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AI in the Feedback Loop: A Survey of Alternative Approaches
Copyright © IFAC Artificial Intelligence in Real-Time Control, Bled. Slovenia, 1995
AI IN THE FEEDBACK LOOP: A SURVEY OF ALTERNATIVE APPROACHES Karl-Erik Arzen Department of Automatic Control Lund Institute of Technology PO. Box 118 S-221 00 Lund, Sweden Email: [email protected]
Abstract: An overview of different ways of using AI techniques in feedback control is presented. The paper gives special attention to fuzzy control and expert control.
1. INTRODUCTION
work in AI in control has developed. Direct-level AI based control is the subject of Section 3. The emphasis here is on fuzzy control. Section 4 describes supervisory-level AI applications with focus on expert control.
The aim of this paper is to present some different Artificial Intelligence (AI) based paradigms that can be used on the feedback control loop level. Main focus will be given to rule-based control, specially fuzzy control and to expert control. The majority of the AI-techniques in use in control today are either based on trying to mimic the behaviour and expertise of a human expert, e.g., a process operator, or based on machine learning. In the first field we find most of the work in fuzzy control and expert control whereas in the second field we find the work in learning control and neuro control. This paper focuses on the first approach.
2. A HISTORICAL SURVEY Ever since Artificial Intelligence (AI) was established as a research discipline in the end of the 1950s it has had strong connection to control engineering. In the 1950s and 1960s the area of Cybernetics was popular. This comprised among several subjects both AI and control theory. Much work at that time was focused on learning using systems modelled after the neuron-based structure of the brain. Systems such as the Perceptron (Rosenblatt, 1962) and ADALINE (Widrow, 1962) belong here. These ideas are also the origin of what is called learning control (Tsypkin, 1971). Other names for this are intelligent control, e.g. (Fu, 1971), and self-organizing control (Saridis, 1977) .
Another way of classifying different approaches is based on whether the AI part of the controller is used on the direct feedback control level or if it used on a supervisory control level orchestrating a set of conventional control algorithms. In the first class we find the majority of the rulebased approaches whereas in the second class we find expert control.
In learning control systems the controller should be able to estimate unknown information during its operation and determine an optimal control action from the estimated information. Different learning schemes have been proposed, e.g., pattern classification with adaptable decision threshold, bayesian estimation and stochastic approximation. Much of the
Common to all the approaches described here is that there are parallel, but strongly related, activities taking place in the conventional control community. For example, there are relations between learning control and adaptive control and between expert control and multi-controllers. Section 2 gives a historical overview of how the 185
work in learning control systems have strong relationships to conventional adaptive control.
During the 1970s much of the activities in AI were focused on expert systems or knowledgebased systems. These systems were designed to solve heuristic problems in an organized way mainly by mimicing the way an experienced human expert solves the problem. The key idea is to organize knowledge in terms of rules. Expert systems have been applied to a wide variety of problems with varying success. Some commonly given criteria for success are that the problem is nontrivial and sufficiently complex, that the problem can be solved by human experts and that experts are available.
The development of the backpropagatwn training algorithm for multi-layer feedforward neural networks (Werbos, 1974) , (Rumelhart et al. , 1986) led to a renewed interest in neural networks and neural control during the 1980s. A number of different net models are used, e.g., feedforward networks such as the multi-layer perceptron, CMAC and radial basis function networks and recurrent networks such as, e.g. , Hopfield networks and Kohonen nets. Also other training or learning strategies have been developed. Backpropagation is an example of supervised learning where it is assumed that inputoutput data pairs are available. In unsupervised learning the network is only presented with input data. The network then organizes itself to best map the input data. One an example of this is the Self-Organizing Feature Maps used in the context of Kohonen nets. Reinforcement learning is an approach that can be used when output data only are available in the form of delayed, partial information that gives credit to a series of actions over time. A large number of textbooks and survey papers on neural networks for control are available, e.g., (Miller et al., 1990) and (Hunt et al., 1992)
The expert system approach soon found its way into control application. The idea of trying to combine expert systems with controllers is, however, not new. Already in 1962, a Heuristic Decision program was discussed (Crossman and Cooke, 1962) in the context of manual control. In control engineering, the main application area is supervisory applications such as process monitoring and diagnosis, scheduling and planning. However, the approach is also used in expert control and several fuzzy controllers can be seen as expert systems. Expert systems were originally developed to solve static problems, i.e., situations where the premises do not change with time. Control problems are generally not static. A statement may, e.g., suddenly switch from true to false because of a change in the physical system being controlled. Reasoning with time is a very complicated problem where many theoretical problems are unresolved (Laffey et al., 1988). Some pragmatic approaches are taken to deal with these issues. One method is to replace the dynamic problem by a static problem by assuming that all premises hold over a small sliding time-window. Another method is to keep track of the chain of reasoning so that all conclusions drawn from a statement can be withdrawn when the statement ceases to be true. It is also important that conclusions are reached in a reasonable time.
Fuzzy set theory (Zadeh, 1965) was developed in the beginning of the 1960s as a way of expressing non-probabilistic uncertainties. Since then, fuzzy set theory has developed and found applications in database management, decision support systems, signal processing, data classifications, computer vision, etc. In 1974, the first successful control application was reported (Mamdani, 1974). Control of cement kilns was an early industrial application (Holmblad and Ostergaard, 1982). Since the first consumer product using fuzzy logic was marketed in 1987, fuzzy control has received enormous attention (Zadeh, 1994). A number of CAD environments for fuzzy control design have emerged (Conner, 1993) and VLSI hardware for fast execution has been developed (Zhang and Mayer, 1993). Fuzzy control is being applied industrially in an increasing number of cases, e.g., (Froese, 1993).
Commercial programming environments for knowledge-based real-time systems have been available for around 10 years now. A good example is G2 from Gensym Corporation, (Moore et al., 1990), which is aimed at supervisory control applications. In G2 the developer can combine a number of programming paradigms, e.g., object-oriented programming, procedural programming, rule-based programming, and event-driven programming. It is also possible to implement Petri Nets, state machines, and neural networks in G2, e.g., (Arzen, 1994a).
Fuzzy control has strong relationships to neural networks. A fuzzy inference system can be expressed in network form, e.g., (Jang, 1993), and can be trained with gradient methods equivalent to the back-propagation procedure. This is utilized in so called "neuro-fuzzy" models where the prior knowledge available about the function to be approximated is expressed as fuzzy rules and used to initialize the structure and weights of the network. After training the resulting net can be reinterpreted in terms of fuzzy sets and rules.
In the late 1980s the focus in expert system and knowledge-based systems shifted from rulebased, heuristics-based systems to model-based 186
approaches that are based on a deeper understanding of the application. In control engineering this led to the interest in Qualitative Methods for, e.g., control and diagnosis.
3.1 Fuzzy control The early work in fuzzy control was motivated by a desire to directly express the control actions of an experienced human operator in the controller, i.e., to mimic his behavior, and to obtain smooth interpolation between discrete controller outputs. Since then the application range of fuzzy control has widened substantially. In most cases a fuzzy controller is used for direct feedback control. However, it can also be used on the supervisory level as, e.g., a self-tuning device in a conventional PID controller. Also, fuzzy control is no longer only used to directly express a priori process knowledge. For example, a fuzzy controller can be derived from a fuzzy model obtained through system identification. Therefore, it is difficult to define what a fuzzy controller is. A very general definition is:
3. DffiECT-LEVEL AI-BASED CONTROL On the direct level an AI-based controller is used to directly calculate the control signals to the process according to Fig. 1. The AI-
Fig. L
Direct level AI-based controller.
based controllers used on this level are either based on the connectionist AI approach, e.g., neural networks, or based on the symbolical AI approach. Here we will concentrate on the latter case.
A Fuzzy Controller is a controller that contains an, often non-linear, mapping that has been defined using fuzzy logic-based rules.
The majority of the symbolical AI that is being applied to direct feedback control is rule-based. The basic knowledge representation formalism is rules of the type
The key issues in this definition are the nonlinear mapping and the fuzzy logic-based rules. Numerous textbooks and survey papers have been written about fuzzy control. A few examples are (Driankov et aI., 1993), (Jang and Sun, 1995), (Wang, 1992), (Mendel, 1995).
If Then
where the conditions are conditions on the input signals to the rule-based controller and the actions determine the value of the output from the rule-based controller. The conditions can either be expressed using ordinary, binary logic or using some interval-valued logic such as fuzzy logic. Examples of conditions could be
3.2 Fuzzy Sets, Rules, and Inference Systems A fuzzy controller consists of a set of rules, each stating the control action to be taken in a certain process state. The process states and control actions are expressed on linguistic form, e.g., IF Error IS Large THEN Control Action IS Large
"-0.2 < control error < 0.2"
in the binary logic case or "y is Large" where Large is a fuzzy set in the fuzzy logic case. In the same way the actions could either be expressed using crisp functions, e.g., "u is 1.5", or "u is f (x)" or using fuzzy sets, e.g., "u is Zero" where Zero is a fuzzy set.
