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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.

Wolfram Language code: 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];
Wolfram Language code: Simplify[StateResponse[ssm, {HeavisideTheta[t - 1]}, t] /. t -> 1]
If the system is controllable, almost any state feedback controller will eliminate the impulses:
Wolfram Language code: ControllableModelQ[ssm]
Wolfram Language code: closedloop = SystemsModelStateFeedbackConnect[ssm, {{0, 0, k, 0, 0}}]
Wolfram Language code: Simplify[StateResponse[closedloop, {HeavisideTheta[t - 1]}, t] /. t -> 1]