# GA-Based Fuzzy Sliding Mode Controller for Nonlinear Systems

Authors: P. C. Chen; C. W. Chen; W. L. Chiang
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Generally, the greatest difficulty encountered when designing a fuzzy sliding mode controller (FSMC) or an adaptive fuzzy sliding mode controller (AFSMC) capable of rapidly and efficiently controlling complex and nonlinear systems is how to select the most appropriate initial values for the parameter vector. In this paper, we describe a method of stability analysis for a GA-based reference adaptive fuzzy sliding model controller capable of handling these types of problems for a nonlinear system. First, we approximate and describe an uncertain and nonlinear plant for the tracking of a reference trajectory via a fuzzy model incorporating fuzzy logic control rules. Next, the initial values of the consequent parameter vector are decided via a genetic algorithm. After this, an adaptive fuzzy sliding model controller, designed to simultaneously stabilize and control the system, is derived. The stability of the nonlinear system is ensured by the derivation of the stability criterion based uponLyapunov's direct method. Finally, an example, a numerical simulation, is provided to demonstrate the control methodology.


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## Body

## 1. Introduction

Over the past few years, fuzzy control (FC) can be designed
without needing an exact mathematical model of the system to be controlled, and can efficiently control complex continuous unmodeled or partially modeled processes [1, 2]. There have
been significant research efforts devoted to the analysis and control designs for
fuzzy systems (see [3, 4] and the
references therein). The main motivation for this development has been applied to
practical nonlinear systems and engineering problems (see [5–7] and the
references therein). Undoubtedly, Lyapunov’s theory is one of the most common approaches
for dealing with the stability analysis of systems. However, to overcome the
conservatism that arises from the use of Lyapunov’s methods, it has been
necessary to develop a number of more effective methods, for example, fuzzy
Lyapunov functions [8, 9]. There are  also many important issues that have advanced results for T-S fuzzy control systems, such as
time delays [10–13], H∞ performance [3–15], robustness [16, 17], neural
networks (NNs), and genetic algorithms (GAs) [18–21]. Furthermore,
much work has been published on the design of fuzzy sliding mode controllers
(FSMCs) [22, 23]. An FSMC is composed of an FC and a sliding mode controller
(SMC) [24–26]. An FSMC is a powerful and robust control strategy for the
treatment of modeling uncertainties and external disturbances. Although control
performance is good, one still has to decide on the parameters. This is one of
the most important issues in their design.In the
so-calledadaptive FSMC (AFSMC), [27–29], an adaptive
algorithm is utilized to find the best high-performance parameters for the FSMC
[30, 31]. In recent years, adaptive fuzzy control system designs have attracted
a good deal of attention as a promising way to approach nonlinear control
problems [30, 31]. For adaptive fuzzy control, one initially constructs a fuzzy
model to describe the dynamic characteristics of the controlled system; then,
an FSMC is designed based on the fuzzy model to achieve the control objectives.
After this, adaptive laws are designed (with Lyapunov’s synthesis approach) for
tuning the adjustable parameters of the fuzzy models, and analyzing the
stability of the overall system.Deciding on the fuzzy rules and the initial parameter vector values for the AFSMC is very important. A genetic algorithm [32–34] is usually
used as an optimization technique in the self-learning or training strategy for
deciding on the fuzzy control rules and the initial values of the parameter
vector. This GA-based AFSMC should improve the immediate response, the
stability, and the robustness of the control system.Another common
problem encountered when switching the control input of the FSMC system is the
so-called “chattering” phenomenon. Chattering is eliminated by smoothing the
control discontinuity inside a thin boundary layer, which essentially acts as a
low-pass filter structure for the local dynamics [25]. The boundary-layer
function is introduced into these updated laws to cover parameter and modeling
errors, and to guarantee that the state errors converge within a specified
error bound.In this study,
we focus on the design of robust tracking control for a class of nonlinear
uncertain system involving plant uncertainties and external disturbances.
First, the nonlinear system for the tracking of a reference trajectory for the
plant [35] is described via fuzzy models with fuzzy rules. A genetic algorithm
is used to find the initial values of the parameter vector. Then the designed
adaptive control laws of the reference adaptive fuzzy sliding mode controller (RAFSMC)
are updated. This GA-based RAFSMC would improve the immediate response, the
stability, and the robustness of the control system. Finally, both the tracking
error and the modeling error approach zero.