Each linguistic relation is represented by a fuzzy set. A classical set is a set with a crisp boundary. A fuzzy set is a set with a non-crisp boundary. An element is a member of a fuzzy set to a degree between and 1. A fuzzy set A is characterized by its membership function
°
If the condition expressions partition the input space in an non-overlapping way it is only one rule that applies and it is this rule that decides the control action. If the input partition contains overlaps, which is normally the case, then several rules apply and the resulting control action is obtained by interpolation. This could be based on fuzzy logic, i.e. fuzzy interpolation or it could be based on some analytical interpolation method. An example of the latter is Rule-based Interpolating (RIP) control (Drechsel and Pandit, 1994). The approach that has gained most popularity is, however, the fuzzy logic approach
f.iA :X~
[0,1]
(1)
The main idea of fuzzy set theory is that statements like Error IS Large are not just either true or false, but can be fulfilled to any degree in the range [0,1]. For a given observation x, the membership function determines to what extent the corresponding linguistic relation applies. Fuzzy logic generalizes the boolean set operators, i.e. union (OR), intersection (AND), etc, to operate on fuzzy sets. The most common operator definitions are: 187
Operation
Definition
Name
)J.AANDB(X)
)J.A(X) · )J.B(X) min[)J.A (X),)J.B (x))
Product Minimum
)J.AOR B(X)
min[l, )J.A(x) +)J.B(X)) max[)J.A(X) , )J.B (x))
Bounded Sum Maximum
The aggregation-OR is typically defined as pointwise maximization or as summation, i.e. , m
fJ.a(U) = LfJ.(;i(u)
(4)
i =l
During defuzzification the fuzzy set is transformed to a numerical value. The most common defuzzification strategy is center of gravity
The process of reasoning with information given by fuzzy sets is called approximate reasoning. A detailed presentation of this can be found in (Lee, 1990) or (J ang and Sun, 1995). A key issue is the fact that a fuzzy rule, e.g. if x is A then y is B where A and B are linguistic values defined by fuzzy sets on X and Y, respectively, is defined as a binary fuzzy relation R on the space Xx Y. The reasoning principle applied is called generalized modus ponens, an approximation of ordinary modus ponens.
(5)
Thus, U is calculated as the ratio between the moment and the area of its fuzzy set.
In a fuzzy controller the above formulae become equivalent to the following calculations:
In most fuzzy controllers the inputs are regarded as exact and represented as crisp values. This simplifies the computations substantially and leads to the two fuzzy inference systems most frequently used in fuzzy control:
1. The fuzzy sets of all inputs are evaluated.
2. The degree of fulfillment for each rule is determined by applying the fuzzy set operators AND, OR and NOT. 3. The contribution of each rule to the control signal is determined by fuzzy implication .
• Mamdani Inference System, and • Takagi-Sugeno Inference System.
4. The output fuzzy set of the controller is formed by aggregating the individual contributions.
3.3 Mamdani inference systems In a Mamdani system the rules are of the form: IF
Xl
IS A~ AND ... AND
Xn
5. The output of the controller is obtained by defuzzification of the output fuzzy set.
IS A~ THEN u IS Bi
The calculations are either performed on-line at each sampling instant or using a precomputed look-up table.
where i is the number of the rule and A~ and B i are fuzzy sets. Each rule defines a fuzzy implication
In principle any combination of implication method and fuzzy set operators can be used (Mizumoto, 1994) . However, in practice only a few combinations are used. Common choices are to use min and max as fuzzy AND and OR operators and min as implication method, or to use product and bounded sum as fuzzy AND and OR operators and product as implication method. The above steps are summarized in Fig. 2 for the latter case.
(2)
fJ.(;i(U) = fJ.AI X... xAn-+B(X,U)
The most common implication rules are:
Definition
Name Product
An internal block diagram view of a Mamdani fuzzy inference system is shown in Fig. 3.
Here, fJ.AI X... xAn(x) is called the degree of fulfillment, d, of the rule.
3.4 Takagi-Sugeno inference systems In the Takagi-Sugeno inference system a rule has the form
All m rules are evaluated in parallel and the total control action is computed by aggregating the control recommendation from each rule using the union operator, i.e., OR.
if x is A and y is B then u=f(x,y)
where A and B are fuzzy sets and U = f(x , y) is a crisp function, often a polynomial, that describes the output of the controller in the range
m
(3) 188
Rule 1: IF e is Zero and ~e is Zero THEN u is Zero
!
1 ___
RIM 1
0.75
X
~----~ ~'. e
is
..,
A1
r- II U
6e
'"
= f1(x)
Rule 2: IF e is Posrtive and !le is Positive THEN u is Positive
x
• ••
R.... n •• 0.32
Fig. 2.
~ . o.oe
-
X
-
-;::::
•
u
Weighted -I-average
Wn
is An I
I
IU =
fJx)
Un
The calculations involved in fuzzy controL ____ • Crisp value
"""- ,
Fig. 4. The internal view of a Takagi-Sugeno fuzzy system with crisp inputs and output. Note that x is vectorvalued.
~-"- uisB , : ' _---' : I..-....;:--..;.J_ - , -
·•••
~~ ~r~.oon ~__ t t
r'::'"_~-I---"""-_-....,, : ; x
i~ An
I
I
d.
!_u is 8 n
I
x
u
_ C r i s p valut - - - - - - Fuzzy value
Fig. 3. The internal view of a Mamdani fuzzy system with crisp inputs and output. Note that x is vector-valued.
Fig. 5. The external view of a fuzzy inference system with crisp inputs and output.
defined by the fuzzy sets of the antecedent of the rule. A zero-order Takagi -Sugeno fuzzy model is obtained if f is a constant. This is a special case of the Mamdani model if the output of each rule is defined by a crisp number (i.e., a singleton fuzzy set). The aggregation and defuzzification of Fig. 3 is now replaced with the more efficient weighted average in Fig. 4.
design a fuzzy controller with linear behavior for small control errors and nonlinear behavior for large errors. It is then possible to make use of all the tuning and design rules for linear controllers in the linear region. Affine fuzzy controllers are easiest to obtain when product and summation is used as fuzzy AND and OR and product is used as implication method. However, it is possible to obtain it also for other choices. For a more complete treatment, the reader is referred to (Meyer-Gramann, 1993) or (Johansson, 1993).
3.5 Non-Linearities Characteristic for fuzzy controllers is that they contain a static nonlinear mapping defined by a fuzzy inference system. The external inputoutput view of a fuzzy inference system is given by Fig. 5
With fuzzy logic the user has great flexibility with respect to which types of non-linearities that can be generated. Nonlinearities that contain areas with constant output can be used to implement dead-zones and to take actuator saturations into account. Utilizing this, fuzzy controllers are often designed to approximate time-optimal "bang-bang" controllers (Kawaji and Matsunga, 1994) . Smooth nonlinearities can be used to compensate for plant nonlinearities similar to a conventional feedback linearization scheme. Finally, jump discontinuities can be used to implement controllers that behave similarly to multi-level relay controllers.
The way fuzzy logic is applied in most commercial fuzzy controllers, each rule defines one point in the mapping. The interpolation between these points is taken care of by the fuzzy logic. Thus, interpolation is the main reason for using fuzzy logic instead of classical logic. In order to fully understand fuzzy control it is important to understand the nature of this interpolation. A particular question is under which conditions the mapping is affine or piecewise affine. If one knows how to design a controller that is affine in a certain region it is, e.g., possible to 189
3.6 Fuzzy Controller Structures The fuzzy inference system in a fuzzy controller defines a non-linear mapping. The inputs and outputs to this mapping determine the structure of the fuzzy controller. The dynamics of the controller are outside the fuzzy system, as shown in Fig. 6 The linear filters on the input are used to
is deemphasized is sometimes regarded as "unscientific." In (Elkan, 1993) the success of fuzzy logic and particularly fuzzy control is presented as a paradox. Still, fuzzy control has had an undisputable success. There are several reasons for this. Fuzzy control is a direct approach to nonlinear control design. The rule-based formalism is intuitive and easy to understand for non-control engineers. Each rule represents local process knowledge about how the control signal should be selected for certain input signals . The local nature of the rules makes it possible to build up a controller in a step-wise fashion . Fuzzy control makes it easy to implement nonlinear control elements. Nonlinear controllers can, potentially, give better control performance than linear controllers both for nonlinear and linear processes. Therefore it is not surprising that fuzzy control has outperformed, e.g., PID control in different comparative studies. However, several of the comparative simulation studies are also unfair because they only compare the control system performance at one point, e.g., for one size of the change in reference value. In (Thomas and Armstrong-Helouvry, 1995) an investigation of a large number of fuzzy control applications is presented. The investigation tries to answer the questions: "What benefits can a control system designer obtain through the use of fuzzy control?" and "What aspects must the application possess for the designer to obtain these benefits?". Some of the benefits found were the possibility to capture operator knowledge in the controller and the possibility to handle exception cases. The latter refers to the possibility to represent special control policies for exceptional cases. An additional benefit was the fast set-point response that could be achieved by designing a fuzzy controller such that it behaves in a manner similar to timeoptimal, "bang-bang" control. This give faster response for large set-point changes when there are bounds on the control authority then what can be achieved with a linear controller.
Fuzzy controller
Fig. 6.
