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In real control problems, there exited many uncertainties, like model structure, measurement, external disturbance and so on, tradition PID and type-1 fuzzy controller can’t deal with these uncertainties [1,2,3,4,5]. Type-2 fuzzy controller can handle uncertainties more robust than PID and type-1 fuzzy controller for it was described by type-2 fuzzy sets proposed by Zadeh in 1975 [6]. Type-2 fuzzy sets mainly included interval type-2 fuzzy sets whose secondary membership degree was 1 and general type-2 fuzzy sets whose secondary membership degree was decided by a function, such as triangular, Gaussian, trapezoid. As the secondary membership degree of interval type-2 fuzzy sets was 1, so it was easily to be implemented and Karnik-Mendel (KM) algorithm was the most widely applied type reduction for interval type-2 fuzzy sets [7]. Interval type-2 fuzzy logic systems has been applied in many applications, like face recognition [8], prediction problems [9,10,11], pattern recognition [12], clustering [13], intelligent control [14], industrial [15], neuro-fuzzy systems [16], interval type-2 fuzzy PID controller [17,18], sculpting the state space [19], peer-to-peer e-commerce [20], classification [21,22], regression [23], diagnosis problems [24], metaheuristics [25], gravitational search algorithm [26], healthcare problem [27], unmanned aerial vehicles [28], deep neural network [29], pursuit evasion game [30], analytical structure of interval type-2 fuzzy controller [31,32,33] and so on.
As the secondary membership degree of general type-2 fuzzy sets was determined by a function rather than 1, so general type-2 fuzzy sets contained more uncertain information than interval type-2 fuzzy sets. And general type-2 fuzzy logic systems had more design parameters when describing reality. Thus, general type-2 fuzzy systems can obtain a better performance in some control systems with high uncertainties. Now there existed some efficient type reduction algorithms for general type-2 fuzzy sets, for example, α-plane representation method [34,35,36], zSlices-based representation method [37,38], sample method [39], geometric method [40,41], hierarchical collapsing method [42] and so on. In these algorithms, α-plane representation method was widely applied in general type-2 fuzzy sets type reduction. By α-plane representation, general type-2 fuzzy sets will be assembled by some interval type-2 fuzzy sets (α-planes). Some exiting interval type-2 fuzzy sets type reduction algorithms can be applied to these α-planes, like KM, EKM [46], IASC [47] or EIASC [48]. General type-2 fuzzy logic systems have been applied in many situations, like: mobile robots [38,46,47,48,49,50,51], water tank [52], traffic signal scheduling [53], inverted pendulum plant [54], 5-agents system [55], nonlinear power systems [56], water level and DC motor speed [57], aerospace [58], airplane flight [59], steam temperature [60], power-line inspection robots [61,62], fractional order general type-2 fuzzy controller [63,64], medical diagnosis [65,66,67], fuzzy classifier and clustering [68,69], sculpting the state space [70], similarity measures [71], forecasting [72], brain-machine interface [73] and so on. [74,75,76] made a detailed introduction on type 2 fuzzy logic applications.
The type reduction of general type-2 fuzzy sets was converted to type reduction of several interval type-2 fuzzy sets. And KM type reduction algorithm was applied to these interval type-2 fuzzy sets in most applications. KM algorithm was an iterative process without analytic solution. The number of α-planes and iterative process of KM algorithm decided the execution time of general type-2 fuzzy sets type reduction. Thus the real time of general type-2 fuzzy controller was weaker than type-1 and interval type-2 fuzzy controller. In according with these problems, a simplified general type-2 fuzzy PID (SGT2F-PID) controller is studied. The SGT2F-PID controller applies triangular function as the primary and secondary membership function. The inputs of SGT2F-PID controller are error and error derivative, and each input defines 2 fuzzy membership functions in fuzzy domains, thus only 4 rules will be derived in this SGT2F-PID controller. This paper mainly contains the following 3 contributions:
Ⅰ). The primary membership degree of apex for secondary membership degree is applied to get the centroid of SGT2F-PID controller. Then the real time of SGT2F-PID controller is almost the same as conventional type-1 fuzzy PID (T1F-PID) controller and better than interval and general type-2 fuzzy PID controller.
Ⅱ). The primary membership degree of apex for secondary membership degree is decided by the up and low bounds of footprint of uncurtains, which inherits the benefits of type-2 fuzzy PID controller. So the SGT2F-PID controller contains more design freedom and handles uncertainties better than PID or type-1 fuzzy PID controller.
Ⅲ). The accurate mathematical expression of SGT2F-PID controller is obtained and compared with mathematical expressions of interval type-2 fuzzy PID controller (IT2F-PID) and conventional T1F-PID controller. The mathematical expressions indicate that these 3 fuzzy PID controllers are all PID type controller. Furthermore, we obtain the relationship of controller gains and explain why SGT2F-PID controller can get better controlling effects.
A type-1 fuzzy set in the universe X is characterized by a membership function μA(x) as Eq (1).
A={(x,μA(x))|x∈X} | (1) |
where
A general type-2 fuzzy sets
˜A={(x,u),μ˜A(x,u)|∀x∈X,∀u∈[0,1]} | (2) |
u is the primary membership degree and
If the secondary membership degrees
˜A={(x,u),1|∀x∈X,∀u∈[0,1]} | (3) |
Figure 1 shows the definition of type-1 fuzzy sets, interval type-2 fuzzy sets and general type-2 fuzzy sets whose secondary membership function is triangular.
Liu introduced an α-plane representation for general type-2 fuzz sets [34], and pointed that α-plane denoted as
˜Aα={(x,u),μ˜A(x,u)⩾ | (4) |
If assemble all α-planes
\boldsymbol{\tilde A} = \bigcup\limits_{\alpha \in [0, 1]} {FOU({{\boldsymbol{\tilde A}}_\alpha })} | (5) |
The centroid of general type-2 fuzzy sets can be calculated by the centroids of its all α-planes
{C_{\boldsymbol{\tilde A}(x)}} = \bigcup\limits_{\alpha \in [0, 1]} {\alpha /{C_{{{\boldsymbol{\tilde A}}_\alpha }(x)}}} | (6) |
{C_{{{\boldsymbol{\tilde A}}_\alpha }(x)}} = [{l_{{{\boldsymbol{\tilde A}}_\alpha }}}, {r_{{{\boldsymbol{\tilde A}}_\alpha }}}] | (7) |
The general structure of fuzzy PID controller can be depicted as Figure 2 [76]. The antecedent parts can be type-1, interval type-2 or general type-2 fuzzy sets and the consequent parameters are crisp values.
In this paper, triangular primary function is applied. The inputs of general type-2 fuzzy PID controller are normalize error (E) defined in [-de-d1, de+d1] and error derivative (
The consequent parameters are symmetric and from Figure 3, 4 rules will be generated as follows, here H1 > H2 > -H2 > -H1.
Rule 1: If
Rule 2: If
Rule 3: If
Rule 4: If
Around the steady state, that is in interval [-de+d1, de-d1] for error and
\left\{ \begin{gathered} \bar \mu _E^{\tilde P} = \frac{{E + de + d1}}{{2 \times de}} \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _E^{\tilde P} = \frac{{E + de - d1}}{{2 \times de}} \\ \end{gathered} \right. | (8) |
\left\{ \begin{gathered} \bar \mu _E^{\tilde N} = \frac{{de + d1 - E}}{{2 \times de}} \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _E^{\tilde N} = \frac{{de - d1 - E}}{{2 \times de}} \\ \end{gathered} \right. | (9) |
\left\{ \begin{gathered} \bar \mu _{\dot E}^{\tilde P} = \frac{{\dot E + d\dot e + d2}}{{2 \times d\dot e}} \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{\dot E}^{\tilde P} = \frac{{\dot E + d\dot e - d2}}{{2 \times d\dot e}} \\ \end{gathered} \right. | (10) |
\left\{ \begin{gathered} \bar \mu _{\dot E}^{\tilde N} = \frac{{d\dot e + d2 - \dot E}}{{2 \times d\dot e}} \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{\dot E}^{\tilde N} = \frac{{d\dot e - d2 - \dot E}}{{2 \times d\dot e}} \\ \end{gathered} \right. | (11) |
By fuzzy inference of interval type-2 fuzzy logic systems and product operation, the fired membership degrees of fuzzy rules can be described as Eq (12).
\text{Rule 1: }\ [{\bar f_1}, {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} _1}] = [\bar \mu _{\dot E}^{\tilde N} \times \bar \mu _E^{\tilde N}, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{\dot E}^{\tilde N} \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _E^{\tilde N}] | (12.1) |
\text{Rule 2:}\ [{\bar f_2}, {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} _2}] = [\bar \mu _{\dot E}^{\tilde P} \times \bar \mu _E^{\tilde N}, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{\dot E}^{\tilde P} \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _E^{\tilde N}] | (12.1) |
\text{Rule 3:}\ [{\bar f_3}, {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} _3}] = [\bar \mu _{\dot E}^{\tilde N} \times \bar \mu _E^{\tilde P}, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{\dot E}^{\tilde N} \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _E^{\tilde P}] | (12.1) |
\text{Rule 4: }\ [{\bar f_4}, {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} _4}] = [\bar \mu _{\dot E}^{\tilde P} \times \bar \mu _E^{\tilde P}, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _{\dot E}^{\tilde P} \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\mu } _E^{\tilde P}] | (12.1) |
The triangular membership function of type-1 fuzzy PID controller is depicted as Figure 4, also for simplify,
In interval [-de, de] for error and
\mu _E^P = \frac{{E + de}}{{2 \times de}} | (13) |
\mu _E^N = \frac{{ - E + de}}{{2 \times de}} | (14) |
\mu _{\dot E}^P = \frac{{\dot E + d\dot e}}{{2 \times d\dot e}} | (15) |
\mu _{\dot E}^N = \frac{{ - \dot E + d\dot e}}{{2 \times d\dot e}} | (16) |
So the fired membership degrees of fuzzy rules for type-1 fuzzy PID controller can be described as Eq (17).
