Adaptive Fuzzy Control

Adaptive Fuzzy Control
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Published Date:25-10-2017
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CHAPTER 6 Adaptive Fuzzy Control Adaptive control is a method of designing a controller with some adjustable parameters and an embedded mechanism for adjusting these parameters. Adaptive controllers have been used mainly to improve the controller’s performance online. For each control cycle, the adaptive algorithm is normally implemented in three basic steps: (i) observable data are collected to calculate the controller’s performance, (ii) the controller’s performance is used as a guidance to calculate the adjustment to a set of controller parameters, and (iii) the controller’s parameters are then adjusted to improve the performance of the controller in the next cycle. Normally, an adaptive controller is designed based on one of the available techniques. Each technique is originally designed for a specific class of dynamic systems. The controller is then adjusted as data are collected during run time to extend its effectiveness to control a larger class of dynamic systems. A typical application of adaptive control was to calibrate a system at startup. In this case, a controller is also designed for a specific class of dynamic systems. However, the parameters that characterize the dynamic behavior of a particular system might not be known in advance. A controller is then designed and arbitrary values are assigned to initialize these parameters. After a few control cycles, parameters are adjusted to converge to the actual parameters of the system. This approach is often used for cases in which a system is designed to handle a variable payload. The payload is different each time, e.g., a crane is used to pick up a sizeable object. The payload will alter the basic dynamic behavior of a dynamic system. Adaptive control is normally used to calibrate these parameters that characterize such dynamic behavior. Traditionally, there are four basic approaches for adaptive control: (i) gain scheduling, (ii) model reference adaptive system, (iii) self-tuning regulator, and (iv) dual control. Gain scheduling is a method of adjusting the control signal based on a known look-up table describing changes of a dynamic system. The model reference adaptive system is a method of comparing the performance of the actual system against an assumed mathematical model that describes the actual system, and designing control input to drive this comparison error to zero. Self-tuning regulator is a method of updating the parameters of a model that describes the plant based on observed data, and channeling the updated information into the controller that is designed based on these parameters. Dual control is a method of extending adaptive control to stochastic model dealing with uncertainties. This chapter seeks to extend the principle of adaptive control to the fuzzy control techniques developed in earlier chapters. In this setting, a dynamic system is controlled to ensure tracking and stability. A controller is designed240 Adaptive Fuzzy Control •• 6 based on a fuzzy representation of that system, and is therefore called an adaptive fuzzy controller. This fuzzy controller is parameterized by a set of constants. These constants are adjusted in each instance of the adaptation to reflect improvement in effectiveness of the controller. Four traditional adaptive approaches (gain scheduling, model reference adaptive system, self- tuning regulator, and dual control) are applied to fuzzy controllers in this chapter. In addition, an alternative approach to adaptive control is included: a sub- optimal fuzzy control. In this approach, a fuzzy controller is optimally designed for use only during a short period of time. Then, as new data are collected to reflect more knowledge of the system, the optimal fuzzy control problem is adjusted and solved for a new solution. Simple applications are given for each method mainly to serve as illustrative examples. Practical and industrial examples are deferred to the next chapter to show how fuzzy logic can be applied to real-life control systems. I. FUNDAMENTAL ADAPTIVE FUZZY CONTROL CONCEPT Adaptive fuzzy control is an extension of fuzzy control theory to allow the fuzzy controller extending its applicability, either to a wider class of uncertain systems or to fine-tune the parameters of a system to accuracy. In this scheme, a fuzzy controller is designed based on knowledge of a dynamic system. This fuzzy controller is characterized by a set of parameters. These parameters are either the controller constants or functions of a model’s constants. A. Operational Concepts A controller is designed based on an assumed mathematical model representing a real system. It must be understood that the mathematical model does not completely match the real system to be controlled. Rather, the mathematical model is seen as an approximation of the real system. A controller designed based on this model is assumed to work effectively with the real system if the error (difference) between the actual system and its mathematical representation is relatively insignificant. However, there exists a threshold constant that sets a boundary for the effectiveness of a controller. An error (difference) above this threshold will render the controller ineffective toward the real system. An adaptive controller is set up to take advantage of additional data collected at run time for better effectiveness. At run time, data are collected periodically at the beginning of each constant time interval, t = t + ∆ t, n n−1 where ∆ t is a constant measurement of time, and t ,t ) is a duration between n n+1 data collection. Let D be a set of data collected at time t = t . It is assumed n n that at any particular time, t = t , a history of data D ,D ,...,D is always n 0 1 n6•• Adaptive Fuzzy Control 241 – + Controller Real System ε u(t) x(t) (t) + – Math Model xdesired y(t) Figure 6.1 Tracking the error function between outputs of a real system and mathematical model. available. The