Abstract We consider a class of nonlinear optimal control problems in which the aim is to minimize control variation subject to an upper bound on the system cost. This idea of sacrificing some cost in exchange for less control volatility—thereby making the control signal easier and safer to implement—is explored in only a handful of papers in the literature, and then mainly for piecewise-constant (discontinuous) controls. Here we consider the case of smooth continuously differentiable controls, which are more suitable in some applications, including robotics and motion control. In general, the control signal’s total variation—the objective to be minimized in the optimal control problem—cannot be expressed in closed form. Thus, we introduce a smooth piecewise-quadratic discretization scheme and derive an analytical expression, which turns out to be rational and non-smooth, for computing the total variation of the approximate piecewise-quadratic control. This leads to a non-smooth dynamic optimization problem in which the decision variables are the knot points and shape parameters for the approximate control. We then prove that this non-smooth problem can be transformed into an equivalent smooth problem, which is readily solvable using gradient-based numerical optimization techniques. The paper includes a numerical example to verify the proposed approach.

中文翻译：

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最小化具有平滑分段二次输入信号的非线性系统的控制波动

摘要 我们考虑一类非线性最优控制问题，其目的是在系统成本的上限下最小化控制变化。这种牺牲一些成本以换取较少控制波动的想法——从而使控制信号更容易和更安全地实施——在文献中只有少数论文进行了探讨，然后主要用于分段恒定（不连续）控制。在这里，我们考虑平滑连续可微控制的情况，它更适合某些应用，包括机器人和运动控制。一般来说，控制信号的总变化——在最优控制问题中要最小化的目标——不能用封闭形式表达。因此，我们引入了平滑的分段二次离散化方案并推导出解析表达式，结果证明它是有理的和非平滑的，用于计算近似分段二次控制的总变化。这导致了一个非平滑动态优化问题，其中决策变量是用于近似控制的结点和形状参数。然后我们证明，这个非光滑问题可以转化为等效的光滑问题，使用基于梯度的数值优化技术可以很容易地解决这个问题。本文包括一个数值例子来验证所提出的方法。然后我们证明这个非光滑问题可以转化为等效的光滑问题，使用基于梯度的数值优化技术可以很容易地解决这个问题。本文包括一个数值例子来验证所提出的方法。然后我们证明这个非光滑问题可以转化为等效的光滑问题，使用基于梯度的数值优化技术可以很容易地解决这个问题。本文包括一个数值例子来验证所提出的方法。