The fuzzy controller structure
generate the inputs to the fuzzy system. These are typically the process output, controller error, error integral or error derivative. The output linear filter consists of an integrator if the fuzzy system output is the control signal increment. Depending on which input and output signals that are used different controller structures can be implemented. For example, if the fuzzy block is chosen as u = :J( e, e) the controller is structurally equivalent to a PD-controller. If the fuzzy mapping is designed to be linear, exact equivalence is obtained. Similarly, it is possible to define fuzzy controllers that are structurally equivalent to all forms of conventional controllers, e.g., PID controllers on position or velocity form, state feedback controllers or general polynomial controllers. Several lessons can be learned from conventional PID control (Astrom and Hagglund, 1995) . To reduce the sensitivity to measurement noise it is important to limit the high frequency gain in the derivative part. In process control it is common practice to take derivatives on the process output instead of the error. This avoids large overshoots at set point changes. If the fuzzy controller uses integrators, an antiwindup scheme should be included.
Another reason for the success is the way the technique is packaged. CAD environments for fuzzy controller development have user-friendly, graphical environments. They are available on industrially accepted hardware and they can automatically generate C code. All this makes fuzzy control straight-forward to apply in industry.
3.7 Fuzzy control advantages Fuzzy control has always been a controversial subject, e.g., (IEEE, 1993a), (IEEE, 1993b) . This owes partly to lack of mutual understanding between the fuzzy control community and the traditional control community and partly to exaggerated claims in certain papers on fuzzy control. Many people active in fuzzy control have no classical control background. This typically leads to reinventions of the wheel. At the same time many classical control engineers have a very "fuzzy" idea of what fuzzy control really is. The empirical nature of fuzzy control where the importance of mathematical models
3.8 Relationships to conventional control Since the fuzzy mapping in a fuzzy controller defines a non-linear input/output mapping fuzzy control has strong relationships to non-linear 190
Knowledgebased system Operator
Process Engineer
6 P =parameter changes Fig. 7.
AI·based performance adaptation
control. Fuzzy controllers are sometimes designed to achieve feedback compensation or feedback linearization.
Fig. 8.
If we instead concentrate on fuzzy logic systems as a modelling methods then there are strong relationships analytical non-linear function approximation methods such as splines or wavelets, and to neural networks (Benveniste et al., 1994). It has been proved that fuzzy models are universal approximants (Wang, 1992), (Castro, 1995) similar to neural networks.
Expert controller structure
4.1 Expert Control
Expert control or autonomous control is an example of how an AI-based system can be used to supervise a number of conventional control algorithms. The notion of expert control was originally introduced as an attempt to obtain a good structure for a system that processes both signal and symbols. The idea was to obtain a strict separation of signal processing and logic where the logic was implemented in an expert system. See (Astrom et al., 1986) . Several different prototypes have been implemented in the expert control framework such as automatic tuners, safety networks for adaptive control and systems for diagnosis. A summary of several experiences is given in (Astrom and Arzen, 1992).
4. SUPERVISORY LEVEL AI-BASED CONTROL On the supervisory level an AI-based control system is either used to adjust the parameters and/or to adjust the structure in a conventional control algorithm or to supervise the operation and application of a number of conventional control algorithms.
An example of an expert controller is shown in Fig. 8. The system has a collection of signal processing algorithms for control, control design, parameter estimation, excitation, diagnosis and logging. There is also a knowledge-based system that coordinates the operation of the algorithms. The system shown in Fig. 8 can be used to implement an advanced controller with facilities for automatic tuning, gain scheduling and adaptation. The system also has features to assist diagnosis, loop assessment, and performance assessment.
Using an AI-based module for parameter adjustments is often done in the context of performance adaptive control. The AI module receives its inputs from an performance monitoring module that, e.g., measures the damping and the response time of the control system. The AI module then adjusts the parameters in, e.g., a PID controller to obtain the specified values of the performance parameters, see Fig. 7 One can find examples of both connectionist and symbolical AI approaches for AI-based adaptive controllers . An early example was the Exact controller from Foxboro (Bristol, 1986), (Kraus and Myron, 1984), a performance adaptive PID controller based on pattern recognition. The controller monitors the control error searching for signs of load disturbances. The ratio of the peaks of the error signal at two consecutive peaks and the time period of between the peaks are used to adjust the PID parameters. The controller contained a reasonable amount of expertize about PID settings and could in that respect be considered as an expert system. However, the system was implemented using conventional programming techniques.
The system in Fig. 8 contains an ordinary feedback loop with a process and a controller. There are, however, a number of auxiliary functions for parameter estimation, control design, supervision, fault detection and diagnosis.There may also be several alternative algorithms for the same task. For example there may be several different controllers. This is indicated by the different layers in the figure. For example, the controller may be a simple PI controller or a more complicated algorithm based on an observer and state feedback. There are also algorithms for generating perturbation signals to excite the process. The fault detection and di191
• which the user can maintain a dialogue with and get information about, e.g., process dynamics, control performance statistics, the factors that delimit control performance, explanations for the controllers actions, etc., and
agnosis tasks are aimed at finding faults that are local to the control loop that the expert controller is part of. This differs from the plantwide approach to diagnosis taken by the majority of the work in diagnosis, e.g., (Prasad and Davis, 1992) . The signal processing algorithms can communicate with the expert system by sending and receiving data. The parameters of the algorithms can be changed and algorithms can be replaced. The expert system, or knowledge based system, decides what algorithm to use when. The knowledge-based system also interacts with the operator or the process engineer.
• where the underlying control knowledge and heuristics is transparently stored in a way that easily allows for modifications and extensions.
This definition of an expert controller is not distinct. Several conventional controllers fulfill some of the items to a certain degree. The total definition is however outside the scope of any existing system.
The system in Fig. 8 is general. Many different systems can be implemented in this framework. A controller with gain scheduling can, e.g., be obtained by simply having one controller whose parameters are changed based on an measured signal. The table for the different parameters can easily be represented by a collection of rules. A controller with automatic tuning, of the type discussed in (Astrom and Hagglund, 1984) can be implemented by using two control algorithms, a PID controller and a relay feedback with sequencing and logic. Such a controller can be represented very conveniently as is discussed in (Arzen, 1993). An adaptive controller based on recursive parameter estimation and control design, see (Astrom and Wittenmark, 1995) is obtained by combining the functions of estimation, excitation, controller design and controller selection. The supervision of the adaptive controller can also very conveniently implemented in the system.
The way to reach the VISIOnary goal can be metaphorically described as an attempt to include an experienced control engineer in the control loop and provide him with an algorithm tool box. Ideally, the controller should be able to act and reason in the same way as the control engineer when he designs a controller and evaluates its performance. The knowledge involved includes both theoretical control knowledge and heuristics or "rules-of-thumb". The knowledge is both procedural and declarative. By employing its general knowledge the controller should be able to extract the specific process knowledge needed automatically through various dynamic experiments with the process. The operation of an expert controller can be divided into two operation phases: tuning and online control. The first stage of the tuning phase is an interrogation where the user may supply his prior process knowledge and the closed loop specifications. The closed loop specifications may be both quantitative and qualitative. Examples of qualitative specifications are, e.g., if the system should have a fast response or that no overshoot is allowed, if the controller should be optimized for set-point following, load disturbance rejection, or noise rejection, etc.
4.2 V£sionary goals of expert control The visionary goal of expert control is a controller • that can satisfactorily control arbitrary processes which are time-varying, nonlinear, and exposed to various disturbances, • which requires a minimal amount of prior process knowledge, • which can take advantage of available prior knowledge,
After the interrogation the controller performs different tuning experiments that returns information about the process dynamics. This information is used to design a suitable controller. During the on-line phase the system monitors and, if needed, changes the controller. The changes may be small parameter adjustments or a completely new controller design. Both parameter adaptation and performance adaptation schemes are conceivable. Also both continuous adaptation and adaptation on demand are possible.
• where the user can enter specifications on the closed loop performance in qualitative terms, e.g., "as fast as possible", "small overshoot" etc., • that successively increases its amount of process knowledge and accordingly improves the control, • that performs diagnosis of the control performance and loop components including detection of actuator and sensor problems, 192
4.4 Integrated control and diagnosis
An important functionality of an expert con-
.,..
troller is the possibility to monitor the components of the control loop and to be able to send out an alarm to the operator or to some supervisory control system when a fault has been detected. On-line fault diagnosis always requires knowledge about the process time scale. This information is naturally available in an expert controller.
-1 0 _ - - : : - ; , . - - . _
A common cause of control problems are the control valves. A control valve with too high friction in combination with a controller with integral actions give rise to oscillations. In (Hagglund, 1995) an automatic procedure for detecting closed loop oscillations is presented. The procedure does not require any additional parameters than the normal controller parameters. The procedure has been implemented in Alfa Laval Automation ECA400 industrial controller, and field tests on industrial plants have verified that it works properly, and manages to detect undesired oscillations in the control loops.
,-Fig. 9. The screen layout for a Grafchart·based expert controller.
4.3 Structuring principles Expert control contains two major problem areas. The first problem considers what process knowledge must be known in order to automatically tune and supervise a controller. This also includes the question of how the knowledge should be obtained, i.e., the nature of the tuning procedures needed.
