关于部门的边界和部门指数

打开实时脚本

Conic Sectors

以最简单的形式，圆锥扇区是由两条线界定的2-D区域， $y = a u$ 和 $y = b u$ 。

阴影区域的特征是不等式 $(y - a u) (y - b u) < 0$ 。更普遍，any such sector can be parameterized as:

${(\begin{array}{c} y \\ u \end{array})}^{T} Q (\begin{array}{c} y \\ u \end{array}) < 0,$

where $Q$ is a 2x2 symmetric indefinite matrix ( $Q$ 具有一个正和一个负特征值）。我们称之为 $Q$ thesector matrix。This concept generalizes to higher dimensions. In an N-dimensional space, a conic sector is a set:

$S = {z \in R^{N} : z^{T} Q z < 0},$

where $Q$ 再次是对称的无限矩阵。

Sector Bounds

部门界限是对系统行为的约束。增益限制和消极限制是部门界限的特殊情况。如果对于所有非零输入轨迹 $u (t)$ ，输出轨迹 $z (t) = (H u) (t)$ of a linear system $H (s)$ 满足：

$\int_{0}^{T} z^{T} (t) Q z (t) d t < 0, \forall T > 0,$

然后的输出轨迹 $H$ lie in the conic sector with matrix $Q$ 。Selecting different $Q$ matrices imposes different conditions on the system's response. For example, consider trajectories $y (t) = (G u) (t)$ 和the following values:

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} 0 & - I \\ - I & 0 \end{array}) 。$

这se values correspond to the sector bound:

$\int_{0}^{T} {(\begin{array}{c} y (t) \\ u (t) \end{array})}^{T} (\begin{array}{cc} 0 & - I \\ - I & 0 \end{array}) (\begin{array}{c} y (t) \\ u (t) \end{array}) d t < 0, \forall T > 0 。$

This sector bound is equivalent to the passivity condition for $G (s)$ :

$\int_{0}^{T} y^{T} (t) u (t) d t > 0, \forall T > 0 。$

换句话说，被动性是一个绑定在以下系统上的特定扇区：

$H = (\begin{array}{c} G \\ I \end{array}) 。$

Frequency-Domain Condition

Because the time-domain condition must hold for all $T > 0$ ，得出等效的频域界限需要一点谨慎，并且并非总是可能的。让以下内容：

$Q = W_{1}^{T} W_{1} - W_{2}^{T} W_{2}$

是（任何）不确定基质的分解 $Q$ 进入其正面和负面部分。什么时候 $W_{2}^{T} H (s)$ is square and minimum phase (has no unstable zeros), the time-domain condition:

$\int_{0}^{T} (H u) (t)^{T} Q (H u) (t) d t < 0, \forall T > 0$

is equivalent to the frequency-domain condition:

$H (j ω)^{H} Q H (j ω) < 0 \forall ω \in R 。$

因此，足以检查扇区不等式的实际频率。使用的分解 $Q$ ，这也等同于：

${‖ (W_{1}^{T} H) (W_{2}^{T} H)^{- 1} ‖}_{\infty} < 1 。$

Note that $W_{2}^{T} H$ is square when $Q$ 具有与输入通道一样多的负特征值 $H (s)$ 。If this condition is not met, it is no longer enough (in general) to just look at real frequencies. Note also that if $W_{2}^{T} H (s)$ is square, then it must be minimum phase for the sector bound to hold.

This frequency-domain characterization is the basis for行业平台。具体来说，行业平台plots the singular values of $(W_{1}^{T} H (j ω)) (W_{2}^{T} H (j ω))^{- 1}$ as a function of frequency. The sector bound is satisfied if and only if the largest singular value stays below 1. Moreover, the plot contains useful information about the frequency bands where the sector bound is satisfied or violated, and the degree to which it is satisfied or violated.

For instance, examine the sector plot of a 2-output, 2-input system for a particular sector.

RNG（4，'twister'）；H = RSS（3,4,2）;Q = [-5.12 2.16 -2.04 2.17 2.16 -1.22 -0.28 -1.11 -2.04 -0.28 -3.35 0.00 2.17 -1.11 0.00 0.00 0.18];sectorplot（H，Q）

Figure contains an axes object. The axes object contains 2 objects of type line. This object represents H.

