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DiscreteRiccatiSolve[{a,b},{q,r}]

離散代数リッカチ(Riccati)方程式 TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0の安定化解である行列 を与える．

DiscreteRiccatiSolve[{a,b},{q,r,p}]

TemplateBox[{a}, ConjugateTranspose].x.a-x-(TemplateBox[{a}, ConjugateTranspose].x.b+p).TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].(TemplateBox[{b}, ConjugateTranspose].x.a+TemplateBox[{p}, ConjugateTranspose])+q=0を解く．

詳細とオプション
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例  
スコープ  
オプション  
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引用書式

DiscreteRiccatiSolve

DiscreteRiccatiSolve[{a,b},{q,r}]

離散代数リッカチ(Riccati)方程式 TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0の安定化解である行列 を与える．

DiscreteRiccatiSolve[{a,b},{q,r,p}]

TemplateBox[{a}, ConjugateTranspose].x.a-x-(TemplateBox[{a}, ConjugateTranspose].x.b+p).TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].(TemplateBox[{b}, ConjugateTranspose].x.a+TemplateBox[{p}, ConjugateTranspose])+q=0を解く．

詳細とオプション

  • TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0における「」は共役転置を表す．
  • が安定可能，が検出可能で，かつ のときのみ，方程式 TemplateBox[{a}, ConjugateTranspose].x.a-x-TemplateBox[{a}, ConjugateTranspose].x.b.TemplateBox[{{(, {r, +, {TemplateBox[{b}, ConjugateTranspose], ., x, ., b}}, )}}, Inverse].TemplateBox[{b}, ConjugateTranspose].x.a+q=0 には一意的で対称な半正定値解がある．したがって，行列 の固有値はすべて単位円内にあり解は安定している．
  • が可制御でが可観測であれば解は正定値である．
  • DiscreteRiccatiSolveはMethodオプションをサポートする．次は，指定可能な設定値である．
  • Automatic自動的に決定されたメソッド
    "Eigensystem"固有値分解に基づく
    "GeneralizedEigensystem"一般化された固有値分解に基づく
    "GeneralizedSchur"一般化されたシューア(Schur)分解に基づく
    "InverseFree""GeneralizedSchur"の変化形
    "MatrixSign"行列符号関数を使った反復法
    "Newton"反復ニュートン(Newton)法
    "Schur"シューア分解に基づく
  • どのメソッドも，近似数値行列に適用される．"Eigensystem"は厳密行列に適用される．

例題

すべて開く すべて閉じる

例  (1)

離散代数リッカチ方程式を解く：

Wolfram Language code: {a, b, q, r} = {(⁠| | | | | - | -- | - | | 1 | -1 | 1 | | 0 | 1 | 1 | | 0 | 0 | 1 |⁠), (⁠| | | | - | - | | 1 | 0 | | 1 | 0 | | 0 | 1 |⁠), (⁠| | | | | -- | - | --- | | 10 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 0.1 |⁠), (⁠| | | | -- | --- | | 10 | 0 | | 0 | 0.1 |⁠)};
Wolfram Language code: (x = DiscreteRiccatiSolve[{a, b}, {q, r}])//MatrixForm

解を確かめる：

Wolfram Language code: Transpose[a].x.a - x - Transpose[a].x.b.Inverse[r + Transpose[b].x.b].Transpose[b].x.a + q//Chop

スコープ  (3)

離散リッカチ方程式を解く：

Wolfram Language code: DiscreteRiccatiSolve[{(| | | | ---- | --- | | -1 | 0.5 | | -0.5 | -2 |), (| | | --- | | 1 | | 0.5 |)}, {(| | | | - | - | | 1 | 0 | | 0 | 2 |), {{1}}}]//MatrixForm

クロスカップリング行列を含む離散リッカチ方程式を解く：

Wolfram Language code: DiscreteRiccatiSolve[{(| | | | | - | ------ | --- | | 0 | 1 | 0 | | 0 | -0.01 | 0.3 | | 0 | -0.003 | -10 |), (| | | | --- | -- | | 0 | 0 | | 0 | -1 | | 0.1 | 0 |)}, {(| | | | | - | - | - | | 1 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 1 |), (| | | | - | - | | 1 | 0 | | 0 | 1 |), (| | | | ---- | --- | | 0.1 | 0 | | -0.2 | 0 | | 0.15 | 0.2 |)}]//MatrixForm

