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StateSpaceTransform[sys,{p,q}]

使用矩阵 p 和 q，变换状态空间模型 sys.

StateSpaceTransform[sys,{{x1p1[z],…},{z1q1[x],…}}]

使用变量转换 {x1p1[z],…} 和 {z1q1[x],…} 进行转换.

更多信息和选项
Details and Options Details and Options
范例  
基本范例  
范围  
线性变换  
Nonlinear Transformations:  
推广和延伸  
选项  
DescriptorStateSpace  
应用  
属性和关系  
可能存在的问题  
参见
相关指南
历史
按以下格式引用：

StateSpaceTransform

StateSpaceTransform[sys,{p,q}]

使用矩阵 p 和 q，变换状态空间模型 sys.

StateSpaceTransform[sys,{{x1p1[z],…},{z1q1[x],…}}]

使用变量转换 {x1p1[z],…} 和 {z1q1[x],…} 进行转换.

更多信息和选项

范例

打开所有单元 关闭所有单元

基本范例  (1)

相似变换 ：

Wolfram Language code: {ssm, p} = {StateSpaceModel[{{{0.5195, 0.2959}, {0.3081, 0.9136}}, {{0.0987}, {-1}}, {{1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], p = (| | | | --- | -- | | 0.5 | 0 | | 1 | -1 |)};
Wolfram Language code: StateSpaceTransform[ssm, p]

使用矩阵对：

Wolfram Language code: StateSpaceTransform[ssm, {p, Inverse[p]}]

范围  (15)

线性变换  (10)

相似变换 :

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

指定矩阵对：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{{p}}, {{(1/p)}}}]

将变换指定为 :

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {Automatic, {{(1/p)}}}]

描述器系统的相似变换 ：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{e}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{{p}}, {{(1/p)}}}]

描述器系统的等价变换 ：

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

相应的矩阵对指定：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{e}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{{p}}, {{1}}}]

获取等价描述器系统，其中状态方程提前乘以矩阵 q：

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

新状态就是旧状态，即 :

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{e}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{{1}}, {{q}}}]

对于描述器系统，具有可逆 p 和 q 的变换给出等价系统：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 21], Subscript[a, 22]}}, {{Subscript[b, 11]}, {Subscript[b, 21]}}, {{Subscript[c, 11], Subscript[c, 12]}}, {{Subscript[d, 11]}}, {{Subscript[e, 11], Subscript[e, 12]}, {Subscript[e, 21], Subscript[e, 22]}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {(| | | | --- | --- | | p11 | p12 | | p21 | p22 |), (| | | | --- | --- | | q11 | q12 | | q21 | q22 |)}]

原始系统：

Wolfram Language code: StateSpaceTransform[%, Inverse /@ {(| | | | --- | --- | | p11 | p12 | | p21 | p22 |), (| | | | --- | --- | | q11 | q12 | | q21 | q22 |)}]

不可逆的 p 或 q 给出有受限等价的系统：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 21], Subscript[a, 22]}}, {{Subscript[b, 11]}, {Subscript[b, 21]}}, {{Subscript[c, 11], Subscript[c, 12]}}, {{Subscript[d, 11]}}, {{Subscript[e, 11], Subscript[e, 12]}, {Subscript[e, 21], Subscript[e, 22]}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {(| | | | --- | --- | | p11 | p12 | | 0 | 0 |), (| | | | --- | - | | q11 | 0 | | q21 | 0 |)}]

在矩阵变换下，线性离散系统时间与连续时间系统的行为类似：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> τ, SystemsModelLabels -> None], {{p}}]

根据 变换一个 AffineStateSpaceModel：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{p}}]

按照新变量 z 来指定变换：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p z}, {z}}]

指定变换为 ：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {Automatic, {{(1/p)}}}]

用变量：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{x}, {z -> (1/p)x}}]

指定变换及它的逆：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{{p}}, {{(1/p)}}}]
Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p z}, {z -> (1/p)x}}]

根据 变换 NonlinearStateSpaceModel：

Wolfram Language code: StateSpaceTransform[NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None], {{p}}]