## 2. Reference Modeling of a Nonlinear Dynamic System

The plant is a single-input/single-outnth-order system with n≥1: (2.1)x˙1=x2,⋮x˙n−1=xn,x˙n=f(x)+g(x)⁢⋅u+d,y=x1, where x=[x1,x2,…,xn−1,xn]T∈Rn is the state vector of the system; u∈R is the control signal; f,  g are smooth nonlinear functions; d denotes the external disturbance d(t) which is unknown but usually bounded.The statesx=[x1,x2,…,xn−1,xn]T are assumed to be available. For example, a
single robot can be represented in the form of (2.1), with n=2 and x(x1=θ,x2=θ˙) being measurable. Differentiating the output
with respect to time for n times
(till the control input u appears),
one obtains the input/output form of (2.1): (2.2)y(n)=f(x)+g(x)⁢⋅u+d(t).The system is said to have a relative
degree n, if g(x) is bounded away from zero.Assumption 2.1.
g(x) is bounded away from zero over a
compact setζ⊂Rn, (2.3)|g(x)|≥b>0,∀x∈ζ.
If the control
goal is for the plant outputy to
track a reference trajectory yr,
the reference control input r can be
defined by the following reference model: (2.4)r=yr(n)+αn−1yr(n−1)+αn−2yr(n−2)+⋯+α1y˙r+α0yr,where αn−1,αn−2,…,α1,α0 are chosen such that the polynomial ℓn+αn−1ℓn−1+αn−2ℓn−2+⋯+α1ℓ+α0 is Hurwitz, and ℓ here denotes the complex Laplace  variable.Iff(x), g(x) are known, and assumption 2.1 is satisfied, the
control law can defined by (2.5)u=−f(x)−d(x)−(αn−1y(n−1)+⋯+α1y˙+α0y)+rg(x),∀x∈S.Substituting (2.5)
into (2.1), the linearized system becomes (2.6)(yr(n)−y(n))+αn−1(yr(n−1)−y(n−1))+⋯+α1(y˙r−y˙)+α0(yr−y)=0.If we definee=yr−y as the tracking error, then the reference
control input (2.4)
results
in  the following error equation: (2.7)e(n)+αn−1e(n−1)+⋯+α1e˙+α0e=0.It is clear thate will approach zero if αn−1,αn−2,…,α1,α0 are chosen, such that the polynomial ℓn+αn−1ℓn−1+αn−2ℓn−2+⋯+α1ℓ+α0 is Hurwitz.


