<Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field>

<Text-field layout="Heading 1" style="Heading 1">Example of a Fuzzy Controller for an Inverted Pendulum</Text-field>

The classic example for demonstrating fuzzy sets is the control of an inverted pendulum mounted on a carriage. We have control of the velocity of the carriage and the object is to control the system to keep the inverted pendulum from falling down.

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dz1fRTpBQmo7QkY7Qj06OjpCSllJeUlSSlVIQnd2VElUUmNbOj5aQ2diSF9iaFBiWk82Sg==

Let the angle NiMlJnRoZXRhRw== be represented by the fuzzy sets NiMlKGFuZ2xlWFlH, the derivatives of NiMlJnRoZXRhRw== by the fuzzy sets NiMlKGRlcml2WFlH and the resulting acceleration by the fuzzy sets NiMlL2FjY2VsZXJhdGlvblhZRw== where NiMlI1hZRw== may be one of:

NL negative large NM negative medium NS negative small AZ approximately zero PS positive small PM positive medium PL positive large

restart;

with( FuzzySets[RealDomain] ):

We define the following fuzzy sets describing the angles in degrees:

( angleNL, angleNM, angleNS, angleAZ, anglePS, anglePM, anglePL ) := ( L( -30, -20 ), Partition( -30, -20, -10, 0, 10, 20, 30 ), Gamma( 20, 30 ) ):

plot( [ angleNL, angleNM, angleNS, angleAZ, anglePS, anglePM, anglePL ], -40..40 );

Next, we define the following sets which describe the rate of change of the angle in degrees per second:

( derivNL, derivNM, derivNS, derivAZ, derivPS, derivPM, derivPL ) := ( L( -30, -20 ), Partition( -30, -20, -10, 0, 10, 20, 30 ), Gamma( 20, 30 ) ):

plot( [ derivNL, derivNM, derivNS, derivAZ, derivPS, derivPM, derivPL ], -45..45 );

Finally, we define the acceleration of the carriage carrying the pendulum in metres per second squared:

( accelerationNL, accelerationNM, accelerationNS, accelerationAZ, accelerationPS, accelerationPM, accelerationPL ) := Partition( -16, -12, -8, -4, 0, 4, 8, 12, 16 ):

plot( [ accelerationNL, accelerationNM, accelerationNS, accelerationAZ, accelerationPS, accelerationPM, accelerationPL ], -17..17 );

Now that we have partition the observable and control variables into fuzzy sets, we must generate a list of rules which map the observables onto the control variables. For example, the first rule says: If the angle is negative and medium and the rate of change of angle is approximately zero, then accelerate the carriage in the negative direction approximately NiMiIik= m/sNiMqJCUhRyIiIw==.

Rules := Controller( { [angleNM, derivAZ] = accelerationNM, [angleNS, derivAZ] = accelerationNS, [angleAZ, derivAZ] = accelerationAZ, [anglePS, derivAZ] = accelerationPS, [anglePM, derivAZ] = accelerationPM, [angleNM, derivPS] = accelerationNS, [angleNS, derivPS] = accelerationAZ, [angleAZ, derivPS] = accelerationPS, [anglePS, derivPS] = accelerationPM, [angleNS, derivNS] = accelerationNM, [angleAZ, derivNS] = accelerationNS, [anglePS, derivNS] = accelerationAZ, [anglePM, derivNS] = accelerationPS }, inference = minimum ):

Suppose we have an angle of NiMsJCkiIzglIm9HISIi and the rate of change of the angle is NiMsJCktJSZGbG9hdEc2JCIjQCEiIiUib0dGKQ==/s. By applying the Mamdani form of fuzzy rules, the returned fuzzy set suggests what the acceleration should be.

soln := Rules( -13, -2.1 );

plot( soln, -17..1, view=[DEFAULT, 0..1], numpoints = 1000 );

We cannot use a fuzzy set to control the acceleration of the carriage, so we must interpret this fuzzy set as a single real value which we may use as the acceleration. We can convert this fuzzy set into a real value using the routine Defuzzify:

Defuzzify( soln );

Thus, we should accelerate the carriage backward at -5.34 m/sNiMqJCUhRyIiIw==.

The following routine simulates an inverted pendulum and returns an animation of the pendulum over time.

