Fig. 1. Two problem formulations for boundary control of a stable string pinned on one end: top—classical formulation with the actuation/sensing collocated on the right end (Neuman/force actuation and Dirichlet/displacement sensing). Bottom (Guo & Xu, 2007)—Neumann actuation on the right end and Neumann sensing on the left (pinned) end.

Fig. 2. Two problem formulations for boundary control of an unstable string with a destabilizing force (boundary condition) on the free end: top (Sections 2–5)—Dirichlet actuation on the right end, Dirichlet sensing on the left end. Bottom (Section 6)—collocated actuation/sensing on the right end (Neumann/force actuation and Dirichlet/displacement sensing).

Fig. 3. Bode plots of the plant (dashed) and the compensator (solid) for the non-collocated design.

Fig. 4. String response w(x,t). Uncontrolled case, zero Dirichlet boundary condition at x=1.

Fig. 5. Observer error in the uncontrolled case.

Fig. 6. String response w(x,t) with observer-based control applied at x=1.

Fig. 7. Time trace of control u(t).

Fig. 8. String response w(x,t) with reduced order (N=6) observer-based control applied at x=1.

Fig. 9. String response w(x,t) with q=1 in the observer mismatching the value q=1.1 in the plant.

Fig. 10. Closed-loop poles: ×—nominal plant and compensator (q=1), —plant with q=1.1, but compensator is designed for q=1.