## Code for Figure 1.9

### main.m

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201% Applies offset-free linear MPC to the nonlinear CSTR.
% See Pannocchia and Rawlings, AIChE J, 2002.

mpc = import_mpctools();

% Parameters and sizes for the nonlinear system
Delta = 1;
Nx = 3;
Nu = 2;
Ny = Nx;
Np = 1;
small = 1e-5; % Small number.

% Parameters.
pars = struct();
pars.T0 = 350; % K
pars.c0 =  1;  % kmol/m^3
pars.r = 0.219; % m
pars.k0 = 7.2e10; % min^-1
pars.E =  8750; % K
pars.U =  54.94; % kJ/(min m^2 K)
pars.rho = 1e3;   % kg/m^3
pars.Cp = 0.239;  % kJ/(kg K)
pars.DeltaH = -5e4; % kJ/kmol
pars.A = pi()*pars.r.^2;
pars.rhoCp = pars.rho*pars.Cp;

ode = @(x, u, p) cstrode(x, u, p, pars);

{'x', 'u', 'p'}, {'ode'});
cstrsim = mpc.getCasadiIntegrator(ode, Delta, [Nx, Nu, Np], ...
{'x', 'u', 'p'}, {'cstr'});

cs = 0.878; % kmol/m^3
Ts = 324.5; % K
hs = 0.659; % m
Fs = 0.1; % m^3/min
Tcs = 300; % K
F0s = 0.1; % m^3/min

xs = [cs; Ts; hs];
us = [Tcs; Fs];
ps = [F0s];

CVs = [1, 3]; % Control Concentration and height.

% Simulate a few steps so we actually get dx/dt = 0 at steady state.
for i = 1:10
xs = full(cstrsim(xs, us, ps));
end
cs = xs(1);
Ts = xs(2);
hs = xs(3);

% Get linearized model and linear controller.
model = mpc.getLinearizedModel(ode_casadi, {xs, us, ps}, ...
{'A', 'B', 'Bp'}, Delta);
A = model.A;
B = model.B;
C = eye(Ny);
Bp = model.Bp;
Q = diag(1./xs.^2);
R = diag(1./us.^2);
K = -dlqr(A, B, Q, R);

% Pick whether to use good disturbance model.
disturbancemodels = {'Good', 'Offset', 'Undetectable', 'No'};
data = struct();
for dmodel = 1:length(disturbancemodels)
dmodel = disturbancemodels{dmodel};
fprintf('Choosing %s disturbance model.\n', dmodel);
switch dmodel
case 'Good'
% disturbance model 6; no offset
Nd = 3;
Bd = zeros(Nx, Nd);
Bd(:,3) = B(:,2);
Cd = [1 0 0; 0 0 0; 0 1 0];
case 'Offset'
% disturbance model with offset
Nd = 2;
Bd = zeros(Nx, Nd);
Cd = [1 0; 0 0; 0 1];
case 'Undetectable'
Nd = 3;
Bd = zeros(Nx, Nd);
Cd = eye(Ny, Nd);
case 'No'
Nd = 0;
Bd = zeros(Nx, Nd);
Cd = zeros(Ny, Nd);
otherwise
error('Unknown choice for disturbance model: %s', dmodel);
end

Aaug = [A, Bd; zeros(Nd, Nx), eye(Nd)];
Baug = [B; zeros(Nd, Nu)];
Caug = [C, Cd];
Naug = size(Aaug,1);

% Detectability test of disturbance model
detec = rank([eye(Naug) - Aaug; Caug]);
if detec < Nx + Nd
fprintf(' * Augmented system is not detectable!\n')
break
end

% Set up state estimator; use KF
Qw = zeros(Naug);
Qw(1:Nx,1:Nx) = small*eye(Nx);
Qw(Nx+1:end,Nx+1:end) = small*eye(Nd);
Qw(end,end) = 1.0;
Rv = small*diag(xs.^2);
[L, ~, Pe] = dlqe(Aaug, eye(Naug), Caug, Qw, Rv);
Lx = L(1:Nx,:);
Ld = L(Nx+1:end,:);