4.5 Relationships to conventional control Expert control has strong relationships to adaptive control. The approach was originally motivated by shortcomings of adaptive controllers such as the requirements for prior process knowledge, poor user understanding, and complex safety nets. The recursive identification algorithms and the control algorithms of a conventional adaptive controller can be seen as the final algorithmic representation of a large amount of underlying theoretical as well as practical control knowledge, (Astrom and Wittenmark, 1995) 'Tb work in practice, it has to be combined with logic safety jackets or safety nets (Isermann and Lachmann, 1985), (Warvick, 1988) often heuristically derived, that assures the controller performance under nonstandard conditions such as, e.g., switching between different operating modes, insufficient process excitation, control signal saturation etc. This is often the dominating part of the controller and experience has shown that design and testing is quite time consuming.
The second problem concerns the representation of the knowledge involved. The expert controller contains a large number of algorithms and methods of quite different nature, e.g., online time critical filtering and control, on-line calculations of less time-critical nature, e.g., for recursive identification and performance assessment, off-line numerical calculations for identification and experiment analysis, automata describing the overall behaviour of the controller, sequential descriptions of the different tuning experiments and loop assessment experiments involved, and heuristic or semi-heuristic methods for controller choice. Several programming paradigms or software architectures (Shaw, 1994) can be used for implementing an expert controller. An architecture where a rule-based system orchestrates a number of numerical algorithms is described in (Arzen, 1986) . A blackboard-based system containing knowledge sources on either procedural or rule-based form is described in (Arzen, 1987) (Arzen, 1989) . Currently, Grafchart, a Grafcetbased G2 toolbox (Arzen, 1994b), is used to structure the implementation of an expert controller (Wallen, 1995) . Here the sequential function charts represent the tuning experiments in the controller and the steps contain actions and rules that are performed and active in a step. Fig. 9 shows a screen dump from this system.
More recently hybrid control and specially the theory of switching controllers, e.g. (Morse, 1995), has attracted a lot of attention in the conventional control community. These systems are based on the notion of multi-controllers according to Fig. 10. The systems selects between a number of controllers based on some performance index derived by the supervisor. The selection can either select one of the controllers or mix the outputs of several control algorithms. An example of a simple binary selector is a min193
Gainscheduling
Supervisor
Con1n>1lor-1
Con1n>lIor-2
I Con1n>1lor-3
I Con1n>1lor-4
.
Operating
point
Fig. 13. Gain scheduling as a multi-controller
The final example of when a multi-controller can be motivated is Heterogeneous Control (Kuipers and Astrom, 1994) . In (Malmborg, 1995) a Heterogeneous controller is described where it is the task to be performed that decides which controller to be used. The controller combines timeoptimal "bang-bang"control with PID control. The time optimal controller is used for large setpoint changes and the PID controller is used for load disturbance rejection in the vicinity of the setpoint according to Fig. 14. The outputs of the
Fig. 10. A multi-controller
max selector. An example of a mixing selector could be a simple linear interpolation or a fuzzy interpolation. The similarities between a multicontroller and an expert controller according to Fig. 8 are obvious. There are a number of reasons why one would like to change between a number of control algorithms. These are summarized in Fig. 11. In
Heterogenous Control SeM> ContJoI-
Regulation Control-
Time Optimal Controu.r
Task
PlO controller
Fig. 14. Heterogeneous control as a multi-controller
controllers are weighted together using fuzzy interpolation according to Fig. 15. Control
t
Con' Tuk""
Fig. 11. Reasons for multi-controller structures
the original formulation of expert control it was the amount of process knowledge available in relation to the specifications of the closed loop behaviour that decideed which controller to use. During the initialization the controller gradually refines it process knowledge and is able to design a more complex controller. The simpler, hopefully more robust, controller is used as back-up controller in the case the controller error should suddenly become much large than anticipated with the advanced controller. The situation is shown in Fig. 12
Measurements
Fig. 15. Heterogeneous controller combining PID and time-optimal control
5. CONCLUSIONS AI-based techniques have several applications in the feedback control loop either for direct control or for supervision of numerical control algorithms. There are also strong relationships between recent developments in conventional control theory and AI-based controllers. Examples of this are analytical non-linear function approximation methods and logic-based switching controllers. This is promising for the future and can hopefully unify much of the parallel development that currently takes place. After all, it is the functionality of the controllers that is of importance rather than which implementation technique that is used.
Expen Control
pt
controU.r
PlO controller
tUgh...-ordef controhr
Process
Knowledge
Fig. 12. Expert control as a multi-controller
A gainscheduling controller can also be interpreted as a multi-controller. Here, it is the operating point that decides which controller to use (A single controller with a number of different parameter sets can be interpreted a set of controllers with fixed parameters.) according to Fig. 13. 194
6. REFERENCES
Drechsel, D. and M. Pandit (1994) : "RIP control in knowledge-based systems." volume Preprints of the 2nd IFAC Workshop on Computer Software Structures Integrating AIIKBS Systems in Process Control, Lund, Sweden, pp. 129-134. Driankov, D., H . Hellendoorn, and M. Reinfrank (1993) : An Introduction to Fuzzy Control. Springer-Verlag. Elkan,. C~ (1993) : "The paradoxical success of fuzzy lOgIC. In Proceedings of AAAI. Froese, T. (1993): "Applying offuzzy control and neuronal networks to modem process control systems." In Proceedings of the EUFIT '93, volume II, pp. 559-568, Aachen. Fu, K-S. (1971) : "Learning control systems and intelligent control systems: An intersection of artificial intelligence and automatic control." IEEE Transactions on Automatic Control, 16, pp. 70-73. Hagglund, T. (1995): "A control-loop performance monitor." Control Engineering Practice, 3:11 . Holmblad, L. and J . Ostergaard (1982) : "Control of a cement kiln by fuzzy logic." In Gupta and Sanchez, Eds., Fuzzy Information and Decision Processes. North-Holland, Amsterdam. Hunt, K , D. Sbarbaro, R. Zbikowski, and P. Gawthrop (1992): "Neural networks for control systems - a survey." Automatica, 28:6, pp. 10831112. IEEE (1993a): "Reader's forum." IEEE Control Systems Magazine, June. IEEE (1993b) : "Reader's forum." IEEE Control Systems Magazine, August. Isermann, R. and K Lachmann (1985): "Parameter adaptive control with configuration aids and supervision functions." Automatica, 21, pp. 623638. Jang, J.-S. R. (1993): "ANFIS: Adaptive-networkbased fuzzy inference systems." IEEE Trans. on Systems, man, and Cybemetics, 23:03, pp. 665685. Jang, J.-S. R. and C.-T. Sun (1995): "Neuro-fuzzy modeling and control." Proceedings of the IEEE, March. Johansson, M. (1993) : "Nonlinearities and interpolation in fuzzy control." Master thesis ISRN LUTFD2ITFRT--5491--SE, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden. Kawaji, S . and N. Matsunga (1994): "Fuzzy control of VSS type and its robustness." In Kandel and Langholz, Eds., Fuzzy Control Systems. CRC Press. Kraus, T. W. and T. J . Myron (1984) : "Self-tuning PID controller uses pattern recognition approach." Control Engineering, June, pp. l06-11I. Kuipers, B. and K J . Astrom (1994) : "The composition and validation of heterogeneous control laws ." Automatica, 30:2, pp. 233-249. Laffey, T. J ., P. A Cox, J . L. Schmidt, S . M. Kao, and J . Y. Read (1988): "Real-time knowledge-based systems." AI Magazine, 9:1, pp. 27-45. Lee, C. (1990) : "Fuzzy logic in control systems: Fuzzy logic controller part I and II." IEEE Transactions on Systems, Man, and Cybemetics, 20:2, pp. 404435. Malmborg, J . (1995): "An approach to hybrid sys-
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Applied Control: Current Trends and Modem Methodologies. Marcel Dekker Inc. Arzer:, K-E . (1994a) : "Grafcet for intelligent superVISOry control applications ." Automatica, 30:10. Arzer:' K-E. (1994b): "Grafcet for intelligent superVIsory control applications." A utomatica, 30:10, pp. 1513-1526. Astrom, K J. , J. J. Anton, and K-E . Arzen (1986) : • "Expert control." Automatica, 22:3, pp. 277-286. Astrom, K J . and K-E. Arzen (1992) : "Expert control." In Passino and Antsaklis, Eds., An Intro-
duction To Intelligent and Autonomous Control. Kluwer Academic Publishers. Astrom, K J . and T. Hagglund (1984) : "Automatic tuning of simple regulators with specifications on phase and amplitude margins." Automatica 20 pp. 645-651. ' , Astrom, K J . and T. Hagglund (1995): PID Controllers: Theory, Design, and Tuning. Instrument Society of America, Research Triangle Park NC second edition. ' , Astr~m, K J . and B. Wittenmark (1995): AdaptIVe Control. Addison-Wesley, Reading, Massachusetts, second edition. Benveniste, A, A Juditsky, B. Delyon, Q. Zhang, and P.-Y. Glorennec (1994) : "Wavelets in idenitification." In Proc. of 10th !FAC Symposium on System Identification, SYSID '94, volume 2, pp. 2748. Bristol, E. H . (1986) : "The EXACT pattern recognition adaptive controller, a user-oriented commercial success." In Narendra, Ed., Adaptive and Learning Systems, pp. 149-163, New York. Plenum Press. Brockett, R. W. (1990): "Formal languages for motion description and map making." In Brocket, Ed., Robotics, Providence, Rhode Island. American Math. Soc. Brockett, R. W. (1993) : "Hybrid models for motion control systems." In Trentelman and Willems Eds ., Essays on Control: Perspectives in the The~ ory and its Application. Birkhauser. Castro, J . (1995) : "Fuzzy logic controllers are universal approximators ." IEEE Transactions on Systems, Man and Cybemetics, April, pp. 629-635. Conner, D. (1993) : "Designing a fuzzy-logic control system." EDN, March. Crossman, E. R. and J. Cooke (1962) : "Manual control of slow-response systems." In Edwards and Lees Eds ., The Human Operator in Process Contro/ Taylor and Francis Ltd., London, 1974. 195
