This paper proposes a new adaptive neural network based control scheme for switched linear systems with parametric uncertainty and external disturbance. A key feature of this scheme is that the prior information of the possible upper bound of the uncertainty is not required. A feedforward neural network is employed to learn this upper bound. The adaptive learning algorithm is derived from Lyapunov stability analysis so that the system response under arbitrary switching laws is guaranteed uniformly ultimately bounded. A comparative simulation study with robust controller given in [Zhang L, Lu Y, Chen Y, Mastorakis NE. Robust uniformly ultimate boundedness control for uncertain switched linear systems. Computers and Mathematics with Applications 2008; 56: 1709–14] is presented.

1. ISA Transactions 51 (2012) 105–110
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Simulation and stability analysis of neural network based control scheme for
switched linear systems
H.P. Singh∗
, N. Sukavanam
Department of Mathematics, Indian Institute of Technology Roorkee (IITR), Roorkee-247667, Uttarakhand, India
a r t i c l e i n f o
Article history:
Received 7 April 2011
Received in revised form
18 July 2011
Accepted 18 August 2011
Available online 18 September 2011
Keywords:
Switched systems
Stability analysis
Feedforward neural network
Lyapunov function
Uniformly ultimate boundedness
Uncertainty
a b s t r a c t
This paper proposes a new adaptive neural network based control scheme for switched linear systems
with parametric uncertainty and external disturbance. A key feature of this scheme is that the prior
information of the possible upper bound of the uncertainty is not required. A feedforward neural network
is employed to learn this upper bound. The adaptive learning algorithm is derived from Lyapunov
stability analysis so that the system response under arbitrary switching laws is guaranteed uniformly
ultimately bounded. A comparative simulation study with robust controller given in [Zhang L, Lu Y, Chen
Y, Mastorakis NE. Robust uniformly ultimate boundedness control for uncertain switched linear systems.
Computers and Mathematics with Applications 2008; 56: 1709–14] is presented.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
A switched system is an important class of hybrid dynamical
system which consists of a collection of subsystems and a
switching law that decides which of the subsystem is active at
each moment. There are many applications of switched systems
in control of mechanical systems, process control, power systems,
traffic management, and many other practical control fields.
The motivation for studying switched systems is from the fact
that many practical systems are inherently multi-models. In last
two decades, there have been numerous control design methods
and stability analysis available in the literature for switched
systems [1–7].
A main problem which is always inherent in all dynamical
systems is the presence of uncertainties and external disturbances.
Even for switched systems this problem has been an active
area of research for many years. In [8], a robust H∞ control
scheme is proposed for switched linear systems with norm-
bounded time-varying uncertainties by using a multiple Lyapunov
function method. In [9], an adaptive robust stabilizing controller is
designed for a class of uncertain switched linear systems, in which
uncertainties satisfying the matching condition and switching
among subsystems are determined by using a nominal system.
∗ Corresponding author. Tel.: +91 8859866804; fax: +91 1332 273560.
E-mail addresses: harendramaths@gmail.com, hpsmaths@gmail.com
(H.P. Singh), nsukvfma@iitr.ernet.in (N. Sukavanam).
A new class of uncertain switched fuzzy systems is proposed
in [10]. In [11], an algorithm for robust adaptive control for
parametric-strict output feedback switched systems is developed.
This control scheme guarantees system stability for bounded
disturbances and parameters without requiring a priori knowledge
on such parameters or disturbances. In [12], L2-gain analysis and
control synthesis problem is designed for a class of uncertain
switched linear systems subject to actuator saturation and external
disturbances. The problem of optimal switching for a class of
switched systems with parameter uncertainty is proposed in [13].
The conditions for robust stabilization of this class of switched
systems with parameter uncertainty are presented based on a
multi-Lyapunov function technique and a Linear matrix inequality
technique. In [14], a continuous state feedback control scheme for
uncertain switched linear systems is proposed in which external
disturbance is not considered and the exact knowledge of the
possible upper bound of uncertainty is known a priori. However in
practical applications the calculation of this bound is difficult and
time consuming. Therefor further improvement is needed for this
type of control scheme.
Artificial neural network (ANN) can be defined as nonlinear
systems consisting of a number of interconnected processing
units or neurons. Due to their versatile features such as learning
capability, nonlinear mapping and parallel processing, ANN is
successfully applicable in real life problems arising in the areas
such as dynamics of economy, system control, robotics, etc. The
limitations of ANN come from its architecture and the accuracy of
the results vary depending on network quality. A European study
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.08.004

