Fig. 3. Closed-loop poles for body # I and the stability chart of the total systems. In this example, a conventional LQ controller is designed for the uncoupled system S|, and it turns out that the designed closed-loop poles trace the Butterworth pattern as r (Jlr is the weight of control u) travels to infinity, and the closed-loop poles approach -a±ja, asymptotically. By investigating the total closed-loop stability region, the aggregate system shows that among the closed-loop poles -a±ja, some parts of the higher gain feedback controller, corresponding to the case of the large value of a, yields the total system stability (shaded area in Fig. 3). It is concluded that the underlying system has the decentrally stabilizable structure in this meaning. The following stability chart depicted in Fig. 4 shows the effect of the coupling magnitude of parameters A12 and A21. Suppose in a case when the coupling degrees are changeable in a manner of eA12 and eA21, it is shown that the stability region depends on the parameter e. In case of strongly coupling system (e is large), the closed-loop poles of the body # 1 are required to be large, i.e., far from the origin, to stabilize the total system. In other words, the local controller must be a high-gain feedback to guarantee the total system stability condition. On the contrary, for the weak coupling system (e is small), the total system can be stabilized by small gain feeding back. Finally, Fig. 5 shows the stability region in the case that both bodies #1, #2 are locally controlled by the same structure of controller. It is noted that the region of the total system stability is broadened compared with Fig. 3 in which only one body is controlled.
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