With the system equation \(\dot{x}=Ax+Bu\) the input is set to a constant gain \(u=-Kx\). This is the same as setting the regulator reference $R$ to zero.
$$u=K(R-x)=-Kx$$
For a SISO system, \(K\) is a vector. For a MIMO system \(K\) is a matrix. When this control value is substituted into the system equation we get.
$$\dot{x}=Ax+B(-Kx)=(A-BK)x$$
This gives us the new closed-loop state matrix
$$A_{CL}=A-BK$$
The gain matrix/vector \(K\) is then set to ensure stability as well as bandwidth requirements, overshoot requirements, and rise time. The downside of state-feedback is that we need to have access to all of the states. If you use a state-estimator then you can set the feedback gains from the result of the estimated states.
$$u=-K\hat{x}$$
This allows you to perform full state feedback when you don’t know the states. Static state feedback does not apply to problems where the dynamics and targeted operating point are uncertain. Static-feedback changes the operating conditions of the open-loop system and may result in wasted control effort.
State-Space Estimators – can be used to estimate the states x
Eurofighter Flight Control Laws – uses full state feedback
Dual Active Bridge Converter – uses state feedback to suppress resonance.
[[Disadvantages of Dynamic Inversion Controllers]] – requires full state feedback
[[Honeywell Multi-Application Control (MACH)]] – assumes full state feedback
[[Modified Mu-Synthesis]] – uses state feedback to determine the weights
[[Pole Placement for Helicopter Flight Control Systems]] – state feedback can be used to place poles for
[[Kleinman’s optimal control model]] – include state-feedback
[[Flight Path Angle Controller]] – based on 3-state feedback
LQR-PI Controller – preserves the stability properties of state feedback
[[Foiling Catamaran]] – can use a state-feedback controller
Sources
- [1] D. Bevly, “MECH 4420 Lecture: State Space Control”.
Backlinks
[[Closed-Loop Tracking]]
[[Flight Path Angle Controller]]
LQR-PI Controller
MIMO State-Space Models
[[State-Space Model]]
[[Types of Feedback Controllers]]
X-29 Flight Control System