6 Chapter Tuning Fuzzy Logic Controllers for Robust Control System Design
6.1 Introduction The limitations of conventional controllers for application to complicated, dynamical, systems have motivated research into "intelligent" control systems. A popular technique is fuzzy control, in which expert knowledge can be incorporated into the design. An introduction to fuzzy logic and fuzzy control is provided in the Appendix A. Fuzzy logic initially proved attractive because of its capability to mimic human intelligence, and thus the concept of fuzzy systems was soon associated with many practical applications. The real boom of fuzzy technology started in the mid 80s-early 90s in Japan, where a whole range of industrial products based on approximate reasoning was developed. Well-known examples are the transport system control for the Sendai Subway, the control of robots at Fuji Electric, and a TV set based on fuzzy technology built by Sony. Although Japan can be credited with the birth of the fuzzy logic industry, elsewhere the new technology gradually gained acceptance. Alongside this historical perspective of fuzzy systems is the role played by this type of expert system in science and technology. Fuzzy systems were not intended to replace conventional methods that had wellestablished theories and many successful applications, but rather to complement them. The ultimate purpose of fuzzy systems was to tackle problems that did not have feasible or straightforward solutions using traditional methods, and to make use of human experience. In industrial examples such as the one studied here, an appeal of fuzzy control is its ability to act as a sophisticated nonlinear controller founded on some simple rules, readily understood and/or proposed by practising industrial engineers. This Chapter describes how genetic algorithms can assist in the design of fuzzy controllers and demonstrates this through a Case Study example. In fuzzy control, a genetic algorithm can be used, off-line, for (a) tuning of the membership functions, and (b) elicitation of the rule-base in addition to tuning. Method (a) is demonstrated in this Chapter. Practitioners of this approach tend to argue that the form of the rules is likely to be known a priori, and that most uncertainty lies in the development of the associated membership functions. Use of a static rule-base also reduces the necessary level of computational complexity, which may be a further reason for the popularity of this approach. Section 6.2 introduces the subject of fuzzy control and provides useful background literature references. Aspects of genetic tuning are introduced and discussed in 6.3. The control of gas turbine engines (GTEs) forms the basis of the example study used to demonstrate the application of genetic tuning. In 6.4, following an overview of GTEs and an introduction to their different types, the nature of the control problem is explained. In Section 6.5, fuzzy controllers are first derived using heuristic methods and then refined with the aid of genetic tuning. Section 6.6 concludes by explaining how the versatility of GAs enables more complex problems to be addressed, which build upon the simple example described in 6.5. Other applications of GAs in fuzzy control are also described. 6.2 Fuzzy Control When dealing with multivariable, non-linear systems, or when the information about the process is scarce/unavailable, traditional systems theory may not be able to provide efficient control. Through local linearisation, the true character of the system may be misrepresented and the control method inadequate. If the system can be approximated with local linear functions, the theory behind the control system is valid
over a limited scope and the system itself is subject to restrictive assumptions. Moreover, when the analysis and the design of a system are performed using linear control theory, an evaluation of the resulting controller on the original non-linear system is normally required and a redesign process may be necessary (Kuo, 1995). To overcome these deficiencies of classical control, and also to incorporate human skills into the control strategy, fuzzy systems may be employed as "expert", linguistic controllers. Driankov et al (1993), and Passino et al (1998) provide introductions to the subject of fuzzy control, which cover salient aspects of this discipline. The competence of fuzzy systems in control is demonstrated by the vast number of successful applications in diverse topics in control. In Kosaki et al (1997), for example, fuzzy control is applied to a magnetic bearing system. A TSK - TSK and Mamdani systems are discussed in Section 6.3 - fuzzy controller is designed by means of Lyapunov stability condition, and the performance of the resulting system is assessed through computer simulation. A similar approach is used in Leung et al (1998). Here, Lyapunov theory is used for the design of a fuzzy controller capable of stabilising a non-linear mass-spring-damper system with unknown parameters. In fact, the issue of designing stable fuzzy controllers using the Lyapunov method is widely studied (see Palm et al, 1997). Mamdani-systems are equally applicable and used in control problems. For instance, Oriolo et al (1999) use Mamdani fuzzy mappings for the real-time control of mobile robots. In this case, simple linguistic rules dictate the robot motion in a certain environment so that collisions with obstacles implanted in the robot neighbourhood are avoided. These examples are but a few in a large and growing area of research. The first major survey of fuzzy logic control and its applications was undertaken by Lee (1990). More recently, Isermann (1998) has summarised and published some of the latest achievements in this area. Before concluding this section it is also worth mentioning that fuzzy logic has proved very useful for modelling systems that are subject to uncertainty, are ill-defined or possess unmodelled dynamics. Hellendoorn and Driankov (1997) have collected and published some of the most important papers in this area and also present a very informative overview of fuzzy model identification techniques. While our example study in this Chapter focuses on fuzzy control, GAs have also proved effective in tuning fuzzy models. 6.3 Genetic tuning of fuzzy control systems A fuzzy system is comprised of three major components:
a) Fuzzification: An input interface, which realises the conversion of crisp inputs into fuzzy inputs,
by means of fuzzy sets.
