Linear controllability

November 10, 2014

Controllability notion refers to state transfer in the state-space. In a linear system, this

property assures arbitrary pole-placement using linear state-feedback control law. We begin

this brief write-up with the definition of controllability for a linear system. 1

Definition 0.1 Let

x = Ax+ bu (1)

be a SISO system, where x ∈ IRn. The pair (A, b) is completely controllable if and only if for

each x1, x2 ∈ IRn and for each T ≥ 0, there exists an admissible input u : [0, T ] −→ IR with

the property that if x(t) is the solution of (1) with x(0) = x1, then x(T ) = x2.

We next present the result by Kalman, Ho and Narendra (1963) that provides a method to

verify the linear controllability property for a given pair (A, b).

Theorem 0.1 The continuous-time system (1) is completely controllable if and only if the

controllability matrix

C(A, b)△

= [b Ab · · ·An−1b]

has rank n.

Proof: For t > 0 define P (t) =∫t

0eAτbb⊤eA

⊤τdτ , P (t) is termed the controllability gram-

mian. We first show that C(A, b) is invertible ⇐⇒ P (t) is positive definite for all t > 0. Since

P (t) is positive semi-definite (x⊤P (t)x =∫t

0x⊤eAτbb⊤eA

⊤τxdτ =∫

t

0(x⊤eAτb)(x⊤eAτb)⊤dτ =∫

t

0|x⊤eAτb|2dτ ≥ 0) we show that C(A, b) is invertible ⇐⇒ P (t) is invertible for all t > 0.

Let C(A, b) be not invertible. Then there exists a non-zero vector v ∈ IRn such that

v⊤C(A, b) = 0 =⇒ v⊤[b Ab · · ·An−1b] = 0. Now by Cayley-Hamilton theorem

An = α1I + α2A+ · · ·+ αnAn−1

1Note that the notion of controllability is not tied to the nature of the system (linear/nonlinear).

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where, the α′

is are not identically equal to zero. On pre and post multiplying the above by

v⊤ and b respectively, we have

v⊤Anb = α1v⊤b+ α2v

⊤Ab+ · · ·+ αnv⊤An−1b

or v⊤Akb = 0 ∀ k ≥ 0 which can be rewritten as z(t)△

= v⊤eAtb = 0 ∀ t ≥ 0. Thus

v⊤eAtbb⊤eA⊤tv = 0 ∀ t ∈ [0, T ] or

∫t

0v⊤(eAτbb⊤eA

⊤τ )vdτ = 0, thereby proving P (t) is not

invertible.

Conversely, let P (t) be not invertible. Then there exists a non-zero vector p ∈ IRn such

that

p⊤eAtb = 0, t ∈ [0, T ]. (2)

Differentiating (2) with respect to time t, (n − 1) times and then evaluating at t = 0, we

get p⊤Ab = 0, p⊤A2b = 0, . . . , p⊤An−1b = 0. Rearranging, we have p⊤[b Ab · · ·An−1b] = 0.

Hence the claim that C(A, b) is not invertible. It now follows that C(A, b) is invertible if

and only if P (t) is invertible.

The controllability property is exhibited by constructing an u which steers x1 to x2. One

such control is

u(t) = −b⊤eA⊤(T−t)P (T )−1(eATx1 − x2)

that results in the solution of (1) satisfying x(T ) = x2.

Finally, suppose that C(A, b) is not invertible, which implies that P (t) is not invertible.

Thus there exists a non-zero vector q ∈ IRn such that q⊤eAtb = 0 for t ∈ [0, T ]. Let

x1△

= e−AT q, then

x(T ) = eATx1 +

∫T

0

eA(T−τ)bu(τ)dτ

q⊤x(T ) = q⊤q −

∫T

0

q⊤eA(T−τ)buτ)dτ.

Since the second term in the above is zero, we have q⊤(x(T )− q) = 0 which further implies

that x(t) cannot be steered to the origin in time T .

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