按照新变量 z：

Wolfram Language code: StateSpaceTransform[NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p z}, {z}}]

根据逆变换：

Wolfram Language code: StateSpaceTransform[NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None], {{x}, {z -> (1/p)x}}]

变换和它的逆：

Wolfram Language code: StateSpaceTransform[NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p z}, {z}}]

对于正交矩阵，可使用 Transpose，而不是 Automatic 或者 Inverse:

Wolfram Language code: p = (| | | | ----------- | ------------ | | (1/Sqrt[2]) | (1/Sqrt[2]) | | (1/Sqrt[2]) | -(1/Sqrt[2]) |);
Wolfram Language code: p.p//MatrixForm
Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{Subscript[a, 1, 1], Subscript[a, 1, 2]}, {Subscript[a, 2, 1], Subscript[a, 2, 2]}}, {{Subscript[b, 1, 1]}, {Subscript[b, 2, 1]}}, {{Subscript[c, 1, 1], Subscript[c, 1, 2]}}, {{Subscript[d, 1, 1]}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {p, p}]

Nonlinear Transformations:  (5)

StateSpaceModel 的非线性变换：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{x -> p[z]}, {z}}]
Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{x}, {z -> q[x]}}]

AffineStateSpaceModel 的非线性变换：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p[z]}, {z}}]
Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {x}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{x}, {z -> q[x]}}]

NonlinearStateSpaceModel 的非线性变换：

Wolfram Language code: StateSpaceTransform[NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p[z]}, {z}}]
Wolfram Language code: StateSpaceTransform[NonlinearStateSpaceModel[{{f[x, u]}, {g[x, u]}}, {x}, {u}, {Automatic}, Automatic, SamplingPeriod -> None], {{x}, {z -> q[x]}}]

指定一个非线性变换和它的逆：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{Subscript[a, 1, 1], Subscript[a, 1, 2]}, {Subscript[a, 2, 1], Subscript[a, 2, 2]}}, {{Subscript[b, 1, 1]}, {Subscript[b, 2, 1]}}, {{Subscript[c, 1, 1], Subscript[c, 1, 2]}}, {{Subscript[d, 1, 1]}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{Subscript[x, 1] -> Subscript[z, 1] + Subscript[z, 1]Subscript[z, 1], Subscript[x, 2] -> Subscript[z, 2] + Subscript[z, 1]}, {Subscript[z, 1] -> (1/2) (-1 + Sqrt[1 + 4 Subscript[x, 1]]), Subscript[z, 2] -> (1/2) (1 - Sqrt[1 + 4 Subscript[x, 1]] + 2 Subscript[x, 2])}}]

只在工作点被指定的时候需要非线性变换的逆：

Wolfram Language code: StateSpaceTransform[AffineStateSpaceModel[{{a[x]}, {{b[x]}}, {c[x]}, {{d[x]}}}, {{x, Subscript[x, 0]}}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None], {{x -> p z}, {z -> (1/p)x}}]

推广和延伸  (2)

非正交方阵选择一个子系统：

Wolfram Language code: ssm = StateSpaceModel[{{{Subscript[a, 1], 0, 0}, {0, Subscript[a, 2], 0}, {0, 0, Subscript[a, 3]}}, {{0, 0}, {Subscript[b, 1], 0}, {0, Subscript[b, 2]}}, {{Subscript[c, 1], 0, 0}, {0, 0, Subscript[c, 3]}}, {{Subscript[d, 11], Subscript[d, 12]}, {Subscript[d, 21], Subscript[d, 22]}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: p = {{0, 0}, {1, 0}, {0, 1}};
Wolfram Language code: StateSpaceTransform[ssm, p]

相应的矩阵对：

Wolfram Language code: StateSpaceTransform[ssm, {p, p}]

