Accepted Manuscript

Robust Control of Uncertain Nonlinear Switched Genetic Regulatory Networks with Time Delays: A Redesign Approach Hojjatullah Moradi , Vahid Johari Majd PII: DOI: Reference:

S0025-5564(16)00037-7 10.1016/j.mbs.2016.02.006 MBS 7753

To appear in:

Mathematical Biosciences

Received date: Revised date: Accepted date:

23 December 2014 10 February 2016 17 February 2016

Please cite this article as: Hojjatullah Moradi , Vahid Johari Majd , Robust Control of Uncertain Nonlinear Switched Genetic Regulatory Networks with Time Delays: A Redesign Approach, Mathematical Biosciences (2016), doi: 10.1016/j.mbs.2016.02.006

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ACCEPTED MANUSCRIPT Highlights Dissipative robust stability of nonlinear switching sys with unknown switching law. New robust dissipative redesign method combined with sliding mode control theory. Synthesizing a switching control law for the nonlinear switching GRN. Considering matched and unmatched norm-bounded uncertainties and perturbations. Lyapunov stability of perturbed switched system in finite time.

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ACCEPTED MANUSCRIPT

Robust Control of Uncertain Nonlinear Switched Genetic Regulatory Networks with Time Delays: A Redesign Approach Hojjatullah Moradi, Vahid Johari Majd*

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Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran.

Abstract:

In this paper, the problem of robust stability of nonlinear genetic regulatory networks (GRNs) is investigated. The developed method is an integral sliding mode control based redesign for a class of perturbed dissipative switched GRNs with time delays. The control law is redesigned by modifying the dissipativity-based control

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law that was designed for the unperturbed GRNs with time delays. The switched GRNs are composed of m modes that are switched from one to another based on time, state, etc. Although, the active subsystem is known in any instance, but the switching law such as transition probabilities are not known. The model for each mode is considered affine with matched and unmatched perturbations. The redesigned control law forces the GRN to always remain on the sliding surface and the dissipativity is maintained from the initial

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time in presence of the norm-bounded perturbations. The global stability of the perturbed GRNs is maintained if the unperturbed model is globally dissipative. The designed control law for the perturbed

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GRNs guarantees robust exponential or asymptotic stability of the closed-loop network depending on the type of stability of the unperturbed model. The results are applied to a nonlinear switched GRN, and the convergence of the state vector to the origin is verified by simulation in presence of nonlinear perturbations. Genetic regulatory networks (GRNs); perturbation; integral sliding mode control; control

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Keywords:

Introduction

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1.

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redesign.

The genetic regulatory networks (GRNs) as a fundamental biological system have received an

increasing attention in the few past decades [1-4] and play a key role in systems biology as they explain the interactions between the genes and the proteins. Analyzing of these interactions is important in order to understand and control the complex mechanisms that regulate biological *

Corresponding Author, Address: P.O. Box 14115-194, Tehran, Iran, E-mail address: [email protected] (V.J. Majd), web page: http://www.modares.ac.ir/~majd, Tel./Fax: +98 21 8288-3353. 2 of 25

ACCEPTED MANUSCRIPT functions in the living organisms. Additionally, the theoretical studies on GRNs are important in engineering applications, such as developing circuits with biotechnological design principles of synthetic of GRNs, like neurochips [5], learnted from biological neural networks [6,7]. Therefore, many results on the stability analysis of GRNs have been proposed in the literature [8-10]. In this regard, many mathematical models are developed to provide a framework for integrating data and

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gaining insights into the dynamic behavior of GRNs, such as Boolean networks, Bayesian networks, differential equation models, stochastic models, etc [11]. Differential equation models are of the most important models describing GRNs reported in the literature [12,13]. In most models, the time derivative of the concentration for each substance is composed of a linear part and a nonlinear part.

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The nonlinear part contains a saturation function, such as the Hill function, which is usually sum of regulatory or product regulatory functions as reported in [14] and the references therein. There are many genes in a cell which have different regulatory networks and can be respectively

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activated under inside and outside variational conditions. Therefore, the GRN is a complex system that may have several subsystems and should be considered as a hybrid system under some biologic

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environmental conditions, i.e. see [15-17]. The network switches from one mode to another with

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uncertain transition probabilities.

