Formal Verification and Modeling in Human-Machine Systems: Papers from the AAAI Spring Symposium
Piecewise Affine Hybrid Automata Representation
of a Multistage Fuzzy PID Controller
Matthew A. Clark
Kuldip S. Rattan
Air Force Research Laboratory
Wright-Patterson AFB, Ohio 45433
Wright State University
Dayton, Ohio 45435
Abstract
completely match the designers intent and the complete satisfaction of those requirements at every level of system design. This can be a challenging endeavor for offline V&V
of such systems. But if the system changes behavior at run
time, an additional challenge is to link the design time requirements in such a way that dynamic changes in the learning system can be verified automatically and later inspection of such changes is humanly readable. Fuzzy logic is
a method of synthesizing natural language and intuitive requirements and specifications into an abstract model of desired behavior. Fuzzy logic can also be considered a special
type of hybrid system (Fierro, Lewis, and Liu 1998).
Over the past fifteen years, hybrid systems applicability has
inspired a great deal of research in the areas of both control
theory and theoretical computer science (Clark et al. 2013).
Piecewise linear hybrid systems may only cover a subset of
possible hybrid system representations, but has potential in
representing an over-approximation of the allowable states
of a constrained adaptive system (Snyder 2013). It is proposed that a fuzzy logic algorithm can be represented as a
piecewise affine hybrid system, provided the following conditions are satisfied: the fuzzification process uses a triangular membership function, the sum of the membership values
is one, the input/output is normalized, and the defuzzification method uses a modified center of area method (Brehm
and Rattan 1994).
In this paper, a simple linear plant is controlled by cascading
Proportional + Integral (PI) and Proportional + Derivative
(PD) controllers, referred to as multistage PID controller.
It is shown that a pseudo-linear multistage PID controller
is more effective than the traditional PID controllers. It is
also demonstrated that if implemented using fuzzy logic, the
multistage PID controller has an even greater advantage in
that it can be represented as a piecewise hybrid system. A
method to automatically translate a fuzzy control system to
a hybrid automata is presented. This translation is then used
to implement a multistage fuzzy PID controller. Future work
plans to build on this example, applying satisfiability solvers
to check or verify system constraints at run-time.
As cyber physical systems become more complex, modeling for the purposes of verification and validation
(V&V) becomes a primary barrier. Additionally, this
complexity drives an increased disconnect between the
system implementation and the original human generated requirements. Therefore, a key challenge is to
model cyber physical systems at the appropriate level
of abstraction required for V&V while maintaining a
clear and understandable linkage between the human
designer and the system under design. Fuzzy logic is
a method to synthesize linguistic, natural language requirements into real world models. It has also been
shown that fuzzy logic is a suitable method to learn
the behavior of a nonlinear system when a model is not
present. A fuzzy logic controller is a special class of hybrid system and provides a mechanism to relate a system model to human specified requirements. Leveraging the wealth of verification techniques for hybrid systems, the goal of this paper is to provide a mechanism
capable of modeling and analyzing the safety and stability of learning and adaptive systems while maintaining a
direct relationship to the original system requirements.
This paper focuses on the model synthesis of a multistage PID fuzzy controller designed for a second-order
plant, converting it from linguistic rule based requirements into a piecewise affine hybrid system and simulated using Matlab.
Introduction
As higher levels of self-governing systems become a reality,
the notion that traditional testing to achieve an acceptable
level of trust is becoming an impossible task, especially for
software within safety critical or flight critical applications
(Hayhurst and Center 2001). As fully autonomous systems
are expected to be safe and resilient in a near infinite set
of unknown environmental stimuli, a fundamental change
in both design and testing of these autonomous systems is
needed. One approach is to shift the burden of proof to a
more run time verification mechanism that constrains the
software to an acceptable and certifiable range of behaviors.
