Real-Time Systems, 21, 219±239, 2001 # 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

Implementation of Adaptive Fuzzy Control for a Real-Time Control Demo-Model DA RUAN Belgian Nuclear Research Centre (SCK ? CEN), Boeretang 200, B-2400 Mol, Belgium

[email protected]

Abstract. From 1995±1999 a R&D project on fuzzy control applications to the Belgian Reactor 1 (BR1) was conducted at the Belgian Nuclear Research Centre (SCK ? CEN). Due to the safety regulations of the nuclear reactor, it is not realistic to perform many experiments at BR1. In this situation, part of the pre-processing experiments had to be carried outside the reactor (e.g., comparisons of different methods and the preliminary choices of the parameters). Therefore a water-level control system, referred to as a real-time control demomodel, was designed and constructed. In this paper, the construction of the demo-model and related hardware aspects is ®rstly outlined, then the results of a fuzzy control (Mamdani-type) and an adaptive fuzzy control are presented. The adaptive fuzzy control is a fuzzy control with an adaptive function that can self-regulate the fuzzy control rules. Finally, an implementation of a computer simulation is introduced with an adaptive fuzzy control for this real-time control demo-model. Keywords: fuzzy control, adaptive fuzzy control, real-time control, adaptive function, computer simulation

1.

Introduction

In many real-world situations, there is a need to automatically control a real-time system such as a car, a nuclear power reactor, or a physical plant. In some situations, the exact model of the controlled system is known, how this system will react to different controls is also exactly known, and the objective of the controlÐusually to optimize a certain characteristic such as the plant's output can be precisely described. In such cases, this optimization problem can be explicitly solved by searching for the optimal control strategy that can be re-formulated as a precisely formulated mathematical problem. In many other situations, however, the exact description of the controlled system does not exist (or the exact description may exist, but the corresponding optimization problem is too dif®cult to be solved). In such situations, the expertise of skilled operators who have experience of controlling this system may play an important role: for example, the experience of reactor operators who have successfully controlled nuclear power reactors. It is desirable to transform this expert experience into an automatic control strategy. It is often dif®cult to come up with such a transformation, because expert operators are sometimes unable to describe their experience in precise terms. Instead, they describe their control by using words of natural language like high, low, etc., these words are usually represented as fuzzy sets [1]. Therefore fuzzy control, a methodology that would translate such fuzzy rules into a precise control strategy, is needed, which can be viewed as complementary to traditional control methods.

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In the framework of the R&D project on fuzzy control applications to the BR1 reactor, it is dif®cult to select the most suitable method for experiments at BR1 and to compare with some other algorithms [2]. Moreover, due to the safety regulations of the nuclear reactor, it is not realistic to perform many experiments at BR1. One solution is to make a simulation program in a computer, but this has the disadvantage that the real-time property cannot be well re¯ected. Therefore a water-level control system, referred to as the real-time control demo-model, was designed and constructed. In particular, this demo-model (see Figure 1) is designed to simulate the power control principle of BR1 [3]. In this demo-model, the goal is to control the water level in the tower T1 at a desired level by means of tuning VL (the valve for the large control tower T2 ) and VS (the valve for the small control tower T3 ). The pump keeps on working to supply water to T2 and T3 . All taps are manually set at this time. Valves V1 and V2 are used to control the water level in T2 and T3 , respectively. For example, when the water level in T2 is lower than the photoelectric switch sensor 1 then the on±off valve V1 will be opened (on), and when the water level in T2 is higher than the photoelectric switch sensor 2 then the on±off valve V1 will be closed (off ). The same is true for V2 . Only when both V1 and V2 are closed will V3 be opened, because it can decrease the pressure of the pump and thereby prolong its working life. The pressure sensor is used to detect the height of the water level in T1 . So T1 is a dynamic real-time system with two entrances and one exit for a water ¯ow.

Figure 1. The working principle of the demo-model.

IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

221

Figure 2. The demo-model is made to mimic the control principle of the BR1 reactor.

As shown in Figure 2, the main tower T1 symbolically represents the nuclear reactor itself. The water level could be interpreted as the power level of the reactor. In the reactor, this power level is controlled by inserting or withdrawing control rods (A-rod as ®ne-tuning, C-rods as main-tuning in the left side of Figure 2) at different speeds [4]. In the demo-model two control towers are used, a big one T2 (corresponding to the C-rods) and a small one T3 (corresponding to the A-rod), which are connected to the main tower T1 with two linear valves, VL and VS. In reality both valves are the same, but in the software the maximal ¯ow through VL is twice the maximal ¯ow through VS. In this paper research results of fuzzy control and adaptive fuzzy-control experiments for the demo-model are mainly reported. Section 2 introduces some hardware con®guration of the demo-model. Section 3 presents fuzzy control results. Section 4 presents an adaptive fuzzy control that is able to self-regulate the fuzzy control rules. Section 5 presents an implementation of a computer simulation using the adaptive fuzzy control in the demo-model. The Appendix gives some detailed mathematical analysis of the demo-model system.