\text{Rule 1:}\ {f_1} = \mu _{\dot E}^N \times \mu _E^N | (17.1) |
\text{Rule 2:}\ {f_2} = \mu _{\dot E}^P \times \mu _E^N | (17.1) |
\text{Rule 3:}\ {f_3} = \mu _{\dot E}^N \times \mu _E^P | (17.1) |
\text{Rule 4:}\ {f_4} = \mu _{\dot E}^P \times \mu _E^P | (17.1) |
Figure 5 shows an example of membership degrees for fuzzy rules corresponding to consequent parameters using TIF-PID.
From Figure 5 and defuzzification process of type-1 fuzzy sets, the output of type-1 fuzzy inference U(t) in Figure 2 can be calculated as Eq (18).
{U_{T1}}{\rm{ = }}\frac{{\sum\limits_{i = 1}^4 {{f_i} \times {y_i}} }}{{\sum\limits_{i = 1}^4 {{f_i}} }} | (18) |
where, fi is described as Eq (17) and yi = [-H1, -H2, H2, H1]. By the mathematical expression of Eqs (13–17) and yi, the final solution of UT1 can be expressed as Eq (19).
\begin{gathered} {U_{T1}}{\rm{ = }}\frac{{({H_1} - {H_2}) \times \dot E + ({H_1} + {H_2}) \times E}}{{2de}} \\ {\rm{ = }}\frac{{({H_1} - {H_2}) \times {G_{CE}} \times \dot e + ({H_1} + {H_2}) \times {G_E} \times e}}{{2de}} \\ \end{gathered} | (19) |
According to Eq (19) and Figure 2, the final output of T1F-PID controller can be expressed as Eq (20).
{u_{T1}} = {G_{PD}} \times {U_{T1}} + {G_{PI}} \times \int {{U_{T1}}} | (20) |
Combine Eq (19) and Eq (20), the output of T1F-PID controller is a PID type controller as Eq (21).
{u_{T1}} = K_P^{T1} \times e + K_I^{T1} \times \int e + K_D^{T1} \times \dot e | (21) |
where:
K_P^{T1} = \frac{{{G_{PD}}({H_1} + {H_2}) \times {G_E} + {G_{PI}}({H_1} - {H_2}) \times {G_{CE}}}}{{2de}} |
K_I^{T1} = \frac{{{G_{PI}}({H_1} + {H_2}) \times {G_E}}}{{2de}} |
K_D^{T1} = \frac{{{G_{PD}}({H_1} - {H_2}) \times {G_{CE}}}}{{2de}} |
Figure 6 shows the shape of control surface of type-1 fuzzy controller, here H1 = 1 and H2 = 0.
For KM algorithm didn’t have analytic solution, so NT type reduction [78,79] algorithm will be applied to get the mathematical expression of IT2F-PID controller. Figure 7 shows an example of upper and lower bounds for fuzzy rules corresponding to consequent parameters using IT2F-PID controller.
From Figure 7, by defuzzification process and NT algorithm, the output of interval type-2 fuzzy inference U(t) in Figure 2 can be calculated as Eq (22).
{U_{IT2}}{\rm{ = }}\frac{{\sum\limits_{i = 1}^4 {({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} }_i} + {{\bar f}_i}) \times {y_i}} }}{{\sum\limits_{i = 1}^4 {({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} }_i} + {{\bar f}_i})} }} | (22) |
where,
\begin{gathered} {U_{IT2}}{\rm{ = }}\frac{{de[({H_1} - {H_2})\dot E + ({H_1} + {H_2})E]}}{{2(d{e^2}{\rm{ + }}d{1^2})}} \\ {\rm{ = }}\frac{{de[({H_1} - {H_2}){G_{CE}} \times \dot e + ({H_1} + {H_2}){G_E} \times e]}}{{2(d{e^2}{\rm{ + }}d{1^2})}} \\ \end{gathered} | (23) |
According to Eq (23) and Figure 2, the final output of IT2F-PID controller can be expressed as Equation (24).
{u_{IT2}} = {G_{PD}} \times {U_{IT2}} + {G_{PI}} \times \int {{U_{IT2}}} | (24) |
Combine Eq (23) and Eq (24), the output of IT2F-PID controller can be calculated as Eq (25).
{u_{IT2}} = K_P^{IT2} \times e + K_I^{IT2} \times \int e + K_D^{IT2} \times \dot e | (25) |
where:
K_P^{IT2} = \frac{{de \times [{G_{PD}}({H_1} + {H_2}) \times {G_E} + {G_{PI}}({H_1} - {H_2}) \times {G_{CE}}]}}{{2(d{e^2}{\rm{ + }}d{1^2})}} |
K_I^{IT2} = \frac{{de \times {G_{PI}}({H_1} + {H_2}) \times {G_E}}}{{2(d{e^2}{\rm{ + }}d{1^2})}} |
K_D^{IT2} = \frac{{de \times {G_{PD}}({H_1} - {H_2}) \times {G_{CE}}}}{{2(d{e^2}{\rm{ + }}d{1^2})}} |
Figure 8 shows the shape of control surface of interval type-2 fuzzy controller, here H1 = 1 and H2 = 0.
For type reduction of general type-2 fuzzy sets was converted to type reduction of several interval type-2 fuzzy sets, so the number of α-planes will affect the real time of GT2F-PID controller.
Figure 9 shows an example of membership degrees for fuzzy rules corresponding to consequent parameters using GT2F-PID controller.
The differences of GT2F-PID and SGT2F-PID controller can be seen from Figure 10.
From Figure 10, GT2F-PID controller firstly fixes the number of α-planes, that is D. Then derives D intervals
In this paper, the SGT2F-PID controller adapts the primary membership degree of α-plane (α = 1) as the membership degree of fuzzy rules, which is calculated as Eq (26).
{f_i}(1) = {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} _i} + w({\bar f_i} - {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} _i}) | (26) |
here, w is an adjustable parameter.
The output of simplified general type-2 fuzzy inference U(t) in Figure 2 can be calculated as Equation (27).
{U_{SGT2}}{\rm{ = }}\frac{{\sum\limits_{i = 1}^4 {{f_i}(1) \times {y_i}} }}{{\sum\limits_{i = 1}^4 {{f_i}(1)} }} = \frac{{\sum\limits_{i = 1}^4 {({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} }_i} + w({{\bar f}_i} - {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} }_i})) \times {y_i}} }}{{\sum\limits_{i = 1}^4 {({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} }_i} + w({{\bar f}_i} - {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{f} }_i}))} }} | (27) |
where,
\begin{gathered} {U_{SGT2}} = \frac{{(de - d1 + 2d1 \times w)[({H_1} - {H_2})\dot E + ({H_1} + {H_2})E]}}{{2(d{e^2} - 2de \times d1{\rm{ + }}d{1^2} + 4de \times d1 \times w)}} \\ {\rm{ }} = \frac{{(de - d1 + 2d1 \times w)[({H_1} - {H_2}){G_{CE}} \times \dot e + ({H_1} + {H_2}){G_E} \times e]}}{{2(d{e^2} - 2de \times d1{\rm{ + }}d{1^2} + 4de \times d1 \times w)}} \\ \end{gathered} | (28) |
According to Eq (28) and Figure 2, the final output of SGT2F-PID controller can be expressed as Eq (29).
{u_{SGT2}} = {G_{PD}} \times {U_{SGT2}} + {G_{PI}} \times \int {{U_{SGT2}}} | (29) |
Combine Eq (28) and Eq (29), the output of SGT2F-PID controller can be calculated as Eq (30).
{u_{SGT2}} = K_P^{SGT2} \times e + K_I^{SGT2} \times \int e + K_D^{SGT2} \times \dot e | (30) |
where:
K_P^{SGT2} = \frac{{(de - d1 + 2d1 \times w) \times [{G_{PD}}({H_1} + {H_2}) \times {G_E} + {G_{PI}}({H_1} - {H_2}) \times {G_{CE}}]}}{{2(d{e^2} - 2de \times d1{\rm{ + }}d{1^2} + 4de \times d1 \times w)}} |
K_I^{SGT2} = \frac{{(de - d1 + 2d1 \times w) \times {G_{PI}}({H_1} + {H_2}) \times {G_E}}}{{2(d{e^2} - 2de \times d1{\rm{ + }}d{1^2} + 4de \times d1 \times w)}} |
K_D^{SGT2} = \frac{{(de - d1 + 2d1 \times w) \times {G_{PD}}({H_1} - {H_2}) \times {G_{CE}}}}{{2(d{e^2} - 2de \times d1{\rm{ + }}d{1^2} + 4de \times d1 \times w)}} |
Figure 11 shows the shape of control surface of simplified general type-2 fuzzy controller, here H1 = 1, H2 = 0 and w = 0.
From the control surface curve of T1-FPID, IT2-FPID and SGT2-FPID controller, when the system error is near the endpoint, the output of SGT2-FPID controller is larger than T1-FPID and IT2-FPID, so the SGT2-FPID controller has the faster rising time. When the error is near zero, the output of SGT2-FPID controller is smoother than T1-FPID and IT2-FPID, so the SGT2-FPID controller has faster steady time and smaller overshoot.
In summary, the unified T1-FPID, IT2F-PID and SGT2F-PID controller mathematical expressions can be indicated as Eq (31).
\begin{gathered} {u_{FPID}} = K({G_{PD}}({H_1} + {H_2}) \times {G_E} + {G_{PI}}({H_1} - {H_2}) \times {G_{CE}}) \times e \\ {\rm{ }} + K{G_{PI}}({H_1} + {H_2}){G_E} \times \int e + K{G_{PD}}({H_1} - {H_2}) \times {G_{CE}} \times \dot e \\ \end{gathered} | (31) |
where:
{K_{T1}} = \frac{1}{{2de}} | (32.1) |
{K_{IT2}} = \frac{{de}}{{2(d{e^2}{\rm{ + }}d{1^2})}} | (32.2) |
{K_{SGT2}} = \frac{{(de - d1 + 2d1 \times w)}}{{2(d{e^2} - 2de \times d1{\rm{ + }}d{1^2} + 4de \times d1 \times w)}} | (32.3) |
If calculate the derivative of KSGT2 to w, then the partial derivative is Eq (33).
