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# Stabilization of uncertain linear dynamics: an offline-online strategy

• *Corresponding author: Sérgio S. Rodrigues
• A strategy is proposed for adaptive stabilization of linear systems, depending on an uncertain parameter. Offline, the Riccati stabilizing feedback input control operators, corresponding to parameters in a finite training set of chosen candidates for the uncertain parameter, are solved and stored in a library. A uniform partition of the infinite time interval is chosen. In each of these subintervals, the input is given by one of the stored parameter dependent Riccati feedback operators. This parameter is updated online, at the end of each subinterval, based on input and output data, where the true data, corresponding to the true parameter, is compared to fictitious data that one would obtain in case the parameter was in a selected subset of the training set. The auxiliary data can be computed in parallel, so that the parameter update can be performed in real time. The focus is put on the case that the unknown parameter is constant and that the free dynamics is time-periodic. The stabilizing performance of the input obtained by the proposed strategy is illustrated by numerical simulations, for both constant and switching parameters.

Mathematics Subject Classification: 93C40, 93B52, 49N10, 93B51.

 Citation:

• Figure 1.  $\mathcal{A}_\sigma$ as in (4.1). No parameter update, $(N_{\rm{g}},\gamma) = (0,0)$

Figure 2.  $\mathcal{A}_\sigma$ as in (4.1). $\varSigma = \varSigma_{11}$ as in (4.2), $(N_{\rm{g}},\gamma) = (0,0.1)$

Figure 3.  $\mathcal{A}_\sigma$ as in (4.1). $\varSigma = \varSigma_{11}$ as in (4.2), $(N_{\rm{g}},\gamma) = (10,0.1)$

Figure 4.  $\mathcal{A}_\sigma$ as in (4.1). $\varSigma = \varSigma_{21}$ as in (4.2), $(N_{\rm{g}},\gamma) = (0,0.1)$

Figure 5.  $\mathcal{A}_\sigma$ as in (4.3). Instability with no control and with no feedback control update

Figure 6.  $\mathcal{A}_\sigma$ as in (4.3). $\sigma\notin\varSigma = \varSigma_{(10,30)}$ as in (4.5), $(N_{\rm{g}},\gamma) = (0,0.02)$

Figure 7.  $\mathcal{A}_\sigma$ as in (4.3). $\sigma\notin\varSigma = \varSigma_{(20,60)}$ as in (4.5), $(N_{\rm{g}},\gamma) = (0,0.02)$

Figure 8.  Dynamics identification. $\sigma\notin\varSigma_N$ as in (4.5), $(N_{\rm{g}},\gamma) = (0,0.02)$

Figure 9.  $\mathcal{A}_\sigma$ as in (4.3). $\sigma\in\varSigma = \varSigma_{(20,60)}$ as in (4.5), $(N_{\rm{g}},\gamma) = (0,0.02)$

Figure 10.  $\mathcal{A}_\sigma$ as in (4.3). $\sigma\notin\varSigma = \varSigma_{(20,60)}$ as in (4.5), $(N_{\rm{g}},\gamma) = (0,0.02)$

Figure 11.  Spatial triangulations and supports of actuators

Figure 12.  $\mathcal{A}_\sigma$ as in (4.7). Instability with no control, $K = 0$, and with no feedback control update, $K = K_{\sigma^{\rm{e}}(0)}$

Figure 13.  With updated parameter, $(N_{\rm{g}},\gamma) = (0,1)$. $K = K_{\sigma^{\rm{e}}(t)}$

Figure 14.  Piecewise constant, slowly switching, parameters $\sigma(t)\notin\varSigma$. $K = K_{\sigma^{\rm{e}}(t)}$

Figure 15.  Piecewise constant, quickly switching, parameters $\sigma(t)\in\varSigma$. $K = K_{\sigma^{\rm{e}}(t)}$

Figure 16.  Comparison of adaptive, optimal, and robust feedback with $\mathcal{A}_\sigma$ as in (5.1)

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