Compensation of Mismatched Disturbances for Nonlinear Plants with
Distributed Time-delay
Igor Furtat
1,2 a
, Pavel Gushchin
1,3 b
, Dmitrii Konovalov
2
and Sergey Vrazhevsky
1,2 c
1
Institute of Problems of Mechanical Engineering Russian Academy of Sciences, Bolshoj pr. V.O., 61,
St. Petersburg, Russia
2
ITMO University, Kronverksky pr., 49, St. Petersburg, Russia
3
Gubkin Russian State University of Oil and Gas (National Research University), Leninsky pr., 65, Moscow, Russia
Keywords: Backstepping, Disturbance, Time-delay.
Abstract: The paper deals with the robust algorithm for compensation of unknown mismatched disturbances
depending on a state vector of plants with distributed time-delay. The algorithm based on generalization of a
backstepping method and disturbance compensation method. The proposed control system compensates
disturbances with required accuracy. The simulation results illustrate the efficiency and robustness of the
suggested control system. System". MIMO System".
a
https://orcid.org/0000-0003-4679-5884
b
https://orcid.org/0000-0002-8813-2723
c
https://orcid.org/0000-0001-9725-5330
1 INTRODUCTION
The paper is devoted to a new robust scheme for
compensation of unknown mismatched disturbances.
It is known, that the backstepping method is
effective method for control of plant under
mismatched condition.
The first backstepping method is proposed in
(Kokotovic, 1992). In more detail the backstepping
method is presented in (Fradkov, Miroshnik, and
Nikiforov, 1999; Khalil, 2002). Currently there are a
lot of modification of the backstepping algorithm.
In (Chang and Cheng, 2010) a methodology of
designing the block backstepping controller for a
class of multi-input systems with mismatched
perturbations is proposed. Some adaptive
mechanisms are embedded both in the virtual input
controller and in the backstepping controllers such
that some part knowledge of the upper bound of
perturbation is not required.
The paper (Ma, Schilling, and Schmid, 2005) is
concerned with the adaptive sliding-mode control of
a class of nonlinear systems in nonlinear parametric-
pure-feedback form with mismatched uncertainties.
Backstepping design procedure is applied, which
leads to a new adaptive sliding-mode control.
Gaussian radial-basis-function networks are used to
approximate the unknown system dynamics. More
nodes are added to the networks progressively in
order to improve the transient behaviour. With ideal
sliding mode, asymptotic stability is reached.
In (Xu and Min, 2010) for a class of strict-
feedback nonlinear systems with mismatched
uncertainties, an adaptive backstepping fuzzy
controller design is presented. By applying
backstepping design strategy and online approaching
uncertainties with fuzzy approximator, the control
inputs and adaptive tuning rules are derived from the
Lyapunov stability theory. To deal with the problem
of extreme expanded operation quantity of
backstepping method, a nonlinear tracking
differentiator is introduced. By choosing suitable
design parameters, the developed control scheme
guarantees that all the signals of the closed-loop
system are uniformly ultimately bounded and the
system tracking error can reach to a very small
region around zero.
However, the above mentoined algorithms
cannot compensate disturbances with distributed
time-delay. For disturbances compensation there are
a lot of methods, for example (Bobtsov and
Kremlev, 2005; Tsykunov, 2007). The main idea of
Furtat, I., Gushchin, P., Konovalov, D. and Vrazhevsky, S.
Compensation of Mismatched Disturbances for Nonlinear Plants with Distributed Time-delay.
DOI: 10.5220/0007915202690275
In Proceedings of the 16th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2019), pages 269-275
ISBN: 978-989-758-380-3
Copyright
c
2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
269
disturbances compensation consists representation of
uncontrollable disturbances as a new function in the
control system which we can use for design of the
control system. Among of them the auxiliary loop
algorithm (Tsykunov, 2007.) is one of the
effectiveness algorithm. The auxiliary loop
algorithm based on parallel reference model
(auxiliary loop) to the plant. The auxiliary loop is
used for obtaining the uncertainties acting on the
plant. The idea of this method consists in
implementing an auxiliary loop with desired
parameters parallel to the plant. The difference
between the output of the plant and the output of the
auxiliary loop gives a function which depends on
parametric and external disturbances. This function
gives the control law that guarantees required
accuracy of the control system. The proposed
algorithm provides the tracking by the output of the
plant of the reference output with the required
accuracy. However, mismatched disturbances may
control system get unstable. It will be demonstrated
in Section 4.