\frac{{\partial {K_{SGT2}}}}{{\partial w}} = \frac{{ - d1(d{e^2} - d{1^2})}}{{{{(d{e^2} - 2dede{1^2} + 4ded1w)}^2}}} \lt 0 | (33) |
From Eq (33), KSGT2 is a decreasing function of w and in general, w is in range [0, 1]. So the ranges of KSGT2 is denoted as Eq (34).
\left\{ \begin{gathered} K_{SGT2}^{\min } = \frac{1}{{2(de + d1)}}, w = 1 \\ K_{SGT2}^{\max } = \frac{1}{{2(de - d1)}}, w = 0 \\ \end{gathered} \right. | (34) |
For de > d1, so,
when w = w1, KSGT2 = KT1 and w = w2, KSGT2 = KIT2, then w1 = (de-d1)/(2de) and w2 = 0.5.
According to the characteristics of PID controller, the advantage of proportional action is timely. If increase proportional gain, then the system response speed will be enhanced (that is reducing the rising time and steady time) but the system overshoot will be increased. The integral action can eliminate static error, if increase integral gain, the system overshoot will be decreased. The differential action also has the advantage of timely, which is belonging to ‘future control’. If increase differential gain, the steady time and system overshoot will be reduced.
From above analysis, if a control system maintains both faster response speed and smaller overshoot, the PID controller should chose larger proportional gain, integral gain and differential gain. Figure 12 shows that if w < w1, then the proportional gain, integral gain and differential gain of SGTF-PID are larger than T1F-PID and IT2F-PID.Thus the controlling efforts of SGTF-PID will be better than T1F-PID and IT2F-PID, which is proved by section 5 of four simulation examples.
In simulations, 3 plants and a practical inverted pendulum system are tested to demonstrate the robustness and efficiency of SGT2F-PID. The controlling efforts of SGTF-PID are also compared with PID, T1F-PID, and IT2F-PID controller using NT type reduction algorithm.
G(s) = \frac{1}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}{e^{ - Ls}} | (35) |
The tuning PID controller parameters are KP = 0.4088, KI = 0.1084, KD = 0.3547 under case 1 plant parameters. Fuzzy PID controller parameters are GE = 0.7757, GCE = 0.7442, GPD = 3.5336, GPI = 0.6996,
Case 1: ζ = 1.125, ωn = 0.45, L = 0.4.
Case 2: ζ = 1.6875, ωn = 0.225, L = 0.4.
Case 3: ζ = 0.5624, ωn = 0.675, L = 0.4.
Case 4: ζ = 1.6875, ωn = 0.675, L = 0.6.
Table 1 summarizes some controlling performance comparisons of SGT2F-PID controller with other 3 controllers. In Table 1, ts is steady state time, tris rising time, OS is system overshoot and three error integral criterions ISE, ITSE, ITAE.
ISE = \int_0^{ts} {e{{(t)}^2}dt} |
ITSE = \int_0^{ts} {t \times e{{(t)}^2}dt} |
ITAE = \int_0^{ts} {t \times \left| {e(t)} \right|dt} |
P1 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 12.3 | 6.1 | 6.69 | 4.96 | |
tr(s) | 5.05 | 2.01 | 2.3 | 2.07 | ||
OS (%) | 5.1 | 22.2 | 19.1 | 14.9 | ||
ISE | 1.44 | 1.15 | 1.24 | 1.13 | ||
ITSE | 1.49 | 0.84 | 0.96 | 0.74 | ||
ITAE | 5.37 | 2.23 | 2.53 | 1.58 | ||
case 2 | ts(s) | 17.68 | 8.5 | 9.3 | 7.01 | |
tr(s) | 3.18 | 1.84 | 2.07 | 1.88 | ||
OS (%) | 29.8 | 39.2 | 35.1 | 28.0 | ||
ISE | 1.59 | 1.26 | 1.32 | 1.15 | ||
ITSE | 3.27 | 1.26 | 1.35 | 0.87 | ||
ITAE | 14.77 | 4.07 | 4.49 | 2.55 | ||
Case3 | ts(s) | > 20 | 9.18 | 9.72 | 7.95 | |
tr(s) | > 20 | 1.94 | 2.25 | 2.01 | ||
OS (%) | - | 17.8 | 11.7 | 11.0 | ||
ISE | 1.74 | 1.13 | 1.21 | 1.11 | ||
ITSE | 3.78 | 0.86 | 0.93 | 0.76 | ||
ITAE | 20.26 | 2.98 | 3.04 | 2.27 | ||
case 4 | ts(s) | > 20 | 8.85 | 9.61 | 8.0 | |
tr(s) | 12.06 | 3.05 | 3.59 | 3.06 | ||
OS (%) | 2.3 | 9.9 | 9.9 | 7.1 | ||
ISE | 2.67 | 1.47 | 1.61 | 1.46 | ||
ITSE | 5.73 | 1.30 | 1.60 | 1.23 | ||
ITAE | 17.08 | 3.87 | 4.77 | 3.10 |
G(s) = \frac{K}{{Ts - 1}}{e^{ - Ls}} | (36) |
The tuning PID controller parameters are KP = 9.999, KI = 0.9483, KD = 0.2785 under case 1 plant parameters. Fuzzy PID controller parameters are GE = 1.9956, GCE = 0.9387, GPD = 0.2532, GPI = 20.0573,
Case 1: K = 1, T = 10, L = 0.2.
Case 2: K = 1, T = 10, L = 0.4.
Case 3: K = 1, T = 20, L = 0.2.
Case 4: K = 2, T = 20, L = 0.35.
Table 2. shows the P2 controlling performance comparisons of SGT2F-PID controller with other 3 controllers.
P2 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 22.9 | 9.36 | 10.49 | 3.94 | |
tr(s) | 1.78 | 2.0 | 2.31 | 2.49 | ||
OS (%) | 16.7 | 27.9 | 28.2 | 5.2 | ||
ISE | 0.71 | 1.09 | 1.23 | 1.09 | ||
ITSE | 1.53 | 0.96 | 1.23 | 0.72 | ||
ITAE | 16.02 | 3.62 | 4.70 | 1.37 | ||
case 2 | ts(s) | 22.71 | 13.44 | 14.91 | 9.72 | |
tr(s) | 1.56 | 1.95 | 2.2 | 2.14 | ||
OS (%) | 19.1 | 44.9 | 46.9 | 30.5 | ||
ISE | 0.90 | 1.44 | 1.60 | 1.26 | ||
ITSE | 1.64 | 1.92 | 2.51 | 1.10 | ||
ITAE | 15.86 | 7.56 | 10.00 | 3.77 | ||
case 3 | ts(s) | 26.3 | 18.33 | 19.76 | 8.14 | |
tr(s) | 3.37 | 2.62 | 3.01 | 2.96 | ||
OS (%) | 20.4 | 43.5 | 40.04 | 17.01 | ||
ISE | 1.27 | 1.78 | 1.91 | 1.44 | ||
ITSE | 4.13 | 3.63 | 4.02 | 1.36 | ||
ITAE | 29.76 | 15.44 | 17.88 | 3.69 | ||
case 4 | ts(s) | 20.94 | 9.25 | 12.28 | 5.68 | |
tr(s) | 1.71 | 1.98 | 2.27 | 2.25 | ||
OS (%) | 12.8 | 35.3 | 36.2 | 16.7 | ||
ISE | 0.75 | 1.27 | 1.41 | 1.17 | ||
ITSE | 0.96 | 1.30 | 1.65 | 0.82 | ||
ITAE | 11.26 | 4.60 | 6.28 | 1.84 |
\frac{{{d^2}y(t)}}{{d{t^2}}} + 2\varepsilon \sigma \frac{{dy(t)}}{{dt}} + {\sigma ^2}{y^2}(t) = {\sigma ^2}u(t - L) | (37) |
PID controller parameters are KP = 0.8028, KI = 1.8548, KD = 0.4609 selected from article [1] optimized by hybridized ABC-GA algorithm. Fuzzy PID controller parameters are GE = 0.8359, GCE = 0.1944, GPD = 20.5501, GPI = 20.2681,
Case 1: ε = 1, σ = 1, L = 0.
Case 2: ε = 1, σ = 1, L = 0.1.
Case 3: ε = 1, σ = 0.7, L = 0.
Case 4: ε = 1.3, σ = 1, L = 0.
Table 3 shows the P3 controlling performance comparisons of SGT2F-PID controller with other 3 controllers.
P3 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 6.32 | 2.69 | 2.91 | 0.52 | |
tr(s) | 1.41 | 0.62 | 0.69 | 0.6 | ||
OS (%) | 16.3 | 21.9 | 20.6 | 13.1 | ||
ISE | 0.52 | 0.28 | 0.31 | 0.25 | ||
ITSE | 0.26 | 0.07 | 0.08 | 0.04 | ||
ITAE | 1.34 | 0.33 | 0.38 | 0.15 | ||
case 2 | ts(s) | 6.65 | 4.08 | 4.34 | 2.84 | |
tr(s) | 1.42 | 0.61 | 0.69 | 0.58 | ||
OS (%) | 20.1 | 38.5 | 34.0 | 27.2 | ||
ISE | 0.62 | 0.41 | 0.43 | 0.34 | ||
ITSE | 0.36 | 0.16 | 0.16 | 0.08 | ||
ITAE | 1.73 | 0.64 | 0.7 | 0.30 | ||
case 3 | ts(s) | 7.52 | 3.15 | 3.44 | 2.07 | |
tr(s) | 1.3 | 0.6 | 0.67 | 0.58 | ||
OS (%) | 22.6 | 27.2 | 26.0 | 16.1 | ||
ISE | 0.54 | 0.29 | 0.33 | 0.25 | ||
ITSE | 0.33 | 0.09 | 0.11 | 0.04 | ||
ITAE | 1.93 | 0.43 | 0.51 | 0.18 | ||
case 4 | ts(s) | 2.73 | 2.19 | 2.37 | 1.56 | |
tr(s) | 1.27 | 0.53 | 0.6 | 0.55 | ||
OS (%) | 5.7 | 12.1 | 11.1 | 4.1 | ||
ISE | 0.36 | 0.21 | 0.23 | 0.20 | ||
ITSE | 0.10 | 0.03 | 0.04 | 0.02 | ||
ITAE | 0.33 | 0.15 | 0.17 | 0.07 |
The inverted pendulum system was often applied to demonstrate the reliability of a new controller, as shown in Figure 25.
The inverted pendulum system is consisted of a cart and a pendulum, the controlling aim is to keep pendulum angle at a certain value under external force. Equation (38) describes the state equations of the inverted pendulum system [80].
\left[ \begin{gathered} {{\dot x}_1} \\ {{\dot x}_2} \\ \end{gathered} \right] = \left[ \begin{gathered} {\rm{ }}{x_2} \\ \frac{{g\sin ({x_1}) - \frac{{({m_p} + \Delta {m_p})lx_2^2\sin ({x_1})\cos ({x_1})}}{{({m_p} + \Delta {m_p} + {m_c})}}}}{{\frac{{4l}}{3} - \frac{{(({m_p} + \Delta {m_p})l\cos {{({x_1})}^2}}}{{({m_p} + \Delta {m_p} + {m_c})}}}} \\ \end{gathered} \right] + \Delta A\left[ \begin{gathered} {x_1} \\ {x_2} \\ \end{gathered} \right] + \left[ \begin{gathered} {\rm{ }}0 \\ \frac{{\frac{{\cos ({x_1})}}{{({m_p} + \Delta {m_p} + {m_c})}}}}{{\frac{{4l}}{3} - \frac{{(({m_p} + \Delta {m_p})l\cos {{({x_1})}^2}}}{{({m_p} + \Delta {m_p} + {m_c})}}}} \\ \end{gathered} \right]u | (38) |
In (38), x1is the pendulum angle θ and x2 is the pendulum angular velocity
PID controller parameters are KP = 40, KI = 100, KD = 8. Fuzzy PID controller parameters are GE = 0.1009, GCE = 0.1944, GPD = 30.5501, GPI = 30.2681,
Case 1: Normal case.
The initial conditions x1 = 0.1rad and x2 = 0rad/s, the setting value is x1 = 0rad. In normal case, Δmp = 0 and
Case 2: Normal case.
The initial conditions x1 = 0.4rad and x2 = 0rad/s, the setting value is x1 = 0rad. In normal case, Δmp = 0 and
From case 3 to case 6, we will indicate the controlling effects of SGT2-FPID controller when the system adding uncertainties.
Case 3: Pendulum mass uncertainty.
Here, we will add pendulum mass uncertainty (Δmp = 2.7kg) at 2s.
Case 4: Measurement uncertainty in pendulum angle.
Here, we will add measurement uncertainty in pendulum angle θ (∆x1 = 0.052) at 3s.
Case 5: Structure uncertainty. Here, we will add structural uncertainty in the inverted pendulum as 2s (
Case 6: External disturbance uncertainty.
Here, we will add an external disturbance of controlling force at 2s (∆d = 29N).
Table 4 shows the P4 controlling performance comparisons of SGT2F-PID controller with other 5 controllers for case 1 and case 2. As compares with controlling performances of [80] and [81], another two error integral criterions are added as follows.
RMSE = \sqrt {\frac{1}{N}\sum\limits_{i = 1}^N {e{{(i)}^2}} } |
IAE = \int_0^{ts} {\left| {e(t)} \right|dt} |
P4 | IT2F-PID [80] | IT2F-PD+I [81] | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ISE | 0.036 | - | 2.78 × 10-4 | 3.32 × 10-4 | 2.53 × 10-4 | 1.9 × 10-4 | |
ITSE | - | - | 2.34 × 10-5 | 2.0 × 10-5 | 7.22 × 10-6 | 3.64 × 10-6 | ||
ITAE | - | - | 0.0036 | 9.35 × 10-4 | 4.52 × 10-4 | 1.62 × 10-4 | ||
RMSE | 0.0085 | - | 0.0118 | 0.0129 | 0.013 | 0.0097 | ||
IAE | 1.8001 | - | 0.0101 | 0.0076 | 0.0051 | 0.0033 | ||
case 2 | ISE | - | 1.5844 | 0.0045 | 0.0064 | 0.0069 | 0.005 | |
ITSE | - | - | 3.33 × 10-4 | 2.34 × 10-4 | 2.43 × 10-4 | 1.23 × 10-4 | ||
ITAE | - | - | 0.0119 | 0.003 | 0.0029 | 0.0014 | ||
RMSE | - | 0.0514 | 0.0472 | 0.0568 | 0.0586 | 0.0499 | ||
IAE | - | 7.4692 | 0.0379 | 0.029 | 0.0287 | 0.0191 |
Table 5 shows the P4 controlling performance comparisons of SGT2F-PID controller with other 4 controllers for case 3 to case 6.
P4 | IT2F-PD+I [81] | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 3 | ISE | 1.9203 | 0.0045 | 0.0064 | 0.0069 | 0.005 | |
ITSE | - | 3.56 × 10-4 | 2.34 × 10-4 | 2.43 × 10-4 | 1.25 × 10-4 | ||
ITAE | - | 0.0288 | 0.0035 | 0.0037 | 0.0019 | ||
RMSE | 0.04 | 0.0273 | 0.0328 | 0.0338 | 0.0288 | ||
IAE | 14.7056 | 0.0419 | 0.0292 | 0.0290 | 0.0192 | ||
case 4 | ISE | 2.527 | 0.0046 | 0.0066 | 0.007 | 0.0051 | |
ITSE | - | 6.47 × 10-4 | 5.9 × 10-4 | 5.5 × 10-4 | 4.0 × 10-4 | ||
ITAE | - | 0.0316 | 0.0172 | 0.0154 | 0.011 | ||
RMSE | 0.0649 | 0.0276 | 0.0331 | 0.0341 | 0.0291 | ||
IAE | 13.3876 | 0.044 | 0.033 | 0.032 | 0.022 | ||
case 5 | ISE | 0.094 | 2.78 × 10-4 | 3.32 × 10-4 | 2.53 × 10-4 | 1.90 × 10-4 | |
ITSE | - | 2.34 × 10-5 | 2.04 × 10-5 | 7.73 × 10-6 | 3.64 × 10-6 | ||
ITAE | - | 0.0036 | 0.0010 | 5.08 × 10-4 | 1.89 × 10-4 | ||
RMSE | 0.0125 | 0.0068 | 0.0074 | 0.0065 | 0.0056 | ||
IAE | 2.2475 | 0.0101 | 0.0077 | 0.0052 | 0.0033 | ||
case 6 | ISE | - | 0.0046 | 0.0065 | 0.0069 | 0.005 | |
ITSE | - | 7.53 × 10-4 | 3.67 × 10-4 | 3.79 × 10-4 | 1.75 × 10-4 | ||
ITAE | - | 0.0447 | 0.0169 | 0.0185 | 0.0091 | ||
RMSE | - | 0.0278 | 0.0329 | 0.0340 | 0.0289 | ||
IAE | - | 0.0508 | 0.0345 | 0.0347 | 0.022 |
We discuss 3 kinds of fuzzy PID controllers and derive the mathematical expressions of TIF-PID, IT2F-PID and SGT2F-PID described by Eq (21), Eq (25) and Eq (30). The SGT2F-PID controller contains more adjustable parameters and only 4 fuzzy rules are generated. For the primary membership degree of α-plane (α = 1) is used to get the defuzzification result of SGT2F-PID controller, thus the SGT2F-PID controller maintains the ability of handing uncertainties as general type-2 fuzzy controller and higher real-time. By the mathematical expressions of TIF-PID, IT2F-PID and SGT2F-PID controller, the controlling performance is discussed and explains why SGT2F-PID controller has better controlling effects than TIF-PID and IT2F-PID controller.
And 4 simulations including a second order linear plant, an unstable first order linear plant and two second order nonlinear plants are tested. In addition, the controller parameters of each plant are the same when the plant parameters are changed, which demonstrate the robustness of SGT2F-PID controller. From the 4 simulation results, when the controlled object changes, the SGT2F-PID controller can still maintain small overshoot, faster response time and stable time. Also the controller performance evaluation indexes (ISE, ITSE, ITAE) of SGT2F-PID controller are better than other 3 compared controllers. The results of simulation 4 indicates that, when the controlled object exists uncertainties of measurement, structure and external disturbance, the SGT2F-PID controller can handle these uncertainties more robust than PID, TIF-PID and IT2F-PID controller.
The next researches will focus on the following 4 aspects:
Ⅰ). Although SGT2F-PID controller can achieve better control performances, but the determined parameters are more than other controllers. How to determine the appropriate parameters will be a major work.
Ⅱ). Triangular function is applied as primary and secondary membership function, other membership function like Gaussian, trapezoid will be discussed in the future.
Ⅲ). In this paper, we fix the parameters de and d1 and discuss the influence of w on the controller parameters gains. In the future, we will study the influence of de and d1 on the controller parameters gains.
Ⅳ). The fractional order simplified general type-2 fuzzy PID controller will be investigated and compared with existing PID and fuzzy PID controllers.
This study was funded by the scientific research fund project of Nanjing Institute of Technology (YKJ201523, QKJ201802).
The authors declare there is no conflict of interest.
