Input delay compensation of nonlinear stochastic systems with both state and input delays by prediction approach

Abstract

In this study, we present an extension of the prediction scheme (dynamic control) for nonlinear stochastic systems with both input and state delays. The stochastic system includes multiplicative noise and it is modelled as Ito stochastic differential equation. Input delay is considered to be equal or less than state delay and both delays are considered to be constant. At first, a new formula for prediction of the system's state is presented and then by means of this prediction vector, control input is constructed. To calculate the stabilising gain of the predictive controller, some sufficient delay-independent conditions in the form of linear matrix inequality (LMI) are presented. Finally, simulation examples are given to illustrate the effectiveness of the proposed approach.

Full text

May 1, 2017 International Journal of Systems Science tSYSguide
To appear in the International Journal of Systems Science
Vol. 00, No. 00, Month 20XX, 1–18
Input delay compensation of nonlinear stochastic systems with both state and
input delays by prediction approach
A. Javadi, M.R. Jahed-Motlagh∗and A.A. Jalali
Electrical Engineering Department, Iran University of Science and Technology, Narmak, Tehran, Iran
(Received 00 Month 20XX; final version received 00 Month 20XX)
In this study, we present an extension of the prediction scheme (dynamic control) for nonlinear stochastic
systems with both input and state delays. The stochastic system includes multiplicative noise and it is
modelled as Ito stochastic differential equation. Input delay is considered to be equal or less than state
delay and both delays are considered to be constant. At first, a new formula for prediction of the system’s
state is presented and then by means of this prediction vector, control input is constructed. To calculate
the stabilising gain of the predictive controller, some sufficient delay-independent conditions in the form
of linear matrix inequality (LMI) are presented. Finally, simulation examples are given to illustrate the
effectiveness of the proposed approach.
Keywords: Stochastic control; Delay compensation; Predictive control; Lyapunov stability; Linear
control systems
1. Introduction
Existing random effects as well as time delay in wide range of systems, justify the use of stochastic
delay differential equations (SDDE) for modeling their behavior. There is an intensive literature
in the area of stochastic systems, for example B. Chen, Niu, and Zou (2013a, 2013b); Song and
Niu (2016) a few to name. In the past years, control of stochastic delayed systems has been the
subject of many studies, and a great number of results on this subject have been reported in the
literature (see Gershon (2013); Ji and Qiu (2014); Li, Guan, and Luo (2011); Liang, Wang, and
Liu (2013); Y. Liu, Wang, and Liu (2007); Zhao and Xie (2014)).
Time delay can be considered in the state or input of the system. State delay indicates intrinsic
lag of the system’s behaviour and has been used to model variety of systems such as genetic
regulatory networks Y. Wang, Zhang, and Hu (2015) and biological systems M. Liu (2015). On
the other hand, input delay arise from external factors. Some of the main sources of input delay
are communication delay between the sensor and controller J. Liu, Liu, Xie, and Zhang (2011),
computational delay Liacu, Koru, Niculescu, and Andriot (2013) and actuator delay Du, Zhang,
and Lam (2008). Both state and input delay have destructive effect on system stability, however
input delay is more difficult to deal with than state delay Krstic (2009).
There are two general methods to confront the input delay compensation problem: static
(memory-less) and dynamic (memory) control. The simplest method used in the literature is static
controller which leads to transformation of the input delay to the state delay system and hence,
many results available in the field of state delay compensation can be utilised. It is well known
that for long input delay, delay-dependent LMIs are not feasible and therefore there are many
papers whose main goal is increasing the maximum allowable time delay Y. Chen, Xue, Zhao, and
∗Corresponding author. Email: jahedmr@iust.ac.ir

