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LyapunovSolve[a,c]

行列リャプノフ(Lyapunov)方程式 の解 を求める．

LyapunovSolve[a,b,c]

を解く．

LyapunovSolve[{a,d},c]

を解く．

LyapunovSolve[{a,d},{b,e},c]

を解く．

詳細
Details and Options Details and Options
例題  
例  
スコープ  
アプリケーション  
特性と関係  
関連項目
関連するガイド
履歴
引用書式

LyapunovSolve

LyapunovSolve[a,c]

行列リャプノフ(Lyapunov)方程式 の解 を求める．

LyapunovSolve[a,b,c]

を解く．

LyapunovSolve[{a,d},c]

を解く．

LyapunovSolve[{a,d},{b,e},c]

を解く．

詳細

  • LyapunovSolveは連続時間のリャプノフとシルベスター(Sylvester)の方程式を解く．
  • LyapunovSolve数値行列と記号行列の両方に使うことができる．

例題

すべて開く すべて閉じる

例  (1)

リャプノフ方程式 を解く：

Wolfram Language code: a = (⁠| | | | - | - | | 1 | 3 | | 2 | 1 |⁠);c = (⁠| | | | -- | -- | | -2 | -1 | | -1 | -4 |⁠);
Wolfram Language code: LyapunovSolve[a, c]

スコープ  (7)

リャプノフ方程式を解く：

Wolfram Language code: {a, c} = {(| | | | -- | -- | | -3 | 2 | | -1 | -1 |), (| | | | - | - | | 1 | 0 | | 0 | 1 |)}; LyapunovSolve[a, c]//MatrixForm

解を証明する：

Wolfram Language code: a.% + %.a - c

を解く：

Wolfram Language code: LyapunovSolve[(| | | | | -- | -- | - | | -1 | 1 | 0 | | -1 | -1 | 1 | | 1 | 1 | 0 |), (| | | | | -- | -- | -- | | -2 | 0 | 0 | | 0 | -1 | 0 | | 0 | 0 | -3 |), -(| | | | | -- | -- | -- | | c1 | 0 | 0 | | 0 | c2 | 0 | | 0 | 0 | c3 |)]

次元が異なる係数行列について を解く：

Wolfram Language code: N@LyapunovSolve[a = (| | | | | | - | - | - | - | | 1 | 2 | 3 | 4 | | 4 | 5 | 6 | 7 | | 7 | 8 | 9 | 1 | | 0 | 0 | 0 | 0 |), b = (| | | | | - | -- | - | | 1 | -1 | 0 | | 1 | 1 | 0 | | 0 | 0 | 2 |), c = (| | | | | -- | -- | -- | | 12 | 10 | 12 | | 24 | 22 | 24 | | 27 | 25 | 27 | | 12 | 10 | 12 |)]//MatrixForm

を解く：

Wolfram Language code: LyapunovSolve[{a = (| | | | | ------ | ------ | ------ | | -0.49 | -0.371 | -0.193 | | 0.967 | 0.87 | 0.134 | | -0.464 | -0.86 | 0.204 |), d = (| | | | | ------ | ------- | ------ | | 1.853 | -0.0728 | 0.5857 | | 0.922 | 1.775 | -1.583 | | -0.185 | -1.237 | -0.069 |)}, c = (| | | | | ----- | ------ | ------ | | 0.014 | -0.259 | -0.117 | | 0.418 | -0.202 | 0.484 | | -0.2 | 0.44 | -0.074 |)]//MatrixForm

を解く：

Wolfram Language code: LyapunovSolve[{a = (| | | | ----- | ----- | | 1.543 | 0.566 | | 1.795 | 1.361 |), d = (| | | | ----- | ----- | | 0.225 | 0.093 | | 0.167 | 0.336 |)}, {b = (| | | | ----- | ----- | | 0.432 | 0.173 | | 0.579 | 1.221 |), e = (| | | | ----- | ----- | | 2.561 | 2.719 | | 1 | 1.47 |)}, c = (| | | | ----- | ----- | | 1.495 | 1.445 | | 1.188 | 1.343 |)]//MatrixForm

記号係数を持つリャプノフ方程式 を解く：

Wolfram Language code: LyapunovSolve[(⁠a⁠), (⁠c⁠)]

の記号解を得る：

Wolfram Language code: LyapunovSolve[{(⁠a⁠), (⁠d⁠)}, {(⁠b⁠), (⁠e⁠)}, (⁠c⁠)]

アプリケーション  (7)

の解が負定値 について正定値であるかどうかを確かめることで の安定性を検証する：

Wolfram Language code: a = (⁠| | | | | ------- | ------- | ------- | | -2.03 | -1.197 | -0.9514 | | -0.1513 | -1.98 | 0.186 | | 0.06049 | -0.6359 | -2.63 |⁠);
Wolfram Language code: LyapunovSolve[a, -(⁠| | | | | - | - | - | | 1 | 0 | 0 | | 0 | 2 | 0 | | 0 | 0 | 3 |⁠)]//PositiveDefiniteMatrixQ