サンプルデータの伝達関数モデルのリッカチ方程式を解く：

Wolfram Language code: tfm = TransferFunctionModel[{{{1}}, (-1 + z)*(-0.605 + z)}, z, SamplingPeriod -> 0.1];
Wolfram Language code: DiscreteRiccatiSolve[Normal[StateSpaceModel[tfm]][[1 ;; 2]], {(| | | | -- | --- | | 10 | 0 | | 0 | 0.1 |), (⁠1⁠)}]

オプション  (7)

Method  (7)

厳密な系には，Automaticおよび"Eigensystem"メソッドが使われる：

Wolfram Language code: x1 = DiscreteRiccatiSolve[{(| | | | - | - | | 0 | 2 | | 1 | 0 |), (| | | - | | 0 | | 1 |)}, {(| | | | - | ----- | | 1 | 0 | | 0 | (4/9) |), (⁠1⁠)}, Method -> Automatic]
Wolfram Language code: x2 = DiscreteRiccatiSolve[{(| | | | - | - | | 0 | 2 | | 1 | 0 |), (| | | - | | 0 | | 1 |)}, {(| | | | - | ----- | | 1 | 0 | | 0 | (4/9) |), (⁠1⁠)}, Method -> "Eigensystem"]
Wolfram Language code: x1 == x2

厳密ではない系にも用いられる：

Wolfram Language code: x1 = DiscreteRiccatiSolve[N@{(| | | | - | - | | 0 | 2 | | 1 | 0 |), (| | | - | | 0 | | 1 |)}, {(| | | | - | ----- | | 1 | 0 | | 0 | (4/9) |), (⁠1⁠)}, Method -> Automatic]//Chop
Wolfram Language code: x2 = DiscreteRiccatiSolve[N@{(| | | | - | - | | 0 | 2 | | 1 | 0 |), (| | | - | | 0 | | 1 |)}, {(| | | | - | ----- | | 1 | 0 | | 0 | (4/9) |), (⁠1⁠)}, Method -> "Eigensystem"]//Chop
Wolfram Language code: x1 == x2

厳密ではない系には"Schur"メソッドを使うことができる：

Wolfram Language code: DiscreteRiccatiSolve[{(| | | | - | - | | 0 | 2 | | 1 | 1 |), (| | | - | | 0 | | 1 |)}, {(| | | | ---- | ---- | | 2.25 | -1.5 | | -1.5 | 1 |), (⁠2⁠)}, Method -> "Schur"]

"Newton"は非厳密系に適用され，Automaticよりも正確な場合がある：

Wolfram Language code: a = SparseArray[{{1, 1} -> 0.8, {1, 5} -> -0.8, {2, 1} -> 0.8, {2, 5} -> -0.5, {5, 1} -> 0.25}]; b = Transpose[{{1, 0, 0, 1, 0}}]; r = {{1}}; q = IdentityMatrix[5];

解と絶対誤差を計算する：

Wolfram Language code: x1 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "Newton"];
Wolfram Language code: Norm[a^.x1.a - x1 - a^.x1.b.Inverse[r + b^.x1.b].b^.x1.a + q]

デフォルトのメソッドと比較する：

Wolfram Language code: x2 = DiscreteRiccatiSolve[{a, b}, {q, r}];
Wolfram Language code: Norm[a^.x2.a - x2 - a^.x2.b.Inverse[r + b^.x2.b].b^.x2.a + q]

"Newton"は，サブオプションとして，"StartingMatrix"，"MaxIterations"，"Tolerance"を取る：

Wolfram Language code: x3 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> {"Newton", "StartingMatrix" -> x2, "Tolerance" -> 10^-15, "MaxIterations" -> 10}];
Wolfram Language code: Norm[a^.x3.a - x3 - a^.x3.b.Inverse[r + b^.x3.b].b^.x3.a + q]

"Newton"法は，たとえそれが存在する場合でも，安定化解を計算しないことがある：

Wolfram Language code: a = {{0., -1.}, {0., 2.}}; b = {{1, 0}, {1, 1}}; q = {{1, 0}, {0, 0}}; r = {{4, 2}, {2, 1}}; x1 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "Newton"];

