ObservabilityMatrix - Maple Help

DynamicSystems

 ObservabilityMatrix
 compute the observability matrix

 Calling Sequence ObservabilityMatrix( sys ) ObservabilityMatrix( Amat, Cmat )

Parameters

 sys - System; state-space system Amat - Matrix; state-space matrix A Cmat - Matrix; state-space matrix C

Description

 • The ObservabilityMatrix command computes the observability matrix of a state-space system.
 • If the parameter sys is a state-space System, then the A and C Matrices are sys:-a and sys:-c, respectively.
 • If the parameters Amat and Cmat are Matrices, then they are the A and C Matrices, respectively.
 • The observability matrix has dimensions o*n x n, where n is the number of states (dimension of A) and o is the number of outputs (row dimension of C) It has the form <, , , , ..., >.

Examples

 > $\mathrm{with}\left(\mathrm{DynamicSystems}\right):$
 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $\mathrm{sys1}≔\mathrm{StateSpace}\left(\frac{1}{{s}^{2}+s+10}\right):$
 > $\mathrm{sys1}:-a,\mathrm{sys1}:-c$
 $\left[\begin{array}{cc}{0}& {1}\\ {-10}& {-1}\end{array}\right]{,}\left[\begin{array}{cc}{1}& {0}\end{array}\right]$ (1)
 > $\mathrm{ObservabilityMatrix}\left(\mathrm{sys1}\right)$
 $\left[\begin{array}{cc}{1}& {0}\\ {0}& {1}\end{array}\right]$ (2)
 > $\mathrm{sys2}≔\mathrm{StateSpace}\left(⟨⟨-3|1|0⟩,⟨-5|0|1⟩,⟨-3|0|0⟩⟩,⟨⟨1,2,3⟩⟩,⟨⟨1|0|0⟩⟩,⟨⟨0⟩⟩\right):$
 > $\mathrm{sys2}:-a,\mathrm{sys2}:-c$
 $\left[\begin{array}{ccc}{-3}& {1}& {0}\\ {-5}& {0}& {1}\\ {-3}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{ccc}{1}& {0}& {0}\end{array}\right]$ (3)
 > $\mathrm{ObservabilityMatrix}\left(\mathrm{sys2}:-a,\mathrm{sys2}:-c\right)$
 $\left[\begin{array}{ccc}{1}& {0}& {0}\\ {-3}& {1}& {0}\\ {4}& {-3}& {1}\end{array}\right]$ (4)
 > $\mathrm{sys3}≔\mathrm{StateSpace}\left(\mathrm{DiagonalMatrix}\left(\left[{a}_{1},{a}_{2},{a}_{3}\right]\right),⟨⟨0|0⟩,⟨{b}_{1}|0⟩,⟨0|{b}_{2}⟩⟩,⟨⟨{c}_{1}|0|0⟩,⟨0|0|{c}_{3}⟩⟩,⟨⟨0|0⟩,⟨0|0⟩⟩\right):$
 > $\mathrm{sys3}:-a,\mathrm{sys3}:-c$
 $\left[\begin{array}{ccc}{{a}}_{{1}}& {0}& {0}\\ {0}& {{a}}_{{2}}& {0}\\ {0}& {0}& {{a}}_{{3}}\end{array}\right]{,}\left[\begin{array}{ccc}{{c}}_{{1}}& {0}& {0}\\ {0}& {0}& {{c}}_{{3}}\end{array}\right]$ (5)
 > $\mathrm{ObservabilityMatrix}\left(\mathrm{sys3}\right)$
 $\left[\begin{array}{ccc}{{c}}_{{1}}& {0}& {0}\\ {0}& {0}& {{c}}_{{3}}\\ {{a}}_{{1}}{}{{c}}_{{1}}& {0}& {0}\\ {0}& {0}& {{a}}_{{3}}{}{{c}}_{{3}}\\ {{a}}_{{1}}^{{2}}{}{{c}}_{{1}}& {0}& {0}\\ {0}& {0}& {{a}}_{{3}}^{{2}}{}{{c}}_{{3}}\end{array}\right]$ (6)