May 1, 2017 International Journal of Systems Science tSYSguide
Zhou (2009); He, Zhang, Wu, and She (2010); O. M. Kwon (2011). Static controllers with gains
coming from a delay-dependent LMI are not suitable for systems with long input delay due to the
infeasibility of LMI conditions.
The first alternative for overcoming this problem is smith predictor whose modified versions are
introduced in different forms such as finite spectrum assignment (FSA) Manitius and Olbrot
(1979), model reduction Artstein (1982); W. Kwon and Pearson (1980), truncated predictor
feedback Zhou, Lin, and Duan (2012) and prediction-based control using PDEs Krstic and
Smyshlyaev (2008). The main drawback of FSA and model reduction approaches is the lack of
Lyapunov stability analysis. This problem is solved with PDE modelling of input delay and using
backstepping transformation in Krstic and Smyshlyaev (2008). PDE-based predictive control
is easily extended to the systems with time-varying delay, nonlinear systems, output feedback
systems and adaptive systems, since exponential stability is achieved via a suitable Lyapunov
function Krstic (2009). It is noted that the structure of the FSA, model reduction and PDE-based
predictor controllers are the same and only their stability analysis are different. Distinguished
from these three methods, the structure of the controller in TPF approach does not involve infinite
dimensional integral term.
All aforementioned predictor controllers are presented only for deterministic systems. Very recently
a predictor-based controller is proposed for stochastic linear systems with input delay (constant
and time-varying) and external disturbance in Gershon, Fridman, and Shaked (2017). So far, this
is the only reported work on memory control of stochastic systems with input delay. In spite of its
name, the prediction of the state vector is not obtained in this paper and the traditional memory
controller for deterministic systems is used to deal with the delay compensation problem. On the
other hand, in Huang and Mao (2009), a robust static controller is suggested for stabilisation
of uncertain stochastic systems with state and input delay. Although considered delay can be
generalised to multiple and distributed delays, state and input delays are supposed to be equal. In
general, these delays may be different for an special system and hence the results in Huang and
Mao (2009) cannot be used.
To the best of our knowledge there is no related work (even static) on stabilisation of nonlinear
stochastic systems with state and input delays. The first contribution of this paper is the
introduction of a new formula which can be used to predict the state of nonlinear stochastic
systems with state and input delay. The second contribution of current work is appending an
extra term to the controller which adds some degrees of freedom to the problem and improves
the feasibility of the LMI conditions. Overall, the major novelty of this paper is extending the
idea of memory control to the nonlinear stochastic systems of Ito type. As mentioned earlier, the
only existing work on memory control of stochastic systems Gershon et al. (2017) has merely
considered linear systems with input delay.
In the following, we first introduce a new formula for obtaining the prediction vector of the
nonlinear stochastic system. Then, we present some new delay-independent LMI criteria for
nonlinear stochastic systems with sector-bounded nonlinearities and with both state and input de-
lay. Finally, three numerical examples are conducted to show the usefulness of the proposed method.
1.1. Notation
Through this paper, the following notations will be used. Let E(·) represents the mathematical
expectation and {Ω,F,Ft,P} be a complete probability space with a filtration {Ft}t≥0satisfying
the usual condition that it is right continuous and F0contains all P-null sets. Rnand R+
denotes the n-dimensional Euclidean space and the set of all positive real numbers respectively.
The superscript ”T” stands for matrix transposition. Iand 0are used to show the identity
and zero matrices with appropriate dimensions. The notation X > Y (respectively X≥Y)
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means that X−Yis positive definite (respectively positive semi-definite). |·|stands for the
Euclidean norm. ∇xyrepresents the derivative of vector ywith respect to vector xin denomina-
tor layout. N(µ, σ) denotes a stochastic variable with normal distribution of mean µand variance σ.
2. Problem Formulation
Consider the following delayed nonlinear stochastic differential equation:
dx(t)=[Ax(t) + D1x(t−τ1) + B1u(t−τ2) + f1(x(t)) + f2(x(t−τ1))] dt
+ [Cx(t) + D2x(t−τ1) + B2u(t−τ2) + g1(x(t)) + g2(x(t−τ1))] dw(t)
x(s) = φ(s), s ∈[−τ1,0] (1)
where x(t)∈Rnis the state vector, u(t)∈Rmis the input vector, τ1is the state delay, τ2is the
input delay, φ(t) is a real-valued initial function on [−τ1,0], matrices A,C,D1,D2,B1,B2are
constant real matrices with appropriate dimensions and w(t) is a scalar Wiener process defined on
the space {Ω,F,P} with
E[w(t)] = 0 , E[w2(t)] = t
In system (1), f1(.),f2(.),g1(.),g2(.) : Rn→Rnare sector-bounded nonlinearities satisfying the
following conditions:
[f1(x(t)) −F11x(t)]T[f1(x(t)) −F12 x(t)] 60
[f2(x(t−τ1)) −F21x(t−τ1)]T[f2(x(t−τ1)) −F22 x(t−τ1)] 60
[g1(x(t)) −G11x(t)]T[g1(x(t)) −G12 x(t)] 60
[g2(x(t−τ1)) −G21x(t−τ1)]T[g2(x(t−τ1)) −G22 x(t−τ1)] 60 (2)
where Fij and Gij for i, j = 1,2 are known constant matrices of appropriate dimensions. In addi-
tion, matrices F11 −F12,F21 −F22 ,G11 −G12 and G21 −G22 are all symmetric positive definite.
Remark 1: The sector-bounded nonlinear functions defined in (2) are more general than Lipschitz
nonlinearities and hence, cover a wider range of nonlinearities. Filtering and control problems for
time-delayed systems with sector bounded nonlinearities are extensively studied in the literature
Y. Liu et al. (2007); Y. Liu, Wang, and Wang (2011); Z. Wang, Liu, and Liu (2008).
Condition 1 : In this paper, we assume that input time delay is equal or less than state delay
i.e., τ2≤τ1.
In the literature, generally a static controller in the form of u(t) = Ksx(t) is chosen so that
delayed input becomes u(t−τ2) = Ksx(t−τ2), whose substitution in system (1) results in a
state delayed SDE. Then the gain Ksis determined such that the resulting closed loop system is
stabilised.
Notwithstanding the simplicity of static controllers’ structure, they are not applicable to the
systems with long input delay. In order to stabilise the system with long input delay, it is logical
to predict the state of the system and use it for feedback. Generally, prediction of the state vector
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is obtained through Variation of Constants Formula (VCF). In the case of deterministic systems,
VCF is valid for different system parameters and can be used for prediction of the state vector
as done in Krstic (2009). However, As mentioned in Mao (2011), VCF for stochastic system
(1) is valid only when matrices Aand Ccommute namely, AC =CA. In the following, we
present a new formula to predict the future of the state for system (1) without the limitation above.
