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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 1558
Domain group: Interdisciplinary & Synthetic
Origin domain: Systems Thinking & Cybernetics
Subdomains: Control Theory, Geometric Control of Linear Systems → Systems Thinking & Cybernetics
Aliases: (A,B)-invariant subspace, Controlled-invariant subspace

Core Idea¶

Consider a finite-dimensional linear time-invariant system over the real or complex numbers,

\[ \dot x(t)=Ax(t)+Bu(t) \]

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 \(V\).

How would you explain it like I'm…

Keep the Marble on the Line

Imagine a marble rolling on a table with a painted straight line on it. The marble tends to wander off the line, but you have a straw to blow on it. If, from any starting spot on the line, you can always blow just right to keep the marble on the line forever, that line is a 'controlled invariant' line.

Stay-Inside Steering Zone

Engineers describe a machine by its 'state', a list of numbers like position and speed, and the state changes over time by fixed rules plus the pushes you give it, called inputs. Some special flat regions of all possible states, like a line or a flat sheet through the center, are interesting. A region is a controlled invariant subspace if, whenever the machine starts anywhere in it, you can choose inputs that keep it inside that region forever. Left alone, the machine might drift out, but your inputs cancel the drift. Engineers can even find one fixed rule, 'push this much depending on where you are', that keeps it inside.

Feedback-Invariant State Subspace

Consider a linear system whose state x changes according to dx/dt = Ax + Bu, where u is a control input you choose (or the step-by-step version x_{k+1} = Ax_k + Bu_k). A subspace V of states is controlled invariant if, starting anywhere in V, some input keeps the state in V for all future time. The control is used to cancel whatever part of the natural motion Ax would push the state out of V. In the standard setting, three statements are equivalent: (1) such inputs exist for every start in V; (2) AV ⊆ V + im B, meaning any escape direction can be offset by an available control direction; (3) there is a linear feedback u = Fx with (A + BF)V ⊆ V. This generalizes an ordinary invariant subspace, where no control is needed.

Consider a finite-dimensional linear time-invariant system dx/dt = Ax + Bu (or x_{k+1} = Ax_k + Bu_k) over the reals or complexes. A subspace V of the state space X is controlled invariant for the pair (A, B) if every initial state in V admits an input keeping the trajectory in V for all future time; the input counteracts the component of Ax that would leave V. In the standard finite-dimensional, unrestricted-input setting three conditions are equivalent: (1) this open-loop existence property; (2) the algebraic inclusion AV ⊆ V + im B; and (3) the existence of a linear feedback F: X → U with (A + BF)V ⊆ V. With the convention u = Fx, such an F is called a friend of V; authors using u = -Kx write A - BK instead. The equivalence of an open-loop existential property, a local subspace inclusion, and a closed-loop synthesis condition is the core of the concept. Introduced by Basile and Marro as a generalization of ordinary (A-)invariant subspaces, it became a basic object of geometric control because it turns feedback-design existence questions into subspace construction and inclusion tests.

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) — outgoing typed relations

Controlled Invariant Subspace is a kind of Invariance Prime

Controlled Invariant Subspace strictly instantiates prime:invariance in a control-mediated form.

Hierarchy path (1) — routes to 1 parentless root

Controlled Invariant Subspace → is a kind of Invariance

Neighborhood in Abstraction Space¶

Controlled Invariant Subspace sits in a sparse region of the domain-specific corpus (82^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25