Automaticメソッドと比較する：

Wolfram Language code: x2 = DiscreteRiccatiSolve[{a, b}, {q, r}]; Norm[a^.x2.a - x2 - a^.x2.b.Inverse[r + b^.x2.b].b^.x2.a + q]

"MatrixSign"は，一般に，"Newton"法の初期近似として使われる：

Wolfram Language code: {a, b} = {{{-0.1, 0}, {0, -0.02}}, {{0.1, 0}, {0.001, 0.01}}}; {q, r} = {{{100, 1000}, {1000, 10^4}}, {{1. + 10^-6, 1.}, {1., 1.}}}; xinit = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "MatrixSign"];

xinitを使って"Newton"法を初期化する：

Wolfram Language code: x = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> {"Newton", "StartingMatrix" -> xinit, "Tolerance" -> 10^-12}];

誤差を比較する：

Wolfram Language code: Map[Norm[a^.#.a - # - a^.#.b.Inverse[r + b^.#.b].b^.#.a + q]&, {xinit, x}]

"MatrixSign"は，サブオプションとして，"MaxIterations"および"Tolerance"を取る：

Wolfram Language code: x3 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> {"MatrixSign", "Tolerance" -> 10^-12, "MaxIterations" -> 10}];
Wolfram Language code: Norm[a^.x3.a - x3 - a^.x3.b.Inverse[r + b^.x3.b].b^.x3.a + q]

a が特異値の場合は，"GeneralizedSchur"あるいは"GeneralizedEigensystem"が適用される：

Wolfram Language code: a = SparseArray[{{1, 1} -> 0.8, {1, 5} -> -0.8, {2, 1} -> 0.8, {2, 5} -> -0.5, {5, 1} -> 0.25}]; b = Transpose[{{1, 0, 0, 1, 0}}]; r = {{1}}; q = IdentityMatrix[5];

行列 a は特異行列である：

Wolfram Language code: Det[a]

これら2つのメソッドは，特異行列 a に使うことができる：

Wolfram Language code: x1 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "GeneralizedSchur"];
Wolfram Language code: x2 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "GeneralizedEigensystem"];

誤差を確かめる：

Wolfram Language code: Map[Norm[a^.#.a - # - a^.#.b.Inverse[r + b^.#.b].b^.#.a + q]&, {x1, x2}]

"Eigensystem"は，a が特異行列のときは使うことができない：

Wolfram Language code: x3 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "Eigensystem"];

"InverseFree"は，r の条件がよくない場合に使うことができる：

Wolfram Language code: {a, b} = {{{-0.1, 0}, {0, -0.02}}, {{0.1, 0}, {0.001, 0.01}}}; {q, r} = {{{100, 1000}, {1000, 10^4}}, {{1. + 10^-6, 1.}, {1., 1.}}};

行列 r の条件番号は高い：

Wolfram Language code: Norm[r] Norm[Inverse[r]]

このメソッドを使う：

Wolfram Language code: x1 = DiscreteRiccatiSolve[{a, b}, {q, r}, Method -> "InverseFree"]

デフォルトのメソッドと比較する：

Wolfram Language code: x2 = DiscreteRiccatiSolve[{a, b}, {q, r}]

この場合は，デフォルトメソッドの方が絶対誤差が大きい：

Wolfram Language code: {Norm[a^.x1.a - x1 - a^.x1.b.Inverse[r + b^.x1.b].b^.x1.a + q], Norm[a^.x2.a - x2 - a^.x2.b.Inverse[r + b^.x2.b].b^.x2.a + q]}

アプリケーション  (2)

線形化されたターボ発電機モデルには状態と入力の行列がある：

Wolfram Language code: {a, b} = {(⁠| | | | | | | | ------- | ------- | ------- | ------- | ------- | ------- | | 0.1346 | 0.1236 | -0.0361 | 0.0037 | 0.0004 | -0.0003 | | -0.1091 | 0.5412 | 0.3851 | -0.0631 | -0.052 | 0.0152 | | 0.0426 | 0.1052 | 0.7915 | 0.07 | 0.0504 | -0.0172 | | -0.0045 | 0.0205 | -0.0542 | 0.7932 | -0.5678 | 0.0025 | | 0.0022 | -0.015 | 0.0384 | 0.5681 | 0.7526 | 0.0357 | | 0.0002 | -0.0162 | 0.0142 | 0.0004 | -0.0299 | 0.9784 |⁠), (⁠| | | | ------- | ------- | | -0.0121 | 0.0117 | | -0.0046 | -0.496 | | -0.015 | 0.5517 | | 0.0095 | -0.1763 | | -0.0055 | -1.0216 | | 0.0025 | 1.3944 |⁠)};

