# State Feedback

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This chapter describes how feedback can be used 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

• 1. Introduction
• 2. Reachability
• 3. Stabilization by State Feedback
• 4. State Feedback Design Issues (to be written)
• 5. Integral Action
• 8. Exercises

## 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 reachability 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 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 .