# Control Systems/Standard Forms

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## Companion Form

A companion form contains the coefficients of a corresponding characteristic polynomial along one of its far rows or columns. For example, one companion form matrix is:

$\begin{bmatrix} 0 & 0 & 0 & \cdots & 0 & -a_0 \\ 1 & 0 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & 0 & \cdots & 0 & -a_2 \\ 0 & 0 & 1 & \cdots & 0 & -a_3 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & -a_{n-1} \end{bmatrix}$

and another is:

$\begin{bmatrix} -a_{n-1} & -a_{n-2} & -a_{n-3} & \cdots & -a_1 & -a_0 \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}$

There are two companion forms that are convenient to use in control theory, namely the observable canonical form and the controllable canonical form. These two forms are roughly transposes of each other (just as observability and controllability are dual ideas). When placed in one of these forms, the design of controllers or observers is simplified because the structure of the system is made apparent (and is easily modified with the desired control).

### Observable Canonical Form

Observable-Canonical Form is useful in a number of cases, especially for designing observers.

The observable-canonical form is as follows:

$A = \begin{bmatrix} -a_1 & 1 & 0 & \cdots & 0 \\ -a_2 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -a_{n-1} & 0 & 0 & \cdots & 1 \\ -a_n & 0 & 0 & \cdots & 0 \end{bmatrix}$
$B = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$
$C = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix}$

### Controllable Canonical Form

Controllable-Canonical Form is useful in a number of cases, especially for designing controllers when the full state of the system is known.

The controllable-canonical form is as follows:

$A = \begin{bmatrix} -a_1 & -a_2 & -a_3 & \cdots & -a_{n-1} & -a_n \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 0 \end{bmatrix}$
$B = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$
$C = \begin{bmatrix} b_1 & b_2 & b_3 & \cdots & b_n \end{bmatrix}$
$D = \begin{bmatrix} b_0 \end{bmatrix}$

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 (Aw, Bw, Cw, Dw) using a transformation matrix Tw. We define this transformation matrix as:

$T = \zeta_v \zeta_w^{-1}$

Where ζ is the controlability matrix.

Notice that we know beforehand Aw and Bw, since we know both the form of the matrices and the coefficients of the equation (e.g. a linear ODE with constant coefficients or a transfer function).

If we know these two matrices, then we can form ζw. We can then use this matrix to create our transformation matrix.

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

## Modal Form

In this form, the state matrix is a diagonal matrix of its (non-repeated) eigenvalues. The control has a unitary influence on each eigenspace, and the output is a linear combination of the contributions from the eigenspaces (where the weights are the complex residuals at each pole).

$A = \begin{bmatrix} -p_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & -p_2 & 0 & \cdots & 0 & 0 \\ 0 & 0 & -p_3 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 0 & -p_n \end{bmatrix}$
$B = \begin{bmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}$
$C = \begin{bmatrix} c_1 & c_2 & \cdots & c_n \end{bmatrix}$

### Jordan Form

This "almost diagonal" form handles the case where eigenvalues are repeated. The repeated eigenvalues represent a multi-dimensional eigenspace, and so the control only enters the eigenspace once and its integrated through the other states of that small subsystem.

$A = \begin{bmatrix} -p_1 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & -p_1 & 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & -p_1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & 0 & -p_4 & 0 &\cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots &\ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & \cdots & 0 & -p_n \end{bmatrix}$
$B = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \\ \vdots \\ 1 \end{bmatrix}$
$C = \begin{bmatrix} c_1 & c_2 & \cdots & c_n \end{bmatrix}$

## Computing Standard Forms in MATLAB

This operation can be performed using this MATLAB command:
canon

MATLAB contains a function for automatically transforming a state space equation into a companion (e.g., controllable or observable canonical form) form.

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

This operation can be performed using this MATLAB command:
compan

Moving from one companion form to the other usually involves elementary operations on matrices and vectors (e.g., transposes or interchanging rows). Given a vector with the coefficients of a characteristic polynomial, MATLAB can compute the corresponding companion form.

compan(P)


Given another vector with the coefficients of a transfer function's numerator polynomial, the canon command can do the same.

[Ap, Bp, Cp, Dp, P] = canon(tf(Pnum,Pden), 'companion');


To transform a state space equation into a modal (e.g., diagonal) form, the same command can be used.

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

This operation can be performed using this MATLAB command:
jordan

However, MATLAB also includes a command to compute the Jordan form of a matrix, which is a modified modal form suited for matrices that have repeated eigenvalues.

jordan(A)