指定された重み行列{q,r}および初期状態 s0についての離散線形二次調整器費用 Jopt：

Wolfram Language code: {q, r} = {(⁠| | | | | | | | - | - | - | - | - | - | | 1 | 0 | 0 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | 0 | 0 | | 0 | 0 | 0 | 1 | 0 | 0 | | 0 | 0 | 0 | 0 | 1 | 0 | | 0 | 0 | 0 | 0 | 0 | 1 |⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)};
Wolfram Language code: Subscript[s, 0] = {{0.1, 0, 0, 0, 0, 0}};
Wolfram Language code: x = DiscreteRiccatiSolve[{a, b}, {q, r}];
Wolfram Language code: Subscript[J, opt] = ( Subscript[s, 0].x.Transpose[Subscript[s, 0]])[[1, 1]]

すべての閉ループ極が半径 の円内にあることを保証する最適状態フィードバックゲインを計算する：

Wolfram Language code: {a, b, q, r} = {(⁠| | | | | - | -------- | ---------- | | 1 | 0.095162 | 0.00555872 | | 0 | 0.904837 | 0.117397 | | 0 | 0 | 1.49182 |⁠), (⁠| | | ----------- | | 0.000180325 | | 0.00555872 | | 0.122956 |⁠), (⁠| | | | | -- | - | - | | 10 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 1 |⁠), (⁠1⁠)};
Wolfram Language code: {Subscript[a, 1], Subscript[b, 1]} = ({a, b}/α) /. α -> 0.7;
Wolfram Language code: x = DiscreteRiccatiSolve[{Subscript[a, 1], Subscript[b, 1]}, {q, r}];
Wolfram Language code: Eigenvalues[a - b.Inverse[Transpose[Subscript[b, 1]].x.Subscript[b, 1] + r].Transpose[Subscript[b, 1]].x.Subscript[a, 1]]

規定の安定度を持たない閉ループの極：

Wolfram Language code: x = DiscreteRiccatiSolve[{a, b}, {q, r}];
Wolfram Language code: Eigenvalues[a - b.Inverse[Transpose[b].x.b + r].Transpose[b].x.a]

特性と関係  (11)

DiscreteRiccatiSolve[{a,b},{q,r,p}]はDiscreteRiccatiSolve[{a-b.r^( -1).p,b},{q-p.r -1.p,r}]に等しい：

Wolfram Language code: {a, b, q, r, p} = {(⁠| | | | | --------------- | ----------------- | -------------- | | -0.25 - 0.011 I | 0.0947  + 0.089 I | 0.35  + 0.16 I | | 0.35  + 0.22 I | -0.08 - 0.22 I | -0.4 - 0.53 I | | -0.16 - 0.07 I | 0.05  + 0.1 I | 0.22  + 0.05 I |⁠), (⁠| | | | --------- | ----------- | | 1 + 0.5 I | -1 - 0.5 I | | -0.5 - 1I | -0.5 + 1.5I | | 0.1 I | -0.5I |⁠), (⁠| | | | | - | - | - | | 1 | 0 | 0 | | 0 | 1 | 0 | | 0 | 0 | 1 |⁠), (⁠| | | | - | -- | | 3 | -I | | I | -3 |⁠), (⁠| | | | ---- | --- | | 0.1 | 0 | | -0.2 | 0 | | 0.15 | 0.2 |⁠)};
Wolfram Language code: x = DiscreteRiccatiSolve[{a, b}, {q, r, p}];

p を行列 a および q に組み込んでも同じ結果を得ることができる：

Wolfram Language code: xp = DiscreteRiccatiSolve[{a - b .Inverse[r].ConjugateTranspose[p], b}, {q - p.Inverse[r].ConjugateTranspose[p], r}];
Wolfram Language code: Chop[x - xp]