tems ." In Morse, Ed., Preprints of the Block Island Workshop on Control Using Logic-Based Switching, Rhode Island. Mamdani, E. (1974) : "Application of fuzzy algorithm for control of simple dynamic plant." Proc. lEE, No 121, pp. 1585-1588. Mendel, J . (1995) : "Fuzzy logic systems for engineering: A tutorial." Proceedings of the IEEE, 83:3, pp. 345-377. Meyer-Gramann, K (1993) : "Easy implementation of fuzzy controller with a smooth control surface." In Proceedings of the EUFIT93, volume I, pp. 117123, Aachen. Miller, w., R. Sutton, and P. Werbos (1990): Neural Networks for Control. The MlT Press, Cambridge, MA. Mizumoto, M. (1994): "Fuzzy controls under productsum-gravity methods and new fuzzy control methods." In Kandel and Langholz, Eds., Fuzzy Control Systems, pp. 275-294. CRC Press. Moore, R., H . Rosenof, and G. Stanley (1990): "Process control using a real time expert system." In Preprints 11th IFAC World Congress, Tallinn, Estonia. Morse, A. (1995): "Control using logic-based switching." In Isidori, Ed., Trends in Control: A European Perspective, pp. 69-114. Springer. Prasad, P. R. and J . F. Davis (1992) : "A framework for knowledge-based diagnosis in process operations." In Passino and Antsaklis, Eds., An Introduction to Intelligent and Autonomous Control. Kluwer Academic Publishers. Rosenblatt, F. (1962): Principles of Neurodynamics. Spartan Books, New York. Rumelhart, D., G. Hinton, and R. Williams (1986): "Learning representations by back-propagating errors." Nature, No 323, pp. 533-536. Saridis, G. N. (1977): Self-Organizing Control of Stochastic Systems. Marcel Dekker, New York. Shaw, M. (1994): "Beyond objects: A software design paradigm based on process control." Technical Report CMU/SEI-94-TR-15, Software Engineering Institute, Carnegie-Mellon University. Thomas, D. E . and B. Armstrong-Helouvry (1995) : "Fuzzy logic control- a taxonomy of demonstrated benefits." Proceedings of the IEEE, 83:3, pp. 40742l. Tsypkin, Y. (1971) : Adaptation and Leaming in Automatic Systems. Academic Press, New York. Wallen, A. (1995): "Using Grafcet to structure control algorithms ." In Proceedings of the Third European Control Conference (ECC 95), !Wma, Italy, pp. 3160-3165. Wang, L. (1992) : "Fuzzy systems are universal approximators." In Proc. First IEEE Conference on Fuzzy Systems, pp. 1163-1169, San Diego. Warvick, K (1988) : Implementation of self-tuning controllers. Peter Peregrinus, London. Werbos, P. (1974) : Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. PhD thesis, Harvard University. Widrow, B. (1962) : "Generalization and information storage in network of Adaline neurons ." In Yovits et al., Eds., Self-Organizing Systems. Spartan Books, Washington, DC. Zadeh, L. (1965): "Fuzzy sets." Information and Con-
trol, No 8, pp. 338-353. Zadeh, L. (1994): "Foreword." In Kandel and Langholz, Eds., Fuzzy Control Systems. CRC Press. Zhang, J. J . R. and F. Mayer (1993) : "Development of a fuzzy co-processor for real-time control." In Proceedings of the EUFIT '93, volume I, pp. 9197, Aachen.
196
AI IN THE FEEDBACK LOOP: A SURVEY OF ALTERNATIVE APPROACHES Karl-Erik Arzen Department of Automatic Control Lund Institute of Technology PO. Box 118 S-221 00 Lund, Sweden Email: [email protected]
Abstract: An overview of different ways of using AI techniques in feedback control is presented. The paper gives special attention to fuzzy control and expert control.
1. INTRODUCTION
work in AI in control has developed. Direct-level AI based control is the subject of Section 3. The emphasis here is on fuzzy control. Section 4 describes supervisory-level AI applications with focus on expert control.
The aim of this paper is to present some different Artificial Intelligence (AI) based paradigms that can be used on the feedback control loop level. Main focus will be given to rule-based control, specially fuzzy control and to expert control. The majority of the AI-techniques in use in control today are either based on trying to mimic the behaviour and expertise of a human expert, e.g., a process operator, or based on machine learning. In the first field we find most of the work in fuzzy control and expert control whereas in the second field we find the work in learning control and neuro control. This paper focuses on the first approach.
2. A HISTORICAL SURVEY Ever since Artificial Intelligence (AI) was established as a research discipline in the end of the 1950s it has had strong connection to control engineering. In the 1950s and 1960s the area of Cybernetics was popular. This comprised among several subjects both AI and control theory. Much work at that time was focused on learning using systems modelled after the neuron-based structure of the brain. Systems such as the Perceptron (Rosenblatt, 1962) and ADALINE (Widrow, 1962) belong here. These ideas are also the origin of what is called learning control (Tsypkin, 1971). Other names for this are intelligent control, e.g. (Fu, 1971), and self-organizing control (Saridis, 1977) .
Another way of classifying different approaches is based on whether the AI part of the controller is used on the direct feedback control level or if it used on a supervisory control level orchestrating a set of conventional control algorithms. In the first class we find the majority of the rulebased approaches whereas in the second class we find expert control.
In learning control systems the controller should be able to estimate unknown information during its operation and determine an optimal control action from the estimated information. Different learning schemes have been proposed, e.g., pattern classification with adaptable decision threshold, bayesian estimation and stochastic approximation. Much of the
Common to all the approaches described here is that there are parallel, but strongly related, activities taking place in the conventional control community. For example, there are relations between learning control and adaptive control and between expert control and multi-controllers. Section 2 gives a historical overview of how the 185
work in learning control systems have strong relationships to conventional adaptive control.
During the 1970s much of the activities in AI were focused on expert systems or knowledgebased systems. These systems were designed to solve heuristic problems in an organized way mainly by mimicing the way an experienced human expert solves the problem. The key idea is to organize knowledge in terms of rules. Expert systems have been applied to a wide variety of problems with varying success. Some commonly given criteria for success are that the problem is nontrivial and sufficiently complex, that the problem can be solved by human experts and that experts are available.
The development of the backpropagatwn training algorithm for multi-layer feedforward neural networks (Werbos, 1974) , (Rumelhart et al. , 1986) led to a renewed interest in neural networks and neural control during the 1980s. A number of different net models are used, e.g., feedforward networks such as the multi-layer perceptron, CMAC and radial basis function networks and recurrent networks such as, e.g. , Hopfield networks and Kohonen nets. Also other training or learning strategies have been developed. Backpropagation is an example of supervised learning where it is assumed that inputoutput data pairs are available. In unsupervised learning the network is only presented with input data. The network then organizes itself to best map the input data. One an example of this is the Self-Organizing Feature Maps used in the context of Kohonen nets. Reinforcement learning is an approach that can be used when output data only are available in the form of delayed, partial information that gives credit to a series of actions over time. A large number of textbooks and survey papers on neural networks for control are available, e.g., (Miller et al., 1990) and (Hunt et al., 1992)
The expert system approach soon found its way into control application. The idea of trying to combine expert systems with controllers is, however, not new. Already in 1962, a Heuristic Decision program was discussed (Crossman and Cooke, 1962) in the context of manual control. In control engineering, the main application area is supervisory applications such as process monitoring and diagnosis, scheduling and planning. However, the approach is also used in expert control and several fuzzy controllers can be seen as expert systems. Expert systems were originally developed to solve static problems, i.e., situations where the premises do not change with time. Control problems are generally not static. A statement may, e.g., suddenly switch from true to false because of a change in the physical system being controlled. Reasoning with time is a very complicated problem where many theoretical problems are unresolved (Laffey et al., 1988). Some pragmatic approaches are taken to deal with these issues. One method is to replace the dynamic problem by a static problem by assuming that all premises hold over a small sliding time-window. Another method is to keep track of the chain of reasoning so that all conclusions drawn from a statement can be withdrawn when the statement ceases to be true. It is also important that conclusions are reached in a reasonable time.