b) Inference engine: A collection of fuzzy rules in the form "if <antecedent> then <conclusion>",
where knowledge about the problem is acquired. A mechanism of inference then deals with the fuzzy rules in order to generate fuzzy conclusions (consequents) from the fuzzy premises (antecedents),
c) Defuzzification: An output interface, which transforms fuzzy outputs into crisp outputs.
The general structure of a linguistic fuzzy system is shown in Fig. A.15. Fuzzy systems in which both the antecedent and the consequent rule part are expressed through fuzzy predicates are known as Mamdani or linguistic fuzzy systems. When fuzzy systems embody fuzzy and nonfuzzy descriptions of the variables of interest, the systems are referred to as Takagi-Sugeno-Kang (TSK) fuzzy systems. The latter system emerged as an alternative to Mamdani's linguistic formulation, and the idea was to alter the rule structure so that qualitative and quantitative knowledge can be equally incorporated into the knowledge base. This dual representation is realized by expressing the consequence of
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the rule in an analytical form, whilst the fuzzy antecedent part of the rule retains the linguistic formulation from the Mamdani case. The defuzzification process is not required for TSK systems, as the outputs are represented in the rule-base as mathematical expressions (usually regressions), which produce directly crisp data. The overall output of the TSK model is calculated as a weighted sum of the outputs resulting from all the contributor rules. The Mamdani formulation is adopted in the Example Study in this Chapter. A contributing factor to the success of fuzzy technology is the transparency exhibited by these systems. The human operator can understand and interpret the information contained in the rule-base, and can relate this information back to the physical proprieties of the process. All these characteristics are the result of an accumulation of factors, which include: the linguistic character of the system and the generalization of information by means of fuzzy sets, the use of heuristic rules to affect the system behaviour, and the interpretation given to the rules contained in the knowledge-base through: - the size and number of partitions of the universe of discourse, - the shape of individual membership functions, - the type of reasoning mechanism applied to the rule-base (the inference engine). Fuzzy systems are rapidly embraced by many industrial environments, as they are not only flexible and structurally simple, but also easy to understand and cheap to implement. This transparency does not extend, however, to parameters defining the precise details of the membership functions and selection of these is a prime candidate for the use of genetic search (genetic tuning). In a fuzzy system the main tuning variables are: The scaling factors (if used) similar to gain tuning in PID controllers, The position of the membership functions a few options are available here, one or more parameters can be tuned for each function. For all types of membership function, the only tuning parameter could be the crossover point (the intersection points between adjacent membership functions). For triangles or trapeziums, for instance, only the centre of gravity can be encoded and used in the optimisation process. Conversely, for these cases, the three (four) vertices can be also encoded, if applicable, The rule-base the entire rule structure, or only the consequence, typically through numerical identifiers. The number of variables to be tuned is determined by the designer, according to the knowledge of the problem and the rigour with which the search should be conducted (i.e. coarse or fine tuning). 6.4 Gas Turbine Engine Control The gas turbine engine will be the vehicle for our example study. It is a good example of an industrial nonlinear multivariable control problem and has been much studied in the literature. This Section describes some of the features of gas turbine engines and the nature of the control problem to be studied. 6.4.1 Gas turbine engines an overview Gas turbine engines (GTEs) are internal combustion heat engines (machines that convert heat energy into mechanical energy), which use air as working fluid in order to produce a propulsive jet. A GTE is made up of three main components: a compressor and a turbine placed on a common shaft, and a combustion chamber, Fig. 6.1. The role of the GTE in the aircraft is to generate thrust, by imparting momentum to the fluid passing through it (Hill and Peterson, 1992). This means that the working fluid should be expanded in order to produce a propulsive jet. Hence, the air entering the engine body is compressed by the compressor unit and
