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DynamicSystems

 SSTransformation
 perform similarity transformations on state-space matrices

 Calling Sequence SSTransformation(Amat, Bmat, Cmat, Dmat, opts )

Parameters

 Amat - Matrix; system matrix of a state-space system Bmat - Matrix; input matrix of a state-space system Cmat - Matrix; output matrix of a state-space system Dmat - (optional) Matrix; direct-through matrix of a state-space system. If not specified, a zero matrix of appropriate dimension is used as the default (if D is requested for the output) opts - (optional) equation(s) of the form option = value; specify options for the SSTransformation command

Options

 • form = ControlStaircase, ObserveStaircase, ControlCanon, ObserveCanon, ModalCanon, or Balanced
 Specifies the form of the transformed Matrix. The default is ControlStaircase.
 • output = T, Tinv, A, B, C, D, r, or list of same
 Specifies the values returned.  The default is T. If output is assigned a name, then the value corresponding to that name is returned. If output is a assigned a list of names, then the values corresponding to each of the names in the list is returned, in that order.
 – The values for T and Tinv are the transformation matrix and its inverse, respectively.
 – The values for A, B, C, and D are the transformed state-space matrices.
 – The value for r is the dimension of the system Matrix. For the ControlStaircase and ObserveStaircase transforms, r is the dimension of the controllable and observable subsystems, respectively. For the other transforms, r is the dimension of A.
 • discrete = truefalse
 True means the system is discrete, false means it is continuous. The default is the value of the variable discrete in the DynamicSystems[SystemOptions] command.
 • returnlist = truefalse
 True means return a list; false means return an expression sequence. The default is false.

Description

 • The SSTransformation command performs a selected similarity transformation on state-space matrices.
 • The similarity transformations are ControlStaircase, ObserveStaircase, ControlCanon, ObserveCanon, ModalCanon, and Balanced. The transformation is selected with the named option form.
 • For a state-space system defined by the state and output equations $\frac{ⅆ}{ⅆt}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}x\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$ and $y\left(t\right)=Cx\left(t\right)+\mathrm{D}u\left(t\right)$, the computed similarity transformation matrix T transforms the input state-space system as follows:

x[new](t) = Tinv . x(t)

A[new] = Tinv . A . T

B[new] = Tinv . B

C[new] = C . T

D[new] = D

where Tinv is the inverse of the matrix T.

ControlStaircase Transformation

 • The ControlStaircase transformation transforms the system matrix A into a lower-triangle staircase form

$A=\mathrm{Matrix}\left(\left[\left[{A}_{\mathrm{uc}},0\right],\left[{A}_{2,1},{A}_{c}\right]\right]\right)$

$B=\mathrm{Matrix}\left(\left[\left[0\right],\left[{B}_{c}\right]\right]\right)$

$C=\mathrm{Matrix}\left(\left[\left[{C}_{\mathrm{uc}},{C}_{c}\right]\right]\right)$

The submatrices ${A}_{c},{B}_{c},{C}_{c}$ are the controllable subsystem of $A,B,C$. The dimension of ${A}_{c}$ is assigned the output option r.

ObserveStaircase Transforms

 • The ObserveStaircase transformation transforms the system matrix A into an upper-triangle staircase form

$A=\mathrm{Matrix}\left(\left[\left[{A}_{\mathrm{uo}},{A}_{1,2}\right],\left[0,{A}_{o}\right]\right]\right)$

$B=\mathrm{Matrix}\left(\left[\left[{B}_{\mathrm{uo}}\right],\left[{B}_{o}\right]\right]\right)$

$C=\mathrm{Matrix}\left(\left[\left[0,{C}_{o}\right]\right]\right)$

The submatrices ${A}_{o},{B}_{o},{C}_{o}$ are the observable subsystem of $A,B,C$. The dimension of ${A}_{o}$ is assigned the output option r.

ControlCanon Transformation

 • The ControlCanon transformation transforms the input system into the controllable canonical form where the system matrix A has the form:

$A=\mathrm{Matrix}\left(\left[\left[0,1,0,0\right],\left[0,0,1,0\right],\left[0,0,0,1\right],\left[-{a}_{0},-{a}_{1},-{a}_{2},-{a}_{3}\right]\right]\right)$

The ${a}_{i}$ are the coefficients of the DynamicSystems[CharacteristicPolynomial] of the n x n system matrix A: s^n + a[n-1]*s^(n-1) + ... + a. The ControlCanon transformation applies only to a controllable system.

