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This example shows how to use a 2-D Lookup Table block to implement a fuzzy inference system for nonlinear fuzzy PID control.

The example uses Control System Toolbox™ and Simulink®.

A fuzzy inference system (FIS) maps given inputs to outputs using fuzzy logic. For example, a typical mapping of a two-input one-output fuzzy controller can be depicted in a 3-D plot. The plot is often referred to as the `control surface`

plot such as the one shown below.

The typical FIS inputs are the signals of error (`e(k)`

) and change of error (`e(k)-e(k-1)`

). The FIS output is the control action inferred from the fuzzy rules. Fuzzy Logic Toolbox™ provides commands and GUI tools to design a FIS for a desired control surface. The designed FIS can then be simulated using the Fuzzy Logic Controller block in Simulink®.

Nonlinear control surfaces can often be approximated by lookup tables to simplify the generated code and improve execution speed. For example, a Fuzzy Logic Controller block in Simulink can be replaced by a set of Lookup Table blocks, one lookup table for each output defined in the FIS. Fuzzy Logic Toolbox provides command such as `evalfis`

to compute data used in those lookup tables.

In this example, we design a nonlinear fuzzy PID controller for a plant in Simulink. The plant is a single-input single-output system in discrete time and our design goal is simply to achieve good reference tracking performance.

Ts = 0.1; Plant = c2d(zpk([],[-1 -3 -5],1),Ts);

We also show how to implement the fuzzy inference system with a 2-D lookup table that properly approximates the control surface and achieves the same control performance.

The fuzzy controller in this example is in the feedback loop and computes PID-like actions through fuzzy inference. The loop structure is displayed below in the Simulink diagram.

```
open_system('sllookuptable')
```

The fuzzy PID controller uses a parallel structure[1] as shown below. It is a combination of fuzzy PI control and fuzzy PD control.

```
open_system('sllookuptable/Fuzzy PID')
```

We use the change of measurement `-(y(k)-y(k-1))`

, instead of change of error `e(k)-e(k-1)`

, as the second input signal to FIS to prevent the step change in reference signal from directly triggering the derivative action. Two gain blocks, `GCE`

and `GCU`

in the feed forward path from `r`

to `u`

, are used to ensure that the error signal `e`

is used in proportional action when the fuzzy PID controller is linear.

Designing a fuzzy PID controller involves configuring the fuzzy inference system and setting the four scaling factors: `GE`

, `GCE`

, `GCU`

and `GU`

. In this example we applied the following design steps[2]:

1. Design a conventional linear PID controller 2. Design an equivalent linear fuzzy PID controller 3. Adjust the fuzzy inference system to achieve nonlinear control surface 4. Fine-tune the nonlinear fuzzy PID controller

The conventional PID controller is a discrete time PID controller with Backward Euler numerical integration method used in both the integral and derivative actions. The controller gains are `Kp`

, `Ki`

and `Kd`

. The controller is implemented in Simulink as below:

```
open_system('sllookuptable/Conventional PID')
```

Similar to the fuzzy PID controller, the input signal to the derivative action is `-y(k)`

, instead of `e(k)`

.

PID controller gains can be tuned either manually or using tuning formulas. In this example, we use the `pidtune`

command from Control System Toolbox™ to obtain an initial PID design.

C0 = pid(1,1,1,'Ts',Ts,'IF','B','DF','B'); % define PID structure C = pidtune(Plant,C0) %#ok<*NOPTS> % design PID [Kp, Ki, Kd] = piddata(C); % obtain PID gains

C = Ts*z z-1 Kp + Ki * ------ + Kd * ------ z-1 Ts*z with Kp = 30.6, Ki = 25.2, Kd = 9.02, Ts = 0.1 Sample time: 0.1 seconds Discrete-time PID controller in parallel form.

By configuring the FIS and selecting four scaling factors, we obtain a linear fuzzy PID control that reproduces the exact control performance as the conventional PID controller does.

First, configure the fuzzy inference system so that it produces a linear control surface from inputs `E`

and `CE`

to output `u`

. The FIS settings summarized below are based on design choices described in [2]:

Use Mamdani style fuzzy inference system.

Use algebraic product for AND connective.

The ranges of both inputs are normalized to [-10 10].

