Use Custom Constraints in Blending Process

This example shows how to design an MPC controller for a blending process using custom input and output constraints.

Blending Process

A continuous blending process combines three feeds in a well-mixed container to produce a blend having desired properties. The dimensionless governing equations are:

\frac{{dv}}{{d\tau }} = \sum\limits_{i = 1}^3 {{\phi _i}}  - \phi \\
V\frac{{d{\gamma _j}}}{{d\tau }} = \sum\limits_{i = 1}^3
{\left( {{\gamma _{ij}} - {\gamma _j}} \right){\phi _i}}


  • $V$ is the mixture inventory (in the container).

  • $\phi_i$ is the plow rate for the ith feed.

  • $\phi$ is the rate at which the blend is being removed from inventory, that is the demand.

  • $\gamma _{ij}$ is the concentration of constituent $j$ in feed $i$.

  • $\gamma _j$ is the concentration of constituent $j$ in the blend.

  • $\tau$ is time.

In this example, there are two important constituents, $j$ = 1 and 2.

The control objectives are targets for the two constituent concentrations in the blend, and the mixture inventory. The challenge is that the demand, $\phi$, and feed compositions, $\gamma _{ij}$, vary. The inventory, blend compositions, and demand are measured, but the feed compositions are unmeasured.

At the nominal operating condition:

  • Feed 1, $\phi _1$, (mostly constituent 1) is 80% of the total inflow.

  • Feed 2, $\phi _2$, (mostly constituent 2) is 20%.

  • Feed 3, $\phi _3$, (pure constituent 1) is not used.

The process design allows manipulation of the total feed entering the mixing chamber, $\phi _T$, and the individual rates of feeds 2 and 3. In other words, the rate of feed 1 is:

$$\phi _1=\phi _T-\phi _2-\phi _2$$

Each feed has limited availability:

$$0 \le {\phi _i} \le {\phi _{i,\max }}$$

The equations are normalized such that, at the nominal steady state, the mean residence time in the mixing container is $\tau=1$.

The constraint $\phi _{1,\max }=0.8$ is imposed by an upstream process, and the constraints $\phi _{2,\max }=\phi _{3,\max }=0.6$ are imposed by physical limits.

Define Linear Plant Model

The blending process is mildly nonlinear, however you can derive a linear model at the nominal steady state. This approach is quite accurate unless the (unmeasured) feed compositions change. If the change is sufficiently large, the steady-state gains of the nonlinear process change sign and the closed-loop system can become unstable.

Specify the number of feeds, ni, and the number of constituents, nc.

ni = 3;
nc = 2;

Specify the nominal flow rates for the three input streams and the output stream, or demand. At the nominal operating condition, the output flow rate is equal to the sum of the input flow rates.

Fin_nom = [1.6,0.4,0];
F_nom  = sum(Fin_nom);

Define the nominal constituent compositions for the input feeds, where cin_nom(i,j) represents the composition of constituent i in feed j.

cin_nom = [0.7 0.2 0.8;0.3 0.8 0];

Define the nominal constituent compositions in the output feed.

cout_nom = cin_nom*Fin_nom'/F_nom;

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 a state-space model with feed flows F1, F2, and F3 as MVs:

A = [zeros(1,nc+1); zeros(nc,1) -eye(nc)];
Bu = [ones(1,ni); gij-1];

Change the MV definition to [FT, F2, F3] where F1 = FT - F2 - F3

Bu = [Bu(:,1), Bu(:,2)-Bu(:,1), Bu(:,3)-Bu(:,1)];

Add the measured disturbance, blend demand, as the 4th model input.

Bv = [-1; zeros(nc,1)];
B = [Bu Bv];

Define all of the states as measurable. The states consist of the mixture inventory and the constituent concentrations.

C = eye(nc+1);

Specify that there is no direct feed-through from the inputs to the outputs.

D = zeros(nc+1,ni+1);

Construct the linear 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'};

Create MPC Controller

Specify the sample time, prediction horizon, and control horizon.

Ts = 0.1;
p = 10;
m = 3;

Create the controller.

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 for all outputs.

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

Specify the normalized nominal values the manipulated variables, u(1), u(2) and u(3), and the measured disturbance, u(4).

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 want to pay more attention to controlling the inventory, and the composition of the first constituent.

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;

The total feed rate and the rates of feed 2 and feed 3 have upper bounds. Feed 1 also has an upper bound, determined by the upstream unit supplying it.

Specify Custom Constraints

Given the specified upper bounds on the feed 2 and 3 rates (0.6), it is possible that their sum could be as much as 1.2. Since the nominal total feed rate is 1.0, the controller can request a physically impossible condition, where the sum of feeds 2 and 3 exceeds the total feed rate, which implies a negative feed 1 rate.

The following constraint prevents the controller from requesting an unrealistic $\phi _1$ value.

$$0 \le {\phi _1} = {\phi _T} - {\phi _2} - {\phi _3} \le 0.8$$

Specify this constraint in the form $Eu + Fy \le g$.

E = [-1 1 1; 1 -1 -1];
g = [0;0.8];

Since no outputs are specified in the mixed constraints, set their coefficients to zero.

F = zeros(2,3);

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

v = zeros(2,1);

Specify zero coefficients for the measured disturbance.

h = zeros(2,1);

Set the custom constraints in the MPC controller.


Open and Simulate Model in Simulink

sys = 'mpc_blendingprocess';
-->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 MPC controller controls the blending process. The block labeled Blending incorporates the previously described model equations and includes an unmeasured step disturbance in the constituent 1 feed composition.

The Demand, $\phi$, is modeled as a measured disturbance. The operator can vary the demand value, and the resulting signal goes to both the process and the controller.

The model simulates the following scenario:

  • At $\tau=0$, the process is operating at steady state.

  • At $\tau=1$, the Total Demand decreases from $\phi=1.0$ to $\phi=0.9$.

  • At $\tau=2$, there is a large step increase in the concentration of constituent 1 in feed 1, from 1.17 to 2.17.

The controller maintains the inventory very close to its setpoint, but the severe disturbance in the feed composition causes a prediction error and a large disturbance in the blend composition, especially for constituent 1, c_1. However, the controller recovers and drives the blend composition back to its setpoint.

Verify Effect of Custom Constraints

Plot the feed rate signals.

plot(MVs.time,[MVs.signals(1).values(:,2), ...
    (MVs.signals(2).values + MVs.signals(3).values), ...

The total feed rate, FT, and the sum of feed rates F2 and F3 coincide for $1.7 \le \tau \le 2.2$. If the custom input constraints had not been included, the controller would have requested an impossible negative feed 1 rate, F1, during this period.


See Also

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