2.

Hardware Implementation

The controller platform selected to implement our fuzzy controller is the Sysmac C200HS programmable logic controller (PLC) of Omron [5]. Among the reasons to choose for this professional platform are the demand to meet the real-time requirements of the real-world system and its proven robustness in an industrial environment. An additional advantage is that our fuzzy controller can be directly mapped onto the fuzzy inference unit, based on the Omron FP3000 processor, which has 12-bit resolution throughout and has a high-speed inference engine incorporated (typical performance: inference of 20 rules, each with ®ve antecedents and two consequences in 650 ms, including defuzzi®cation) without requiring capacity of the main PLC processor. A ®nal point to note is the availability of programming tools under Windows. In this way the ¯exibility of a software implementation can be combined with the robustness of an

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industrial PLC. The PLC is programmed with the SYSWIN programming package, whereas a special package Fuzzy Support Software (FSS) is used to program the fuzzy logic unit C200H-FZ001 via its RS-232C link. The same serial link can be used for online monitoring of the actions of the fuzzy controller. As reported in the progress of the work [2], the FLC con®guration offers great ¯exibility.

3.

Fuzzy Control

The fuzzy control algorithm in this demo-model is a Mamdani-type algorithm. Two fuzzy controllers (denoted as FLC1 and FLC2) are used to control VL and VS separately (note: it is possible to use one fuzzy logic controller with two outputs to control VL and VS and the related result can be referred to in Li and Ruan [3]). Let D be the difference between the actual value (P) of water level and the set value (S) and DD be the derivative of D, in other words, the speed and direction of the change of water level. VL and VS represent the control signal to VL and VS, respectively. When D is too big, FLC1 controls VL (main-tuning). When D is small, FLC2 controls VS (®ne-tuning). D and DD are as inputs of the fuzzy logic controller, and VL or VS as the output of the fuzzy logic controller. D and DD are fuzzi®ed before fuzzy inference. Suppose the universes of discourse (or input variables' intervals) of D and DD are ‰ d; dŠ and ‰ dd; ddŠ, respectively. Five fuzzy sets are to partition them, i.e., negative large (NL), negative small (NS), zero (ZE), positive small (PS), and positive large (PL). As for VL and VS, because the result of fuzzy reasoning is also a fuzzy linguistic value, the universes of discourse of VL and VS also need to be fuzzi®ed. The same ®ve fuzzy linguistic terms are used. Symmetrical triangular-shaped functions are used to de®ne the membership functions for input variables, and singletons are used for output variables [5]. In order to get an ideal control effect, much time must be taken to regulate some parameters. That is perhaps the reason to slow down further applications of fuzzy logic control. Table 1 contains all control rules. In this table, for example, PL/ZE at row 2 and column 3 means: if D is NS and DD is NL then VL is PL and VS is ZE. In other words, if the practical water level is a little lower (NS) than the desired level and the speed of the water level falling down is large (NL) then VL will open largely (PL) and VS will not change (ZE). Table 1. Control rule table. DD/D

NL

NS

ZE

PS

PL

NL NS ZE PS PL

PL/ZE PL/ZE PL/ZE PS/ZE ZE/ZE

PL/ZE PL/ZE PS/PS ZE/NS NL/NL

PL/ZE PS/NS ZE/ZE NL/ZE NL/ZE

PS/ZE PS/PS NS/PS NL/NL NL/NL

PS/ZE PS/NS NS/PS NL/NL NL/NL

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Figure 3. The trajectory of fuzzy logic control water level.

Each fuzzy controller has one rule base that contains 25 fuzzy control rules. The i-th rule can be represented as the following form: if D is Ai and DD is Bi ; then VL (or VS) is Ci where Ai , Bi , and Ci are fuzzy linguistic values, such as NL, PS, and so on. The above rule is sometimes abbreviated as …Ai ; Bi : Ci †. With the help of the fuzzy control rules in Table 1, the control effect illustrated in Figure 3 (the thick curve) is obtained. In this ®gure, the thick curve records the trajectory of the water level in T1 . From 0±5 min, the set value is 15 cm …S1 ˆ 15†; from 5±10 min, the set value is 25 cm …S2 ˆ 25†; from 10±15 min, the set value is 15 cm …S3 ˆ 15†. This is the best result of all experimental tests. Before this result, the control effect is normally either curve a or b. Curve a means a big overshot but with a fast response. Curve b means no overshot but with a slow response. It is well known that it is dif®cult to achieve a control result with a fast response and no overshot. The current result has already overcome this dilemma. In view of this ®gure, the fuzzy control has quick responses (quickly approaching the set value) and small overshoot (almost invisible), but with a small steady error (not so smooth in steady state). 4.