Validation and verification (V&V) of cyber physical systems
is fundamentally linked to requirements that correctly and
Multistage PID Controller Representation
PD and PI controllers are effective for many control problems but lack the advantages of a PID controller. A PD controller could add damping to a system but does not effect
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Case 88ABW-2013-4425
104
the steady-state error of the system. The PI controller on
the other hand improves the steady-state error of the system but reduces the speed of response and stability of the
system(Kuo and Golnaraghi 2003). This leads to the motivation of using a Proportional + Integral + Derivative (PID)
controller. However, in the classical PID controller, the PI
controller is active during the transient phase of the system
response, which tends to reduce the stability of the system.
The transfer function of a PID controller can be written as
Ki
Gc (s) = (Kp + Kd s)(Kpi +
)
(1)
s
In (1), both the PD and PI controllers are always active. To
input
u
PD Controller
K p + Kd s
set is defined as a range, it is possible to have a small error
when the controller action is small. To reduce the steadystate error, the ZO fuzzy set for the error can be made small
so that there is more contribution from the control action of
the adjacent rules of the matrix. Narrowing the ZO fuzzy set
range corresponds to increasing the gain of the controller
around the zero error point. However, there is a limit on
how small the ZO fuzzy set range can be made, To eliminate this small error, a PI fuzzy controller (PIFLC) can be
cascaded with the PD fuzzy controller (PDFLC) to create a
multistage fuzzy PID controller. The fuzzy PID multistage
fuzzy controller can, therefore, eliminate the steady-state
error quickly without overshoot along with provide better
speed of response, damping and bandwidth when compared
to a classical PID multistage controller. Another advantage
of the multistage fuzzy PID controller over the fuzzy PID
controller with three inputs is that the number of rules are
significantly reduced, thus significantly reducing the design
complexity and implementation. The block diagram of the
multistage fuzzy PID controller is shown in Fig. 2.
output
d
dt
.
. if abs(u) < δ, set output = 1
u
X
otherwise set output=0
PI Controller
K pi1 + Ks i
Figure 1: A multistage PID controller.
utilize the best features of both the controllers, the PI controller in (1) should not be active (output of the PI portion
of the controller should be zero) until the system reaches
the steady-state. Conversely, the output of the PD portion
of the controller in (1) should be constant when the system
reaches steady-state and the PI controller should be active.
To achieve this, the PID controller can be implemented as a
multistage PID controller. The transfer function of a multistage PID controller can be obtained by modifying the PID
controller transfer function given in (1) as
G0c (s)
Error
Change in Error
(Kp + Kd s)(1 + Kpi1 +
=
(Kp + Kd s) + (Kp + Kd s)(Kpi1 +
gce
Fuzzy
PD Controller
(PDFLC)
u
Gain output
K
PIFLC Controller
Gain
Ku
Fuzzy
P Controller
PI
Controller
Figure 2: A multistage fuzzy PID controller.
There are three processes involved in the implementation
of an FLC; fuzzification of inputs, a rule base or an inference engine, and defuzzification to obtain a “crisp output”
as shown in Fig 3.
Ki
)
s
=
ge
Ki
) (2)
s
Input
The block diagram of the multistage PID controller is shown
in Fig. 1. Note that the second term in (2) is zero during the
transient phase of the system response.
Input
Scaling
(Gain)
Fuzzification
Module
Inference Defuzzification
Engine
Module
Output
Scaling
(Gain)
Output
Knowledge
Base
Multistage Fuzzy PID Controller
Figure 3: Components of a general FLC.
One of the problems with the classical multistage PID controller shown in Fig. 1 is the difficulty in switching from a
PD to a PID controller, i.e., the difficulty in selecting the
value of δ. This problem can be solved by a multistage PID
fuzzy logic controller (PIDFLC) by selecting an appropriate gain Ku (see Fig. 2) based on the Zero (ZO) fuzzy set
of the fuzzy PD controller as discussed later in this section.