## 3. Development of a GA-Based FSMC

In general,
people describe the decision-making process using linguistic statements, such
as “IF something happens, THEN do a certain action.” For example, let us look
at a rule: “IF the temperature is high, THEN the power of the heater is low.”
In this statement both “high” and “low” are linguistic terms. Although this
kind of linguistic rule is not precise, humans can use them to make correct
decisions. To utilize such fuzzy information in a scientific way, mathematical
representation of the fuzzy information is needed. Fuzzy set theory and
approximate reasoning are two ways that such linguistic information can be
dealt with mathematically. A review of the literature provides the theoretical
foundation for the  developed fuzzy logic controller. The configuration of the
fuzzy logic controller is shown in Figure1.Figure 1
The fuzzy logic controller system.The basic
concepts for fuzzy sets and fuzzy logic are briefly described below.(1)Fuzzy set,  fuzzifier,   and
membership function.    Let X denote the universe of discourse. A fuzzy set A in X is characterized by a membership function μA:X→[0,1],
with μA(x) representing the grade of membership of x∈X in fuzzy set A.
For example, the Gaussian-shaped membership function is represented as μA(x)=exp(−((x−m)/σ)2), where m is the center and σ denotes the spread of the membership function.(2)Fuzzy rule base and fuzzy
inference engine. Each rule Rj in the fuzzy rule base can be expressed as (3.1)Rj:IFx1⁢isA1jand⋯xnisAnj,THENy⁢isBj;and⁢μRj(χ)=⋂i=1nμAij(xi).(3)Deffuzzifier.  The defuzzifier
maps a fuzzy set A in X to a crisp point x∈X.
There are several defuzzification methods described in the literature. The most
popular is the weighted average defuzzification method defined as y=∑j=1Nθj⋅μRj(χ)/∑j=1NμRj(χ).The FSMC is
composed of a sliding mode controller and an FLC. This makes it a powerful and
robust control strategy for the treatment of modeling uncertainties and
external disturbances. The sliding
mode  plant combined with
the FLC is shown in Figure2.Figure 2
The sliding mode plant combined with the FLC.Genetic
algorithms (GAs) are parallel, global search techniques derived from the
concepts of evolutionary theory and natural genetics. They emulate biological
evolution by means of genetic operations such as reproduction, crossover, and
mutation. GAs are usually used as optimization techniques and it has been shown
that they also perform well with multimodal functions (i.e., functions which
have multiple local optima).Genetic
algorithms work with a set of artificial elements (binary strings, e.g.,0101010101) called a population. An individual (string)
is referred to as a chromosome, and a single bit in the string is called a
gene. A new population (called offspring) is generated by the application of
genetic operators to the chromosomes in the old population (called parents). Each
iteration of the genetic operation is referred to as a generation.A fitness
function, specifically the function to be maximized, is used to evaluate the
fitness of an individual. The offspring may have better fitness than their
parents. Consequently, the value of the fitness function increases from
generation to generation. In most genetic algorithms, mutation is a random-work
mechanism to avoid the problem of being trapped in a local optimum. Theoretically,
a global optimal solution can be found.Offspring are
generated from the parents until the size of the new population is equal to
that of the old population. This evolutionary procedure continues until the
fitness reaches the desired specifications.
However, in a specific application, the fitness specification might be used to
stop the evolutionary process. In most applications, the optimal fitness value
is totally unknown. In this case, the evolutionary process is interrupted
either by stabilization of the fitness value (the variation is below a specific
value) or by reaching the maximum number of generations.Knowledge
acquisition is the most important task in the fuzzy sliding mode controller design.
The initial values of the entries in the consequent parameter vector are
decided by the self-organizing
of FSMC system which developed based on GA.  The configuration of this system is shown
in Figure3.Figure 3
GA-based FSMC.The learning
procedure for the GA-based FSMC is
summarized as follows.(1) The fuzzy rule base of FSMC
(with fixed premise parts and random consequence parts) is constructed. For
example, FSMC for system (2.1): (3.2)FSMC:{R1(i):IFSisPB(4,0.424)THENuisu^1(i)(θ^1(i)),R2(i):IFS⁢isPM(3.2,0.424)THENuisu^2(i)(θ^2(i)),⋮RN(i):IFSisPB(−4,0.424)THENuisu^N(i)(θ^N(i)),  where u^j(i) is an unknown linguistic label for the control u; θ^j(i) is the adjustable parameter, which have to be encoded
as binary strings for genetic operations.(2) Encode each parameter,θ^j(i) (i=1,2,…,M;j=1,2,…,N), to a d-bit binary code, Pj(i)(h)=(bj1bj2⋯bjd)(h)=enc(θ^j(i)(h)),
where bj1,bj2,…,bjd∈{0,1} and enc(*)  denote the encoding operator which encodes the
real  values to the corresponding binary codes and synthesizes the chromosome of
the ith individual.(3) Establish the population for generationh, Pj(h)={Pj(1)(h),Pj(2)(h),…,Pj(M)(h)},
where M is the population size, and every individual Pj(i)(h) corresponds to a binary-code parameter of an
FSMC candidate.(4) Evaluate the fitness value of each individual. The fitness functionF is defined as F=1/(w∥s(k)∥+v∥u(k)∥+ε0),
where k=int(t/Δt) denotes the iteration instance; Δt is the sampling period;
int(*)
is the rounding off operator; w and v are positive weights; ε0 is a very small positive constant used to
avoid the numerical error of dividing by zero.(5)   Based on the fitness value of the
individual, keep the best and apply the genetic operators. Assuming that the
population sizeM is 12, pick the top
ten-fitted individuals in Pj(h) to apply as genetic operators, that is,
reproduction, crossover, mutation (assuming the mutation rate is 0.03125), and
keep the top two fitted individuals to generate a new population Pj(h+1),
as the offspring of Pj(h).(6)  Decode each binary code to its real
value and use this to calculate the controlu,
then apply u to the system (2.1).(7) Seth=h+1;
go to Step 2, and repeat the aforementioned procedure until F≥FM or h≥H,
where FM and H denote an acceptable specific fitness value and the top generation number,
respectively, as specified by the designer.In general, there
are at least four methods for the construction of a fuzzy rule base: (1) from
expert knowledge or operator experience; (2) modeling an operator’s control
action; (3) modeling a process; (4) generating fuzzy rules by training,
self-organizing, and self-learning algorithms. In Figure3, GA is used as the
learning and training mechanism. The use of the GA means that the second, third,
and fourth approaches also provide an efficient way to obtain a fuzzy rule
base. Although there are several methods that can provide excellent results in this kind of modeling [36–38], we are convinced that GAs are the most
advantageous way to extract an optimal, or at least suboptimal fuzzy rule base for
the initial values of the consequent parameter vector of the FSMC or AFSMC.