[1] | M. F. Abdelmalek and A. M. Diehl, Nonalcoholic fatty liver disease as a complication of insulin resistance, Med. Clin. North Am., 91 (2007), 1125-1149. |
[2] | L. A. Adams, P. Angulo and K. D. Lindor, Nonalcoholic fatty liver disease, CMAJ, 172 (2005), 899-905. |
[3] | C. M. Anderson and A. Stahl, SLC27 fatty acid transport proteins, Mol. Aspects Med., 34 (2013), 516-528. |
[4] | Y. G. Anissimov and M. S. Roberts, A compartmental model of hepatic disposition kinetics: 1. Model development and application to linear kinetics, J. Pharmacokinet. Pharmacodyn., 29 (2002), 131-156. |
[5] | J. P. Arab, M. Arrese and M. Trauner, Recent insights into the pathogenesis of nonalcoholic fatty liver disease, Annu. Rev. Pathol. Mech. Dis., 13 (2018), 321-350. |
[6] | W. B. Ashworth, N. A. Davis and I. D. L. Bogle, A computational model of hepatic energy metabolism: understanding zonated damage and steatosis in NAFLD, PLoS Comput. Biol., 12 (2016), e1005105. |
[7] | W. B. Ashworth, C. Perez-Galvan, N. A. Davies and I. D. L. Bogle, Liver function as an engineering system, AIChE J., 62 (2016), 3285-3297. |
[8] | B. A. Banini and A. J. Sanyal, Nonalcoholic fatty liver disease: epidemiology, pathogenesis, natural history, diagnosis, and current treatment options, Clin. Med. Insights Ther., 8 (2016), 75-84. |
[9] | H. T. Banks, Modeling and Control in the Biomedical Sciences, Lecture Notes in Biomathematics, Vol 6, Springer-Verlag, New York, 1975. |
[10] | H.T. Banks and H.T. Tran, Mathematical and Experimental Modling of Physical and Biological Processes, Chapman & Hall/CRC Press, Taylor & Francis Group, Boca Raton, 2009. |
[11] | L. Bass, S. Keiding, K. Winkler and N. Tygstrup, Enzymatic elimination of substrates flowing through the intact liver, J. Theor. Biol., 61 (1976), 393-409. |
[12] | L. Bass, P. Robinson and A. J. Bracken, Hepatic elimination of flowing substrates: the distributed model, J. Theor. Biol., 72 (1978), 161-184. |
[13] | N. Berndt, M. S. Horger, S. Bulik and H. G. Holzhütter, A multiscale modelling approach to assess the impact of metabolic zonation and microperfusion on the hepatic carbohydrate metabolism, PLoS Comput. Biol., 14 (2018), e1006005. |
[14] | S. N. Bhatia, M. Toner, B. D. Foy, A. Rotem, K. M. O'Neil, R. G. Tompkins and M. L. Yarmush, Zonal liver cell heterogeneity: effects of oxygen on metabolic functions of hepatocytes, Cell. Eng., 1 (1996), 125-135. |
[15] | P. N. Black, C. Ahowesso, D. Montefusco, N. Saini and C. C. DiRusso, Fatty acid transport proteins: targeting FATP2 as a gatekeeper involved in the transport of exogenous fatty acids, Medchemcomm, 7 (2016), 612-622. |
[16] | P. N. Black, A. Sandoval, E. Arias-Barrau and C. C. DiRusso, Targeting the fatty acid transport proteins (FATP) to understand the mechanisms linking fatty acid transport to metabolism, Immunol. Endocr. Metab. Agents Med. Chem., 9 (2009), 11-17. |
[17] | E. M. Brunt, Pathology of nonalcoholic fatty liver disease, Nat. Rev. Gastroenterol. Hepatol., 7 (2010), 195-203. |
[18] | E. M. Brunt, Pathology of fatty liver disease, Mod. Pathol., 20 (2007), S40-S48. |
[19] | E. M. Brunt, C. G. Janney, A. M. Di Bisceglie, B. A. Neuschwander-Tetri and B. R. Bacon, Nonalcoholic steatohepatitis: a proposal for grading and staging the histological lesions, Am. J. Gastroenterol., 94 (1999), 2467-2474. |
[20] | X. Buqué, A. Cano, M. E. Miquilena-Colina, C. García-Monzón, B. Ochoa and P. Aspichueta, High insulin levels are required for FAT/CD36 plasma membrane translocation and enhanced fatty acid uptake in obese Zucker rat hepatocytes, Am. J. Physiol. Endocrinol. Metab., 303 (2012), E504-E514. |
[21] | D. Calvetti, A. Kuceyeski and E. Somersalo, Sampling-based analysis of a spatially distributed model for liver metabolism at steady state, Multiscale Model. Simul., 7 (2008), 407-431. |
[22] | G. D. Cartee, Mechanisms for greater insulin-stimulated glucose uptake in normal and insulin-resistant skeletal muscle after acute exercise, Am. J. Physiol. Endocrinol. Metab., 309 (2015), E949-E959. |
[23] | N. Chalasani, L. Wilson, D. E. Kleiner, O. W. Cummings, E. M. Brunt and A. Ünalp, Relationship of steatosis grade and zonal location to histological features of steatohepatitis in adult patients with non-alcoholic fatty liver disease, J. Hepatol., 48 (2008), 829-834. |
[24] | E. Chalhoub, L. Xie, V. Balasubramanian, J. Kim and J. Belovich, A distributed model of carbohydrate transport and metabolism in the liver during rest and high-intensity exercise, Ann. Biomed. Eng., 35 (2007), 474-491. |
[25] | M. Colletti, C. Cicchini, A. Conigliaro, L. Santangelo, T. Alonzi, E. Pasquini, M. Tripodi and L. Amicone, Convergence of Wnt signaling on the HNF4α-driven transcription in controlling liver zonation, Gastroenterology, 137 (2009), 660-672. |
[26] | S. W. Coppack, R. M. Fisher, G. F. Gibbons, S. M. Humphreys, M. J. McDonough, J. L. Potts and K. N. Frayn, Postprandial substrate deposition in human forearm and adipose tissues in vivo, Clin. Sci. (Lond.), 79 (1990), 339-348. |
[27] | M. E. Daly, C. Vale, M. Walker, A. Littlefield, K. G. Alberti and J. C. Mathers, Acute effects of insulin sensitivity and diurnal metabolic profiles of a high-sucrose compared with a high-starch diet, Am. J. Clin. Nutr., 67 (1998), 1186-1196. |
[28] | R. A. DeFronzo and E. Ferrannini, Influence of plasma glucose and insulin concentration on plasma glucose clearance in man, Diabetes, 31 (1982), 683-688. |
[29] | A. Deussen and J. B. Bassingthwaighte, Modeling [15O] oxygen tracer data for estimating oxygen consumption, Am. J. Physiol., 270 (1996), H1115-H1130. |
[30] | Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Muller and Birgitt Schonfisch, A Course in Mathematical Biology: Quantitative Modelling with Mathematical and Computational Methods, SIAM, Philadephia, 2006. |
[31] | G. Dimitriadis, P. Mitrou, V. Lambadiari, E. Boutati, E. Maratou, E. Koukkou, M. Tzanela, N. Thalassinos and S. A. Raptis, Glucose and lipid fluxes in the adipose tissue after meal ingestion in hyperthyroidism, J. Clin. Endocrinol. Metab., 91 (2006), 1112-1118. |
[32] | H. Doege, R. A. Baillie, A. M. Ortegon, B. Tsang, Q. Wu, S. Punreddy, D. Hirsch, N. Watson, R. Gimeno and A. Stahl, Targeted deletion of FATP5 reveals multiple functions in liver metabolism: alterations in hepatic lipid homeostasis, Gastroenterology, 130 (2006), 1245-1258. |
[33] | H. Doege, D. Grimm, A. Falcon, B. Tsang, T. A. Storm, H. Xu, A. M. Ortegon, M. Kazantzis, M. A. Kay and A. Stahl, Silencing of hepatic fatty acid transporter protein 5in vivo reverses diet-induced non-alcoholic fatty liver disease and improves hyperglycemia, J. Biol. Chem., 283 (2008), 22186-22192. |
[34] | B. Erdogmus, A. Tamer, R. Buyukkaya, B. Yazici, A. Buyukkaya, E. Korkut, A. Alcelik and U. Korkmaz, Portal vein hemodynamics in patients with non-alcoholic fatty liver disease, Tohoku J. Exp. Med., 215 (2008), 89-93. |
