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A new sliding control strategy for nonlinear system solved by the Lie-group differential algebraic equation method

机译:李群微分代数方程法求解的非线性系统滑模控制新策略

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For a control problem of nonlinear system x = f(x, u, t), the optimal control by minimizing a performance index is reformulated to be a set of differential algebraic equations (DAEs) with the Lagrangian being partially replaced by an exponentially time-decaying constraint: L_1 (x, t) = A_0e~(-xt), and meanwhile the control force is bounded by |u| ≤ u_(max). Then, we develop an implicit GL(n, R) Lie-group DAE (LGDAE) method to find u(t) by solving the DAEs: x = f(x, u, r) and L_1 (x, t) - A_0e~(-xt) = 0. Similarly, we propose a new sliding mode control (SMC) strategy by using the LGDAE to solve the control force, where in addition to the equivalent control force we add a compensated control force which is used to quickly steer and continuously enforce the state trajectory on the sliding surface. This novel SMC is robust and is chattering-free for regulator problem and finite-time tracking problem of nonlinear systems. Furthermore, we combine the above two methods as being a two-stage controller for the forced nonlinear Duffing oscillator by stabilizing it to an equilibrium point. The present SMC together with the LGDAE is also used to stabilize the state trajectory of some uncertain chaotic systems to a desired state point. Its robustness against uncertainty is obvious.
机译:对于非线性系统x = f(x,u,t)的控制问题,将性能指标最小化的最优控制被重新构造为一组微分代数方程组(DAE),其中拉格朗日方程被指数时间部分替换。衰减约束:L_1(x,t)= A_0e〜(-xt),同时控制力受| u |限制。 ≤u_(max)。然后,我们开发了一个隐式GL(n,R)李群DAE(LGDAE)方法,通过求解DAE来找到u(t):x = f(x,u,r)和L_1(x,t)-A_0e 〜(-xt)=0。类似地,我们通过使用LGDAE提出了一种新的滑模控制(SMC)策略来求解控制力,其中,在等效控制力之外,我们还添加了补偿控制力,该控制力用于快速引导并持续在滑动表面上执行状态轨迹。这种新颖的SMC具有鲁棒性,并且对于非线性系统的调节器问题和有限时间跟踪问题没有抖动。此外,我们将上述两种方法结合起来,通过将其稳定到一个平衡点作为强制非线性Duffing振荡器的两级控制器。本发明的SMC与LGDAE一起还用于将某些不确定混沌系统的状态轨迹稳定到期望的状态点。它对不确定性的鲁棒性是显而易见的。

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