{a,b}が安定化可能，{a,g}が検出可能で q=Transpose[g].g であれば，離散リッカチ方程式の解は半正定値である：

Wolfram Language code: {a, b, g, r} = {(⁠| | | | | ---- | ---- | --- | | 0.45 | 0 | 0 | | 1 | -0.9 | 1 | | 0 | 0 | 0.7 |⁠), (⁠| | | - | | 1 | | 1 | | 0 |⁠), (⁠1 0 0⁠), (⁠1⁠)};
Wolfram Language code: JordanModelDecomposition[StateSpaceModel[{a, b, g}]]//Last
Wolfram Language code: PositiveSemidefiniteMatrixQ@DiscreteRiccatiSolve[{a, b}, {Transpose[g].g, r}]

{a,b}が可制御で{a,g}が可観測であり q=Transpose[g].g であれば，離散リッカチ方程式の解は正定値である：

Wolfram Language code: {a, b, g, r} = {(⁠| | | | | | | ---- | ----- | ------ | ------ | ----- | | 0.98 | 0.048 | 0.0025 | 0.0046 | 0.008 | | 0 | 0.95 | 0.08 | 0.2 | 0.533 | | 0 | 0 | 0.23 | 1.26 | 6.52 | | 0 | 0 | 0 | 0.082 | 1.428 | | 0 | 0 | 0 | 0 | 0.367 |⁠), (⁠| | | ------ | | 0.0059 | | 0.509 | | 10.15 | | 3.97 | | 1.89 |⁠), (⁠1 0 0 0 0⁠), (⁠1⁠)};
Wolfram Language code: ControllableModelQ[StateSpaceModel[{a, b}]]
Wolfram Language code: ControllableModelQ[StateSpaceModel[{Transpose[a], Transpose[g]}]]
Wolfram Language code: PositiveDefiniteMatrixQ[DiscreteRiccatiSolve[{a, b}, {Transpose[g].g, r}]]

離散代数リッカチ方程式に関連する行列はシンプレクティックである：

Wolfram Language code: With[{a = (⁠| | | | | - | --- | ------- | | 1 | 0.5 | 0.4 | | 0 | 1 | 0.09932 | | 0 | 0 | 0.6738 |⁠), b = (⁠| | | ------- | | 0.0085 | | 0.04 | | 0.09932 |⁠), q = (⁠| | | | | - | - | - | | 5 | 0 | 0 | | 0 | 3 | 0 | | 0 | 0 | 1 |⁠), r = (⁠1⁠)}, M = (| | | | ---------------------------------- | ----------------------------- | | a + b.Inverse[r].b.Inverse[a^].q | -b.Inverse[r].b.Inverse[a^] | | -Inverse[a^].q | Inverse[a^] |)//ArrayFlatten];
Wolfram Language code: Ω = ArrayFlatten[(| | | | ------------------------ | ------------------------ | | ConstantArray[0, {3, 3}] | IdentityMatrix[3] | | -IdentityMatrix[3] | ConstantArray[0, {3, 3}] |)];
Wolfram Language code: Chop[ConjugateTranspose[M].Ω.M] == Ω

シンプレクティック行列の固有値は の形のペアである：

Wolfram Language code: With[{a = (⁠| | | | | ------------------ | ------------------ | ------------------ | | 0.0661 - 0.2022 I | -0.8839 - 0.4677 I | -0.9869 - 0.2443 I | | -0.7635 - 1.0728 I | 0.312  - 1.66 I | -0.8182 - 1.546 I | | 1.311 + 1.3799 I | 1.0445  + 2.1423 I | 2.2958 + 1.8621 I |⁠), b = (⁠| | | ------------------ | | 0.0015  + 0.0062 I | | 0.0098 + 0.01223I | | -0.004 - 0.01371I |⁠), q = (⁠| | | | | - | - | - | | 5 | 0 | 0 | | 0 | 3 | 0 | | 0 | 0 | 1 |⁠), r = (⁠1⁠)}, M = (| | | | --------------------------------- | ---------------------------- | | a + b.Inverse[r].b.Inverse[a].q | -b.Inverse[r].b.Inverse[a] | | -Inverse[a].q | Inverse[a] |)//ArrayFlatten];
Wolfram Language code: ListPlot[Eigenvalues[M] /. Complex[x_, y_] :> {x, y}, PlotStyle -> PointSize[Medium], Epilog -> {Red, Circle[{0, 0}]}, AxesOrigin -> {0, 0}]

