New Nonlinear Optimal Control Method Stabilizes the Tricky 4-DOF Planar Inverted Pendulum

Controlling an inverted pendulum is one of the classic benchmark challenges in nonlinear control theory, and a new study published in Autonomous Intelligent Systems extends that challenge to a genuinely difficult case: the four-degree-of-freedom planar inverted pendulum. G. Rigatos and colleagues propose a nonlinear optimal control method, built around an H-infinity feedback controller, that keeps this underactuated, strongly nonlinear mechanical system balanced and tracking desired trajectories. The work is theoretical and simulation-based, but it offers a complete pipeline from dynamic modelling through stability proof to numerical demonstration.

The planar inverted pendulum studied here is not the familiar cart-pole of classroom demonstrations. Its mechanical structure consists of a pendulum rod carrying a payload at its end, a rail along the x-axis on which the cart moves under a driving force, and two rails parallel to the y-axis along which that first rail slides under a second driving force. The result is a mechanism whose rod can rotate about two axes and whose cart can reach any point of the horizontal plane. With only two control inputs acting on an eight-dimensional state space, the system is underactuated, and its dynamics are strongly nonlinear. The authors note that it cannot be directly linearized through feedback linearization, which rules out several standard control approaches.

The first step of the method is to derive the dynamic model rigorously. Using Euler-Lagrange analysis, the authors formulate the kinetic and potential energies of the cart and payload, construct the Lagrangian, and obtain four coupled differential equations describing the cart positions and the two rod angles. Written compactly, the model takes the standard robotic form in which an inertia matrix multiplies the acceleration vector, augmented by Coriolis and gravitational force terms, all equal to the applied forces. Inverting the inertia matrix yields an affine-in-the-input state-space description with eight state variables: the two cart coordinates, the two rod angles, and their velocities.

The core computational trick of the proposed controller is sequential approximate linearization. At every sampling instance, the nonlinear model is linearized around a temporary operating point defined by the present value of the state vector and the last sampled value of the control inputs. The linearization uses a first-order Taylor series expansion, with the Jacobian matrices of the state-space description computed explicitly. Because the expansion happens around the current state rather than around points of a pre-planned trajectory, the authors argue that the span of the Taylor expansion stays small and the induced linearization error remains negligible. Whatever modelling error does arise from truncating higher-order terms is treated as a perturbation to be compensated by the robustness of the control loop.

For the approximately linearized model, the authors design an H-infinity feedback controller. The H-infinity formulation treats control as a min-max differential game: the control inputs try to minimize a quadratic cost function of the state tracking error, while model uncertainty and external disturbances try to maximize it. Solving this game yields a controller that rejects the worst-case disturbance the system might face, up to a bound set by a tunable attenuation coefficient. The feedback gains are obtained by solving an algebraic Riccati equation at each time step of the control algorithm, using the matrices of the locally linearized model. The authors report that this Riccati solution completes in a few milliseconds, shorter than the sampling period, so the method remains computationally tractable.

A key contribution of the paper is the formal stability analysis. Through Lyapunov analysis, the authors show that the H-infinity tracking performance criterion is satisfied, meaning the control loop is robust to model uncertainty and perturbations. Under moderate conditions and bounded disturbances, they further prove that the closed-loop system is globally asymptotically stable, with the tracking error converging to zero via Barbalat’s Lemma. The authors contrast this with nonlinear model predictive control, whose iterative optimum search can depend on initialization and ad-hoc parametrization, and with linear parameter varying schemes that require transforming the plant into LPV form. Compared with fuzzy multi-model approaches, the method avoids an exponential explosion of local models: with eight state variables each partitioned into five fuzzy sets, a fuzzy design would need 390,625 local models and Riccati equations, whereas the proposed method solves one Riccati equation per iteration.

To avoid measuring the entire state vector, the authors pair the controller with an H-infinity Kalman Filter as a robust state estimator. Only the four measured outputs, the cart positions and the rod angles, need to be sensed; the filter reconstructs the four velocity states. When a specific parameter of the filter is set to zero, it reduces to the standard Kalman Filter, but the generalized version retains robustness guarantees for state-estimation-based control.

The performance of the complete scheme is demonstrated through simulation experiments rather than physical hardware. Using model parameters of a 1.0 kg cart, a 0.1 kg payload, a 0.5 m rod and standard gravity, the authors ran eight sets of tracking tests, presented across twenty-four figures. In all test cases, they report fast and accurate convergence of the state variables to their setpoints under moderate variations of the control inputs. Accompanying tables quantify root-mean-square tracking errors for all eight states in disturbance-free and disturbed conditions, the estimation accuracy of the Kalman filter, and convergence times. Robustness to parametric uncertainty was also examined by allowing the rod length in the model to deviate by up to thirty percent.

The authors also discuss how tuning shapes behaviour. Smaller values of the attenuation coefficient strengthen disturbance rejection, with the smallest value still admitting a valid positive-definite Riccati solution providing maximum robustness. Larger diagonal entries in the state-weighting matrix accelerate convergence to the target setpoints. These knobs determine the transient performance of the loop, giving practitioners a systematic way to trade off speed against robustness.

Beyond the pendulum itself, the authors argue the method applies to any nonlinear system expressible in the generic state-space form, and they point to future applications in underactuated robotic manipulators, vehicles, flexible-link robots, and the postural control of humanoid robots, where balancing problems structurally resemble the planar inverted pendulum.

Subject of Research: Nonlinear optimal control and stabilization of the 4-DOF planar inverted pendulum

Article Title: Nonlinear optimal control of the planar inverted pendulum

Article References: Rigatos, G., Cuccurullo, G., Siano, P., Abbaszadeh, M., Gao, Z., & Zouari, F. (2026). Nonlinear optimal control of the planar inverted pendulum. Autonomous Intelligent Systems, 6(1), Article 11. https://doi.org/10.1007/s43684-026-00133-4

Image Credits: AI Generated

DOI: 10.1007/s43684-026-00133-4

Keywords: nonlinear control, H-infinity control, inverted pendulum, underactuated systems, Lyapunov stability, Riccati equation, Taylor series linearization, Kalman filtering, robotics, simulation, disturbance rejection, Euler-Lagrange model