Adaptive Hybrid Intelligent Control for Uncertain
Nonlinear Dynamical Systems
Chi-Hsu Wang, Senior Member, IEEE, Tsung-Chih Lin, Tsu-Tian Lee, Fellow, IEEE, and Han-Leih Liu
Abstract—A new hybrid direct/indirect adaptive fuzzy neural
network (FNN) controller with state observer and supervisory controller for a class of uncertain nonlinear dynamic systems is developed in this paper. The hybrid adaptive FNN controller, the free parameters of which can be tuned on-line by observer-based output feedback control law and adaptive law, is a combination of direct and indirect adaptive FNN controllers. A weighting factor, which can be adjusted by the tradeoff between plant knowledge and control knowledge, is adopted to sum together the control efforts from indirect adaptive FNN controller and direct adaptive FNN controller. Furthermore, a supervisory controller is appended into the FNN controller to force the state to be within the constraint set. Therefore, if the FNN controller cannot maintain the stability, the supervisory controller starts working to guarantee stability. On the other hand, if the FNN controller works well, the supervisory controller will be deactivated. The overall adaptive scheme guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. Two nonlinear systems, namely, inverted pendulum system and Chua’s chaotic circuit, are fully illustrated to track sinusoidal signals. The resulting hybrid direct/indirect FNN control systems show better performances, i.e., tracking error and control effort can be made smaller and it is more flexible during the design process.
Index Terms—Adaptive control, fuzzy neural networks (FNNs),
nonlinear systems, state observer, supervisory control.
I. INTRODUCTION
M
OST current techniques for designing control systems are based on a good understanding of the plant under consideration and its environment. However, in a number of instances, the plant to be controlled is too complex and the basic physical processes in it are not fully understood. Hence, control design methods need to be augmented with an identification technique aimed at obtaining a progressively better understanding of the plant to be controlled. Adaptive control is a technique of applying some system identification techniques to obtain a model of the process and its environment from input/output experiment and using this model to designManuscript received August 3, 2001; revised January 18, 2002. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC91-2213-E009-067. This paper was recommended by Associate Ed-itor C. T. Lin.
C.-H. Wang and H.-L. Liu are with the School of Microelectronic En-gineering, Griffith University, Nathan, Brisbane Q4111, Australia (e-mail: c.wang@me.gu.edu.au).
T.-C. Lin is with the Department of Electronic Engineering, Feng-Chia Uni-versity, Taichung, Taiwan, R.O.C., and the School of Microelectronic Engi-neering, Griffith University, Nathan, Brisbane Q4111, Australia.
T.-T. Lee is with the Department of Electrical and Control Engineering, Na-tional Chiao-Tung University, Hsinchu 300, Taiwan, R.O.C.
Publisher Item Identifier S 1083-4419(02)05144-0.
a controller. The adaptive control for feedback linearizable nonlinear systems is an approach to nonlinear control design that has attracted a great deal of interest in the nonlinear control community for at least a quarter of a century. By using feedback linearization [1]–[3], the nonlinear adaptive control problem is transformed into a linear adaptive control problem, then the linear control methods can be applied to acquire the desired performance. The adaptive control methodologies include direct adaptive control (DAC) and indirect adaptive control (IAC) algorithms [4]–[9].
Recently, an important adaptive fuzzy neural network (FNN) control system [4]–[14] has been proposed to incorporate with the expert information systematically, and the stability can be guaranteed by universal approximation theorem [15]. For sys-tems with a high degree of nonlinear uncertainty, such as chem-ical process, aircraft, etc., they are very difficult to control using the conventional control theory. However, human operators can often successfully control them. Based on the fact that FNN logic systems are capable of uniformly approximating a non-linear function over a compact set to any degree of accuracy, a globally stable adaptive FNN controller is defined as an FNN logic system equipped with an adaptation algorithm. Moreover, FNN is constructed from a collect of fuzzy IF–THEN rules using fuzzy logic principles, and the adaptation algorithm ad-justs the free parameters of the FNN based on the numerical ex-periment data. Like the conventional adaptive control, the adap-tive FNN control has direct and indirect FNN adapadap-tive control categories [7], [8]. Direct adaptive FNN control has been dis-cussed in [4] and [7], in which the adaptive FNN controller uses fuzzy logic systems as controller. Hence, linguistic fuzzy con-trol rules can be directly incorporated into the concon-troller. Also, indirect adaptive FNN control has been proposed in [4] and [7], in which the indirect FNN controller uses fuzzy descriptions to model the plant. Hence, fuzzy IF–THEN rules describing the plant can be directly incorporated into the indirect FNN con-troller.