Many results on GRNs with the switching parameters have been reported in the literature. In

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[18], the mono stability and multi stability of the GRNs are analyzed considering the multiple time-

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varying delays and different types of regulation functions. A hybrid GRN model based on Markov chain is proposed in [19] considering the structure variations at discrete time instances of gene regulation. In [20,21], an average dwell time approach is purposed to study the stability of a hybrid GRN model based on unknown deterministic switching law with time delays. GRN models are unavoidably affected by the uncertainties due to inexact measurement of the coefficients and the saturation functions of a GRN. Another reason is that a family of GRNs are often considered to study the behavior of a class of organisms and not just a specific one.

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ACCEPTED MANUSCRIPT Additionally, the communication in dynamical networks are often subject to external noisy environments and therefore, disturbances are inevitably to be considered for modeling practical networks [22-24]. This means that GRN models contain uncertain terms constrained in some bounded sets. Moreover, the switched systems are sensitive to the uncertainties and perturbations which may destabilize the system. Therefore, robust stability analysis and control design of

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perturbed switched GRNs have attracted the researchers. For example, the robust state estimation of uncertain time-delay Markovian jumping GRNs with SUM logic is investigated in [25], where the uncertainties are considered in the network parameters and the mode transition rate, while the mode

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transition rates in [19,25] are supposed to be known.

In most works in the stability analysis of stochastic dynamical GRNs, the switching law is defined by the probability of transition rate [10,26,27] and the robust controller is designed [28,29]. On the other hand, employing the deterministic switching models only with the information of the

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active subsystem yields less conservative design for GRNs, and received attention in few recent works. More specifically, the exponential stability problem for the delayed GRNs with the

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switching parameters and unknown mode transition rate was proposed in [20,21] using the average

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dwell time approach. However, these works offer only analysis of stability of the switched GRNs, and do not consider the controller design and uncertainties.

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The existence of feedback in the biological systems and especially in GRNs is obvious and is

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usually described by nonlinear and saturation functions. Although many works consider only the stability analysis of autonomous GRNs, the control design is considered in some recent works by considering the control signal as multiplicative coefficient [9], or in affine form [30]. In practice, there exist circuits and bench marks that show the feasibility of control design for managing the behavior a biological network [31]. In [8], the Pulse-width modulation (PWM) is used to control a GRN, which show that the control of a biological system is practical.

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ACCEPTED MANUSCRIPT Dissipativity and passivity, as important system properties, offer effective tools for the system’s stability analysis and the controller synthesis. The dissipativity and its special form, passivity, imply that the dissipated energy inside a dynamical system is larger than the supplied energy from outside, which ensure the stability of system. The dissipativity and passivity notions have been employed in the stability analysis and synthesis of many types of complex dynamical systems, including the

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biological systems, such as the neural systems [32-34], GRNs [10], cell metabolism [35], etc. Similar to the problem of Lyapunov stabilization for a dynamical system, the dissipativity can be used to design a controller to ensure the stability of the closed-loop system. In the literature, there are many studies on dissipativity, passivity and passification to design the stabilizing control signals

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[10,36-38].

In this paper, by considering this fact that the GRNs and cellular metabolism are dissipative or passive [10,35], we introduce a new robust control design based on the dissipativity for the

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nonlinear dynamical GRNs with model uncertainties. Since the measurements and gathering the information of biological systems are inexact, difficult ,expensive, and time consuming, finding the

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information of switching rates and their probabilities are a challenging problem. Therefore, it is

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assumed that the switching law is unknown, and only it is known that which subsystem is active in any instance of time. The uncertainty terms are considered norm bounded. The proposed method is

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a dissipative-based redesign method combined with an integral sliding mode control to guarantee the preservation of the dissipativity of the closed-loop nonlinear switched system in the presence of

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norm bounded perturbations. The major advantage of this method is that it robustifies the available nominal controllers while reducing the complexity of the robust controller design of the nonlinear switched GRNs with time delays. With this method, the switching law is not necessary to be designable. Finally, a numerical example for a perturbed switched GRN is considered and its stability conditions are derived. The simulation results of the perturbed switched GRN show the effectiveness of the proposed method in preserving the stability of the closed-loop system.