Fuzzy set theory (Zadeh, 1965) was developed in the beginning of the 1960s as a way of expressing non-probabilistic uncertainties. Since then, fuzzy set theory has developed and found applications in database management, decision support systems, signal processing, data classifications, computer vision, etc. In 1974, the first successful control application was reported (Mamdani, 1974). Control of cement kilns was an early industrial application (Holmblad and Ostergaard, 1982). Since the first consumer product using fuzzy logic was marketed in 1987, fuzzy control has received enormous attention (Zadeh, 1994). A number of CAD environments for fuzzy control design have emerged (Conner, 1993) and VLSI hardware for fast execution has been developed (Zhang and Mayer, 1993). Fuzzy control is being applied industrially in an increasing number of cases, e.g., (Froese, 1993).
Commercial programming environments for knowledge-based real-time systems have been available for around 10 years now. A good example is G2 from Gensym Corporation, (Moore et al., 1990), which is aimed at supervisory control applications. In G2 the developer can combine a number of programming paradigms, e.g., object-oriented programming, procedural programming, rule-based programming, and event-driven programming. It is also possible to implement Petri Nets, state machines, and neural networks in G2, e.g., (Arzen, 1994a).
Fuzzy control has strong relationships to neural networks. A fuzzy inference system can be expressed in network form, e.g., (Jang, 1993), and can be trained with gradient methods equivalent to the back-propagation procedure. This is utilized in so called "neuro-fuzzy" models where the prior knowledge available about the function to be approximated is expressed as fuzzy rules and used to initialize the structure and weights of the network. After training the resulting net can be reinterpreted in terms of fuzzy sets and rules.
In the late 1980s the focus in expert system and knowledge-based systems shifted from rulebased, heuristics-based systems to model-based 186
approaches that are based on a deeper understanding of the application. In control engineering this led to the interest in Qualitative Methods for, e.g., control and diagnosis.
3.1 Fuzzy control The early work in fuzzy control was motivated by a desire to directly express the control actions of an experienced human operator in the controller, i.e., to mimic his behavior, and to obtain smooth interpolation between discrete controller outputs. Since then the application range of fuzzy control has widened substantially. In most cases a fuzzy controller is used for direct feedback control. However, it can also be used on the supervisory level as, e.g., a self-tuning device in a conventional PID controller. Also, fuzzy control is no longer only used to directly express a priori process knowledge. For example, a fuzzy controller can be derived from a fuzzy model obtained through system identification. Therefore, it is difficult to define what a fuzzy controller is. A very general definition is:
3. DffiECT-LEVEL AI-BASED CONTROL On the direct level an AI-based controller is used to directly calculate the control signals to the process according to Fig. 1. The AI-
Fig. L
Direct level AI-based controller.
based controllers used on this level are either based on the connectionist AI approach, e.g., neural networks, or based on the symbolical AI approach. Here we will concentrate on the latter case.
A Fuzzy Controller is a controller that contains an, often non-linear, mapping that has been defined using fuzzy logic-based rules.
The majority of the symbolical AI that is being applied to direct feedback control is rule-based. The basic knowledge representation formalism is rules of the type
The key issues in this definition are the nonlinear mapping and the fuzzy logic-based rules. Numerous textbooks and survey papers have been written about fuzzy control. A few examples are (Driankov et aI., 1993), (Jang and Sun, 1995), (Wang, 1992), (Mendel, 1995).
If
where the conditions are conditions on the input signals to the rule-based controller and the actions determine the value of the output from the rule-based controller. The conditions can either be expressed using ordinary, binary logic or using some interval-valued logic such as fuzzy logic. Examples of conditions could be
3.2 Fuzzy Sets, Rules, and Inference Systems A fuzzy controller consists of a set of rules, each stating the control action to be taken in a certain process state. The process states and control actions are expressed on linguistic form, e.g., IF Error IS Large THEN Control Action IS Large
"-0.2 < control error < 0.2"
in the binary logic case or "y is Large" where Large is a fuzzy set in the fuzzy logic case. In the same way the actions could either be expressed using crisp functions, e.g., "u is 1.5", or "u is f (x)" or using fuzzy sets, e.g., "u is Zero" where Zero is a fuzzy set.
Each linguistic relation is represented by a fuzzy set. A classical set is a set with a crisp boundary. A fuzzy set is a set with a non-crisp boundary. An element is a member of a fuzzy set to a degree between and 1. A fuzzy set A is characterized by its membership function
°
If the condition expressions partition the input space in an non-overlapping way it is only one rule that applies and it is this rule that decides the control action. If the input partition contains overlaps, which is normally the case, then several rules apply and the resulting control action is obtained by interpolation. This could be based on fuzzy logic, i.e. fuzzy interpolation or it could be based on some analytical interpolation method. An example of the latter is Rule-based Interpolating (RIP) control (Drechsel and Pandit, 1994). The approach that has gained most popularity is, however, the fuzzy logic approach
f.iA :X~
[0,1]
(1)
The main idea of fuzzy set theory is that statements like Error IS Large are not just either true or false, but can be fulfilled to any degree in the range [0,1]. For a given observation x, the membership function determines to what extent the corresponding linguistic relation applies. Fuzzy logic generalizes the boolean set operators, i.e. union (OR), intersection (AND), etc, to operate on fuzzy sets. The most common operator definitions are: 187
Operation
Definition
Name
)J.AANDB(X)
)J.A(X) · )J.B(X) min[)J.A (X),)J.B (x))
Product Minimum
)J.AOR B(X)
min[l, )J.A(x) +)J.B(X)) max[)J.A(X) , )J.B (x))
Bounded Sum Maximum
The aggregation-OR is typically defined as pointwise maximization or as summation, i.e. , m
fJ.a(U) = LfJ.(;i(u)
(4)
i =l
During defuzzification the fuzzy set is transformed to a numerical value. The most common defuzzification strategy is center of gravity
The process of reasoning with information given by fuzzy sets is called approximate reasoning. A detailed presentation of this can be found in (Lee, 1990) or (J ang and Sun, 1995). A key issue is the fact that a fuzzy rule, e.g. if x is A then y is B where A and B are linguistic values defined by fuzzy sets on X and Y, respectively, is defined as a binary fuzzy relation R on the space Xx Y. The reasoning principle applied is called generalized modus ponens, an approximation of ordinary modus ponens.
(5)
Thus, U is calculated as the ratio between the moment and the area of its fuzzy set.
In a fuzzy controller the above formulae become equivalent to the following calculations:
In most fuzzy controllers the inputs are regarded as exact and represented as crisp values. This simplifies the computations substantially and leads to the two fuzzy inference systems most frequently used in fuzzy control:
1. The fuzzy sets of all inputs are evaluated.
2. The degree of fulfillment for each rule is determined by applying the fuzzy set operators AND, OR and NOT. 3. The contribution of each rule to the control signal is determined by fuzzy implication .
• Mamdani Inference System, and • Takagi-Sugeno Inference System.
4. The output fuzzy set of the controller is formed by aggregating the individual contributions.
3.3 Mamdani inference systems In a Mamdani system the rules are of the form: IF
Xl
IS A~ AND ... AND
Xn
5. The output of the controller is obtained by defuzzification of the output fuzzy set.
IS A~ THEN u IS Bi
The calculations are either performed on-line at each sampling instant or using a precomputed look-up table.
where i is the number of the rule and A~ and B i are fuzzy sets. Each rule defines a fuzzy implication
In principle any combination of implication method and fuzzy set operators can be used (Mizumoto, 1994) . However, in practice only a few combinations are used. Common choices are to use min and max as fuzzy AND and OR operators and min as implication method, or to use product and bounded sum as fuzzy AND and OR operators and product as implication method. The above steps are summarized in Fig. 2 for the latter case.
(2)
fJ.(;i(U) = fJ.AI X... xAn-+B(X,U)
The most common implication rules are:
Definition
Name Product
An internal block diagram view of a Mamdani fuzzy inference system is shown in Fig. 3.
Here, fJ.AI X... xAn(x) is called the degree of fulfillment, d, of the rule.
3.4 Takagi-Sugeno inference systems In the Takagi-Sugeno inference system a rule has the form
All m rules are evaluated in parallel and the total control action is computed by aggregating the control recommendation from each rule using the union operator, i.e., OR.
if x is A and y is B then u=f(x,y)
where A and B are fuzzy sets and U = f(x , y) is a crisp function, often a polynomial, that describes the output of the controller in the range
m
(3) 188
Rule 1: IF e is Zero and ~e is Zero THEN u is Zero
!
1 ___
RIM 1
0.75
X
~----~ ~'. e
is
..,
A1
r- II U
6e
'"
= f1(x)
Rule 2: IF e is Posrtive and !le is Positive THEN u is Positive
x
• ••
R.... n •• 0.32
Fig. 2.
~ . o.oe
-
X
-
-;::::
•
u
Weighted -I-average
Wn
is An I
I
IU =
fJx)
Un
The calculations involved in fuzzy controL ____ • Crisp value
"""- ,
Fig. 4. The internal view of a Takagi-Sugeno fuzzy system with crisp inputs and output. Note that x is vectorvalued.
