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in the experiment-protocol script and then assign
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MATRIX MB = mb1 * mb2 | MATRIX MMB = mb * mb3
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in a DYNAMIC program segment 7-9 Example: Fuzzy-logic Control of a Servomechanism
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(a) Problem Statement
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Recalling the servomechanism model in Section 1-14, we replace its linear controller function
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voltage = k * error r * xdot
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If min/max fuzzy-set logic is preferred, the DESIRE matrix assignment MATRIX MB = mb1 & mb2 produces matrix elements min[M(E1i | x1), M(E1k | x1)] But these joint membership functions would have to be renormalized
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Modeling Fuzzy-logic Function Generators
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by a nonlinear fuzzy-logic controller function voltage(e, xdot) of the servo error e and the output rate xdot We define N1 = 5 fuzzy sets (very negative, negative, small, positive, and very positive) for e and N2 = 5 fuzzy sets for xdot with triangle membership functions such as those in Section 7-7b We will use the N1N2 = 25 products of these triangle functions as joint fuzzy-set membership functions for e and xdot, assign heuristic rule-table values voltage[k] to each fuzzy set, and invoke Eq(7-1) to produce the controller output voltage(e, xdot)
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(b) Experiment Protocol and Rule Table
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The experiment-protocol script in Figure 7-5a first defines the triangle-function submodel described in Section 7-7b We then declare triangle-peakabscissa vectors xx1, xx2 and membership-function vectors mb1, mb2 for the servo error e and the output rate xdot with
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N1 = 5 ARRAY xx1[N1] ARRAY mb1[N1] -N2 = 5 ARRAY xx2[N2] ARRAY mb2[N2] | | --peak locations for e membership functions for e
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peak locations for xdot membership functions for xdot
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We next declare the N1 N2 joint-membership matrix M12 and an equivalent N1N2-dimensional joint-membership vector m12, as in Section 7-8:
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ARRAY M12[N1, N2] = m12 | -joint memberships
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The N1 N2 rule-table vector ruletabl is declared with
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ARRAY ruletabl[N1 * N2] xx2 with the values 2emax, 005emax, 0, 005emax, 2emax for e 2xdotmax, -05xdotmax, 0, 05 dotmax, 2xdotmax for xdot | -controller rule table
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We use data/read assignments to fill the triangle-peak-location arrays xx1,
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where emax = xdotmax = 1 We fill the rule-table array rultabl as follows:
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if e is very negative if e is negative if e is small if e is positive if e is very positive 8k 8r, 8k r, 8k, 8k+r, 8k+8r 2k 2r, 2k r, 5k, 2k+r, 2k+2r 2r, 008r, 0, 008r, 2r 2k 2r, 2k r, 5k, 2k+r, 2k+2r 8k 8r, 8k r, 8k, 8k+r, 8k+8r
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Successive entries in each row refer to xdot = very negative, negative, small, positive, very positive, and k = 035 and r = 2 Note that we wrote each rule-table entry in the form k + r k is our intuitive guess at the controller-output contribution due to e, and r is our idea of the contribution due to xdot Our choices of peak-location abscissas and rule-table entries express a heuristic guess for a controller design In this example, we decided to use larger-than-linear controller gains for large servo errors and little or no damping for very small servo errors Our results (Fig 7-6a) did produce a better noise-following and step-input response than a linear controller The remainder of the experiment-protocol script in Figure 7-5a sets system parameters for the fuzzy-logic-controlled servomechanism and also for a similar servo using a linear controller The script then calls a simulation run to display the time histories of both servomechanisms for comparison (Fig 7-6a) Another simulation run exercises a second DYNAMIC program segment to display the fuzzy-set membership functions for the servo error e
(c) DYNAMIC Program Segment and Results
The DYNAMIC program segment in Figure 7-5b invokes the triangle-function submodel described in Section 7-7b twice to generate the fuzzy-set membership functions mb1[k] and mb2[k] for e and xdot The desired controller output voltage voltage(e, xdot) is then produced as a DOT (Section 3-7a):
DOT Voltage = ruletabl * m12
Figure 7-6a shows the servo response to a random-noise input together with that obtained with an optimized linear controller Results are comparable to those produced with an early version of DESIRE in References [4,5], but our new program is simpler and faster In practice, these experiments must be repeated with different signal amplitudes, since the control system is nonlinear
FIGURE 7-5a The experiment-protocol script for the fuzzy-logic-controlled servomechanism defines the triangle-function submodel, sets up triangle-peak abscissas, rule table, and system parameters, and calls a simulation run Another simulation run uses a second DYNAMIC program called members to display the fuzzy-set membership functions
-FUZZY-LOGIC-CONTROLLED SERVOMECHANISM -also simulates a similar linear servo for comparison ---------------------------------------------------------------------------------------------------triangle-function partition ARRAY X$[1], mb$[1] | -- dummy-argument arrays SUBMODEL fuzzmemb(N$, X$, mb$, input$) Vector mb$ = SAT((X$ - input$)/(X$ - X${1})) mbb = mb$[1] | mcc = mb$[N$ - 1] Vector mb$ = mb${-1} - mb$ mb$[1] = 1 - mbb | mb$[N$] = mcc end ---------------------------------------------------------------------------------------------------declare arrays for e, xdot fuzzy-set membership functions -N1 = 5 ARRAY xx1[N1] | -peak locations for e ARRAY mb1[N1] | -membership functions for e -N2 = 5 ARRAY xx2[N2] | -peak locations for xdot ARRAY mb2[N2] | -membership functions for xdot -ARRAY M12[N1, N2] = m12 | -- joint memberships ARRAY ruletabl[N1 * N2] | -controller rule table ---------------------------------------------------------------------------------------------------read membership-peak abscissas emax = 1 | xdotmax = 1 data -2*emax, -005 * emax, 0, 005 * emax, 2 * emax data -2*xdotmax, -05*xdotmax, 0, 05*xdotmax, 2*xdotmax read xx1,xx2 --------------------------------------------------------------------------------------------------A = 15 | w = 1 B = 300 | maxtrq = 1 | g1 = 10000 | -- servo parameters g2 = 2 | R = 06 k = 03500 | r = 2 | -fuzzy-controller parameters kk = 10 | rr = 01500 | -linear-controller parameters ---------------------------------------------------------------------------------------------------rule table data -8*k-8*r, -8*k-r, -8*k, -8*k+r, -8*k+8*r | -high gain data -2*k-2*r, -2*k-r, -5*k, -2*k+r, -2*k+2*r | -for large errors data -2*r, -008*r, 0, 008 * r, 2*r | -- and no damping data 2*k-2*r, 2*k-r, 5*k, 2*k+r, 2*k+2*r | -for small errors data 8*k-8*r, 8*k-r, 8*k, 8*k+r, 8*k+8*r read ruletabl ---------------------------------------------------------------------------------------NN = 4000 | TMAX = 10 | DT = 0001 | scale = 008 p = A * ran() | -- must initialize noise! drun | -make a run write type go to see membership functions | STOP ---------------------------------------------------------------------------------------DT = 000001 | NN = 40000 scale = 5 | TMAX = 05 e = -25 | -start of display sweep drun members | -show the membership functions