State Feedback

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Textbook Contents

State Feedback (pdf, 28Sep12)

  • 1. Reachability
  • 2. Stabilization by State Feedback
  • 3. State Feedback Design Issues
  • 4. Integral Action
  • 5. Further Reading
  • Exercises

Lecture Materials

Supplemental Information

Chapter Summary


  1. A linear system with dynamics


    is said to be reachable if we can find an input math defined on the interval math that can steer the system from a given final point math to a desired final point math.

  2. The reachability matrix for a linear system is given by


    A linear system is reachable if and only if the reachability matrix math is invertible (assuming a single intput/single output system). Systems that are not reachable have states that are constrained to have a fixed relationship with each other.

  3. A linear system of the form




    A reachable linear system can be transformed into reachable canonical form through the use of a coordinate transformation math.

  4. A state feedback law has the form


    where math is the reference value for the output. The closed loop dynamics for the system are given by


    The stability of the system is determined by the stability of the matrix math. The equilibrium point and steady state output (assuming the systems is stable) are given by


    Choosing math as

    gives math.




  6. Integral feedback can be used to provide zero steady state error instead of careful calibration of the gain math. An integral feedback controller has the form




    is the integral error. The gains math, math and math can be found by designing a stabilizing state feedback for the system dynamics augmented by the integrator dynamics.

  7. A linear quadratic regulator minimizes the cost function







Additional Exercises


Frequently Asked Questions





Additional Information
