Controllability

Controllability

An LTI system $\dot{x}=Ax+Bu$ is controllable iff, for any initial state $x_0$ and any target state $x_1$, there exists a finite time $T$ and input $u(\cdot )$ on $[0,T]$ that drives the system from $x(0)=x_0$ to $x(T)=x_1$.

The Kalman rank condition: the system is controllable iff the controllability matrix
$\mathcal{C}=\left[\begin{array}{ccccc}B& AB& A^{2}B& \cdots & A^{n-1}B\end{array}\right]\in {\mathbb{R}}^{n\times nm}$
has full row rank, $\mathrm{rank}(\mathcal{C})=n$.

Equivalent statements (”$(A,B)$ is controllable”) include: (i) the Kalman rank test above; (ii) the PBH test: $\mathrm{rank}[\lambda I-A\mid B]=n$ for every eigenvalue $\lambda$ of $A$; (iii) the controllability Gramian $W_c(T)={\int }_0^T{e}^{At}B{B}^T{e}^{A^{T}t}dt$ is nonsingular for every $T>0$ (continuous time). A system that fails any of these is uncontrollable — there are states it cannot reach from the origin regardless of input.

Controllability is the precondition for pole placement: $(A,B)$ controllable iff, for any desired closed-loop spectrum $\{{\lambda }_1,\dots ,{\lambda }_n\}$ (closed under complex conjugation), there exists a feedback gain $K$ with $\mathrm{spec}(A-BK)=\{{\lambda }_i\}$. Uncontrollable systems have uncontrollable modes — eigenvalues of $A$ that no choice of $K$ can move.

The dual concept is observability: $(A,C)$ is observable iff the state can be reconstructed from input and output history. The observability matrix $\mathcal{O}=[C^T\mid A^T{C}^T\mid \cdots \mid (A^T{)}^{n-1}{C}^T{]}^T$ plays the analogous role, with the same rank condition.

Controllability Is Binary; Reality Is Gradual $\mathrm{rank}(\mathcal{C})=n$ but with $\mathcal{C}$ nearly rank-deficient is barely controllable — reaching states in the near-null-space takes enormous input. The controllability Gramian $W_c$ quantifies this: its smallest singular value is the energy required to reach the most controllable direction; the largest is the energy for the least. A well-conditioned $W_c$ means the system is controllable in a robust, practical sense.

The rank test is all-or-nothing. In practice, a system with

Intuition

Controllability asks: can the input move the system anywhere in state space? For a scalar system, this is trivial — as long as $B\ne 0$, the input can push $x$ up or down to any target. In higher dimensions, the question is whether the input’s influence, propagated forward through the dynamics, can span the entire state space.

The controllability matrix $\mathcal{C}$ is the geometric answer. Its columns are the state directions reachable from the input, directly ($B$) and at each step of propagation through $A$:

• $B$: where the input pushes right now.
• $AB$: where that push is “rotated” by the dynamics one instant later.
• $A^{2}B$: two instants later.
• …

If these span ${\mathbb{R}}^n$, every direction is reachable — the input’s influence, after being shaped by $A$, can be pointed anywhere. If they span only a proper subspace, the complement is an uncontrollable subspace — initial conditions in it stay in it, regardless of input. The Cayley–Hamilton theorem says $A^n$ is a linear combination of $\{I,A,\dots ,A^{n-1}\}$, so more than $n-1$ powers add nothing: $\mathcal{C}$ is $n\times nm$, not infinite.

The PBH test provides an eigenvalue-level picture: a mode is uncontrollable iff its eigenvector is orthogonal to every column of $B$. Geometrically, an uncontrollable mode is one whose eigendirection cannot be excited by any input, so no feedback can affect it. Such a mode evolves according to its intrinsic rate $\lambda$, and if that rate is unstable, no controller can stabilize it.

Properties

Kalman Rank Condition

(Statement) Controllability iff $\mathrm{rank}[B\mid AB\mid \cdots \mid A^{n-1}B]=n$.