2. 106 H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110
on the Stimulation Initiative for European Neural Applications
(SIENA project) indicates ANN is used in 39% of the production or
manufacturing sectors, with 35% of the usage related to control,
monitoring, modeling and optimization [15]. In [16], the possibility
to apply neural network modeling for simulation and prediction
of the EU-27 global index of production and domestic output
price index behavior is investigated. The most useful property
of neural network in control is their ability to approximate
arbitrary linear or nonlinear functions through learning. Due to
this property neural networks have proven to be a suitable tool
for controlling complex nonlinear dynamical systems. The basic
idea behind neural network based control is to learn unknown
nonlinear dynamics and compensate for uncertainties existing in
the dynamic model. There are many NN based control schemes
available for a class of switched systems [17–19]. But the case of
uncertainty bound estimation using neural network is not found
in the literature for switched systems.
In this paper, we propose a new adaptive neural network
based control scheme for switched linear systems with parametric
uncertainties and external disturbances. The proposed control
scheme has the following salient features:
(i) prior information of the upper bound of the uncertainty is not
required,
(ii) the feedforward neural network is able to learn the upper
bound of uncertainty, and
(iii) uniformly ultimately bounded (UUB) stability of system re-
sponse and NN weight error are guaranteed under arbitrary
switching laws.
This paper is organized as follows. Some basics of stability analysis
are given in Section 2. In Section 3, a review of feedforward neural
network and a new robust adaptive controller is proposed. A
common Lyapunov function approach is used to show that system
response and NN weight error are all UUB. A simulation example is
provided in Section 4 to illustrate the effectiveness of the proposed
control scheme. Section 5 gives conclusions.
2. Preliminaries
Consider a switched linear system represented by a differential
equation of the form
˙x(t) = Aσ(t)(ω)x(t) + Bσ(t)(ω)u(t), (1)
σ(t) : R+
−→ S = {1, . . . , N}
where t ≥ 0, x = (x1, x2, . . . , xn) ∈ Rn
denotes the state vector of
the system. u(t) ∈ Rm
is the control input and R+
denotes the set
of nonnegative real numbers. σ(t) is a piecewise constant function
called a switching law, which indicates the active subsystem at
each instant. Ai(ω), Bi(ω), i = 1, . . . , N are matrices whose
elements are continuous functions of a time-varying vector ω on
a compact set Ω ⊂ Rq
. When switched system (1) has parametric
uncertainty and external disturbance then it can be written in the
following form
˙x(t) = ( ¯Ai + Ai(ω))x(t) + ( ¯Bi + Bi(ω))u(t) + d(t), (2)
where d(t) is the bounded external disturbance, ¯Ai, i =
1, . . . , N are commuting Hurwitz matrices and Ai(ω), Bi(ω) are
uncertainty terms which satisfy the following conditions [20,21]
Ai(ω) = ¯BDi(ω) (3)
Bi(ω) = ¯BEi(ω) (4)
I +
1
2
(Ei(ω) + ET
i (ω)) ≥ δI (5)
where δ is a positive constant, Di : Ω −→ Rm×n
and Ei : Ω −→
Rm×m
are continuous matrix functions.
Now consider the nominal case with u(t) = 0 and d(t) = 0
˙x(t) = Aσ(t)(ω)x(t). (6)
Then stability conditions for system (6) are given by the following
theorems [22].
Theorem 2.1. If {Ai : i = 1, . . . , N} is a set of commuting Hurwitz
matrices, then the switched linear system (6) is globally uniform
asymptotic stable for any arbitrary switching sequence between Ai.
Theorem 2.2. For a given symmetric positive definite matrix Q ,
let P1, P2, . . . , PN > 0 be the unique symmetric positive definite
solutions of the following equations
AT
1 P1 + P1A1 = −Q , (7)
AT
i Pi + PiAi = −Pi−1, (8)
with the condition of Theorem 2.1, then the function L = xT
PN x is a
common Lyapunov function for the switched linear system (6).
Because of the parametric uncertainty and external disturbance in
switched linear system (2), we cannot derive the state response
x(t) which exactly converges to the equilibrium point. Therefore
it is only reasonable to expect that the system response converges
into a neighborhood of the equilibrium point and remains within it
thereafter, which is the so-called uniformly ultimate boundedness.
In [23], the following uniformly ultimate boundedness is proposed
for dynamical systems.
Definition 2.1. The solution of a dynamical system is uniformly
ultimately bounded (UUB) if there exists a compact set S ⊂ Rn
such that for all x(t0) = x0 ∈ S, there exists a λ > 0 and a number
T(λ, x0) such that ‖x(t)‖ < λ for all t ≥ t0 + T.
Theorem 2.3. If there exists a function L with continuous partial
derivatives such that for x in a compact set S ⊂ Rn
, L is positive
definite and ˙L < 0 for ‖x‖ > R, R > 0 such that the ball of radius R
is contained in S, then the system is UUB and the norm of the state is
bounded to within some neighborhood of R.
3. Neural network based control design
In this section, our aim is to design a control input u(t) in such
a way that the switched linear system (2) is uniformly ultimately
bounded (UUB). The following robust control scheme
u(t) =