In this paper we propose a new modified
backstepping algorithm with mismatched
disturbances compensation (MBADC), where
disturbances are presented nonlinear functions
depending on external bounded disturbances,
parametric uncertainties and state vector with
distributed time-delay. This algorithm is a
generalization of results (Khalil, 2002; Tsykunov,
2007) for robust control of nonlinear plants under
mismatched parametric uncertainties and external
disturbances. It is assumed that input signal and state
vector of the plant are available for measurement.
The proposed algorithm guarantees stabilization by
the plant state vector with the required accuracy.
The paper is organized as follows. The problem
statement is presented in Section 2. The MBADC for
control of nonlinear plants under mismatched
perturbations is proposed in Section 3. In Section 4
the efficient of MBADC is illustrated by modeling
of an unstable nonlinear plant. Also, in Section 4 the
comparison of the simulation results for the
MBADC, the backstepping algorithm and auxiliary
loop algorithm are presented. Concluding remarks
are given in Section 5. Appendix A gives the proof
of the MBADC.
2 PROBLEM STATEMENT
Consider the plant model with distributed time-delay
in the form
,,)()(
,,,)(...,,)(),(),...,(
)()(
,,,)(...,,)(,),...,(
)()(
2
00
332
32
00
221
21
221
111
tftbutx
tfdtxdtxtxtx
txtx
tfdtxdtxxtx
txtx
n
h
n
h
n
h
n
h
n
n
n
(1)
where x(t) = [x
1
(t), x
1
(t), …, x
n
(t)]
T
is a state vector,
u(t) R is an input,
i
() R, i = 1, 2, …, n are
unknown functions depending on parametric
uncertainties and external disturbances, f is a vector
of unknown parameters, b > 0 is unknown constant
and [f
T
, b]
T
, is a known bounded set, h
ij
,
i, j = 1, 2, …, n are unknown time delay.
We assume that signal x(t) is available for
measurement.
Additionally assume that the functions
i
() R,
i = 1, 2, …, n are bounded or bounded on t and f, and
Lipchitz in x(t),
0
2
)(
ij
h
dtx
,
0
2
)(
ij
h
dtx
, ...,
0
)(
ij
h
n
dtx
, i, j = 1, 2, …, n.
The problem is to synthesis a control law such
that the goal condition holds
)(tx
for t > T, (2)
where
> 0 is a prespecified required accuracy,
T > 0 is a transient time, || is Euclidean norm of the
corresponding vector.
3 MAIN RESULT
Let us denote
.,,)(...,,)(),(
00
1
1
tfdtxdtxtx
ini
h
n
h
ii
The synthesis of the control system is split into n
steps. Auxiliary loops and auxiliary controls will are
designed on 1, …, n – 1 steps for compensation of
unknown function
i
. The control law u(t) will be
introduced on the n-th step.
Step 1. Introduce the first auxiliary loop in the
form
),()()(
2111
txtzctz
(3)
where c
1
> 0 is a coefficient chosen by a designer.
ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics
270
Taking into account the first equation of system
(1) and equation (3), rewrite the function
1
(t) = x
1
(t) – z
1
(t) as follows
.)()(
1111
tzct
(4)
From (4) it follows that the function
1
() may be
rewritten in the form
).()(
1111
tzct
(5)
Substituting (5) to the first equation of (1), we
have
).()()()(
11121
tzcttxtx
(6)
Assume that the function x
2
(t) is a control signal
in (6) and define x
2
(t) in the form x
2
(t) = v
1
(t). Since
the function
is not available for measurement,
we introduce the first auxiliary control law v
1
(t) as
follows
),(
ˆ
)()(
1111
ttctv
(7)
where
is an estimate of the function
.
Substituting (7) to (6), we get
),()()(
1111
ttxctx
(8)
where
)(
ˆ
)()(
111
ttt
is an error estimate.