The input sets are triangular and cross neighbor sets at membership value of 0.5.

The output range is [-20 20].

Use singletons as output, determined by the sum of the peak positions of the input sets.

Use the center of gravity method (COG) for defuzzification.

Construct fuzzy inference system:

FIS = newfis('FIS','FISType','mamdani','AndMethod','prod','OrMethod','probor',... 'ImplicationMethod','prod','AggregationMethod','sum');

Define input `E`

:

FIS = addvar(FIS,'input','E',[-10 10]); FIS = addmf(FIS,'input',1,'Negative','trimf',[-20 -10 0]); FIS = addmf(FIS,'input',1,'Zero','trimf',[-10 0 10]); FIS = addmf(FIS,'input',1,'Positive','trimf',[0 10 20]);

Define input `CE`

:

FIS = addvar(FIS,'input','CE',[-10 10]); FIS = addmf(FIS,'input',2,'Negative','trimf',[-20 -10 0]); FIS = addmf(FIS,'input',2,'Zero','trimf',[-10 0 10]); FIS = addmf(FIS,'input',2,'Positive','trimf',[0 10 20]);

Define output `u`

:

FIS = addvar(FIS,'output','u',[-20 20]); FIS = addmf(FIS,'output',1,'LargeNegative','trimf',[-20 -20 -20]); FIS = addmf(FIS,'output',1,'SmallNegative','trimf',[-10 -10 -10]); FIS = addmf(FIS,'output',1,'Zero','trimf',[0 0 0]); FIS = addmf(FIS,'output',1,'SmallPositive','trimf',[10 10 10]); FIS = addmf(FIS,'output',1,'LargePositive','trimf',[20 20 20]);

Define the rules:

If

`E`

is Negative and`CE`

is Negative then`u`

is -20If

`E`

is Negative and`CE`

is Zero then`u`

is -10If

`E`

is Negative and`CE`

is Positive then`u`

is 0If

`E`

is Zero and`CE`

is Negative then`u`

is -10If

`E`

is Zero and`CE`

is Zero then`u`

is 0If

`E`

is Zero and`CE`

is Positive then`u`

is 10If

`E`

is Positive and`CE`

is Negative then`u`

is 0If

`E`

is Positive and`CE`

is Zero then`u`

is 10If

`E`

is Positive and`CE`

is Positive then`u`

is 20

ruleList = [1 1 1 1 1;... % Rule 1 1 2 2 1 1;... % Rule 2 1 3 3 1 1;... % Rule 3 2 1 2 1 1;... % Rule 4 2 2 3 1 1;... % Rule 5 2 3 4 1 1;... % Rule 6 3 1 3 1 1;... % Rule 7 3 2 4 1 1;... % Rule 8 3 3 5 1 1]; % Rule 9 FIS = addrule(FIS,ruleList);

In this example, we build the FIS using commands. You can also use the FIS Editor GUI tool to design the system.

The linear control surface is plotted in 3-D below.

gensurf(FIS)

Next, we determine scaling factors `GE`

, `GCE`

, `GCU`

and `GU`

from the `Kp`

, `Ki`

, `Kd`

gains used by the conventional PID controller. By comparing the expressions of the traditional PID and the linear fuzzy PID, the variables are related as:

Kp = GCU * GCE + GU * GE

Ki = GCU * GE

Kd = GU * GCE

Assume the maximum reference step is 1, whereby the maximum error `e`

is 1. Since the input range of `E`

is [-10 10], we first fix `GE`

at 10. `GCE`

, `GCU`

and `GCU`

are then solved from the above equations.

GE = 10; GCE = GE*(Kp-sqrt(Kp^2-4*Ki*Kd))/2/Ki; GCU = Ki/GE; GU = Kd/GCE;

The fuzzy controller block has two inputs (`E`

and `CE`

) and one output (`u`

) that can be replaced by a 2-D lookup table.

The 2-D lookup table for FIS is generated by looping through the input universe and computing the output with the `evalfis`

command.