Adaptive Fuzzy Control

The kernel part of the fuzzy logic control is the fuzzy rule base with linguistic terms, though the membership functions and scale factors also have an important effect on the fuzzy logic controller. There are some papers that discuss how to adjust membership functions and/or scale factors [6, 7]. This section focuses on rules. Normally the methods of deriving rules can be broadly divided into two types, sourceable and non-sourceable. The term ``sourceable method'' means that the rules are obtained from some information source, such as human experience or historical input±output data. Experience has been widely used by the fuzzy engineers, especially by the early fuzzy engineers. The problem of using human

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experience is that it is time-consuming, and to some degree subjective. To overcome these problems, particularly avoiding the subjectivity, historical input±output data if available can be used. To obtain rules from such data, many methods are used, one of the popular approaches is neural networks (NN) [8±14]. One problem of the sourceable method is that it depends strictly on the source that will be transformed into rules. In the case that the source is noisy, then the rules might be biased. Another problem of the sourceable method is that it is usually non-adaptive, i.e., all the rules are ®xed, and therefore it cannot perform well in a dynamic environment. The non-sourceable methods are source-free and they produce and choose rules according to a performance measurement of the controller, such as genetic algorithms (GAs) [15,16] (mostly also generating membership functions and scale factors) and self-organizing controllers (SOC) [17±22]. With a GA it is possible to ®nd integrated optimal parameters but the GA is very computation rich. Perhaps the SOC is the only method that has the following advantages: objective, adaptive, less computation required, more error-tolerant, and simple. The general principle of the SOC is that the controller monitors its own performance and adjusts its control rules to improve performance for time-varying and unknown plants. The problem of the SOC is how to perform the performance measurement. The basic way is to design a performance measurement table that looks like a fuzzy control rule table and to use it to assess the performance of the controller (rules) [19], but to design such a performance measurement table is also very dif®cult [23] and it is systemdependent. Based on the SOC, this section will introduce an adaptive method that uses a set of new norms to replace the former performance measurement. The new norms are very simple and system-independent, therefore they can be easily applied to most fuzzy controllers. In this section, the adaptive fuzzy control means the above SOC, in other words, a fuzzy control with an adaptive function, where the adaptive function contains two steps: performance judgement and changing fuzzy control rules. Figure 4 illustrates how an adaptive function is incorporated into the fuzzy control system. At the beginning of each cycle, the controller's last behavior is judged and then the rule base is changed

Figure 4. An adaptive function is incorporated into a fuzzy control system.

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IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

accordingly. In this cycle, the controller will use the new rule base and output the result to the controlled object. The behavior of the new rule base will be judged and changed again in the next cycle.

4.1.

The Principle of the Adaptive Function

Let D and DD represent the difference between the actual value and the desired value and the derivative of D, respectively. Let D…t† and DD…t† represent D and DD at time t, respectively. They are two input variables. Let U be an output variable, and assume the total number of the rules is n, then every rule has the following form: if D is Ai and DD is Bi ;

then U is Ci ;

i ˆ 1; 2; . . . ; n

where Ai , Bi , and Ci are fuzzy linguistic values and i is an index pointing out each rule's position in the rule table (or the rule data ®le). Use r‰iŠ to represent the fuzzy control magnitude (conclusion fuzzy set) of the i-th rule, and let simply r‰iŠ ˆ 1; 2; 3; 4; 5; 6; 7 where 1 ˆ NL, 2 ˆ NM, 3 ˆ NS, 4 ˆ ZE, 5 ˆ PS, 6 ˆ PM, 7 ˆ PL (note here two more fuzzy sets NM (negative middle) and PN ( positive middle) are added). In general, a control locus may be expressed as in Figure 5, and it can be regarded as having up to four feature sections and four feature points. For each feature part, a norm to guide the regulation of the fuzzy control rules is offered. For example, the current water level P…t† is in the feature part (1), then after the fuzzy controlling using the current control rules, the water level P…t ‡ 1† is measured at the next time which has three possibilities: *

P…t†5P…t ‡ 1†  S

*

P…t ‡ 1†5S and P…t ‡ 1†  P…t†

Figure 5. Any trajectory has up to four feature sections and four feature points.