Another advantage of the fuzzy controller is its ability to increase performance by increasing the proportional gain of
the system as the error decreases [2]. It has been shown that
the fuzzy PD controller is able to reduce the steady-state error while maintaining the speed and damping of the system
(Brehm and Rattan 1993) (Brehm and Rattan 1994) (Brehm
and Rattan 1995) (Gurpreet S. Sandhu and Rattan 1996).
However, like the classical PD controller, its fuzzy counterpart can not completely eliminate the steady-state error. The
fuzzy control action drives the plant output to the ranges of
the ZO fuzzy set for both the error and change in error (the
center rule of the rule matrix). However, since the ZO fuzzy
Fuzzification involves dividing each input variables’ universe of discourse into ranges called fuzzy sets (also called
the membership function). In this effort all membership
functions are assumed to be triangular with seven fuzzy sets
from -1 to 1 and Zero (ZO) as the center fuzzy set centered at
zero. The partitions are also assumed to be symmetric about
the ZO fuzzy set as shown in Fig. 4. This approach simplifies
the computation while typically giving robust and satisfactory results. The remaining parts of the partition are Negative Big (NB), Negative Medium (NM), Negative Small
(NS), Positive Small (PS), Positive Medium (PM) and positive big (PB). Note that A and B are the design parameters of
an FLC as shown in figures 4 and 5. An input applied across
each range determines the membership of the variables’ current value in the fuzzy sets. The value at which the membership is maximum is called the center point of the fuzzy sets.
Note that in Fig. 4 the membership value in the PB and NB
fuzzy sets is 1 if the input variables’ current value is greater
105
NB
NM
NS ZO PS
−1
−B
−A 0 A
PM
overshoot ≤ 10 % and 2) settling time to be ≤ 0.2 s. The
values of proportional gain Kp = 10 and the derivative gain
10
were obtained to satisfy these specifications. The
Kd = 37
response of the PD controlled system is shown in Fig. 6.
It can be seen from this figure that both the specifications
are satisfied. However, there is substantial steady-state error
(10%). To reduce this error, a single stage PID can be designed. The problem with the single stage PID controller is
that either it takes longer to eliminate the steady-state error
or the settling time specification is not satisfied. To overcome this, a PDFLC was designed using the method described in (Gurpreet S. Sandhu and Rattan 1996). The values of “A” and “B” parameters for the error and change in
error fuzzy sets like the one shown in Fig. 5 were obtained
as shown in Table 2. The values of the position and velocity scaling gain ge and gce along with the output gain K
1
were selected as 1.0, 37
and 10, respectively. The response
of the PDFLC controlled system is shown in Fig. 6. It can
be seen from this figure that along with satisfying both the
specifications, the steady-state error is substantially reduced
(≤ 3%. To eliminate this error quickly without effecting the
settling time of the system, a multistage PIDFLC is designed
in which the PIFLC is active only when the response of the
system compensated when the PDFLC controller reaches
the steady-state. This can be accomplished by making the
PIFLC controller active only during the PDFLC ZO error
fuzzy set. This can be accomplished by scaling the output
of the PDFLC by the inverse of the width of the zero error input fuzzy set (i.e. 1\0.08 =12.5), using the the fuzzy P
controller (“A” = 0.33 and “B” = 0.66) as shown in Fig. 5. A
classical PI controller is designed with the proportional gain
Kpi = 3.3 and the integral gain Ki = 10. The response of the
multistage PIDFLC is shown in Fig. 6. It can be seen from
this figure that PIFLC portion of the controller is active only
after the error reaches the ZO fuzzy set of the PDFLC. Once
the PIFLC becomes active, it quickly removes the steadystate error and the system response reaches the final value of
one.
PB
1.0
B
1 input variable
Figure 4: Triangular fuzzy sets with membership value of 1
if the input variable is greater than 1 or less than -1.
than 1 or less than -1, respectively. If it is necessary that the
values of the membership value in PB and NB are zero for
the input variables’ current value greater than 1 or less than
-1, respectively, the partitions can be determined as shown
in Fig. 5.