Lemma 2.1: Under condition 1, prediction vector for system (1) in input time delay horizon is
given by:
z(t) = x(t+τ2) = eAτ2x(t)
+Zt
t−τ2
eA(t−s)[D1x(s−τ1+τ2) + B1u(s)] ds
+Zt
t−τ2
eA(t−s)[f1(z(s)) + f2(x(s+τ2−τ1))] ds
+Zt
t−τ2
eA(t−s)[Cz(s) + B2u(s) + D2x(s−τ1+τ2)] dw(s+τ2)
+Zt
t−τ2
eA(t−s)[g1(z(s)) + g2(x(s+τ2−τ1))] dw(s+τ2) (3)
Proof. Pre-multiplying equation (1) by e−Atand using stochastic product rule, we have:
d[e−Atx(t)] = e−Atdx(t)−e−AtAx(t)dt =
=e−At[D1x(t−τ1) + B1u(t−τ2)] d(t)
+e−At[f1(x(t)) + f2(x(t−τ1))] d(t)
+e−At[Cx(t) + D2x(t−τ1) + B2u(t−τ2)] dw(t)
+e−At[g1(x(t)) + g2(x(t−τ1))] dw(t) (4)
Integrating both sides of equation (4) yields:
Zt+τ2
t
d[e−Asx(s)] = e−A(t+τ2)x(t+τ2)−e−Atx(t)
=Zt+τ2
t
e−As[D1x(s−τ1) + B1u(s−τ2)]d(s)
+Zt+τ2
t
e−As[f1(x(s)) + f2(x(s−τ1))]d(s)
+Zt+τ2
t
e−As[Cx(s) + D2x(s−τ1) + B2u(s−τ2)]dw(t)
+Zt+τ2
t
e−As[g1(x(s)) + g2(x(s−τ1))]dw(t) (5)
Eventually, changing the limits of the integrals and pre-multiplying both sides of equation (4) by
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eA(t+τ2), we get:
x(t+τ2) = eAτ2x(t) + Zt
t−τ2
eA(t−s)[D1x(s−τ1+τ2) + B1u(s)]ds
+Zt
t−τ2
eA(t−s)[f1(x(s+τ2)) + f2(x(s−τ1+τ2))]ds
+Zt
t−τ2
eA(t−s)[Cx(s+τ2) + D2x(s−τ1+τ2) + B2u(s)]dw(s+τ2)
+Zt
t−τ2
eA(t−s)[g1(x(s+τ2)) + g2(x(s−τ1+τ2))]dw(s+τ2) (6)
Consequently, the proof is complete.
Remark 2: Consider a special case of system (1) without delayed nonlinear terms i.e.,
f2(x(t−τ1)) = g2(x(t−τ1)) = 0. With the same lines of Lemma 2.1, it is easy to show that
in this case, the prediction vector can be obtained from the following formula:
z(t) = x(t+τ2) = eAτ2x(t)
+Zt
t−τ2
eA(t−s)[D1x(s−τ1+τ2) + B1u(s) + f1(z(s))] ds
+Zt
t−τ2
eA(t−s)[Cz(s) + B2u(s) + D2x(s−τ1+τ2) + g1(z(s))] dw(s+τ2) (7)
Remark 3: Another special case of stochastic system (1) is linear stochastic systems with state
and input delays i.e., f1(x(t)) = g1(x(t)) = f2(x(t−τ1)) = g2(x(t−τ1)) = 0. In this case,
prediction vector is given by:
z(t) = x(t+τ2) = eAτ2x(t) + Zt
t−τ2
eA(t−s)[D1x(s−τ1+τ2) + B1u(s)]ds
+Zt
t−τ2
eA(t−s)[Cz(s) + B2u(s) + D2x(s−τ1+τ2)] dw(s+τ2) (8)
Now we are ready to construct the control input structure. Let the control input to have the
following structure:
u(t) = Kz(t) + Lx(t−τ1+τ2) (9)
Second term in (9) is included to help the feasibility of the LMI, as the gain Ladds some degrees
of freedom to the problem. Moreover, Condition 1 must be held so that the predictive control law
(9) can be implemented. Note that Condition 1 is also necessary for prediction vectors (3),(7) and
(8) to be implementable.
With respect to Lemma 2.1, delayed control input is
u(t−τ2) = Kx(t) + Lx(t−τ1) (10)
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Substituting (10) in (1), we have:
dx(t) = [(A+B1K)x(t)+(D1+B1L)x(t−τ1) + f1(x(t)) + f2(x(t−τ1))] dt
+ [(C+B2K)x(t)+(D2+B2L)x(t−τ1) + g1(x(t)) + g2(x(t−τ1))] dw(t) (11)
As equation (11) indicates, the input delay in (1) is eliminated instead, prediction vector introduces
a new dynamic to the system which should be figured out. Corresponding SDE of prediction vector
(3) can be calculated by Ito formula as:
dz(t) =∇tz(t)dt +∇T
xz(t)dx(t)
+1
2dxT(t)∇2
xz(t)dx(t)(12)
where
∇T
xz(t) = eAτ2,∇2
xz(t) = 0 (13)
∇tz(t) = D1x(t−τ1+τ2) + B1u(t) + f1(z(t)) + f2(x(t−τ1+τ2))
−eAτ2[D1x(t−τ1) + B1u(t−τ2) + f1(z(t−τ2)) + f2(x(t−τ1))]
+AZt
t−τ2
eA(t−s)[D1x(s−τ1+τ2) + B1u(s)] ds
+AZt
t−τ2
eA(t−s)[f1(z(s)) + f2(x(s−τ1+τ2))] ds
+ [Cz(t) + D2x(t−τ1+τ2) + B2u(t)] dw(t+τ)
dt
+ [g1(z(t)) + g2(x(t−τ1+τ2))] dw(t+τ)
dt
−eAτ2[Cz(t−τ2) + D2x(t−τ1) + B2u(t−τ2)] dw(t)
dt
−eAτ2[g1(z(t−τ2)) + g2(x(t−τ1))] dw(t)
dt
+AZt
t−τ
eA(t−s)[Cz(s) + D2x(s−τ1+τ2) + B2u(s)] dw(s+τ2)
+AZt
t−τ
eA(t−s)[g1(z(s)) + g2(x(s−τ1+τ2))] dw(s+τ2) (14)
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May 1, 2017 International Journal of Systems Science tSYSguide
Using Lemma 2.1 and substituting (9) and (10) in (14), we get
∇tz(t)=(A+B1K)z(t)−eAτ2(A+B1K)x(t)
+ (D1+B1L)x(t−τ1+τ2)−eAτ2(D1+B1L)x(t−τ1)
+f1(z(t)) + f2(x(t−τ1+τ2))
−eAτ2[f1(x(t)) + f2(x(t−τ1))]
+ [(C+B2K)z(t)+(D2+B2L)x(t−τ1+τ2)] dw(t+τ2)
dt
+ [g1(z(t)) + g2(x(t−τ1+τ2))] dw(t+τ2)
dt
−eAτ2[(C+B2K)x(t)+(D2+B2L)x(t−τ1)] dw(t)
dt
−eAτ2[g1(x(t)) + g2(x(t−τ1))] dw(t)
dt (15)
Knowing that z(t−τ1) = x(t−τ1+τ2) and by substituting (11), (13) and (15) in (12), stochastic
differential equation of prediction vector is obtained as
dz(t) = [(A+B1K)z(t)+(D1+B1L)z(t−τ1) + f1(z(t)) + f2(z(t−τ1))] dt
+ [(C+B2K)z(t)+(D2+B2L)z(t−τ1) + g1(z(t)) + g2(z(t−τ1))] dw(t+τ2) (16)
SDE of the prediction vector (16) is very similar to the SDE of the state vector (12) except
that the time index is shifted τ2seconds. This means that stability analysis of these two SDEs
can be incorporated using a single Lyapunov function. In other words, if a suitable Lyapunov
function is found to insure the stability of the SDE (12), then the same Lyapunov function
can be used to guarantee the stability of the SDE (16). Consequently, only one of these SDEs
will be used for the stability analysis which insures both state and prediction vector to be bounded.
Remark 4: Although the prediction vectors (3),(7) and (8) include the future values of Wiener
process however, it is not required to compute the future values of Wiener process. It is well
known that the Wiener process is a stochastic process with stationary independent increments i.e.
w(t+ ∆t)−w(t)∼N(0,∆t) for any t∈R+Allen (2007). In other words, time shifting does not
affect the probability distribution function of the Wiener difference:
dw(s+τ2) = w(s+τ2+ ∆t)−w(s+τ2)∼N(0,∆t)
dw(s) = w(s+ ∆t)−w(s)∼N(0,∆t)
where ∆tis a sufficiently short time interval for numerical evaluation of stochastic integral. As
a result, stochastic integral in (3),(7) and (8) can be numerically computed by dividing the time
interval into the subintervals of equal width ∆tand producing a standard Wiener process commen-
surate with the ∆tand then using a prevalent algorithm for numerical evaluation of the definite
integral such as rectangle method.
Before continuing, we present the following definition and lemma which will be used in our
derivation.
Lemma 2.2: [Zhang (2005)] Given constant matrices Ω1,Ω2and Ω3satisfying Ω1= ΩT
1and
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May 1, 2017 International Journal of Systems Science tSYSguide
Ω2≥0, then
Ω1+ ΩT
3Ω−1
2Ω3<0
if and only if
Ω1ΩT
3
Ω3−Ω2<0
Definition 1: [Kolmanovskii and Myshkis (1999)] The trivial solution of system (11) is said to be
asymptotically stable in mean square, if for each ε > 0 there is a δ > 0 such that x(t) exists for
t0≤t < ∞and E|x(t)|< ε, if only
sup
−τ1≤s≤0
E|φ(s)|2< δ(ε) , sup
−τ1≤s≤0
E|φ(s)|4<∞
and for all initial functions satisfying above conditions we have
lim
t→∞ E|x(t)|2= 0
3. Main results
To this end, we converted the stability problem of SDDE (1) with memory controller (9) to the
stability analysis of SDDE (11) or (16). In this section, we study the asymptotic stability of the
system (11) (and therefore system (1)) based on Lyapunov technique.
Theorem 3.1: Under condition 1, the nonlinear stochastic delayed system (1) with memory con-
troller (9) using prediction vector (3) is asymptotically stable in mean square if there exist positive
definite matrices Xand Pand matrices Y1and Y2such that the following LMI holds
Ψ =