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consequently delivered to the turbines, at higher pressure. In order to increase the energy of the working fluid prior to expansion through the turbine, fuel is mixed with the compressed air and ignited at virtually constant pressure, in a combustion chamber. The burning of the air-fuel mixture results in a rise in temperature and thus an expansion of the gases. The power developed in this manner is sufficient to actuate the turbine and also to create a propulsive jet, or thrust. GTE compressors may experience surge, a destructive phenomenon that can cause excessive aerodynamic pulsations, which are transmitted through the whole machine and must be avoided at all costs. Over the entire operational range of the engine, a surge line is normally defined and used as a measure of aerodynamic stability. The surge line is made up of a number of individual surge points, corresponding to different engine speeds, which span the entire working space. 6.4.2 GTE types Engine types can be classified according to the number of compressor-turbine pairs (or spools) they employ. The three main groups are: single-spool, twin-spool or three-spool engines. The twin-spool structure is the most commonly used. In these engines, the compressor is divided in two separate units (Fig. 6.2), each of these achieving a different level of air compression. The first compressor absorbs the atmospheric air and raises its pressure by a certain extent this is the low-pressure (LP) compressor or fan. The second compressor, which is termed the high-pressure (HP) compressor, increases the air pressure still further prior to reaching the combustion chamber. A variable inlet guide vane (IGV) is used to match the air from the fan to the HP compressor characteristics. Each compressor is driven by a turbine mounted on the same shaft. The engine type employed in this case study is an example of a twin-spool turbofan engine, which is one of the most common types of GTE used for aircraft propulsion. In a turbofan engine, a portion of the air flow bypasses the engine core and is then mixed with the combusted gas from the turbine exit before being ejected through the jet pipe and nozzle area (NOZZ) to produce thrust. The thrust generated in this manner will thus have two components: a main hot thrust component coming from the engine core, and a cold stream (or fan) thrust component, resulting from the by-pass flow. The by-pass duct was designed with the aim of reducing the overall jet velocity by allowing cold air to be added to the main hot jet, and reduce its temperature. (The decrease in the propulsive speed due to by-pass air leads to better propulsive efficiency, lower noise levels and improved specific fuel consumption). Fig. 6.2 shows the mechanical layout of a typical twin spool gas turbine engine. 6.4.3 The GTE control problem From a control perspective, a GTE is a complex plant, subject to stringent constraints and strict performance requirements. Over its entire flight envelope, the engine is required to meet a number of performance specifications, while maintaining stability and safe operation with minimum overall cost. The control problem is further complicated by cross-coupling between different engine parameters, where a variation in one will disturb other control variables. The inherent non-linear dynamics, added to the multivariable nature of the engine, augment the difficulty of the control problem. The characteristics of operation of a fixed cycle gas turbine engine, such as specific thrust and specific fuel consumption, are fundamental to engine design. The design thus becomes a compromise between meeting the conflicting requirements for performance at different points in the flight envelope and the achievement of low life-cycle costs, whilst maintaining structural integrity. However, variable geometry components, such as the inlet guide vanes and nozzle area, may be used to optimise the engine cycle over a range of flight conditions with regard to thrust, specific fuel consumption, and engine life, assisting in the reduction of life-cycle costs. . Dry-engine control of a conventional engine is normally based on a single closed-loop of fuel flow for thrust rating, engine idle and maximum limiting, and acceleration control. The closed-loop control concept provides accuracy and repeatability of defined engine parameters under all operating conditions, and automatically compensates for the effects of engine and fuel system degradation. Fig. 6.3 shows the baseline configuration for a typical engine controller block. A non-linear thermodynamic model of the
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engine (realised in SIMULINK), with inputs for fuel flow (WFE), HP inlet guide vane angle (IGV) and exhaust nozzle area (NOZZ), is used to simulate dynamic behaviour. Further inputs for flight conditions (altitude, Mach number and temperature) allow the engine operation to be simulated over the full flight envelope. Sensors provided from the engine outputs are high and low pressure spool speeds (NH and NL), bypass duct Mach number (DPUP) and turbine and jet pipe temperatures (TBT and JPT). Other outputs, such as the (fan) low-pressure surge margin (LPSM) and gross thrust (XGN), are calculated directly from internal engine parameters. For the purposes of this study, we have two controllers, Fuel Flow (WFE) and Nozzle Area (NOZZ), based around a conventional proportional plus integral (PI) controller structure, see Fig. 6.3. A single input, NHDem, derived from the pilot's lever angle, is used to determine the thrust setting. The WFE controller uses this and the measured HP compressor speed, NH, to determine the required fuel flow demand. To protect the engine from over-acceleration, NHDem is rate limited. A third input, air data, is required to correct the NH value for changes in flight conditions, i.e. temperature and pressure, so that the controller can operate over the full-flight envelope. As the controller is also required to operate over a range of engine conditions, such as idle, cruise and full power, the fuel flow controller uses gain schedules to accommodate