Acta Polytechnica Hungarica
Vol. 6, No. 2, 2009
The Adaptive Control of Nonlinear Systems
Using the T-S-K Fuzzy Logic
Martin Kratmüller
SIEMENS PSE sro Slovakia
Dúbravská cesta 4, 845 37 Bratislava, Slovak Republic
E-mail: martin.kratmueller@siemens.com
Abstract: Fuzzy adaptive tracking controllers for a class of uncertain nonlinear dynamical
systems are proposed and analyzed. The controller consists of adaptive and robustifying
components whose role is to nullify the effect of uncertainties and achieve a desired
tracking performance. The interactions between the two components have been
investigated. We use the Takagi-Sugeno-Kang type of the fuzzy logic system to approximate
the controller. It is proved that the closed-loop system using this adaptive fuzzy controller
is globally stable in the sense that all signals involved are bounded. Finally, we apply the
method of direct adaptive fuzzy controllers to control an inverted pendulum and the
simulation results are included.
Keywords: adaptive fuzzy approach, nonlinear plant, T-S-K fuzzy logic
1
Introduction
Designing a control law requires a model of the plant to be controlled. However, a
plant model of a physical system usually contains uncertainties. There are
currently two approaches to deal with uncertainties-adaptive control and robust
control. The adaptive control scheme reduces uncertainties by learning them. Most
of the adaptive controllers involve certain types of function approximators in their
learning mechanism (see, for example [1]-[5]). In this proposed controller, we use
fuzzy logic based approximators. The learning rate of adaptation law is relatively
slow and this can lead to a prolonged transcient response. In contrast, a robust
control scheme focuses on compensating the uncertainties by employing high
gain. By combining the two control schemes, the resulting adaptive robust control
techniques have all the advantages of the robust and adaptive control. The
adaptive control scheme attempts to determine all of the uncertainties whose
characteristics cannot be captured by the learning mechanism of the adaptive
control part. Examples of such approaches are reported in the papers [6]-[10],
among others.
–5–
M. Kratmüller
The Adaptive Control of Nonlinear Systems Using the T-S-K Fuzzy Logic
This paper also combines the adaptive and robust approaches in the proposed
tracking controller architecture. A fuzzy logic-based function approximator is used
for the purpose of learning system uncertainties. The robustifying term of the
controller compensates for the modeling inaccuracies. We derive a bound on the
steady-state tracking error as a function of the controller’s gain. Specifically, We
give an expression for an uncertainty region of the tracking error. The proposed
controller guarantees the uniform ultimate boundedness of the tracking error. We
use other fuzzy logic systems that are different from those used in ([2] and [3]).
This form of the fuzzy logic system is called the Takagi-Sugeno-Kang type. And
the consequent part of the fuzzy rules is based on the locally linear feedback
control theory. In Section 2, the considered problem formulation is shown. In
Section 3, the utilized fuzzy logic system is described. The main result is
presented is Section 5. In Section 6, the proposed design steps of the direct
adaptive fuzzy controller are used to control an inverted pendulum system. Some
conclusions of this paper are given in last Section.
2
Problem Formulation
Consider the n-th order nonlinear systems of the form
x (n ) = f (x, t) + bu
y=x
(1)
where f is unknown but bounded function, b is positive unknown constant and
u ∈ R , y ∈ R are the input and output of the system, respectively. Let
x = [x, x, , x (n −1) ]T = [x1 , x 2 , , x n ]T ∈ R n be the state vector of the system
which is assumed to be available.
