## Robust Control Toolbox |

This example shows how to design and tune a gain-scheduled controller for a chemical reactor transitioning from low to high conversion rate. For background, see Seborg, D.E. et al., "Process Dynamics and Control", 2nd Ed., 2004, Wiley, pp. 34-36.

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**Continuous Stirred Tank Reactor**

The process considered here is a continuous stirred tank reactor (CSTR) during transition from low to high conversion rate (high to low residual concentration). Because the chemical reaction is exothermic (produces heat), the reactor temperature must be controlled to prevent a thermal runaway. The control task is complicated by the fact that the process dynamics are nonlinear and transition from stable to unstable and back to stable as the conversion rate increases. The reactor dynamics are modeled in Simulink. The controlled variables are the residual concentration `Cr` and the reactor temperature `Tr`, and the manipulated variable is the temperature `Tc` of the coolant circulating in the reactor's cooling jacket.

```
open_system('rct_CSTR_OL')
```

We want to transition from a residual concentration of 8.57 kmol/m^3 initially down to 2 kmol/m^3. To understand how the process dynamics evolve with the residual concentration `Cr`, find the equilibrium conditions for five values of `Cr` between 8.57 and 2 and linearize the process dynamics around each equilibrium. Log the reactor and coolant temperatures at each equilibrium point.

clear G CrEQ = linspace(8.57,2,5)'; % concentrations TrEQ = zeros(5,1); % reactor temperatures TcEQ = zeros(5,1); % coolant temperatures for k=1:5 opspec = operspec('rct_CSTR_OL'); % Set desired residual concentration opspec.Outputs(1).y = CrEQ(k); opspec.Outputs(1).Known = true; % Compute equilibrium condition [op,report] = findop('rct_CSTR_OL',opspec,... findopOptions('DisplayReport','off')); % Log temperatures TrEQ(k) = report.Outputs(2).y; TcEQ(k) = op.Inputs.u; % Linearize process dynamics G(:,:,k) = linearize('rct_CSTR_OL', 'rct_CSTR_OL/CSTR', op); end G.InputName = {'Cf','Tf','Tc'}; G.OutputName = {'Cr','Tr'}; G.SamplingGrid = struct('Cr',CrEQ); G.TimeUnit = 'min';

Plot the reactor and coolant temperatures at equilibrium as a function of concentration.

subplot(311), plot(CrEQ,'b-*'), grid, title('Residual concentration'), ylabel('CrEQ') subplot(312), plot(TrEQ,'b-*'), grid, title('Reactor temperature'), ylabel('TrEQ') subplot(313), plot(TcEQ,'b-*'), grid, title('Coolant temperature'), ylabel('TcEQ')

An open-loop control strategy consists of following the coolant temperature profile above to smoothly transition between the `Cr`=8.57 and `Cr`=2 equilibria. However, this strategy is doomed by the fact that the reaction is unstable in the mid range and must be properly cooled to avoid thermal runaway. This is confirmed by inspecting the poles of the linearized models for the five equilibrium points considered above (three out of the five models are unstable).

pole(G)

ans(:,:,1) = -0.5225 + 0.0000i -0.8952 + 0.0000i ans(:,:,2) = 0.1733 + 0.0000i -0.8866 + 0.0000i ans(:,:,3) = 0.5114 + 0.0000i -0.8229 + 0.0000i ans(:,:,4) = 0.0453 + 0.0000i -0.4991 + 0.0000i ans(:,:,5) = -1.1077 + 1.0901i -1.1077 - 1.0901i

To prevent thermal runaway while ramping down the residual concentration, use feedback control to adjust the coolant temperature `Tc` based on measurements of the residual concentration `Cr` and reactor temperature `Tr`. For this application, we use a cascade control architecture where the inner loop regulates the reactor temperature and the outer loop tracks the concentration setpoint. Both feedback loops are digital with a sampling period of 0.5 minutes.

```
open_system('rct_CSTR')
```

