By Nikolai Michailovich Filatov, Heinz Unbehauen

This monograph demonstrates how the functionality of assorted famous adaptive controllers may be superior considerably utilizing the twin impact. The alterations to include twin keep an eye on are learned individually and independently of the most adaptive controller with no complicating the algorithms. a brand new bicriterial strategy for twin keep watch over is constructed and utilized to varied forms of well known linear and nonlinear adaptive controllers. sensible purposes of the designed controllers to numerous real-time difficulties are provided. This monograph is the 1st e-book supplying a whole exposition at the twin keep watch over challenge from the inception within the early Sixties to the current state-of-the-art aiming at scholars and researchers in adaptive regulate in addition to layout engineers in undefined.

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**Example text**

It was suggested by Zhivoglyadov (1965) to use P(k), the covariance of the unknown parameters. 21) was used, where P(k) is the covariance matrix of the estimation error. 23) while Wittenmark and Elevitch (1985) as well as Allison et al. (1995a) used the covariance of the estimate of the first parameter b1 of the ARX model of the system. Instead of the well-known innovational cost measure of Milito et al. (1982), see also eq. 24) can be used in the bicriterial design method. 25) can be used instead.

The RLS algorithm with forgetting factor 0 < α ≤ 1 will be obtained when eq. 15) takes the form P (k + 1) = 1 [ P (k ) − q (k + 1)m T (k ) P (k )] . 16) Thus eq. 15) has the same form as eq. 15) if Qξ (k ) = 1−α [ P (k ) − q (k + 1) m T (k ) P (k )] . 17) The constant trace modification, where tr [ P (k + 1)] = tr [ P (k )] , can be obtained by appropriately choosing the matrix Qξ (k ) in eq. 15). 1. Parameter Estimation Qξ (k ) = tr [q(k + 1)m T (k ) P (k )] T tr [ P (k ) − q(k + 1)m (k ) P (k )] [ P (k ) − k (k + 1)m T (k ) P (k )] .

The dual controller is obtained after the bicriterial optimization problem is solved with the constraints Ω k for eq. 25) u ( k )∈Ω k u (k ) where uc (k ) is the cautious control, which is obtained after the minimization of eq. 19). The constraints Ω k define the value of the excitation signal for each time step k. They are symmetrically located around the optimal solution according to eq. 4. Therefore, according to eqs. 24), the amplitude of the excitation depends on the value of the parameter η and the trace of the covariance matrix P(k).