Can these two adaptive FNN controllers be combined to-gether to yield stable and robust adaptive control laws with su-pervisory controller? The answer is “yes.” A hybrid direct/in-direct adaptive FNN controller can be constructed by incorpo-rating both fuzzy description and fuzzy control rules using a weighting factor to sum together the control efforts from in-direct adaptive FNN controller and in-direct adaptive FNN con-troller. The weighting factor can be adjusted by the tradeoff between plant knowledge and control knowledge. We let if pure indirect adaptive FNN controller is required and when pure direct adaptive FNN controller is chosen. If fuzzy control rules are more important and reliable than fuzzy 1083-4419/02$17.00 © 2002 IEEE
descriptions of the plant, choose smaller ; otherwise choose larger . In [4], [7], and [8], the full state must be assumed to be available for measurement. This assumption may not hold in practice because either the state variables are not accessible for direct connection or because sensing devices or transducers are not available. In this paper, our main objective is to create a tech-nique for designing a state observer-based [12] hybrid direct/in-direct adaptive FNN control for a class of uncertain nonlinear systems in which only the system output is measurable. Based on the Lyapunov synthesis approach, the free parameters of hy-brid direct/indirect adaptive FNN controller can be tuned on-line by an observer-based output feedback control law and adaptive law. Also, a supervisory controller is designed to cascade with FNN controller. If the nonlinear system tends to unstable by the FNN controller, especially in the transient period, the supervi-sory controller will be activated to work with the FNN controller to stabilize the whole system. On the other hand, if the FNN controller works well, the supervisory controller will be deacti-vated. This will result in a smaller control effort (energy). There-fore, the overall adaptive scheme guarantees that the global sta-bility of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. We have successfully designed the FNN adaptive controllers with supervisory con-trol to concon-trol the inverted pendulum and Chua’s chaotic circuit [16] to track reference sinusoidal signals. The resulting hybrid direct/indirect FNN control systems show better performances, i.e., both tracking error and control effort can be made smaller. This paper is organized as follows. Problem formulation is described in Section II. A brief description of the T–S FNN is presented in Section III. The observer-based hybrid direct/indi-rect FNN controller appended with a supervisory controller is constructed in Section IV. Simulation examples to demonstrate the performances of the proposed method are provided in Sec-tion V. SecSec-tion VI lists the conclusions of the advocated design methodology.
II. PROBLEMFORMULATION
Consider the th-order nonlinear dynamical system of the form [1], [17]
(1) or equivalently the form
(2) where
and unknown but bounded functions;
and control input and output of the system, respectively;
external bounded disturbance.
Equation (1) [or (2)] is actually the Isidori–Byrnes canonical form [1], [17] for certain nonlinear systems. We consider only the nonlinear systems which can be represented by (1) or (2). The state space representation of (2) is expressed as
(3) where .. . ... ... ... . .. ... ... .. . ... (4) and is
a state vector where not all are assumed to be available for measurement. Only the system output is assumed to be mea-surable. In order for (2) to be controllable, it is required that for in a certain controllability region .
Without loss of generality, we assume that for
. The control objective is to force the system output to follow a given bounded reference signal , under the constraint that all signals involved must be bounded.
To begin with, the reference signal vector , the tracking error vector , and estimation error vector will be defined as
where and denote the estimates of and , respectively. If the functions ) and ) are known and the system is free of external disturbance , then we can choose the controller to cancel the nonlinearity and design controller. In particular,
let be chosen such that all roots
of the polynomial are in the
open-left half-plane and control law of the certainty equivalent controller is obtained as [8]
(5) Substituting (5) into (2), we obtain the closed-loop system gov-erned by
where the main objective of the control is .
cannot be implemented, and not all system states can be mea-sured. We have to design an observer to estimate the state vector
in the following context.