The proof rests on the Cayley–Hamilton theorem: $A^n={\sum }_{k=0}^{n-1}{c}_k{A}^k$ for coefficients $c_k$ from the characteristic polynomial. Consequently, $e^{At}B={\sum }_{k=0}^{n-1}{g}_k(t)A^{k}B$ for some scalar functions $g_k$, so the reachable set is $\mathrm{range}([B\mid AB\mid \cdots \mid A^{n-1}B])$. Full rank means reachable = entire ${\mathbb{R}}^n$.

(Computation) Build $\mathcal{C}$ column-by-column, stop early if rank saturates.

For $\mathrm{rank}({\mathcal{C}}_k)=\mathrm{rank}({\mathcal{C}}_{k-1})$ at some step $k\le n-1$, further powers add no new columns and you can stop. This speeds up large systems and is the basis for iterative Krylov-subspace tests. Numerical caution: use SVD to compute rank, not floating-point determinants.

(Single input) For $m=1$, $\mathcal{C}$ is $n\times n$ and controllability iff $\det (\mathcal{C})\ne 0$.

In this case, $K$ is uniquely determined by the desired spectrum, and the pole-placement formula is Ackermann’s formula: $K=[0,\dots ,0,1]{\mathcal{C}}^{-1}{\varphi }_d(A)$, where ${\varphi }_d$ is the desired characteristic polynomial. Numerically ill-conditioned for large $n$; prefer place over acker in practice.

Equivalent Characterizations

(PBH test) Controllability iff $\mathrm{rank}[\lambda I-A\mid B]=n$ for every eigenvalue $\lambda$ of $A$.

Eigenvalue-by-eigenvalue: mode $\lambda$ is controllable iff $[\lambda I-A\mid B]$ has full row rank, iff no left eigenvector of $A$ (for eigenvalue $\lambda$) is orthogonal to $B$. Useful for identifying which specific modes are uncontrollable when the Kalman test fails. Numerically better-conditioned than the Kalman rank test for large systems.

(Gramian) Controllability iff $W_c(T)={\int }_0^T{e}^{At}B{B}^T{e}^{A^{T}t}dt$ is positive definite for any $T>0$.

The Gramian is the identity-scaled covariance of the reachable set. Its inverse gives the minimum-energy input to reach a target: $u^{\ast }(t)=B^T{e}^{A^T(T-t)}{W}_c^{-1}(T)(x_1-e^{AT}{x}_0)$, with total energy $\Vert u^{\ast }{\Vert }^2=(x_1-e^{AT}{x}_0{)}^T{W}_c^{-1}(T)(x_1-e^{AT}{x}_0)$. Eigenvalues of $W_c$ that are small indicate directions that require large control effort.

(Modal decomposition) $A=V\Lambda V^{-1}$; $(A,B)$ controllable iff every row of $V^{-1}B$ is nonzero.

In eigenbasis coordinates $z=V^{-1}x$, the system decouples: ${\dot{z}}_i={\lambda }_i{z}_i+(V^{-1}B{)}_{i}u$. Mode $i$ is controllable iff its input-channel row is nonzero. An all-zero row means the input has no direct handle on that mode, and its eigenvalue ${\lambda }_i$ is an uncontrollable pole.

Uncontrollability

(Uncontrollable modes) Eigenvalues of $A$ not appearing as closed-loop eigenvalues of $A-BK$ for any $K$.

An uncontrollable eigenvalue ${\lambda }_i$ is fixed — it always appears in the spectrum of $A-BK$, regardless of $K$. If $\mathrm{Re}({\lambda }_i)\ge 0$, the system cannot be stabilized. Stabilizability is the weaker condition that all unstable modes are controllable; a stabilizable system can be driven to the origin (but cannot have arbitrary closed-loop dynamics).

(Kalman decomposition) Any $(A,B)$ decomposes into controllable and uncontrollable parts.

Change of coordinates partitions the state into $x=(x_c,x_{\bar{c}})$ where $x_c$ is controllable and $x_{\bar{c}}$ is not, with $A=\left[\begin{array}{cc}{A}_c& A_{12}\\ 0& A_{\bar{c}}\end{array}\right]$ and $B=\left[\begin{array}{c}{B}_c\\ 0\end{array}\right]$. The uncontrollable subsystem ${\dot{x}}_{\bar{c}}=A_{\bar{c}}{x}_{\bar{c}}$ evolves autonomously; only $A_c$‘s eigenvalues can be moved by feedback.

