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Advances in PID Control Part 10

Tham khảo tài liệu 'advances in pid control part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | PID Tuning Robust and Intelligent Multi-Objective Approaches 169 Since the ideal differentiator used in 1 3 and 4 is unrealizable a real differentiator should be applied in practice. Although most of PID controllers in use have the derivative part switched off proper use of the derivative action can improve the stability and help maximize the integral gain for a better performance. For real implementation ideal differintiator kDs can be approximated as kDs ẤkDs 1 where Ấ is a small number. The effect of real and approximated differentiator on the closed-loop dynamics are discussed in PID control literature. 2.2 ILMI-based H2 H SOF design A general control scheme using mixed H2 H control technique is shown in Fig. 2. G s is a linear time invariant system with the given state-space realization in 5 . The matrix coeificients are constants and it is assumed the system to be stabilizable via a SOF system. Here x is the state variable vector w is disturbance and other external input vector y is the augmented measured output vector and K is the controller. The output channel z2 is associated with the LQG aspects H2 performance while the output channel zTO is associated with the H performance. x Ax B B2u z Cx D w. D u TO TO TO1 i to2 z2 C2x D21w D22u y Cyx Dy 5 Assume Tz w and Tz w are the transfer functions from w to zTO and w to z2 respectively and consider the following state-space realization for the closed-loop system. After defining the appropriate H and H2 control outputs zTO and z2 for the system it will be easy to determine matrix coefficients Cro D I D 2 and C2 D21 D22 . x Ax. B w c i 1c z CTOcx D w z2 C2cx D2 w b y Cycx Dyc A mixed H2 H SOF control design can be expressed as following optimization problem Optimization problem Determine an admissible SOF law K belong to a family of internally stabilizing SOF gains Ksof u Ky K e Kf 7 such that iL Ik subject to lT L 1 KeKsof 8 The following lemma gives the necessary and sufficient condition for the existence of the