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Advances in PID Control Part 8

Tham khảo tài liệu 'advances in pid control part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 130 Advances in PID Control a Reference input r t and output x t Fig. 9. Output response of the system 59 with controller 57 for a smooth reference input r t and a step disturbance w t where b1d aị the reference model is a system of type 2 specifications given by 12 . By a Z-transform of 12 preceded by a ZOH the desired pulse transfer function b Control u t and disturbance w t Hr z ZU-1 1 T s s 1 T t kTs 1- e-T T z e-Ts T 62 follows. Hence from 62 the desired stable difference equation xk Xk-1 Tsa Ts rk-1 Xk 1 63 results where a Ts - --------- lim a Ts i Ts TsZo k s T and the output response of 63 corresponds to the assigned output transient performance indices. Let us rewrite for short the desired difference equation 63 as xk F xk-1 rk-1 64 where we have rk Xk at the equilibrium of 64 for rk const V k. Denote ek F xk-1 rk-1 - xk 65 where ekF is the realization error of the desired dynamics assigned by 64 . Accordingly if for all k o 1 . . . the condition eFk 0 66 holds then the desired behavior of Xk with the prescribed dynamics of 64 is fulfilled. The expression 66 is the insensitivity condition for the output transient performance with respect to the external disturbances and varying parameters of the plant model given by 60 . In other words the control design problem 61 has been reformulated as the requirement 66 . PI PID Control for Nonlinear Systems via Singular Perturbation Technique 131 The insensitivity condition given by 66 is the discrete-time counterpart of 15 which was introduced for the continuous-time system 9 . 7.2 Discrete-time counterpart of PI controller Let us consider the following control law uk uk 1 Ấ0 F xk 1 rk 1 - xk 67 where Ả0 Ts 1Ả and the reference model of the desired output behavior is given by 63 . In accordance with 63 and 65 the control law 67 can be rewritten as the difference equation uk uk 1 Ả a Ts rk 1 - xk 1 - xk xk- Ts 68 The control law 68 is the discrete-time counterpart of the conventional continuous-time PI controller .