(Practical causes of uncontrollability) Hidden symmetry, un-actuated modes, or model simplifications.

Examples: a symmetric mechanical system with inputs only at symmetric pairs of joints cannot excite asymmetric modes; a robot with one missing joint torque loses controllability of modes requiring that axis; model reduction that averages out fast modes makes them uncontrollable in the reduced model. Identifying the physical cause of uncontrollability is often more useful than re-designing the controller.

Duality with Observability

(Dual pairs) $(A,B)$ controllable iff $(A^T,B^T)$ observable.

The duality swaps inputs and outputs, states and co-states. Every theorem about controllability has a dual for observability: $\mathcal{O}$ plays the role of $\mathcal{C}$; the observability Gramian plays the role of $W_c$; PBH becomes $\mathrm{rank}[(\lambda I-A{)}^T\mid C^T]=n$. This duality underlies observer design (a feedback in the dual space) and the separation principle.

(Minimal realization) A transfer function $G(s)$ corresponds to a unique minimal (controllable and observable) state-space realization up to coordinate change.

Non-minimal realizations contain uncontrollable or unobservable modes that do not appear in $G(s)$. The minimal realization has dimension equal to the McMillan degree of $G(s)$. Pole-zero cancellation in $G(s)$ is the transfer-function signature of uncontrollable or unobservable modes.

Examples

Controllability of a Double Integrator

Is $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ controllable?

Solution $AB=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right]$.

$\mathcal{C}=[B\mid AB]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$, $\det =-1\ne 0$.

Controllable. Any desired pole placement is achievable (consistent with the example in 7.6. State Feedback).

Uncontrollable System

Is $A=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$, $B=\left[\begin{array}{c}1\\ 0\end{array}\right]$ controllable?

Solution $AB=\left[\begin{array}{c}1\\ 0\end{array}\right]$, so $\mathcal{C}=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$, rank $1<2$.

Uncontrollable. Modally, $B$ has a zero component in the eigendirection of $\lambda =2$: the $x_2$ mode has no input coupling. By the PBH test, $[\lambda I-A\mid B]=\left[\begin{array}{ccc}\lambda -1& 0& 1\\ 0& \lambda -2& 0\end{array}\right]$ at $\lambda =2$ becomes $\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 0\end{array}\right]$, rank $1$ — confirming $\lambda =2$ is the uncontrollable eigenvalue.

Because $\lambda =2>0$, this system is also unstabilizable: no $K$ can move $\lambda =2$ into the LHP.

Under-Actuated Two-Mass System

Two carts connected by a spring, with force applied only to cart $1$: ${\ddot{q}}_1=(u-k(q_1-q_2))/m$, ${\ddot{q}}_2=k(q_1-q_2)/m$. State $x=(q_1,q_2,{\dot{q}}_1,{\dot{q}}_2)$. Is this controllable?

Solution $A=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -k/m& k/m& 0& 0\\ k/m& -k/m& 0& 0\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 0\\ 1/m\\ 0\end{array}\right]$.

$AB=(1/m)(0,0,0,0{)}^T$… wait, let me redo: $AB$‘s fourth entry is $A_{4,:}\cdot B=(k/m)(0)+(-k/m)(0)+(0)(1/m)+(0)(0)=0$. The third entry is $(-k/m)(0)+(k/m)(0)+(0)(1/m)+(0)(0)=0$. First two entries come from the identity rows: $AB=(1/m,0,0,0{)}^T$.

$A^{2}B=A\cdot AB$; proceeding gives successive nonzero rows in previously-zero positions.

Working it out, $\mathcal{C}=[B\mid AB\mid A^{2}B\mid A^{3}B]$ has rank $4$: the system is controllable. Intuition: even though only cart $1$ is actuated, the spring coupling propagates the input to cart $2$ through the dynamics, and four derivatives of the input reach every state. Under-actuation does not, in itself, imply loss of controllability.

See Also

• 7.6. State Feedback — the design technique that relies on controllability
• 7.5. Stability — stabilizability is a weaker version: only unstable modes need be controllable
• 7.1. State-Space Representation — the $(A,B)$ pair whose controllability is being tested
• eigenvalues and eigenvectors — the PBH test’s modal interpretation