A. Observer-Based Hybrid Direct/Indirect FNN Controller With Supervisory Control Scheme
Here, we will develop the observer-based hybrid direct/in-direct FNN controller with supervisory control scheme. The overall control law is constructed as
(6) where
indirect FNN controller [see (8)];
output of the Takagi–Sugeno (T–S)-based DAC FNN controller (described in Sections III and IV);
supervisory control (described in Section IV) to force the state within the constraint set;
weighting factor.
If the plant knowledge is more important and reliable than the control knowledge, we should choose a larger ; otherwise, a smaller should be chosen. Since cannot be available and
and are unknown, we replace the functions ,
, and error vector in (5) by estimation functions and (described in Section III), and . The certainty equivalent controller can be rewritten as
(7) The indirect control law is written as
(8) Applying (6) and (7) to (3), and after some simple manipula-tions, we can obtain the error dynamic equation
(9)
where .
From (9), the following observer that estimates the state error vector in (9)
(10)
where is the observer gain
vector.
The observation errors are defined as: and
.
Subtracting (10) from (9), we can obtain the error dynamics
(11)
where . Since ( ) pair is observable, the
observer gain vector can be chosen such that the character-istic polynomial of is strictly Hurwitz (i.e., the roots of the closed-loop system are in the open-left half-plane) and we know that there exists a positive definite symmetric matrix which satisfies the Lyapunov equation
(12) where is an arbitrary positive definite matrix.
Let us rewrite (10) as
(13)
where is a strictly Hurwitz matrix. Therefore,
there exists a positive definite symmetric matrix which satisfies the Lyapunov equation
(14) where is an arbitrary positive definite matrix. Let
, then by using (13) and (14), we have
(15) Since and are determined by the designer, we can choose
and , such that . Hence, is a bounded function
and there exists a constant value , such that . III. THETAKAGI–SUGENO(T–S) FNN SYSTEMS Fuzzy logic systems address the imprecision of the input and output variables directly by defining them with fuzzy numbers (and fuzzy sets) that can be expressed in linguistic terms (e.g.,
small, medium, and large). The basic configuration the T–S
FNN system [18]–[22] includes a fuzzy rule base, which con-sists of a collection of fuzzy IF–THEN rules in the following form:
IF is and and is
THEN (16)
where are fuzzy sets and is a vector
of the adjustable factors of the consequence part of the fuzzy rule. Furthermore, is a linguistic variable, and a fuzzy infer-ence engine to combine the fuzzy IF–THEN rules in the fuzzy rule base into a mapping from an input linguistic vector
to an output variable . Let
be the number of the fuzzy IF–THEN rules. The output of the fuzzy logic systems with central average defuzzifier, product in-ference, and singleton fuzzifier can be expressed as
Fig. 1. Configuration of the T–S FNNs.
where is the membership function value of the fuzzy
variable and is the true value of the th
implication. Equation (17) can be rewritten as
(18)
where is an adjustable parameter vector
and is a fuzzy basis
function vector defined as
(19)
When the inputs are fed into the T–S FNN, the true value of the th implication is computed. Applying the common de-fuzzification strategy, the output of the NNs expressed as (17) is pumped out. The overall configuration of the T–S FNN is shown in Fig. 1.
Based on the universal approximation theorem [15], the aforementioned fuzzy logic system is capable of uniformly approximating any well-defined nonlinear function over a com-pact set to any degree of accuracy. It is also straightforward to show that a multi-output system can always be approximated by a group of single-output approximation systems.
IV. HYBRIDDIRECT/INDIRECTADAPTIVEFNN CONTROLLER
WITHOBSERVER ANDSUPERVISORYCONTROLLER
An adaptive fuzzy system is a fuzzy logic system equipped with a training algorithm to maintain a consistent performance under plant uncertainties. The most important advantage of the adaptive FNN control over conventional adaptive control is that adaptive FNN controllers are capable of incorporating linguistic fuzzy information from a human operator, whereas the conven-tional adaptive controller is not. The adaptive FNN control is divided into two categories. One is called the indirect adaptive
FNN control and the other is called the direct adaptive FNN con-trol [7], [8]. An adaptive FNN concon-troller that uses fuzzy logic
systems as a model of the plant is an indirect adaptive FNN con-troller. An adaptive FNN controller that directly uses fuzzy logic systems as controller is a direct adaptive FNN controller. There-fore, the indirect adaptive FNN controller can incorporate fuzzy descriptions but cannot incorporate fuzzy control rules. On the other hand, the direct adaptive FNN controller can incorporate fuzzy control rules but cannot incorporate fuzzy descriptions. In this section, we will develop the hybrid direct/indirect adap-tive FNN controller that can incorporate linguistic information and design an adaptive law for the adjustable parameters in the controller, such that the closed loop output follows the
ref-erence output .