Π∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
XDT
1+YT
2BT
1−P∗ ∗ ∗ ∗ ∗ ∗ ∗
I−¯
FT
12X 0 −I∗ ∗ ∗ ∗ ∗ ∗
I−¯
FT
22X 0 −I∗ ∗ ∗ ∗ ∗
−¯
GT
12X 0 0 0 −I∗ ∗ ∗ ∗
−¯
GT
22X 0 0 0 0 −I∗ ∗ ∗
CX +B2Y1D2X+B2Y20 0 I I −X∗ ∗
X 0 0 0 0 0 0 Λ∗
0 X 0 0 0 0 0 0 Σ














<0(17)
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where
Π = AX +XAT+B1Y1+YT
1BT
1+P
Λ = ¯
F11 +¯
G11−1
Σ = ¯
F21 +¯
G21−1
¯
F11 =FT
11F12 +FT
12F11 2
¯
F12 =−FT
11 +FT
122
¯
F21 =FT
21F22 +FT
22F21 2 (18)
¯
F22 =−FT
21 +FT
222
¯
G11 =GT
11G12 +GT
12G11 2
¯
G12 =−GT
11 +GT
122
¯
G21 =GT
21G22 +GT
22G21 2
¯
G22 =−GT
21 +GT
222
In this case, the stabilising gains of the memory controller (9) are obtained as
K=Y1X−1,L=Y2X−1
Proof. Consider the following Lyapunov function for the closed loop system
V(t) = xT(t)Qx(t) + Zt
t−τ1
xT(s)¯
Px(s)ds (19)
Differential of the Lyapunov function (19) can be calculated by stochastic calculus as
dV (t) = LV(t)dt
+ 2xT(t)Q[(C+B2K)x(t)+(D2+B2L)x(t−τ1) + g1(x(t)) + g1(x(t−τ1))] dw(t) (20)
Infinitesimal generator of Lyapunov function (19) along the closed loop system (11) can be obtained
as
LV(t) = xT(t)Q[(A+B1K)x(t)+(D1+B1L)x(t−τ1)]
+xT(t)Q[f1(x(t)) + f2(x(t−τ1))]
+ [(A+B1K)x(t)+(D1+B1L)x(t−τ1)]TQx(t)
+ [f1(x(t)) + f2(x(t−τ1))]TQx(t)
+(C+B2K)x(t)+(D2+B2L)x(t−τ1)
+g1(x(t)) + g2(x(t−τ1)) T
Q
×(C+B2K)x(t)+(D2+B2L)x(t−τ1)
+g1(x(t)) + g2(x(t−τ1)) 
+xT(t)¯
Px(t)−xT(t−τ1)¯
Px(t−τ1) (21)
Defining new vector ξ=xT(t)xT(t−τ1)fT
1(x(t)) fT
2(x(t−τ1)) gT
1(x(t)) gT
2(x(t−τ1)) T,
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May 1, 2017 International Journal of Systems Science tSYSguide
infinitesimal generator can be rewritten in the quadratic form LV(t) = ξT(t)Ψ1ξ(t) where
Ψ1=