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P…t ‡ 1†4S

*

The related norm to guide how to change rules is the following: i. if D…t ‡ 1†  0 and DD…t ‡ 1†40, that is, P…t†5P…t ‡ 1†  S, then r‰iŠ ˆ r‰iŠ; ii. iii.

if D…t ‡ 1†50 and DD…t ‡ 1†  0, that is, P…t ‡ 1†5S and P…t ‡ 1†  P…t†, then r‰iŠ ˆ r‰iŠ ‡ a; if D…t ‡ 1†40, that is, P…t ‡ 1†4S, then r‰iŠ ˆ r‰iŠ

a;

where a is a step size and a ˆ 1, 2, 3, 4, 5, 6. In case i, the fuzzy controller makes the water level P…t ‡ 1† closer to the set value S, therefore the behavior of the fuzzy controller is good, no rules should be changed. In case ii, the fuzzy controller makes the water level P…t ‡ 1† go further away from the set value S, therefore the behavior of the fuzzy controller is not good, the strength is too weak and the action of the corresponding rules should be stronger. In case iii, the fuzzy controller makes the water level P…t ‡ 1† overpass the set value S, therefore the behavior of the fuzzy controller is not good, the strength is too strong and the action of the corresponding rules should be weakened. Not all rules but some of those that are activated in last cycle should be regulated. The following formula describes which rule should be adjusted:  ^ _  ^ Ai …D† Bi …DD† ˆ Aj …D† Bj …DD† …1† Cj ˆ Ci

which means the ith rule is changed only if it is the largest activated among those activated rules that have the same conclusion part. For example, (NL, NM: PL) and (NM, NM: PL) are two activated rules and have the same conclusion part, PL. Comparing NL…D†6NM…DD† with NM…D†6NM…DD†, the larger one corresponds to the rule which should be adjusted.

4.2.

An Experimental Result

To guarantee no overshoot, the best way is to initialize all rules as the same conclusion part: NL, as shown in Table 2. In this table, for example, NL at the row 2 and column 3 means: if D is NM and DD is NL then VL or VS is NL. All rules have the same conclusion part though condition parts are different. Figure 6 illustrates the comparison result between adaptive fuzzy control and fuzzy control with the above rule base. In this example, the set value is 20 cm. Both start from 0 cm. During the ®rst stage, i.e., increasing from zero, some analytic rules manipulate the valves and not fuzzy control. Only after the water level reaches 18 cm does the fuzzy controllers start to operate VL and VS. Apparently, the adaptive fuzzy control has a much better result by self-regulating gradually fuzzy control rules. The

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IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

Table 2. The initial rule table for both FLC1 and FLC2. DD/D

NL

NM

NS

ZE

PS

PM

PL

NL NM NS ZE PS PM PL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

normal fuzzy control without adaptive function cannot self-regulate rules, therefore it cannot draw up the water level. About 10 minutes later, the rule tables on the screen is observed and both rule tables have been changed a lot. Table 3 gives the result of FLC1 and Table 4 gives the result of FLC2, where the regulated rules are marked by bold.

4.3. *

Comments for the Adaptive Function The initial idea about the previously described norms of the adaptive function, which was published in Li et al. [18], and where a simulated inverted pendulum system and a real industrial heating system were used to make testing, gave satisfactory results.

Figure 6. Comparison between adaptive fuzzy control and fuzzy control.

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Table 3. The regulated rule table for FLC1. DD/D

NL

NM

NS

ZE

PS

PM

PL

NL NM NS ZE PS PM PL

NL NM NL NS NL NL NL

NL NL NS PL NL NL NL

NL NL PM PL NM NL NL

NL NL NL ZE NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

*

The parameter a is in¯uential on the overshoot and response time (rise time). When a is too big, there will be a large overshoot possibly; when a is too small, possibly there will be a long response time [18].

*

The adaptive function considers only the last value, i.e., it uses P…t 1† not P…t t† (t is the delay) to decide P…t†, but our experimental results show the effect is good, although the valves of the demo model have a maximum delay of 90 seconds.

*

The adaptive function selects only some of the rules according to the formula (1) for adjustment, not all activated rules like Lin et al. [12]. This makes the transition of the rules smoother, i.e., without or with less resonance.

*

Selecting initial rules appropriately will bene®t the control effect. For example, if the overshoot is strictly limited, we may initialize all rules with the conclusion part of NL, as was done in the previous experiment. Once some experience has been obtained, it can be transformed into the initial rules of the adaptive function, the advantage being that the rise time will be shorter [24].

*

The rule, ``if D is ZE and DD is ZE then U is ZE,'' should be ®xed, and this will help the system to become stable.

