Page 153 - IJOCTA-15-3

Proportional integral derivative plus control for nonlinear discrete-time state-dependent parameter. . .
SVF/SDP-PID+ control gain vector, as defined This equation is used to determine the SDP-
+
in Equation (25). PID+ compensators, k , in Equation (25). It
k
For simplicity, a frozen parameter system is worth noting that the order of characteris-
′
′
{F , g } can be employed, where the frozen sys- tic Equation (32) is n + m + δ − 1, ∀ n ≥ 2,
k
k
tem is defined as a single sample instance from with a minimum order of three for a discrete-
the family of {F k , g }. 36,37 In this scenario, the time SDP-TF of triad {≤ 2, 1, 1}. The polyno-
k
matrix P becomes time-invariant, and Equation mials corresponding to the proportional plus com-
(28) is simplified to pensator, K + z −1 , and input plus compensator,
k

K u,k z −1 , are defined as in Equation (33),

−1
′ T
′ T
′ T
′
′
P (i) = F k P (i+1) F k − g k g k P (i+1) ′ g k P (i+1) F k + Q
g k + R
n+1
(29) + −1 −1 −2 X −(i−1)
K k z = k 1,k − k 2,k z + k 3,k z + k i,k z
After obtaining the matrix P from Equation i=4
+
(29), the SDP gain vector k can be derived from
k m+δ−2
X
Equation (28) as follows: K u,k z −1 = 1 + k u j ,k z −j

j=1
−1

T
T
+

k = g P g k + R g P F k (30) (33)
k k k
Note that the system matrices {F k , g } re- where,
k
main unfrozen when implementing Equation (30).
A straightforward trial and error approach is em- k 1.k = k P 1 ,k + k I,k + k D,k
ployed to evaluate the gain vector in Equation k 2.k = k P 1 ,k + 2k D,k
(30). The process begins with unity weights to
k 3.k = k D,k + k P i−1 ,k i = 3 if n ≥ 3
ensure a balanced influence of each parameter, (34)
preventing any single parameter from dominating . . .
the others. This approach allows for consistent k i,k = −k P i−2 ,k + k P i−1 ,k 4 ≤ i ≤ n + 1
and logical experimentation.
(i = 3, ..., n+1)
The plus compensators, k P i−1
3.3.2. SDP-PID+/pole placement tuning (j = 1, . . . m + δ − 2) for
for n ≥ 3 and k u j
approach m+δ > 2, in Equations (33) and (34) allow for ex-
The design methodology for discrete SDP- ploiting additional poles to suit the discrete-time
PID+/pole placement was introduced by Sha- SDP-TF in Equation (4) that exceeds second-
ban et al. 11 It has been shown that the pro- order dynamics, i.e. n ≥ 3, and has time delay
cess is relatively straightforward, provided the and/or numerator order greater than unity, i.e.
performance of the closed-loop control system is m + δ > 2. Applying polynomial algebra to the
predictable, i.e., if the SDP-TF in Equation (4) characteristic Equation (32) as Equation (35)
is controllable. Here, the nonlinear SDP-PID+
compensators, as defined in Equation (25), can ∆ K u,k z −1 A k χ k , z −1 + K + z −1 B k χ k , z −1
k
be evaluated by modifying the location of the =(1 − p 1 z −1 ) (1 − p 2 z −1 )
poles of the closed-loop SDP-PID+ control sys- −1 −1
(1 − p 3 z ) . . . (1 − p i z )
tem. In this regard, the regulator structure of
i = 4, . . . , n + m + δ − 1 ∀ n ≥ 2
the SDP-PID+ controller, depicted in Figure 2, i = 4, . . . , n + m + δ ∀ n = 1
needs to be reduced to the unity feedback closed- (35)
loop SDP-PID+ control system in TF form, as
gives SDP-PID+ gains {k 1,k , k 2,k , k 3,k , ..., k i,k
illustrated in Figure 3 as follows in Equation (i = 4, ..., n + 1), k u j ,k (j = 1, ..., m + δ − 2)}
(31),
at pre-determined pole locations {p 1 , p 2 , p 3 , ..., p i
(i = 4, ..., n + m + δ − 1 ∀ n ≥ 2)} in the complex
+
K (z ) B(χ k , z ) z-plane. The control law can subsequently be ex-
−1
−1
y k = k r k
+
∆ K u,k (z −1 ) A(χ k , z −1 )+K (z −1 ) B(χ k , z −1 ) tracted from Figure 3 in Equation (36).
k
(31)
given that ∆ = 1 − z −1 represents the differ-
u k =u k−1 + k 1,k e k − k 2,k e k−1 + k 3,k e k−2
ence operator. The characteristic equation for
n+1
the SDP-PID+ control system is extracted from + X k i,k e
Equation (31) as Equation (32). k−(i−1)
i=4 (36)
m+δ−2
∆ K u,k z −1 A χ k ,z −1 + K + z −1 B χ k ,z −1 − X k u j ,k u k−j − u
k k−(j+1)
(32) j=1
525