Controllability and reachability Gramians

For ẋ=Ax+Bu on $\mathbb R^n$, start with the columns of B, then the columns of AB, A²B and so on. They identify directions available directly and through repeated action of the dynamics. The Kalman controllability matrix stops at A^{n−1}B because Cayley–Hamilton expresses all higher powers using those already listed. The pair is controllable exactly when this matrix has rank n. For continuous-time LTI systems with unrestricted inputs, this is equivalent to reaching every final state from zero over any prescribed positive horizon.

Reaching any final state from any initial state is equivalent here, because the required displacement is x_f−e^{AT}x₀ and e^{AT} is invertible. This equivalence requires care in discrete time: if A is singular, driving an initial state to zero and reaching every state from zero need not be the same property. Throughout this lesson, controllability refers to the continuous-time finite-dimensional convention. A rank-deficient pair may still be stabilizable if every uncontrollable mode is already strictly stable; full state transfer is a stronger demand than feedback stabilization.

On horizon T>0, the endpoint map sends an input u to the integral of e^{A(T−t)}Bu(t). Its adjoint sends a state-space vector v to Bᵀe^{Aᵀ(T−t)}v. Composing endpoint map with adjoint gives the reachability Gramian W_T. The quadratic form vᵀW_Tv is the integral of the squared norm of the adjoint signal, hence is nonnegative. It is zero precisely when that direction is orthogonal to every endpoint reachable through the input. The Gramian is positive definite exactly for a controllable pair.

When W_T is invertible, the minimum-energy input is the adjoint signal associated with W_T^{-1} times the required endpoint displacement. Substituting it into the endpoint map verifies that it reaches the target. Any other reaching input differs by an element of the endpoint map’s kernel, which is orthogonal to this minimum-energy signal. Pythagoras in the input function space then proves minimality, not merely stationarity. The resulting energy is a quadratic form in the target displacement. Small Gramian eigenvalues correspond to expensive state directions.

A nearly singular Gramian creates a distinction between exact algebra and numerical practice. A theoretical rank calculation may say all directions are reachable, while finite precision makes an inverse unreliable and the corresponding input extremely large. State and input units also affect raw Gramian magnitudes, so compare energies only after choosing physically meaningful scaling. If the target lies in a proper reachable subspace, a pseudoinverse can compute minimum energy within that subspace; it cannot make an unreachable component disappear while still satisfying the original endpoint requirement.

For a Hurwitz A, the infinite-horizon Gramian converges and solves AW+WAᵀ+BBᵀ=0. Hurwitz stability is a sufficient convergence hypothesis, not a necessary one for arbitrary B: an unreachable unstable mode can contribute zero to the integrand. Finite-horizon reachability itself does not require stable dynamics. A longer horizon often reduces minimum energy because a feasible earlier transfer can be embedded in a longer zero-start experiment, but constraints and a nonzero prescribed initial state change the comparison one must make. In design, inspect both structural rank and energy geometry before concluding that an actuator placement is adequate.