Step i (2 ≤ i ≤ n – 1). Since v
i
(t) is not real
control law, we consider the i-th error function e
i -
1
(t) = x
i
(t) – v
i - 1
(t) as follows
),(
~
)()(
11
ttxte
iii
(9)
where
).()(
~
1
tvt
iii
Introduce the i-th auxiliary loop in the form
),()()( txtzctz
iiii
(10)
where c
i
> 0 is a coefficient chosen by a designer.
Taking into account (9) and (10), rewrite the
function
i
(t) = e
i - 1
(t) – z
i
(t) as follows
).(
~
)()( ttzct
iiii
(11)
From (11) it follows that the function
)(
~
t
i
may
be rewritten in the form
).()()(
~
tzctt
iiii
(12)
Substituting (12) to (9), we have
).()()()(
11
tzcttxte
iiiii
(13)
Assume that function x
i + 1
(t) is a control signal in
(13) and define x
i + 1
(t) in the form x
i + 1
(t) = v
i
(t).
Since the function
is not available for
measurement, we introduce the second auxiliary
control law v
i
(t) as follows
),(
ˆ
)()( ttctv
iiii
(14)
where
is an estimate of the function
.
Substituting (14) to (13), we get
),()()(
11
ttecte
iiii
(15)
where
)(
ˆ
)()( ttt
iii
is an error estimate.
Step n. Since v
n
(t) is not real control law,
consider the n – 1-th error function e
n - 1
(t) = x
n
(t) –
v
n - 1
(t) in the form
),(
~
)()(
1
ttbute
nn
(16)
where
).()(
~
1
tvt
nnn
Introduce the n-th auxiliary loop in the form
),()()( tutzctz
nnn
(17)
where c
n
> 0 and
> 0 are coefficients chosen by a
designer.
Taking into account (16) and (17), rewrite the
function
n
(t) = e
n - 1
(t) – z
n
(t) as follows
),()(
~
)()( tubttzct
nnnn
(18)
From (18) it follows that the function
)(
~
t
n
may
be rewritten in the form
).()()(
~
tzctt
nnnn
(19)
Substituting (19) to (16), we get
).()()()(
1
tutzctte
nnnn
(20)
Since the function
)(t
n
is not available for
measurement, we introduce the control law u(t) as
follows
,)(
ˆ
)(
1
)( ttctu
nnn
(21)
where
)(
ˆ
t
n
is an estimate of the function
)(t
n
.
Substituting (21) to (20), we get
),()()(
11
ttecte
nnnn
(22)
where
)(
ˆ
)()( ttt
nnn
is an error estimate.
The signals
, i = 1, 2, ..., n are not available
for measurement, because it depends on derivatives
of x(t). Therefore, we introduced the estimate
functions
, i = 1, 2, ..., n of the functions
,
Compensation of Mismatched Disturbances for Nonlinear Plants with Distributed Time-delay
271
i = 1, 2, ..., n at each step. For implementation of
, i = 1, 2, ..., n use the following observers
)()(
ˆ
)(
ˆ
1
1
1
1
1
ttt
iii
,
,...,,2,1 ni
(23)
where
> 0 is enough small number.
Theorem: There exist constants c
i
> 0, i = 1, 2, ..., n
and µ
0
> 0 such that for µ ≤ µ
0
the control system
consisting of auxiliary loops (3), (10), (17), auxiliary
control laws (7), (14), control law (21), observers
(23) provides goal (2) for plant (1).
The proof of Theorem is given in Appendix.
It is shown that the proposed algorithm based on
multi-agent system design, where each equation is
associated as appropriate agent. The additional
investigation have shown that the proposed
algorithm is effective for dynamical networks with
unknown distributed time-delays and mismatched
parametric and external disturbances.
4 EXAMPLE
Consider the plant model with distributed time-delay
in the form
,,)()(
,,,)(),()()(
22
0
22121
12
tftbutx
tfdtxtxtxtx
h
(24)
where f = [f
1
, f
2
]
T
The set is defined by the
following inequalities:
–10 ≤ f
1
≤ 10, –10 ≤ f
2
≤ 10.
The problem is to synthesis the control system
providing goal condition (2).