Table 4. The regulated rule table for FLC2. DD/D

NL

NM

NS

ZE

PS

PM

PL

NL NM NS ZE PS PM PL

NM PL PM PL PL PM ZE

PM PL PL PL PL PL ZE

PL PM PL PS PM PL NM

PL PL PS ZE NS NM NL

NM PL PL NL NL NL NL

NL NL NL NL NL NL NL

NL NL NL NL NL NL NL

IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

229

*

The adaptive function is very helpful in keeping the system stable in steady state. It cannot guarantee any overshoot if the initial rules are randomly selected.

*

The adaptive function cannot adjust membership functions and scale factors.

5.

Implementation of a Computer Simulation with Adaptive Fuzzy Control

So far, all control is done by an early version of a PLC and fuzzy logic unit provided by Omron, Belgium, both for the nuclear reactor [2, 4] and the demo-model. Unfortunately, this early hardware has no facilities for adaptive fuzzy logic control. This is the main reason why this demo-model is needed. The other option was to keep using the real demo-model and running the control software on a PC connected with the PLC (that gets degraded to an IO unit). For every tower and the tank we have a differential equation representing the mass balance for that part. The linear valves with a differential equation are modelled. Let us ®rst de®ne some ¯ows: Flow A: from T2 to T1 Flow B: from T3 to T1 Flow C: from T1 to the tank Flow D: from the tank, via the pump to T2 Flow E: from the tank, via the pump to T3 And some constants: h1 , A1 : water level and area of T1 h2 , A2 : water level and area of T2 h3 , A3 : water level and area of T3 h4 , A4 : water level and area of tank VL,VS: position of linear valves ‰0; 1Š V speed: the speed of the linear valves

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Now the system of differential equations can be written down: dh1 dt dh2 dt dh3 dt dh4 dt dVL dt dVS dt

ˆ …Flow A ‡ Flow B

Flow C†=A1

ˆ …Flow D

Flow A†=A2

ˆ …Flow E

Flow B†=A3

ˆ …Flow C

Flow D

Flow E†=A4

ˆ + V speed or 0 ˆ + V speed or 0

Note: The formulas for the various ¯ows are omitted here. They are standard hydraulic equations. For example, s D12 h Flow A ˆ VL* *a KdA* LA with D12 h the difference in water level between T2 and T1 , VL the position of the big valve, KdA the K-factor of the connection, LA the length of the connection, and a a correction factor obtained by comparing experiments and the simulation. To simulate the complete behavior of the demo-model and the control actions that are coordinated by a PLC and a fuzzy logic unit, every step in the simulation consists of two parts: Simulate the evolution of the hydraulic system for a certain time period Tsample and simulate t he fuzzy logic control action afterwards. The time period Tsample resembles the real time needed for the PLC to ®nish its program (loop time). In the current case this is approximately 10 ms. The evolution of the system is simulated using the system equations given above. This set of differential equations from a certain moment in time T to T ‡ Tsample is integrated using numerical techniques for solving ordinary differential equations. The ¯ow-chart on page 231 shows the overall structure of the simulation process. Here the ®rst results of using adaptive fuzzy logic control on the simulated demomodel are presented. Starting with an ``empty'' rule base, i.e., for every state of the error and the change of error (two input parameters), both valves are completely closed. This means no control action is taken at all. During the process of simulation and control, the adaptive function tries to change the rule base so the system performs well. As seen in Figure 7 that for both states (steady at 20 cm and steady at 15 cm) a learning phase (oscillations) is ®rstly obtained, but after some time the system performs well. The use of simulation helped to study the application of adaptive fuzzy logic control to a real hydraulic system. The effort of implementing the simulation was worth it for several reasons. The main reason is that it is easier to change system parameters and see the effect of these changes. Secondly, in such case, simulations are faster than the real demo-model.

IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

Figure 7. Using adaptive fuzzy logic control on the simulated demo-model.

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The next step in this research is to connect the real demo-model to the software. It is hoped that the adaptive fuzzy control developed based on a computer simulation can be easily transferred to the real-time system [25]. Furthermore, the adaptive fuzzy control is able to aid the designer in ®nding the fuzzy control rules, especially for systems possessing much of dynamical uncertainty in real-time systems.

Appendix Analysis of the Demo-Model For the tower T1 (see Figure 8), it has an in¯ow and an out¯ow. Suppose the height of the water is h, the size (area) of water is A, and the size of output hole is a, the basic relationship between the in¯ow and the out¯ow is: Adh ˆ inflow…t† outflow…t† dt p Adh ˆ f …t† ka 2gh dt p where outflow…t† ˆ ka 2gh and k is a constant coef®cient.

Figure 8. The dynamic analysis of the tower.

IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

233

Substitute

p yˆ h dy 1 dh ˆ dt 2y dt

then, dh dy ˆ 2y dt dt So, (2) becomes p dy ˆ f …t† ka 2gy dt p dy f …t† ka 2g ‡ ˆ0 dt 2Ay 2A p let m ˆ 1=2A, c ˆ ka 2g=2A 2Ay

dy dt

mf …t† ‡cˆ0 y

…3†

Without f …t† If f …t† ˆ 0, (3) becomes dy ˆ dt y…t† ˆ

c ct ‡ y…0†

for h ˆ y2 , h…t† ˆ y2 …t† ˆ ‰

ct ‡

p 2 h…0†Š

…4†

f …t† is a constant If f …t† ˆ p and mf …t† ˆ k, dy dt y

Z 

k ‡cˆ0 y dy cy k ˆ dt y

dy ˆ cy k  k=c 1‡ dy ˆ y …k=c†   k k ˆ y ‡ ln y c c

dt Z cdt ct ‡ C

…5†

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f …t† ˆ pt If f …t† ˆ pt and mf …t† ˆ kt, dy dt

kt ‡cˆ0 y

…6†

substitute

r c2 y…t† ˆ tz…t† k ‡ 4

ct 2

then the function z satis®es the differential equation 1

z…t†2 ˆ tz …t†…z…t† 0

gk †

where c gk ˆ s ‡ 4k c This can be seen as follows, from r c2 ct y…t† ˆ tz…t† k ‡ 2 4 one sees

r r c2 c c2 0 y …t† ‡ c ˆ z…t† k ‡ ‡ ‡ tz …t† k ‡ 2 4 4 0

and hence 0 ˆ …y0 …t† ‡ c†y…t† kt ( r r) r c2 c c2 c2 0 ˆ t z…t† k ‡ ‡ ‡ tz …t† k ‡ z…t† k ‡ 4 2 4 4   c2 0 1 ‡ tz …t†…z…t† gk †g ˆt k‡ fz…t†2 4

c 2

! kt

Consequently, for t=0, 1

2

0

z…t† ˆ tz …t†…z…t†

gk † p Suppose in the regime y…t† > t k ‡ …c2 =4† z…t†41, one differentiates the function t ° t2 …z…t†

1†

1

gk

…z…t† ‡ 1†

1 ‡ gk

…ct=2†, or what amounts to the same

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IMPLEMENTATION OF ADAPTIVE FUZZY CONTROL

From Leibniz' rule and the chain rule one gets o dn2 t …z…t† 1†1 gk …z…t† ‡ 1†1 ‡ gk dt ˆ 2t…z…t† 1†1 gk …z…t† ‡ 1†1 ‡ gk gk †t2 z0 …t†…z…t†

1†

‡ …1 ‡ gk †t2 z0 …t†…z…t†

1†

‡ …1 ˆ 2t…z…t†

gk

1†

gk 1

…z…t† ‡ 1†

gk

1 ‡ gk

…z…t† ‡ 1†

…z…t† ‡ 1†gk fz…t†2

gk

1 ‡ tz0 …t†…z…t†

gk †g

So that, if y…t† is a solution to 1

2

z…t† ˆ tz0 …t†…z…t†

gk †

and hence d 2 ft …z…t† dt

1†1

gk

…z…t† ‡ 1†1 ‡ gk g ˆ 0

The latter means that the function t ° t2 …z…t†

1†

1

gk

…z…t† ‡ 1†

1 ‡ gk

is a constant, say ck. Always under the regime z…t†41; t40, one sees that the function z0 …t†50 and hence it decreases. (Here we used 1

z…t†2 ˆ tz0 …t†…z…t†

gk †

together with gk 51.) From the equality ck ˆ t2 …z…t†

1†

1

gk

…z…t† ‡ 1†

1 ‡ gk

one sees that z…t† decreases to the constant function 1 and that z…t† increases to ? if t decreases to 0. A similar reasoning applies under the regime z…t†5 1. The situation is somewhat more complicated for 15z…t†5 ‡ 1. In this case ck;0 ˆ t2 …1

z…t††1

gk

…1 ‡ z…t††1 ‡ gk

and hence 05

ck;0 521 ‡ gk t2

A consequence is that solution y…t† for which r r c2 ct c2 ct 5y…t†5t k ‡ t k‡ 2 2 4 4 only can exist for p t > ck;0 2

1=2

gk =2

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In the previous reasoning we used the following argument. Suppose above it follows that the derivative of the function t ° t2 …1

z…t††

1

gk

…1 ‡ z…t††

15z…t†51. As

1 ‡ gk

is equal to zero. In other words the function t ° t2 …1

z…t††

1

gk

…1 ‡ z…t††

1 ‡ gk

is constant. Of course this constant ck;0 may depend on gk . If z…0† > 1, then one should consider the function t2 …z…t† and if z…0†5 t ° t2 …1

