From (34), (46), and (47), the inequality)relation)below (can 2 obtained: e(k 2 y
From (34), (46), and (47), the inequality)relation)below (can two obtained: e(k two y(k two u k ) be two [-1 V (k ) = inequality From (34), (46), and (47), thee (k) y(k)relation) below (k) 2be obtained: 2 u(k two W can (48) 2 () ()+1e(k ) 2 y(k ) two u(k ) two ] y(k ) 2 u(k ) 2 W (k )When V (k ) 0, satisfies: 02 sup sup(49)Florfenicol amine MedChemExpress Mathematics 2021, 9,10 of5. Equivalent Constructive Feedback Good feedback is made use of in APNF. The simplest notion to understand the combination in the two control modes in the model is usually to establish the framework of the two manage modes in the controller, after which use diverse frameworks for handle at distinct instances by means of situation judgment. Having said that, in practical study, recognizing the way to make situation judgment is very tough. Right here, we get inspiration in the equivalence of good feedback and adverse feedback. Within a single framework, we do not have to have to create conditional judgment, and only recognize the state transition by means of the adjust of parameters. In line with the control theory, when the method is actually a negative feedback method, e(k) = yd (k) – y(k ), and when the method can be a optimistic feedback program, e(k ) = yd (k ) + y(k ). The PID controller might be cited as an example here to introduce how negative feedback is equivalent to constructive feedback, that is associated for the realization of optimistic feedback in APNF. Within this section, a theoretical proof of equivalent constructive feedback is provided. The PID controller is controlled by three adjustable parameters, which ascertain the performance from the controller: the proportional achieve (K P ), integral get (K I ), and Eperisone Epigenetics differential gain (Kd ). The PID controller operates by continuously monitoring the distinction between the set value plus the measured method variable. The 3 aforementioned elements are made use of to calculate the manage quantity and to control the target perform. The generalized PID handle strategy is usually expressed by (50):k u(k ) = K P e(k) + K I i=0 e(i ) + K D [e(k) – e(k – 1)](50)where the 3 parameters K P , Ki , and Kd are all non-negative, and e(k) could be the error involving the set worth and the actual feedback worth. Some systems may possibly need to keep only some parameters and set the other parameters to 0 to supply acceptable control, including the PI, PD, and P controllers. Theorem 3. In APNF, when w (k ), w (k ), w (k ) 0, the corresponding gains of -w (k), -w (k), and – w (k) of adverse feedback is often equivalent for the corresponding gains of w (k), w (k ), and w (k) of optimistic feedback. Proof. When w (k) 0, the adverse feedback proportional obtain might be equivalent for the following:-w (k)[y(k) – yd (k)] = w (k)[yd (k) – y(k)] = w (k)Y (k) Y (k = [y (k)+) (k)] w (k)[yd (k) + y(k)] y d = w (k)[yd (k) + y(k)] = w ( k ) e ( k )(51)where Y (k) = yd (k) – y(k ) would be the modest distinction amongst y(k) and yd (k), and w (k) could be the proportional obtain of the equivalent optimistic feedback. When lim w (k0 ) exists, accordingkkto (51), the -w (k) acquire under adverse feedback might be transformed into w (k) gain beneath optimistic feedback. The updated formula is shown under: w (k) -Y (k) w k [yd (k)+y(k)] ( )(52)Meanwhile, according to the constructive feedback handle principle, yd (k) increases with time, so when time t tends to infinity, w (k) converges to w (k). That is certainly:klim w (k) = limY (k)k [yd (k )+y(k )]w ( k ) = w ( k )(53)Mathematics 2021, 9,11 ofWhen w (k), w (k ) is damaging, the integral and differential gains of negative feedback are equivalent, as follows:-w (k) ik=[y(i ) – yd (.