ObserveCanon Transformation

 • The ObserveCanon transformation transforms the input system into the observable canonical form where the system matrix A has the form:

$A=\mathrm{Matrix}\left(\left[\left[0,0,0,-{a}_{0}\right],\left[1,0,0,-{a}_{1}\right],\left[0,1,0,-{a}_{2}\right],\left[0,0,1,-{a}_{3}\right]\right]\right)$

The ${a}_{i}$ are the coefficients of the DynamicSystems[CharacteristicPolynomial] of the n x n system matrix A: s^n + a[n-1]*s^(n-1) + ... + a. The ObserveCanon transformation only applies to observable system. The observable canonical form is also known as the companion canonical form.

ModalCanon Transformation

 • The ModalCanon transformation transforms the input system into the modal canonical form, which has a block-diagonal system matrix A. The block-diagonal entries of the transformed A are the distinct eigenvalues of the given A. Real eigenvalues correspond to a 1 x 1 block while complex eigenvalues constitute a 2 x 2 block of the form $\mathrm{Matrix}\left(\left[\left[\mathrm{\sigma },\mathrm{\omega }\right],\left[-\mathrm{\omega },\mathrm{\sigma }\right]\right]\right)$ where $\mathrm{\sigma }$ and $\mathrm{\omega }$ are the real and imaginary parts, respectively, of the complex eigenvalue. The ModalCanon transformation only applies to numeric systems with distinct eigenvalues (that is, no repeated eigenvalues). The modal canonical form is also known as the diagonal canonical form.

Balanced Transformation

 • The Balanced transformation transforms the input system such that the controllability and observability grammians of the system are diagonal and equal to each other. The Balanced transformation is only applicable to stable systems that are both controllable and observable.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{Amat}≔⟨⟨-5|1|0⟩,⟨0|-2|1⟩,⟨0|0|-1⟩⟩:$
 > $\mathrm{Bmat}≔⟨⟨0,0,1⟩⟩:$
 > $\mathrm{Cmat}≔⟨⟨1|0|0⟩⟩:$
 > $\mathrm{Dmat}≔⟨⟨0⟩⟩:$
 > $\mathrm{SSTransformation}\left(\mathrm{Amat},\mathrm{Bmat},\mathrm{Cmat},\mathrm{Dmat},\mathrm{form}=\mathrm{ControlCanon},\mathrm{output}=\left['A','B','C','\mathrm{D}'\right]\right)$
 $\left[\begin{array}{ccc}{0}& {1}& {0}\\ {0}& {0}& {1}\\ {-10}& {-17}& {-8}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{c}{0}\end{array}\right]$ (1)
 > $\mathrm{SSTransformation}\left(\mathrm{Amat},\mathrm{Bmat},\mathrm{Cmat},\mathrm{Dmat},\mathrm{form}=\mathrm{ObserveCanon},\mathrm{output}=\left['A','B','C','\mathrm{D}'\right]\right)$
 $\left[\begin{array}{ccc}{0}& {0}& {-10}\\ {1}& {0}& {-17}\\ {0}& {1}& {-8}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {0}\\ {0}\end{array}\right]{,}\left[\begin{array}{ccc}{0}& {0}& {1}\end{array}\right]{,}\left[\begin{array}{c}{0}\end{array}\right]$ (2)
 > $\mathrm{SSTransformation}\left(\mathrm{Amat},\mathrm{Bmat},\mathrm{Cmat},\mathrm{Dmat},\mathrm{form}=\mathrm{ModalCanon},\mathrm{output}=\left['A','B','C','\mathrm{D}','T'\right]\right)$
 $\left[\begin{array}{ccc}{-5}& {0}& {0}\\ {0}& {-2}& {0}\\ {0}& {0}& {-1}\end{array}\right]{,}\left[\begin{array}{c}\frac{{1}}{{12}}\\ \frac{{1}}{{3}}\\ \frac{{1}}{{4}}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {-1}& {1}\end{array}\right]{,}\left[\begin{array}{c}{0}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {-1}& {1}\\ {0}& {-3}& {4}\\ {0}& {0}& {4}\end{array}\right]$ (3)