1†1

gk

…1 ‡ z…t††1 ‡ gk

1, then one should consider z…t††1

gk

… 1

z…t††1 ‡ gk

In case one considers 15z…t†51, then solutions do not exist for small positive t, and 0 for large t they either increase to y‡ …t† (because z …t† > 0 if z…t† > gk or they decrease to 0 y …t† (because z …t†50 if z…t†5gk ). So the dichotomy z…t† increases to 1, or it decreases to 1. The statements are correct. Talking about z…0† means that one implicitly assumes that y…0† ˆ 0 and that y0 …0† exists. From the previous discussion it follows that this situation might not happen. Also notice that the functions ! ! r r c2 c c2 c y ‡ …t† ˆ t k‡ k‡ and y …t† ˆ 4 2 4 2 are particular solutions. They correspond to z…t†:1 and z…t†:1, respectively. If a solution y lies somewhere below y (or somewhere between y and y‡ , or somewhere above y‡ ), it possesses everywhere this property. Hence solutions cross neither y nor y‡ . Remark In order to study the behavior of the solutions, one always considers the equalities 1

z…t†2 ˆ tz0 …t†…z…t†

gk †

and ck ˆ t2 …z…t†

1†1

gk

…z…t† ‡ 1†1 ‡ gk

together. One also uses gk 51. Here assuming that in the situation y…t† > y‡ …t†. In the other cases, y…t†5y …t† (which is equivalent to z…t†5 1† and y …t†5y…t†5y‡ …t† (which is equivalent to 15z…t†51†, using similar expressions. Another equality which might be interesting is the following:  g z…t† ‡ 1 k 2 2 ck ˆ t …z…t† 1† z…t† 1 Here considering again the case y…t† > y‡ …t†, or, what is equivalent, z…t†41.

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Remark One observes that solutions between y …t† and y‡ …t† do not exist for small t40. Solutions y…t† above y‡ …t† have the property that y…t†=t increase to ? if t decreases to 0. By the same token, solutions below y …t† have the property that y…t†=t decrease to ? if t decreases to 0. In fact, f …t† ˆ pt is the normal condition. Because f …t† ˆ f1 …t† ‡ f2 …t†

…7†

where f1 …t† is the out¯ow of T2 and f2 …t† is the out¯ow of T3 , and they are nonlinear variables. So the system is a nonlinear and time variable system. It is dif®cult to control it with classical strategy like PID method. Some similar work can be refereed to Jansen et al. [26]. They reported that they could not use PID to control it because of the large dif®culty. Their fuzzy control is not a real-time system but a computer simulation. So this real-time fuzzy logic control system will be very important to show how fuzzy logic control can solve some complex dynamic control systems.

Acknowledgments The author would like to thank the following colleagues: A. J. van der Wal, J. van Casteren, X. Li, and G. Van den Eynde for their assistance and support for the project reported in this paper.

References 1. Zadeh, L. A. 1965. Fuzzy sets. Information and Control 8(3): 338±353. 2. Ruan, D., and van der Wal, A. J. 1998. Controlling the power output of a nuclear reactor with fuzzy logic. Information Sciences 110: 151±177. 3. Li, X., and Ruan, D. 1997. Constructing a fuzzy logic control demo model at SCK ? CEN. In Proceedings of the 5th European Congress on Intelligent Techniques and Soft Computing. Aachen, Germany, 1408±1412. 4. Ruan, D., and Li, X., 1998. Fuzzy-logic control applications to the Belgian Reactor 1 (BR1). Computers and Arti®cial Intelligence 17(2±3): 127±150. 5. Omron. 1992. C200H-FZ001 Fuzzy Logic Unit, Operation Manual. Omron company. 6. Batur, C., and Kasparian, V. 1991. Adaptive expert control. International Journal of Control 54(4): 867± 881. 7. Tonshoff, H. K., and Walter, A. 1994. Self-tuning fuzzy-controller for process control in internal grinding. Fuzzy Sets and Systems 63: 359±373. 8. Berenji, H. R., and Khedkar, P. 1992. Learning and tuning fuzzy logic controllers through reinforcements. IEEE Trans. Neural Networks 3: 724±740. 9. Halgamuge, K., and Glesner, M. 1994. Neural networks in designing fuzzy systems for real world applications. Fuzzy Sets and Systems 65: 1±12. 10. Jang, J. S. R. 1992. Self-learning fuzzy controllers based on temporal back propagation. IEEE Trans. Neural Networks 3(5): 714±723. 11. Kosko, B. 1992. Fuzzy Systems and Neural Networks. Englewood Cliffs, NJ: Prentice-Hall. 12. Lin, C. T., Lin, C.J., and Lee, G. C. S. 1995. Fuzzy adaptive learning control network with on-line neural learning. Fuzzy Sets and Systems 71: 25±45

