The stabilization of the inverted pendulum-cart system (IPCS) is a classical problem in the control engineering. The study of IPCS is motivated by its applications to the balancing of rocket boosters and bipedal robots. IPCS represents a class of nonlinear, under-actuated, and unstable system hard to be controlled in real-time. In this paper, a novel nonlinear time-frequency control (NTFC) strategy is applied to stabilize an inverted pendulum mounted on a cart. The proposed controller design is adaptive and employs discrete the wavelet transform and filtered-x least-mean-square (Fx-LMS) algorithm to realize the control in real-time. Using the wavelet transform, the adaptive controller is demonstrated to inhibit the deteriorations of the time and frequency responses simultaneously before the residual oscillation is too broadband to be controlled. The presented controller consists of two adaptive finite impulse response filers that operate on the wavelet coefficients: the first one realizes the online identification and provides a priori information in real-time while the second one realizes a feed-forward control and rejects the uncontrollable input signal based on the first FIR filter. The equation of motion is derived based on the Newton's Second law of motion and the model id simulated in MATLAB for verification. A number of commonly used control methods for the stabilization of the IPCS are investigated and evaluated against the proposed NTFC strategy. The simulation results show that the proposed control strategy is feasible for balancing the IPCS for a large, tilted initial angle within a short time interval and strongly robust to external impact and perturbation in real-time.
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