State Feedback

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This chapter describes how feedback can be used to shape the local behavior of a system. The concept of reachability is introduced and used to investigate how to "design" the dynamics of a system through placement of its eigenvalues. In particular, it will be shown that under certain conditions it is possible to assign the system eigenvalues to arbitrary values by appropriate feedback of the system state.

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

This chapter describes how state feedback can be used to design the (closed loop) dynamics of the system:

  1. A linear system with dynamics

    math

    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

    math

    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

    math

    is said to be in reachable canonical form. A system in this form is always reachable and has a characteristic polynomial given by

    math

    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

    math

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

    math

    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

    math

    Choosing math as

    math
    gives math.

  5. If a system is reachable, then there exists a feedback law of the form

    math

    the gives a closed loop system with an arbitrary characteristic polynomial. Hence the eigenvalues of a reachable linear system can be placed arbitrarily through the use of an appropriate feedback control law.

  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

    math

    where

    math

    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

    math

    The solution to the LQR problem is given by a linear control law of the form

    math

    where math is a positive definite, symmetric matrix that satisfies the equation

    math

    This equation is called the algebraic Riccati equation and can be solved numerically.

The following exercises cover some of the topics introduced in this chapter. Exercises marked with a * appear in the printed text.

Additional Exercises

Frequently Asked Questions

Errata

MATLAB code

The following MATLAB scripts are available for producing figures that appear in this chapter.

See the software page for more information on how to run these scripts.

Additional Information

More information on optimal control and the linear quadratic regulator can be found in the Optimization-Based Control supplement: