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ControllabilityMatrix[ssm]

给出状态空间模型 ssm 的可控性矩阵.

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ControllabilityMatrix

ControllabilityMatrix[ssm]

给出状态空间模型 ssm 的可控性矩阵.

更多信息

  • 对于具有状态方程的标准状态空间模型：
  • 连续时间系统
    离散时间系统
  • 可控性矩阵根据 计算，其中 是 的维度.
  • 对于具有状态方程的描述符状态空间模型：
  • 连续时间系统
    离散时间系统
  • 慢速和快速子系统可以以 KroneckerModelDecomposition 中描述的方法解耦：
  • 慢速子系统
    快速子系统
  • ControllabilityMatrix 根据解耦型慢速和快速子系统返回矩阵对 {q1,q2}. 矩阵 q1 和 q2 定义如下，其中 是 的维度，而 是 的幂零指数.
  • 慢速子系统
    快速子系统
  • 只有当描述符系统对于某个 λ 有 Det[λ e-a]≠0 成立时，可控性矩阵存在.

范例

打开所有单元 关闭所有单元

基本范例  (2)

状态空间模型的可控性矩阵：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{Subscript[a, 1], 0, 0}, {0, Subscript[a, 2], 0}, {0, 0, Subscript[a, 3]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}, {Subscript[b, 3]}}, {{Subscript[c, 1], Subscript[c, 2], Subscript[c, 3]}}, {{Subscript[d, 1]}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

可控性单输入状态空间模型：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{-1, 1, 0}, {0, -1, 1}, {0, 0, -1}}, {{1, 0}, {1, 0}, {0, 1}}, {{1, 0, 0}}, {{0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

范围  (5)

符号式单输入系统的可控性矩阵：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{Subscript[a, 1], Subscript[a, 2]}, {Subscript[a, 3], Subscript[a, 4]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}}, {{Subscript[c, 1], Subscript[c, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

双输入系统的可控性矩阵有两倍的列：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{Subscript[a, 1], Subscript[a, 2]}, {Subscript[a, 3], Subscript[a, 4]}}, {{Subscript[b, 1], 0}, {0, Subscript[b, 2]}}, {{Subscript[c, 1], Subscript[c, 2]}}, {{0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

不可控单输入系统的可控性矩阵：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{Subscript[λ, 1], 0, 0}, {0, Subscript[λ, 2], 0}, {0, 0, Subscript[λ, 3]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

对角多输入系统的可控性矩阵：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{Subscript[λ, 1], 0, 0}, {0, Subscript[λ, 2], 0}, {0, 0, Subscript[λ, 3]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

连续时间系统的可控性矩阵：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{1, 1, 1}, {-2, 1, 1}, {0, 1, 2}}, {{0}, {1}, {1}}, {{1, 0, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

奇异描述符系统返回两个矩阵：

Wolfram Language code: cm = ControllabilityMatrix[StateSpaceModel[{{{1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, -5}}, {{0}, {0}, {0}, {1}, {1}}, {{-1, 17, -139, 803, 4096}}, {{0}}, {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 1}}}, SamplingPeriod -> None, SystemsModelLabels -> None]];
Wolfram Language code: MatrixForm /@ cm

属性和关系  (8)

计算仅仅取决于状态和输入矩阵：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{a[1, 1], a[1, 2]}, {a[2, 1], a[2, 2]}}, {{b[1, 1]}, {b[2, 1]}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]//MatrixForm

一个系统是可控的，当且仅当它的可控性矩阵是满秩：

Wolfram Language code: ssm = StateSpaceModel[{{{1, -1, 1}, {-2, 1, 3}, {0, 2, 1}}, {{1}, {-1}, {1}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: MatrixRank[ControllabilityMatrix[ssm]] ControllableModelQ[ssm]

该系统不是可控的，但是是输出可控的：

Wolfram Language code: ssm = StateSpaceModel[{{{-3, 1, 0}, {-5, 0, 1}, {-3, 0, 0}}, {{1}, {2}, {3}}, {{1, 0, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]
Wolfram Language code: MatrixRank[ControllabilityMatrix[ssm]] ControllableModelQ[ssm]
Wolfram Language code: OutputControllableModelQ[ssm]

该系统是可控的，但是不是输出可控的：

Wolfram Language code: ssm = StateSpaceModel[{{{-2.5, -1, -0.5}, {-0.5, -2., 0.5}, {0.5, 1., -1.5}}, {{1.}, {1.}, {-3.}}, {{1, 0, -1}, {-1, 1, 0}, {3, -2, -1}}, {{0}, {0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: MatrixRank[ControllabilityMatrix[ssm]] ControllableModelQ[ssm]
Wolfram Language code: OutputControllableModelQ[ssm]

离散时间系统的可控性矩阵不依赖于采样周期：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{1, 1, 1}, {-2, 1, 1}, {0, 1, 2}}, {{0}, {1}, {1}}, {{1, 0, 0}}, {{0}}}, SamplingPeriod -> τ, SystemsModelLabels -> None]]//MatrixForm

对于描述符系统，慢速和快速系统矩阵需要是满秩的，以得到可控性：

Wolfram Language code: ssm = StateSpaceModel[{{{-1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, {{1}, {2}, {0}}, {{1, 1, 1}}, {{0}}, {{1, 0, 0}, {0, 0, 1}, {0, 0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]
Wolfram Language code: {qs, qf} = ControllabilityMatrix[ssm]
Wolfram Language code: ControllableModelQ[ssm]

慢速子系统的可控性由第一个矩阵决定：

Wolfram Language code: ControllableModelQ[{ssm, "Slow"}] MatrixRank[qs] === Length[qs]

对非奇异描述符系统，快速系统矩阵是空的：

Wolfram Language code: ControllabilityMatrix[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{e}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

每个矩阵都与来自 Kronecker 分解的子系统相关联：

Wolfram Language code: ssm = StateSpaceModel[{{{0, -1, 0, 0}, {-1, 0, 0, 0}, {10, 0, 0, -1}, {0, 1, 1, 1}}, {{0}, {0}, {0}, {-1}}, {{0, 0, 1, 0}}, {{0}}, {{-1, 0, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; kssm = Last[KroneckerModelDecomposition[ssm]]

可控性矩阵与元素系统的匹配：

Wolfram Language code: slowssm = SystemsModelExtract[kssm, All, All, {1, 2}]; fastssm = SystemsModelExtract[kssm, All, All, {3, 4}];
Wolfram Language code: Simplify[ControllabilityMatrix /@ {slowssm, fastssm}] Simplify[ControllabilityMatrix[ssm]]
Wolfram Research (2010)，ControllabilityMatrix，Wolfram 语言函数，https://reference.wolfram.com/language/ref/ControllabilityMatrix.html （更新于 2012 年）.

文本

Wolfram Research (2010)，ControllabilityMatrix，Wolfram 语言函数，https://reference.wolfram.com/language/ref/ControllabilityMatrix.html （更新于 2012 年）.

CMS

Wolfram 语言. 2010. "ControllabilityMatrix." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2012. https://reference.wolfram.com/language/ref/ControllabilityMatrix.html.

APA

Wolfram 语言. (2010). ControllabilityMatrix. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/ControllabilityMatrix.html 年

BibTeX

@misc{reference.wolfram_2026_controllabilitymatrix, author="Wolfram Research", title="{ControllabilityMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}", note=[Accessed: 13-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_controllabilitymatrix, organization={Wolfram Research}, title={ControllabilityMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/ControllabilityMatrix.html}, note=[Accessed: 13-September-2026]}