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DynamicSystems

 ControllabilityMatrix
 compute the controllability matrix

 Calling Sequence ControllabilityMatrix( sys ) ControllabilityMatrix( Amat, Bmat )

Parameters

 sys - System(ss); state-space system Amat - Matrix; state-space Matrix A Bmat - Matrix; state-space Matrix B

Description

 • The ControllabilityMatrix command computes the controllability matrix of a state-space system.
 • If the parameter sys is a state-space System, then the A and B Matrices are sys:-a and sys:-b, respectively.
 • If the parameters Amat and Bmat are Matrices, then they are the A and B Matrices, respectively.
 • The controllability matrix has dimensions n x n*m, where n is the number of states (dimension of A) and m is the number of inputs (column dimension of B). It has the form << B | A . B | A^2 . B | A^3 . B | ... | A^(n-1) . B >>.

Examples

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