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