these non-linearities in the system dynamics. For the NOZZ controller, the demand input is NL, the fan speed, which is mapped to a desired value of DPUP. A measurement of DPUP is the second input and a comparison of the two is used to determine the position of the variable geometry in components the jet pipe and hence the nozzle area. Adjustment of NOZZ may be used to alter the pressure distributions in the engine and thus to trim the LSPM or thrust level. As for the PI fuel flow controller, gain schedules are used to cover the range of operating conditions. (However, we are only considering a single operating point in this example.) In recent years, a considerable amount of research has been directed towards the design of controllers for GTEs in an attempt to improve performance and, at the same time, allowing reduced production costs. In the next section, an Example investigates the use of fuzzy systems for the gas turbine engine control, and looks at the capability of the proposed control configuration to improve performance. Genetic algorithms, as tools for search and optimisation, assist the design process. 6.5 Fuzzy Control System Design Example Study 6.5.1 Problem formulation Here, we demonstrate, through a simple example, the feasibility of fuzzy controllers for GTEs, using a genetic algorithm to tune the desired fuzzy control law. For simplicity, a single 85% high-pressure spool speed (NH) operating point is considered at sea level static conditions. The control options considered are the replacement of the PI-based control loops for the WFE and NOZZ loops with Mamdani fuzzy equivalents. As we have seen, the control of the fuel loop is based on measurement of NH, whilst the nozzle area loop uses measurements of the bypass duct Mach number (DPUP) and fan speed (NL). For this particular operation, the system is required to meet the following design objectives: (1) 70% NH rise-time (2) 10% NH settling time < a (3) XGN x (4) JPT (5) LPSM E G sec r H sec
where objectives (1) and (2) are in response to a change in high pressure spool speed demand of 85% to 90% and represent typical dynamic performance requirements for a military engine. Engine thrust, XGN, is the main control parameter to be optimised; recall that it is not measured, rather it is derived from internal engine parameters. JPT is the maximum temperature of the jet pipe and is employed as a measure of thermodynamic stress. A lower value of JPT indicates less stress and therefore a longer engine life. Finally, LPSM is the fan surge margin and is employed as a measure of aerodynamic stability and hence safety;
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again it is not measured, instead it is also derived from internal engine parameters. The design process considers the replacement of the PI WFE and NOZZ controllers simultaneously. Suitable fuzzy structures are found by analysing the shapes and the domains of the controllers' inputs and outputs, thereby formulating linguistic rules that describe their behaviour. The resulting hand-tuned fuzzy controllers should be able to replicate accurately the input-output behaviour of the engine's PI controllers. In this study, then, first fuzzy controllers are obtained via a heuristic approach (Section 6.5.2) and next the controllers are tuned using a GA search technique (Section 6.5.3). 6.5.2 Heuristic design of the fuzzy controllers The design objective is the construction of fuzzy logic mappings that emulate the input-output behaviour of the baseline PI controllers. Hence, the fuzzy partitions and rule base are derived by observing the initial relationships between the inputs and outputs of each engine controller. The first design step is to set the limits of the universe of discourse for each fuzzy parameter. In the original model, for example, NHDem varies within the interval [85, 92], Fig. 6.4. The fuzzy universe of discourse is defined so that it includes this range with a certain tolerance in either sides of the real axis: [80, 95], Fig. 6.5. In general, if { a, b} are the limits of the original variable, then the fuzzy universe of discourse for the corresponding fuzzy variable has the form: {a tola.a, b + tolb.b}. Here tola and tolb represent the lower/upper interval tolerance, expressed as a percentage. The shapes of the WFE input-output variables, Fig. 6.4 (a) are relatively simple and follow a similar pattern. These observations suggest that three membership functions for each variable should be sufficient to build the fuzzy system. The membership functions are positioned intuitively at this stage of the design process, Fig. 6.5. In the WFE controller, the NH demand input is rate limited to provide protection from over-acceleration and thus has a slower dynamic response than the measurement of NH. This remark, added to observations of the signals characteristic, is used to formulate the membership functions and the rule base, Table 6.1. For example, in Table 6.1 and Fig. 6.5, at the extremes of the manoeuvre when both inputs are either low or high, the fuel flow should be low or high accordingly. Likewise, the fuzzy nozzle area controller is constructed from observation of the input-output relationship of the PI controller, Fig. 6.6 and Table 6.2. Additionally, it may be noted that the number of rules required for adequate control is small, as the manoeuvre is only around a small area of the full flight envelope. The fuzzy controllers are not therefore necessarily