予想通り，固有値は左半平面にある：

Wolfram Language code: Eigenvalues[a]

不安定な系：

Wolfram Language code: a = (⁠| | | | | ------- | ------- | ------- | | 1.29 | -0.2947 | 0.573 | | -0.2275 | 1.036 | -0.3158 | | 0.3037 | -0.5616 | 1.714 |⁠);
Wolfram Language code: LyapunovSolve[a, -(⁠| | | | | - | - | - | | 1 | 0 | 0 | | 0 | 2 | 0 | | 0 | 0 | 3 |⁠)]//PositiveDefiniteMatrixQ
Wolfram Language code: Eigenvalues[a]

安定した連続時間系の可制御性グラミアンを計算する：

Wolfram Language code: {a, b} = {(⁠| | | | -- | -- | | -1 | 1 | | 1 | -2 |⁠), (⁠| | | - | | 1 | | 0 |⁠)};
Wolfram Language code: LyapunovSolve[a, -b.b]
Wolfram Language code: ControllabilityGramian[StateSpaceModel[{a, b}]]

安定した連続時間系の可観測性グラミアンを計算する：

Wolfram Language code: {a, c} = {(⁠| | | | ------- | ------- | | -(7/40) | (3/40) | | (3/40) | -(7/40) |⁠), (⁠1 1⁠)};
Wolfram Language code: LyapunovSolve[a, -c.c]
Wolfram Language code: ObservabilityGramian[StateSpaceModel[{a, {{Subscript[b, 1]}, {Subscript[b, 2]}}, c}]]

漸近的に安定した連続時間系の ノルムを計算する：

Wolfram Language code: {a, b, c} = {(⁠| | | | | -- | -- | -- | | -1 | 2 | 3 | | 0 | -2 | 2 | | 0 | 0 | -3 |⁠), (⁠| | | | - | - | | 1 | 1 | | 1 | 1 | | 1 | 1 |⁠), (⁠| | | | | - | - | - | | 1 | 1 | 1 | | 1 | 1 | 1 |⁠)};
Wolfram Language code: x = LyapunovSolve[a, -b.b];
Wolfram Language code: h2norm = Sqrt[Tr[c.x.c]]//N

極を所望の位置に置くフィードバックゲインを計算する：

Wolfram Language code: {a, b} = {(⁠| | | | | -- | ---- | ---- | | -3 | 0 | 0 | | 0 | -0.5 | -0.1 | | 0 | 0.1 | -0.5 |⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 | | 0 | 0 |⁠)};
Wolfram Language code: poles = {-6, -3 - 0.2I, -3 + 0.2I};
Wolfram Language code: With[{kk = ConstantArray[1, {2, 3}], aa = DiagonalMatrix[poles]}, kk.Inverse[LyapunovSolve[a, -aa, b.kk]]]//Chop

解を検証する：

Wolfram Language code: Eigenvalues[a - b.%]

多重入力多重出力の系の場合，フィードバックゲインは一意的ではない：

Wolfram Language code: With[{kk = {{1, 0, 0}, {0, 1, 1}}, aa = DiagonalMatrix[poles]}, kk.Inverse[LyapunovSolve[a, -aa, b.kk]]]//Chop
Wolfram Language code: Eigenvalues[a - b.%]

StateSpaceModelの観測器を構築する：

Wolfram Language code: {a, b, c} = {(⁠| | | | -- | -- | | -1 | 3 | | -3 | -1 |⁠), (⁠| | | - | | 1 | | 0 |⁠), (⁠1 0⁠)};
Wolfram Language code: ssm = StateSpaceModel[{a, b, c}]

まず，リャプノフ方程式 が非特異解を与えるような と を選ぶ：

Wolfram Language code: f = (⁠| | | | -- | -- | | -1 | 0 | | 0 | -3 |⁠);g = (⁠| | | - | | 1 | | 3 |⁠);
Wolfram Language code: Subscript[x, L] = LyapunovSolve[-f, a, g.c]//N; MatrixRank[Subscript[x, L]] == 2

次に，で x^^(t)=TemplateBox[{{x, _, L}}, Inverse]z(t)となる観測器を構築する．ただし， は観測器の状態ベクトル，は出力，は入力，は推定状態ベクトルである：

Wolfram Language code: obs = StateSpaceModel[{f, ArrayFlatten[{{g, Subscript[x, L].b}}], Inverse[Subscript[x, L]]}, SystemsModelLabels -> {{"y", "u"}, {"x1hat", "x2hat"}, {"z1", "z2"}}]

UnitStep入力について，推定された状態軌跡を計算する：

Wolfram Language code: y = OutputResponse[ssm, UnitStep[t], t];
Wolfram Language code: Overscript[x, ^ ] = OutputResponse[obs, Flatten@{y, UnitStep[t]}, t]

UnitStep 入力について，実際の状態軌跡を計算する：

Wolfram Language code: x = StateResponse[{ssm, {0.2, 0.1}}, UnitStep[t], t]

