To control a system we first need a clean mathematical picture of how it moves. The
state-space model is that picture: collect into one vector
x(t) everything needed to predict the system's future, and
write its evolution as a first-order
system of
differential equations driven by the control:
\dot{x} = f(x, u, t), \qquad x \in \mathbb{R}^n, \quad u \in \mathbb{R}^m.
The word state is precise: x(t) is a complete summary of
the past. Given x(t) and the future controls, the entire future is
determined — nothing about the history before t can add anything.
The linear time-invariant case
The workhorse of the whole theory is the linear time-invariant (LTI) model,
where f is linear in the state and control and the coefficient
matrices do not depend on time:
\dot{x} = A x + B u, \qquad y = C x + D u.
Here x \in \mathbb{R}^n is the state,
u \in \mathbb{R}^m the control, and
y \in \mathbb{R}^p the output — the part we
actually measure. The four matrices each carry one job, and reading
Ax as a
matrix acting on a
vector makes them concrete:
- A (size n \times n) — the
dynamics: how the state drives itself.
- B (size n \times m) — the
input map: how the control pushes the state.
- C (size p \times n) — the
output map: which combinations of the state we observe.
- D (size p \times m) — direct
feedthrough from input to output (often zero).
From a high-order ODE to first-order vector form
Most physical laws arrive as a single high-order equation. The headline example is the
spring–mass–damper: a mass m on a spring of
stiffness k with damping c, pushed by a
force F, obeying Newton's second law
m\,\ddot{q} + c\,\dot{q} + k\,q = F.
This is second order, but state space wants first order. The trick is universal:
promote each derivative up to one below the highest into its own state variable.
Step 1 — name the states. Let
x_1 = q (position) and
x_2 = \dot{q} (velocity). The state vector is
x = \begin{bmatrix} q \\ \dot{q} \end{bmatrix}, and the control is
the force, u = F.
Step 2 — differentiate the definitions. The first row is free, just the
definition of x_2:
\dot{x}_1 = \dot{q} = x_2.
Step 3 — use the physics for the last row. Solve Newton's law for the
highest derivative \ddot{q} and substitute the state names:
\dot{x}_2 = \ddot{q} = \frac{1}{m}\big(F - c\,\dot{q} - k\,q\big) = -\frac{k}{m}\,x_1 - \frac{c}{m}\,x_2 + \frac{1}{m}\,u.
Step 4 — stack the rows into matrices. The two scalar equations are exactly
one matrix equation \dot{x} = Ax + Bu:
\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \underbrace{\begin{bmatrix} 0 & 1 \\ -\dfrac{k}{m} & -\dfrac{c}{m} \end{bmatrix}}_{A} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \underbrace{\begin{bmatrix} 0 \\ \dfrac{1}{m} \end{bmatrix}}_{B} u.
If we only measure position, the output map picks off the first coordinate,
y = q = \begin{bmatrix} 1 & 0 \end{bmatrix} x, so
C = \begin{bmatrix} 1 & 0 \end{bmatrix} and
D = 0. The same recipe turns any order-n
scalar ODE into an n-state first-order system — a
companion form with 1s on the super-diagonal and the
equation's coefficients along the bottom row.
- An LTI system is \dot{x} = Ax + Bu,
y = Cx + Du, with state
x \in \mathbb{R}^n, input
u \in \mathbb{R}^m, output
y \in \mathbb{R}^p.
- The matrices have sizes A: n\times n,
B: n\times m, C: p\times n,
D: p\times m.
- Any order-n scalar ODE becomes an
n-state system by taking the function and its first
n-1 derivatives as the state.
- For m\ddot{q}+c\dot{q}+kq=F with
x=[q,\dot{q}]^{\mathsf{T}}:
A = \begin{bmatrix} 0 & 1 \\ -k/m & -c/m \end{bmatrix},
B = \begin{bmatrix} 0 \\ 1/m \end{bmatrix}.
Watch the state evolve
Set the unforced spring–mass–damper going from a stretched, still start
(q, \dot{q}) = (2, 0) with m = 1 and
u = 0, and trace its trajectory in the phase plane —
position across, velocity up. The whole motion is one curve produced by
A = \begin{bmatrix} 0 & 1 \\ -k & -c \end{bmatrix}. Add damping
c and the curve spirals inward to the origin (energy bleeding away);
stiffen the spring k and it winds faster. With
c = 0 it closes into a loop — undamped oscillation forever.
Phase space is older than control theory — Poincaré and the dynamicists drew trajectories of
planets in it a century earlier. Its modern triumph in engineering, due largely to Rudolf
Kálmán around 1960, was the realisation that recasting a tangle of high-order
input–output relations as one clean first-order vector equation
\dot{x}=Ax+Bu makes the full machinery of
eigenvalues
and matrices available. Stability, controllability, optimal feedback — every question in this
branch becomes a question about the matrix A. That single change of
viewpoint is why the rest of this stage is linear algebra.