Background

The inverted pendulum is a classic control problem: the upright equilibrium is open-loop unstable, so keeping the rod balanced requires continuous feedback. This project was done as part of a two-part lab sequence where the goal was to simultaneously control the cart position x and pendulum angle θ using only the motor voltage (a SIMO system).

Because the lab focuses on small deviations about upright, the plant is linearized using the small-angle approximation (sinθ ≈ θ, cosθ ≈ 1).

System overview

• Measured outputs: cart position x and pendulum angle θ (encoders).
• Control input: motor voltage V (force on cart through motor dynamics).
• State definition: \(\mathbf{x} = \begin{bmatrix} x & \dot{x} & \theta & \dot{\theta} \end{bmatrix}^T\)

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Modeling (high level)

Under the small-angle approximation, the linearized equations of motion can be written as: \((M+m)\ddot{x} + mL_p\ddot{\theta} = F_a\) \(mL_p\ddot{x} + \frac{4mL_p^2}{3}\ddot{\theta} - mgL_p\theta = 0\)

These dynamics are combined with the motor dynamics (force as a function of motor voltage and cart motion) to produce a state-space model (A, B, C, D) with outputs y = [x, θ]^T.

We can notice that the upright configuration is unstable in open loop (one eigenvalue lies in the right half-plane), which motivates feedback control.

Lab 6a — Pole placement with full-state feedback

Goal

Design and implement a full-state feedback controller that stabilizes the upright equilibrium and enables cart position tracking (including sinusoidal references).

Control law (concept)

For design, we assume the full state is available: \(u = -K\,\mathbf{x}\)

The gain K = [k1 k2 k3 k4] is chosen so the closed-loop eigenvalues of (A − BK) match desired pole locations . In our implementation, we solved for K in MATLAB and verified it via pole placement.

Reference tracking structure

To track position commands, we used: \(u = K(r - \mathbf{x})\) so that the reference r has the same dimension as the state.

For experiments, we used a sinusoidal cart position reference of the form: \(r = [M\sin(\omega t),\ 0,\ 0,\ 0]^T\)

Observations/Limitations

We noticed that numerical differentiation amplified noise, producing spiky, low-quality velocity estimates. This noise propagated into the control input and could manifest as jittery motion and audible high-frequency behavior. We also saw persistent small oscillations about equilibrium due to modeling linearization, sensor noise, and derivative-based state estimates.

Lab 6b — Upgraded design with a Luenberger observer

We improved the controller by replacing derivative blocks with a state estimator, improving velocity estimates and control smoothness—especially in the presence of measurement noise.

Observer equations (concept)

A Luenberger observer has the form: \(\dot{\hat{\mathbf{x}}} = A\hat{\mathbf{x}} + Bu + L(y - \hat{y})\) where y = Cx and ŷ = Cx̂.

Intuitively:

• Predictor: A x̂ + B u rolls the model forward
• Corrector: L(y − ŷ) pulls the estimate toward the measured outputs

The controller then uses the estimated state: \(u(t) = K\big(r(t) - \hat{\mathbf{x}}(t)\big)\)

Choosing observer poles (design idea)

The observer gain L is selected so that (A − LC) is stable, and typically faster than the closed-loop plant so that the estimate converges quickly. We used MATLAB pole placement to compute L for a desired set of observer poles (provided by the lab).

Qualitatively, the observer-based controller produced smoother motion and reduced the “step-like” jitter observed with derivative-based state estimation, especially during sinusoidal tracking and small perturbations.

We also compared our controllers under injected sensor noise and evaluated performance using metrics like:

• Position deviation (e.g., standard deviation of cart position about its reference)
• High-frequency noise power (relative spectral power above a chosen cutoff)

We noticed that faster observer poles improve responsiveness but can pass more measurement noise into the estimate, while slower poles smooth noise but can lag the true state.