InvertedPendulum := proc( initial_angle, initial_angular_velocity, controller, iterations ) local position, velocity, angle, angular_velocity, dt, plotvehicle, plotlist, i, acceleration, angular_acceleration; position := 0; velocity := 0; angle := evalf( Pi*initial_angle/180 ); angular_velocity := evalf( Pi*initial_angular_velocity/180 ); dt := 60.0/1000.0; # animations run at approximately 1000 frames per minute plotvehicle := proc( posn, a, angular_velocity ) local x, y; x := sin(a); y := cos(a); plots[display]( plots[pointplot]( [[posn, 0], [posn + x, y]], symbol = circle, symbolsize = 20 ), plots[pointplot]( [[-1, 0], [1, 0]] ), plottools[line]( [posn, 0], [posn + x, y] ), plots[textplot]( [0.5, 1.2, sprintf( "% 3d deg, % 3d deg/s", round( evalf( a*180/Pi ) ), round( evalf( angular_velocity*180/Pi ) ) )] ), scaling = constrained ); end proc; plotlist := Vector( 1..iterations + 1 ); plotlist[1] := plotvehicle( position, angle, angular_velocity ); for i from 2 to iterations + 1 do try acceleration := Defuzzify( controller( angle*180/3.14, angular_velocity*180/3.14 ), -100..100, evalf ); catch: acceleration := 0; end try; angular_acceleration := 9.8 * sin( angle ) - acceleration * cos( angle ); angular_velocity := angular_velocity + angular_acceleration * dt - 0.02 * angular_velocity; angle := angle + angular_velocity * dt; if i mod 100 = 0 then print( i ); end if; velocity := velocity + acceleration * dt; position := position + velocity * dt; plotlist[i] := plotvehicle( position, angle, angular_velocity ); end do; plots[display]( seq( plotlist[i], i = 1..(i - 1) ), insequence = true, scaling = constrained ); end proc:

The following function call generates a simulation where the pendulum begins at -10 degrees and its initial angular speed is 40 degrees per second.

InvertedPendulum( -10, 40, Rules, 100 );

This next example uses the product Mamdani inference.

ProductRules := Controller( { [angleNM, derivAZ] = accelerationNL, [angleNS, derivAZ] = accelerationNS, [angleAZ, derivAZ] = accelerationAZ, [anglePS, derivAZ] = accelerationPS, [anglePM, derivAZ] = accelerationPL, [angleNM, derivPS] = accelerationNS, [angleNS, derivPS] = accelerationAZ, [angleAZ, derivPS] = accelerationPS, [anglePS, derivPS] = accelerationPL, [angleNS, derivNS] = accelerationNL, [angleAZ, derivNS] = accelerationNS, [anglePS, derivNS] = accelerationAZ, [anglePM, derivNS] = accelerationPS }, inference = product ):

InvertedPendulum( -10, 40, ProductRules, 100 );

solnP := ProductRules( -13, -2.1 );

plot( solnP, -20..10, 0..1 );

This next example uses the Sugeno form of fuzzy rules. The first rule may be interpreted as if the angle is negative and medium and the derivative is approximately zero, accelerate the cart at -12 m/sNiMqJCUhRyIiIw==.

SugenoRules := Controller( { [angleNM, derivAZ] = -12, [angleNS, derivAZ] = -4, [angleAZ, derivAZ] = 0, [anglePS, derivAZ] = 4, [anglePM, derivAZ] = 12, [angleNM, derivPS] = -4, [angleNS, derivPS] = 0, [angleAZ, derivPS] = 4, [anglePS, derivPS] = 12, [angleNS, derivNS] = -12, [angleAZ, derivNS] = -4, [anglePS, derivNS] = 0, [anglePM, derivNS] = 4 }, rules = Sugeno ):

InvertedPendulum := proc( initial_angle, initial_angular_velocity, controller, iterations ) local position, velocity, angle, angular_velocity, dt, plotvehicle, plotlist, i, acceleration, angular_acceleration; position := 0; velocity := 0; angle := evalf( Pi*initial_angle/180 ); angular_velocity := evalf( Pi*initial_angular_velocity/180 ); dt := 60.0/1000.0; # animations run at approximately 1000 frames per minute plotvehicle := proc( posn, a, angular_velocity ) local x, y; x := sin(a); y := cos(a); plots[display]( plots[pointplot]( [[posn, 0], [posn + x, y]], symbol = circle, symbolsize = 20 ), plots[pointplot]( [[-1, 0], [1, 0]] ), plottools[line]( [posn, 0], [posn + x, y] ), plots[textplot]( [0.5, 1.2, sprintf( "% 3d deg, % 3d deg/s", round( evalf( a*180/Pi ) ), round( evalf( angular_velocity*180/Pi ) ) )] ), scaling = constrained ); end proc; plotlist := Vector( 1..iterations + 1 ); plotlist[1] := plotvehicle( position, angle, angular_velocity ); for i from 2 to iterations + 1 do acceleration := controller( angle*180/3.14, angular_velocity*180/3.14 ); if acceleration::'undefined' then acceleration := 0; end if; angular_acceleration := 9.8 * sin( angle ) - acceleration * cos( angle ); angular_velocity := angular_velocity + angular_acceleration * dt - 0.02 * angular_velocity; angle := angle + angular_velocity * dt; if i mod 100 = 0 then print( i ); end if; velocity := velocity + acceleration * dt; position := position + velocity * dt; plotlist[i] := plotvehicle( position, angle, angular_velocity ); end do; plots[display]( seq( plotlist[i], i = 1..(i - 1) ), insequence = true, scaling = constrained ); end proc:

