Nonlinear Control of a Two-Wheeled Inverted Pendulum (Segway)

Modeling, nonlinear controller design, Lyapunov stability analysis, and MATLAB simulation for a two-wheeled inverted pendulum (TWIP / Segway) — an underactuated, inherently unstable nonlinear system. Two controllers, Sliding Mode Control (SMC) and Feedback Linearization (FL), are derived, proven stable, and compared on settling time, overshoot, and disturbance rejection.

MEEN 655 — Design of Nonlinear Control Systems · Texas A&M University

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🎯 Problem

A Segway is a wheeled inverted pendulum (WIP): a body balanced on a wheeled base that must actively stabilize its upright tilt. It is:

• Nonlinear — tilt/translation coupling, gravity, trigonometric terms;
• Underactuated — fewer independent actuators than degrees of freedom;
• Open-loop unstable — the upright equilibrium ($\theta = 0$) diverges under the slightest perturbation.

Open-loop response — with no control, tilt and position blow up.

🧮 Modeling

Using a Lagrangian formulation with state $\mathbf{x} = [x,\ \dot{x},\ \theta,\ \dot{\theta}]^\top$ (cart position, velocity, tilt, tilt rate), the equations of motion are:

$$\left(2m + M + \tfrac{2J_w}{R^2}\right)\ddot{x} + Ml\ddot{\theta}\cos\theta - Ml\dot{\theta}^2\sin\theta = \tfrac{\tau_L+\tau_R}{R}$$

$$Ml\ddot{x}\cos\theta + (J_b + Ml^2)\ddot{\theta} - Ml\dot{x}\dot{\theta}\sin\theta + Mgl\sin\theta = 0$$

Jacobian linearization reveals two equilibria: the upright position ($\theta=0$) is an unstable saddle, while the hanging position ($\theta=\pi$) is stable — confirming that active control is required to balance.

🛠️ Controllers

1 · Sliding Mode Control (SMC)

Sliding surface and reaching law:

$$s = \dot{\theta} + \lambda\theta, \qquad \dot{s} = -\eta,\mathrm{sign}(s),\quad \eta>0$$

$$\tau = \frac{-f(x) - \lambda\dot{\theta} - \eta,\mathrm{sat}(s/\phi)}{g(x)}$$

A saturation function with boundary layer $\phi$ replaces $\mathrm{sign}(s)$ to suppress chattering.

2 · Feedback Linearization (FL)

Output $y=\theta$ has relative degree 2; since the system is 4th order with full relative degree, it is input-state linearizable. With virtual control $v = -k_1\theta - k_2\dot{\theta}$:

$$u = \frac{1}{k(x)}\big[-k_1\theta - k_2\dot{\theta} - h(x)\big] ;\Rightarrow; \ddot{\theta} = v$$

📐 Lyapunov Stability

Controller	Lyapunov function	Derivative	Result
SMC	$V = \tfrac{1}{2}s^2$	$\dot{V} = s\dot{s} = -\eta\lvert s\rvert \le 0$	Asymptotically stable; $s\to 0 \Rightarrow \theta\to 0$
FL	$V = \tfrac{1}{2}\theta^2 + \tfrac{1}{2}\dot{\theta}^2$	$\dot{V} = \dot{\theta}\big((1-k_2)\theta - k_1\dot{\theta}\big)$	Stable for $k_2>1,\ k_1>0$

📊 Results

SMC — tilt driven to upright	Feedback Linearization — smooth convergence
SMC rejects a 20 N·m disturbance spike at t = 3 s	SMC vs FL — head-to-head

Controller comparison

Feature	Sliding Mode Control	Feedback Linearization
Settling time	Often faster	Slightly longer
Overshoot	Lower (sliding behavior)	Slightly higher
Disturbance rejection	Very strong (inherent robustness)	Weaker

Conclusion: both stabilize the upright equilibrium, but SMC outperforms Feedback Linearization, primarily due to its inherent robustness to disturbances. Note the underactuation: tilt is stabilized while the wheel position $x$ drifts (see 06-position-drift-underactuation.png).

💻 Code

All simulations are pure MATLAB (ode45), no toolboxes required.

File	Purpose	Generates
TWIP_OpenLoop.m	Uncontrolled dynamics (motivation)	Open-loop instability
segway_smc_main.m + segway_dynamics.m	Sliding Mode Controller w/ boundary-layer saturation	SMC stabilization
TWIP_FL.m	Feedback Linearization controller	FL stabilization
ProjectCompare.m	SMC vs FL + disturbance-rejection test	Comparison plots

Run it

cd matlab
TWIP_OpenLoop      % see why control is needed
segway_smc_main    % sliding mode stabilization
TWIP_FL            % feedback linearization
ProjectCompare     % compare both + disturbance rejection

📄 Report

The full IEEE-format report — derivations, Lyapunov proofs, and discussion — is in docs/Segway_Nonlinear_Control_Report.pdf.

📜 License

Code released under the MIT License. The report is © 2026 Ishan Sharma.

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Authored by Ishan Sharma · M.S. Electrical Engineering, Texas A&M University · Control Systems & Robotics.