Π1∗ ∗∗∗∗
(D1+B1L)TQ−¯
P∗ ∗ ∗ ∗
Q 0 0 ∗∗∗
Q 0 0 0 ∗ ∗
0 0 0 0 0 ∗
0 0 0 0 0 0








+ ΥTQΥ (22)
Υ = C+B2K D2+B2L00II
Π1= (A+B1K)TQ+Q(A+B1K) + ¯
P
The sector bounded nonlinearities defined in (2) can be written in the following form:
xT(t)fT
1(x(t)) ¯
F11 ¯
F12
¯
FT
12 I x(t)
f1(x(t)) 60
xT(t)gT
1(x(t)) ¯
G11 ¯
G12
¯
GT
12 I x(t)
g1(x(t)) 60
(23)
xT(t−τ1)fT
2(x(t−τ1)) ¯
F21 ¯
F22
¯
FT
22 I x(t−τ1)
f2(x(t−τ1)) 60
xT(t−τ1)gT
2(x(t−τ1)) ¯
G21 ¯
G22
¯
GT
22 I x(t−τ1)
g2(x(t−τ1)) 60
(24)
where ¯
Fij and ¯
Gij for i, j = 1,2 are defined in (18).
Now, from (23) and (24) we can write
LV(t)6LV(t)−xT(t)fT
1(x(t)) ¯
F11 ¯
F12
¯
FT
12 I x(t)
f1(x(t)) 
−xT(t−τ1)fT
2(x(t−τ1)) ¯
F21 ¯
F22
¯
FT
22 I x(t−τ1)
f2(x(t−τ1)) 
−xT(t)gT
1(x(t)) ¯
G11 ¯
G12
¯
GT
12 I x(t)
g1(x(t)) 
−xT(t−τ1)gT
2(x(t−τ1)) ¯
G21 ¯
G22
¯
GT
22 I x(t−τ1)
g2(x(t−τ1)) 
=ξTΨ2ξ(25)
where
Ψ2=