Let x d = x d ( t ) = [x d ( t ) , x d ( t ) ,
and let e = x − x d = [e, e,
, x (dn −1) ( t )]T denote the desired state trajectory
, e(n −1) ]T . We wish to construct a controller u such that
lim e ( t ) = 0 .
t →∞
3
Description of the T-S-K Fuzzy Logic System
The fuzzy rule base contains a collection of fuzzy IF-THEN rules
R l : IF x1 is F1l and
u = K1l x1 + K l2 x 2 +
and x n is Fnl
THEN
+ K ln x n
(2)
–6–
Acta Polytechnica Hungarica
Vol. 6, No. 2, 2009
where Fil is the label of the fuzzy set in i, K il is the constant coefficient of the
consequent part of the fuzzy rule for l = 1, 2,… . The product operation for the
fuzzy implication and the singleton fuzzifier technology, the final output values is
⎛
M
u (x) =
l =1
⎝
⎞
n
∑ ⎜⎜ ∏ μ
i =1
Fil
( x i ) ⎟⎟ ⋅ ( K1l x1 + K l2 x 2 +
⎠
⎛
⎜
⎜
l =1 ⎝
M
n
∑∏
i =1
+ K ln x n
)
(3)
⎞
μ Fl ( x i ) ⎟
i
⎟
⎠
where μ Fl ( x i ) are the membership function for i = 1, 2,… , n and l = 1, 2,… , M .
i
If the μ Fl ( x i ) ‘s are fixed and the K il ‘s is viewed as adjustable parameter, then
i
(3) can be rewritten as
u ( x ) = θT ξ ( x )
(4)
where θ = ⎡ K11 ,… , K1n , K12 ,… , K 2n ,… , K1n ,… , K nn ⎤
⎣
⎦
(
T
is a parameter vector and
ξ ( x ) = ξ11 ( x ) ,… , ξ1n ( x ) , ξ12 ( x ) ,… , ξn2 ( x ) ,… , ξ1n ( x ) ,… , ξ nn ( x )
)
T
is a regressive
vector with the regressor ξlj ( x ) defined as [1]
⎛
⎜
⎜
ξlj ( x ) = ⎝
⎞
n
∏μ
i =1
⎛
⎜
⎜
l =1 ⎝
M
Fil
n
∑∏
4
i =1
( x i ) ⎟⎟ ⋅ x j
⎠
(5)
⎞
μ Fl ( x i ) ⎟
i
⎟
⎠
Projection Operator
Consider a vector-valued function of time
T
θ ( t ) = ⎡ K11 ( t ) ,… , K1n ( t ) , K12 ( t ) ,… , K 2n ( t ) ,… , K1n ( t ) ,… , K nn ( t ) ⎤ ∈ R m
⎣
⎦
(6)
where i = 1,… , m . We wish the components θi to be between θiL and θiU , that
is, θiL ≤ θi ( t ) ≤ θiU . One way how to achieve this goal is by setting θi ( t ) = 0
when θi ( t ) reaches either of the bounds and tends to go beyond the bound, that is
[11],
–7–
M. Kratmüller
The Adaptive Control of Nonlinear Systems Using the T-S-K Fuzzy Logic
⎧ 0
θi = θiL
and θi ( t ) < 0
⎪⎪
U
θi ( t ) = ⎨
θi = θi
and θi ( t ) > 0
⎪θ ( t ) otherwise
⎪⎩ i
(7)
For compactness, we use the notation Proj for the right hand side of (7). Thus, we
( )
()
can write θi ( t ) = Pr ojθi θi , i = 1,… , m , or in vector notation, θ ( t ) = Pr ojθ θ .