more data available, the more accurate the approximation of the system will become. At run time, the control input is fed into both the real system and the mathematical model representing that system. The output of the real system and the output of that mathematical model are collected and an error representing the difference between these two outputs are calculated. Let x(t) be the output of the real system, and y(t) the output of the mathematical model. The error ε(t) is defined as ε(t) = x(t)– y(t). (6.1) Figure 6.1 depicts this tracking of the difference between the mathematical model and the real system it represents. Notice that in the context of this chapter, the mathematical model will be in the form of fuzzy IF-THEN rules. In this case, the output of the mathematical model is still crisp numerical data because of the defuzzification process that converts a fuzzy representation to a numerical one. An adaptive controller will be adjusted based on the error function ε(t). This calculated data will be fed into either the mathematical model or the controller for adjustment. Since the error function ε(t) is available only at run time, an adjusting mechanism must be designed to accept this error as it becomes available, i.e., it must evolve with the accumulation of data in time. At any time, t = t , the set of calculated data in the form of a time series n ε(t ),ε(t ),ε(t ),…,ε(t ) is available and must be used by the adjusting 0 1 2 n mechanism to update appropriate parameters. In normal practice, instead of doing re-calculation based on a lengthy set of data, the adjusting algorithm is reformulated to be based on two entities: (i) sufficient information, and (ii) newly collected data. The sufficient information is a numerical variable representing the set of data ε(t ),ε(t ),ε(t ),…,ε(t ) collected from the initial time t to the previous 0 1 2 n–1 0 collecting cycle starting at time t = t . The new datum ε(t ) is collected in the n–1 n242 Adaptive Fuzzy Control •• 6 current cycle starting at time t = t . Let ξ(t) be the sufficient information n representing data from the initial time t to present time t. Then, in every 0 collecting cycle, ξ(t) is iteratively calculated as ξ(t ) = Ξ(ξ(t ),ε(t ) ). (6.2) n n–1 n An adaptive controller will operate as follows. The controller is initially designed as a function of a parameter set and state variables of a mathematical model. The parameters can be updated any time during operation and the controller will adjust itself to the newly updated parameters. The time frame is usually divided into a series of equally spaced intervals t ,t )n=0,1,2,...; n n+1 t = t +∆ t. At the beginning of each time interval t ,t ), observable data n+1 n n n+1 are collected and the error function ε(t ) is calculated. This error is used to n calculate the adjustment in the parameters of the controller. New control input u(t ) for the time interval t ,t ) is then calculated based on the newly n n n+1 calculated parameters and fed into both the real dynamic system under control and the mathematical model upon which the controller is designed. This completes one control cycle. The next control cycle will consist of the same steps repeated for the next time interval t ,t ), and so on. n+1 n+2 B. System Parameterization As discussed in Chapter 3, a dynamic system can be described by a set of fuzzy IF-THEN statements that correlate the input and the output. These statements include a set of parameters that is used to uniquely calculate the estimated output of a system given the inputs and current states of the system. A dynamic system is mathematically modeled as a set of IF-THEN rules: 1 R:IF x (t )∈ X AND ... AND x (t )∈ X 1 n 11 i n 1i THEN y (t )= α x (t ) + α x (t ) + ... + α x (t ) 1 n+1 11 1 n 12 2 n 1i i n +β u (t ) + β u (t ) + ... + β u (t ), 11 1 n 12 2 n 1i j n 2 R:IF x (t )∈ X AND ... AND x (t )∈ X 1 n 21 i n 2i THEN y (t )= α x (t ) + α x (t ) + ... + α x (t ) 2 n+1 21 1 n 22 2 n 2i i n +β u (t ) + β u (t ) + ... + β u (t ), 21 1 n 22 2 n 2i j n M k R:IF x (t )∈ X AND ... AND x (t )∈ X 1 n k1 i n ki THEN y (t )= α x (t ) + α x (t ) + ... + α x (t ) k n+1 k1 1 n k2 2 n ki i n +β u (t ) + β u (t ) + ... + β u (t ), k1 1 n k2 2 n ki j n where x , x , ..., x are the observable state variables of the real system, y , y , 1 2 i 1 2 ..., y the calculated state variables of the mathematical model, and u , u , ..., u k 1 2 j the control inputs to both the real system and the mathematical model. In addition, a set of fuzzy membership functions µ (⋅), µ (⋅), ..., µ (⋅) is X X X 1 2 i given. The above fuzzy mathematical model is characterized by a set of parameters α ,β k’=1,2,...,k;i’=1,2,...,i;j’=1,2,...,j. These parameters will k’i’ k’j’ determine the behavior of the mathematical model. Theoretically, the model will approximate the real system. The more accurately the model6•• Adaptive Fuzzy Control 243 approximates the real dynamic system, the better the controller designed based on this model behaves. Generally, there can be more than one set of fuzzy rules describing a system. This is a special feature of fuzzy modeling: different (and sometimes contradicting) rules for a same set of conditions can co-exist. In case of having more than one rule applicable to a particular condition, the membership functions will be used to determine a unique result. This process is defuzzification, which resolves difference in a compromising fashion. The most popular defuzzification technique is the weighted average formula which assigns a weight for each result according to a given fuzzy membership function. Similarly, a fuzzy controller, if designed based on the fuzzy mathematical model above, will have the following form of fuzzy rules: 1 R:IF x (t )∈ X AND ... AND x (t )∈ X 1 n 11 i n 1i THEN u (t)= κ x (t ) + κ x (t ) + ... + κ x (t ) 1 n 11 1 n 12 2 n 1i i n 2 R:IF x (t )∈ X AND ... AND x (t )∈ X 1 n 21 i n 2i THEN u (t)= κ x (t ) + κ x (t ) + ... + κ x (t ) 2 n 21 1 n 22 2 n 2i i n M j R:IF x (t )∈ X AND ... AND x (t )∈ X 1 n k1 i n ki THEN u (t)= κ x (t ) + κ x (t ) + ... + κ x (t ). j n