Π2∗ ∗∗∗∗
(D1+B1L)TQ−¯
P−¯
FT
21 −¯
GT
21 ∗∗∗∗
Q−¯
FT
12 0−I∗∗∗
Q−¯
FT
22 0−I∗ ∗
−¯
GT
12 0 0 0 −I∗
−¯
GT
22 0 0 0 0 −I








+ ΥTQΥ
Π2= (A+B1K)TQ+Q(A+B1K) + ¯
P−¯
F11 −¯
G11 (26)
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May 1, 2017 International Journal of Systems Science tSYSguide
According to Lemma 2.2, negative definitness of Ψ2is equivalent to negative definiteness of
Ψ3=










Π2∗ ∗ ∗ ∗ ∗ ∗
(D1+B1L)TQ−¯
P−¯
FT
21 −¯
GT
21 ∗∗∗∗ ∗
Q−¯
FT
12 0−I∗ ∗ ∗ ∗
Q−¯
FT
22 0−I∗ ∗ ∗
−¯
GT
12 0 0 0 −I∗ ∗
−¯
GT
22 0 0 0 0 −I∗
C+B2K D2+B2L 0 0 I I −Q−1










(27)
Denote Θ = diag Q−1,Q−1,I,I,I,I,Iand let
Ψ4= ΘTΨ3Θ
=










Π3∗ ∗ ∗ ∗ ∗ ∗
Q−1(D1+B1L)TΠ4∗∗∗∗ ∗
I−¯
FT
12Q−10−I∗ ∗ ∗ ∗
I−¯
FT
22Q−10−I∗ ∗ ∗
−¯
GT
12Q−10 0 0 −I∗ ∗
−¯
GT
22Q−10 0 0 0 −I∗
(C+B2K)Q−1(D2+B2L)Q−10 0 I I −Q−1










(28)
where
Π3=Q−1(A+B1K)T+ (A+B1K)Q−1+Q−1¯
PQ−1−Q−1¯
F11 +¯
G11Q−1
Π4=−Q−1¯
PQ−1−Q−1¯
FT
21 +¯
GT
21Q−1
Defining new matrices X=Q−1,P=Q−1¯
PQ−1,Y1=KQ−1and Y2=LQ−1, (28) can be
rewritten as
Ψ4=










Π∗ ∗ ∗ ∗ ∗ ∗
XDT
1+YT
2BT
1−P∗∗∗∗ ∗
I−¯
FT
12X 0 −I∗ ∗ ∗ ∗
I−¯
FT
22X 0 −I∗ ∗ ∗
−¯
GT
12X 0 0 0 −I∗ ∗
−¯
GT
22X 0 0 0 0 −I∗
CX +B2Y1D2X+B2Y20 0 I I −X










−∆T
1¯
F11 +¯
G11∆1−∆T
2¯
FT
21 +¯
GT
21∆2(29)
where
∆1=X000000
∆2=0X00000
Applying Lemma 2.2 to Ψ4<0 two times directly leads to the LMI (17) which means that negative
definiteness of Ψ is equivalent to negative definiteness of Ψ4. Ψ4<0 is equivalent to Ψ3<0 and
this is equivalent to Ψ2<0. Ψ2<0 results in LV(t)<0 and it then follows from Kolmanovskii and
Myshkis (1999) that the closed loop system is asymptotically stable in mean square. As discussed
earlier, stability of closed loop system (11) guarantees stability of the original system (1) controlled
by memory feedback (9) with prediction vector (3) and hence, the proof is complete.
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May 1, 2017 International Journal of Systems Science tSYSguide
Corollary 3.2: Under condition 1 and assuming f2(x(t−τ1)) = 0and g2(x(t−τ1)) = 0, the
nonlinear stochastic delayed system (1) with memory controller (9) using prediction vector (7) is
asymptotically stable in mean square if there exist positive definite matrices Xand Pand matrices
Y1and Y2such that the following LMI holds
Ξ =








Π∗ ∗ ∗ ∗ ∗
XDT
1+YT
2BT
1−P∗ ∗ ∗ ∗
I−¯
FT
12X 0 −I∗ ∗ ∗
−¯
GT
12X 0 0 −I∗ ∗
CX +B2Y1D2X+B2Y20 I −X∗
X 0 0 0 0 Λ








<0(30)
In this case, the stabilising gains of the memory controller (9) are obtained as
K=Y1X−1,L=Y2X−1
Proof. Since we have F21 =F22 =G21 =G22 =0, from (18) we see that ¯
F21 =¯
G21 =¯
F22 =
¯
G22 =0. Keeping these in mind, the proof is similar to the proof of Theorem 3.1 except that after
equation (21), ξshould be defined as ξ=xT(t)xT(t−τ1)fT
1(x(t)) gT
1(x(t)) T. In addition,
only inequalities (23) are subtracted from the right side of inequality (25). Continuing the rest of
the proof analogous to the procedure in the proof of Theorem 3.1, leads to the LMI (30) and hence
the proof is complete.
Corollary 3.3: Under condition 1 and assuming the linear case i.e., f1(x(t)) = 0,f2(x(t−τ1)) =
0,g1(x(t)) = 0,g2(x(t−τ1)) = 0, the stochastic delayed system (1) with memory controller (9)
using prediction vector (8) is asymptotically stable in mean square if there exist positive definite
matrices Xand Pand matrices Y1and Y2such that the following LMI holds
Γ = 

