Full state feedback

Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in pre-determined locations in the s-plane.[1] Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. The system must be considered controllable in order to implement this method.

Principle[2]

If the closed-loop input-output transfer function can be represented by a state space equation, see State space (controls),

$\dot{\underline{x}}=\mathbf{A}\underline{x}+\mathbf{B}\underline{u};$
$\underline{y} = \mathbf{C}\underline{x}+\mathbf{D}\underline{u}$

then the poles of the system are the roots of the characteristic equation given by

$\left|s\textbf{I}-\textbf{A}\right|=0.$

Full state feedback is utilized by commanding the input vector $\underline{u}$. Consider an input proportional (in the matrix sense) to the state vector,

$\underline{u}=-\mathbf{K}\underline{x}$.

Substituting into the state space equations above,

$\dot{\underline{x}}=(\mathbf{A}-\mathbf{B}\mathbf{K})\underline{x};$
$\underline{y} = (\mathbf{C}-\mathbf{D}\mathbf{K})\underline{x}.$

The roots of the FSF system are given by the characteristic equation, $\det\left[s\textbf{I}-\left(\textbf{A}-\textbf{B}\textbf{K}\right)\right]$. Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix $\textbf{K}$ which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation.

Example of FSF

Consider a control system given by the following state space equations

$\dot{\underline{x}}=\begin{bmatrix}0 & 1 \\ -2 & -3\end{bmatrix}\underline{x}+\begin{bmatrix} 0 \\ 1\end{bmatrix}\underline{u}$

The uncontrolled system has closed-loop poles at $s=-1$ and $s=-2$. Suppose, for considerations of the response, we wish the controlled system eigenvalues to be located at $s=-1$ and $s=-5$. The desired characteristic equation is then $s^2+6s+5=0$.

Following the procedure given above, $\mathbf{K}=\begin{bmatrix} k_1 & k_2\end{bmatrix}$, and the FSF controlled system characteristic equation is

$\left|s\mathbf{I}-\left(\mathbf{A}-\mathbf{B}\mathbf{K}\right)\right|=\det\begin{bmatrix}s & -1 \\ 2+k_1 & s+3+k_2 \end{bmatrix}=s^2+(3+k_2)s+(2+k_1)$.

Upon setting this characteristic equation equal to the desired characteristic equation, we find

$\mathbf{K}=\begin{bmatrix}3 & 3\end{bmatrix}$.

Therefore, setting $\underline{u}=-\mathbf{K}\underline{x}$ forces the closed-loop poles to the desired locations, affecting the response as desired.

NOTE: This only works for Single-Input systems. Multiple input systems will have a K matrix that is not unique. Choosing, therefore, the best K values is not trivial. Recommend using a linear-quadratic regulator for such applications.

References

1. ^ *Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second Edition. Springer. ISBN 0-387-98489-5.
2. ^ Control Design Using Pole Placement