State Feedback

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Chapter 6 - 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 Example: Vectored thrust aircraft 3. State Feedback Design Issues 4. Integral Action 5. Further Reading Exercises

Lecture Materials Lecture: State Feedback Additional Exercises Supplemental Information Frequently Asked Questions Errata Wikipedia entries: Controllability (reachability), state feedback, LQR Additional 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

   

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

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

   

   A linear system is reachable if and only if the reachability matrix 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

   

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

   

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

4. A state feedback law has the form

   

   where 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 . The equilibrium point and steady state output (assuming the systems is stable) are given by

   

   Choosing as

   gives .

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

   

   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 . An integral feedback controller has the form

   

   where

   

   is the integral error. The gains , and 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

   

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

   

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

   

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

Additional Exercises The following exercises cover some of the topics introduced in this chapter. Exercises marked with a * appear in the printed text. Exercise: State feedback control of an AFM in contact mode Frequently Asked Questions FAQ: In equation (6.23), shouldn't the B matrix have omega 0^2? FAQ: In Section 6.4 (integral action), doesn't the value of z e depend on K? FAQ: What are the parameters for the Whipple bicycle model? FAQ: What is closed form? Errata Errata: In Example 6.7, the open loop eigenvalues are incorrect Errata: In Example 6.10, the reference gain should be k r = a 1/b No errata reported in most recent version of this chapter Full list of errata starting from first printing Submit bug report

MATLAB code The following MATLAB scripts are available for producing figures that appear in this chapter. Figure 6.6: steering_place.m Figure 6.7: predprey_control.m, predprey.m, predprey_rh.m Figure 6.8: secord_stepresp.m Figure 6.9: secord_bode.m Figure 6.10: compartment_statefbk.m Figure 6.11: pvtol_lqr.m, pvtol/ (help) Figure 6.12: balance_place.m, balance_params.m Figure 6.14: webserver_control.m, apache_lin.m Figure 6.15: cruise_integral.m, cruise_conpar.m, cruise_lin.m 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: OBC Chapter 2 - Optimal Control