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ACCEPTED MANUSCRIPT The rest of the paper is organized as follows: In Section 2, the model of GRNs is introduced. In Section 3, an integral sliding surface is presented which preserves the dissipativity property of the closed-loop system on the sliding surface. Then, a variable structure control is introduced which always places the switched GRNs on the sliding mode. In Section 4, the theoretical results are

2.

The preliminaries and problem statement

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illustrated in an example. Finally, the conclusion of the paper is presented in section 5.

There exist different mathematical framework for the GRNs. Since the gene networks have a hybrid nature in the practice [39,40], a hybrid modeling with a set of subsystems like what was

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considered for Markov jumping systems, will provide a more precise and practical model [10, 19, 25]. However, in this paper, similar to [30], it is further considered that the transition probabilities of subsystems are unknown. A typical autonomous hybrid differential equation model for nonlinear

(1)

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m   Ai m(t )  Bi fi  p(t   )   Li ,   p  Ci p(t )  Di m(t   )

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switched GRNs without perturbations can be considered as [30]:

n n where m(t )  m1 (t ),..., mn (t )   and p(t )   p1 (t ),..., pn (t )   with elements m j (t ) and T

T

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p j (t ) , j  1,..., n , being the concentrations of mRNA and protein of the j -th gene or node at time t

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, respectively. Moreover, i  1,2,, M  with M being the number of subsystems, is the piecewiseconstant right-hand-side-continuous switching signal which may depend on time, state, or input













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nn nn n n signals; Ai  diag ai ,1 ,..., ai ,n   , Ci  diag ci ,1 ,..., ci ,n   and Di  diag di ,1,..., di , n  

are subsystem matrices with ai , j , ci. j and d i , j representing the decay rates of mRNA, protein and

 

the translation rate of the j -th node of the i -th subsystem, respectively; Bi  bi , jk coupling matrix of GRNs whose elements are defined by:

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nn

 nn is the

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bi , jk

 li , jk   0  l  i , jk

if transcription factor k is an activator of gene j if there is no link from gene k to gene j

.

(2)

if transcription factor k is a repressor of gene j

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The nonlinear vector function fi  fi ,1 ,..., fi , n



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represents the feedback regulation of the proteins

on the transcriptions whose elements are monotonic increasing or decreasing regulatory functions.

f i , j ( x) 

x

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In this paper we consider the regulatory function f i , j of the Hill form as [41]: hi , j

1 x

hi , j

,

(3)

where hi , j is the Hill coefficient of the j -th gene in the i -th mode or subsystem. The

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Li  [li ,1 , ..., li ,n ]T is a constant vector, where li , j  kI li , jk with I i , j being the set of all nodes i, j

operating as the repressor of the j -th gene in the i -th mode. The  and  are the time delays of



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the switching system. Let m* , p *

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T

the equilibrium point m* , p *



T T



(5)

of Eq. (1) shifts to the origin and the Eq. (1) can be rewritten as:



* *   x1   Ai x1 (t )  Ai m  Bi f i x2 (t   )  p  Li .  * *   x   C x ( t )  C p  D x ( t   )  D m i 2 i i 1 i  2

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(4)

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 x1 (t )  m(t )  m* ,   x2 (t )  p(t )  p *

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Using the transformation:

be the equilibrium point of Eq. (1), then:

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 

   0   Ai m  Bi f i p  Li .     0   C p  D m i i 



T T

(6)

By considering Eq. (4) and using some manipulations, Eq. (6) and Eq. (1) can be transformed into the following form:  x1   Ai x1 (t )  Bi g i x2 (t   )  ,   x 2  Ci x2 (t )  Di x1 (t   )

(7)

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ACCEPTED MANUSCRIPT g i x   g i ,1 x ,..., g i ,n x 

T

where

with





 

g i , j x   f i , j x2  p *  f i , j p * .

Since

fi, j

is

a

monotonically increasing function with saturation, it satisfies that [12]:

0

f i , j ( x)  f i , j ( y ) x y

 ri , j ,

(8)

for all x, y   , x  y , where ri , j is a positive constant. Therefore, the function g i , j ( x) satisfies

0

g i , j ( x) x

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the sector condition:

 ri , j ,

and respectively:

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gi , j ( x)gi , j ( x)  ri. j x   0 .

(9)

(10)

In this paper, the nonautonomous hybrid framework of dynamical Eq. (7) with control inputs can be

 x1   Ai x1 (t )  Bi g x2 (t   )   Ei u1,i ,   x2  Ci x2 (t )  Di x1 (t   )  Fi u2,i



T



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

where u1,i  u11,i ,..., u1n,i

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consider in the form of [10]:

and u2,i  u21,i ,..., u2n,i

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(11)

are system control inputs of i -th subsystem; Ai ,

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Bi , Ei , Ci , Di and Fi are known constant matrices with appropriate dimension for i -th subsystem.

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Except the new input matrices Ei and Fi , the matrices are the same as ones of the model (1).

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, T

any subsystem of the Eq. (11) can be represented in the form of

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Considering x  x1T , x2T

piecewise-affine gene network model offered in [30]:

x   ( x, u )  ( x, u ) x ,

(12)

where  ( x, u )  n is the production vector and ( x, u ) is the degradation rate matrix. Extending the unperturbed model (11) , the structure of the perturbed gene regulatory network used in this paper is as follows:

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ACCEPTED MANUSCRIPT  x1   Ai x1 (t )  Bi g x2 (t   )   Ai x, t   Ei I n  Ei x, t u1,i  u1,i x, ui , t  ,   x2  Ci x2 (t )  Di x1 (t   )  Ci x, t   Fi I n  Fi x, t u2,i  u2,i x, u2,i , t 



(13)



where I n is the identity matrix with dimension n n , xt   x1T , x2T , and for i  1,2,, M , the T

bounded terms:

Ai x, t    a,i x, t 

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(14)

and:

Ci x, t   c,i x, t  are unmatched perturbations, and:

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Ei x, t   e,i x, t   1, Fi x, t    f ,i x, t   1 , u1,i x, u1,i , t   u1,i x, t  ,

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u2,i x, u2,i , t   u 2,i x, t  ,

(15)

(16) (17) (18) (19)

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are matched perturbations, where the nonnegative functions  a ,i x, t  ,  c,i x, t  ,  e,i x, t  ,  f ,i x, t  ,

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u1,i x, t  and u 2,i x, t  are known. The initial condition of GRN (13) is assumed to be: x1 t   1 t , x2 t   2 t  for  T  t  0, T  max ,  ,

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where the 1 t  and 2 t  are known vector functions with proper dimension.

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Remark 1: In this paper, every subsystem is known with some uncertainties but the law of switching between subsystems, such as transition probabilities like what is considered in Markov jump GRNs, are not known and can be considered as an uncertainty. Therefore, this model and consequently the stability analysis will be less conservative than most recent results. In this paper, we only need to know that which subsystem is active in any instance of time. The following assumptions are used in the stability analyses and control redesign, based on multiple storage functions.

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ACCEPTED MANUSCRIPT Assumption 1: The states of the switched system do not jump in the switching instances, and thus,

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the trajectory xt   x1T , x2T



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is always continuous.

Assumption 2: The unperturbed model of GRN (11) is dissipative and stable [37], i.e. for all





subsystems (i  1,2,, M ) , there exist state feedback inputs ui x, t   u1,i x, t T , u2,i x, t T , positive T

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definite continuous storage functions Si x  C1 , nonpositive supply rate functions wi ui , hi  , cross  supply rate functions wij x, ui , hi , t  , and upper bound functions ij t  L1 0,  ( j  1,2,, m, i  j )

such that when the i-th subsystem is active, the following conditions hold for all j  1,2,, M , i  j :

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i) Si  wi ui , hi  , ii) S j  wij x, ui , hi , t  ,

Bi g 0  Ei u1,i 0, t   0 iii)  ,   F u 0 , t  0 i 2 , i 

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iv) wij xt , ui t , t   ij t  .