Let us replace , , and in (11) by the fuzzy
logic system , , and , respectively.
Therefore, the error dynamics (11) can be rewritten as
(20)
Let , then using (12) and (20) we have
(21) In order to design such that , we need the following assumption.
Assumption I: We can determine functions , ,
and such that and
for , where
, , and
for . This is due to the fact that we can choose in (10) to let . Furthermore, external disturbance is bounded, i.e.,
where is the upper bound of noise .
From Assumption I, and by observing (21), we choose the supervisory control as
where if (which is a constant chosen by the
designer), if , and if
. Considering the case and substituting (22) into (21), we obtain
(23) Therefore, we always have , by using the supervisory
control [see (22)]. Because , the bound of implies
the bound of , which in turn implies the bound of . Moreover, it implies the bound of . It is obvious that the supervisory con-trol is nonzero when is greater than a positive value . Therefore, if the closed-loop system with the fuzzy controller as
(24) works well in the sense that the error is not too large, i.e.,
, then the supervisory control is zero. On the other hand, if the system tends to diverge, i.e., , then the supervisory
control begins to operate to force .
We replace , , and in specific
fuzzy logic systems as (18), i.e.,
(25) (26) (27)
where is a vector of fuzzy base, and and are the
corresponding parameters of fuzzy logic systems. Also, is a vector of fuzzy base, and is the corresponding parameters of fuzzy logic systems. In order to adjust the parameters in the fuzzy logic systems, we have to derive adaptive laws. Hence, the optimal parameter estimations , , and are defined as
(28)
(29) and
(30)
where and are compact sets of suitable
bounds on and , respectively, and they are
de-fined as
and
where and are
positive constants.
Define the minimum approximation errors as
(31) The error dynamics (20) can be expressed as
(32) Substituting (25)–(27) into (32), the above equation can be rewritten as
(33)
where , , and .
Now consider the Lyapunov function
(34) The time derivative of is
(35)
Since , , and , and by using (12)
and (34), (35) can be rewritten as
According to (22) and , we have . If the adaptive laws are chosen as
(37) (38) (39) Substituting (37)–(39) into (36), we have
(40) Since the term is of the order of the minimum approx-imation error, this is the best we can hope to obtain. If , from (40) we have
If is not equal to zero, we can expect to be small based on the universal approximation theorem. From (28) to (30), the
constraint sets and of the optimal parameters ,
and respectively, if we can constrain and
within the sets, then in (24) and in (22) will be bounded due to the fact that, in this case, , , and are bounded, and it should be reminded that is bounded because of the supervi-sory control . Obviously, the adaptive laws in (37)–(39) are
unable to guarantee that , , and .
Therefore, all of the adaptive laws have to be modified by using the parameters projection algorithm [4], [8], [12], such that the parameter vectors will remain inside the constraints. The mod-ified adaptive laws are given as follows.
• Use the following adaptive law to adjust the parameter vector : if or and if and (41) where the projection operator is defined as
(42)
• Use the following adaptive law to adjust the parameter vector :
Whenever an element in (16) of , use
if
if (43)
where is the th component of .
Otherwise, use if or and if and (44) where the projection operator is defined as
(45) • Use the following adaptive law to adjust the parameter
vector : if or and if and (46) where the projection operator is defined as
(47) Following the preceding consideration, we obtain the following theorem.
Theorem 1: Consider the plant (2) with control (24), where
is given by (8) and is given by (22), and the fuzzy logic systems , , and are represented in (27) form. Let
Assump-tion I be true and the parameter vectors , , and be adjusted by the adaptive laws (41)–(47). Then, the overall ob-server-based control scheme as shown in Fig. 2 guarantees the following properties:
1) , , , and all of the
elements in (16) of
(48) and
Fig. 2. Overall scheme of the observer-based hybrid direct/indirect adaptive FNN control.
for all , where is the minimum eigenvalue
of , , , and
. 2)
(50)
for all , where and are constants and is the
minimum approximation error defined in (31).