j1 1 n j2 2 n ji i n In this case, the set of parameters κ i’=1,2,...,i;j’=1,2,...,j characterizes j’i’ the controller. It can be understood intuitively that the parameter set κ is j’i’ designed as a function of the parameter set α ,β , so that the state k’i’ k’j’ variables of the fuzzy mathematical model are driven to a target point with stability. Therefore, it is natural for an engineer who designs the controller to either adjust the system parameters α ,β , which will lead to the adjustment of k’i’ k’j’ the control parameters κ , or directly adjust the controller parameters j’i’ κ as a scheme to implement adaptive strategies. The adjustment of the j’i’ system parameters is sometimes called system identification. This system identification process is often applied to a system that can be described by a time-varying parameter set in the fuzzy model. Sometimes the identification process can be used to establish the parameters of a dynamically changing system, such as a flying jet that continuously changes its mass due to fuel consumption. C. Adjusting Mechanism Adjusting the system parameters is normally carried out to improve the performance of the overall closed-loop system, or to improve accuracy of the assumed model upon which the controller is designed. The process of calculating the adjustment to the parameters is the adjusting mechanism mentioned as part of the adaptive control problem. (n) Let θ be a set of parameters to be adjusted at time t = t . The adjustment n can be the recalculation of the parameters, (n) θ = Θ(D ,D ,...,D ), 0 1 n244 Adaptive Fuzzy Control •• 6 where D is the data collected at time t = t . For numerical efficiency, this m m formulation is normally reformulated by the update of the parameter set based on its previous value, namely, (n) (n−1) θ = Φ(θ ,D ). n A special case of such an update formulation is the separation of the function Φ into a recursive form: (n) (n−1) θ = θ + Λ(D ,Λ(D )), (6.3) n n−1 whereΛ(D ) is known as the update (or adjustment) of the parameter set. This n quantity is sometimes referred to as sufficient information. This formulation improves clarity and efficiency. It is easy to see the new parameters as the sum of the old parameters and the changes in these parameters. It is also computationally efficient not having to repeat the same type of calculation based on n sets of data. When accuracy of system identification is considered, the problem of calculating the adjustment to the parameters can be formulated as an optimization: min ε(t), (6.4) θ where ε(t) is the error function representing the difference between the real system and its approximated model. The real system is given in terms of observable data collected over time. The approximated model is given in terms of a mathematical expression. The expression ⋅ represents an adequate norm function of a vector. For numerical simplicity, it is convenient to use an l -norm defined as 2 2 2 2 ε(t) = ε (t )+ε (t )+ ...+ε (t ) . 2 1 2 n The optimization problem in this l -normed space is often known as the least- 2 squares problem. Given a set of linear constraints, this problem will lead to a standard linear solution (see Chapter 3). This solution can be reformulated to become an efficient calculation of the update of the parameters in the form of (6.3). When the performance of the overall closed-loop system is considered, the problem of calculating the adjustment to the parameters can be formulated as an optimization problem minimizing the rate of convergence to a desired stable point: min x(t)−x . (6.5) f θ There are normally variations of the objective function to be optimized. It is common to select a norm formulation that will lead directly to a closed- form solution. This closed-form solution is important because it will allow rapid calculation of the updated parameters in a short time interval, t ,t +∆ t), n n before the next cycle begins. In this time interval, the algorithm to compute the control input u(t ) is also performed, increasing the demand for numerical n efficiency. The capability to calculate everything required in a small time6•• Adaptive Fuzzy Control 245 Table 6.1 Criteria for Selecting an Adaptive Fuzzy Controller Adaptive Type System Description Fuzzy STR Time invariant with unknown parameters Fuzzy STR, MRAFS Time varying Sub-optimal Fuzzy Dynamic environment is changing with no prior Control knowledge Dual Control Highly nonlinear Fuzzy Gain Scheduling Only modeled accurately piecewise and represented by multiple models each for a specific range of time interval (with a predefined duration ∆ t) is sometimes called the real-time processing capability. D. Guidelines for Selecting an Adaptive Fuzzy Controller It should be noted that an adaptive fuzzy controller, while providing increasing accuracy and effectiveness, should be carefully selected according to the criteria it was designed for. Otherwise, the increase in calculation workload for implementation of an adaptive fuzzy controller might not be offset by the gain in accuracy or effectiveness. In order to determine whether to apply adaptive fuzzy control to a system, one must understand the nature of that system and the mathematical fuzzy model used to design a controller. To demonstrate an understanding of a system, one may try to answer the following questions: 1. Can the system be approximated entirely by a fuzzy model? 2. If a system can be approximated entirely by a fuzzy model, are the parameters of this fuzzy model readily available or must they be determined online? 3. If a system cannot be approximated entirely by a fuzzy model, can it be approximated piecewise by a set of fuzzy models? 4. If a system can be approximated by a set of fuzzy models, are these models having the same format with different parameters or are they having different formats? 