v-Gap Metric–Based Simultaneous Frequency-Shaping Stabilization for Unstable Multi-Input Multi-Output Plants

Gap Reduced Minimum Error Robust Simultaneous Estimation For Unstable Nano Air Vehicle

“…where ∆A il is the difference between the system matrices, A l and A i of P l (s) and P i (s), respectively, x i (t) is the state vector of P i (s), B e = I ∈ R 11×11 is the input matrix of error dynamics, and u e (t) = −L l e y il (t). Applying this u e (t) in (12) results in…”

“…We now describe a method that induce the characteristics of a dynamic compensator in the error dynamics by solving LMIs similar to (33). In this method, the dynamic compensator is in the form of pre and post compensators that will be cascaded with the plant models belonging to P. Now, let us consider the pre compensator, W ei (s), and the post compensator, W eo (s) which would be cascaded with the plants belonging to P. The W ei (s) and W eo (s) are defined by ( 13) and ( 14), respectively, of [12] with m = 3 and r = 5 (( 13) and ( 14) are also given in supporting material). Also, the elements of W ei (s) and W eo (s) are given by ( 15) and ( 16), respectively of [12] ((15) and ( 16) are also given in supporting material).…”

“…In this method, the dynamic compensator is in the form of pre and post compensators that will be cascaded with the plant models belonging to P. Now, let us consider the pre compensator, W ei (s), and the post compensator, W eo (s) which would be cascaded with the plants belonging to P. The W ei (s) and W eo (s) are defined by ( 13) and ( 14), respectively, of [12] with m = 3 and r = 5 (( 13) and ( 14) are also given in supporting material). Also, the elements of W ei (s) and W eo (s) are given by ( 15) and ( 16), respectively of [12] ((15) and ( 16) are also given in supporting material). The cascading of these compensators modifies the characteristics of all the plants of P. This in turn alters the characteristics of the estimation error dynamics.…”

“…, N } \ {j} and G(P j (s)), it is necessary to compute the gap between the graphs. Let δ v (P 1 (jω), P 2 (jω)) ∈ [0, 1] be the v-gap metric (see [12] for details) between two plants,P 1 (s) and P 2 (s). Then, the gap between two graphs is given by [13] gap(G(P 1 ), G(P 2 )) = δ v (P 1 (jω), P 2 (jω))…”

“…In [11], algebraic geometry tools were presented to characterize the simultaneous observability of a set of linear single-input single-output plants and also to design a simultaneous state observer for the same. The methods presented in [8]- [11] are not suitable for synthesizing a simultaneous estimator for a NAV due to the following reasons: All the plants of the NAV may be unstable and also can have zeros on the right half of the s-plane [12]. Also, the number of outputs of the NAV is more than the number of inputs violating the condition mentioned in [10].…”