3) If is squared integrable, i.e., , then
.
Proof:
I. i). To prove :
A) Let , if the first line of (41) is true, we
have either or,
for , i.e., we always have .
B) If the second line of (41), we have , and
i.e.,
Therefore, we prove that , .
ii) Use the similar method to show that ,
, .
From (43), we see that if in (16) , then ; that is,
we have for all elements of .
iii) To prove (48).
In the above description, we prove that ; therefore, ; i.e.
Since , we have
iv) To prove (49).
Since , and are weighted
aver-ages of the elements of , , and , respectively, we have (51) (52) (53)
and [since in (16) ]. Therefore, from
(8) we obtain
(54)
According to (22) and (51)–(54), we manipulate them and have
(55)
By combining (53)–(55) and substituting into (6), we can prove (49).
II. From (36), and by using the modified adaptive laws in (41)–(47), we have
(56)
Since and from (22), we have .
Hence, (56) can be simplified as
(57) where is the minimum eigenvalue of . By integrating
both sides of (57) and assuming that (since is
specified by the designer, we can choose such a ), after some simple manipulations, we can obtain
(58)
Defining and
, we can prove (50) by substituting and into (58).
III. From (50), if , we have . We have
, because we have proven that all variables in the right-hand side of (33) are bounded. Using Barbalat’s lemma [23] [if
and , then ] we have
. This completes the proof.
Remark I: It is obvious that we need to know before-hand in adaptive law (39), i.e., in the above theorem the adap-tive FNN control works under those nonlinear systems of which is well known. If the dynamics can be split into a
well-known nominal part , plus an uncertain part ,
then can be considered as a part of the external distur-bance. In the meantime, it can be attenuated by the proposed methodology.
To summarize the above analysis, the design algorithm for observer-based hybrid direct/indirect adaptive FNN control is proposed as follows.
Step 1) Specify the feedback and observer gain vector and , such that the characteristic matrices
and are strictly Hurwitz matrices,
respectively.
Step 2) Specify a positive definite matrix and solve the Lyapunov equation (12) to obtain a positive
def-inite symmetric matrix .
Step 3) Solve the state error equation (10) to obtain estimate
state vector .
Step 4) Specify the parameters , , , , , ,
, , and based on the practical constraints. Al-though is any given constant, we let be the same as (described at the end of Section II), which
Fig. 3. Inverted pendulum system.
can be determined from , and of
in (48). This is to match the magnitude scale of the system so that the designer is free from supplying at random to the system.
Step 5) Define the membership function for
and compute the fuzzy basis functions . Then, fuzzy logic control systems are con-structed as
Similarly, define the other membership functions and compute . Then, fuzzy logic control system is constructed as
Step 6) Obtain the control and apply it to the plant, then com-pute the adaptive laws (41)–(47) to adjust the param-eter vectors , , and . Following Remark I,
we let the unknown be in (46) and (47).
V. EXAMPLES
In this section, we will apply our observer-based hybrid di-rect/indirect adaptive FNN controller to control inverted pen-dulum and Chua’s chaotic circuit to track a sine-wave trajectory.
Example 1: Consider the inverted pendulum system shown
in Fig. 3. Let be the angle of the pendulum with respect to the vertical line.
The dynamic equations of the inverted pendulum system [4], [8], [12], [23] are
(59) where
and m/s is the acceleration due to gravity; is the
mass of the cart; is the half-length of the pole; is the mass of the pole; and is the control input. In this example, we assume
that kg, kg, and m. It is obvious that so that Assumption I in Section IV is satisfied. This
is due to and
. We also have to determine the bounds and as
follows:
(60)
and if we require that , then
(61) The control object is to control the state of the system to
track the reference trajectory if only the
system output is measurable. Also, the external disturbance is assumed to be a square-wave with amplitude 0.1, period
, and the parameters are chosen as , ,
, and step size . The choices of s and
are to improve the convergence rate of the closed-loop system controlled by our proposed controller.
According to the design procedure, the design is given in the following steps.
Step 1) The observer and feedback gain vectors are chosen
as and , respectively.
Step 2) We select in (12) as , then after solving (12), the positive definite symmetric 2 2 matrix
in (12) is . The minimum eigenvalue of
, i.e., is 3.19, which satisfies the transition from (56) to (57).