## 4. GA-Based RAFSMC for Nonlinear Systems

A schematic representation of the GA_RAFSMC system is shown
in Figure4. If f(x), g(x) are known, we can design the
FLC (4.1)
to approximate u:
(4.1)u¯(θ¯)=∑k=1mRk(−(‖Si−Cki‖β)2)⋅θ¯k,
where m is the sum
of the  fuzzy rules, θ¯k, that is, |θ¯k|≤θmax indicate   the adjustable consequent parameters
of the FLC, and R(S)=[R1(S),R2(S),…,Rm(S)]T is the vector of fuzzy basis function [23]
which is defined as (4.2)Rk(S)=Rk(‖Si−Cki‖)=∏i=1nμk(‖Si−Cki‖)∑k=1m[∏i=1nμk(‖Si−Cki‖)],where k=1,…,m and i=1,…,n with μk represent the degree of membership. The Si in μk can be chosen by (4.3)μk(‖Si−Cki‖)=exp⁡(−(‖Si−Cki‖β)2). Since here n, the sum of input variables, is only one, we know that (4.4)Rk(S)=μk(S−Ck)∑k=1mμk(S−Ck),  where k=1,…,m with μk represent the degree of membership. The S in μk can be chosen by μk(∥S−Ck∥)=exp(−(∥S−Ck∥/β)2).Figure 4
GA_RAFSMC system.From the
approximation property of the fuzzy system, an uncertain and nonlinear plant
can be well approximated and described via a fuzzy model with FLC rules to
achieve the control object [14, 39, 40].Assumption 4.1.
Forx∈ζ⊂Rn,
there exists an adjustable parameter vector θ¯=[θ¯1,θ¯2,…,θ¯m]T such that the fuzzy system u¯(S,θ¯)=θ¯TR(S) can approximate a continuous function u with accuracy εmax over the set ζ,
that is, ∃θ¯,
such that (4.5)sup⁡|u¯(S,θ¯)−u(S)|≤εmax⁡,∀S∈ζ.
Letθ^ denote the estimate of θ¯ at time t. Now, we can define the estimated
control output u^(S,θ^) by (4.6)u^(S,θ^)=∑k=1mθ^k⁢⋅Rk(S)=θ^TR(S),and decide on the initial values of the consequent parameter
vector θ^=[θ^1,θ^2,…,θ^m]T based on the genetic algorithm.First, define
the parameter error vector at timet by θ=θ¯−θ^, and then (4.7)θTR(S)=u¯(S,θ¯)−u^(S,θ^).According to assumption 4.1, we can define the
modeling error (4.8)ε=u−u¯(S,θ¯),where |ε|≤εmax.We can say that(4.9)u=u^(S,θ^)+θTR(S)+ε.Now, by substituting (4.9) into (2.5), we
obtain the error dynamic equation: (4.10)e(n)+αn−1e(n−1)+⋯+α1e˙+α0e=g(x)⋅(θTR(S)+ε).We now define the augmented error
as (4.11)S=βn−1e(n−1)+⋯+β1e˙+β0e, where βn−1,…,β1,β0 in (4.11), and αn−1,…,α1,α0 in (4.10) are chosen such that (4.12)M^(ℓ)=βn−1ℓn−1+⋯+β1ℓ+β0ℓn+αn−1ℓn−1+⋯+α1ℓ+α0=N(ℓ)D(ℓ) is   strictly positive
real (SPR)  transfer function,
and N(ℓ) and D(ℓ) are coprime. Now, S and g(x)⋅(θTR(S)+ε) can be  related by (4.13)L{S(t)}=M^(ℓ)⋅L{g(x)⋅(θTR(S)+ε)},where L{⋅} is the Laplace transform of
the function, and ℓ denotes the complex Laplace  transform variable.If we defineem=[e,…,e(n−1)]T as the states of (4.10), then (4.10) can be