NB
NM
NS ZO PS
PM
−B
−A 0 A
B
PB
1.0
−1
1 input variable
Figure 5: Triangular fuzzy sets with membership value of 0
if the input variable is greater than 1 or less than -1.
Linguistic rules express the relationship between the input
and output variables. Table 1 is an example of a matrix of
rules that covers all possible combinations of fuzzy sets for
two input variables. The rules describe a proportional plus
derivative FLC (PDFLC). The rule matrix is just a convenient way to represent all the rules in “English” of the form:
Rn : if error (e) is Ej and change in error (∆e) is ∆Ek then
the output is Uk,j
where 1 ≤ j ≤ number of fuzzy sets for error, 1 ≤ k ≤
number of fuzzy sets for change in error and 1 ≤ n ≤
(number of fuzzy sets for error multiplied by the number of
fuzzy sets for change in error). Ej and ∆Ek are the fuzzy
sets for the error and change in error, respectively and Ukj
are the output fuzzy sets. In this case, if each variable has
seven fuzzy sets, there are 49 rules represented as 7 × 7
matrix.
Table 2: “a” and “b” parameters of the error, change in error
and output fuzzy sets for PDFLC
Error (e)
Change in Error (∆e)
Output
Table 1: Rule matrix for PDFLC
NB
NM
Change NS
in
ZO
Error
PS
(∆e)
PM
PB
∆Ek
NB
NM
NS
NB
NB
NB
NB
NM
NS
ZO
NB
NB
NB
NM
NS
ZO
PS
NB
NB
NM
NS
ZO
PS
PM
Error (e)
ZO
Ej
NB
NM
NS
Uk,j =ZO
PS
PM
PB
PS
PM
PB
NM
NS
ZO
PS
PM
PB
PB
NS
ZO
PS
PM
PB
PB
PB
ZO
PS
PM
PB
PB
PB
PB
A
0.08
0.33
0.3
B
0.5
0.66
0.7
Piecewise Affine Representation of Fuzzy PID
Controller
Unlike the classical approach to PID control design, fuzzy
logic control systems lack methods to assess stability and
robustness. The following sections demonstrate an approach
to translate a fuzzy logic controller into a formally verifiable construct. A piecewise hybrid representation of a
fuzzy logic controller can be generated provided the following constraints on the fuzzy logic implementation are satisfied; The fuzzification process uses a triangular membership function, the sum of the membership values is one, the
The PD portion of the multistage PID controller shown in
Fig. 1 was designed for the system with the transfer function
45
to satisfy the specifications: 1) percentage
s2 + 18 s + 45
106
1.2
Leveraging the above fuzzy to piecewise affine transformations enables the construction of a Hybrid Automaton. A
Hybrid Automaton is a modeling formalism for hybrid systems which is defined in (Lygeros et al. 2001) by the octuple:
H = (Q, X, f, Init, D, E, G, R), where
1
Output
0.8
PD controller
Fuzzy PD controller
Fuzzy Multistage PID controller
0.6
• Q is a finite set of discrete variables;
• X is a finite set of continuous variables;
• f : Q × X → X is a vector field defining the dynamics
of the continuous variables;
• Init ⊂ Q × X is the set of initial states;
• D : Q → P (X) defines the domain of the discrete modes
• E ⊂ Q × Q is a set of edges;
• G : E → P (X) defines guard conditions for discrete
transitions;
• R : E × X → P (X) is a reset map defining discontinuities in the continuous state of the system during discrete
transitions.
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time
Figure 6: Response of the PD portion of the PID controller
with PI inactive.
input/output is normalized, and the defuzzification method
uses a modified center of area method (Brehm and Rattan
1994). Given these constraints, a two-input one-output fuzzy
controller (PDFLC) can be represented as piecewise affine
gains Kp (k, j), Kd (k, j), and constant β(k, j) given by (35).