这plot shows that the largest singular value of $(W_{1}^{T} H (j ω)) (W_{2}^{T} H (j ω))^{- 1}$ 超过1个低于0.5 rad/s的1，在3 rad/s的狭窄带中。所以，Hdoes not satisfy the sector bound represented byQ。

相对部门指数

We can extend the notion of relative passivity index to arbitrary sectors. Let $H (s)$ 成为LTI系统，并让：

$Q = W_{1}^{T} W_{1} - W_{2}^{T} W_{2}, W_{1}^{T} W_{2} = 0$

成为正交的分解 $Q$ 像从Schur分解中获得的正分和负分。 $Q$ 。这relative sector index $R$ ，或r-index定义为最小 $r > 0$ 因此对于所有输出轨迹 $z (t) = (H u) (t)$ :

$\int_{0}^{T} z^{T} (t) (W_{1}^{T} W_{1} - r^{2} W_{2}^{T} W_{2}) z (t) d t < 0, \forall T > 0 。$

因为增加 $r$ makes $W_{1}^{T} W_{1} - r^{2} W_{2}^{T} W_{2}$ more negative, the inequality is usually satisfied for $r$ large enough. However, there are cases when it can never be satisfied, in which case the R-index is $R = + \infty$ 。显然，原始部门的界限是满足的，并且只有 $R \leq 1$ 。

To understand the geometrical interpretation of the R-index, consider the family of cones with matrix $Q (r) = W_{1}^{T} W_{1} - r^{2} W_{2}^{T} W_{2}$ 。在2D中，圆锥角倾斜角 $θ$ 与 $r$ by

$棕褐色 (θ) = r \frac{‖ W_{2} ‖}{‖ W_{1} ‖}$

（请参见下图）。更普遍， $棕褐色 (θ)$ 与成比例 $R$ 。Thus, given a conic sector with matrix $Q$ ，一个R-指数值 $R < 1$ 意味着我们可以减少 $棕褐色 (θ)$ （缩小圆锥）的因素 $R$ 在某些输出轨迹之前 $H$ 离开圆锥部门。同样，一个值 $R > 1$ means that we must increase $棕褐色 (θ)$ （将锥体扩大）因子 $R$ 包括所有的输出轨迹 $H$ 。这显然使R-INDEX成为对响应的相对度量 $H$ fits in a particular conic sector.

In the diagram,

$d_{1} \frac{| W_{1}^{T} z |}{‖ W_{1} ‖}, d_{2} \frac{| W_{2}^{T} z |}{‖ W_{2} ‖}, R = \frac{| W_{1}^{T} z |}{| W_{2}^{T} z |},$

和

$棕褐色 (θ) = \frac{d_{1}}{d_{2}} = R \frac{‖ W_{2} ‖}{‖ W_{1} ‖} 。$

什么时候 $W_{2}^{T} H (s)$ is square and minimum phase, the R-index can also be characterized in the frequency domain as the smallest $r > 0$ 这样：

$H (j ω)^{H} (W_{1}^{T} W_{1} - r^{2} W_{2}^{T} W_{2}) H (j ω) < 0 \forall ω \in R 。$

使用基本代数，这导致：

$R = \underset{ω}{最大限度} ‖ (W_{1}^{T} H (j ω)) (W_{2}^{T} H (j ω))^{- 1} ‖ 。$

In other words, the R-index is the peak gain of the (stable) transfer function $φ (s) : = (W_{1}^{T} H (s)) (W_{2}^{T} H (s))^{- 1}$ ，以及奇异的值 $φ (j w)$ can be seen as the "principal" R-indices at each frequency. This also explains why plotting the R-index vs. frequency looks like a singular value plot (see行业平台）。相对部门指数和系统增益之间有一个完全的类比。但是请注意，此类比只有在 $W_{2}^{T} H (s)$ 是正方形和最小相。

方向扇区指数

同样，我们可以将定向被动指数的概念扩展到任意部门。给定一个具有矩阵的圆锥形扇区 $Q$ 和方向 $δ Q$ ，定向部门指数是最大的 $τ$ 因此对于所有输出轨迹 $z (t) = (H u) (t)$ :