5
Control Law Development
We assume that function f ( x ) in unknown to us. We approximate f ( x ) using
fuzzy logic system θTf ξ
f
( x ) . Let
θ*f = arg min sup f ( x ) − θTf ξ
θ*f be „optimal“ constant vectors such that
f
(x)
(8)
θ*g = arg min sup g ( x ) − θTg ξ
(x)
(9)
θf
x∈Ω
θg
x∈Ω
g
where Ω ⊆ R n is a region in which the state x is constrained to reside. We
assume that
f ( x ) − θ*f ξf ( x ) ≤ d f
∀
x ∈Ω
(10)
g ( x ) − θ*g ξ
∀
x ∈Ω
(11)
T
T
g
( x ) ≤ dg
where d f > 0 , d g > 0 and each element of θ*f is a constant and bounded below
and above as follows
θfLi ≤ θ*fi ≤ θfUi
∀
0 < θgLj ≤ θ*g j ≤ θgUj
i = 1,… , rf
(12)
∀
(13)
j = 1,… , rg
or in vector notation
θfL ≤ θ*f ≤ θfU and θgL ≤ θ*g ≤ θgU
(14)
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Acta Polytechnica Hungarica
Vol. 6, No. 2, 2009
We assume that the lower and upper bounds are known to us. This assumption
may seem unrealistic and restrictive. However, one can just choose rough bounds
based on experience or any knowledge about the nonlinearities.
We define the adaptation parameter errors as
φ = θf − θ*f and φ = θg − θ*g
(15)
g
f
To proceed, we analyze the tracking error dynamics. We write the error dynamics
as
⎡0 1
⎢0 0
e=⎢
⎢
⎢
⎣0
0
1
0⎤
⎡0 ⎤
⎥
⎢ ⎥
0⎥
n
(n)
e + ⎢ ⎥ x( ) − xd
⎥
⎢0 ⎥
⎥
⎢ ⎥
0⎦
⎣1 ⎦
(
0
(
n
(n)
= Ae − bke + bke + b x ( ) − x d
(
)
(n)
= ( A − bk ) e + b ke + f + gu − x d
)
(16)
)
where k is chosen so that A m = A − bk is asymptotically stable.
We use the adaptation laws
(
φf ( t ) = θf ( t ) = Pr ojθf Γf σξf
)
(
φg ( t ) = θg ( t ) = Pr ojθg Γ g σξg u a
(17)
)
r ×rg
where Γf ∈ R rf ×rf and Γ g ∈ R g
(18)
are diagonal symmetric positive definite
matrices and σ = eT P b . The matrix P is the solution of A Tm P + PA m = −2Q for
Q = QT > 0 . It is easy to verify, using the definition of Proj and the fact that for
each i we have θfLi ≤ θ*fi ≤ θfUi and θgLi ≤ θ*gi ≤ θgUi , that
(
(Γ
(
)
)
φTf Γ f−1 Pr ojθf Γ f σξf − σξf ≤ 0
φTg
−1
g Pr ojθg
(19)
( Γ σξ u ) − σξ u ) ≤ 0
g
f
a
f
(20)
a
where
ua =
1
θTg ξ
g
( −θ ξ + x( ) − ke)
T
f f
n
d
(21)
–9–
M. Kratmüller
The Adaptive Control of Nonlinear Systems Using the T-S-K Fuzzy Logic
The initial values of the components of θg are chosen to satisfy (13), where for
each j = 1,… , rg , we have ωgj > 0 . This ensures that θTg ξ > 0 because we choose
g
fuzzy sets so that at least one rule fires at any time and so at least one component
of ξ is nonzero and in fact positive.
g
We use the following fuzzy adaptive control law [11]
u = ua + us
(22)
where u s satisfies the following conditions
(
(n)
σ ke + f + gu a + gu s − x d
)≤ε
(23)
σu s ≤ 0
where ε > 0 is a design parameter. There are several different ways to select u s
so that it satisfies (23).