Step 3) Solve (10) to obtain .
Step 4) We select , , ,
, and , and in (14) is chosen as
and in (14). Therefore, the positive
definite symmetric 2 2 matrix in (14) can be solved as . The minimum eigenvalue of value , i.e., the in (48), is 2.93. Therefore, we can have from (48) as 0.257.
Step 5) The following membership functions for , are selected as:
To cover whole cases, we apply 25 fuzzy rules. For
simplification, we let .
Hence, and are constructed.
TABLE I
FOURCASES OF THEINITIALSTATES
Fig. 4. Trajectories of the statesx (solid line) and ^x (dashed line) of four cases.
Fig. 5. Output trajectoriesy of four cases and reference y with = 0:9. Step 6) Compute the adaptive laws (41)–(47). From (60) and
(61), we can let to replace the
un-known in (46) and (47). This has been ex-plained in Remark I.
According to the initial states, four cases are simulated, as shown in Table I.
Fig. 4 shows the trajectories of the states and of four cases if is chosen and it shows that the estimation state
takes very short time to catch up to the system state . The tracking performances of four cases are also very good, as shown in Fig. 5, in which is the reference trajectory and is the system output trajectory. This result is better, as shown in [4] and [12].
Fig. 6. Trajectory of the control input (include supervisory control) of Case 1 with = 0:9 (time = 0 1 s).
Fig. 7. Trajectory of the control input (include supervisory control) of Case 1 with = 0:9 (time = 1 15 s).
We show the control input of
Case 1 with in Figs. 6 and 7.
Fig. 8 shows the supervisory control , and one can obvi-ously see that it is activated in four periods: [0, 0.0057], [0.4361, 0.4389], [0.4475, 0.4503], and [0.4703, 0.4760]. After time
s, the FNN controller can stabilize the system and the supervisory controller will never be activated again. The spikes in Figs. 6 and 8 are caused by the fact that must maintain a larger initial value to stabilize the system when the system tends to be unstable. Therefore, the adaptive controller can perform successful control and the desired performance can be achieved. Applying the different weighting factor , the tracking error performance of Case 1 is shown in Fig. 9.
Example 2: The typical Chua’s chaotic circuit in Fig. 10
con-sists of one linear resistor ( ), two capacitors ( ), one inductor, and one piecewise-linear resistor ( ) [16], [24]. It has been shown to own very rich nonlinear dynamics such as chaos and bifurcations.
Fig. 8. Trajectory of the supervisory controlu of Case 1 with = 0:9 (time
= 0 0:6 s).
Fig. 9. Tracking performance e (t) dt for Case 1 with different .
Fig. 10. Chua’s chaotic circuit.
The dynamic equations of Chua’s chaotic circuit are written as
(62) where voltages and and current are state variables; is a constant; and denotes the nonlinear resistor, which is a function of the voltage across the two terminals of . Here,
Fig. 11. Nonlinear resistor characteristics.
we define as a cubic function as in (63), and its diagram is shown in Fig. 11 [24]
(63) The system can be rewritten as
(64) where
and
The above state space equations are not in the standard canon-ical form defined in (3). Therefore, we need to perform a linear transformation to transform them into the form of (3). Let us
de-fine or where is a
trans-formation matrix. Using the transtrans-formation in [25] and [26], the transformed system can be obtained as
(65) where, as shown in the equation at the bottom of the page,
, .
Choose the parameters as follows:
Therefore, after computation, we get the transformed system as follows:
(66) For comparison, the simulation results of Chua’s chaotic circuit and its transformed system are shown in Fig. 12.
We will design the hybrid FNN adaptive controller to domi-nate the transformed system to track a reference signal. For con-venience, we let replace in the above transformed system. Therefore, the closed-loop configuration of (66) can be repre-sented by
and
(67)
where
and is the external disturbance. Although the above is well defined since the Chua’s circuit is well specified, we do not apply it in the adaptive law. However, we can indeed use it to estimate the upper bound of , which is required in our design
Fig. 12. (a)V of Chua’s circuit. (b)V of Chua’s circuit. (c)i of Chua’s circuit. (d) Phase-plane trajectory of Chua’s circuit. (e) z of transformed system. (f)z of transformed system. (g) z of transformed system. (h) Phase-plane trajectory of transformed system.
procedure. The bounds and can be estimated as
fol-lows:
(68)
The above estimation comes from several simulation runs of the uncontrolled and transformed Chua’s circuit in (66). Since
, we let
(69) and
(70)
The control object is to control the state of the system
to track the reference trajectory if
only the system output is measurable. Therefore, in the phase plane, this reference trajectory is a circle with radius . Also the external disturbance is assumed to be a square-wave with amplitude 0.5, period and the
parameters are chosen as , ,
, and step size . The choices of s
and are to improve the convergence rate of the closed-loop system controlled by our proposed controller.