[35] | A. Falcon, H. Doege, A. Fluitt, B. Tsang, N. Watson, M. A. Kay and A. Stahl, FATP2 is a hepatic fatty acid transporter and peroxisomal very long-chain acyl-CoA synthetase, Am. J. Physiol. Endocrinol. Metab., 299 (2010), E384-E393. |
[36] | S. Fogler, Elements of Chemical Reaction Engineering, 3rd edition, Prentice Hall, New Jersey, 2001. |
[37] | E. L. Forker and B. Luxon, Hepatic transport kinetics and plasma disappearance curves: distributed modeling vs. conventional approach, Am. J. Physiol., 235 (1978), E648-E660. |
[38] | R. L. Fournier, Basic Transport Phenomena in Biomedical Engineering, Taylor & Francis, New York, 1998. |
[39] | X. Fu, J. P. Sluka, S. G. Clendenon1, K. W. Dunn, Z. Wang, J. E. Klaunig and J. A. Glazier, Modeling of Xenobiotic Transport and Metabolism in Virtual Hepatic Lobule Models, PLOS ONE, 13 (2018), e0198060. |
[40] | R. Gebhardt, Metabolic zonation of the liver: regulation and implications for liver function, Pharmacol. Ther., 53 (1992), 275-354. |
[41] | Z. Gong, E. Tas, S. Yakar and R. Muzumdar, Hepatic lipid metabolism and non-alcoholic fatty liver disease in aging, Mol. Cell. Endocrinol., 455 (2017), 115-130. |
[42] | M. R. Gray and Y. K. Tam, The series-compartment model for hepatic elimination, Drug Metab. Dispos., 15 (1987), 27-31. |
[43] | J. T. Haas, S. Francque and B. Staels, Pathophysiology and mechanisms of nonalcoholic fatty liver disease. Annu. Rev. Physiol., 78 (2016), 181-205. |
[44] | K. C. Hames, A. Vella, B. J. Kemp and M. D. Jensen, Free fatty acid uptake in humans with CD36 deficiency. Diabetes, 63 (2014), 3606-3614. |
[45] | T. Hardy, F. Oakley, Q. M. Anstee and C. P. Day, Nonalcoholic fatty liver disease: pathogenesis and disease spectrum. Annu. Rev. Pathol. Mech. Dis., 11 (2016), 451-496. |
[46] | B. S. Hijmans, A. Grefhorst, M. H. Oosterveer and A. K. Groen, Zonation of glucose and fatty acid metabolism in the liver: mechanism and metabolic consequences, Biochimie, 96 (2014), 121-129. |
[47] | W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, Berlin, Germany, 2003. |
[48] | K. Jungermann, Metabolic zonation of liver parenchyma: significance for the regulation of glycogen metabolism, gluconeogenesis, and glycolysis, Diabetes Metab. Rev., 3 (1987), 269-293. |
[49] | K. Jungermann and N. Katz, Functional hepatocellular heterogeneity, Hepatology, 2 (1982), 385-395. |
[50] | K. Jungermann and N. Katz, Functional specialization of different hepatocyte populations, Physiol. Rev., 69 (1989), 708-764. |
[51] | K. Jungermann and T. Kietzmann, Oxygen: modulator of metabolic zonation and disease of the liver, Hepatology, 31 (2000), 255-260. |
[52] | K. Jungermann and T. Kietzmann, Role of oxygen in the zonation of carbohydrate metabolism and gene expression in liver, Kidney Int., 51 (1997), 402-412. |
[53] | K. Jungermann and T. Kietzmann, Zonation of parenchymal and nonparenchymal metabolism in liver, Annu. Rev. Nutr., 16 (1996), 179-203. |
[54] | K. Jungermann and D. Sasse, Heterogeneity of liver parenchymal cells, Trends Biochem. Sci., 3 (1978), P198-P202. |
[55] | N. R. Katz, Metabolic heterogeneity of hepatocytes across the liver acinus, J. Nutr., 122 (1992), 843-849. |
[56] | T. Kietzmann, Metabolic zonation of the liver: the oxygen gradient revisited, Redox Biol.,11 (2017), 622-630. |
[57] | T. Kietzmann, E.Y. Dimova, D. Flúgel and J. G. Scharf, Oxygen: modulator of physiological and pathophysiological processes in the liver, Z. Gastroenterol., 44 (2006), 67-76. |
[58] | T. Kietzmann and K. Jungermann, Modulation by oxygen of zonal gene expression in liver studied in primary rat hepatocyte cultures, Cell. Biol. Toxicol., 13 (1997), 243-255. |
[59] | M. Kot, Elements of Mathematical Biology, Cambridge University Press, Cambridge, UK, 2001. |
[60] | Y. Li, C. C. Chow, A. B. Courville, A. D. Sumner and V. Periwal, Modeling glucose and free fatty acid kinetics in glucose and meal tolerance test, Theor. Biol. Med. Model., 13 (2016). |
[61] | D. Magalotti, G. Marchesini, S. Ramilli, A. Berzigotti, G. Bianchi and M. Zoli, Splanchnic haemodynamics in non-alcoholic fatty liver disease: effect of a dietary/pharmacological treatment: a pilot study, Dig. Liver Dis., 36 (2004), 406-411. |
[62] | D. G. Mashek, Hepatic fatty acid trafficking: multiple forks in the road, Adv. Nut., 4 (2013), 697-710. |
[63] | M. E. Miquilena-Colina, E. Lima-Cabello, S. Sánchez-Campos, M. V. García-Mediavilla, M. Fernández-Bermejo, T. Lozano-Rodríguez, J. Vargas-Castrillón, X. Buqe, B. Ochoa, P. Aspichueta, J. González-Gallego and C. García-Monzón, Hepatic fatty acid translocase CD36 upregulation is associated with insulin resistance, hyperinsulinaemia and increased steatosis in non-alcoholic steatohepatitis and chronic hepatitis C, Gut, 60 (2011), 1394-1402. |
[64] | H. Mitsuyoshi, K. Yasui, Y. Harano, M. Endo, K. Tsuji, M. Minami, Y. Itoh, T. Okanoue and T. Yoshikawa, Analysis of hepatic genes involved in the metabolism of fatty acids and iron in nonalcoholic fatty liver disease, Hepatol. Res., 39 (2009), 366-373. |
[65] | A. Mohammadi, M. Ghasemi-rad, H. Zahedi, G. Toldi and T. Alinia, Effect of severity of steatosis as assessed ultrasonographically on hepatic vascular indices in non-alcoholic fatty liver disease, Med. Ultrason., 13 (2011), 200-206. |
[66] | E. P. Newberry, Y. Xie, S. Kennedy, X. Han, K. K. Buhman, J. Luo, R. W. Gross and N. O. Davidson, Decreased hepatic triglyceride accumulation and altered fatty acid uptake in mice with deletion of the liver fatty acid-binding protein gene, J. Biol. Chem., 278 (2003), 51664-51672. |
[67] | H. Ohno, Y. Naito, H. Nakajima and M. Tomita, Construction of a biological tissue model based on a single-cell model: a computer simulation at metabolic heterogeneity in the liver lobule, Artif. Life, 14 (2008), 3-28. |
[68] | A. Okubo, Difffusion and Ecological Problems: Mathematical Models, Springer-Verlag, Berlin, Heidelberg, New York, 1980. |
[69] | K. S. Pang, M. Weiss and P. Macheras, Advanced pharmacokinetic models based on organ clearance, circulatory, and fractal concepts, AAPS J., 9 (2007), E268-E283. |
[70] | S. Park, S. H. J. Kim, G. E. P. Ropella, M. S. Roberts and C. A. Hunt, Tracing multiscale mechanisms of drug disposition in normal and diseased livers, J. Pharmacol. Exp. Ther., 334 (2010), 124-136. |
[71] | V. Periwal, C. C. Chow, R. N. Bergman, M. Rick, G. L. Vega and A. E. Sumner, Evaluation of quantitative models of the effect of insulin on lipolysis and glucose disposal, Am. J. Physiol. Regul. Integr. Comp. Physiol., 295 (2008), R1089-R1096. |
[72] | I. Probst, P. Schwartz and K. Jungermann, Induction in primary culture of `gluconeogenic' and `glycolytic' hepatocytes resembling periportal and perivenous cells, Eur. J. Biochem., 126 (1982), 271-278. |