% Closed-loop simulation.
Nsim = 50;
x = NaN(Nx, Nsim + 1);
x(:, 1) = 0;
y = NaN(Ny, Nsim + 1);
u = NaN(Nu, Nsim);

randn('seed', 927);
v = zeros(Ny, Nsim + 1);

xhatm = NaN(Nx, Nsim + 1);
xhatm(:,1) = 0;
dhatm = NaN(Nd, Nsim + 1);
dhatm(:,1) = 0;

xhat = NaN(Nx, Nsim);
dhat = NaN(Nd, Nsim);

xtarg = zeros(Nx, Nsim);
utarg = zeros(Nu, Nsim);

% Disturbance and setpoint.
p = zeros(Np, Nsim);
p(10:end) = 0.1*F0s;
ysp = zeros(Ny, Nsim);

if length(CVs) > Nu
error('At most 2 CVs can be specified!');
end
H = full(sparse(1:length(CVs), CVs, 1, length(CVs), Ny));
Ginv = [eye(Nx) - A, -B; H*C, zeros(size(H,1), Nu)]\eye(Nx + Nu);

% Start loop.
for i = 1:(Nsim + 1)
% Take measurement.
y(:,i) = C*x(:,i) + v(:,i);

e = y(:,i) - C*xhatm(:,i) - Cd*dhatm(:,i);
xhat(:,i) = xhatm(:,i) + Lx*e;
dhat(:,i) = dhatm(:,i) + Ld*e;

% Stop if at last time.
if i == Nsim + 1
break
end

targ = Ginv*[Bd*dhat(:,i); H*(ysp(:,i) - Cd*dhat(:,i))];
xtarg(:,i) = targ(1:Nx);
utarg(:,i) = targ((Nx + 1):end);

% Apply control law.
u(:,i) = K*(xhat(:,i) - xtarg(:,i)) + utarg(:,i);

% Evolve plant. Our variables are deviation but cstrsim needs positional.
x(:,i + 1) = full(cstrsim(x(:,i) + xs, u(:,i) + us, p(:,i) + ps)) - xs;

xhatm(:,i + 1) = A*xhat(:,i) + B*u(:,i) + Bd*dhat(:,i);
dhatm(:,i + 1) = dhat(:,i);
end

% Convert to positional units.
u = bsxfun(@plus, u, us);
x = bsxfun(@plus, x, xs);
ysp = bsxfun(@plus, ysp, C*xs);
ysp(setdiff(1:Ny, CVs),:) = NaN(); % Mask out uncontrolled variables.

% Make a plot.
mpc.mpcplot('x', x, 'u', u, 'xsp', ysp, 'xnames', {'c', 'T', 'h'}, ...
'unames', {'T_c', 'F'}, 'spcolor', 'r', ...
'title', sprintf('%s Disturbance Model', dmodel));

% Store data.
u = [u, u(:,end)]; % Duplicate final point.
data.(dmodel) = struct('x', [0:Nsim; x]', 'u', [0:Nsim; u]');
end

% Save data.
gnuplotsave('cstr.dat', data);

```

### cstrode.m

```
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21function rhs = cstrode(x, u, p, pars)
% Nonlinear ODE model for reactor.
c = x(1);
T = x(2);
h = x(3) + eps(); % Avoids division by zero.

Tc = u(1);
F = u(2);

F0 = p(1);

k = pars.k0*exp(-pars.E/T);
rate = k*c;

dcdt = F0*(pars.c0 - c)/(pars.A*h) - rate;
dTdt = F0*(pars.T0 - T)/(pars.A*h) ...
- pars.DeltaH/pars.rhoCp*rate ...
+ 2*pars.U/(pars.r*pars.rhoCp)*(Tc - T);
dhdt = (F0 - F)/pars.A;

rhs = [dcdt; dTdt; dhdt];

```