Let $\dot{x} = A x + B u$, $y = x$ be a BIBO (bounded input, bounded output) stable system. Given an output bound $y_l \leq y(t) \leq y_h$, how can we determine the maximum input bound $u_l \leq u(t) \leq u_h$ so that any such bounded input $u(t)$ yields an output bounded by $y_l$, $y_t$?

Conversely, what is the minimum output bound $y_l \leq y(t) \leq y_h$ for a given input bound $u_l \leq u(t) \leq u_h$?

Pointers to literature are welcome.

  • $\begingroup$ The bound will also depend on the initial conditions. Is the system one-dimensional? Or are the bounds vector bounds? $\endgroup$
    – RTJ
    Apr 3 '15 at 20:18
  • $\begingroup$ It is fine to assume a fixed initial value for $x$, e.g. 0. Even better would be an approach where $x_0$ is symbolic. I am interested in 1-dimensional and vector bounds. In fact, I would like to know if this is a standard problem and if there is literature available on it. How to decide BIBO stability is mentioned in many books about control but I couldn't find a reference for the problem above. $\endgroup$
    – Mathabc
    Apr 3 '15 at 20:46

Based on the system response you can calculate the general bound $$\|x(t)\|\leq \|e^{At}x(0)\|+\left[\int_0^t{\|e^{As}B\|ds}\right]\sup_{t}\|u(t)\|$$ Assuming that $A$ is stable and $x(0)=0$ then $$ \sup_t\|x(t)\|\leq \left[\int_0^{\infty}{\|e^{As}B\|ds}\right]\sup_{t}\|u(t)\|$$ Note that $g(t):=e^{At}B$ is the impulse response matrix and the above condition results in the standard textbook condition $$\sup_t\|y(t)\|\leq \left[\int_0^{\infty}{\|g(s)\|ds}\right]\sup_{t}\|u(t)\|$$


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