Let us design the control system. Consider
auxiliary loops in the following form
,,
2222111
uzczxzcz
(25)
where c
1
= c
2
= 1 and
= 1.
Introduce the auxiliary control law v
1
(t) and the
control law u(t) as follows
,
ˆ
1
,
ˆ
2221111
cucv
(26)
where
1
(t) = x
1
(t) – z
1
(t),
2
(t) = e
1
(t) – z
2
(t) and
e
1
(t) = x
2
(t) – v
1
(t).
Introduce the observers in the forms
),(
1
)(
ˆ
),(
1
)(
ˆ
2211
t
p
p
tt
p
p
t
(27)
where p = d / dt, µ = 0.01.
Let all initial conditions be zero in the control
system.
Let us choose the functions of (24) as follows
.2,)(3.1sin)()(
,1,)()(sin5.01
22
0
2
2
2
2
122
12
0
1111
22
12
hdtxttxtxf
hdtxtxtf
h
h
(28)
For comparison we also consider the synthesis of
control systems using the backstepping algorithm
(Khalil, 2002) and the auxiliary loop algorithm
(Tsykunov, 2007). According to (Khalil, 2002) the
backstepping algorithm is presented by the
following equations
,)(
,
11212
1
12
1111
xcbecu
xcv
(29)
where e
1
(t) = x
2
(t) – v
1
(t). Algorithm (29) depends on
functions
i
, i = 1, 2, therefore, implementations of
(29) requires that functions
i
, i = 1, 2 must be
known.
According to (Tsykunov, 2007), the auxiliary
loop algorithm is presented by the following
equations:
equation of the auxiliary loop
),()(
21
10
)( tutxtx
aa
(30)
equation of the control law
),(10
1
)(21)( t
p
p
ttu
(31)
where
(t) = x(t) – x
a
(t). Algorithm (30), (31) can
compensate only function
2
and coefficient b and
cannot compensate function
1
. Therefore, algorithm
(30), (31) may be unstable for appropriate values of
the function
1
.
Consider two cases.
Case 1. Let f
1
= 1 and f
2
= 1 in (24) and functions
i
, i = 1, 2 are known.
Case 2. Let f
1
= 2 and f
2
= 3 in (24) and functions
i
, i = 1, 2 are unknown.
In Fig. 1-6 the transients are presented for the state
vector x(t) which is obtained by proposed MBADC,
the backstepping algorithm and the auxiliary loop
algorithm for each of two cases. In Fig.1-6 black
curve and red curve correspond to the signals x
1
(t)
and x
2
(t) respectively.
ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics
272
Figure 1: The transients of the vector state x(t) with
MBADC for case 1.
Figure 2: The transients of the vector state x(t) with
MBADC for case 2.
Figure 3: The transients of the vector state x(t) with the
backstepping algorithm for case 1.
Figure 4: The transients of the vector state x(t) with the
backstepping algorithm for case 2.
The analysis of simulation results shows that the
proposed MBADC guaranties the fulfillment of goal
(2). The backstepping algorithm is stable if all
functions in (24) are known, but the control system
is sensitive to parametric uncertainty and
external disturbances.
Figure 5: The transients of the vector state x(t) with the
auxiliary loop algorithm for case 1.
Figure 6: The transients of the vector state x(t) with the
auxiliary loop algorithm for case 2.
The auxiliary loop algorithm may be unstable
even we known all functions in the plant model.
We also can note that the proposed algorithm
(25)-(27) compensates parametric uncertainties and
external disturbances with the required accuracy
= 0.04 achieved after 1 s for any uncertainties from
the set . Additional investigation under saturated
control input is shown that the proposed algorithm is
stable while algorithms (Khalil, 2002; Tsykunov,
2007) lose stability. These results are similar for
multi-agent systems.