5. If a system can be approximated by a set of fuzzy models having the same format, each with a different set of parameters, are these parameter sets readily available or must they be determined online? Satisfactorily answering the above questions will help determine whether an adaptive controller is necessary and, if so, which type of adaptive controller is appropriate. Table 6.1 summarizes some criteria to determine whether an adaptive control design is necessary and, if so, which type of adaptive control method is appropriate. In the table, STR stands for self-tuning regulator and MRAFS is a model-reference adaptive fuzzy system. If a system can be approximated piecewise by a set of fuzzy models, with the parameters of each model readily known, then it is best to implement a246 Adaptive Fuzzy Control •• 6 fuzzy gain scheduling control algorithm. This algorithm is similar to the fuzzy PD/PI/PI+D controllers presented in Chapter 5. If a system can be approximated entirely by a set of fuzzy models, but the parameters of these models are not readily available, then it is best to implement a STR fuzzy controller based on these parameters. These parameters will be constantly evaluated and updated, consequently changing the controller parameters. If a system can be approximated piecewise by a set of fuzzy models, with all models having the same format but different parameter sets, then it is best to implement a fuzzy sub-optimal controller. In this approach, as soon as the controller moves the closed-loop system into a particular area in the control space, an optimal fuzzy control problem is solved only for a short time duration. As soon as control action is pushing the system into another area in the control space, the sub-optimal fuzzy control problem is solved again with a different set of parameters. II. GAIN SCHEDULING Gain scheduling is a method of combining several controllers together to control a particular dynamic system. In this approach, each controller is the most effective in a particular region. Joining them together will require a condition checking procedure to determine which controller is activated. In order to design a gain scheduling fuzzy controller, one needs to design two sets of rules: (i) a set of several fuzzy controllers that are applicable to a dynamic system, each effective under certain conditions, and (ii) a set of meta rules to determine when to activate which controller for an appropriate action. This type of fuzzy controller is normally useful for applying rule-based expert systems to control a complicated dynamic system. The expert system will contain fuzzy rules on how to control that dynamic system. The fuzzy rules are usually extracted from several human operators. In this setting, the fuzzy system will contain several rules: either each will be consistently applicable to a scenario or some might contradict one another for a scenario. In both cases, a meta rule set is required, either to determine when to activate a rule for the case of having many exclusively effective controllers, or to provide a compromised solution for the case of having contradicting rules. Another application of interest is the case of a piecewise linear system that behaves linearly under a unique set of parameters in each region. A piecewise linear mathematical model is normally used to approximate a nonlinear system. An example of this type of application is the dynamic behavior of a moving object in a gravity field generated by a large planetary body such as the earth. Near the surface of this planetary body, the gravity field can be approximated as a cause that results in a constant downward acceleration of 2 9.8 m/sec .6•• Adaptive Fuzzy Control 247 A fuzzy gain scheduling consists of a series of rule sets, each consisting of several IF-THEN fuzzy logic rules. Each rule set is in the form of N N M (k1) Rule Set k: R : (x∈X ) ⇒ y = a x + b u ∧ i k1i 1 ∑ ∑ k1i i k1i i i=1 i=1 i=1 M N N M (kN) R : (x∈X ) ⇒ y = a x + b u ∧ i kNi N ∑ ∑ kNi i kNi i i=1 i=1 i=1 for k = 1,2,...,K. Fuzzy membership functions µ (⋅) for appropriate values of X kni k, n, i are available either in closed-form or in look-up tables. In order to determine when to activate a rule set, one must implement a set of meta rules that provides selection conditions. The meta rules should be of the form: K N (j) M : (x∈X ) ⇒ Rule Set j for j = 1,2,...,J. ∨ ∧ i jki k=1 i=1 The above meta rules are cumbersome and detailing. One might prefer to simplify the above meta rule by approximating it with N (j) M : (x∈Y ) ⇒ Rule Set j for j = 1,2,...,J, ∧ i ji i=1 where Y is defined as the union of the sets X : ki kji K Y = I X , ji jki k=1 and the fuzzy membership function associated with Y is ji µ (x) = maxµ (x),...,µ (x). Y X X ji j1i jKi There are two difficulties associated with a fuzzy gain scheduling scheme: (i) it is often time-consuming to define a set of meta rules determining when to activate which set of fuzzy rules, and (ii) the meta rule set is used in a feed- forward fashion without feedback. There is no formal method of defining a set of meta rules. As a result, one has to design the rules heuristically, most of the time exhaustively. The meta rules are used to select a set of fuzzy rules to activate. This selection does not include feedback to confirm whether the right set of rules has been used. Example 6.1. Controller for a moving body is normally based on a set of equations of motion. These equations include dynamic description of an object moving in a gravitational field, accelerating toward the ground at a constant rate called gravity effect. From Newtonian physics law, this gravitational effect is described as an interactional force between two bodies, one of which is the earth: GM F = m, 2 R where G is the gravitational constant, M the mass of the earth, m the mass of the moving object, and R the distance between the object and the center of the 248 Adaptive Fuzzy Control •• 6 g R e R R i i+1 Figure 6.2 Approximation of the constant g. earth. Near the surface of the earth, where R = R , the radius of the earth, the E force is estimated as GM F = m≈ m g, 2 R E 2 where g is the gravity on earth, often estimated as GM / R , which yields a E 2 constant value of 9.8 m/sec . An object with mass m falling near the surface of the earth will obey the equation of motion which can be described by the following fuzzy rules: (1) & R :IF x (t)∈ S THEN x (t)∈ S , 2 k k 1 (2) & R :IF u(t)∈ S THEN x (t)∈ S , f f/m+g 2 where x (t) is the displacement, x (t) the velocity, and u(t) the external force 1 2 acting on the object. The fuzzy set S is defined as k S = (k−1)σ,(k+1)σ), k x ⎧ − k + 1 for (k −1)σ≤ x ≤ kσ ⎪ σ µ (x) = S ⎨ k x ⎪ k + 1− for kσ≤ x (k + 1)σ, ⎩ σ where σ is half the width of the fuzzy set S . Notice that a real line is k partitioned into a series of fuzzy sets, each indexed by an integer k, where k = ...,−2,−1,0,1,2,... (see Chapter 3). A typical fuzzy controller for the system of two rules representing the equations of motion for a moving object near earth’s surface is (1) C :IF x (t)∈ S AND x (t)∈ S THEN u(t)∈ S . 