realized as (4.14)e˙m(t)=Λ⁢⋅em(t)+b⁢⋅[g(x)⋅(θTR(S)+ε)],(4.15)S(t)=cTem(t),where (4.16)Λ=[010⋯00001⋯00⋮⋮⋮⋱⋮⋮000⋯10000⋯01−α0−α1−α2⋯−αn−2−αn−1]n×nc=[β0β1⋯βn−1]T,letβn−1=1.,b=[00⋮001]n×1, According to the Kalman-Yakubovich lemma, when M^(ℓ) is SPR, there exist symmetric and positive
definite matrices P and Q such that (4.17)PΛ+ΛTP=−Q,Pb=c,fori=1,…,p.Next, we
investigate the asymptotic stability of the origin usingLyapunov’s function candidates. First, define a Lyapunov candidate function as (4.18)V(em,θ)=η⁢⋅emTPem+θTH11θ,where η is a positive constant representing the
learning rate (4.19)θ=[θ1θ2⋯θm]T,H11=g(x)⁢⋅Im×m,θTH11=[g(x)⁢⋅θ10⋯00g(x)⋅θ2⋯0⋮⋮⋱000⋯g(x)⁢⋅θ]m×m,m:thesumofthefuzzyrules.IfemTPem>ϕ2,
the derivate of V(em,θ) along the trajectories of the system should be
negative definite for all nonlinearities that satisfy a given sector condition
(Lyapunov’s stability):(4.20)V˙(em,θ)=η⁢⋅(e˙mTPem+emTPe˙m)+2θTH11θ˙. As mentioned above θ=θ¯−θ^,
and we can infer that θ˙=−θ^˙,
and (4.21)V˙=η⁢⋅(emTΛTPem+emTPΛem)+2η⁢⋅emTPb⋅[g(x)⁢⋅(θTR(S)+ε)]+2⁢⋅θTH11(−θ^˙)=η⁢⋅emT(−Q)em+2η⁢⋅S⁢⋅[g(x)⁢⋅(θTR(S)+ε)]+2⁢⋅θTH11(−θ^˙).In general,
chattering must be eliminated for the controller to perform properly. This can
be achieved by smoothing out control discontinuity in a thin boundary layer
neighboring the switching surface. To amend the modeling errorε and the chattering phenomenon, we propose a
modified adaptive law (4.22) with which
to tune the adjustable consequent
parameters of the RAFSMC:(4.22)θ^˙=η⁢⋅|S|⁢⋅R(S)⋅sat(SΦ). The thin boundary layer function sat(S/Φ) is defined as (4.23)sat(SΦ)={1,if(SΦ)>1,(SΦ),if−1≤(SΦ)≤1,−1,if(SΦ)<−1,  where Φ>0 is the thickness of the boundary layer.If we  substitute (4.22) into (4.21),
then (4.21) becomes (4.24)V˙=−η⁢⋅emTQem+2η⁢⋅S⁢⋅[g(x)⋅(θTR(S)+ε)]−2η⁢⋅|S|⁢⋅[g(x)⁢⋅θTR(S)]⋅sat(SΦ).When|S|>Φ,
then (4.25)V˙=−η⁢⋅emTQem+2η⁢⋅emTc⁢⋅(g(x)⁢⋅ε)≤−η⁢⋅‖em‖2⁢⋅Q+2η⁢⋅‖em‖⋅‖c‖⋅‖g(x)⁢⋅ε‖≤−η⁢⋅‖em‖⁢⋅[‖em‖⁢⋅Q−2‖c‖⋅‖g(x)⁢⋅ε‖].Ifμ is positive and  small enough, then ϕ>0 and σ>0,
such that (4.26){ϕQP−2‖c‖⁢⋅‖g(x)⋅ε‖}>σ, where emTPem>ϕ2.It is real thatV˙≤−η⋅∥em∥⋅σ if emTPem>ϕ2 and |S|>Φ,
and hence V˙<0.
Thus V will gradually converge to zero as all the ς.Based on the
above inference andLyapunov’s stability theory, em will gradually converge inside the bounded
zone |em|≤(ϕ/P,Φ/β0).
The tracking error and the modeling error will then both approach zero.Theorem 4.2.
Consider a nonlinear uncertain systemy(n)=f(x)+g(x)⋅u+d that satisfies the assumptions (θ¯,θ^).
Suppose that the unknown control input u can be approximated by u^(S,θ^) as in (4.6). Now, S is given by (4.15), and Q is a symmetric positive definite weighting
matrix.


