Conference on Fuzzy Systems, ISBN 0-7803-0615-5, p. 52 - 58. |
[9] | Takahashi, H., Ikeura, K. and Yamamori, T., "5-speed automatic transmission installed fuzzy reasoning", IFES'91 - Fuzzy Engineering toward Human Friendly Systems, p. 1136 - 1137. |
[10] | von Altrock, C., Krause, B. and Zimmermann, H.-J. "Advanced fuzzy logic control of a model car in extreme situations", Fuzzy Sets and Systems, Vol 48, Nr 1 (1992), p. 41 - 52. |
[11] | von Altrock, C. and Krause, B., "On-Line-Development Tools for Fuzzy Knowledge-Base Systems of Higher Order", 2nd Int'l Conference on Fuzzy Logic and Neural Networks Proceedings, IIZUKA, Japan (1992), ISBN 4-938717-01-8. |
[12] | von Altrock, C., Arend, H.-O., Krause, B., Steffens, C., and Behrens-Rommler, E., "Customer-Adaptive Fuzzy Control of Home Heating System", IEEE Conference on Fuzzy Systems in Orlando (1994). |
[13] | von Altrock, "Fuzzy Logic and NeuroFuzzy Applications Explained", ISBN 0-1336-8465-2, Prentice Hall 1995. |
[14] | Zimmermann, H.-J. and Thole, U., "On the suitability of minimum and product operators for the intersection of fuzzy sets", Fuzzy Sets and Systems, 2, p. 173-186. |