(21) (22)

(23)

(24)

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Inequalities (21), (22) and (24) imply the existence of a control law ui that decreases the energy

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of the i -th active subsystem and also guarantees the boundedness of the stored energy in the j -th inactive subsystem supplied by the i -th active subsystem. The equations in (23) guarantee that the

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origin is the equilibrium point of the unperturbed closed loop GRN.

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Remark 2: The condition (23) for the closed loop perturbed GRN (13) becomes:

Bi g 0  Ai 0, t   Ei I n  Ei 0, t u1,i 0, t   u1,i 0, u1,i 0, t , t   0 .                 C 0 , t  F I   F 0 , t u 0 , t   u 0 , u 0 , t , t  0 i i n i 2 , i 2 , i 2 , i 

(25)

The above conditions (25) are always satisfied if the perturbation terms are vanishing, but in this paper, the perturbations (14) up to (19) may be non-vanishing in general.

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ACCEPTED MANUSCRIPT Remark 3: In Assumption 2, for radially unbounded storage functions, the primary control signal



ui t   u1,i xt , t  , u2,i xt , t 



T T

T

globally stabilizes the unperturbed switched GRN (11).

Remark 4: By setting the supply and cross supply rate functions to zero, the dissipativity-based stability analysis reduces to Lyapunov-based stability analysis, and in this case, the storage

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functions can be considered as Lyapunov functions. A Lyapunov-based stability analysis is a subset of dissipativity-based stability analysis with zero supply and cross supply rate functions.

Any perturbation and uncertainty in the GRN may cause the violation of the dissipativity conditions in Assumption 2, and thus, may destabilize the perturbed GRN. The problem is to

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redesign the control law and to provide the sufficient conditions that preserve the dissipativity of the perturbed GRN. The purpose of this paper is to robustify the controller using a sliding mode approach to compensate the side effects of the perturbations (14)-(19) on the GRN, so as to preserve the dissipativity and stability of the switched GRN (13) in the presence of norm bounded

Main results

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3.

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perturbations.

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The structure of final closed loop switched GRN is as one depicted in Fig.1. The system has an inner loop containing a primary controller for each subsystem and an outer loop integral sliding

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mode controller which acts as a robustifying control signal to guarantee the dissipativity of the

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overall closed loop system.

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Perturbations

The switching GRN

+ _

+

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Predesigned inner loop control law

Redesigned outer loop control law

Fig.1. The block diagram of the robust closed loop system for the perturbed GRN.

In the rest of this section, we first drive the sufficient conditions for preserving the dissipativity

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of the perturbed switching GRN (13) and discuss the feasibility of these conditions. Then, we design an integral sliding mode robustifier term as an outer loop for each subsystem in the closed loop switched GRN that satisfy the mentioned conditions.

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3.1. Redesign analysis

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The following lemma gives the sufficient condition for preserving the dissipativity of perturbed GRN (13).

x1

A x, t   E I i

i

n

 Ei x, t u1,i  u1,i x, t   u1,i x, u1,i , t 

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S j

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Lemma: Assuming that the Assumption 2 and the equations in (25) hold, then, satisfying:

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

S j x 2

C x, t   F I i

i

n

 Fi x, t u 2,i  u 2,i x, t   u 2,i x, u 2,i , t    ij t ,

(26)

 for all i, j  1,2,, M and for any  ij t  L1 0,  with  ii t   0 , guarantees that the perturbed

GRN (13) is dissipative with the same storage, supply and cross supply rate functions as the ones used for the dissipativity proof of the unperturbed GRN (11). Proof: Differentiating the storage functions S i x  , i  1,..., M , along with the trajectories of the perturbed GRN (13) when the i -th subsystem is active, yields:

