Control Systems/Standard Forms

Companion Form

We can create a companion form matrix in one of two ways:

$\left[\begin{array}{cccccc}0& 0& 0& \cdots & 0& -a_0\\ 1& 0& 0& \cdots & 0& -a_1\\ 0& 1& 0& \cdots & 0& -a_2\\ 0& 0& 1& \cdots & 0& -a_3\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& -a_{n-1}\end{array}\right]$

Or, we can also write it as:

$\left[\begin{array}{cccccc}-a_{n-1}& -a_{n-2}& -a_{n-3}& \cdots & -a_1& -a_0\\ 1& 0& 0& \cdots & 0& 0\\ 0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& 0\end{array}\right]$

Controllable-Canonical Form

Controllable-Canonical Form is useful in a number of cases, especially for simplifying certain equations.

The controllable-canonical form is as follows:

$A=\left[\begin{array}{cccccc}{a}_{n-1}& a_{n-2}& a_{n-3}& \cdots & a_1& a_0\\ 0& 0& 0& \cdots & 0& 0\\ 1& 0& 0& \cdots & 0& 0\\ 0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & 1& 0\end{array}\right]$

$B=\left[\begin{array}{c}1\\ 0\\ ⋮\\ 0\end{array}\right]$

If we have two spaces, space v which is the original space of the system (A, B, C, and D), then we can transform our system into the w space which is in controllable-canonical form (A_w, B_w, C_w, D_w) using a transformation matrix T_w. We define this transformation matrix as:

$T={\zeta }_v{\zeta }_w^{-1}$

Where ζ is the controllability matrix. We know before hand what the matrix A_w and B_w. This is because we know the form of the matrices, and we know the coefficients of the characteristic equation. If we know these two matrices, then we can form ζ_w. We can then use this matrix to create our transformation matrix.

We will discuss controllable canonical form later when we discuss state-feedback and closed-loop systems.

MATLAB Standard Forms

MATLAB contains a function for automatically transforming a state space equation into one of the above listed canonical forms. At the moment, the canon function only supports the "companion" and "modal" forms. These can be obtained by using the following respective lines of code:

[Ap, Bp, Cp, Dp, P] = canon(A, B, C, D, 'companion');
[Ap, Bp, Cp, Dp, P] = canon(A, B, C, D, 'modal');