−
w
‖w‖
ρ, if ‖ω‖ > ϵ
−
w
ϵ
ρ, if ‖ω‖ ≤ ϵ
(9)
is proposed in [14], where w = ¯BT
PN ρx, ϵ > 0 is any constant and
ρ denotes the possible upper bound of the parametric uncertainty
defined as
ρ =
1
δ
max
i
max
ωϵΩ
‖Di(ω)‖ ‖x‖ (10)
which is assumed to be known.
However, there are some potential difficulties associated with
the robust controller (9). (i) Implementation requires a precise
knowledge of the uncertainty bound, (ii) if an uncertain switched
system has many subsystems, the computation of the uncertainty
bound will be a complex and time consuming task, and (iii) if a
new switched system comes in, the whole procedure has to start
again. Therefore, we introduce a new adaptive control approach to
avoid the requirement of a prior knowledge of the upper bound
ρ in the expression (10) by using a feedforward neural network
(FFNN). Mathematically, a two-layer feedforward neural network
(Fig. 1) with n input units, m output units and L units in the hidden
layer is given as

3. H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 107
zl =
L−
j=1

wljσ

n−
k=1
vjkxk + θvj

+ θwl

l = 1, 2, . . . , m (11)
where σ(.) are the activation functions of the neurons of the
hidden-layer [23]. The inputs-to-hidden-layer interconnection
weights are denoted by vjk and the hidden-layer-to-outputs
interconnection weights by wlj. The bias weights are denoted by
θvj, θwl. There are many classes of activation functions e.g. sigmoid,
hyperbolic tangent and Gaussian. The usual choice is the sigmoid
activation function, defined as
σ(x) =
1
1 + e−x
. (12)
By collecting all the NN weights vjk and wlj into matrices of weights
VT
and WT
, we can write the NN equation in terms of vectors as
z = WT
σ(VT
x) (13)
with the vector of activation functions defined by σ(y) = [σ(y1)
· · · σ(yn)]T
for a vector y ∈ Rn
. Without losing generality first-layer
weights are fixed and only the second-layer weights are tuned,
then the NN has only one layer of tunable weights. Then according
to [23–26], ρ can be written in neural network form as
ρ = WT
φ(x) + ρ
and an estimate of ρ can be given by
ˆρ = ˆWT
φ(x)
where ρ denotes NN approximation error, φ(.) is a basis matrix
and ˆW is the estimation of the weight matrix W provided by
some on-line weight tuning algorithms. For further analysis, the
following assumptions are selected as in [24,26,27].
Assumption 3.1. Given an arbitrary small positive constant ξ and
a continuous function ρ on a compact set Ω, there exist an optimal
vector W∗
such that
‖ ρ‖ = ‖W∗T
φ(x) − ρ‖ < ξ.