Π∗ ∗
XDT
1+YT
2BT
1−P∗
CX +B2Y1D2X+B2Y2−X

<0(31)
In this case, the stabilising gains of the memory controller (9) are obtained as
K=Y1X−1,L=Y2X−1
Proof. The proof is the same as proof of Theorem 3.1 except the following changes. Since all
nonlinear functions are set to zero, after equation (21), we define ξ=xT(t)xT(t−τ1)T. In
addition, subtracting step defined in (25) is eliminated. Applying these changes, with the same
lines of the proof of Theorem 3.1, it easy to obtain LMI (31).
Remark 5: The static controller has a simple structure and hence can be easily implemented in
practice. The predictor feedback, however, involves stochastic integration and saving past data to
be used for predicting state’s future values. In cases where high performance is required or the
input time delay is longer (and therefore delay-dependent LMI for static controller is not feasible),
the dynamic controller may be a better choice whereas for general purposes or short time delays
simple static controller has higher priority in the case of LMI’s feasibility.
Remark 6: The sufficient conditions (17), (30) and (31) are delay-independent and hence they
result in conservative designs for their corresponding problems.
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May 1, 2017 International Journal of Systems Science tSYSguide
Remark 7: In this paper both state and input delays are considered to be constant however, in
practice, time delay may vary in time. Extension of the proposed approach to the systems with
time varying delays is not very straightforward. The main challenge is obtaining the prediction
vector for this case. If input delay varies with time, the limits of the integrals cannot be changed as
done from equation (5) to (6). In addition, existence of time varying state delay in the Lyapunov
function will lead to an infinitesimal generator containing derivative of the state delay. This means
that the resulting sufficient condition will be rate-dependent LMI condition rather than delay- and
rate-independent conditions obtained in this paper.
4. Numerical Examples
In this section, we provide some numerical examples to illustrate the theoretical results.
Example 1: Consider a nonlinear stochastic system with the following parameters:
A=

−4.5 1.1 0.7
−0.3−4.5 0.5
0 0.6 0

,D1=

1.2−0.5 0.7
0.3−1.2 0.4
−0.5−1.2−0.3


C=

−0.4 0.5 0.6
−0.5 0.7 0.6
−0.5−0.6 0.8

,D2=

−0.9 0.4 0.6
−0.5 0.6 0.4
0.7−0.6 0.7


B1=

1 0.2
0.2 0.1
0.3−6

,B2=

0.2 0.1
0.2 0.4
0.3−0.2


f1(x) = 

0.5x1−0.2x2+ 0.1x3+ 0.2x1sin x2
−0.2x1+ 0.5x2+ 0.1x3+ 0.2x3sin x2
0.1x1+ 0.1x2+ 0.4x3+ 0.2x2sin x3


f2(x(t−τ1)) = g1(x(t)) = g2(x(t−τ1)) = 0
(32)
These parameters are taken from Y. Liu et al. (2007). Uncertainties of the system matrices, dis-
turbance and distributed delay considered in Y. Liu et al. (2007) are omitted here instead, input
delay is added to the dynamic. It is easy to check that the nonlinear vector function f1(x) in (32)
belongs to the sector [F11,F12] where
F11 =

0.6−0.2 0.1
−0.1 0.5 0.1
0.1 0.1 0.4

,F12 =

−0.4 0.2−0.1
0.3−0.5−0.1
−0.1−0.1−0.4

,G11 =G12 =0(33)
Substituting matrices (33) in (18) we get
¯
F11 =

−0.28 0.19 −0.07
0.19 −0.3−0.07
−0.07 −0.07 −0.18

,¯
F12 =

−0.1−0.1 0
0 0 0
0 0 0


Λ = 

−12.532 −9.9796 8.7545
−9.9796 −11.613 8.3971
8.7545 8.3971 −12.226


(34)
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May 1, 2017 International Journal of Systems Science tSYSguide
Applying Corollary 3.2 and solving LMI (30) using parameters (32) and (34), we obtain
X=