Kp (k, j)
∆e(k + 1) ∗ (U (k, j + 1) − U (k, j))
DEN
∆e(k) ∗ (U (k + 1, j + 1) − U (k + 1, j))
DEN
e(j + 1) ∗ (U (k + 1, j) − U (k, j))
DEN
e(j) ∗ (U (k, j + 1) − U (k + 1, j + 1))
DEN
(e(j + 1)
∗ {∆e(k + 1) ∗ U (k, j)}
DEN
(e(j + 1)
∗ {∆e(k) ∗ U (k + 1, j)}
DEN
e(j)
∗ {∆e(k) ∗ U (k + 1, j + 1)}
DEN
e(j)
∗ {∆e(k + 1) ∗ U (k, j + 1)}
DEN
=
−
Kd (k, j)
=
+
β(k, j)
=
−
+
−
For each fuzzy inference rule there exists an equivalent
affine controller G0c (k, j)(s) that can be represented within
a closed loop system as the vector field of continuous dynamics ẋ = A(k, j)x + b(k, j) + B(k, j)r. The finite set
of discrete variables are represented by the regions between
each of fuzzy input inference vectors. The guard conditions
are represented by the center points of the fuzzy input inference vectors.
(3)
(4)
PWA Hybrid Representation of Fuzzy
Controlled System
As noted in the previous section, a fuzzy controller can be
represented as a collection of piecewise affine hybrid automata where each automaton contains the continuous time
representation of the controller and the transitions are governed by the boundary between the centers of the fuzzy input
membership functions. To illustrate this point, consider the
second-order, type-1 system used in section III whose openloop dynamics
are represented
as ẋ =Ao x + Bo u, where
0
1
0
Ao =
and Bo =
. Additionally, con−45 −18
45
sider a simple 3 membership one-input, one-output proportional fuzzy controller applied to the above plant as shown
in figure 7. Let the input membership function defined by
e = [N egative, Zero, P ositive] = [−1, 0, 1]. Additionally, let the output rule vector u = [N, Z, P ] = [−2, 0, 1].
As stated previously, the ‘english’ representation of the relationship between the input and output of the PFLC would
be; R1 : if e is N then output is u is N , R2 : if e is ZO then
the output u is ZO and R3 : if e is P then the output u is P .
The set of proportional controllers can be represented by the
affine gains and offsets Kp1 (j) and β1 (j), respectively. The
index j denotes the equivalent gain and offset values associated with a partition of the state-space governed by the fuzzy
input rules. The closed-loop system is represented in figure
7.
It is important to note that the fuzzy controller causes the
system to go open-loop (driving the system to the maximum
(5)
where DEN = (e(j + 1) − e(j)) ∗ (∆e(k + 1) − ∆e(k)) and e and ∆e
are the error and change in error input vectors, respectively
and U is the output rule matrix. It should be noted that these
values only change at the transition between fuzzy membership functions. For a one-input, one-output fuzzy logic
controller (PFLC) with e as the input vector and U as the
output rule matrix where U (k + 1, j) = U (k, j) = U (j)
and U (k + 1, j + 1) = U (k, j + 1) = U (j + 1). Substituting
these in (3-5), the piecewise affine gain Kp1 (j) and β1 (j)
are given by (6-7).
Kp1 (j)
=
U (j + 1) − U (j)
e(j + 1) − e(j)
(6)
β1 (j)
=
(e(j + 1) ∗ U (j)) − (e(j) ∗ U (j + 1))
e(j + 1) − e(j)
(7)
Therefore, the transfer function G0c (s) of the piecewise
affine multistage PID controller shown in Fig. 2 can be written as
0
Gc (k, j)(s)
=
(Kp (k, j) + Kd ki, j) s + β(k, j))
+(Kpe1 (i, j) + Kde (i, j) s + β(k, j))
×(Kpi 1(k, j) + β1 (k, j) +
Ki
)
s
(8)
107
Proportional Fuzzy Controller
β1 (j)
.