Assumption 3.2. The norm of parametric uncertainty µ defined as
µ = Di(ω)x and its upper bound satisfy the following relationship
ρ − ‖µ‖ > ξ.
For the design of a neural network based control input u(t) and the
weight vector ˆW, we have the following theorem.
Theorem 3.1. Consider the switched linear systems (2). If u(t)
in (2) is designed as
−
ˆρ2
w
ˆρ‖w‖ + ϵ
, (14)
Singh et al. [24], where ˙ϵ = −γ ϵ, ϵ(0) > 0, γ is a positive constant,
w = ¯BT
PN x and the adaptive neural network weight law is given as
˙ˆW = F‖w‖(φ(x) − κ ˆW),
with a positive definite diagonal gain matrix F, and a positive constant
κ, then the system response x and ˜W are uniformly ultimately
bounded (UUB), and x can be made small by a suitable choice of F
and κ.
Proof. Consider the following Lyapunov function candidate
L =
1
2
xT
PN x +
1
2
tr( ˜WT
F−1 ˜W) + k−1
ϵ (15)
where ˜W = W∗
− ˆW,
˙˜W = − ˙ˆW. Then, the time derivative of the
Lyapunov function is given as
˙L =
1
2
˙xT
PN x +
1
2
xT
PN ˙x + tr( ˜WT
F−1 ˙˜W) + k−1
˙ϵ
˙L =
1
2
[xT
( ¯Ai
T
+ AT
i ) + uT
(¯BT
+ BT
i )]PN x
+
1
2
xT
PN [( ¯Ai + Ai)x + (¯B + Bi)u] +
1
2
dT
PN x
+
1
2
xT
PN d + tr( ˜WT
F−1 ˙˜W) + k−1
˙ϵ.
Now using condition (3) and (4), we get
˙L =
1
2
[xT
( ¯Ai
T
PN + PN
¯Ai) + (¯BDi)T
PN + PN (¯BDi)]x
+
1
2
[uT
(¯BT
+ Ei
T ¯BT
)PN x + xT
PN (¯B + ¯BEi)u]
+
1
2
dT
PN x +
1
2
xT
PN d + tr( ˜WT
F−1 ˙˜W) + k−1
˙ϵ
˙L =
1
2
xT
( ¯Ai
T
PN + PN
¯Ai)x
+
1
2
xT
[(¯BDi)T
PN + PN (¯BDi)]x
−
ˆρ2
wT
2(ˆρ‖w‖ + ϵ)
(I + Ei)T ¯BT
PN x
−
ˆρ2
2(ˆρ‖w‖ + ϵ)
xT
PN
¯B(I + Ei)w +
1
2
dT
PN x
+
1
2
xT
PN d + tr( ˜WT
F−1 ˙˜W) + k−1
˙ϵ
using condition (5) with δ = 1 and condition (8) with ¯Ai
T
PN +
PN
¯Ai = −Ri, where Ri is a positive definite symmetric matrix, then
we have
˙L ≤ −
1
2
λmin(Ri)‖x‖2
−
ˆρ2
‖w‖2
ˆρ‖w‖ + ϵ
+ ‖w‖(‖µ‖ − ρ) + ‖w‖(W∗T
φ(x) + ρ)
+ tr( ˜WT
F−1 ˙˜W) + η‖PN ‖ ‖x‖ − ϵ
where η is the disturbance bound.
˙L ≤ −
1
2
λmin(Ri)‖x‖2
−
ˆρ2
‖w‖2
ˆρ‖w‖ + ϵ
+ ‖w‖(−ξ + ρ)
+ ‖w‖W∗T
φ(x) + tr( ˜WT
F−1 ˙˜W) + η‖PN ‖ ‖x‖ − ϵ
˙L ≤ −
1
2
λmin(Ri)‖x‖2
−
ˆρ2
‖w‖2
ˆρ‖w‖ + ϵ
+ ˆρ‖w‖
+ tr(‖w‖W∗T
φ(x) − ‖w‖W∗T
φ(x))
+ κ‖w‖tr ˜WT
(W∗
− ˜W) + η‖PN ‖ ‖x‖ − ϵ
˙L ≤ −
1
2
λmin(Ri)‖x‖2
+ κ‖w‖tr ˜WT
(W∗
− ˜W) + η‖PN ‖ ‖x‖.
Let the optimal weight W∗
is bounded by ‖W∗
‖F < Wmax on a
compact set Ω, where Wmax > 0. Then the following identity is
used in the derivation of the above inequality
tr ˜WT
(W∗
− ˜W) = ⟨ ˜W, W∗
⟩F − ‖ ˜W‖2
F
≤ ‖ ˜W‖F ‖W∗
‖F − ‖ ˜W‖2
F .
Then
˙L ≤ −
1
2
λmin(Ri)‖x‖2
+ κ‖¯BT
PN x‖ ‖ ˜W‖F (‖W∗
‖F − ‖ ˜W‖F )
+ η‖PN ‖ ‖x‖
˙L ≤ −‖x‖
[
1
2
λmin(Ri)‖x‖ + κ‖¯B‖ ‖PN ‖ ‖ ˜W‖F (‖ ˜W‖F
− ‖W∗
‖F ) − η‖PN ‖
]
.

