Controlled Invariant Subspace¶
A controlled invariant subspace is a linear-system state subspace whose trajectories can be kept inside it by suitable control input, equivalently one satisfying AV ⊆ V + im B or made invariant by some static state feedback.
Core Idea¶
Consider a finite-dimensional linear time-invariant system over the real or complex numbers,
in continuous time, or \(x_{k+1}=Ax_k+Bu_k\) in discrete time. A linear state subspace \(V\subseteq X\) is controlled invariant for the pair \((A,B)\) when every initial state \(x_0\in V\) admits an input that keeps the resulting state trajectory in \(V\) for all future time. The input is allowed to counteract the component of \(Ax\) that would otherwise leave \(V\).
Scope of Application¶
The nucleus is finite-dimensional LTI geometric control with unrestricted real- or complex-valued inputs and ordinary state-space dynamics. It applies in continuous time and, through the same algebraic inclusion, in discrete time. The closed-loop characterization presumes that full state is available for static feedback; if only outputs are measured, observer or dynamic-compensator questions must be solved separately.
Within this scope, controlled invariant subspaces organize disturbance decoupling, output-nulling problems, constrained pole assignment, regulation, system decomposition, and computation of the largest subspace contained in a safety or output kernel constraint.
Clarity¶
A useful diagnostic is: what causes the next-state or velocity component normal to \(V\) to vanish?
- If \(A\) alone maps every \(v\in V\) back into \(V\), the subspace is ordinarily invariant and therefore controlled invariant trivially.
- If the normal component of \(Av\) can be canceled by some vector in \(\operatorname{im}B\) for every \(v\in V\), it is controlled invariant nontrivially.
- If even one \(v\in V\) has a normal drift component outside the projected input image, the candidate fails.
- If containment must hold for every admissible input, the relevant property is strong invariance, not existential controlled invariance.
Manages Complexity¶
The abstraction replaces an infinite family of trajectory-existence questions—one for every initial state in \(V\) and every future time—with a single finite-dimensional inclusion. Instead of directly searching over time functions \(u(\cdot)\), one checks a span or rank relation. Instead of searching blindly over all feedback matrices, one first constructs a subspace with the right containment geometry and then solves linear equations for a friend.
Abstract Reasoning¶
The inclusion licenses several deductions.
Feedback existence. If \(AV\subseteq V+\operatorname{im}B\), choose for each basis vector \(q_i\) of \(V\) an input correction so that \(Aq_i+Bf_i\in V\). Linear extension gives a single feedback \(F\) working for every state in \(V\). Thus pointwise correctability does not require a different nonlinear policy for each initial state.
Knowledge Transfer¶
Knowledge transfers exactly across finite-dimensional continuous- and discrete-time LTI models because the operative test is the same pair/subspace inclusion. A proof or algorithm expressed in coordinates can be transported under similarity transformations, changes of input basis, or state feedback, then recovered as a geometric statement about the corresponding subspaces.
Transfer among geometric-control problems proceeds by changing the containing constraint \(K\). For output nulling, one searches within an output kernel. For disturbance decoupling, one additionally asks that disturbance directions lie inside the selected subspace.
Relationships to Other Abstractions¶
Current abstraction Controlled Invariant Subspace Domain-specific
Parents (1) — more general patterns this builds on
-
Controlled Invariant Subspace is a kind of Invariance Prime
Controlled Invariant Subspace strictly instantiates
prime:invariancein a control-mediated form.
Hierarchy path (1) — routes to 1 parentless root
- Controlled Invariant Subspace → Invariance
Neighborhood in Abstraction Space¶
Controlled Invariant Subspace sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Quantum Operation — 0.83
- Lyapunov Exponent — 0.83
- Hartman–Grobman Theorem — 0.82
- Vector Addition System — 0.82
- Compact Operator — 0.81
Computed from structural-signature embeddings · 2026-09-08