+
r +
e K (j) +
p1
K
u B
o
+
−
X
1
s
X
C
X1
+
Ao
Figure 7: Simple Proportional Fuzzy Controller and Plant
Model.
value of the fuzzy controller output) when the input error
is beyond the established input range, i.e., Kp1 = 0; and
β1 = 1 if the input to the PFLC e is ≥ 1 and β1 = −1 if the
input to the PFLC e is ≤ -1.
The hybrid automata chosen to represent the system has four
discrete modes, two of which cover the region of the statespace not included in the fuzzy universe of discourse. In
the above example this would account for all values outside of the region [−1 : 1]. The first discrete mode (denoted by N ) represents the region of space outside the negative input limit (-1) represented by R = [−∞ : −1]. The
second discrete mode N Z represents the region of statespace between the N fuzzy set and the ZO fuzzy set denoted by R = [−1 : 0]. The third discrete mode ZP , represents the region of state-space between the ZO fuzzy set
and the P fuzzy set denoted by R = [0 : 1]. The fourth
and final discrete mode (denoted by P ) represents the region of space outside the positive input limit (1) represented
by R = [1 : ∞]. The state of the system is the collection
of continuous and discrete states and will be denoted by
(x, q) ∈ X × Q. The formal hybrid automaton that defines
the simple proportional fuzzy controller and plant model is
Figure 8: Simple Three Membership Fuzzy Controller
Model Hybrid Representation.
The discrete modes Q = {qZ , qNZ , qZP , qP } directly correspond to the index value j = [0, 1, 2, 3]. The equivalent gain
and offset value for each mode within the universe of discourse (j = 1, 2) can be found using (6) and (7), respectively. For the error outside the universe of discourse (j =
0, 3), the equivalent gain values are zero and the equivalent
offset values correspond to the maximum fuzzy controller
output. The subsequent gain and offset values in the above
example are Kp1 (j) = [0, 2, 1, 0] and β1 (j) = [−1, 0, 0, 1].
The closed-loop system dynamics within each mode correspond to f (Q, x) = A(j)x + b(j) + B(j)r, where
"
A(j)
=
b(j)
=
B(j)
=
0
−45 − 45 × K × Kp1 (j)
"
#
0
45 × K × β1 (j)
"
#
0
45 × K × Kp1 (j)
#
1
−18
(9)
(10)
(11)
where K is the output gain of the PFLC. As the number of
membership functions increase, the hybrid system representation increases in the number of guard conditions. For each
additional fuzzy membership function, 2 guards are created,
identifying the transition between the adjacent membership
functions.
• Q = {qN , qNZ , qZP , qP };
• X = R, e = r − x1 ;
• f (qN , x) = A(qN )x + b(qN ) + Br,
f (qNZ , x) = A(qNZ )x + b(qNZ ) + Br,
f (qZP , x) = A(qZP )x + b(qZP ) + Br,
f (qP , x) = A(qP )x + b(qP ) + Br;
• Init = qNZ × {e ∈ X : e ≤ 0};
PWA Multistage Hybrid PID Fuzzy Controller
• D(qN ) = {e ∈ X : e ≤ −1},
D(qNZ ) = {e ∈ X : −1 ≥ e ≤ 0},
D(qZP ) = {e ∈ X : 0 ≥ e ≤ 1},
D(qP ) = {e ∈ X : e ≥ 1};
From the example in the previous section, one can see that
the multistage fuzzy controller in section can be converted
to a hybrid automaton using (3) - (7). Adding just the PD
portion of the multistage fuzzy controller (with the PIFLC
inactive) does not increase the continuous states of the system and each hybrid mode has continuous dynamics corresponding to f (Q, x) = A(k, j)x+b(k, j)+B(k, j)r, where
• E = {(qN , qNZ ), (qNZ , qN ),
(qNZ , qZP ), (qZP , qNZ ),
(qZP , qP ), (qP , qZP };
"
• G(qN , qNZ ) = {e ∈ R : e > −1},
G(qNZ , qN ) = {e ∈ R : e ≤ −1},
G(qNZ , qZP ) = {e ∈ R : e > 0},
G(qZP , qNZ ) = {e ∈ R : e ≤ 0},
G(qZP , qP ) = {e ∈ R : e > 1},
G(qP , qZP ) = {e ∈ R : e ≤ 1};
A(k, j) =
• R(qN , qNZ,e) = R(qNZ , qN,e) =
R(qNZ , qZP,e) = R(qZP , qNZ , e) =
R(qZP , qP,e) = R(qP , qZP , e) = e
#
1
−18 − 45 K gce Kd (k, j)
"
#
0
b(k, j) =
45 × K × β(k, j)
"
#
0
B(k, j) =
45 × K × Kp (k, j)
0
−45 − 45 K Kp (k, j)
(12)
(13)
(14)
K is the output gain and gce is the normalization constant of
the change in error of the PDFLC. The output of the PDFLC
108
ysis and verification of fuzzy based learning systems. Future work will concentrate on proving reachability and safety
properties of the fuzzy logic hybrid automaton before and
after learning takes place.