4. 108 H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110
Fig. 1. Feedforward neural network.
Thus ˙L is negative as long as the term in brackets is positive.
Completing square terms, it is obtained that
1
2
λmin(Ri)‖x‖ + κ‖¯B‖ ‖PN ‖ ‖ ˜W‖F (‖ ˜W‖F − ‖W∗
‖F ) − η‖PN ‖
=
1
2
λmin(Ri)‖x‖ + κ‖¯B‖ ‖PN ‖

‖ ˜W‖F −
Wmax
2
2
−
κ
4
‖¯B‖ ‖PN ‖W2
max − η‖PN ‖.
Then RHS is positive if
‖x‖ >
κ‖¯B‖ ‖PN ‖W2
max + 4η‖PN ‖
2λmin(Ri)
(16)
or
‖ ˜W‖F >
Wmax
2
+

η
‖¯B‖κ
+
W2
max
4
. (17)
Thus ˙L is negative outside a compact set. According to the
Theorem 2.3, x and ˜W are UUB. Moreover, x can be made small by
a suitable choice of F and κ.
4. Simulation results
In this section, we consider a numerical example to demon-
strate the utility of the proposed control scheme for the switched
linear system (2) with the following two subsystems similar to [14]
A1(ω) =
[
0 1
−6 + ω2(t) −3 + ω1(t)
]
,
B1(ω) =
[
0
1.4387 + ω3(t)
]
,
A2(ω) =
[
0 1
−6.225 + ω2(t) −3 + ω1(t)
]
,
B2(ω) =
[
0
0.5613 + ω3(t)
]
,
where ω1(t) = 0.5 cos(t), ω2(t) = sin(t), ω3(t) = 0.25 sin(t).
¯Ai =
[
0 1
−6 −3
]
, ¯B =
[
0
1
]
,
d(t) =
[
0.1 sin(2t)
0.1 cos(2t)
]
,
D1(w) = [w2(t) w1(t)],
D2(w) = [−0.225 + w2(t) w1(t)],
Fig. 2. Response of state variable x1 with control design (9).
Fig. 3. Response of state variable x2 with control design (9).
E1(w) = 0.4387 + w3(t), E2(w) = −0.4387 + w3(t),
A1(ω) =
[
0 0
ω2(t) ω1(t)
]
,
B1(ω) =
[
0
0.4387 + ω3(t)
]
,
A2(ω) =
[
0 0
−0.225 + ω2(t) ω1(t)
]
,
B2(ω) =
[
0
−0.4387 + ω3(t)
]
which satisfies the condition (3) and (4). We choose a positive
definite matrix Q as
Q =
[
36 3
3 24
]
(18)
then the common Lyapunov solution matrix
PN =
[
17 3
3 5
]