238

RUAN

13. Takagi, H., and Hayashi, I. 1991. NN-driven fuzzy reasoning. International Journal of Approximate Reasoning 5: 191±212. 14. Wang, L. X., and Mendel, J. M. 1992. Generating fuzzy rules by learning from examples. IEEE Trans. Systems, Man and Cybernetics 22: 1414±1427. 15. Lim, M. H. Rahardja, S., and Gwee. B. H. 1995. A GA paradigm for learning fuzzy rules. Fuzzy Sets and Systems 82: 177±186. 16. Qi, X. M., and Chin, T. C. 1997. Genetic algorithms based fuzzy controller for high order systems. Fuzzy Sets and Systems 91: 279±284. 17. He, S. Z., Tan, S. H., Hang, C. C., and Wang, P. Z. 1993. Control of dynamical processes using an on-line rule-adaptive fuzzy control system. Fuzzy Sets and Systems 54: 11±22. 18. Li, X., Bai, S., and Zhang, Z. 1996. An introduction to a fuzzy adaptive control algorithm. Chinese Journal of Advanced Software Research 3(1): 1±11. 19. Procyk, T. J., and Mamdani, E. H. 1979. A linguistic self-organizing process controller. Automatica 15(1): 15±30. 20. Shao, S. 1988. Fuzzy self-organizing controller and its application for dynamic processes. Fuzzy Sets and Systems 26: 151±164. 21. Tanscheit, R., and Scharf, E. M. 1988. Experiments with the use of a rule based self-organizing controller for robotic applications. Fuzzy Sets and Systems 26: 151±164. 22. Wu, Z. Q., Wang, P. Z., Teh, H. H. and Song, S. S. 1992. A rule self-regulating fuzzy controller. Fuzzy Sets and Systems 47: 13±21. 23. Chung, B. M., and Oh, J. H. 1993. Control of dynamic systems using fuzzy learning algorithm. Fuzzy Sets and Systems 59: 1±14. 24. Li, X, and Ruan, D. 1998. Comparative study of fuzzy control, PID control, and advanced fuzzy control for simulating a nuclear reactor operation. In Proceedinds of the 3rd International FLINS Workshop. Antwerp, Belgium, 424±434. 25. Ruan, D., and Van den Eynde, G. 1999. Adaptive fuzzy control for simulating a nuclear reactor operation. In Proceedinds of the 13th European Simulation Multiconference. Warsaw, Poland, 311±315. 26. Jansen, W. J., Venema, R. S., ter Brugge, M. H., Diepenhorst. M, Nijhuis, J. A. G., and Spaanenburg, L. 1995. Stability properties of fuzzy controllers. Benelux Quarterly Journal on Automatic Control, 36(3): 27± 37.

Da Ruan received a BSc Degree in Applied Mathematics from Fudan University, Shanghai in 1983, a Certi®cate in Management from Leuven University, Belgium in 1990 and a PhD Degree in Mathematics from Ghent University, Belgium in 1990. He held the positions of Teaching Research Assistant at Fudan University from 1983±1986, a PhD Researcher at Ghent University from 1987± 1990, a Post-doctoral Researcher at the Belgian Nuclear Research Centre (SCK ? CEN) from 1991± 1993 and since 1994 has been the Senior Researcher and Project Leader at SCK ? CEN. He is a principal investigator for research project on intelligent control for nuclear reactors at SCK ? CEN. His major research interests lie in the areas of mathematical modelling, computational methods, fuzzy logic and intelligent systems in various applications of the

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nuclear science and engineering ®elds. He is the editor of four books on fuzzy logic, intelligent systems and nuclear engineering applications and has authored and co-authored over 100 scienti®c publications. Dr Ruan is a member of the International Fuzzy Systems Association (IFSA), the Belgian Nuclear Society (BNS), and the chair of Fuzzy logic and Intelligent technologies in Nuclear Science (FLINS), Belgium (a founding editor of the FLINS forum). He received an honorary doctoral Degree from the Nuclear Power Institute of China in 1995. He has been appointed to be a concurrent professor at the Institute of Resource and Environmental Science of the Beijing Normal University from 1995±1998 and since 1997 has become the international editor-inchief of an international book series on ``Advances in Fuzzy Mathematics and Engineering'' from Beijing Normal University Press, Beijing. He has been recently rewarded as an advisory professor of the Department of Applied Mathematics of Southwest Jiaotong University in Sichuan (China) since 1998.