Controlled invariant subspace

In control theory, a controlled invariant subspace of the state space representation of some system is a subspace such that, if the state of the system is initially in the subspace, it is possible to control the system so that the state is in the subspace at all times. This concept was introduced by Giuseppe Basile and Giovanni Marro (Basile & Marro 1969).

Definition

Consider a linear system described by the differential equation

$\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + B\mathbf{u}(t).$

Here, x(t) ∈ Rⁿ denotes the state of the system and u(t) ∈ R^p is the input. The matrices A and B have size n × n and n × p respectively.

A subspace V ⊂ Rⁿ is a controlled invariant subspace if for any x(0) ∈ V, there is an input u(t) such that x(t) ∈ V for all nonnegative t.

Properties

A subspace V ⊂ Rⁿ is a controlled invariant subspace if and only if AV ⊂ V + Im B. If V is a controlled invariant subspace, then there exists a matrix K such that the input u(t) = Kx(t) keeps the state within V; this is a simple feedback control (Ghosh 1985, Thm 1.1).

References

• Basile, Giuseppe; Marro, Giovanni (1969), "Controlled and conditioned invariant subspaces in linear system theory", Journal of Optimization Theory and Applications 3 (5): 306–315, doi:10.1007/BF00931370 .
• Ghosh, Bijoy K. (1985), "Controlled invariant and feedback controlled invariant subspaces in the design of a generalized dynamical system", Proceedings of the 24th IEEE Conference on Decision and Control, IEEE, pp. 872–873, doi:10.1109/CDC.1985.268620 .
• Basile, Giuseppe; Marro, Giovanni (1992), Controlled and Conditioned Invariants in Linear System Theory, Englewood Cliffs : Prentice-Hall .