Renie W A, Murphy E A
Am J Med Genet. 1984 Jun;18(2):295-310. doi: 10.1002/ajmg.1320180214.
The properties are explored of a cybernetic process with lag L and force of restoration equal to the size of displacement one lag unit earlier, raised to any arbitrary positive power (w greater than 1), multiplied by the restoration constant b, the sign being opposite to that of the displacement at that time. No closed-form solution is available, but a converging Taylor series expansion is presented that gives a solution of arbitrary precision. Unlike the linear system previously discussed, the amplitude of the displacement (A) is germane, and it is shown that the criterion that the effect of a perturbation be damped out is whether or not LbAw-1 exceeds a constant, kw, that depends only on w. The counterpart of this pattern of higher-order homeostasis is closer to the biological behavior of many forms of homeostasis than the scaled invariance characteristic of the linear process. kw is shown to be bounded between pi/2 and 2, and increases monotonically with w. Exactly computed values and large-sample approximations are given for the critical values below which no overshoot at all occurs. It is shown that where it does not, the minimum expected cost decreases with increasing w and reaches a minimum at infinite w. However, this evolutionary gain is vitiated where large displacements are involved, and may need to be redressed by a change in the restoration constant. Thus, the dynamic balance becomes much more subtle than with lower-order processes. Several brief clinical illustrations of these ideas are presented.
研究了一种控制论过程的性质，该过程具有滞后L，恢复力等于前一个滞后单位位移大小的任意正幂（w大于1），再乘以恢复常数b，其符号与当时位移的符号相反。没有封闭形式的解，但给出了一个收敛的泰勒级数展开式，它给出了任意精度的解。与之前讨论的线性系统不同，位移的幅度（A）是相关的，并且表明扰动的影响被衰减的判据是LbAw - 1是否超过一个仅取决于w的常数kw。这种高阶稳态模式的对应物比线性过程的尺度不变性特征更接近多种稳态形式的生物学行为。kw被证明介于π/2和2之间，并且随w单调增加。给出了完全不会出现过冲的临界值的精确计算值和大样本近似值。结果表明，在不存在过冲的情况下，最小预期成本随w的增加而降低，并在w趋于无穷大时达到最小值。然而，在涉及大位移的情况下，这种进化优势会被削弱，可能需要通过改变恢复常数来纠正。因此，动态平衡比低阶过程变得更加微妙。给出了这些观点的几个简要临床实例。