[73] | G. Rajaraman, M. S. Roberts, D. Hung, G. Q. Wang and F. J. Burczynski, Membrane binding proteins are the major determinants for the hepatocellular transmembrane flux of long-chain fatty acids bound to albumin, Pharm. Res., 22 (2005), 1793-1804. |
[74] | V Rezania, D. Coombe and J. A. Tuszynski, A physiologically-based flow network model for hepatic drug elimination III: 2D/3D DLA lobule models, Theor. Biol. Med. Model., 13 (2016). |
[75] | T. Ricken, U. Dahmen and O. Dirsch, A biphasic model for sinusoidal liver perfusion remodeling after outflow obstruction. Biomech. Model. Mechanobiol., 9 (2010), 435-450. |
[76] | T. Ricken, D. Werner, H. G. Holzhütter, M. König, U. Dahmen and O. Dirsch, Modeling function-perfusion behavior in liver lobules including tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale PDE-ODE approach, Biomech. Model. Mechanobiol., 14 (2015), 515-536. |
[77] | T. Ricken, N. Waschinsky and D. Werner, Simulation of steatosis zonation in liver lobule-a continuummechanical bi-scale, tri-phasic, multi-component approach, inLecture Notes in Applied and Computational Mechanics, Vol. 84, (eds. P. Wriggers and T. Lenarz), Springer, (2018), 15-33. |
[78] | M. S. Roberts and M. Rowland, Correlation between in-vitro microsomal enzyme activity and whole organ hepatic elimination kinetics: analysis with a dispersion model, J. Pharm. Pharmacol., 38 (1986), 177-181. |
[79] | M. S. Roberts and M. Rowland, Hepatic elimination-dispersion model, J. Pharm. Sci., 74 (1985), 585-587. |
[80] | S. I. Rubinow, Introduction to Mathematical Biology, John Wiley & Sons, New York, 1975. |
[81] | J. Schleicher, U. Dahmen, R. Guthke and S. Schuster, Zonation of hepatic fat accumulation: insights from mathematical modelling of nutrient gradients and fatty acid uptake, J. R. Soc. Interface, 14 (2017), 20170443. |
[82] | S. Sheikh-Bahaei, J. J. Maher and C. A. Hunt, Computational experiments reveal plausible mechanisms for changing patterns of hepatic zonation of xenobiotic clearance and hepatotoxicity, J. Theor. Biol., 265 (2010), 718-733. |
[83] | J. Shi and K. V. Kandror, Study of glucose uptake in adipose cells, Methods Mol. Biol., 456 (2008), 307-315. |
[84] | J. P. Sluka, X. Fu, Maciej Swat, J. M. Belmonte, A. Cosmanescu, S. G. Clendeno, J. F. Wambaugh, and J. A. Glazier, A liver-centric multiscale modeling framework for xenobiotics, PLOS ONE (11), 2016: DOI:10.1371/journal.pone.0162428. |
[85] | M. Soresi, L. Giannitrapani, D. Noto, A. Terranova, M. E. Campagna, A. B. Cefal, A. Giammanco and G. Montalto, Effects of steatosis on hepatic hemodynamics in patients with metabolic syndrome, Ultrasound Med. Biol., 41 (2015), 1545-1552. |
[86] | A. W. Thorburn, B. Gumbiner, F. Bulacan, P. Wallace and R. R. Henry, Intracellular glucose oxidation and glycogen synthase activity are reduced in non-insulin-dependent (type II) diabetes independent of impaired glucose uptake, J. Clin. Invest., 85 (1990), 522-529. |
[87] | C. Torre, C. Perret and S. Colnot, Molecular determinants of liver zonation, Prog. Mol. Biol. Transl. Sci., 97 (2010), 127-150. |
[88] | V. van Ginneken, E. de Vries, E. Verheij and J. van der Greef, Potential biomarkers for ``fatty liver'' (hepatic steatosis) and hepatocellular carcinoma (HCC) and an explanation of their pathogenesis, Gastroenterol. Liver Clin. Med., 1 (2017), 001. |
[89] | D. Wölfle and K. Jungermann, Long-term effects of physiological oxygen concentrations on glycolysis and gluconeogenesis in hepatocyte cultures, Eur. J. Biochem., 151, (1985), 299-303. |
[90] | Q. Wu, A. M. Ortegon, B. Tsang, H. Doege, K. R. Feingold and A. Stahl, FATP1 is an insulin-sensitive fatty acid transporter involved in diet-induced obesity, Mol. Cell. Biol., 26 (2006), 3455-3467. |
[91] | M. M. Yeh and E. M. Brunt, Pathological features of fatty liver disease, Gastroenterology, 147 (2014), 754-764. |
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P1 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 12.3 | 6.1 | 6.69 | 4.96 | |
tr(s) | 5.05 | 2.01 | 2.3 | 2.07 | ||
OS (%) | 5.1 | 22.2 | 19.1 | 14.9 | ||
ISE | 1.44 | 1.15 | 1.24 | 1.13 | ||
ITSE | 1.49 | 0.84 | 0.96 | 0.74 | ||
ITAE | 5.37 | 2.23 | 2.53 | 1.58 | ||
case 2 | ts(s) | 17.68 | 8.5 | 9.3 | 7.01 | |
tr(s) | 3.18 | 1.84 | 2.07 | 1.88 | ||
OS (%) | 29.8 | 39.2 | 35.1 | 28.0 | ||
ISE | 1.59 | 1.26 | 1.32 | 1.15 | ||
ITSE | 3.27 | 1.26 | 1.35 | 0.87 | ||
ITAE | 14.77 | 4.07 | 4.49 | 2.55 | ||
Case3 | ts(s) | > 20 | 9.18 | 9.72 | 7.95 | |
tr(s) | > 20 | 1.94 | 2.25 | 2.01 | ||
OS (%) | - | 17.8 | 11.7 | 11.0 | ||
ISE | 1.74 | 1.13 | 1.21 | 1.11 | ||
ITSE | 3.78 | 0.86 | 0.93 | 0.76 | ||
ITAE | 20.26 | 2.98 | 3.04 | 2.27 | ||
case 4 | ts(s) | > 20 | 8.85 | 9.61 | 8.0 | |
tr(s) | 12.06 | 3.05 | 3.59 | 3.06 | ||
OS (%) | 2.3 | 9.9 | 9.9 | 7.1 | ||
ISE | 2.67 | 1.47 | 1.61 | 1.46 | ||
ITSE | 5.73 | 1.30 | 1.60 | 1.23 | ||
ITAE | 17.08 | 3.87 | 4.77 | 3.10 |
P2 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 22.9 | 9.36 | 10.49 | 3.94 | |
tr(s) | 1.78 | 2.0 | 2.31 | 2.49 | ||
OS (%) | 16.7 | 27.9 | 28.2 | 5.2 | ||
ISE | 0.71 | 1.09 | 1.23 | 1.09 | ||
ITSE | 1.53 | 0.96 | 1.23 | 0.72 | ||
ITAE | 16.02 | 3.62 | 4.70 | 1.37 | ||
case 2 | ts(s) | 22.71 | 13.44 | 14.91 | 9.72 | |
tr(s) | 1.56 | 1.95 | 2.2 | 2.14 | ||
OS (%) | 19.1 | 44.9 | 46.9 | 30.5 | ||
ISE | 0.90 | 1.44 | 1.60 | 1.26 | ||
ITSE | 1.64 | 1.92 | 2.51 | 1.10 | ||
ITAE | 15.86 | 7.56 | 10.00 | 3.77 | ||
case 3 | ts(s) | 26.3 | 18.33 | 19.76 | 8.14 | |
tr(s) | 3.37 | 2.62 | 3.01 | 2.96 | ||
OS (%) | 20.4 | 43.5 | 40.04 | 17.01 | ||
ISE | 1.27 | 1.78 | 1.91 | 1.44 | ||
ITSE | 4.13 | 3.63 | 4.02 | 1.36 | ||
ITAE | 29.76 | 15.44 | 17.88 | 3.69 | ||
case 4 | ts(s) | 20.94 | 9.25 | 12.28 | 5.68 | |
tr(s) | 1.71 | 1.98 | 2.27 | 2.25 | ||
OS (%) | 12.8 | 35.3 | 36.2 | 16.7 | ||
ISE | 0.75 | 1.27 | 1.41 | 1.17 | ||
ITSE | 0.96 | 1.30 | 1.65 | 0.82 | ||
ITAE | 11.26 | 4.60 | 6.28 | 1.84 |
P3 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 6.32 | 2.69 | 2.91 | 0.52 | |
tr(s) | 1.41 | 0.62 | 0.69 | 0.6 | ||
OS (%) | 16.3 | 21.9 | 20.6 | 13.1 | ||
ISE | 0.52 | 0.28 | 0.31 | 0.25 | ||
ITSE | 0.26 | 0.07 | 0.08 | 0.04 | ||
ITAE | 1.34 | 0.33 | 0.38 | 0.15 | ||
case 2 | ts(s) | 6.65 | 4.08 | 4.34 | 2.84 | |
tr(s) | 1.42 | 0.61 | 0.69 | 0.58 | ||
OS (%) | 20.1 | 38.5 | 34.0 | 27.2 | ||
ISE | 0.62 | 0.41 | 0.43 | 0.34 | ||
ITSE | 0.36 | 0.16 | 0.16 | 0.08 | ||
ITAE | 1.73 | 0.64 | 0.7 | 0.30 | ||
case 3 | ts(s) | 7.52 | 3.15 | 3.44 | 2.07 | |
tr(s) | 1.3 | 0.6 | 0.67 | 0.58 | ||
OS (%) | 22.6 | 27.2 | 26.0 | 16.1 | ||
ISE | 0.54 | 0.29 | 0.33 | 0.25 | ||
ITSE | 0.33 | 0.09 | 0.11 | 0.04 | ||
ITAE | 1.93 | 0.43 | 0.51 | 0.18 | ||
case 4 | ts(s) | 2.73 | 2.19 | 2.37 | 1.56 | |
tr(s) | 1.27 | 0.53 | 0.6 | 0.55 | ||
OS (%) | 5.7 | 12.1 | 11.1 | 4.1 | ||
ISE | 0.36 | 0.21 | 0.23 | 0.20 | ||
ITSE | 0.10 | 0.03 | 0.04 | 0.02 | ||
ITAE | 0.33 | 0.15 | 0.17 | 0.07 |