$\int_{0}^{T} z^{T} (t) (Q + τ δ Q) z (t) d t < 0, \forall T > 0 。$

系统的定向消极指数 $G (s)$ 对应于：

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} 0 & - I \\ - I & 0 \end{array}) 。$

这directional sector index measures by how much we need to deform the sector in the direction $δ Q$ 为了使其紧密地围绕着输出轨迹 $H$ 。且仅当方向指数为正时，就满足了扇区的结合。

普通部门

有很多方法可以指定扇区界限。接下来，我们审查通常遇到表达式并提供相应的系统 $H$ 和部门矩阵 $Q$ for the standard form used bygetSectorIndex和行业平台:

$\int_{0}^{T} (H u) (t)^{T} Q (H u) (t) d t < 0, \forall T > 0 。$

For simplicity, these descriptions use the notation:

$‖ x ‖_{T} = \int_{0}^{T} ‖ x (t) ‖^{2} d t,$

并省略 $\forall T > 0$ 要求。

被动性

消极性是一个界面，与以下部门结合：

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} 0 & - I \\ - I & 0 \end{array}) 。$

Gain constraint

增益约束 ${‖ G ‖}_{\infty} < γ$ is a sector bound with:

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} I & 0 \\ 0 & - γ^{2} I \end{array}) 。$

距离之比

Consider the "interior" constraint,

$‖ y - c u ‖_{T} < r ‖ u ‖_{T}$

where $c, r$ 是标量 $y (t) = (G u) (t)$ 。这是一个与：

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} I & - c I \\ - c I & (c^{2} - r^{2}) I \end{array}) 。$

这underlying conic sector is symmetric with respect to $y = c u$ 。Similarly, the "exterior" constraint,

$‖ y - c u ‖_{T} > r ‖ u ‖_{T}$

is a sector bound with:

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} - I & c I \\ c I & (r^{2} - c^{2}) I \end{array}) 。$

双不平等

什么时候dealing with static nonlinearities, it is common to consider conic sectors of the form

$a u^{2} < y u < b u^{2},$

where $y = ϕ (u)$ is the nonlinearity output. While this relationship is not a sector bound per se, it clearly implies:

$a \int_{0}^{T} u (t)^{2} d t < \int_{0}^{T} y (t) u (t) d t < b \int_{0}^{T} u (t)^{2} d t$

沿着所有I/O轨迹以及所有 $T > 0$ 。This condition in turn is equivalent to a sector bound with:

$H (s) = (\begin{array}{c} ϕ (。) \\ 1 \end{array}), Q = (\begin{array}{cc} 1 & - (a + b) / 2 \\ - (a + b) / 2 & a b \end{array}) 。$

产品形式

Generalized sector bounds of the form:

$\int_{0}^{T} (y (t) - K_{1} u (t))^{T} (y (t) - K_{2} u (t)) d t < 0$

correspond to:

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = (\begin{array}{cc} 2 I & - (K_{2} + K_{1}^{T}) \\ - (K_{1} + K_{2}^{T}) & K_{1}^{T} K_{2} + K_{2}^{T} K_{1} \end{array}) 。$

和以前一样，静态部门约束：

$(y - K_{1} u)^{T} (y - K_{2} u) < 0$

implies the integral sector bound above.

QSR耗散

一个系统 $y = G u$ 如果满足QSR，则是QSR的。

$\int_{0}^{T} {(\begin{array}{c} y (t) \\ u (t) \end{array})}^{T} (\begin{array}{cc} Q & S \\ S^{T} & R \end{array}) (\begin{array}{c} y (t) \\ u (t) \end{array}) d t > 0, \forall T > 0 。$

这是一个与：

$H (s) = (\begin{array}{c} G (s) \\ I \end{array}), Q = - (\begin{array}{cc} Q & S \\ S^{T} & R \end{array}) 。$

参考

[1] Xia，M.，P。Gahinet，N。Abroug，C。Buhr和E. Laroche。“稳定性分析和控制设计中的部门界限。”International Journal of Robust and Nonlinear Control30，否。18（2020年12月）：7857–82。https://doi.org/10.1002/rnc.5236。

关于部门的边界和部门指数

Conic Sectors

Sector Bounds

Frequency-Domain Condition

相对部门指数

方向扇区指数

普通部门

参考

See Also

相关话题