5. H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110 109
Fig. 4. Response of state variable x1 with control design (14) for F = I10 and κ = 1.
Fig. 5. Response of state variable x2 with control design (14) for F = I10 and κ = 1.
can be easily obtained using the Lyapunov equations (7) and (8).
ρ = 1.15‖x‖ is selected as in [14] for comparison between the
robust controller (9) and our proposed controller (14).
The architecture of the FFNN is composed of 2 input units, 10
sigmoid units and a single output unit. The controller parameters
F, κ and γ in the proposed scheme are selected as I10, 0.01 and 0.01,
respectively. For simulation, we take a particular switched law σ(t)
defined as σ(t) = 1 for t < 30 and σ(t) = 2 for 30 ≤ t ≤ 70. The
simulation of the whole system is shown for 70 s. The effectiveness
of the FFNN to learn the upper bound of the switched linear system
uncertainty for producing the desired system response is shown
in the following figs. Response of state variables x1 and x2 are
shown in Figs. 2 and 3, respectively with known uncertainty bound
(i.e. with robust control design (9)). Figs. 4–5, Figs. 6–7 and Figs. 8–9
depict the response of the state variable x with neural network
based control design (14) for (F = I10, κ = 1), (F = 0.1I10, κ =
0.01) and (F = I10, κ = 0.01), respectively.
Fig. 6. Response of state variable x1 with control design (14) for F = 0.1I10 and
κ = 0.01.
Fig. 7. Response of state variable x2 with control design (14) for F = 0.1I10 and
κ = 0.01.
From Figs. 4–5 and Figs. 6–7, we see that the UUB region of the
state x is increasing for (F = I10, κ = 1) and (F = 0.1I10, κ =
0.01). In Figs. 8 and 9, the UUB region of the state x is small for
F = I10, κ = 0.01 in comparison to Figs. 2–7. In Figs. 4–9, for
the fixed γ , we observe that the state x is improved (decreased) by
increasing F and decreasing κ. Overall, we can say that the FFNN
based controller is quite effective without prior knowledge of the
upper bound of uncertainty, in comparison to the controller using
known upper bound of uncertainty as in [14].
5. Conclusion
In this paper, we investigated the uniformly ultimately bound-
edness of a switched linear system with parametric uncertainty
and external disturbance under arbitrary switched signals using a
Lyapunov function generated by the weighting matrices. A feed-
forward neural network is employed to learn the upper bound of

6. 110 H.P. Singh, N. Sukavanam / ISA Transactions 51 (2012) 105–110
Fig. 8. Response of state variable x1 with control design (14) for F = I10 and
κ = 0.01.
Fig. 9. Response of state variable x2 with control design (14) for F = I10 and
κ = 0.01.
uncertainty. The NN weights may be simply initialized to zero or
randomized. Finally, the numerical simulation is carried out for
two subsystems. The simulation results show that the feedforward
neural network with the on-line updating law can compensate the
switched linear system efficiently.
In our proposed controller (14) any prior knowledge of the
uncertainty bound is not required. This is the sharp contrast to
the control approach in [14], which has some potential difficulties
such as (i) implementation requires a prior knowledge of the
uncertainty bound, (ii) if an uncertain switched system has many
subsystems, the computation of the uncertainty bound will be a
complex and time consuming task, and (iii) if a new switched
system comes in, the whole procedure has to start again.
Acknowledgment
This work is financially supported by the Council of Scientific
and Industrial Research (CSIR), New Delhi, India and Department of
Science and Technology (DST), Government of India through grant
No. DST-347-MTD.
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