Tham khảo tài liệu 'adaptive control system part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6 Adaptive internal model control infinity. Thus minimizing the Hx norm of T 5 does not make much sense since the infimum value of zero is unattainable. On the other hand if we consider the weighted sensitivity minimization problem where we seek to minimize w 5 5 5 for some stable minimum phase rational weighting transfer function w 5 then we have an interesting IF. minimization problem . choose a stable 2 5 to minimize w T l FÁj2Ájll x -The solution to this problem depends on the number of open right half plane zeros of the plant p s and involves the use of Nevanlinna-Pick interpolation when the plant p s has more than one right half plane zero 7 . However when the plant has only one right half plane zero bl and none on the imaginary axis there is only one interpolation constraint and the closed form solution is given by 7 eơ i - .IT Fortunately this case covers a large number of process control applications where plants are typically modelled as minimum phase first or second order transfer functions with time delays. Since approximating a delay using a first order Padé approximation introduces one right half plane zero the resulting rational approximation will satisfy the one right half plane zero assumption. Remark As in the case of IF optimal control the optimal Q s defined by is usually improper. This situation can be handled as in Remark so that the Q s to be implemented becomes 2 A h M I 5 J p-i F s 1-5 where F 5 is a stable IMC filter. In this case however there is more freedom in the choice of F 5 since the IF. optimal controller does not necessarily guarantee any asymptotic tracking properties to start with. Robustness to uncertainties small gain theorem In the next section we will be combining the above schemes with a robust adaptive law to obtain adaptive IMC schemes. If the above IMC schemes are unable to tolerate uncertainty in the case where all the plant parameters are known then there is little or no hope that certainty
