A counterexample for the global separation principle for discrete-time nonlinear systems

In the control systems literature, it is well known that a separation principle holds locally for nonlinear control systems, when exponential feedback stabilizers and exponential observers are used. In this paper, we present a counterexample to show that the global separation principle need not hold for nonlinear control systems. Our example demonstrates that global stability might be lost when an exponential observer is introduced into the nonlinear feedback loop associated with an exponentially stabilizing feedback control law.