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ACCEPTED MANUSCRIPT S S Si  i  i  Ai x1 (t )  Bi g x2 (t   )   Ai x, t   Ei I n  Ei x, t  u1,i  u1,i x, u1,i , t  t x1 S i  Ci x2 (t )  Di x1 (t   )  Ci x, t   Fi I n  Fi x, t u2,i  u2,i x, u2,i , t  x2

 wi ui , hi   

S i Ai x, t   Ei I n  Ei x, t u1,i  u1,i x, t   u1,i x, u1,i , t  x1 Si Ci x, t   Fi I n  Fi x, t u2,i  u2,i x, t   u2,i x, u2,i , t , x2

(27)

CR IP T



S j S j  Ai x1 (t )  Bi g x2 (t   )  Ai x, t   Ei I n  Ei x, t u1,i  u1,i x, u1,i , t  S j   t x1 S j x2

 C x (t )  D x (t   )  C x, t   F I i 2

 wij x, ui , hi , t   

S j x2

S j x1

i

1

i

A x, t   E I i

i

C x, t   F I i

i

n

n

i

n

 Fi x, t  u2,i  u 2,i x, u 2,i , t 

 Ei x, t  u1,i  u1,i x, t   1,i x, u1,i , t 

AN US



 Fi x, t  u 2,i  u 2,i x, t   u2,i x, u2,i , t .

(28)

Therefore one can easily see that if (26) is satisfied for all i, j  1,2,, M and for any

M

 ij t  L1 0,  with  ii t   0 , then, the dissipativity conditions (21), (22) and (24) are preserved

ED

for the perturbed GRN (13) with the same storage, supply and cross supply rate functions as the ones used for the unperturbed GRN (11). Satisfying (25) guarantees that the origin is the stable

PT

equilibrium point of the perturbed system (13). □

CE

Here, we introduce a nonlinear integral sliding surface that robustifies closed loop perturbed system when the system’s trajectory is on the surface. Finding robust feedback control signals

AC

ui x, t  , i  1,2,, M , to satisfy the sliding mode condition will be discussed in the next subsection.

Similar to the nonlinear integral sliding surface in [42], we consider the following sliding surface: t    Ai x1 (t )  Bi g x2 (t   )   Ei  S s  H  xt   xt0      0   C x ( t )  D x ( t   ) i 2 i 1    t  0 

13 of 25

0    ui d , Fi   

(29)

ACCEPTED MANUSCRIPT E where H is an arbitray constant sliding surface matrix with proper dimensions to make H  i 0

0 Fi 

square and invertible. Since this sliding surface depends on the initial time t 0 such that S s t0   0 , the reaching phase is eliminated. Therefore, the sensitivity of the closed loop system against the uncertainties and disturbances, which is related to the reaching phase, is eliminated and the resulting

CR IP T

control signal is more robust than other sliding-mode control systems that contain a reaching phase [42].

The following theorem shows that when the GRN trajectories are on the sliding surface, the

AN US

dissipativity of the perturbed nonlinear switched GRN (13) is guaranteed.

Theorem 1: Under the Assumption 2, when the nonlinear perturbed switched GRN (13) operates on the nonlinear integral sliding surface Ss  0 in (29), the dissipativity and consequently the stability of the perturbed switched GRN is guaranteed with the same switching law, storage, supply

ED

Proof: On the sliding surface, we have:

M

and cross supply rate functions as ones of the unperturbed switched GRN (11).

PT

Ss  0 .

(30)

Differentiating (30) with respect to the time and substituting (13), yields:

CE

  Ai x1 (t )  Bi g x2 (t   )   Ei S s  H  x t     0   C x ( t )  D x ( t   ) i 2 i 1    

0  u1,i    ui  Fi  u 2,i  

AC

  Ai x1 (t )  Bi g x2 (t   )   Ai x, t   Ei  H    Ci x2 (t )  Di x1 (t   )  Ci x, t    0   Ai x1 (t )  Bi g x2 (t   )   Ei    Ci x2 (t )  Di x1 (t   )   0

