Asymptotic Output Tracking 

Design a feedback controller that will maintain the liquid level and product concentration of a continuous stirred-tank reactor at specified values. »

The affine system with states and inputs .

In[1]:=

X pars = {Subscript[c, B1] -> 20.05, Subscript[c, B2] -> 0.2, Subscript[k, 1] -> 0.4, Subscript[k, 2] -> 1};

In[2]:=

X asys = AffineStateSpaceModel[ {{-Subscript[k, 1] Sqrt[Subscript[x, 1]], -((Subscript[k, 2] Subscript[x, 2])/(1 + Subscript[x, 2])^2)}, {{ 1, 1 }, {( Subscript[c, B1] - Subscript[x, 2])/Subscript[x, 1], ( Subscript[c, B2] - Subscript[x, 2])/Subscript[x, 1] }}, {Subscript[ x, 1], Subscript[x, 2]}, {{ 0, 0 }, {0, 0 }}}, {{Subscript[x, 1], 25.05}, {Subscript[x, 2], 9}}, {{Subscript[u, 1], 1}, {Subscript[u, 2], 1}}, {Automatic, Automatic}, Automatic , SamplingPeriod -> None];

The objective is to maintain the outputs at , with error decay rates .

In[3]:=

X pars1 = {Subscript[r, 1] -> 30, Subscript[r, 2] -> 10.5, Subscript[p, 1] -> -1, Subscript[p, 2] -> -1.5};

The feedback controller that will achieve the objective.

In[4]:=

X fb = AsymptoticOutputTracker[ asys, {Subscript[r, 1], Subscript[r, 2]}, {Subscript[p, 1], Subscript[p, 2]}]

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The closed-loop system.

In[5]:=

X csys = SystemsModelStateFeedbackConnect[asys, fb] /. pars /. pars1

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Simulate the closed-loop system to verify the tracking.

In[6]:=

X OutputResponse[csys, {0, 0}, {t, 0, 4}];

In[7]:=

X OutputResponse[csys, {0, 0}, {t, 0, 4}];; Plot[Evaluate[ Join[%, {Subscript[r, 1], Subscript[r, 2]} /. pars1]], {t, 0, 4}, PlotStyle -> {Automatic, Automatic, Dashed, Dashed}, PlotLegends -> {"liquid level", "product concentration", "ref. liquid level", "ref. product concentration"}]

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The control effort.

In[8]:=

X pars = {Subscript[c, B1] -> 20.05, Subscript[c, B2] -> 0.2, Subscript[k, 1] -> 0.4, Subscript[k, 2] -> 1};; Plot[Evaluate[ Join[%, {Subscript[r, 1], Subscript[r, 2]} /. pars1]], {t, 0, 4}, PlotStyle -> {Automatic, Automatic, Dashed, Dashed}, PlotLegends -> {"liquid level", "product concentration", "ref. liquid level", "ref. product concentration"}]; -fb /. Thread[{Subscript[x, 1], Subscript[x, 2]} -> StateResponse[csys, {0, 0}, {t, 0, 4}]] /. pars /. pars1; Plot[Evaluate[%], {t, 0, 4}, PlotRange -> All, AxesOrigin -> {0, 0}, PlotLegends -> {"input 1", "input 2"}]

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