对非线性变换，分隔逆指定选出一个子系统：

Wolfram Language code: assm = AffineStateSpaceModel[{{Subscript[a, 1]*Subscript[x, 1], Subscript[a, 2]*Subscript[x, 2], Subscript[a, 3]*Subscript[x, 3]}, {{Subscript[b, 1]}, {Subscript[b, 2]}, {Subscript[b, 3]}}, {Subscript[c, 1]*Subscript[x, 1], Subscript[c, 2]*Subscript[x, 2]}, {{Subscript[d, 1]}, {Subscript[d, 2]}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3]}, Automatic, {Automatic, Automatic}, Automatic, SamplingPeriod -> None]; pp = {Subscript[x, 1] -> Subscript[z, 1], Subscript[x, 2] -> Subscript[z, 2], Subscript[x, 3] -> Subscript[z, 3]}; qq = {{Subscript[z, 1] -> Subscript[x, 1], Subscript[z, 2] -> Subscript[x, 2]}, {Subscript[z, 3] -> Subscript[x, 3]}};
Wolfram Language code: StateSpaceTransform[assm, {pp, qq}]

完备系：

Wolfram Language code: StateSpaceTransform[assm, {pp, Flatten[qq]}]

选项  (1)

DescriptorStateSpace  (1)

视标准 StateSpaceModel 为广义系统：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{p}}, DescriptorStateSpace -> True]

它给出与显式广义规范相同的结果：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{1}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {{p}}]

相似的结果也可以通过逆变换获取：

Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {Automatic, {{q}}}, DescriptorStateSpace -> True]
Wolfram Language code: StateSpaceTransform[StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}, {{1}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {Automatic, {{q}}}]

应用  (6)

一个获取单输入系统的能控相伴型的函数：

Wolfram Language code: ControllableCompanion[ssm_StateSpaceModel] /; ControllableModelQ[ssm] := Module[{A = First@Normal@ssm, dim, cv, coeffsList, mat, s1}, dim = Last@Dimensions@A; coeffsList = CoefficientList[Det[cv IdentityMatrix[dim] - A], cv]; mat = HankelMatrix[Rest[coeffsList] / Last[coeffsList]]; s1 = ControllabilityMatrix[ssm].mat; StateSpaceTransform[ssm, s1]]
Wolfram Language code: ControllableCompanion[StateSpaceModel[{{{1, -1, 1}, {1, 2, 1}, {-1, 1, 2}}, {{1}, {-1}, {0}}, {{1, -1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

一个获取单输出系统的能观测相伴型的函数：

Wolfram Language code: ObservableCompanion[ssm_StateSpaceModel] /; ObservableModelQ[ssm] := Module[{A = First@Normal@ssm, dim, cv, coeffsList, mat, s1}, dim = Last@Dimensions@A; coeffsList = CoefficientList[Det[cv IdentityMatrix[dim] - A], cv]; mat = HankelMatrix[Rest[coeffsList] / Last[coeffsList]]; s1 = mat.ObservabilityMatrix[ssm]; StateSpaceTransform[ssm, {Inverse@s1, s1}]]
Wolfram Language code: ObservableCompanion[StateSpaceModel[{{{1, -1, 1}, {1, 2, 1}, {-1, 1, 2}}, {{1}, {-1}, {0}}, {{1, -1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

对状态重新排列：

Wolfram Language code: ssm = StateSpaceModel[{{{Subscript[a, 11], Subscript[a, 12], Subscript[a, 13], Subscript[a, 14]}, {Subscript[α, 21], Subscript[α, 22], Subscript[α, 23], Subscript[α, 24]}, {Subscript[Α, 31], Subscript[Α, 32], Subscript[Α, 33], Subscript[Α, 34]}, {Subscript[a, 41], Subscript[a, 42], Subscript[a, 43], Subscript[a, 44]}}, {{Subscript[b, 1]}, {Subscript[β, 2]}, {Subscript[Β, 3]}, {Subscript[b, 4]}}, {{Subscript[c, 11], Subscript[γ, 12], Subscript[Γ, 13], Subscript[c, 14]}, {Subscript[c, 21], Subscript[γ, 22], Subscript[Γ, 23], Subscript[c, 24]}}, {{Subscript[d, 1]}, {Subscript[d, 2]}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: StateSpaceTransform[ssm, (| | | | | | - | - | - | - | | 1 | 0 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 1 | 0 | 0 | | 0 | 0 | 0 | 1 |)]