## 5. Numerical Simulation

In this
section, the proposed GA-based RAFSMC is demonstrated with an example of the
control methodology.Consider the
problem of balancing an inverted pendulum
on a cart as shown in Figure5. The
dynamic equations of motion of the pendulum are given below [27]:Figure 5
Inverted pendulum system.(5.1)x˙1=x2,x˙2=g⋅sin⁡(x1)−aml⁢x22sin⁡(2x1)/2−acos⁡(x1)⋅u4l/3−aml⁢cos⁡2(x1),  where x1 denotes the angle (in radian) of the pendulum
from the vertical; and x2 is the angular vector. Thus the gravity
constant g=9.8  m/s2,
where m is the mass of the pendulum, M is the mass of the cart, l is the length of   F (input force), s is the force  applied to the cart (in Newtons), and a=1/(m+M).
The parameters chosen for the pendulum in this simulation are m =0.1 kg, M =1 kg, and l =0.5 m.The control objective in this example is to balance the
inverted pendulum in the approximate rangex∈(−π/2,π/2).
The GA-based RAFSMC designed based on the procedure discussed above will have
the following steps.Step 1.
Specify the response of the control system by defining a suitable sliding
surface(5.2)S=cTem=5e+e˙[27].Step 2.
Construct the fuzzy rule base (3.2) and the fuzzy models (4.6) based on the
genetic algorithm. After carrying out
the abovementioned genetic-based learning procedure, the number of individual
strings is 10, the size of population M is
12, the crossover rate is 0.8333, the mutation rate is 0.03125, and the maximum number of the generations H is 15. Now, the initial values of the
consequent parameter vector θ^ for the GA-based RAFSMC can be chosen as
follows: (5.3)[1,0.6263,0.4113,0.2100,0.0850,0,−0.0850,−0.2100,−0.4113,−0.6263,−1]T.Step 3.
Apply the controller as given by (4.6) to control the nonlinear system (2.1).
Now, let η=10, Φ=0.3,
and adjust θ^ by the adaptive law as given by (4.22).Therefore,
based on Theorem4.2, the
proposed GA-based RAFSMC can asymptotically stabilize the inverted pendulum.
The simulation results are illustrated in
Figures 6–9. The initial
conditions are x1(0)=30°,60°,
and x2(0)=0.Figure 6
Angle response of the pendulum with the initial conditionx1(0)=30°.Figure 7
Control force in the pendulum system with the initialx1(0)=30°.Figure 8
Angle response of the pendulum with the initial conditionx1(0)=60°.Figure 9
Control force in the pendulum system with the initial conditionx1(0)=60°.Figures6–9 show that the
inverted pendulum system  (compare
with Yoo and Ham  [27]) is rapidly, asymptotically stable because the system trajectory
starts from any nonzero initial state, to rapidly and asymptotically approach the
origin.


## 6. Conclusion

The stability
analysis of a GA-based reference adaptive fuzzy sliding model controller for a
nonlinear system is discussed. First, we track the reference trajectory for an
uncertain and nonlinear plant. We make sure that it is well approximated and
described via the fuzzy model involving FLC rules. Then we decide on the initial
values of the consequent parameter vectorθ^ via a GA. Next, an adaptive fuzzy sliding
model controller is proposed to simultaneously stabilize and control the
system. A stability criterion is also derived from Lyapunov’s direct method to ensure stability of the nonlinear
system. Finally, we discuss an example and provide a numerical simulation. From
this example, we see that the stability of the inverted pendulum system is ensured
because the trajectories from nonzero initial states approach to zero by
proposed controller design, and the results demonstrate that with this control
methodology we can rapidly and efficiently control a complex and nonlinear
system.



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