The target concentration `Cref` ramps down from 8.57 kmol/m^3 at t=10 to 2 kmol/m^3 at t=36 (the transition lasts 26 minutes). The corresponding profile `Tref` for the reactor temperature is obtained by interpolating the equilibrium values `TrEQ` from trim analysis. The controller computes the coolant temperature adjustment `dTc` relative to the initial equilibrium value `TcEQ(1)`=297.98 for `Cr`=8.57. Note that the model is set up so that initially, the output `TrSP` of the "Concentration controller" block matches the reactor temperature, the adjustment `dTc` is zero, and the coolant temperature `Tc` is at its equilibrium value `TcEQ(1)`.

clf t = [0 10:36 45]; C = interp1([0 10 36 45],[8.57 8.57 2 2],t); subplot(211), plot(t,C), grid, set(gca,'ylim',[0 10]) title('Target residual concentration'), ylabel('Cref') subplot(212), plot(t,interp1(CrEQ,TrEQ,C)) title('Corresponding reactor temperature at equilibrium'), ylabel('Tref'), grid

Use `TuningGoal` objects to capture the design requirements. First, `Cr` should follow setpoints `Cref` with a response time of about 5 minutes.

R1 = TuningGoal.Tracking('Cref','Cr',5);

The inner loop (temperature) should stabilize the reaction dynamics with sufficient damping and fast enough decay.

MinDecay = 0.2; MinDamping = 0.5; % Constrain closed-loop poles of inner loop with the outer loop open R2 = TuningGoal.Poles('du',MinDecay,MinDamping); R2.Openings = 'TrSP';

The Rate Limit block at the controller output specifies that the coolant temperature `Tc` cannot vary faster than 10 degrees per minute. This is a severe limitation on the controller authority which, when ignored, can lead to poor performance or instability. To take this rate limit into account, observe that `Cref` varies at a rate of 0.25 kmol/m^3/min. To ensure that `du` does not vary faster than 10 degrees/min, the gain from `Cref` to `du` should be less than 10/0.25=40.

R3 = TuningGoal.Gain('Cref','du',40);

Finally, require at least 7 dB of gain margin and 45 degrees of phase margin at the plant input `du`.

```
R4 = TuningGoal.Margins('du',7,45);
```

To achieve these requirements, we use a PI controller in the outer loop (see Figure 1) and a lead compensator in the inner loop (see Figure 2). Note that due to the slow sampling rate, a pure gain in the inner loop is not enough to adequately stabilize the chemical reaction at the mid-range concentration `Cr` = 5.28 kmol/m^3/min.

**Figure 1: PI controller for concentration loop.**

**Figure 2: Lead compensator for temperature loop.**

Plot the Bode response of the linearized models obtained for 5 concentration values between 8.57 and 2 kmol/m^3. The reaction dynamics vary substantially with concentration, suggesting that the PI controller and lead compensator gains should be scheduled as a function of the residual concentration `Cr`.

clf, bode(G(:,3),{0.01,10})

Because the target bandwidth is within a decade of the Nyquist frequency, it is easier to tune the controller directly in the discrete domain. Discretize the linearized process dynamics with sample time of 0.5 minutes. Use the ZOH method to reflect how the digital controller interacts with the continuous-time plant.

```
Ts = 0.5; % in minutes
Gd = c2d(G,Ts);
```

Use low-order polynomials to model the dependence of the controller gains `Kp,Ki,Kt,a,b` on the scheduling variable `Cr`. First-order polynomials are a natural starting point, but here better results are obtained with quadratic polynomials in `Cr`, for example,

Using `systune`, you can automatically tune the coefficients
to meet the requirements `R1-R4` at the five equilibrium points computed above. This amounts to tuning the gain-scheduled controller at five design points along the `Cref` trajectory. Use `gainsurf` to model the quadratic gain schedules
at the five concentration values `CrEQ`.

Kp = gainsurf('Kp',0,CrEQ,CrEQ.^2); Ki = gainsurf('Ki',-1,CrEQ,CrEQ.^2); Kt = gainsurf('Kt',0,CrEQ,CrEQ.^2); a = gainsurf('a',0,CrEQ,CrEQ.^2); b = gainsurf('b',0,CrEQ,CrEQ.^2);

Use these gains to build the PI and lead controllers. Insert "analysis point" blocks at the controller outputs `TrSP` and `du` to gain access to open-loop responses at these locations.

