Consider a continuous-time LTI system x(t) = Ax(t), t ≥ 0, with no input (such a system is said to be autonomous), and output y(t) = Cx. We wish to evaluate the energy contained in the system's output, as measured by the index

where

1. Show that if the system is stable, then J( xq) < 00, for any given x₀.

2. Show that if the system is stable and there exists a matrixsuch that 

then it holds that J(x₀) ≤ x₀^TPx₀.

3. Explain how to compute a minimal upper bound on the state energy, for the given initial conditions.

Transcribed Image Text:

I(xo) = √ y(t)Ty(t)dt = √° x(t)¹Qx(t)dt, ·00 0