 Ai x, t    Ei  H   Ci x, t   0 

0  I n  Ei x, t u1,i  u1,i x, u1,i , t     Fi  I n  Fi x, t u 2,i  u 2,i x, u 2,i , t 

0  u1,i     Fi  u 2,i  

0  I n  Ei x, t u1,i  u1,i x, u1,i , t   u1,i     . Fi  I n  Fi x, t u 2,i  u 2,i x, u 2,i , t   u 2,i  

On the sliding surface, we have Ss  0 . Thus, (31) yields:

14 of 25

(31)

ACCEPTED MANUSCRIPT  Ai x, t    Ei H   Ci x, t   0 

0  I n  Ei x, t u1,i  u1,i x, u1,i , t   u1,i      0 . Fi  I n  Fi x, t u2,i  u2,i x, u2,i , t   u2,i  

 Ei Based on the condition (16) and (17), the matrix  I 2 n    0 

0   is invertible. Additionally, Fi  

0  is nonsingular. Therefore, Eq. (32) results in a valid equivalent control as: Fi 

 u1,i x, t   u1,i x, u1,i , t      Ei  H uieq x, t   Gi     u 2,i x, t  u 2,i x, u 2,i , t     0      1

E Gi  I 2 n   i  0

0  . Fi 

Since H is full rank, Eq. (32) results in:

1

0  I n  Ei x, t u1,i  u1,i x, u1,i , t   u1,i    0. Fi  I n  Fi x, t u2,i  u2,i x, u2,i , t   u2,i 

(33)

(34)

(35)

M

Ai x, t    Ei C x, t    0  i  

 0 Ai x, t   Gi  H  ,  Fi  Ci x, t  

AN US

where:

CR IP T

E H i  0

(32)

ED

Consequently, the inequality (26) is satisfied when the GRN (13) operates on the integral sliding surface (29). Hence, the perturbed switched GRN (13) is dissipative with the same storage, supply

PT

and cross supply rate functions as unperturbed GRN (11). Notice that, satisfying (23) and (35)

CE

guarantees that the origin is the stable equilibrium point of the perturbed switched GRN (13). Thus, any dissipative-based results about the equilibrium point of the unperturbed nonlinear switched

AC

GRN is still valid for the overall closed-loop GRN in the presence of the perturbations. □

3.2. Redesign synthesis The following theorem provides a sliding mode control law that guarantees the trajectory of the perturbed switched GRN remains on the stabilizing sliding surface from the initial time. Theorem 2: Consider the integral sliding surface (29) for the perturbed switched GRN (13) under the same switching law as the unperturbed GRN (11) with given primary state feedback ui x, t  ,

15 of 25

ACCEPTED MANUSCRIPT i  1,2,, M , and suppose that the norm inequalities in (14)-(19) are satisfied with some given upper bound functions  a ,i t , x  ,  c,i t , x  ,  e,i t , x  ,  f ,i t , x  , u1,i t , x  and u 2,i t , x  . Then, using a redesigned variable structure controller of the form: ui x, t   ui x, t   uiR x, t 

(36)

where:

1   ef ,i

   

  Ei H   0

1   0    sign  H  Ei    H     u x , t    ac,i u ,i i ef ,i i     0 Fi     

T  0    S s ,  Fi   

CR IP T

u iR 

1

 ac,i t , x    a,i t , x    c,i t , x  ,

AN US

u ,i t , x   u1,i t , x   u 2,i t , x 

ef ,i t , x   max e,i t , x ,  f ,i t , x , x ,t

(37)

(38) (39) (40)

guarantees that the perturbed switched nonlinear GRN (13) operates in the sliding mode from the

M

initial time and remains on the sliding surface for all t  t0 . The i ’s are arbitrary non-negative

ED

constant scalars which adjust the convergence rate to the sliding surface. Proof: Considering the Lyapunov function candidate as: 1 T Ss Ss , 2

(41)

PT

V

CE

and considering (31), the time derivative of V along the trajectories of the switched GRN (13) is:

AC

 Ai x, t   E V  S sT  H   H i   Ci x, t  0 

0  I n  Ei x, t u1,i  u1,i x, u1,i , t   u1,i     . Fi  I n  Fi x, t u 2,i  u 2,i x, u 2,i , t   u 2,i  

(42)