Let h f ≥ ξ
f
θfU − θfL
and h g ≥ ξ θgU − θgL u a . Given the design parameter
g
ε , we select positive numbers ε1 , ε 2 and ε3 such that ε = ε1 + ε 2 + ε3 . Then k s
should satisfy the following condition
2
2
h g2
1 ⎛⎜ d f + u a d g h f2
+
+
g⎜
2ε1
4ε 2 4ε3
⎝
2
ks ≥
⎞
⎟
⎟
⎠
(24)
When the gain k s satisfies the above condition, u s = −k s σ guarantees that (23)
are satisfied. This can be shown by completing the squares. An alternative
implementation of the robustifying component is
⎛b⎞ ⎛σ⎞
u s = − ⎜ ⎟ sat ⎜ ⎟
⎜ ⎟
⎝g⎠ ⎝δ⎠
where
sat(x) = sign(x)
(25)
is
x >1
and
b = d f + d g u a + θfU − θfL ξf + θgU − θgL ξg u a
sat(x) = x
and δ =
x ≤ 1,
4ε
. It is easy to
b
verify that the above u s satisfies (23).
We now proceed to analyze the dynamics of the tracking error.
– 10 –
if
Acta Polytechnica Hungarica
Vol. 6, No. 2, 2009
Theorem 1 Consider the closed-loop system
n
x( ) = f ( x ) + g ( x ) u
u = ua + us
( )
( Γ σξ u )
θf = Pr ojθf Γ f σξ
f
θg = Pr ojθf
f
g
(26)
a
where u s satisfies (23). Then we have
eT ( t ) P e ( t ) ≤ exp ( −μt ) eT ( 0 ) P e ( 0 ) +
a)
b) if
there
θ*f
exist
θ*g
and
λ min ( Q )
ε
where μ =
λ max ( P )
μ
such
that
f ( x ) = θ*T
f ξ
f
(x)
and
then the origin of the ⎡ e φ φ ⎤ -space is stable
f
g⎥
⎢⎣
⎦
and hence e , φ ( t ) and φ ( t ) are bounded and e ( t ) → 0 as t → ∞ .
g ( x ) = θ*g T ξ
g
(x) ,
f
g
1 T
e P e . The time derivative of V evaluated
2
on the solutions of the closed-loop system (1), (22) is
Proof: To prove (a), we consider V =
(
(n)
V = eT Pe = −eT Qe + σPb ke + f + gu − x d
)
(27)
Substituting into the above expression for the control law given by (22) and
applying (23) gives
(
(n)
V = −eT Qe + σ ke + f + gu a + gu s − x d
≤ −eT Qe + ε ≤ −λ min ( Q ) e + ε ≤ −
2
where μ =
λ min ( Q )
λ max ( P )
)
λ min ( Q )
λ max ( P )
(28)
eT Pe + ε = −μV + ε
. Invoking now the Comparison lemma (see, for example
[12]), we obtain (a).
To prove (b), we consider the following Lyapunov function candidate
V=
(
1 T
e P e + φT Γ f−1 φ + φT Γg−1 φ
f
f
g
g
2
)
(29)
– 11 –
M. Kratmüller
The Adaptive Control of Nonlinear Systems Using the T-S-K Fuzzy Logic
*T
By assumption, f ( x ) = θ*T
f ξ ( x ) , g ( x ) = θg ξ
f
(20)
and
(n)
x d − ke =
(23)
θ*T
f ξf
hold.
In
+ θ*T
f ξf u a .
addition,
g
it
(x)
as well as conditions (19),
follows
from
(21)
that
Taking into account all of the above mentioned
expressions in evaluating the time derivative of V on the trajectories of the closedloop system (26), we obtain
V = eT Pe + φT Γf−1 φ + φT Γg−1 φ
f
f
g
g
≤ −eT Qe + σgu s ≤ −eT Qe ≤ −
λ min ( Q )
λ max ( P )
eT P e = −μeT P e
(30)
It follows from the above that the closed-loop system (26) is stable and therefore,
e ( t ) , φf ( t ) and φ ( t ) are bounded for t ≥ 0 .
g
To show that e → 0 as t → ∞ , we integrate (30) to obtain
∫ V ( τ) dτ = V ( t ) − V ( 0) ≤ ∫ ( −μe Pe ) dτ
t
t
0
0
T
(31)
or equivalently
t
∫ μe Pedτ ≤ V ( 0) − V ( t ) ≤ V ( 0)
T
(32)
0
∫ ( e P e ) dτ
t
Thus, lim
t →∞ 0
T
exists and is finite. It follows from the first part of the
theorem that eT Pe is bounded and since
d T
e P e ≤ −μeT Pe + ε , we conclude that
dt
d T
e P e is also bounded. Therefore, eT Pe is uniformly continuous. By
dt
Barbalat’s lemma [13], eT Pe → 0 as t → ∞ , which concludes the proof of the
theorem.