5 CONCLUSIONS
The paper describes the robust algorithm for
compensation of mismatched disturbances with
distributed time-delay. The synthesis of control
system based on the backstepping algorithm and the
auxiliary loop algorithm. The proposed algorithm
guarantees stabilization of the plant state vector with
the required accuracy. We also compare the
proposed algorithm with the backstepping algorithm
and auxiliary loop algorithm. The simulation results
illustrate the efficiency and robustness of the
suggested control system.
st,
st,
st,
st,
st,
st,
Compensation of Mismatched Disturbances for Nonlinear Plants with Distributed Time-delay
273
ACKNOWLEDGEMENTS
Research in Sections 1-3 and Appendix supported by
Russian Science Foundation (project no. 18-79-
10104) in IPME RAS. The reported study under
saturated control input in Section 4 was funded by
RFBR according to the research project № 17-08-
01266.
REFERENCES
Bobtsov A., Kremlev A.S., 2005. Observer synthesis in
the compensation problem for a finite-demensional
quasi-harmoniv disturbance. J. Computer and Systems
Sciences International, vol. 44, no. 3, pp. 331-337.
Chang Y.N. Cheng C.C., 2010. Block backstepping
control of multi-input nonlinear systems with
mismatched perturbations for asymptotic stability.
International Journal of Control, vol. 83, no. 10, pp.
2028-2039.
Fradkov A.L., Miroshnik I.V., Nikiforov V.O., 1999.
Nonlinear and Adaptive Control of Complex Systems.
Dordrecht: Kluwer.
Furtat I., Fradkov A., Tsykunov A., 2014. Robust
synchronization of linear dynamical networks with
compensation of disturbances. International Journal
of Robust and Nonlinear Control, vol. 24, no. 17, pp.
2774-2784.
Furtat I.B., Gushchin P.A, 2019. A Control Algorithm for
an Object with Delayed Input Signal Based on
Subpredictors of the Controlled Variable and
Disturbance. Automation and Remote Control, vol. 80,
no. 2, pp. 201-216.
Furtat I., Gushchin P., 2019. Tracking control algorithms
for plants with input time-delays based on state and
disturbance predictors and sub-predictors. Journal of
the Franklin Institute, vol. 356, pp. 4496-4512.
Khalil H.K., 2002. Nonlinear Systems. NJ: Prentice Hall.
Kokotovic P.V., 1992. The joy of feedback: nonlinear and
adaptive. IEEE Control Systems Magazine, vol. 12,
no. 3, pp. 7-17.
Ma L., Schilling K., Schmid C., 2005. Adaptive
backstepping sliding mode control with Gaussian
networks for a class of nonlinear systems with
mismatched uncertainties. 44th IEEE Conference on
Decision and Control & European Control
Conference, Seville, Spain, pp. 5504-5509.
Tsykunov A.M., 2007. “Robust control algorithms with
compensation of bounded perturbations,” Automation
and Remote Control, vol. 71, no.7, pp. 1213-1224.
Xu Z.B.,Min J.Q., 2010. Fuzzy Backstepping Control for
Strict-Feedback Nonlinear Systems with Mismatched
Uncertainties. 8th World Congress on Intelligent
Control and Automation (WCICA), Jinan, pp. 5054-
505.
APPENDIX
Lemma (Furtat, 2014; Furtat and Gushchin,
2019). Let the system be described by the following
differential equation
),,,(
21
txfx
, (32)
where
1
s
Rx
,
2
),(col
21
s
R
, f(x, µ
1
, µ
2
, t)
is Lipchitz continuous function in x. Let (34) have a
bounded closed set of attraction = {x | P(x) ≤ C}
for µ
2
= 0, where P(x) is piecewise-smooth, positive
definite function in
1
s
R
. In addition let there exist
some numbers C
1
> 0 and
0
1
such that the
following condition holds
.)(),0,,(,
)(
sup
11
T
11
CCxPtxf
x
xP
Then there exists µ
0
> 0 such that the system (32)
has the same set of attraction for µ
2
≤ µ
0.
Proof of Theorem. Taking into account (30),
rewrite the equations for the error estimates
)(
ˆ
)()( ttt
iii
, i = 1, 2, ..., n as follows
)()()(
1
ttt
iii
, i = 1, 2, ..., n. (33)
Rewrite (8), (15), (22) and (32) as the following
system
,...,,2,1),()()(
,1...,,2,1),()()(
),()()(
21
11
1111
nittt
njttecte
ttxctx
iii
jjjj
(34)
where µ
1
= µ
2
= µ. To analyze system (34) the
following Lemma is needed.