1 i 2 j −m(6i+5j+g) This controller will yield a stable closed-loop system described by the following rules: (1) & R :IF x (t)∈ S AND x (t)∈ S THEN x (t)∈ S , 1 i 2 j j 1 (2) & R :IF x (t)∈ S AND x (t)∈ S THEN x (t)∈ S , 1 i 2 j −6i−5j 2 which can describe any system whose output is bounded from above and from below by the two bounds, as illustrated by Figure 6.2.6•• Adaptive Fuzzy Control 249 These two bounds characterize the approximation accuracy, implying that all systems bounded by them will also have the same error bounds of model approximation. However, the controller is working only with an accurate estimate of the gravitational effect g. Near earth’s surface, this value g is estimated at 9.8 2 m/sec . For planetary spacecraft, the value of g is dependent on the distance of 2 the spacecraft from earth, to be given in terms of GM/R . The further the craft from earth, the smaller the value of g. A simple solution to the control problem with variation in gravity constant is the use of gain scheduling. The controller is still effective provided an accurate value of g is used. Therefore, one can program a set of possible values of g that the spacecraft will likely encounter in its operating range. For example, a spacecraft can be designed to be functional in many orbital levels, each defined at a specific altitude. In this case, a value g is pre-determined for each orbit. As soon as sensor indicates the altitude x (t), a set of meta rules can 1 be used to determine which value of g to use, thus triggering a new set of control gains to apply to the system. This meta rule set can either be crisp or fuzzy, determined by the designer. For a set of crisp meta rules, the pre- calculated g is used for the controller. For a set of fuzzy meta rules, the final value of g is determined from the defuzzification of the results based on membership values of each individual result. Figure 6.2 shows the actual value of g versus the altitude of an object. Instead of recalculation of the value of g as a function of an altitude and increasing the complexity of the mathematical model used to design the controller, one can use a constant value of g for each pre-defined range (shown in the rectangular box in the figure). Let G be the set of possible values of g, G = g ,g ,...,g . Each value of g 1 2 P i is applicable only for a range R ,R . The meta rules for selecting different i i+1 values of g can be listed as the following rules: (1) M :IF R ≤ x (t)≤ R THEN g = g 1 1 2 1 (2) M :IF R ≤ x (t)≤ R THEN g = g 2 1 3 2 M (P) M :IF R ≤ x (t)≤ R THEN g = g . P 1 P+1 P Example 6.2 (Cash flow analysis of potential of a project). This example discusses the use of fuzzy logic in a set of expert systems. Each expert system will operate according to a set of fuzzy logic rules. It calculates the confidence level of each rule using fuzzy logic processing. A meta rule set will be used to select which expert system (or set of rules) is to be used throughout the entire process. This is the gain-scheduling technique applied to the problem of decision making on investment. Currently, there are several models that a financial manager can use to analyze cash inflow and outflow of a project before making a commitment decision. Each model has its own unique implication and calculation rules. A financial manager must have an ability to analyze the cash flow of a project before committing to it. This analysis is based on the profit returned250 Adaptive Fuzzy Control •• 6 from the investment. There are several factors involved in this analysis, e.g., predicted inflation rate, periodic dividends, initial investment, length of commitment, etc. Using these factors, a financial manager computes some index that shows profits or losses. If the index shows profits, the manager can comfortably commit to the project. When a financial manager has to select one investment project among several, the analysis above can be modified to become the selection of a project with the highest index of profit or lowest index of loss. However, there are several techniques of analyzing a project. All such techniques are consistent with each other in determining the expected profit (or expected loss) situation. Yet, when one ranks projects using these techniques, the rankings are not consistent. Depending on the criteria that each method is designed for, the use of each method will yield different ranking. To illustrate the inconsistencies in ranking of profit index using different techniques of analyzing a cash flow, we summarize the five indicators commonly used to analyze cash flows: profitablity index, internal rate of return, net present value, payback period, and modified internal rate of return. Profitability Index (PI) is an indicator that determines the ratio of a net present value of all cash inflows to net present value of all cash outflows of a project. This indicator has a profit threshold value of one. If the PI of a project is greater than one, then the project returns profit; if it is less than one, then the project shows losses; if it is equal to one, then the project breaks even. In case of a single-project analysis, if the PI is greater than one, a financial manager can commit to the project. The profitability index is calculated as n C (t) in ∑ t (1+ k) t=0 PI = , n C (t) out ∑ t (1+ k) t=0 where C (t) is the cash inflow, C (t) the