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1、Satisfactory Optimization Control Algorithm based onInfinite-norm Performance IndexAbstract This paper investigates the use of fuzzy decision making in predictive control, .the use of fuzzy goals and fuzzy constraints in predictive control allows for a more flexible aggregation of the control object
2、ives than the usual weighting sum of squared errors. By defining the membership degree of the control objective and system constraint, and using the fuzzy interference, the optimal control problem with constraint, multi-objective multi-degree of freedom can be transferred as a convex optimal problem
3、, so as to utilize the efficient optimal algorithm and guarantee the global optimal solution. More importantly, we can increase the freedom degree of control by adjusting the relevant membership degree parameters of control objective and system constraints. The designers experience of control object
4、ive and system constraint can be utilized through the fuzzy inference of language variables, thus can be get better understanding of effect for control performance.I. INTRODUCTIONIt is often difficult to characterize the behavior of the plants in process control systems, which makes the approaches b
5、ased on the exact mathematical model very limited ill the applications, especially for the complex nonlinear system and partial unknown processes. The classical linear control theory can only be applied to the local linear systems and often can not get the global satisfactory control. In addition, t
6、here are various disturbances in industry environment that will affect the dynamic of process greatly in industry environment. As the scale of the whole process, so we pose different performance indexes and optimize these indexes synthetically, thus form the satisfactory control under the dynamic en
7、vironment. The importance for different requirements is defined by decision-maker and guaranteed by the control algorithm, to construct a man-machine cooperative control mode to make user satisfactory. In relevant literature, (see Xi, l995, Xi, et al, l998, and Li 2000), the different performance in
8、dexes and constraints are definitively described, the weights of all kinds of importance for difficult requirements are only expressed with different coefficients, Which undoubtedly make difficult to decide the satisfactory control. In fact, the requirements for the performance indexes and the toler
9、ance for the variable constraints are all fuzzy only using a simple coefficient to describe them clearly is often impossible. In order to overcome the above disadvantage and improve the exactness of importance for various requirements, we introduce the fuzzy inference into the satisfactory control.
10、in this paper, we present a satisfactory optimal control with fuzzy constraints and fuzzy goals to solve the complex industry process with constraints under the fuzzy dynamic environment, thus make the limited horizon optimal problem in the fuzzy environment become the equivalent definite programmin
11、g problem.II. PROBLEM DESCRIPTIONUsually, the constraints in complex process caused by the inherent physical characteristics (such as mechanical, thermodynamic and electricity etc) and all is summed up the constraints of the control variables and its diversification rate and the output variables. Th
12、ey are often in the form of time-invariant with upper and lower boundary: （1）In the traditional constraint programming, the constraint conditions can not be exceeded and changed, but in the satisfactory optimal control, some of constraints are adjustable, called soft constraints. Thus, every constra
13、int variable can be adjusted within a limit boundary and has a function to reflect the fuzziness of constraint variable boundary defined by decision-maker. We can use the fuzzy variable to describe this case. For fuzzy variable we define the membership function, which express the degree of membershi
14、p. indicates the corresponding fuzzy variable belongs to this set, conversely for In fact, we can understand as the degree of satisfactory degree. Fig. l is a kind of fuzzy boundary where the membership function is linear function (of course, we can assume other function) to simplify the computation
15、. Then the degree of membership is expressed as follows: (2)where pl、 p2 is called fuzzy width or tolerant width, ,is the expected boundary of fuzzy variable b. Obviously, when the fuzzy width is zero, it corresponds the hard constraint.Adjusting the soft constraint is based on the man- machine inte
16、raction which is actuarially the interaction of the experience decision and the knowledge base and rules base with computer It is natural to build an expert system - in computer according to the specified industry process, and make the decision on various input and output states real-time and at the
17、 same time adjust the boundary so as to realize the moving-horizon optimization. The decision- maker takes part in the control only in a special case. In other words, at this time, decision-maker makes proposal and order to the whole system at a higher level (such as change the production plan, impl
18、ement a new completely standard, etc), while the simple logical, the knowledge base and rule base needed for this kind of expert system are not very large because they are designed for a special industry environment, and the cost of building and operating is also very feasible.One of the mathematica
19、l approaches to express the rule base of expert system is the fuzzy inference approach. It has advantages of simple computation, explicit implication expressed by natural languages and according with mankinds logical thinking. We can first assume a series of fuzzy variables, usually we need adjust t
20、he constraint boundary the parameters of control algorithm ate in satisfactory control, then according to the known experiences, we can derive the relationship between the fuzzy variables under all kinds of conditions to make the fuzzy rules and fuzzy matrixes. In practical on-line computation, we s
21、hould solve the receding horizon optimal problem when the boundary conditions are fuzzy, so as to make the membership degree of fuzzy boundary maximal. If the membership degree is zero, it represents no feasible solution to the fuzzy boundary and we shouldEquality caseInequality caseOptimize the bou
22、ndary again according to the rule base. This corresponds to the procedure of exchanging information between the decision-maker and comput.FUZZY CONSTRAINTS IN SATISFACTORY CONTROLA.Model-based predictive controlSatisfactory control is actually the predictive control based on the model. It in essence
23、 utilizes systems predictive information to optimize the performance index within a finite horizon. In order to overcome the uncertainty we take the receding horizon strategy in predictive control.The predictive output , is derived from the information at current time t and the future control signal
24、 where is the predictive horizon. The objective to be optimized is: (3)where is the predictive error, is the control increment, k. is the weight coefficient of the control signal.The system can be described by the CARIMA model: (4)whereThe predictive equation is : (5)The control law is:where is the