Step = 10; % use 3 break points for both E and CE inputs E = -10:Step:10; CE = -10:Step:10; N = length(E); LookUpTableData = zeros(N); for i=1:N for j=1:N % compute output u for each combination of break points LookUpTableData(i,j) = evalfis([E(i) CE(j)],FIS); end end

The fuzzy PID controller using 2-D lookup table is shown below. Notice that we only replace the Fuzzy Logic Controller block with a 2-D Lookup Table block in the Simulink diagram.

```
open_system('sllookuptable/Fuzzy PID using Lookup Table')
```

When the control surface is linear as we already designed above, fuzzy PID controller using the 2-D lookup table should produce exactly the same result as the one using fuzzy logic controller block.

In the Simulink model, we use three different sub-systems, namely `Conventional PID`

, `Fuzzy PID`

and `Fuzzy PID using Lookup Table`

, to control the same plant. The closed-loop responses to a step reference change are displayed in the scope and they are exactly the same (three response curves overlap each other).

open_system('sllookuptable/Scope') sim('sllookuptable')

After verifying that the linear fuzzy PID controller is properly designed, adjust FIS settings such as its style, membership functions and rule base to obtain a desired nonlinear control surface.

In this example, we choose to design a `steep`

control surface using Sugeno style of FIS. Each input set has two terms (Positive and Negative) and the number of rules is reduced to four.

Construct fuzzy inference system:

FIS = newfis('FIS','FISType','sugeno');

Define input `E`

:

FIS = addvar(FIS,'input','E',[-10 10]); FIS = addmf(FIS,'input',1,'Negative','gaussmf',[7 -10]); FIS = addmf(FIS,'input',1,'Positive','gaussmf',[7 10]);

Define input `CE`

:

FIS = addvar(FIS,'input','CE',[-10 10]); FIS = addmf(FIS,'input',2,'Negative','gaussmf',[7 -10]); FIS = addmf(FIS,'input',2,'Positive','gaussmf',[7 10]);

Define output `u`

:

FIS = addvar(FIS,'output','u',[-20 20]); FIS = addmf(FIS,'output',1,'Min','constant',-20); FIS = addmf(FIS,'output',1,'Zero','constant',0); FIS = addmf(FIS,'output',1,'Max','constant',20);

Define the rules:

If

`E`

is Negative and`CE`

is Negative then`u`

is -20If

`E`

is Negative and`CE`

is Positive then`u`

is 0If

`E`

is Positive and`CE`

is Negative then`u`

is 0If

`E`

is Positive and`CE`

is Positive then`u`

is 20

ruleList = [1 1 1 1 1;... % Rule 1 1 2 2 1 1;... % Rule 2 2 1 2 1 1;... % Rule 3 2 2 3 1 1]; % Rule 4 FIS = addrule(FIS,ruleList);

The 3-D nonlinear control surface is plotted below. It has higher gain near the center of the `E`

and `CE`

plane than the linear surface has, which helps reduce the error more quickly when the error is small. When the error is large, controller becomes less aggressive so that control action is limited to avoid possible saturation.

gensurf(FIS)

Before starting the simulation, update the lookup table with the new control surface data. Since the surface is nonlinear, add more break points to obtain a sufficient approximation.

Step = 1; % use 21 break points for both E and CE inputs E = -10:Step:10; CE = -10:Step:10; N = length(E); LookUpTableData = zeros(N); for i=1:N for j=1:N % compute output u for each combination of break points LookUpTableData(i,j) = evalfis([E(i) CE(j)],FIS); end end

The closed-loop responses to a step reference change are displayed in the scope.

```
sim('sllookuptable')
```

Compared with the traditional linear PID controller (the response curve with large overshoot), the nonlinear fuzzy PID controller reduces the over shoot by 50%, which is what we expected. The two response curves from the nonlinear fuzzy controllers almost overlap each other, which indicates that the 2-D lookup table approximates the fuzzy inference system very well.

In this example, we design a nonlinear fuzzy PID controller and show that lookup table can be used to approximate the fuzzy inference system. By replacing a Fuzzy Logic Controller block with a Lookup Table block in Simulink, a fuzzy controller can be deployed with simplified generated code and improved execution speed.

[1] Xu, J. X., Hang, C. C., Liu, C. (2000), Parallel structure and tuning of a fuzzy PID controller, Automatica, Vol. 36, page 673-684

[2] Jantzen, J. (1999), Tuning of Fuzzy PID Controllers, Technical Report, Dept. of Automation, Technical University of Denmark

```
bdclose('sllookuptable');
```

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