outflow, k the index of inflation, and in out n the number of time units. The fuzzy rules determining if a project is commitable are (1) R : if (PI − 1) is positive then accept, (2) R : if (PI − 1) is negative then reject. Net Present Value (NPV) is the computation converting all cash flows at different times into a single quantity equivalent to the value at presence. This method is based on the expected inflation rate in the future, the amount of cash received periodically, and the initial investment. The total of these amounts after indexed for inflation rate is the expected profit. If the NPV of a project is positive, the project shows profits and is commitable; if it is negative, the project shows losses and is not commitable; if it is zero, the6•• Adaptive Fuzzy Control 251 project breaks even and the commitment shows no effects. The NPV is calculated as n C (t)− C (t) in out NPV = , ∑ t (1+ k) t=0 with notation as above. The fuzzy rules determining if a project is commitable are (1) R : if NPV is positive then accept, (2) R : if NPV is negative then reject. Internal Rate of Return (IRR) is the minimum rate of return that allows break-even in the project. This is the yield rate that, if used for calculating net present value, shows a total of zero net present value of cash flows. This rate is compared to the current cost of capital, i.e., the interest rate available in the market. If the IRR is greater than the cost of capital, then it is profitable to commit investment to the project; if it is less than the cost of capital, then the project is not profitable and should not be committed to. The IRR is calculated as n C (t)− C (t) in out IRR = r such that = 0, ∑ t (1+ r) t=0 with notation as above. The fuzzy rules determining if a project is commitable are (1) R : if (IRR−k ) is positive then accept, inflation (2) R : if (IRR−k ) is negative then reject. inflation Payback Period (PP) is the period required to collect back all the initial investments. Normally, a company has a threshold for payback period. This threshold is the maximum length of a period that the company is willing to wait before showing profit. If the PP of a project is less than this threshold, then the project is commitable; if it is greater than this threshold, then the project should not be invested. This method has been popular for quick calculation inside an executive’s head. Recently, it has been modified to reflect the inflation rate. This rate is used to calculate the net present value of each cash flow. These net-present-valued cash flows are used to calculate the payback period: n C (t)− C (t) in out PP = n such that = 0, ∑ t (1+ k) t=0 with notation as above. The fuzzy rules determining if a project is commitable are (1) R : if (PP−threshold) is positive then accept, (2) R : if (PP−threshold) is negative then reject. Modified Internal Rate of Return (MIRR) is a slight variation of the IRR. Here, the terminal value, i.e., future value at the time of the final payment of all the cash inflows, is computed. Then, this amount is adjusted to reflect the net present value. The rate that returns such a zero net present value is the252 Adaptive Fuzzy Control •• 6 MIRR. The analysis of this indicator is similar to that of the IRR. The main advantage of using this indicator instead of the IRR is the claim that the MIRR provides an alternative solution of simply investing the company’s fund to saving institutions where the MIRR is smaller than the rate of return offered by such saving institutions. The MIRR is calculated as n n−t C (t)(1+ k) in ∑ n C (t) out t=0 MIRR = r such that = , ∑ t n (1+ k) (1+ r) t=0 with notation as above. The fuzzy rules determining if a project is committable are (1) R : if (MIRR−k ) is positive then accept, inflation (2) R : if (MIRR−k ) is negative then reject. inflation The meta rules used to select which value to use is as follows. If profits are the main determining factor, then the values PI, IRR, and MIRR will be emphasized. If the principle of the investment is protected, then the value NPV should be used. One can vary the emphasis on each value and combine them together using the weighted average formula, which is equivalent to resolving results of different fuzzy rules, each with a different membership value. III. FUZZY SELF-TUNING REGULATOR Fuzzy Self-Tuning Regulator (FSTR) is the extension of traditional self- tuning regulator, which is an adaptive control method to handle systems modeled by fuzzy logic rules. In this approach, a set of fuzzy logic rules is used to describe the dynamics of a system. A fuzzy controller, therefore, is designed based on the system parameters and is characterized by its own controller parameters. The design of a fuzzy controller can be set up so that the controller’s fuzzy parameters are dependent on the system’s fuzzy parameters. In this setup, the fuzzy self-tuning regulator approach is utilized to update the system’s parameters at every control cycle. Then, the controller’s parameters will be updated as a consequence. The new controller (with newly computed parameters) will then calculate the control signal to be sent to the system to control it for tracking with stability. Figure 6.3 illustrates this FSTR concept, where u (t) is the reference input, u(t) and y(t) the input and output of c the system, respectively, a and b are system parameters, and K ij mn pq controller parameters (including the control gains). Notice that in the FSTR approach, the problem is partitioned into two parts: (i) system identification and (ii) adjusting the control gain. The first part, system identification, is an estimation of the parameters in the fuzzy model. The adjusting of the control gains is the recalculation of the control gains based on new parameters of the system model. In this setup, inputs and outputs of the system to be controlled are monitored and profiled at the6•• Adaptive Fuzzy Control 253 Kpq System Calculate Control a ,b ij mn Identification Parameters Dynamic + u(t) u (t) y(t) c System − Fuzzy Controller Figure 6.3 Fuzzy self-tuning regulator. beginning of every