P4 | IT2F-PID [80] | IT2F-PD+I [81] | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ISE | 0.036 | - | 2.78 × 10-4 | 3.32 × 10-4 | 2.53 × 10-4 | 1.9 × 10-4 | |
ITSE | - | - | 2.34 × 10-5 | 2.0 × 10-5 | 7.22 × 10-6 | 3.64 × 10-6 | ||
ITAE | - | - | 0.0036 | 9.35 × 10-4 | 4.52 × 10-4 | 1.62 × 10-4 | ||
RMSE | 0.0085 | - | 0.0118 | 0.0129 | 0.013 | 0.0097 | ||
IAE | 1.8001 | - | 0.0101 | 0.0076 | 0.0051 | 0.0033 | ||
case 2 | ISE | - | 1.5844 | 0.0045 | 0.0064 | 0.0069 | 0.005 | |
ITSE | - | - | 3.33 × 10-4 | 2.34 × 10-4 | 2.43 × 10-4 | 1.23 × 10-4 | ||
ITAE | - | - | 0.0119 | 0.003 | 0.0029 | 0.0014 | ||
RMSE | - | 0.0514 | 0.0472 | 0.0568 | 0.0586 | 0.0499 | ||
IAE | - | 7.4692 | 0.0379 | 0.029 | 0.0287 | 0.0191 |
P4 | IT2F-PD+I [81] | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 3 | ISE | 1.9203 | 0.0045 | 0.0064 | 0.0069 | 0.005 | |
ITSE | - | 3.56 × 10-4 | 2.34 × 10-4 | 2.43 × 10-4 | 1.25 × 10-4 | ||
ITAE | - | 0.0288 | 0.0035 | 0.0037 | 0.0019 | ||
RMSE | 0.04 | 0.0273 | 0.0328 | 0.0338 | 0.0288 | ||
IAE | 14.7056 | 0.0419 | 0.0292 | 0.0290 | 0.0192 | ||
case 4 | ISE | 2.527 | 0.0046 | 0.0066 | 0.007 | 0.0051 | |
ITSE | - | 6.47 × 10-4 | 5.9 × 10-4 | 5.5 × 10-4 | 4.0 × 10-4 | ||
ITAE | - | 0.0316 | 0.0172 | 0.0154 | 0.011 | ||
RMSE | 0.0649 | 0.0276 | 0.0331 | 0.0341 | 0.0291 | ||
IAE | 13.3876 | 0.044 | 0.033 | 0.032 | 0.022 | ||
case 5 | ISE | 0.094 | 2.78 × 10-4 | 3.32 × 10-4 | 2.53 × 10-4 | 1.90 × 10-4 | |
ITSE | - | 2.34 × 10-5 | 2.04 × 10-5 | 7.73 × 10-6 | 3.64 × 10-6 | ||
ITAE | - | 0.0036 | 0.0010 | 5.08 × 10-4 | 1.89 × 10-4 | ||
RMSE | 0.0125 | 0.0068 | 0.0074 | 0.0065 | 0.0056 | ||
IAE | 2.2475 | 0.0101 | 0.0077 | 0.0052 | 0.0033 | ||
case 6 | ISE | - | 0.0046 | 0.0065 | 0.0069 | 0.005 | |
ITSE | - | 7.53 × 10-4 | 3.67 × 10-4 | 3.79 × 10-4 | 1.75 × 10-4 | ||
ITAE | - | 0.0447 | 0.0169 | 0.0185 | 0.0091 | ||
RMSE | - | 0.0278 | 0.0329 | 0.0340 | 0.0289 | ||
IAE | - | 0.0508 | 0.0345 | 0.0347 | 0.022 |
P1 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 12.3 | 6.1 | 6.69 | 4.96 | |
tr(s) | 5.05 | 2.01 | 2.3 | 2.07 | ||
OS (%) | 5.1 | 22.2 | 19.1 | 14.9 | ||
ISE | 1.44 | 1.15 | 1.24 | 1.13 | ||
ITSE | 1.49 | 0.84 | 0.96 | 0.74 | ||
ITAE | 5.37 | 2.23 | 2.53 | 1.58 | ||
case 2 | ts(s) | 17.68 | 8.5 | 9.3 | 7.01 | |
tr(s) | 3.18 | 1.84 | 2.07 | 1.88 | ||
OS (%) | 29.8 | 39.2 | 35.1 | 28.0 | ||
ISE | 1.59 | 1.26 | 1.32 | 1.15 | ||
ITSE | 3.27 | 1.26 | 1.35 | 0.87 | ||
ITAE | 14.77 | 4.07 | 4.49 | 2.55 | ||
Case3 | ts(s) | > 20 | 9.18 | 9.72 | 7.95 | |
tr(s) | > 20 | 1.94 | 2.25 | 2.01 | ||
OS (%) | - | 17.8 | 11.7 | 11.0 | ||
ISE | 1.74 | 1.13 | 1.21 | 1.11 | ||
ITSE | 3.78 | 0.86 | 0.93 | 0.76 | ||
ITAE | 20.26 | 2.98 | 3.04 | 2.27 | ||
case 4 | ts(s) | > 20 | 8.85 | 9.61 | 8.0 | |
tr(s) | 12.06 | 3.05 | 3.59 | 3.06 | ||
OS (%) | 2.3 | 9.9 | 9.9 | 7.1 | ||
ISE | 2.67 | 1.47 | 1.61 | 1.46 | ||
ITSE | 5.73 | 1.30 | 1.60 | 1.23 | ||
ITAE | 17.08 | 3.87 | 4.77 | 3.10 |
P2 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 22.9 | 9.36 | 10.49 | 3.94 | |
tr(s) | 1.78 | 2.0 | 2.31 | 2.49 | ||
OS (%) | 16.7 | 27.9 | 28.2 | 5.2 | ||
ISE | 0.71 | 1.09 | 1.23 | 1.09 | ||
ITSE | 1.53 | 0.96 | 1.23 | 0.72 | ||
ITAE | 16.02 | 3.62 | 4.70 | 1.37 | ||
case 2 | ts(s) | 22.71 | 13.44 | 14.91 | 9.72 | |
tr(s) | 1.56 | 1.95 | 2.2 | 2.14 | ||
OS (%) | 19.1 | 44.9 | 46.9 | 30.5 | ||
ISE | 0.90 | 1.44 | 1.60 | 1.26 | ||
ITSE | 1.64 | 1.92 | 2.51 | 1.10 | ||
ITAE | 15.86 | 7.56 | 10.00 | 3.77 | ||
case 3 | ts(s) | 26.3 | 18.33 | 19.76 | 8.14 | |
tr(s) | 3.37 | 2.62 | 3.01 | 2.96 | ||
OS (%) | 20.4 | 43.5 | 40.04 | 17.01 | ||
ISE | 1.27 | 1.78 | 1.91 | 1.44 | ||
ITSE | 4.13 | 3.63 | 4.02 | 1.36 | ||
ITAE | 29.76 | 15.44 | 17.88 | 3.69 | ||
case 4 | ts(s) | 20.94 | 9.25 | 12.28 | 5.68 | |
tr(s) | 1.71 | 1.98 | 2.27 | 2.25 | ||
OS (%) | 12.8 | 35.3 | 36.2 | 16.7 | ||
ISE | 0.75 | 1.27 | 1.41 | 1.17 | ||
ITSE | 0.96 | 1.30 | 1.65 | 0.82 | ||
ITAE | 11.26 | 4.60 | 6.28 | 1.84 |
P3 | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ts(s) | 6.32 | 2.69 | 2.91 | 0.52 | |
tr(s) | 1.41 | 0.62 | 0.69 | 0.6 | ||
OS (%) | 16.3 | 21.9 | 20.6 | 13.1 | ||
ISE | 0.52 | 0.28 | 0.31 | 0.25 | ||
ITSE | 0.26 | 0.07 | 0.08 | 0.04 | ||
ITAE | 1.34 | 0.33 | 0.38 | 0.15 | ||
case 2 | ts(s) | 6.65 | 4.08 | 4.34 | 2.84 | |
tr(s) | 1.42 | 0.61 | 0.69 | 0.58 | ||
OS (%) | 20.1 | 38.5 | 34.0 | 27.2 | ||
ISE | 0.62 | 0.41 | 0.43 | 0.34 | ||
ITSE | 0.36 | 0.16 | 0.16 | 0.08 | ||
ITAE | 1.73 | 0.64 | 0.7 | 0.30 | ||
case 3 | ts(s) | 7.52 | 3.15 | 3.44 | 2.07 | |
tr(s) | 1.3 | 0.6 | 0.67 | 0.58 | ||
OS (%) | 22.6 | 27.2 | 26.0 | 16.1 | ||
ISE | 0.54 | 0.29 | 0.33 | 0.25 | ||
ITSE | 0.33 | 0.09 | 0.11 | 0.04 | ||
ITAE | 1.93 | 0.43 | 0.51 | 0.18 | ||
case 4 | ts(s) | 2.73 | 2.19 | 2.37 | 1.56 | |
tr(s) | 1.27 | 0.53 | 0.6 | 0.55 | ||
OS (%) | 5.7 | 12.1 | 11.1 | 4.1 | ||
ISE | 0.36 | 0.21 | 0.23 | 0.20 | ||
ITSE | 0.10 | 0.03 | 0.04 | 0.02 | ||
ITAE | 0.33 | 0.15 | 0.17 | 0.07 |
P4 | IT2F-PID [80] | IT2F-PD+I [81] | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 1 | ISE | 0.036 | - | 2.78 × 10-4 | 3.32 × 10-4 | 2.53 × 10-4 | 1.9 × 10-4 | |
ITSE | - | - | 2.34 × 10-5 | 2.0 × 10-5 | 7.22 × 10-6 | 3.64 × 10-6 | ||
ITAE | - | - | 0.0036 | 9.35 × 10-4 | 4.52 × 10-4 | 1.62 × 10-4 | ||
RMSE | 0.0085 | - | 0.0118 | 0.0129 | 0.013 | 0.0097 | ||
IAE | 1.8001 | - | 0.0101 | 0.0076 | 0.0051 | 0.0033 | ||
case 2 | ISE | - | 1.5844 | 0.0045 | 0.0064 | 0.0069 | 0.005 | |
ITSE | - | - | 3.33 × 10-4 | 2.34 × 10-4 | 2.43 × 10-4 | 1.23 × 10-4 | ||
ITAE | - | - | 0.0119 | 0.003 | 0.0029 | 0.0014 | ||
RMSE | - | 0.0514 | 0.0472 | 0.0568 | 0.0586 | 0.0499 | ||
IAE | - | 7.4692 | 0.0379 | 0.029 | 0.0287 | 0.0191 |
P4 | IT2F-PD+I [81] | PID | T1F-PID | IT2F-PID | SGT2F-PID | ||
case 3 | ISE | 1.9203 | 0.0045 | 0.0064 | 0.0069 | 0.005 | |
ITSE | - | 3.56 × 10-4 | 2.34 × 10-4 | 2.43 × 10-4 | 1.25 × 10-4 | ||
ITAE | - | 0.0288 | 0.0035 | 0.0037 | 0.0019 | ||
RMSE | 0.04 | 0.0273 | 0.0328 | 0.0338 | 0.0288 | ||
IAE | 14.7056 | 0.0419 | 0.0292 | 0.0290 | 0.0192 | ||
case 4 | ISE | 2.527 | 0.0046 | 0.0066 | 0.007 | 0.0051 | |
ITSE | - | 6.47 × 10-4 | 5.9 × 10-4 | 5.5 × 10-4 | 4.0 × 10-4 | ||
ITAE | - | 0.0316 | 0.0172 | 0.0154 | 0.011 | ||
RMSE | 0.0649 | 0.0276 | 0.0331 | 0.0341 | 0.0291 | ||
IAE | 13.3876 | 0.044 | 0.033 | 0.032 | 0.022 | ||
case 5 | ISE | 0.094 | 2.78 × 10-4 | 3.32 × 10-4 | 2.53 × 10-4 | 1.90 × 10-4 | |
ITSE | - | 2.34 × 10-5 | 2.04 × 10-5 | 7.73 × 10-6 | 3.64 × 10-6 | ||
ITAE | - | 0.0036 | 0.0010 | 5.08 × 10-4 | 1.89 × 10-4 | ||
RMSE | 0.0125 | 0.0068 | 0.0074 | 0.0065 | 0.0056 | ||
IAE | 2.2475 | 0.0101 | 0.0077 | 0.0052 | 0.0033 | ||
case 6 | ISE | - | 0.0046 | 0.0065 | 0.0069 | 0.005 | |
ITSE | - | 7.53 × 10-4 | 3.67 × 10-4 | 3.79 × 10-4 | 1.75 × 10-4 | ||
ITAE | - | 0.0447 | 0.0169 | 0.0185 | 0.0091 | ||
RMSE | - | 0.0278 | 0.0329 | 0.0340 | 0.0289 | ||
IAE | - | 0.0508 | 0.0345 | 0.0347 | 0.022 |