State feedback vs. output feedback
A feedback controller computes its input from information about the system, which may be more or less complete. State feedback uses the full state, which is rarely measured directly. Output feedback uses only the measured output, and it may need an observer or other dynamics to do what state feedback can.
Definitions
Level:
Undergraduate
State feedback is a control law that uses the full state, $u = k(x)$ instead of only a partial measurement of the state.
Examples
-
$u = -K x$, with the full state measured
$u = -K x$ is state feedback because the input uses the whole state. For a controllable system, $K$ can place the closed-loop eigenvalues anywhere.
-
$u = -K y$, when $y = x$
$u = -K y$ is also state feedback, because here the output is the whole state.
Non-examples
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$u = -K \hat{x}$, where an observer computes $\hat{x}$ from $y$
By the separation principle, a stabilizing state feedback combined with a converging observer stabilizes the system.
- $u = -k y$, for $\dot x_1 = x_2$, $\dot x_2 = u$ with $y = x_1$
Properties
- For a controllable linear system, linear state feedback can place the closed-loop eigenvalues anywhere (pole placement).
Undergraduate
Consider a control system with state $x$ and measured output $y$. Output feedback is a control law that uses only $y$, either directly, as in $u = k(y)$, or through a dynamic controller.
Examples
-
$u = -K y$, when $y = x$
$u = -K y$ is output feedback, because the input is computed from the measured output $y$.
-
$u = -K \hat{x}$, where an observer computes $\hat{x}$ from $y$
$u = -K \hat{x}$ is output feedback because the observer’s estimate $\hat{x}$ is computed from the output $y$, not from the state.
-
$u = -k y$, for $\dot x_1 = x_2$, $\dot x_2 = u$ with $y = x_1$
$u = -k y$ is output feedback because the input uses only the measured position $y = x_1$, not the velocity. The closed loop $\ddot x_1 = -k x_1$ oscillates for every $k > 0$, so no static output feedback stabilizes the double integrator.
Non-examples
Properties
- Some controllable and observable systems cannot be stabilized by static output feedback, but can be by . They can, however, be stabilized by dynamic output feedback, for example by combining an observer with state feedback (the separation principle).
Examples
| Example | state feedback | output feedback | Why |
|---|---|---|---|
| $u = -K x$, with the full state measured | ✓ | ✗ | |
| $u = -k y$, for $\dot x_1 = x_2$, $\dot x_2 = u$ with $y = x_1$ | ✗ | ✓ | |
| $u = -K \hat{x}$, where an observer computes $\hat{x}$ from $y$ | ✗ | ✓ | |
| $u = -K y$, when $y = x$ | ✓ | ✓ |