2.9775 0.9287 0.2064
0.9287 2.7617 −0.426
0.2064 −0.426 3.7984


P=

8.7938 0.6821 −0.811
0.6821 7.0647 −0.7016
−0.811 −0.7016 7.0677


Y1=3.6684 4.661 −10.494
−1.1535 0.67351 0.78053 
Y2=−1.5796 0.9974 −4.4394
−0.49166 −0.5817 −0.3425 
Therefore, the stabilising gains of the predictive controller are obtained as
K=1.146 0.8818 −2.7261
−0.5557 0.4753 0.289 
L=−0.575 0.3857 −1.0942
−0.0971 −0.1944 −0.1067 
Suppose that the state and input delays to be τ1= 1 and τ2= 0.5 respectively. Moreover, assume
that initial function of all states and prediction vector to be one and zero respectively. In this
case, a sample path of the system states is shown in Figure 1. Step size of numerical evaluation for
approximating integrals involved in controller structure is supposed to be ∆t= 0.01.
0 2 4 6 8 10 12 14 16 18 20
−0.5
0
0.5
1
1.5
Time(s)
x1(t)
x2(t)
x3(t)
Figure 1. A sample path of the states of nonlinear system in example 1 controlled by predictive controller. Solid: x1(t),
Dash-dot: x2(t) and Dot: x3(t).
Example 2: Consider a stochastic delayed differential equation with parameters below:
A=0 0
0 1 ,D1=−1−1
0−0.9,B1=−0.2
−0.1
C=0.01 0
0−0.01 ,D2=0.01 0.01
0−0.01 ,B2=0
0
τ1=τ2
(35)
In C. Wang and Shen (2014), a robust static controller is designed for above system however,
some delay free inputs are let to be presented. Here we assume the same parameters except the
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May 1, 2017 International Journal of Systems Science tSYSguide
delay free input vector and aim to find the stabilising predictive controller such that the closed
loop system becomes asymptotically stable in mean square. we will compare our results with a
static controller presented in Huang and Mao (2009) which ensures the mean square exponential
stability of the system (1) albeit, under the condition of equal input and state delays. To have a
fair comparison, upper bounds of uncertainties are set to zero. By Theorem 2 in Huang and Mao
(2009) with δM= 1, δR= 0.0001, δS= 100 and δQ= 3, the maximum allowable time delay is
τ1=τ2= 0.649. Free design parameters δM,δR,δSand δQare changed by trial and error but
there was no improvement in the maximum allowable delay for static controller. For parameters
(35), LMI (31) in Corollary 3.3 is feasible and the corresponding gains of the static and dynamic
controllers are
Ks=−0.27518 3.6448 
K=−288.51 750.36 
L=−2.9321 −8.4863 
where static controller is given by u(t) = Ksx(t). The system response for predictor controller
and memory-less controllers are shown in Figure 2 for τ1=τ2= 0.75. It is evident that transient
response of the dynamic controller is better than static controller. In Figure 3, the state and
prediction vectors for the same simulation are shown. This figure clarifies that the prediction
vector is always 0.75 seconds ahead of the states and hence prediction vector in (3), successfully
predicts the future of the state vector. It is noted that integrations involved in obtaining z(t) are
approximated by left hand rectangle method with sample time ∆t= 0.01. In both figures initial
functions are supposed to be one for states and zero for input.
0 5 10 15 20
−8
−6
−4
−2
0
2
4
x1(t)
(a)
Dynamic
Static
0 5 10 15 20
−3
−2
−1
0
1
2
Time(s)
x2(t)
(b) Dynamic
Static
Figure 2. System response for static (dash-dot) and dynamic (solid) controllers. (a) first state; (b) second state.
Example 3: Let us consider an open-loop almost surely exponentially unstable system with the
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May 1, 2017 International Journal of Systems Science tSYSguide
0 5 10 15 20
−8
−6
−4
−2
0
2
4
(a) x1(t)
z1(t)
0 5 10 15 20
−3
−2
−1
0
1
Time(s)
(b) x2(t)
z2(t)
Figure 3. State (solid) and prediction (dash-dot) vectors of example 1. (a) first state; (b) second state.
following parameters:
A=0 1
0 2 ,D1=0 0
0 0 ,B1=0
1
C=0 0
2 0.5,D2=0 0
0 0 ,B2=0
0
Robust exponential stability of this system with parameter uncertainties is considered in Huang
and Mao (2009), which is an special case of system (1) without state delay and nonlinear terms.
To apply our results, we need to discard the uncertainties. With assumption of zero upper bound
for all uncertainties, the maximum allowable input time delay for static (from Theorem 2 in Huang
and Mao (2009)) controller is τ2= 0.1256. When the time delay is supposed to be τ2= 0.1256,
static and dynamic controller gains are
Ks=−10.4787 −5.4418 
K=−7.6333 −4.7053 
L=0 0 
It is worth mentioning that the first term of the predictive controller (Kz(t)) takes care of the input
delay, whereas the second term (Lx(t−τ1+τ2)) is responsible for the state delay. As expected,
the obtained gain matrix Lis zero since there is no state delay in this example. States and control
input of both static and dynamic controllers are shown in Figure 4. Initial functions are assumed
to be one for both states and zero for control input. It is seen that static controller is in the margin
of instability, while dynamic controller has better transient response and requires less input energy.
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May 1, 2017 International Journal of Systems Science tSYSguide
0 1 2 3 4 5
0
0.5
1
x1(t)
(a) Dynamic
Static
0 1 2 3 4 5
−4
−2
0
x2(t)
(b) Dynamic
Static
0 1 2 3 4 5
−20
−10
0
10
Time(s)
u(t)
(c) Dynamic
Static
Figure 4. States and control input with static (dash-dot) and dynamic (solid) controllers. (a) first state; (b) second state; (c)
control effort.
5. Conclusion
A new technique has been introduced to compensate long input delay for nonlinear stochastic
systems with both input and state delays. To construct the memory controller, a new prediction
formula is given to predict the state and then feedbacked together with carefully chosen delayed
state. The resulting closed loop system is input delay free. Constructed memory controller requires
the past input and state data in time delay horizon. The Sufficient conditions has been provided
in the form of an LMIs, whose feasibility guarantee the mean square asymptotic stability of the
closed loop system. At the end, numerical examples are given to demonstrate the effectiveness of
the proposed approach.
Disclosure
No potential conflict of interest was reported by the authors.
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