control cycle, t ,t +∆ t. These two profiles will contain n n data from the initial time, t = t , to the current time, t = t . Based on the input- 0 n output data, a relation can be established in terms of a fuzzy model that will map the input into the output. Consider an SISO fuzzy model in the fuzzy IF-THEN form (see Chapter 3): (i) & R:IF x∈ S AND u∈ S THEN x∈ S , i=1,...,l, n m an+bm n,m=0,±1,±2,... . Under specific conditions that S = −σ(n−1),σ(n+1)), and that the associated n membership function of this set S is triangular shape of the form n ⎧ 1− x− nσ /σ for x∈(n−1)σ,(n+1)σ), µ (x) = S ⎨ n 0 else, ⎩ it can be shown that x(t) is bounded from both above and below by the bounds x& (t) = ax (t) + bu(t) + a + b σ, upper upper x& (t) = ax (t) + bu(t)− a + b σ. lower lower A typical fuzzy controller for this model is given in terms of the following rules: (i) C:IF x∈ S THEN u = kx, i=1,...,l, n=0,±1,±2,..., n where the constant k is designed so that a + bk 0, in order to have a stable closed-loop system. For purpose of illustration, k can be selected as a k = −2 sign(b) . b For the first part of system identification, given two sets of data profiles u(t )i=0,1,2,....,n and x(t )i=0,1,2,....,n, the problem is to find the two i i constants a and b in the IF-THEN fuzzy model that will produce the two254 Adaptive Fuzzy Control •• 6 profiles, x (t )i=0,1,2,....,n and x (t )i=0,1,2,....,n, to ensure the upper i lower i inequalities x (t )≤ x(t )≤ x (t ) for i=0,1,2,....,n. lower i i upper i The SISO problem above can be extended to a general MIMO problem, where, instead of a scalar parameter a, a matrix A = a is used, and similarly ij a matrix B = b is in place of the scalar b. mn It can be seen that there are more variables than equations, implying that there exist an infinite number of solutions. In order to ensure a unique solution, one may establish an additional constraint: the parameters will change gradually from one instant to the next. This additional constraint can convert the system identification problem to an optimization problem of the form K K K J ⎧ ⎫ ⎪ 2 2 min+ a (t )− a (t ) b (t )− b (t ) ⎨ ∑∑ ∑∑ ⎬ ij k ij k−1 mn k mn k−1 a ,b ij mn ⎪ i== 11 j m== 11 n ⎭ ⎩ subject to K J & x (t ) = a x + b u . ∑ ∑ r k rn n rm m n=1 m=1 This optimization problem would minimize the deviation of the parameters from the previous set of parameters. This minimization provides the smooth change in the values of the parameters, and the constraint would put the estimated transition right in the middle of the two bounds. The control parameters will be updated based on the new values of the model parameters calculated from the above optimization problem. There are several techniques to design a controller that can control the two bounds of the fuzzy model to ensure convergence and stability. The solution to the first part of the FSTR problem is calculated in three steps as follows. Step 1. Estimate the transition of the state variables: In this calculation, the following simple finite difference may be used: x (t )− x (t ) n k n k−1 & x (t ) = . n k t − t k k−1 Step 2. Estimate the parameters of the fuzzy model: The solution to the optimization problem is solved by Calculus in closed form as follows: & x (t )x (t ) j k i k a (t)= a (t ) + ij k ij k−1 K J 2 2 x (t )+ u (t ) ∑∑ m k n k m== 11 n K J x (t ) x (t )a (t )+ u (t )b (t ) ∑∑ j k m k im k−1 n k in k−1 m== 11 n − , K J 2 2 x (t )+ u (t ) ∑∑ m k n k m== 11 n6•• Adaptive Fuzzy Control 255 u (t )x& (t ) n k m k b (t ) = b (t ) + mn k mn k−1 K J 2 2 x (t )+ u (t ) ∑∑ p k q k p== 11 q K J u (t ) x (t )a (t )+ u (t )b (t ) ∑∑ n k p k mp k−1 q k mq k−1 p== 11 q − . K J 2 2 x (t )+ u (t ) ∑∑ p k q k p== 11 q Step 3. Update the control parameters: The control parameters are designed to stabilize the system in terms of the constants a and b for the SISO case. A typical solution was given earlier, for which the control parameter k can be updated as a(t ) k k(t ) = −2 sign(b(t )) . k k b(t ) k The adaptation of the system parameters to input data can be reformulated into a matrix form, as T T x& x A x x k k k−1 k k A = A + − k k−1 2 2 2 2 x + u x + u k k k k T B u x k−1 k k − , 2 2 x + u k k T T x& u A x u k k k−1 k k B = B + − k k−1 2 2 2 2 x + u x + u k k k k T B u u k−1 k k – , 2 2 x + u k k which can be implemented efficiently on a digital computer in the updating scheme. IV. MODEL REFERENCE ADAPTIVE FUZZY SYSTEMS Model Reference Adaptive Fuzzy Systems (MRAFS) is a method of adjusting the gain(s) to a system so that the output of that system tracks the output of a model having the same control input. In this setting, a control signal is being fed into both the model and the system. The model, being fuzzy in this case, is designed to be stable. It is the reference that the system is controlled to match. Figure 6.4 illustrates the concept of MRAFS. Associated with the MRAFS approach are different numerical techniques to calculate the adjusting gain(s). This approach basically establishes a desired dynamic behavior of a system through its model. Then, the system is controlled so that its output matches that of the model. 256 Adaptive Fuzzy Control •• 6 ym(t) Fuzzy Model − Dynamic u(t) y(t) + e(t) ×× u (t) System c γ Adaptive Gain Figure 6.4 Model Reference Adaptive Fuzzy Systems (MRAFS). Even though there is no consideration about stability of the overall system in the discussion of the MRAFS approach below, it is understood that if the model is designed to be stable, i.e., with internal feedback loop that brings its output to a desired set-point in a stable manner, then matching the output of a system with this reference model will ensure the system to be stable as well. Therefore, an important criterion in this design is the requirement that the reference model be stable. In fuzzy logic applications, it is assumed that the model being used is represented by a set of fuzzy logic rules. The adaptive gain is also given in terms of fuzzy logic rules. A basic guideline to calculate the gain constant is the minimization of the error function defined as the difference between the