According to the design procedure, the design is given in the following steps.
Step 1) The observer and feedback gain vectors are chosen
as and ,
respectively.
Step 2) We select in (12) as , then after solving (12), the positive definite symmetric 3 3 matrix in (12) is
The minimum eigenvalue of , i.e., is 6,
which satisfies the transition from (56) to (57). Step 3) Solve (10) to obtain .
Step 4) We select , , ,
, and , and in (14) is chosen as
and in (14) is computed. Therefore, the positive definite symmetric in (14) can be solved as
The minimum eigenvalue of value , i.e., the in (48) is 2.1. Therefore, we can have from (48) as 1.05.
Fig. 13. Trajectories of the statesx (solid line) and ^x (dashed line). Step 5) The following membership functions are selected as:
Let , , , for , . Set
, , , for .
To cover whole cases, we apply 216 fuzzy rules.
For simplification, we let . Hence,
and are constructed.
Step 6) Compute the adaptive laws (41)–(47). From (69) and
(70), it is obvious that we can let to
replace the unknown in (46) and (47). This has been explained in the Remark I.
Fig. 13 shows the trajectories of the states and if is chosen and it shows that the estimation state takes less than 1.4 s to catch up to the system state .
Fig. 14(a)–(c) shows the responses of the transformed Chua’s circuit. Fig. 14(d)–(f) shows the responses of the original Chua’s circuit by restoring the transformed system to its original states.
Fig. 15 shows the phase plane trajectories of the transformed and original Chua’s circuit. Fig. 15 clearly indicates the fact that the tracking performances are guaranteed by our hybrid adaptive FNN controller.
Fig. 16(a) shows the overall control effort for the first 6 s. Fig. 16(b) extends the time-scale in Fig. 16(a) to 25 s. Obviously, the overall control effort in the steady state has its maximum magnitude less than 5 NT. Fig. 16(c) shows the supervisory con-trol with its activation and activation periods in the initial 5 s. After 5 s, the is no longer necessary.
Applying a different weighting factor , the tracking error performance of Example 2 is shown in Fig. 17.
Fig. 14. (a) Output trajectories ofy (dashed line) and y (solid line) with =
0:9. (b) Output trajectories of _y (dashed line) and _y (solid line) with = 0:9.
(c) Output trajectories ofy (dashed line) and y (solid line) with = 0:9. (d) Trajectory ofV . (e) Trajectory of V . (f) Trajectory of i .
Fig. 15. (a) Phase-plane trajectory of transformed Chua’s circuit with =
0:9. (b) Phase-plane trajectory of Chua’s circuit.
Fig. 16. (a) Trajectory of the control input (including supervisory control) with
= 0:9 (time = 0 6 s). (b) Trajectory of the control input (including
supervisory control) with = 0:9 (time = 6 25 s). (c) Trajectory of the supervisory controlu with = 0:9.
Fig. 17. Tracking performance e (t) dt with different . VI. CONCLUSION
An observer-based hybrid direct/indirect adaptive FNN controller appended with a supervisory controller for a class of unknown nonlinear dynamical systems is proposed in this paper. It is a flexible design methodology by the tradeoff between plant knowledge and control knowledge using a weighting factor adopted to sum together the control effort from indirect adaptive FNN controller and direct adaptive FNN controller. If the fuzzy descriptions of the plant are more important and viable, then choose large ; otherwise, choose small . Based on the Lyapunov synthesis approach, the free parameters of the adaptive FNN controller can be tuned on-line by an observer-based output feedback control law and adaptive law. Furthermore, it is a valuable idea that the supervisory control is appended into the FNN controller. The supervisory controller will be activated to force the state to be within the constraint set as long as the system tends to be unstable controlled only by the FNN controller. On the other hand, if the FNN controller works well, the supervisory controller will be deactivated. The simulation results show explicitly that the tracking error of larger is less than smaller i.e., the plant knowledge is more important and viable, and the supervisory controller only works in the beginning period and after that the FNN controller is a main controller. Two nonlinear systems, namely, inverted pendulum system and Chua’s chaotic circuit, are fully illustrated to track sinusoidal signals. Furthermore, it is obvious that the control effort is much less and tracking performance is better than those in previous works.