Let us check conditions of Lemma. Consider
system (34) for µ
2
=0. Let P(x) = V(t), where V(t) is
Lyapunov function defined in the form
n
i
i
n
j
i
ttetxtV
1
2
1
1
22
1
)(5.0)(5.0)(5.0)(
. (35)
Take the derivative of V(t) along the trajectories
(34), we get
.)()()()(
)()()()(
1
21
1
1
1
1
2
1
11
2
11
n
i
i
n
j
jjjj
tttetec
ttxtxctV
(36)
Find upper bounds for the fourth term of (36):
.1...,,2,1,5.05.0
2
10
21
01
njee
jjjj
(37)
Substituting (37) to (36), we get
ICINCO 2019 - 16th International Conference on Informatics in Control, Automation and Robotics
274
,
ˆ
ˆ
2
22
1
1
1
1
1
2
1
2
11
n
i
ii
n
j
jj
decxcV
(38)
where
1
011
5.0
ˆ
jj
cc
and
0
1
1
5.0
ˆ
i
d
.
Obviously, there exist coefficients c
j + 1
,
j = 1, 2, ..., n-1, µ
1
, and µ
0
such that
0
ˆ
1
j
c
,
0
ˆ
i
d
and system (34) is asymptotically stable.
Taking into account to (35), rewrite (38) as
follows
),()( tVtV
(39)
where
nn
ddccc
ˆ
...,,
ˆ
,,
ˆ
...,,
ˆ
,min2
2
1
121
.
Solving inequality (46) with respect to V(t), we
get
t
eVtV
)0()(
. (40)
From (40) it follows that solutions of system (33)
are exponentially tend to zero.
Proof boundendess of all signals in the closed-
loop system.
Taking into account (23), rewrite the first
equation of (4) in the following form
).(
)1)((
)(
1
2
1
11
1
tx
p
pcp
t
Since the function x
1
(t) is asymptotically stable
than the functions
1
(t),
,
and
0
2
1
)(
i
h
dtx
, i = 1, 2, ..., n are bounded. From
boundedness of
1
(t) it follows that the signal
is bounded. Taking into account (23) and system
(34), the proof of boundedness of the signals
i
(t),
,
and
, i = 2, 3, ..., n is same.
Therefore, from (7), (14) and (21) the signals v
i
(t),
i = 1, 2, ..., n – 1 and u(t) are bounded. Hence, the
function x(t) is bounded. From (3), (10) and (17) it
follows that the functions z
i
(t), i = 1, 2, ..., n are
bounded. Therefore, the functions
i
and
i
~
are
bounded. Consequently, all signals in the closed-
loop system are bounded.
According to Lemma there exist
0
> 0 such that
for
1
≤
0
and
2
≤
0
the attraction set is the same
as for
2
= 0. However, system (34) is not
asymptotically stable for
2
0. It will be has some
attraction set. Let us find the set of attraction of
system (34) for
2
0. Taking into account result
(38), take derivative in time of (35) along
trajectories (33) for
1
=
2
=
0
,
~
ˆ
12
22
1
1
0
1
1
2
1
2
11
n
l
ll
n
i
ii
n
j
jj
decxcV
(41)
where
.5.0
ˆ
0
1
0
i
d
Use the following upper bounds:
,)(5.05.0)(5.0
0
21
0
2
0
21
0
tt
lllll
(42)
where
2
...,2,1, ,
sup5.0
l
nlt
.
Taking into account (42), rewrite (41) in the
form
,5.0
ˆ
0
2
22
1
1
0
1
1
2
1
2
11
ndecxcV
n
i
i
n
j
jj
(43)
where
0
1
0
d
Taking into account (35), rewrite (43) as follows
0
)()( ntVtV
, (44)
where
dccc
n
,5.0,
ˆ
...,,
ˆ
,min2
1
021
.
Solving inequality (44) with respect to V(t), we
get
.1)0()(
0
1
neVetV
tt
(45)
From (45) we can note that goal (2) holds. The
theorem is proved.
Compensation of Mismatched Disturbances for Nonlinear Plants with Distributed Time-delay
275