Tham khảo tài liệu 'systems, structure and control 2012 part 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | LQ and H2 Tuning of Fixed-Structure Controller for Continuous Time Invariant System with H Constraints 213 Notice that all three kinds of adaptability characterize structural properties of the control system but not of the plant characterized by the invariant properties called controllability observability stabilizability and detectability. Also denote that the adaptability property can be verified experimentally. The above adaptability definitions can be extended onto linear discrete time invariant systems dynamic systems with static nonlinearities bilinear control systems as well as onto MIMO linear and bilinear control systems Yadykin 1981 1983 1985 1999 Morozov Yadykin 2004 Yadykin Tchaikovsky 2007 . Adaptability matrices 14 possess the following properties Yadykin 1999 1. The adaptability matrix L is the block Toeplitz matrix for MIMO systems. For SISO systems L is the Toeplitz matrix. 2. The adaptability matrix L has maximal column rank if and only if det CpBp 0. 20 Condition 20 is the necessary and sufficient condition of partial adaptability of control system 1 2 as well as the necessary condition of its complete adaptability. 3. Each block NX of the block adaptability matrix N equals to block scalar product of the block row of the matrix L and column vector G where all variables subscripts are added with subscript m in the cases when it is absent and vice versa. 4. Each block of the matrix L is a linear combination of block products of the plant matrices CpAp Bp controller matrices CcmA C mv Bc Dc and products of the coefficients of the characteristic equations of the plant controller and their reference models. 5. Upper and lower square blocks of the adaptability matrix L have upper and lower triangle form respectively. 4. Solutions to LQ and H2 Tuning Problems In this section we consider the solutions of LQ and H2 optimal tuning problems 17 and 18 for fixed-structure controllers formulated in Section 2 and briefly outline an approach to LQ optimal .