将一个笛卡尔坐标中的简单球摆模型转换为极坐标：

在笛卡尔坐标中的模型：

Wolfram Language code: asys = AffineStateSpaceModel[ {{v, (-v^2)*(x/(L^2 - x^2)) - (g*x*Sqrt[L^2 - x^2])/L^2}, {{0}, {Sqrt[L^2 - x^2]/(L^2*m)}}, {ArcSin[x/L]}, {{0}}}, {x, v}, {τ}, {Automatic}, Automatic, SamplingPeriod -> None];

在极坐标中的模型：

Wolfram Language code: StateSpaceTransform[asys, {{x -> L Sin[θ], v -> L Cos[θ] ω}, {θ, ω}}]//PowerExpand

应用一个直接正交变换于一个永久磁铁步进电机模型：

Wolfram Language code: motor = AffineStateSpaceModel[{{((-R)*Subscript[i, a] + ω*Sin[θ*Subscript[n, r]]* Subscript[K, m])/L, ((-R)*Subscript[i, b] - ω*Cos[θ*Subscript[n, r]]* Subscript[K, m])/L, ((-b)*ω - Sin[θ*Subscript[n, r]]* Subscript[i, a]*Subscript[K, m] + Cos[θ*Subscript[n, r]]* Subscript[i, b]*Subscript[K, m])/J, ω}, {{1, 0}, {0, 1}, {0, 0}, {0, 0}}, {θ, ω}, {{0, 0}, {0, 0}}}, {Subscript[i, a], Subscript[i, b], ω, θ}, {Subscript[v, a], Subscript[v, b]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None];

应用直接正交变换于电流 和 ：

Wolfram Language code: StateSpaceTransform[motor, {{Subscript[i, a] -> Subscript[i, d] Cos[Subscript[n, r]θ] - Subscript[i, q] Sin[Subscript[n, r]θ], Subscript[i, b] -> Subscript[i, d] Sin[Subscript[n, r]θ] + Subscript[i, q] Cos[Subscript[n, r]θ], ω -> ω, θ -> θ}, {Subscript[i, d], Subscript[i, q], ω, θ}}]

也应用直接正交变换于输入电压：

Wolfram Language code: SystemsModelSeriesConnect[TransferFunctionModel[{{{Cos[θ*Subscript[n, r]], -Sin[θ*Subscript[n, r]]}, {Sin[θ*Subscript[n, r]], Cos[θ*Subscript[n, r]]}}, 1}, ], %]//Simplify

将飞行器运动方程从体坐标转化到风轴：

Wolfram Language code: nsys = NonlinearStateSpaceModel[{{r*v - q*w + X/m - g*Sin[θ], (-r)*u + p*w + Y/m + g*Cos[θ]*Sin[ϕ], q*u - p*v + Z/m + g*Cos[θ]*Cos[ϕ]}, {}}, {u, v, w}, {X, Y, Z}, Automatic, Automatic, SamplingPeriod -> None, SystemsModelLabels -> {{}, None}];
Wolfram Language code: Simplify[StateSpaceTransform[nsys, {{u -> V Cos[α] Cos[β], v -> V Sin[β], w -> V Sin[α] Cos[β]}, {V, α, β}}]]

属性和关系  (8)

输出相应在相似度转化下不变：

Wolfram Language code: ssm1 = StateSpaceModel[{{{-0.75, 0.25}, {-0.01, -1}}, {{0.005}, {0.05}}, {{1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; ssm2 = StateSpaceTransform[ssm1, {{1, -1}, {0, 1}}];
Wolfram Language code: Grid[{Table[Plot[Evaluate@OutputResponse[s, 1, {t, 10}], {t, 0, 10}, ImageSize -> Small], {s, {ssm1, ssm2}}]}, Spacings -> 2]

相似的系统有完全相同的传递函数：

Wolfram Language code: ssm1 = StateSpaceModel[{{{a}}, {{b}}, {{c}}, {{d}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; ssm2 = StateSpaceTransform[ssm1, {{p}}];
Wolfram Language code: TransferFunctionModel[ssm1][s] == TransferFunctionModel[ssm2][s]