From (a) of the above theorem and the expression for u s given in (25), we can see
that the design parameter ε determines the final accuracy of the tracking error
which can be made arbitrarily small. As expected, the smaller the desired tracking
error, the larger the controller‘s gain is required. This means the tracking error can
be made arbitrarily small by increasing the controller’s authority.
– 12 –
Acta Polytechnica Hungarica
6
Vol. 6, No. 2, 2009
Simulation Example
In this example, we apply the adaptive fuzzy controller to the system
y'' +
1
y' + 1.7y − 0.5u = 0
0.25 + y
(33)
Define six fuzzy sets over interval <-10, 10> with labels N3, N2, N1, P1, P2, P3.
The membership functions are
1
μ N1 ( x ) =
e
μ N2 ( x ) =
μ N3 ( x ) =
1
e(
x +1.5 )
(35)
2
1
1+ e
(36)
5( x + 2 )
1
μ P1 ( x ) =
e
(37)
( x − 0.5)2
1
μ P2 ( x ) =
e
μ P3 ( x ) =
(34)
( x + 0.5 )2
(38)
( x −1.5)2
1
1+ e
(39)
−5( x − 2 )
The reference model is assumed to be
M (s) =
1
s 2 + 2s + 1
(40)
and the reference signal is the square periodic signal of magnitude 1.5 and
frequency 0.01 Hz.
⎡50 30 ⎤
We choose P = ⎢
⎥ , k1 = 2 , k 2 = 1 , and λ min ( P ) = 1.52 . To satisfy the
⎣30 20 ⎦
constraint related to x we choose V = 0.25 , d f = 20 , d g = 2.1 and ε = 0.25 .
At 200th second of simulation the system (33) was switched to another system
⎡
⎤
1
⎥ y' + y − 5u = 0
−
y''' + 5y'' + ⎢
1.7
⎢ ( 0.25 + y )2
⎥
⎣
⎦
– 13 –
(41)
M. Kratmüller
The Adaptive Control of Nonlinear Systems Using the T-S-K Fuzzy Logic
All initial states have been set to zero y ( 0 ) = y' ( 0 ) = y'' ( 0 ) = y''' ( 0 ) = 0 .
As it can be seen from Fig. 1, the simulation results confirm a good adaptation
capability of the proposed control system. The system dynamic changes are in
particular manifested by changes of control input signal (Fig. 2).
[-]
7
6
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400 s
Figure 1
The state x1 (blue dashed line), its desired reference model value y m ( t ) (green solid line) and
reference signal (red solid line)
[-]
50
40
30
20
10
0
-10
0
50
100
150
200
250
Figure 2
Control signal
– 14 –
300
350
400 s
Acta Polytechnica Hungarica
Vol. 6, No. 2, 2009
Conclusions
In this paper, the TSK-type fuzzy logic system is used in the adaptive fuzzy
controller. The major advantage is that the accurate mathematical model of the
system is not required to be known. The proposed method can guarantee the
global stability of the resulting closed-loop system in the sense that all signals
involved are uniformly bounded. Furthermore, the specific formula of the bounds
is also given. Finally, we use the adaptive fuzzy controller to regulate the system
to the origin. We also showed explicitly how the supervisory control forced the
state to be within the constraint set.
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M. Kratmüller
The Adaptive Control of Nonlinear Systems Using the T-S-K Fuzzy Logic
[11]
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– 16 –