25、former m lines of the matrix , the significance of parameters (see also Chen et al, l996)A.Handling the fuzzy constraintsIn this section, we discuss in detail how to deal with the fuzzy boundary optimization. Firstly, consider the control variable u and output variable y in constraint equation. They
26、 are all decided by the control increment Au during the receding horizon optimization in GPC algorithm. The boundary condition (l) can be expressed as: (6)where (7) (8) (9)the matrixes are defined as follows.Notice that for the constr8int of control variable and its change rate, we should consider t
27、he future cycles, which for the output constraint, the cycles are from to. However from the perspective of computing, the computation complexity of solving the optimization with constraints has much to do with the number of constraint conditions. Therefore, sometimes we only consider the constraints
28、 in the near cycles to decrease the computation.Whats more, as in equation (6) is transformed from the boundary expression (l), and this transformation is only a series of displacement and inverse, thus under the fuzzy boundary condition, the derived still has the same form as the non-fuzzy constrai
29、nts, and the fuzzy width of all fuzzy variables are not changed. It can be expressed as: (10)and the vector p represents the fuzzy width of various fuzzy variable of , and where stands for the fuzzy width of fuzzy variable x.IV. OPTIMAL Algorithm BASED ON FUZZY PROGRAMMINGFrom the constr8ints expres
30、sion (l0) and its Optimalperformance index (normally in quadratic form), we can get the optimal control acted on the next time. The main difficulty is that the constraints are fuzzy variables. This is a fuzzy programming problem. In fact, the fuzziness of constraint implies an optimal performance in
31、dex, which is to make the membership degree of fuzzy constraint maximal, that is: (11)where represents the objective function of fuzzy constraint, represents the fuzzy space of constraint, and is the membership degree of fuzzy variable in , and is to minimize the membership degree of all fuzzy varia
32、bles.It is well known that the genera1 step of using fuzzy mathematics to solve problems is to fuzzily the mathematical model of concrete problem and then de- fuzzily the fuzzy variables. We will proceed like this. Firstly, we can obtain from (2): (12)where is the i th line vector of matrix ,is the
33、ith expected value of fuzzy variable ,is the corresponding membership degree. From the performance index (ll), we can define the variable as: (13)Combined with optimal predominance index (l l), we have: (14)from the district we can substitute and eliminate , and then the above inequality can be writ
34、ten as: (15)also (16)Defining matrix ,as: (17)and define the optimal variable as: (18)then the fuzzy programming can be transformed as the following standard programming problem: (19)The main approach of satisfactory control is to unify the control objective and system constraints into fuzzy soft go
35、als and constraints for the multi-objective multi-degree of freedom syst6ms. Due to above transformation, the performance index can be characterized by a expected fuzzy set with a tolerant width as follows: (20)where J is the performance index, and its fuzzy membership function is defined as: (21)So
36、, the original optimization problem can be transformed as: (22)In most case, it used to Optimize a finite horizon quadratic criterion in Model-based predictive Control (MIC), so that the quadric programming (QP) optimization can be used to solve the MPC problem at each sample instant, but, When we c
37、onsider the system input/output constraints as (l), the original MPC results in a nonlinear optimization problem, and cause a heavy computational burden. To avoid such situations, the infinite-norm Performance index is used: (23)Where (24)Based on the infinite-norms character (24) can be transformed
38、 as:. (25)Hence, p linear control objectives are Obtained, and the fuzzy optimization is reformed based on the predictive control with fuzzy goals and constraints: (26)where is the tolerant width for objective, so, the constrained multi-objective optimization is decomposed single-objective linear pr
39、ogram problem within the predictive horizon, and the simplex optimization can be used to solve.V. Simulation RESULTSAccording to the above analysis, we built the simulation structure under the MATLAB, assume the transfer function of the system is a SISO system and has the form :which is a non-minimu
40、m phrase open-loop unstable system, the reference input is the step input. The simulation results are shown as fig.2.The constraints of control variable, control increment and output variable are -l.7, 0, -l, l and 0, l.2 respectively, and the corresponding fuzzy width are all 0.3, Fig. a1 is the st
41、ep respond of the system, b1 is the control signal, and c1 is the satisfactory degree. Under these constraint conditions, there is no constraint for output and collator variable. During the optimal process, the satisfactory degree of the system increases gradually, and arrives l. Under another const
42、raint conditions, the constraint of control is -l.5, 0, the control increment constraint is -l, l, and the output constraint is 0, l.5, and the corresponding fuzzy width are all 0.5, there is a hard constraint -l.5 for the control variable, the system will have no solution if without fuzzy width. Ac
43、cording to the algorithm proposed in this paper, we deal the constraint boundary with fuzziness. In addition, relaxing the system hard constraint gradually during the optimal process. The optimal direction is the one, which makes the membership degree maximal. Fig. a2 is the system output, b2 is the
44、 control variable,c2 represent the fuzzy satisfactory degree. It can be seen that the satisfactory degree is increasingly adjusted during the optimal process and we can get the control variable along the increasing direction of membership degree, which shows the effectiveness of this algorithm.VI. C
45、onclusionBy defining the membership degree of the control objective and system constraint, and using the fuzzy interference, the optimal control problem with constraint ,multi-objective multi-degree of freedom can be transferred as a convex optimal problem, so as to utilize the efficient optimal alg
46、orithm and guarantee the global optimal solution. More importantly we can increase the freedom degree of control by adjusting the relevant membership degree parameters of control objective and system constraints. The designers experience of control objective and system constraint can be utilized thr
47、ough the fuzzy inference of language variables, thus can be get better understanding of effect for control performance.ACKNOWLEDGMENTThis work was supported by National Nature Science Foundation of China (Grant No.f 6O074004) and by the Shu-Guang Plan of Shanghai Municipal Education Commission.REFERENCESl Masatoshi, S. Kosuke, K(l997). Interactive decision-making for large-scale multiobjective linear progra