と は相似行列である：

Wolfram Language code: Complement[Eigenvalues[M], Eigenvalues[Inverse[M]], SameTest -> (Chop[#1 - #2] == 0&)]

リッカチ方程式の安定化解を得るためには，シンプレクティック行列は安定性および相補性の特性を満足しなければならない：

Wolfram Language code: a = b = q = r = (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠);
Wolfram Language code: {vals, vecs} = Eigensystem[(| | | | --------------------------------- | ----------------------------- | | a + b.Inverse[r].b.Inverse[a].q | -b.Inverse[ r].b.Inverse[a] | | -Inverse[a].q | Inverse[a] |)//ArrayFlatten]

安定性特性：

Wolfram Language code: Cases[vals, _ ? (Abs[#] == 1&)]

相補性特性：

Wolfram Language code: stableBasis = Extract[vecs, Position[vals, _ ? (Abs[#] < 1&), {1}]];
Wolfram Language code: {{Subscript[x, 1], Subscript[x, 2]}} = Partition[stableBasis, {2, 2}]
Wolfram Language code: MatrixRank[Subscript[x, 1]] == 2

安定化解：

Wolfram Language code: Subscript[x, 2].Inverse[Subscript[x, 1]]//Simplify
Wolfram Language code: DiscreteRiccatiSolve[{a, b}, {q, r}] == %

のあるフィードバック系の固有値はシンプレクティック行列の安定固有値である：

Wolfram Language code: {a, b, q, r} = {(⁠| | | | - | - | | 1 | 3 | | 1 | 2 |⁠), (⁠| | | -- | | 1 | | -2 |⁠), (⁠| | | | --- | - | | 0.2 | 0 | | 0 | 1 |⁠), (⁠0.5)};
Wolfram Language code: With[{x = DiscreteRiccatiSolve[{a, b}, {q, r}]}, Eigenvalues[a - b.Inverse[Transpose[b].x.b + r].Transpose[b].x.a]]
Wolfram Language code: Eigenvalues[ArrayFlatten[(| | | | --------------------------------- | ----------------------------- | | a + b.Inverse[r].b.Inverse[a].q | -b.Inverse[ r].b.Inverse[a] | | -Inverse[a].q | Inverse[a] |)]]; Select[%, Abs[#] < 1&]

DiscreteRiccatiSolveを使って最適状態フィードバックゲインを計算する：

Wolfram Language code: {a, b, q, r} = {(⁠| | | | --- | --- | | 0.2 | 1 | | -1 | 0.5 |⁠), (⁠| | | - | | 1 | | 0 |⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠), (⁠1⁠)};
Wolfram Language code: With[{x = DiscreteRiccatiSolve[{a, b}, {q, r}]}, Inverse[Transpose[b].x.b + r].Transpose[b].x.a]//N

LQRegulatorGainsを使って同じ結果を直接得る：

Wolfram Language code: LQRegulatorGains[StateSpaceModel[{a, b}, SamplingPeriod -> T], {q, r}]//N

DiscreteRiccatiSolveを使って最適出力フィードバックゲインを計算する：

Wolfram Language code: {a, b, c, q, r} = {(⁠| | | | -- | -- | | -1 | 1 | | 0 | -1 |⁠), (⁠| | | | - | - | | 0 | 1 | | 1 | 0 |⁠), (⁠1 0⁠), (⁠10⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)};
Wolfram Language code: With[{x = DiscreteRiccatiSolve[{a, b}, {Transpose[c].q.c, r}]}, Inverse[Transpose[b].x.b + r].Transpose[b].x.a]//N//MatrixForm

LQOutputRegulatorGainsは同じ結果を与える：

Wolfram Language code: LQOutputRegulatorGains[StateSpaceModel[{a, b, c}, SamplingPeriod -> T], {q, r}]//N//MatrixForm

