Impulse Elimination 

The solution to a StateSpaceModel with a singular descriptor matrix can include differentiated inputs. When the input is discontinuous, the result can be undesirable impulses.

In[1]:=

X ssm = StateSpaceModel[{{{0, 1, 0, 0, 0}, {-1, 0, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}}, {{0}, {1}, {0}, {0}, {1}}, {{-2, 0, -1, 0, 1}}, {{0}}, {{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 1}, {0, 0, 0, 0, 0}}}, SamplingPeriod ->None, SystemsModelLabels -> None];

In[2]:=

X Simplify[StateResponse[ssm, {HeavisideTheta[t - 1]}, t] /. t -> 1]

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If the system is controllable, almost any state feedback controller will eliminate the impulses.

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X ControllableModelQ[ssm]

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In[4]:=

X closedloop = SystemsModelStateFeedbackConnect[ssm, {{0, 0, k, 0, 0}}]

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In[5]:=

X Simplify[StateResponse[closedloop, {HeavisideTheta[t - 1]}, t] /. t -> 1]

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