在相似度变换下，特征值（因此，稳定性）是不变的：

Wolfram Language code: ssm1 = StateSpaceModel[{{{Subscript[a, 11], Subscript[a, 12]}, {Subscript[a, 21], Subscript[a, 22]}}, {{Subscript[b, 1]}, {Subscript[b, 2]}}, {{Subscript[c, 1], Subscript[c, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; ssm2 = StateSpaceTransform[ssm1, {{Subscript[p, 11], Subscript[p, 12]}, {Subscript[p, 21], Subscript[p, 22]}}];
Wolfram Language code: {a1, a2} = First /@ Normal /@ {ssm1, ssm2};
Wolfram Language code: Simplify[Eigenvalues[a1] == Eigenvalues[a2]]

在相似变换下，能控性和能观测性是不变的：

Wolfram Language code: ssm1 = StateSpaceModel[{{{Subscript[a, 11], 0}, {0, Subscript[a, 22]}}, {{Subscript[b, 1]}, {0}}, {{Subscript[c, 1], Subscript[c, 2]}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; ssm2 = StateSpaceTransform[ssm1, {{Subscript[p, 11], Subscript[p, 12]}, {Subscript[p, 21], Subscript[p, 22]}}];
Wolfram Language code: ControllableModelQ /@ {ssm1, ssm2}
Wolfram Language code: ObservableModelQ /@ {ssm1, ssm2}

Hankel 奇异值 在相似度变化下不变：

Wolfram Language code: hankelSingularValues[sys_] := Module[{ns = N[sys]}, wc = ControllabilityGramian[ns];wo = ObservabilityGramian[ns];Sqrt[Eigenvalues[wc.wo]]];
Wolfram Language code: ssm1 = StateSpaceModel[{{{-1, 0, 3}, {2, -1, 3}, {1, -1, -1}}, {{1}, {1}, {2}}, {{0, 1, -1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]; ssm2 = StateSpaceTransform[ssm1, {{1, -1, 1}, {0, 1, 0}, {1, 0, 0}}];
Wolfram Language code: hankelSingularValues[ssm1] - hankelSingularValues[ssm2]//Norm

系统模型分解全部使用状态转化：

Wolfram Language code: ssm = StateSpaceModel[{{{0, 1}, {-12, -7}}, {{0}, {1}}, {{2, 1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];

基于 ControllabilityMatrix 的 ControllableDecomposition：

Wolfram Language code: {p, ssmt} = ControllableDecomposition[ssm]; {StateSpaceTransform[ssm, p], ssmt}

基于 ObservabilityMatrix 的 ObservableDecomposition：

Wolfram Language code: {p, ssmt} = ObservableDecomposition[ssm]; {StateSpaceTransform[ssm, p], ssmt}

基于 ObservabilityGramian 的 InternallyBalancedDecomposition：

Wolfram Language code: {p, ssmt} = InternallyBalancedDecomposition[ssm]; {StateSpaceTransform[ssm, p], ssmt}//N

基于 JordanDecomposition 的 JordanModelDecomposition：

Wolfram Language code: {p, ssmt} = JordanModelDecomposition[ssm]; {StateSpaceTransform[ssm, p], ssmt}

KroneckerModelDecomposition:

Wolfram Language code: {p, ssmt} = KroneckerModelDecomposition[ssm]; {StateSpaceTransform[ssm, p], ssmt}

通过坐标转换的线性化也是与使用 StateSpaceTransform 相关的：

Wolfram Language code: 𝒮 = StateTransformationLinearize[asys = AffineStateSpaceModel[{{4*Subscript[x, 1], 7*Subscript[x, 1]^2 + Subscript[x, 2]}, {{1}, {-1 + 2*Subscript[x, 1]}}, {Subscript[x, 1]}, {{0}}}, {Subscript[x, 1], Subscript[x, 2]}, {{Subscript[, 1], 0}}, {Automatic}, Automatic, SamplingPeriod -> None]];
Wolfram Language code: {StateSpaceTransform[asys, 𝒮[{"StateTransformation", "InverseStateTransformation"}]], 𝒮["TransformedSystem"]}