DiscreteRiccatiSolveを使って最適推定器ゲインを計算する：

Wolfram Language code: {a, b, c} = {(| | | | | | ----- | ----- | ---- | --- | | -0.02 | 0.005 | 2.4 | -32 | | -0.14 | 0.44 | -1.3 | 30 | | 0 | 0.018 | -1.6 | 1.2 | | 0 | 0 | 1 | 0 |), (| | | | ---- | ----- | | 0.14 | -0.12 | | 0.36 | -0.86 | | 0.35 | 0.009 | | 0 | 0 |), (0 1 0 0)};
Wolfram Language code: {w, v} = {(⁠| | | | ----- | ----- | | 10^-2 | 0 | | 0 | 10^-2 |⁠), (⁠10^-4⁠)};
Wolfram Language code: With[{x = DiscreteRiccatiSolve[{Transpose[a], Transpose[c]}, {b.w.Transpose[b], v}]}, x.Transpose[c].Inverse[c.x.Transpose[c] + v]]

LQEstimatorGainsを使う：

Wolfram Language code: LQEstimatorGains[StateSpaceModel[{a, b, c}, SamplingPeriod -> τ], {w, v}]

系の離散近似の最適軌道は高くつく：

Wolfram Language code: ssm = StateSpaceModel[{{{0, 0, 1, 0}, {0, 0, 0, 1}, {10.7275, 0, 0, 0}, {-0.9275, 0, 0, 0}}, {{0}, {0}, {-0.178571}, {0.178571}}, {{1, 0, 0, 0}, {0, 1, 0, 0}}, {{0}, {0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: ssmd = ToDiscreteTimeModel[ssm, 0.1];
Wolfram Language code: {q, r} = {(⁠| | | | | | -- | - | -- | - | | 10 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | | 0 | 0 | 10 | 0 | | 0 | 0 | 0 | 1 |⁠), (⁠1000⁠)};

s0から始まる系の最適線形二次調整器費用：

Wolfram Language code: Subscript[s, 0] = {{0}, {0.1}, {0}, {0}};
Wolfram Language code: J = (Transpose[Subscript[s, 0]].RiccatiSolve[Normal[ssm][[1 ;; 2]], {q, r}].Subscript[s, 0])[[1, 1]];
Wolfram Language code: Subscript[J, d] = (Transpose[Subscript[s, 0]].DiscreteRiccatiSolve[Normal[ssmd][[1 ;; 2]], {q, r}].Subscript[s, 0])[[1, 1]];
Wolfram Language code: Greater[Subscript[J, d], J]

考えられる問題  (1)

{a,b}が安定化不可能で{a,g}が検出不可能であれば，q=g.g のリッカチ方程式には安定化解がない：

Wolfram Language code: {a, b, q, g, r} = {(⁠| | | | | --- | --- | ---- | | 0.3 | 0 | 0 | | 0.3 | 0.4 | 0.06 | | 0 | 0 | 1.2 |⁠), (⁠| | | ----- | | 0.006 | | 0.04 | | 0 |⁠), (⁠| | | | | - | - | - | | 1 | 0 | 0 | | 0 | 0 | 0 | | 0 | 0 | 0 |⁠), (⁠1 0 0⁠), (⁠1⁠)};
Wolfram Language code: {ControllableModelQ[StateSpaceModel[{a, b}]], ControllableModelQ[StateSpaceModel[{Transpose[a], Transpose[g]}]]}
Wolfram Language code: DiscreteRiccatiSolve[{a, b}, {q, r}]
Wolfram Research (2010), DiscreteRiccatiSolve, Wolfram言語関数， https://reference.wolfram.com/language/ref/DiscreteRiccatiSolve.html （2014年に更新）.

テキスト

Wolfram Research (2010), DiscreteRiccatiSolve, Wolfram言語関数， https://reference.wolfram.com/language/ref/DiscreteRiccatiSolve.html （2014年に更新）.

CMS

Wolfram Language. 2010. "DiscreteRiccatiSolve." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/DiscreteRiccatiSolve.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_discretericcatisolve, author="Wolfram Research", title="{DiscreteRiccatiSolve}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteRiccatiSolve.html}", note=[Accessed: 11-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_discretericcatisolve, organization={Wolfram Research}, title={DiscreteRiccatiSolve}, year={2014}, url={https://reference.wolfram.com/language/ref/DiscreteRiccatiSolve.html}, note=[Accessed: 11-September-2026]}