Dynamics & Control · Lab

State-Space Lab

A continuous-time, real-valued, single-input single-output model x' = A x + B u, y = C x + D u. Eigenvalues and stability, controllability and observability, conversion to and from a transfer function, response decomposition and full-state-feedback pole placement.

Model (A, B, C, D)

States n

A matrix

A11

A12

A21

A22

B vector

C vector

Initial state x(0)

Examples

x' = -x + u, y = x. G(s) = 1/(s + 1).

Both eigenvalues at the origin; controllable and observable.

ωn = 2 rad/s, ζ = 0.1: a lightly damped complex pair.

Diagonal A with B touching only one mode: rank(C_ctr) = 1.

Diagonal A with C reading only one mode: rank(O) = 1.

An internal mode at -3 that never reaches the output.

Controllable, but the controllability matrix is badly scaled.

Share and scope

This model is fully encoded in the address bar (40 characters).

Scope: continuous-time, real, single-input single-output, up to 8 states. Discrete-time models, observers, Kalman filtering, LQR, Gramians and model reduction are not implemented. Trajectories come from the validated RK4 propagation used across the platform; no matrix exponential Φ(t) = e^(At) is displayed.

For polynomial G(s) work see the Transfer Function Lab, and for compensator design the Control Design Lab.

Spectrum and stability

Classification	asymptotically stable
Dominant Re(λ)	-1
Right half-plane eigenvalues	0
Imaginary-axis eigenvalues	0
det(sI − A)	s^2 + 3s + 2

Mode	Eigenvalue λ	Eigenvector
λ1	-1	[ 0.707, -0.707 ]
λ2	-2	[ -0.447, 0.894 ]

Eigenvalues in the s-plane

poles = × · zeros = ○ · dashed line = imaginary axis

Numeric poles and zeros
Root	Re(s)	Im(s)	Multiplicity
Pole ×	-1	0	1
Pole ×	-2	0	1

These are the eigenvalues of A — the poles of the system’s internal dynamics. Modes that are uncontrollable or unobservable still appear here, even though they are absent from the transfer function.

Continue with this system

The current model travels with the link; nothing is re-entered and nothing opens on its own.

Transfer functionSame model as the equivalent SISO G(s) = C(sI − A)⁻¹B + D, verified against these matrices · poles, response, frequency and stability analysis.Control DesignThe same converted G(s) taken as the plant · classical output-feedback controllers, which is a different problem from the full-state feedback designed here.

Only the input–output behaviour travels: unobservable or uncontrollable internal modes have no place in G(s) and stay behind in this Lab.

What this Lab models

A linear state-space model ẋ = Ax + Bu, y = Cx + Du: eigenvalues and stability, controllability and observability, the simulated state/output/input response, and pole placement by full-state feedback (Ackermann's formula) against desired closed-loop poles.