output of the reference model and the output of the actual system. In this setup, the error function e(t) is defined as e(t) = y(t)− y (t), m where y(t) is the output of the system, y (t) the output of the reference model. m If the reference model is given in terms of fuzzy logic rules of the form IF x(t)∈ S AND u(t)∈ S THEN x& (t)∈ S , (6.6) n n an+bm it describes a system that has an output bounded from both above and below by two analytical systems (see Chapter 3): & Ξ (t) = aΞ(t) + bu(t) + a + b σ, & ξ (t) = aξ(t) + bu(t)– a + b σ. The task of guiding the dynamic system (to be controlled) to have the output bounded by these two bounds is the goal of the MRAFS. In this case, the error function is slightly modified to reflect these bounds, as the average of the error with respect to the upper bound and the error with respect to the lower bound: 1 1 e(t) = y(t)− y (t) + y(t)− y (t). upper lower 2 26•• Adaptive Fuzzy Control 257 Let J(e) be the objective function for designing the MRAFS gain. This function J(e) is normally the magnitude of the error function e(t). Two popular magnitude functions are the l -norm 2 T J (e) = e (t) e(t) 2 and the l -norm 1 J (e) = e(t) . 1 Each formulation will lead to a specific closed-form solution. The l 2 solution is common because it normally yields a linear solution. It can be shown that the l solution is similar to the traditional bang-bang control. 1 Let θ be the parameter used to control the system so that it tracks the reference model. The approach here is to optimize the objective function J(e) with respect to this parameter θ. The objective function is gradually reduced when the parameter θ is adjusted in the negative gradient of J, i.e., dθ ∂J = −γ , dt ∂θ where γ is a scalar which determines the convergence rate in general. In this approach, the parameter θ is assumed to be continuously adjusted. Depending on the definition of J and the nature of the reference model, the formulation to calculate the adjustment to the parameter θ is derived on a case by case basis. V. DUAL CONTROL The traditional concept of dual control was originally designed to handle stochastic control systems. It consists of a controller and an estimator for the statistical properties of the closed-loop system. The estimator is used to provide statistical information to the controller that will adjust itself accordingly. This concept is usually difficult to implement on an actual system. The same dual control concept can be extended to fuzzy control systems. In this setting, the fuzzy setup consisting of definitions for fuzzy sets and its associated membership functions is re-evaluated during the operational process of the system. These definitions are continuously adjusted and, thus, the controller is continuously adjusted accordingly. The main difficulty of implementing the traditional dual control is also translated into the fuzzy dual control concept. The process of adjusting the fuzzy definitions involves changing the membership functions. This change does not permit the analytical process of simplifying the rules to be converted into two analytical bounds during the design phase. It is therefore difficult to implement this simplification method on a digital computer and operate it on line in synchronization with the processing rate at which data are collected. The second difficulty in the fuzzy dual control concept is in the change of the boundaries of the fuzzy sets that lead to possible widening of the coverage of the area of applicability.258 Adaptive Fuzzy Control •• 6 Basically, the concept of fuzzy dual control is often proposed for application in fuzzy clustering where input and output are analyzed to determine an effective set of fuzzy rules that correlate the two. It shows promising future in mimicking human’s intelligence in observing surrounding environment and making intelligent decisions, but generally not in control applications. VI. SUB-OPTIMAL FUZZY CONTROL An optimal controller is a controller designed to optimize its performance criteria. Normally, the criteria can be either the tracking performance (various specifications), the cost of operation (amount of control input required), or a combination of the two. An optimal controller is usually designed for its operative time duration from the initial time t to some specific ending time t . i f Sometimes the ending time t can be infinite. The optimal controller is a f solution to the following optimization problem: t f min J (x(t),u(t), x )dt f ∫ u(t) t i subject to x&(t) = f(x(t),u(t)), where the function J(⋅,⋅) is the criterion function, x the desired set point, and f u(t) the controller. When the mathematical model is given in terms of fuzzy logic rules, the resulting controller is referred to as the optimal fuzzy controller. There are cases in which the desired set point x is changing with time and f cannot be profiled in advance. This scenario is often seen in the case of a system trying to synchronously track another dynamic system, e.g., a spacecraft docking with another moving spacecraft in orbit. In this case, a prediction scheme can be implemented to provide the moving set point x . f However, the prediction method will result in an error that normally increases with time. It is impractical to design an optimal controller with operative time duration t ,t ) for a large t based on this predictive scheme for the moving set i f f point. In this case, it is reasonable to take an alternate approach, called sub- optimal fuzzy control. The sub-optimal fuzzy control method is a strategy of optimally solving a control problem only over a short time interval when prediction of dynamic environment is still accurate. At the beginning of a new interval, data are collected and assessment of the dynamic environment is revised to give a new prediction. Another optimal control problem is then set up and solved. In this adaptive way, the same problem format is repeatedly used, with only parameters being revised for better accuracy in the prediction of the changing environment.

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