PI = AnalysisPoint('TrSP') * pid(Kp,Ki,'Ts',Ts,'TimeUnit','min'); PI.u = 'ECr'; PI.y = 'TrSP'; LEAD = AnalysisPoint('du') * Kt * tf([1 -a],[1 -b],Ts,'TimeUnit','min'); LEAD.u = 'ETr'; LEAD.y = 'Tc';

Use `connect` to build a closed-loop model of the overall control system at the five design points. Here `T0` is a 5-by-1 array of linear models depending on the tunable coefficients
. Each model is discrete and sampled every half minute.

S1 = sumblk('ECr = Cref - Cr'); S2 = sumblk('ETr = TrSP - Tr'); T0 = connect(Gd(:,'Tc'),LEAD,PI,S1,S2,'Cref','Cr');

You can now use `systune` to tune the controller coefficients against the requirements `R1-R4`. Make the stability margin requirement a hard constraints and optimize the remaining requirements.

T = systune(T0,[R1 R2 R3],R4);

Final: Soft = 1.21, Hard = 0.9999, Iterations = 235

The resulting design satisfies the hard constraint (`Hard<1`) and nearly satisfies the remaining requirements (`Soft` close to 1). To validate this design, simulate the responses to a ramp in concentration with the same slope as `Cref`. Each plot shows the linear responses at the five design points `CrEQ`.

t = 0:Ts:20; uC = interp1([0 2 5 20],(-0.25)*[0 0 3 3],t); subplot(211), lsim(getIOTransfer(T,'Cref','Cr'),uC) grid, set(gca,'ylim',[-1.5 0.5]), title('Residual concentration') subplot(212), lsim(getIOTransfer(T,'Cref','du'),uC) grid, title('Coolant temperature variation')

Note that rate of change of the coolant temperature remains within the physical limits (10 degrees per minute or 5 degrees per sample period).

Each controller gain in the Simulink model is set up as a quadratic function of the scheduling variable `Cr`. For example, Figure 3 shows how the `Kp` block implements the formula

The controller gains are updated at each sampling period based on the measured value of `Cr`, which ensures that the control signal varies smoothly as a function of `Cr`.

**Figure 3: Scheduling of the proportional gain Kp as a function of Cr.**

To validate the gain-scheduled controller in Simulink, extract the tuned coefficients using `gainsurfdata`. Use the same variable names as in the Simulink model, for example, `Kp_0,Kp_1,Kp_2` for the proportional gain
.

[Kp_0,Kp_1,Kp_2] = gainsurfdata(setBlockValue(Kp,T)); [Ki_0,Ki_1,Ki_2] = gainsurfdata(setBlockValue(Ki,T)); [Kt_0,Kt_1,Kt_2] = gainsurfdata(setBlockValue(Kt,T)); [a_0,a_1,a_2] = gainsurfdata(setBlockValue(a,T)); [b_0,b_1,b_2] = gainsurfdata(setBlockValue(b,T));

The simulation results appear in Figure 4. The gain-scheduled controller successfully drives the reaction through the transition with adequate response time and no saturation of the rate limits (controller output `du` matches effective temperature variation `dTc`). The reactor temperature stays close to its equilibrium value `Tref`, indicating that the controller keeps the reaction near equilibrium while preventing thermal runaway.

**Figure 4: Transition with gain-scheduled cascade controller.**

Inspect how each gain varies with `Cr` during the transition.

SG = G.SamplingGrid; subplot(321), Kp.SamplingGrid = SG; view(setBlockValue(Kp,T)), ylabel('Kp') subplot(322), Ki.SamplingGrid = SG; view(setBlockValue(Ki,T)), ylabel('Ki') subplot(323), Kt.SamplingGrid = SG; view(setBlockValue(Kt,T)), ylabel('Kt') subplot(324), a.SamplingGrid = SG; view(setBlockValue(a,T)), ylabel('a') subplot(325), b.SamplingGrid = SG; view(setBlockValue(b,T)), ylabel('b')

You can generate code for this digital gain-scheduled controller "as is". If preferred, you can also convert the quadratic gain formulas into classic gain-scheduling lookup tables.