~-"- uisB , : ' _---' : I..-....;:--..;.J_ - , -
·•••
~~ ~r~.oon ~__ t t
r'::'"_~-I---"""-_-....,, : ; x
i~ An
I
I
d.
!_u is 8 n
I
x
u
_ C r i s p valut - - - - - - Fuzzy value
Fig. 3. The internal view of a Mamdani fuzzy system with crisp inputs and output. Note that x is vector-valued.
Fig. 5. The external view of a fuzzy inference system with crisp inputs and output.
defined by the fuzzy sets of the antecedent of the rule. A zero-order Takagi -Sugeno fuzzy model is obtained if f is a constant. This is a special case of the Mamdani model if the output of each rule is defined by a crisp number (i.e., a singleton fuzzy set). The aggregation and defuzzification of Fig. 3 is now replaced with the more efficient weighted average in Fig. 4.
design a fuzzy controller with linear behavior for small control errors and nonlinear behavior for large errors. It is then possible to make use of all the tuning and design rules for linear controllers in the linear region. Affine fuzzy controllers are easiest to obtain when product and summation is used as fuzzy AND and OR and product is used as implication method. However, it is possible to obtain it also for other choices. For a more complete treatment, the reader is referred to (Meyer-Gramann, 1993) or (Johansson, 1993).
3.5 Non-Linearities Characteristic for fuzzy controllers is that they contain a static nonlinear mapping defined by a fuzzy inference system. The external inputoutput view of a fuzzy inference system is given by Fig. 5
With fuzzy logic the user has great flexibility with respect to which types of non-linearities that can be generated. Nonlinearities that contain areas with constant output can be used to implement dead-zones and to take actuator saturations into account. Utilizing this, fuzzy controllers are often designed to approximate time-optimal "bang-bang" controllers (Kawaji and Matsunga, 1994) . Smooth nonlinearities can be used to compensate for plant nonlinearities similar to a conventional feedback linearization scheme. Finally, jump discontinuities can be used to implement controllers that behave similarly to multi-level relay controllers.
The way fuzzy logic is applied in most commercial fuzzy controllers, each rule defines one point in the mapping. The interpolation between these points is taken care of by the fuzzy logic. Thus, interpolation is the main reason for using fuzzy logic instead of classical logic. In order to fully understand fuzzy control it is important to understand the nature of this interpolation. A particular question is under which conditions the mapping is affine or piecewise affine. If one knows how to design a controller that is affine in a certain region it is, e.g., possible to 189
3.6 Fuzzy Controller Structures The fuzzy inference system in a fuzzy controller defines a non-linear mapping. The inputs and outputs to this mapping determine the structure of the fuzzy controller. The dynamics of the controller are outside the fuzzy system, as shown in Fig. 6 The linear filters on the input are used to
is deemphasized is sometimes regarded as "unscientific." In (Elkan, 1993) the success of fuzzy logic and particularly fuzzy control is presented as a paradox. Still, fuzzy control has had an undisputable success. There are several reasons for this. Fuzzy control is a direct approach to nonlinear control design. The rule-based formalism is intuitive and easy to understand for non-control engineers. Each rule represents local process knowledge about how the control signal should be selected for certain input signals . The local nature of the rules makes it possible to build up a controller in a step-wise fashion . Fuzzy control makes it easy to implement nonlinear control elements. Nonlinear controllers can, potentially, give better control performance than linear controllers both for nonlinear and linear processes. Therefore it is not surprising that fuzzy control has outperformed, e.g., PID control in different comparative studies. However, several of the comparative simulation studies are also unfair because they only compare the control system performance at one point, e.g., for one size of the change in reference value. In (Thomas and Armstrong-Helouvry, 1995) an investigation of a large number of fuzzy control applications is presented. The investigation tries to answer the questions: "What benefits can a control system designer obtain through the use of fuzzy control?" and "What aspects must the application possess for the designer to obtain these benefits?". Some of the benefits found were the possibility to capture operator knowledge in the controller and the possibility to handle exception cases. The latter refers to the possibility to represent special control policies for exceptional cases. An additional benefit was the fast set-point response that could be achieved by designing a fuzzy controller such that it behaves in a manner similar to timeoptimal, "bang-bang" control. This give faster response for large set-point changes when there are bounds on the control authority then what can be achieved with a linear controller.
Fuzzy controller
Fig. 6.
The fuzzy controller structure
generate the inputs to the fuzzy system. These are typically the process output, controller error, error integral or error derivative. The output linear filter consists of an integrator if the fuzzy system output is the control signal increment. Depending on which input and output signals that are used different controller structures can be implemented. For example, if the fuzzy block is chosen as u = :J( e, e) the controller is structurally equivalent to a PD-controller. If the fuzzy mapping is designed to be linear, exact equivalence is obtained. Similarly, it is possible to define fuzzy controllers that are structurally equivalent to all forms of conventional controllers, e.g., PID controllers on position or velocity form, state feedback controllers or general polynomial controllers. Several lessons can be learned from conventional PID control (Astrom and Hagglund, 1995) . To reduce the sensitivity to measurement noise it is important to limit the high frequency gain in the derivative part. In process control it is common practice to take derivatives on the process output instead of the error. This avoids large overshoots at set point changes. If the fuzzy controller uses integrators, an antiwindup scheme should be included.
Another reason for the success is the way the technique is packaged. CAD environments for fuzzy controller development have user-friendly, graphical environments. They are available on industrially accepted hardware and they can automatically generate C code. All this makes fuzzy control straight-forward to apply in industry.
3.7 Fuzzy control advantages Fuzzy control has always been a controversial subject, e.g., (IEEE, 1993a), (IEEE, 1993b) . This owes partly to lack of mutual understanding between the fuzzy control community and the traditional control community and partly to exaggerated claims in certain papers on fuzzy control. Many people active in fuzzy control have no classical control background. This typically leads to reinventions of the wheel. At the same time many classical control engineers have a very "fuzzy" idea of what fuzzy control really is. The empirical nature of fuzzy control where the importance of mathematical models
3.8 Relationships to conventional control Since the fuzzy mapping in a fuzzy controller defines a non-linear input/output mapping fuzzy control has strong relationships to non-linear 190
Knowledgebased system Operator
Process Engineer
6 P =parameter changes Fig. 7.
AI·based performance adaptation
control. Fuzzy controllers are sometimes designed to achieve feedback compensation or feedback linearization.
Fig. 8.
If we instead concentrate on fuzzy logic systems as a modelling methods then there are strong relationships analytical non-linear function approximation methods such as splines or wavelets, and to neural networks (Benveniste et al., 1994). It has been proved that fuzzy models are universal approximants (Wang, 1992), (Castro, 1995) similar to neural networks.
Expert controller structure
4.1 Expert Control
Expert control or autonomous control is an example of how an AI-based system can be used to supervise a number of conventional control algorithms. The notion of expert control was originally introduced as an attempt to obtain a good structure for a system that processes both signal and symbols. The idea was to obtain a strict separation of signal processing and logic where the logic was implemented in an expert system. See (Astrom et al., 1986) . Several different prototypes have been implemented in the expert control framework such as automatic tuners, safety networks for adaptive control and systems for diagnosis. A summary of several experiences is given in (Astrom and Arzen, 1992).
4. SUPERVISORY LEVEL AI-BASED CONTROL On the supervisory level an AI-based control system is either used to adjust the parameters and/or to adjust the structure in a conventional control algorithm or to supervise the operation and application of a number of conventional control algorithms.
An example of an expert controller is shown in Fig. 8. The system has a collection of signal processing algorithms for control, control design, parameter estimation, excitation, diagnosis and logging. There is also a knowledge-based system that coordinates the operation of the algorithms. The system shown in Fig. 8 can be used to implement an advanced controller with facilities for automatic tuning, gain scheduling and adaptation. The system also has features to assist diagnosis, loop assessment, and performance assessment.
Using an AI-based module for parameter adjustments is often done in the context of performance adaptive control. The AI module receives its inputs from an performance monitoring module that, e.g., measures the damping and the response time of the control system. The AI module then adjusts the parameters in, e.g., a PID controller to obtain the specified values of the performance parameters, see Fig. 7 One can find examples of both connectionist and symbolical AI approaches for AI-based adaptive controllers . An early example was the Exact controller from Foxboro (Bristol, 1986), (Kraus and Myron, 1984), a performance adaptive PID controller based on pattern recognition. The controller monitors the control error searching for signs of load disturbances. The ratio of the peaks of the error signal at two consecutive peaks and the time period of between the peaks are used to adjust the PID parameters. The controller contained a reasonable amount of expertize about PID settings and could in that respect be considered as an expert system. However, the system was implemented using conventional programming techniques.
