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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.
This chapter describes how state feedback can be used to design the (closed loop) dynamics of the system:
A linear system with dynamics
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.
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 state feedback law has the form
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.
A linear quadratic regulator minimizes the cost function
The solution to the LQR problem is given by a linear control law of the form
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.
Frequently Asked Questions
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.
More information on optimal control and the linear quadratic regulator can be found in the Optimization-Based Control supplement: