State-Space Control Lab

Plant

p = f(θ₁, θ₂, θ₃) ṗ = J(θ)·θ̇ θ̇ = J⁻¹·ṗ

Joint shown in the lab panels below

Each joint has a 12 V motor with a 20 : 1 gearbox and its own state-feedback loop. The ζ and ωₙ sliders below set the poles of all three joints.

Controller u = −Kx + N·r

You can also drag in the s-plane to move the dominant pole pair. K comes from Ackermann's formula.

Estimate the full state with an observer (uses sensor y₁ only)

The controller then uses x̂ instead of x: u = −K·x̂ + N·r. Observer poles = controller poles × this factor. With the observer off, the page assumes every state is measured perfectly.

Signals & initial state + w(t)

Extra input w(t)Amp. Aω rad/s

u = −Kx + N·r + w(t). With the controller off and r = 0, the plant sees u = w(t) only.

Initial state x₀

or click the state plane

Auto time horizon (from the slowest pole)

Show open-loop response (dashed)

Ping-pong robot · 3 joints, 3-axis target

Scenarios:

Gatekeeper · the arm moves only if every workspace and Jacobian check passes

Where the paddle can go on the hitting plane · click to aim

Joint responses

Each joint follows its inverse-kinematics angle through its own loop u = −K·x

Modes · PBH test on each eigenvalue of A

s-plane

State plane

Time response

Solution split

x(t) = e^(A(t−t₀))·x₀ + ∫ e^(A(t−τ))·B·u(τ) dτ

Show state

Integrating factor k(t)

Scalar case ẋ = a·x + b·u, Lec 4

k̇(t) = −a·k(t)
k(t) = k(t₀)·e^(−a(t−t₀))
x(t) = [k(t₀)/k(t)]·x(t₀) = e^(a(t−t₀))·x(t₀)

Set a from a mode of the current A (a = Re λ):

State-Space Control Lab

Ping-pong robot · 3 joints, 3-axis target

Gatekeeper · the arm moves only if every workspace and Jacobian check passes

Where the paddle can go on the hitting plane · click to aim

Joint responses

Arm joint

How the loop runs, every 1 ms on the microcontroller

Pick a design · step of r, full-state feedback

Modes · PBH test on each eigenvalue of A

s-plane

State plane

Time response

Solution split

Integrating factor k(t)

Matrices

Check it in MATLAB