実際と推定された両方の状態をプロットする：

Wolfram Language code: {Plot[Evaluate[First /@ {x, Overscript[x, ^ ]}], {t, 0, 6}, PlotRange -> All, PlotLegends -> {"x1", "x1hat"}], Plot[Evaluate[Last /@ {x, Overscript[x, ^ ]}], {t, 0, 6}, PlotRange -> All, PlotLegends -> {"x2", "x2hat"}]}

特性と関係  (5)

負定値 を持つ方程式 は， の固有値が閉左半平面にあるときかつそのときに限り一意的な正定値解を与える：

Wolfram Language code: Subscript[a, unstable] = (⁠| | | | --- | ---- | | 0.5 | 0.1 | | 0 | -0.1 |⁠);
Wolfram Language code: LyapunovSolve[Subscript[a, unstable], -(⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)]//PositiveDefiniteMatrixQ
Wolfram Language code: Eigenvalues[Subscript[a, unstable]]

安定した系：

Wolfram Language code: Subscript[a, stable] = (⁠| | | | ------- | ------- | | -1.0782 | -0.1956 | | 0.3913 | -0.0217 |⁠);
Wolfram Language code: LyapunovSolve[Subscript[a, stable], -(⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)]//PositiveDefiniteMatrixQ
Wolfram Language code: Eigenvalues[Subscript[a, stable]]

が漸近的に安定しているとき，定積分 は の解である：

Wolfram Language code: {a, c} = {(⁠| | | | -- | -- | | -1 | 3 | | 0 | -1 |⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)};
Wolfram Language code: -Subsuperscript[∫, 0, ∞]MatrixExp[a t].c.MatrixExp[at]ⅆt//MatrixForm
Wolfram Language code: LyapunovSolve[a, c] == %

漸近的に安定した系 の無限期間二次費用を計算する：

Wolfram Language code: {a, q} = {(⁠| | | | -- | -- | | -1 | 0 | | 0 | -2 |⁠), (⁠| | | | --- | - | | 0.1 | 0 | | 0 | 1 |⁠)};
Wolfram Language code: s = LyapunovSolve[a, -q];
Wolfram Language code: Subscript[x, 0] = (⁠| | | -- | | -1 | | 1 |⁠); ((1/2)Transpose[Subscript[x, 0]].s.Subscript[x, 0])[[1, 1]]

直接積分を使って計算する：

Wolfram Language code: ((1/2)Subsuperscript[∫, 0, ∞](MatrixExp[a t].Subscript[x, 0]).q.(MatrixExp[a t].Subscript[x, 0]) ⅆt)[[1, 1]]

行列方程式 を解く：

Wolfram Language code: LyapunovSolve[a = (| | | | | -- | -- | - | | 1 | 1 | 0 | | -1 | -1 | 4 | | 1 | 5 | 2 |), (| | | | | - | - | - | | 0 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |), (c = (| | | | | -- | - | - | | 1 | 4 | 1 | | 0 | 2 | 0 | | -1 | 0 | 3 |))]//MatrixForm

LyapunovSolveは同じ解を返す：

Wolfram Language code: LinearSolve[a, c]//MatrixForm

LinearSolveを使ってリャプノフ方程式 を解く：

Wolfram Language code: {a, b, c, d, e} = {(⁠| | | | ------ | ------- | | 0.6685 | 0.3482 | | 0.154 | -0.7161 |⁠), (⁠| | | | ------- | ------ | | -0.1552 | 0.1178 | | -0.9785 | 0.2884 |⁠), (⁠| | | | ------- | ------- | | 0.7412 | -0.9487 | | -0.0326 | 0.3568 |⁠), (⁠| | | | ------ | ------- | | 0.8778 | 0.8733 | | 0.3024 | -0.4085 |⁠), (⁠| | | | ------- | ----- | | 0.874 | 0.753 | | -0.4265 | 0.758 |⁠)};
Wolfram Language code: Partition[LinearSolve[KroneckerProduct[a, e] + KroneckerProduct[d, b], Flatten[c]], 2]

LyapunovSolveは同じ解を与える：

Wolfram Language code: LyapunovSolve[{a, d}, {b, e}, c]
Wolfram Research (2010), LyapunovSolve, Wolfram言語関数， https://reference.wolfram.com/language/ref/LyapunovSolve.html.

テキスト

Wolfram Research (2010), LyapunovSolve, Wolfram言語関数， https://reference.wolfram.com/language/ref/LyapunovSolve.html.

CMS

Wolfram Language. 2010. "LyapunovSolve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/LyapunovSolve.html.

APA

Wolfram Language. (2010). LyapunovSolve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LyapunovSolve.html

BibTeX

@misc{reference.wolfram_2026_lyapunovsolve, author="Wolfram Research", title="{LyapunovSolve}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/LyapunovSolve.html}", note=[Accessed: 05-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_lyapunovsolve, organization={Wolfram Research}, title={LyapunovSolve}, year={2010}, url={https://reference.wolfram.com/language/ref/LyapunovSolve.html}, note=[Accessed: 05-November-2026]}