controller is provided as the input to the PIFLC controller as
shown in Fig. 2. The PIFLC controller is scaled with gain
Ku such that the range of values within its universe of discourse are within the range of the PDFLC ZO error membership function. Outside of that range, the PIFLC controller
should be inactive, taking the value of 0 for both the equivalent gain and offset values. Within the range of the ZO error input, the PIFLC should be active. This is done using
the fuzzy sets of a PFLC controller shown in Fig. 5. The
output of the PIFLC is followed by a classical PI controller
with an integration component. This increases the continuous state by one. The corresponding continuous system can
be written as f (Q, x) = A(k, j)x+b(k, j)+B(k, j)r, where
A(k, j), b(k, j), B(k, j) are given by (15) - (17); and Kpi
and Ki are the proportional and integral gains of the classical PI controller. The multistage PID fuzzy hybrid system
was simulated using the same parameters used in section .
The multistage Hybrid PDFLC and PIFLC both contained 7
input/output membership functions. The equivalent closedloop hybrid automata contained 38 hybrid modes. Thirty of
the thirty eight modes contained continuous states of the
system with only the PDFLC controller active, six discrete
modes include both the PDFLC and PIFLC controllers, and
the final two hybrid modes represent the regions of the error input outside [-1 : 1]. The full multistage PIDFLC is active only when the error input, e, of the PIFLC controller is
within ZO membership function (between the N S to P S
center points), i.e., during the six discrete modes. Within
these six modes, the equivalent continuous state equations
took the form as described in (15) - (17). It should be noted
that, when active, the PIFLC controller increased the continuous states of the closed-loop system within the hybrid
modes by one but did not add additional discrete modes
to the hybrid automata. A comparison was made between
the multistage fuzzy PID controller representation in section and the hybrid automaton representation of the same
controller in section . Both simulations yielded identical responses to a unit-step input.
References
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Conclusions and Future Challenges
This paper presents a unique method to cascade PD and PI
fuzzy controllers to create a multistage PID fuzzy controller,
enabling greater performance by eliminating the

0

A(k, j) = −45 − 45KKp (k, j)(1 + Kpi Kp1 (j, k)Ku )
−Ku Kp (k, j)Kp1 (k, j)

1
−18 − 45KKd (k, j)gce (1 + Kpi Kp1 (j, k)Ku )
Kp1 (k.j)Ku Kd (k, j)gce
(15)

0


b(k, j) = 45KKpi β1 (k, j) + 45K(1 + Ku Kpi Kp1 (k, j))β(k, j)
Ku Kp1 (k, j) β1 (k, j)
(16)


0


B(k, j) = 45KKp1 (k, j)(1 + Kpi 
(17)
Ku Kp1 (kj)Kp (k, j)
negative effects of the integration during the transient phase
of the response. This paper also presents a general method
to translate triangular membership, weighted average fuzzy
logic inference systems into a piecewise affine hybrid system. It is thought that this method will enable a direct anal-
109

0

45KKi 
0