Sustained oscillations in MIMO nonlinear systems through limit cycle shaping

Abstract

Summary This paper studies on generating periodic behaviors through shaping stable limit cycles in multiple‐input‐multiple‐output nonlinear systems. For this purpose, first, limit cycles are shaped with respect to the desired sustained oscillations of the system's outputs. Then, the Lyapunov analyses, which are appropriate for stability analysis of invariant sets, are employed to design the control law and conclude the asymptotic convergence toward the predefined limit cycles. The problem is studied in two cases. In the first case, some assumptions in the mathematical model are assumed that leads to simplification in the design procedure. In the second case, the design procedure is discussed in more general cases. Finally, the validity and performance of the proposed method for shaping limit cycles with different geometric shapes are illustrated by computer simulations for practical and numerical examples.

Robust Stable Limit Cycle Generation in Multi-Input Mechanical Systems

“…The above relation assures that all trajectories of the system starting from the sliding surface cannot leave it. 37 Therefore, s(x) = 0 ∀t ≥ 0 which means that the perturbed controlled system (8) follows the controlled nominal system, and therefore the phase trajectories asymptotically converge to the desired limit cycle E. Remark 1. The proposed controller may lead to a chattering problem due to discontinuity in the relation (24).…”

“…Theorem 4. Consider the system (8). The controller u = u nom + u ISM , where u nom is as given in equation ( 14) and u ISM is given in (24), guarantees robust asymptotic convergence to the limit cycle E despite the uncertain term δ(x, t):…”

“…[1][2][3][4] For example, in a walking robot, the foot motion follows a repetitive pattern. This behavior can be described by an attractive limit cycle in the phase trajectories of the dynamical system [5][6][7][8] The limit cycle is an isolated periodic orbit in the phase plane, which is a very rich dynamical behavior in the nonlinear dynamical systems. The authors of 9,10 have shown that walking and running can also be addressed as periodic motions, which are equal to specified limit cycles in the phase trajectories of the robotic system.…”

“…15 In this approach, an appropriate limit cycle is selected according to the amplitude and frequency of the desired output oscillations. 8 Then, the Lyapunov function is shaped according to the desired limit cycle equation and limit cycle controller is designed for shaping this stable limit cycle in the phase trajectories of the controlled system. Backstepping control, 16 synchronization approach, 17 feedback linearization, 18 and state feedback control 5,19 are some of the designing methods employed by the second approach to generate stable oscillations in dynamical systems.…”

Filtering-Based Parameter Identification Methods for Multivariable Stochastic Systems

“…System identification is the theory and methods of establishing the mathematical models of dynamical systems [1][2][3][4][5] and some identification approaches have been proposed for scalar systems and multivariable systems [6][7][8][9][10][11]. Multivariable systems exist more widely in modern large-scale industrial processes, multivariable systems can more accurately describe the characteristics of dynamic processes, and have extensive application prospects to study the identification methods of multivariable systems [12][13][14]. The identification methods of multivariable systems can be regarded as an extension of those of scalar systems [15,16].…”