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Chi-Hsu Wang (M’92–SM’93) was born in Tainan, Taiwan, R.O.C., in 1954. He received the B.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, in 1976, the M.S. in computer science from the National Tsing-Hua University, Bejing, China, in 1978, and the Ph.D. degree in electrical and computer engineering from the University of Wisconsin, Madison, in 1986.
He was appointed Associate Professor in 1986, and Professor in 1990 in the Department of Electrical Engineering, National Taiwan University of Science and Technology, Taipei. He is currently with the School of Microelectronic En-gineering, Griffith University, Nathan, Brisbane, Australia. His current research interests and publications are in the areas of digital control, FNN, intelligent control, adaptive control, and robotics. He has published over 35 international journal articles in these areas and has also served on program committees and chaired technical sessions for several international conferences.
Tsung-Chih Lin was born in Taichung, Taiwan, R.O.C., in 1961. He received the B.S. degree in electrical engineering from the Feng-Chia University, Taichung, in 1984, and the M.S. degree in control engineering from the National Chiao-Tung University, Hsinchu, Taiwan, in 1986. He is currently pursuing the Ph.D. degree at the School of Microelectronic Engineering, Griffith University, Nathan, Brisbane, Australia.
He is currently a Lecturer at the Feng-Chia University. His current research interests and publications are in the areas of adaptive control, FNN, and robust control.
Tsu-Tian Lee (M’87–SM’89–F’97) was born in Taipei, Taiwan, R.O.C., in 1949. He received the B.S. degree in control engineering from the National Chiao-Tung University (NCTU), Hsinchu, Taiwan, in 1970, and the M.S. and Ph.D. degrees in electrical engineering from the University of Oklahoma, Norman, in 1972 and 1975, respectively.
In 1975, he was appointed Associate Professor and in 1978 Professor and Chairman of the Department of Control Engineering at NCTU. In 1981, he be-came Professor and Director of the Institute of Control Engineering, NCTU. In 1986, he was a Visiting Professor and in 1987, a Full Professor of Elec-trical Engineering at the University of Kentucky, Lexington. In 1990, he was a Professor and Chairman of the Department of Electrical Engineering at the National Taiwan University of Science and Technology (NTUST). In 1998, he became the Professor and Dean of the Office of Research and Development, NTUST. Since 2000, he has been with the Department of Electrical and Con-trol Engineering, NCTU, where he is now a Chair Professor. He has published more than 170 refereed journal and conference papers in the areas of automatic control, robotics, fuzzy systems, and NNs. His current research involves motion planning, fuzzy and neural control, optimal control theory and application, and walking machines.
Prof. Lee received the Distinguished Research Award from the National Sci-ence Council, Taiwan, in 1991–1992, 1993–1994, 1995–1996, and 1997–1998, and the Academic Achievement Award in Engineering and Applied Science from the Ministry of Education, R.O.C., in 1998. He was elected to the grade of IEEE Fellow in 1997 and IEE Fellow in 2000. He became a Fellow of New York Academy of Sciences in 2002. His professional activities include serving on the Advisory Board of the Division of Engineering and Applied Science, National Science Council, serving as the Program Director, Automatic Control Research Program, National Science Council, and serving as an Advisor of Ministry of Education, Taiwan, as well as numerous consulting positions.
Han-Leih Liu was born in Taichung, Taiwan, R.O.C., in 1952. He received the B.S. degree in computer science from the Tam-Kang University, Taipei, Taiwan, in 1975, the M.S. degree in computer engineering from the City University of New York, NY, in 1985, and the Ph.D. degree from the School of Microelec-tronic Engineering, Griffith University, Nathan, Brisbane, Australia, in 2001.
His current research interests and publications are in the areas of FNN and adaptive control.