InvertedPendulum( -10, 40, SugenoRules, 100 );

solnP := SugenoRules( -13, -2.1 );

<Text-field layout="Heading 1" style="Heading 1">A Faster Fuzzy Controller for an Inverted Pendulum</Text-field>

The RealDomain package is designed to use symbolic algebra, and therefore will be relatively slow. Generating a package which works with fuzzy subsets of a finite domain is significantly faster. This example reproduces the example shown in the previous section without the commentary.

restart;

FzS := FuzzySets[FiniteDomain]( i, i = -45..45 ):

with( FzS ):

( angleNL, angleNM, angleNS, angleAZ, anglePS, anglePM, anglePL ) := ( L( -30, -20 ), Partition( -30, -20, -10, 0, 10, 20, 30 ), Gamma( 20, 30 ) ):

( derivNL, derivNM, derivNS, derivAZ, derivPS, derivPM, derivPL ) := ( L( -30, -20 ), Partition( -30, -20, -10, 0, 10, 20, 30 ), Gamma( 20, 30 ) ):

( accelerationNL, accelerationNM, accelerationNS, accelerationAZ, accelerationPS, accelerationPM, accelerationPL ) := Partition( -16, -12, -8, -4, 0, 4, 8, 12, 16 ):

soln := Rules( -13, -2.1 ):

plot( soln, -17..1 );

Defuzzify( soln );

The defuzzified value is very close to the value found using fuzzy sets on the real domain: NiMsJC0lJkZsb2F0RzYkIit2SCUpUWAhIiohIiI=.

evalf(%);

InvertedPendulum := proc( initial_angle, initial_angular_velocity, controller, iterations ) local position, velocity, angle, angular_velocity, dt, plotvehicle, plotlist, i, acceleration, angular_acceleration; position := 0; velocity := 0; angle := evalf( Pi*initial_angle/180 ); angular_velocity := evalf( Pi*initial_angular_velocity/180 ); dt := 60.0/1000.0; # animations run at approximately 1000 frames per minute plotvehicle := proc( posn, a, angular_velocity ) local x, y; x := sin(a); y := cos(a); plots[display]( plots[pointplot]( [[posn, 0], [posn + x, y]], symbol = circle, symbolsize = 20 ), plots[pointplot]( [[-1, 0], [1, 0]] ), plottools[line]( [posn, 0], [posn + x, y] ), plots[textplot]( [0.5, 1.2, sprintf( "% 3d deg, % 3d deg/s", round( evalf( a*180/Pi ) ), round( evalf( angular_velocity*180/Pi ) ) )] ), scaling = constrained ); end proc; plotlist := Vector( 1..iterations + 1 ); plotlist[1] := plotvehicle( position, angle, angular_velocity ); for i from 2 to iterations + 1 do try acceleration := 1.5*Defuzzify( controller( angle*180/3.14, angular_velocity*180/3.14 ) ); catch: acceleration := 0; end try; angular_acceleration := 9.8 * sin( angle ) - acceleration * cos( angle ); angular_velocity := angular_velocity + angular_acceleration * dt - 0.02 * angular_velocity; angle := angle + angular_velocity * dt; if i mod 100 = 0 then print( i ); end if; velocity := velocity + acceleration * dt; position := position + velocity * dt; plotlist[i] := plotvehicle( position, angle, angular_velocity ); end do; plots[display]( seq( plotlist[i], i = 1..(i - 1) ), insequence = true, scaling = constrained ); end proc:

This command takes 25% the time of the RealDomain version to run.

InvertedPendulum( -10, 40, Rules, 100 );

On my computer, this takes approximately one minute to calculate and the animation runs for approximately one minute. This should not be taken to imply that this package should be used for real-time systems.

InvertedPendulum( -10, 40, Rules, 100 );