状态转化是反馈线性化的中间步骤：

Wolfram Language code: asys = AffineStateSpaceModel[{{Subscript[x, 2] + Subscript[x, 2]^2, Subscript[x, 3] - Subscript[x, 1]*Subscript[x, 4] + Subscript[x, 4]*Subscript[x, 5], -Subscript[x, 2]^2 + Subscript[x, 2]*Subscript[x, 4] + Subscript[x, 1]*Subscript[x, 5], Subscript[x, 5], Subscript[x, 2]^2}, {{0, 1}, {0, 0}, {Cos[Subscript[x, 1] - Subscript[x, 5]], 1}, {0, 0}, {0, 1}}, {Subscript[x, 1] - Subscript[x, 5], Subscript[x, 4]}, {{0, 0}, {0, 0}}}, {Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[x, 4], Subscript[x, 5]}, {Subscript[u, 1], Subscript[u, 2]}, {Automatic, Automatic}, Automatic, SamplingPeriod -> None]; ℱ = FeedbackLinearize[asys]

应用反馈和状态转换来获取变换系统：

Wolfram Language code: SystemsModelSeriesConnect[ℱ["FeedbackCompensator"], asys];
Wolfram Language code: StateSpaceTransform[%, {Automatic, ℱ["InverseStateTransformation"]}]

"TransformedSystem" 属性给出相同的结果：

Wolfram Language code: NonlinearStateSpaceModel[ℱ["TransformedSystem"]]

可能存在的问题  (2)

变换矩阵必须是标准正交的或者可逆的：

Wolfram Language code: p = {{2, 1, 1}, {1, 0, -1}, {1, 1, 2}};
Wolfram Language code: StateSpaceTransform[StateSpaceModel[{RandomReal[2, {3, 3}], RandomReal[1, {3, 1}], {{1, 0, 0}}}], p]

该矩阵既不是标准正交的，也不是可逆的：

Wolfram Language code: {p.p == IdentityMatrix[3], MatrixRank[p] == 3}

如果有多个反解的话，只有一个被选定：

Wolfram Language code: assm = AffineStateSpaceModel[{{-Subscript[x, 1] + Subscript[x, 2], -Subscript[x, 1]}, {{1 + Subscript[x, 2]}, {Subscript[x, 1]}}, {Subscript[x, 1]}, {{0}}}, {{Subscript[x, 1], 0}, {Subscript[x, 2], 0}}, Automatic, {Automatic}, Automatic, SamplingPeriod -> None];
Wolfram Language code: pp = {Subscript[x, 1] -> Subscript[z, 1] + Subsuperscript[z, 2, 2], Subscript[x, 2] -> Subscript[z, 2] + Subscript[z, 1]Subscript[z, 2]}; qq = {Subscript[z, 1], Subscript[z, 2]};
Wolfram Language code: StateSpaceTransform[assm, {pp, qq}]

有三个反解：

Wolfram Language code: sols = Solve[Equal@@@pp, qq]; Length[sols]

选出期望的反解：

Wolfram Language code: StateSpaceTransform[assm, {pp, sols[[3]]}]
Wolfram Research (2010)，StateSpaceTransform，Wolfram 语言函数，https://reference.wolfram.com/language/ref/StateSpaceTransform.html （更新于 2014 年）.

文本

Wolfram Research (2010)，StateSpaceTransform，Wolfram 语言函数，https://reference.wolfram.com/language/ref/StateSpaceTransform.html （更新于 2014 年）.

CMS

Wolfram 语言. 2010. "StateSpaceTransform." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2014. https://reference.wolfram.com/language/ref/StateSpaceTransform.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_statespacetransform, author="Wolfram Research", title="{StateSpaceTransform}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/StateSpaceTransform.html}", note=[Accessed: 03-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_statespacetransform, organization={Wolfram Research}, title={StateSpaceTransform}, year={2014}, url={https://reference.wolfram.com/language/ref/StateSpaceTransform.html}, note=[Accessed: 03-November-2026]}