State Feedback

Basic information

  • Feedback: Part of the output signal is sent back to the input.
  • Negative feedback:
  • It stabilizes a system and usually speeds up the response but decreases the output gain.
  • The closed-loop transfer function for constant feedback is:

Negative feedback can push poles farther into the LHP (more stable) and increase bandwidth.

  • For example, if

G(s) = 1/(S + 1) and K = 4, Gf(s) = 1/(s + 5).

  • Positive feedback:
  • Destabilizes a system and slows down the response time. Positive feedback should be eliminated in most systems.
  • Used in oscillators to set up oscillations:
  • Positive feedback to start and enhance oscillation.
  • Negative feedback to maintain the output level once the oscillation has reached the desired level. Usually, the negative feedback is controlled by the output level of the oscillator via a nonlinear element. The combined effect of the positive and negative feedback provides a loop gain of (+1).
  • A filter circuit is used to select the oscillation frequency.

SISO State feedback

  • Formulation:

x’ = (A – bK) x + b u = Afx+ b u

y = cx

G(s) = c(sI – A)-1 b

Gf(s) = c(sI – A + bK)-1 b = c(sI – Af)-1 b

  • (A – bK,b) controllable  (A,b) controllable. Although controllability is invariant under state feedback, observability is not necessarily invariant under state feedback.
  • State feedback does not affect the zeroes of G(s). It only shifts the position of the poles.
  • The eigenvalues can be repositioned by applying state feedback with an appropriate gain matrix K.

Example:

  • The direct evaluation technique, however, becomes rather tedious for higher dimensional systems.
  • A more systematic approach is needed. For a controllable system, we can use the controllability matrix to compute the gain matrix (text, p. 235)

Example:

Example:

  • Lyapunov Equation: Every eigenvalue of the feedback system must be different from that of the original system.

1)Select F (nn): Use the modal form of Af

2)Select k (1 n): At least one nonzero entry associated with each diagonal block (text, p. 241)

3)Solve the unique T in the Lyapunov equation: AT – TF=bk

4)k =kT-1

Example: F and k are pre-selected (P. 241)

  •  placement:
  • The controllability condition is the necessary and sufficient condition for the placement of s to any desired locations.
  • Criteria: rise/fall time (response time), settling time, overshoot, …
  • Guide: Place all s inside a region:

- Large  fast response

- Large r  fast response, large input (low gain), and large BW (more noise)

- Large  large overshoot

- If s are too close together, the response will be slow and the required input will be large. Hence, s should be placed evenly along an arc.

  • Discrete-time systems: The discussion applies directly except for the pole placement region, which can be obtained from the z = es transformation (Fig. 8.3b, p. 239).

Regulation and tracking

  • Regulator problem: If the reference signal is zero [r(t) = 0] and the system is disturbed by some nonzero initial conditions, we need to find a state feedback gain to ensure that the response dies out at a desired rate.
  • Tracking problem: If the reference signal is a constant [r(t) = a], we need to find a state feedback gain to ensure that the response approaches r(t) = a as t approaches infinity. This is also called asymptotic tracking of a step reference input.
  • If a = 0 then this becomes the regulator problem.
  • If s are inside a sector the disturbed response will die out quickly. No state feedback is needed. Otherwise, we can use state feedback to shift the s inside a sector.
  • The tracking problem: Tracking is possible if the G(s) has no zero at s = 0.
  • In addition to state feedback, tracking requires a feedforward path u(t) = pr(t) – k x . The calculation of the feedforward gain is illustrated below: