# Nonlinear Blending Process with Custom Constraints

This example shows how to use constraints on a combination of inputs and outputs to control a nonlinear blending process.

The control objective is to adjust the feed rates in order to keep these concentrations near target values. The MPC controller must also control the total inventory of material in the blending vessel.

## Contents

## About the Blending Process

A continuous blending process combines three streams in a well-mixed vessel to produce a blend with desired properties. These properties depend on concentrations of two key constituents:

- Feed rates

- Inventory of materials in the blending vessel.

Let the feed rates be `F1`, `F2`, and `F3`, and the total `FT = F1 + F2 + F3`. The plant configuration allows `F2`, `F3`, and `FT` to be controlled independently. Therefore, F1 = FT - F2 - F3 is determined by difference.

The process operates under the following constraints:

- Physical limitations require
`0 <= FT <= 2`and`0 <= F2, F3 <= 0.6`.

- Each feed has limited availability. For
`F2`and`F3`, maximum availability equals the physical limit 0.6. For F1, a maximum availability of 0.8 is imposed by an upstream process.

During operation, it is important to prevent the controller from requesting `FT < F2 + F3`, which implies `F1 < 0` which is impossible. Similarly, `FT > F2 + F3 + 0.8`, implying `F1 > 0.8` should also be avoided. This can be accomplished using MPC controller with a mixed constraint (constraints on input and output combinations).

System disturbances include the feed stream compositions and the demand for blended material, `F`, defined as the rate at which material leaves the blending vessel. This is set by a downstream process. From the point of view of the blending process, `F` is a measured disturbance. Feed composition disturbances are unmeasured.

Measurements available for feedback control include the constituent concentrations in the vessel and the blend inventory.

## Define Linear Model for the Blending Process

Create a linear approximation -- a state-space model based on the nominal operating point:

ni = 3; % number of feed streams nc = 2; % number of components Fin_nom = [1.6, 0.4, 0]; % Nominal flow rate for the ith feed stream F_nom = sum(Fin_nom); % Nominal flow rate for the exit stream (demand) cin_nom = [0.7 0.2 0.8 % Nominal composition for jth constituent in the ith feed flow 0.3 0.8 0]; cout_nom = cin_nom*Fin_nom'/F_nom; % Nominal product composition

Normalize the linear model such that the target demand is 1 and the product composition is 1:

fin_nom = Fin_nom/F_nom; gij = [cin_nom(1,:)/cout_nom(1) cin_nom(2,:)/cout_nom(2)];

Create the state-space model with feed flows `[F1, F2, F3]` as MVs:

A = [ zeros(1,nc+1) zeros(nc,1) -eye(nc)]; Bu = [ones(1,ni) gij-1]; % Change MV definition to [FT, F2, F3] where F1 = FT - F2 - F3 Bu = [Bu(:,1), Bu(:,2)-Bu(:,1), Bu(:,3)-Bu(:,1)]; % Add the blend demand as the 4th model input, a measured disturbance Bv = [-1 zeros(nc,1)]; B = [Bu Bv]; % All the states (inventory and compositions) are measurable C = eye(nc+1); % No direct feed-through term D = zeros(nc+1,ni+1); % Construct the plant model Model = ss(A, B, C, D); Model.InputName = {'F_T','F_2','F_3','F'}; Model.InputGroup.MV = 1:3; Model.InputGroup.MD = 4; Model.OutputName = {'V','c_1','c_2'};

## Design MPC Controller

Create the controller object with sampling period, prediction and control horizons:

Ts = 0.1; p = 10; m = 3; mpcobj = mpc(Model, Ts, p, m);

-->The "Weights.ManipulatedVariables" property of "mpc" object is empty. Assuming default 0.00000. -->The "Weights.ManipulatedVariablesRate" property of "mpc" object is empty. Assuming default 0.10000. -->The "Weights.OutputVariables" property of "mpc" object is empty. Assuming default 1.00000.

The outputs are the inventory `y(1)` and the constituent concentrations `y(2)` and `y(3)`. Specify nominal values of unity after normalization:

mpcobj.Model.Nominal.Y = [1 1 1];

The manipulated variables are `u1 = FT`, `u2 = F2`, `u3 = F3`. Specify nominal values after normalization:

mpcobj.Model.Nominal.U = [1 fin_nom(2) fin_nom(3) 1];

Specify output tuning weights. Larger weights are assigned to the first two outputs because we pay more attention to controlling the inventory and composition of the first blending material:

mpcobj.Weights.OV = [1 1 0.5];

Specify the hard bounds (physical limits) on the manipulated variables:

umin = [0 0 0]; umax = [2 0.6 0.6]; for i = 1:3 mpcobj.MV(i).Min = umin(i); mpcobj.MV(i).Max = umax(i); mpcobj.MV(i).RateMin = -0.1; mpcobj.MV(i).RateMax = 0.1; end

## Specify Mixed Constraints

Impose the following constrainsts on the control system:

- Physical constraint
`F1 = FT - F2 - F3 >= 0`.

- Availability constraint,
`F1 <= 0.8`.

Putting these in the standard form, you obtain:

`-FT + F2 + F3 <= 0` `FT - F2 - F3 <= 0.8`

This defines the input constraint coefficient matrix:

E = [-1 1 1; 1 -1 -1];

No outputs are specified in the mixed constraints, so set their coefficients to zero:

F = zeros(2,3);

Specify vector g in E*u + F*y <= g:

g = [0; 0.8];

Specify that both constraints are hard (ECR = 0):

v = zeros(2,1);

Specify zero coefficients for the measured disturbance:

h = zeros(2,1);

Include the mixed constraints in the controller object:

setconstraint(mpcobj, E, F, g, v, h);

## Simulate Using Simulink®

To run this example, Simulink® is required.

if ~mpcchecktoolboxinstalled('simulink') disp('Simulink(R) is required to run this example.') return end

Open and simulate the Simulink® model. The model includes a nonlinear model of the blending process. The time scale in normalized such that one time unit is vessel's nominal mean residence time.

```
mdl = 'mpc_blendingprocess';
open_system(mdl);
sim(mdl);
```

-->Converting model to discrete time. Assuming no disturbance added to measured output channel #1. -->Assuming output disturbance added to measured output channel #2 is integrated white noise. -->Assuming output disturbance added to measured output channel #3 is integrated white noise. -->The "Model.Noise" property of the "mpc" object is empty. Assuming white noise on each measured output channel.

The simulation shows:

- At time 0, the plant is operating steadily at the nominal conditions.

- At time 1, the demand decreases 10% (from 1 to 0.9).

- At time 2, there is a large (unmeasured) increase in the concentration of constituent 1 contained in feed 1. This is a nonlinear effect but the linear MPC compensates well (target values are 1 for the three controlled outputs).

## Validate No Mixed Constraint Violations

Plot the input and output signals to check if the mixed constraint is violated during simulation:

figure plot(MVs.time,[MVs.signals(1).values(:,2), ... (MVs.signals(2).values + MVs.signals(3).values), ... (MVs.signals(1).values(:,2)-MVs.signals(2).values-MVs.signals(3).values)]) grid legend('FT','F2+F3','F1')

The plot shows:

- When demand drops, the composition disturbance requires
`F1`to be decreased. During the transient,`F1`becomes zero. If the mixed constraint had not been included, it would have gone negative, i.e., the controller's requests for`FT`,`F2`, and`F3`would have been impossible to satisfy, causing a performance degradation. With the constraint included, the controller does its best given the physical limits.

- The availability constraint
`F1 <= 0.8`is maintained at all times.

bdclose(mdl)