The system in Fig. 8 contains an ordinary feedback loop with a process and a controller. There are, however, a number of auxiliary functions for parameter estimation, control design, supervision, fault detection and diagnosis.There may also be several alternative algorithms for the same task. For example there may be several different controllers. This is indicated by the different layers in the figure. For example, the controller may be a simple PI controller or a more complicated algorithm based on an observer and state feedback. There are also algorithms for generating perturbation signals to excite the process. The fault detection and di191
• which the user can maintain a dialogue with and get information about, e.g., process dynamics, control performance statistics, the factors that delimit control performance, explanations for the controllers actions, etc., and
agnosis tasks are aimed at finding faults that are local to the control loop that the expert controller is part of. This differs from the plantwide approach to diagnosis taken by the majority of the work in diagnosis, e.g., (Prasad and Davis, 1992) . The signal processing algorithms can communicate with the expert system by sending and receiving data. The parameters of the algorithms can be changed and algorithms can be replaced. The expert system, or knowledge based system, decides what algorithm to use when. The knowledge-based system also interacts with the operator or the process engineer.
• where the underlying control knowledge and heuristics is transparently stored in a way that easily allows for modifications and extensions.
This definition of an expert controller is not distinct. Several conventional controllers fulfill some of the items to a certain degree. The total definition is however outside the scope of any existing system.
The system in Fig. 8 is general. Many different systems can be implemented in this framework. A controller with gain scheduling can, e.g., be obtained by simply having one controller whose parameters are changed based on an measured signal. The table for the different parameters can easily be represented by a collection of rules. A controller with automatic tuning, of the type discussed in (Astrom and Hagglund, 1984) can be implemented by using two control algorithms, a PID controller and a relay feedback with sequencing and logic. Such a controller can be represented very conveniently as is discussed in (Arzen, 1993). An adaptive controller based on recursive parameter estimation and control design, see (Astrom and Wittenmark, 1995) is obtained by combining the functions of estimation, excitation, controller design and controller selection. The supervision of the adaptive controller can also very conveniently implemented in the system.
The way to reach the VISIOnary goal can be metaphorically described as an attempt to include an experienced control engineer in the control loop and provide him with an algorithm tool box. Ideally, the controller should be able to act and reason in the same way as the control engineer when he designs a controller and evaluates its performance. The knowledge involved includes both theoretical control knowledge and heuristics or "rules-of-thumb". The knowledge is both procedural and declarative. By employing its general knowledge the controller should be able to extract the specific process knowledge needed automatically through various dynamic experiments with the process. The operation of an expert controller can be divided into two operation phases: tuning and online control. The first stage of the tuning phase is an interrogation where the user may supply his prior process knowledge and the closed loop specifications. The closed loop specifications may be both quantitative and qualitative. Examples of qualitative specifications are, e.g., if the system should have a fast response or that no overshoot is allowed, if the controller should be optimized for set-point following, load disturbance rejection, or noise rejection, etc.
4.2 V£sionary goals of expert control The visionary goal of expert control is a controller • that can satisfactorily control arbitrary processes which are time-varying, nonlinear, and exposed to various disturbances, • which requires a minimal amount of prior process knowledge, • which can take advantage of available prior knowledge,
After the interrogation the controller performs different tuning experiments that returns information about the process dynamics. This information is used to design a suitable controller. During the on-line phase the system monitors and, if needed, changes the controller. The changes may be small parameter adjustments or a completely new controller design. Both parameter adaptation and performance adaptation schemes are conceivable. Also both continuous adaptation and adaptation on demand are possible.
• where the user can enter specifications on the closed loop performance in qualitative terms, e.g., "as fast as possible", "small overshoot" etc., • that successively increases its amount of process knowledge and accordingly improves the control, • that performs diagnosis of the control performance and loop components including detection of actuator and sensor problems, 192
4.4 Integrated control and diagnosis
An important functionality of an expert con-
.,..
troller is the possibility to monitor the components of the control loop and to be able to send out an alarm to the operator or to some supervisory control system when a fault has been detected. On-line fault diagnosis always requires knowledge about the process time scale. This information is naturally available in an expert controller.
-1 0 _ - - : : - ; , . - - . _
A common cause of control problems are the control valves. A control valve with too high friction in combination with a controller with integral actions give rise to oscillations. In (Hagglund, 1995) an automatic procedure for detecting closed loop oscillations is presented. The procedure does not require any additional parameters than the normal controller parameters. The procedure has been implemented in Alfa Laval Automation ECA400 industrial controller, and field tests on industrial plants have verified that it works properly, and manages to detect undesired oscillations in the control loops.
,-Fig. 9. The screen layout for a Grafchart·based expert controller.
4.3 Structuring principles Expert control contains two major problem areas. The first problem considers what process knowledge must be known in order to automatically tune and supervise a controller. This also includes the question of how the knowledge should be obtained, i.e., the nature of the tuning procedures needed.
4.5 Relationships to conventional control Expert control has strong relationships to adaptive control. The approach was originally motivated by shortcomings of adaptive controllers such as the requirements for prior process knowledge, poor user understanding, and complex safety nets. The recursive identification algorithms and the control algorithms of a conventional adaptive controller can be seen as the final algorithmic representation of a large amount of underlying theoretical as well as practical control knowledge, (Astrom and Wittenmark, 1995) 'Tb work in practice, it has to be combined with logic safety jackets or safety nets (Isermann and Lachmann, 1985), (Warvick, 1988) often heuristically derived, that assures the controller performance under nonstandard conditions such as, e.g., switching between different operating modes, insufficient process excitation, control signal saturation etc. This is often the dominating part of the controller and experience has shown that design and testing is quite time consuming.
The second problem concerns the representation of the knowledge involved. The expert controller contains a large number of algorithms and methods of quite different nature, e.g., online time critical filtering and control, on-line calculations of less time-critical nature, e.g., for recursive identification and performance assessment, off-line numerical calculations for identification and experiment analysis, automata describing the overall behaviour of the controller, sequential descriptions of the different tuning experiments and loop assessment experiments involved, and heuristic or semi-heuristic methods for controller choice. Several programming paradigms or software architectures (Shaw, 1994) can be used for implementing an expert controller. An architecture where a rule-based system orchestrates a number of numerical algorithms is described in (Arzen, 1986) . A blackboard-based system containing knowledge sources on either procedural or rule-based form is described in (Arzen, 1987) (Arzen, 1989) . Currently, Grafchart, a Grafcetbased G2 toolbox (Arzen, 1994b), is used to structure the implementation of an expert controller (Wallen, 1995) . Here the sequential function charts represent the tuning experiments in the controller and the steps contain actions and rules that are performed and active in a step. Fig. 9 shows a screen dump from this system.
More recently hybrid control and specially the theory of switching controllers, e.g. (Morse, 1995), has attracted a lot of attention in the conventional control community. These systems are based on the notion of multi-controllers according to Fig. 10. The systems selects between a number of controllers based on some performance index derived by the supervisor. The selection can either select one of the controllers or mix the outputs of several control algorithms. An example of a simple binary selector is a min193
Gainscheduling
Supervisor
Con1n>1lor-1
Con1n>lIor-2
I Con1n>1lor-3
I Con1n>1lor-4
.
Operating
point
Fig. 13. Gain scheduling as a multi-controller
The final example of when a multi-controller can be motivated is Heterogeneous Control (Kuipers and Astrom, 1994) . In (Malmborg, 1995) a Heterogeneous controller is described where it is the task to be performed that decides which controller to be used. The controller combines timeoptimal "bang-bang"control with PID control. The time optimal controller is used for large setpoint changes and the PID controller is used for load disturbance rejection in the vicinity of the setpoint according to Fig. 14. The outputs of the
Fig. 10. A multi-controller
max selector. An example of a mixing selector could be a simple linear interpolation or a fuzzy interpolation. The similarities between a multicontroller and an expert controller according to Fig. 8 are obvious. There are a number of reasons why one would like to change between a number of control algorithms. These are summarized in Fig. 11. In
Heterogenous Control SeM> ContJoI-
Regulation Control-
Time Optimal Controu.r
Task
PlO controller
Fig. 14. Heterogeneous control as a multi-controller
controllers are weighted together using fuzzy interpolation according to Fig. 15. Control
t
Con' Tuk""
Fig. 11. Reasons for multi-controller structures
the original formulation of expert control it was the amount of process knowledge available in relation to the specifications of the closed loop behaviour that decideed which controller to use. During the initialization the controller gradually refines it process knowledge and is able to design a more complex controller. The simpler, hopefully more robust, controller is used as back-up controller in the case the controller error should suddenly become much large than anticipated with the advanced controller. The situation is shown in Fig. 12
Measurements
Fig. 15. Heterogeneous controller combining PID and time-optimal control
5. CONCLUSIONS AI-based techniques have several applications in the feedback control loop either for direct control or for supervision of numerical control algorithms. There are also strong relationships between recent developments in conventional control theory and AI-based controllers. Examples of this are analytical non-linear function approximation methods and logic-based switching controllers. This is promising for the future and can hopefully unify much of the parallel development that currently takes place. After all, it is the functionality of the controllers that is of importance rather than which implementation technique that is used.
Expen Control
pt
controU.r
PlO controller
tUgh...-ordef controhr
Process
Knowledge
Fig. 12. Expert control as a multi-controller
A gainscheduling controller can also be interpreted as a multi-controller. Here, it is the operating point that decides which controller to use (A single controller with a number of different parameter sets can be interpreted a set of controllers with fixed parameters.) according to Fig. 13. 194
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