ABSTRACT This paper considers the well-known problem of deriving a linear model of dynamics of periodically switched circuits w.r.t. small perturbations in duty cycle (or switching instances) as external control inputs. A rigorous approach to this problem is developed and is shown that the linearized model is shift invariant and discrete time in nature. This is at variance with the well-known model, which is linear time invariant continuous time referred as state space averaging (SSA) model. SSA model ignores commutativity conditions in matrices of state space model due to varying parameters over intervals as well as the discrete nature of control input. The proposed method of linearization considers the problem of linearization in a neighbourhood of a periodic solution. The monodromy matrix for state transition over all phases of switching is considered to account for non-commuting matrices of parameters. Similarly discrete nature of the input changing once in every period of switching leads to the discrete model. This methodology is applicable for multiple independently switched circuits and takes into account orders of switching once the nominal periodic solution over which linearization is sought is fixed. This paper gives the detailed theory as well as illustrative examples to prove the usefulness of the proposed methodology.

中文翻译：

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周期性开关电路相对于占空比扰动的严格离散时间线性化

摘要 本文考虑了众所周知的问题，即在占空比（或开关实例）中的小扰动作为外部控制输入的情况下，导出周期性开关电路的动态线性模型。开发了解决这个问题的严格方法，并表明线性化模型本质上是平移不变和离散时间的。这与众所周知的模型不同，后者是线性时不变连续时间，称为状态空间平均 (SSA) 模型。SSA 模型忽略了状态空间模型矩阵中的交换条件，因为参数在区间内变化以及控制输入的离散性质。所提出的线性化方法考虑了周期解邻域中的线性化问题。切换所有阶段的状态转换的单一矩阵被认为是考虑参数的非交换矩阵。类似地，输入的离散性质在每个切换周期中改变一次，导致离散模型。这种方法适用于多个独立切换的电路，并在寻求线性化的标称周期解固定后考虑切换顺序。本文给出了详细的理论以及说明性的例子来证明所提出的方法的有效性。这种方法适用于多个独立切换的电路，并在寻求线性化的标称周期解固定后考虑切换顺序。本文给出了详细的理论以及说明性的例子来证明所提出的方法的有效性。这种方法适用于多个独立切换的电路，并在寻求线性化的标称周期解固定后考虑切换顺序。本文给出了详细的理论以及说明性的例子来证明所提出的方法的有效性。