A synthesis theory for linear time-varying feedback systems with plant uncertainty

Given a feedback system containing a linear, time-varying (LTV) plant with significant plant uncertainty, it is required that the system response to command and disturbance inputs satisfy specified tolerances over the range of plant uncertainty. The synthesis procedure guarantees the latter satisfied, providing that they are of the following form. Let
h(t',\tau)
be the system response at
t'= t - \tau
due to a command input
\delta(t - \tau)
, and
h_{\tau}(s)=\int \liminf{0}\limsup{infty}h(t',\tau)e^-{st'}dt'
is the Laplace transform of
h(t',\tau)
. There is given a set
M_{\tau}(\omega)=\{m_{\tau}(\omega)\} , \omega \in[0, \infty)
, with the requirement that
|h_{\tau}(j\omega)| \in M_{\tau}(\omega)
, over the range of plant uncertainty. The disturbance response tolerances are of the same form, in response to a disturbance input
\delta (t- \tau)
. The acceptable response set
M_{\tau}(\omega)
can depend on τ. The design emerges with a fixed pair of LTV compensation networks and can be considered applicable to time-domain response tolerances, to the extent that a set of bounds on a time function can be translated into an equivalent set on its frequency response. The design procedure utilizes only time-invariant frequency response concepts and is conceptually easy to follow and implement. At any fixed τ, the time-varying system is converted into an equivalent time-invariant one with plant uncertainty, for which an exact solution is available, with "frozen" time-invariant compensation. Schauder's fixed-point theorem is used to prove the equivalence of the two systems. The ensemble over τ of the time-invariant compensation gives the final required LTV compensation. It is proven that the design is stable and nonresonant for all bounded inputs.