# Control Systems/State Feedback

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## State Observation

The state space model of a system is the model of a single plant, not a true feedback system. The feedback mechanism that relates x' to x is a representation of the mechanism internal to the plant, where the state of the plant is related to its derivative. As such, we do not have an A "component" in the sense that we can swap one A "chip" with another A "chip". The entire state-space model, incorporating A, B, C, and D are all part of one device. Frequently, these matrices are immutable, that is that they cannot be altered by the engineer, because they are intrinsic parts of the plant. However, these matrices can change if the plant itself is altered, such as through thermal effects and RF interference.

If the system can be treated as basically immutable (except for effects out of the engineers control), then we need to find a way to modify the system externally. From our studies in classical controls, we know that the best system for such modifications is a feedback loop. What we would like to do, ultimately, is to add an additional feedback element, K that can be used to move the poles of the system to any desired location. Using a technique called "state feedback" on a controllable system, we can do just that.

## State Feedback

In state feedback, the value of the state vector is fed back to the input of the system. We define a new input, r, and define the following relationship:

$u(t) = r(t) + Kx(t)$

K is a constant matrix that is external to the system, and therefore can be modified to adjust the locations of the poles of the system. This technique can only work if the system is controllable.

### Closed-Loop System

If we have an external feedback element K, the system is said to be a closed-loop system. Without this feedback element, the system is said to be an open-loop system. Using the relationship we've outlined above between r and u, we can write the equations for the closed-loop system:

$x' = Ax + B(r + Kx)$
$x' = (A + BK)x + Br$

Now, our closed-loop state equation appears to have the same form as our open loop state equation, except that the sum (A + BK) replaces the matrix A. We can define the closed-loop state matrix as:

$A_{cl} = (A_{ol} + BK)$

Acl is the closed-loop state matrix, and Aol is the open-loop state matrix. By altering K, we can change the eigenvalues of this matrix, and therefore change the locations of the poles of the system. If the system is controllable, we can find the characteristic equation of this system as:

$\alpha(s) = |sI - A_{cl}| = |sI - (A_{ol} + BK)|$

Computing the determinant is not a trivial task, the determinant of that matrix can be very complicated, especially for larger systems. However, if we transform the system into controllable canonical form, the calculations become much easier.