designed to implement a PIlike controller. These observations apply to both fuzzy controllers. Fig. 6.7 (a) and (b) depict the outputs of the fuzzy WFE and NOZZ controllers, which are indicated with a dashed line, plotted against the original PI controllers outputs, represented by a continuous line. The results presented in Fig. 6.7 indicate that the empirically derived fuzzy controllers are capable of mimicking the original controllers' outputs. Furthermore, the responses of the fuzzy systems are smoother in comparison with the PI controller's outputs. The steady-state error of each output is zero, indicating a good system response. After adequate fuzzy structures are obtained, their ability to control the plant is verified on the SIMULINK non-linear thermodynamic engine model. The performance of the PI and fuzzy controllers is compared in Fig. 6.8 for the NH step response. It can be observed that the fuzzy controller, denoted by a dashed line, offers a satisfactory level of control for this manoeuvre, meeting the specified design requirements whilst offering marginally improved LPSM and JPT characteristics. This means that in the fuzzy controller case the LPSM is slightly increased, whilst the JPT has a lower average value that in the PI controller case. 6.5.3 GA tuning of the fuzzy controllers It is to be expected that, following such a heuristic approach to fuzzy control design, it will be commonplace to want to apply some tuning to the selected parameters. Genetic algorithms are natural
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search engines to select for this task. Thus, having designed fuzzy controllers for the operating point of interest that satisfactorily approximate the original PI controllers' responses, the rule base and membership functions are now tuned. The encoding of each fuzzy rule for use in the GA is made up of a coding of the linguistic consequent part of the rule and the position of the membership functions. An individual encodes the parameters of both controllers and is therefore composed of a set of 15 rules (see Tables 6.1 and 6.2) represented by integer identifiers and 6 output membership functions represented with real-values for their co-ordinates in the parameter space, Fig. 6.9. X_WFE_i denotes the encoded co-ordinates in Fig. 6.9. The limits within which each co-ordinate is allowed to vary are defined by the crossover points of the adjacent membership function and are also indicated in the Figure. The universe of discourse of each parameter is lower and upper bounded in order to limit the size of the search space and to ensure that the membership functions cover all possible values of the control parameters. As a result, the lowest and the highest values of the co-ordinates do not change during tuning, as indicated. Simple GAs with populations of 40 real-valued individuals were employed. The individuals were encoded as described above with the addition of constraints on the relative positions of the membership functions being included directly in the representation. The recombination operator was applied with a probability of 0.9 during recombination, and mutation was then employed with a probability of 0.1. The single objective to be maximised was the system thrust resulting from the engine simulation. Thus, this is a reasonably compute-intensive approach, requiring one simulation (using SIMULINK) to evaluate the objective function, thrust, for each individual used in the search. Since the GA is a stochastic search method, a series of 7 runs were performed, for result analysis purposes. Fig 6.10 illustrates the behaviour of the gas turbine engine with GA-tuned fuzzy controllers. The thick line indicates the system performance with the original PI controllers and the alternative GA-tuned fuzzy controllers with fine lines. Each fine line curve indicates a solution resulting from one GA run. Comparing results of Fig.6.8 (heuristic design) with those of Fig. 6.10 (GA-tuned design), it can be seen that the target objective, thrust, is greatly improved in the GA-tuned fuzzy control approach. From the optimisation point of view, the goal of maximising engine thrust is therefore achieved. The considerable difference in engine thrust in the hand-tuned (heuristic) and GA-tuned approaches, as seen in Fig. 6.10, is attributed to different settings of nozzle area. A reduction in the nozzle area improves thrust, but has the shortcoming of decreasing the surge margin and increasing the jet pipe temperature. Therefore, lower surge margins and higher jet pipe temperatures are the price for improving the engine thrust. This observation also points to the fact that conflicts exist between different control objectives; an improvement in one objective leads to degradation in other objectives. This finding suggests that the inherent trade-offs in the GTE should be taken into account in the design and optimisation process. Moreover, a simultaneous consideration of various conflicting objectives could offer the engineer a better insight into the nature of these trade-offs and the choices between optimal design solutions. In Chapter 9, we introduce the multiobjective genetic algorithm, which can be used to address such multiple objectives. 6.6 Applications of GAs for Fuzzy Control The example used in this Chapter has demonstrated an approach to the design of fuzzy controllers for a gas turbine engine based on plant knowledge, and observations of the plants behaviour for a s...