Preamble

This study investigates the dynamics of a nonlinear system described by the differential equation:
$$x'' + f(x) = u(x, x')$$
where the control function is defined as $u(x, x') = -u_0 \text{sign}(x' - \phi(x))$ and the switching surface is given by $\phi(x) = \alpha + \phi_0(x)$. We assume $u_0 > 0$ and $\alpha > 0$. The function $f(x)$ is periodic such that $f(x) = -f(-x)$ and satisfies $f(0) = f(-\pi) = 0$. Furthermore, we assume $\phi_0(0) = \phi_0(-\pi) = 0$, with $f(x) > 0$ and $x \phi_0(x) < 0$ for $0 < |x| < \pi$. The control magnitude $u_0$ is assumed to be sufficiently large such that $u_0 > \max | \Phi(x) |$, where $\Phi(x) = f(x) + [\alpha + \phi_0'(x)]\phi(x)$.

The system can be rewritten as a first-order system:
$$\begin{aligned} \dot{x} &= y \ \dot{y} &= -\alpha y - f(x) - u_0 \text{sign}(y - \phi(x)) \end{aligned}$$
This formulation allows for the analysis of the phase portrait in the $(x, y)$ plane. We consider the behavior of trajectories relative to the switching line $y = \phi(x)$. Following the methods established in \cite{1, 2}, we examine the existence and stability of equilibrium points and limit cycles. For the specific case where $f(x) = \sin x$, the system behavior is characterized by the interaction between the restoring force and the discontinuous control law.

Phase Space Analysis and Sliding Modes

The switching line $y = \phi(x)$ divides the phase plane into two regions. In the region where $y > \phi(x)$, the system follows $\dot{y} = -\alpha y - f(x) - u_0$, while for $y < \phi(x)$, it follows $\dot{y} = -\alpha y - f(x) + u_0$. A sliding mode exists on the segment of the switching line where the vector fields on both sides point toward the line. This condition is satisfied when:
$$\lim_{y \to \phi(x)^+} \dot{R} < 0 \quad \text{and} \quad \lim_{y \to \phi(x)^-} \dot{R} > 0$$
where $R(x, y) = y - \phi(x)$. Substituting the system equations, the sliding mode condition reduces to $u_0 > |\Phi(x)|$. Within the sliding domain, the motion is governed by the reduced-order equation $\dot{x} = \phi(x)$.

As shown in [FIGURE:1], the phase trajectories demonstrate that for $0 < \alpha < -\min \phi_0(x)$, there exist stable equilibrium points. If the damping coefficient $\alpha$ exceeds this threshold, the qualitative nature of the phase portrait changes, potentially leading to the disappearance of certain singular points. We utilize the comparison method \cite{3, 4, 5} to bound the trajectories. Specifically, we define comparison functions $Y(x)$ such that $Y(x) > \phi(x)$, allowing us to prove the convergence of trajectories to the sliding manifold or a specific limit cycle.

Stability and Equilibrium States

The equilibrium states of the system are located at the intersections of the nullclines. For the autonomous case, the points $(x_0, 0)$ are analyzed. Under the condition $u_0 > \max { \max f(x), \max \Phi(x) }$, the system exhibits robust stability. As illustrated in [FIGURE:2], the trajectories starting from an initial state $(x_0, y_0)$ eventually enter the sliding regime on $y = \phi(x)$ and converge to the origin or a periodic orbit.

In the interval $-\pi < x < \pi$, the behavior near the points $(x_1, 0)$ and $(x_2, 0)$ is critical. For $x = x_0 > 0$, the system trajectories are directed towards the switching line. If $\alpha > -\min \phi_0(x)$, the sliding motion is guaranteed to be stable. [FIGURE:3] and [FIGURE:4] provide a detailed visualization of the phase trajectories under varying parameters of $u_0$ and $\alpha$. The global structure of the phase space is determined by the synthesis of these local behaviors, ensuring that for a wide range of initial conditions, the system tracks the desired sliding surface $\phi(x)$.

Conclusion

The analysis demonstrates that the discontinuous control law effectively forces the system state onto the surface $y = \phi(x)$. The existence of a stable sliding mode is contingent upon the control gain $u_0$ overcoming the effective nonlinearity $\Phi(x)$. The results presented in [FIGURE:5] and [FIGURE:6] confirm that the system achieves the desired regulation or tracking performance, even in the presence of nonlinear restoring forces $f(x)$. This approach remains valid for various functional forms of $\phi_0(x)$, provided the structural conditions outlined in the preamble are maintained.

References

1. Barbashin, E. A., Introduction to the Theory of Stability, Nauka, 1967.
2. Tabueva, V. A., On a nonlinear differential equation of the second order, Prikl. Mat. Mekh., Vol. 23, No. 5, pp. 826–835, 1959.
3. Barbashin, E. A., Tabueva, V. A., On the stability of the solution of a third-order differential equation, Prikl. Mat. Mekh., Vol. 24, No. 5, 1960.
4. Barbashin, E. A., On the construction of Lyapunov functions, Differentsial'nye Uravneniya, Vol. 4, No. 2, pp. 377–390, 1963.
5. Eidinov, R. M., On the control of certain nonlinear systems, Prikl. Mat. Mekh., Vol. 24, No. 1, pp. 882–889, 1963.
6. Barbashin, E. A., Methods of constructing Lyapunov functions, 1967.