Substituting the control signal (36) in (42) yields: Ai x, t   E V  S sT H   S sT H  i  0 Ci x, t 

 u1,i x, u1,i , t    0  0  E x, t       i  u i x, t         (43)   Fi   Fi x, t   0 u 2,i x, u 2,i , t    

where

16 of 25

ACCEPTED MANUSCRIPT

 

1 1   ef ,i

   

1   0    sign  H  Ei    H     u x , t    ac,i u ,i i ef ,i i    0 Fi     

  Ei H   0

T  0    S s .    Fi   

(44)

By using the subadditivity property of matrix norms, one can easily see that:

Ai  Ai  0  C          a ,i   c ,i   ac,i , 0  Ci   i

CR IP T

(45)

and

(46)

AN US

u1,i  u1,i  0        u    u1,i   u 2,i   u ,i . 0   2,i  u2,i  From properties of diagonal matrices, we have:

Ei  0 

0   max  Ei , Fi   max  e,i ,  f ,i    ef ,i x ,t x ,t Fi 

(47)

M

Using the matrix norm inequality and substituting the Inequalities (45), (46) and (47) into Eq. (43) yields:

 ef ,i

1   ef ,i

 Ei H   0

 Ei H   0

1   ef ,i

AC

  ef ,i u i  x, t  

 ef ,i

Ai  x, t    S H   i C i  x, t  T s

E  S sT H  i 0

   

2

 Ei H   0

1

 ef ,i  u ,i 0  i  H  ac,i   u i  x, t     Fi   1   ef ,i 1   ef ,i 1   ef ,i

1

   

 ef ,i  ef ,i  ef ,i 0    H  ac,i   u ,i  u i  x, t    i   u ,i  Fi   1   ef ,i 1   ef ,i 1   ef ,i 

0  E  S s  S sT H  i   Fi   0

2

0 Fi 

0  A  x, t   E  H i  S sT H  i    Fi   0 C i  x, t  1

1

 ef ,i  u ,i 0  i  H  ac,i   u i  x, t     Fi   1   ef ,i 1   ef ,i 1   ef ,i

1

T

 Ei H   0

0   E i H Fi   0

 E   i  H i  i  0

 Ei H   0

 Ei H   0

 ef ,i  ef ,i  ef ,i 0    H  ac,i   u ,i  u i  x, t    i   u ,i  Fi   1   ef ,i 1   ef ,i 1   ef ,i 

T   0    1  S    s Fi     1   ef ,i

CE

Ai  x, t    S sT H   C i  x, t 

T   0    1  S    s Fi     1   ef ,i

PT

  ef ,i u i  x, t  

 Ei H   0

ED

Ai  x, t   V  S sT H   C i  x, t 

 Ei H   0 0 Fi 

1

0   H  ac,i Fi  

 Ei H   0

1

0   H  ac,i   i Fi  

 Ei H   0

T

0   Ss Fi  

T

0   Ss . Fi  

(48)

17 of 25

ACCEPTED MANUSCRIPT   Ei Since  H   0

0   is nonsingular, then, S s x   0 results in V  0 . Therefore, since S s xt0   0 , Fi  

the controller (36) preserves the sliding mode for all t  t 0 . The assumptions e,i  1 ,  f ,i  1 and respectively, ef ,i  1 prevents zero division in (36). This completes the proof.□

CR IP T

Remark 5: The i ’s determine the decay rates of the Lyapunov function for each subsystem and can be adjusted by the designer. The greater i ’s yield faster convergence to the sliding surface but at the cost of greater values of the control signal. Notice that as discussed before, the design method eliminates the reaching phase, but due to the existence of uncertainties, the system’s trajectory may

AN US

depart the sliding surface at some instances. However, the Lyapunov-based sliding design is such that it forces the trajectory to return to the sliding surface with the convergence rate (48).

4.

Simulation results

M

To illustrate the results, we study the switched GRN in [21]. The system is described by the

0  m(t )  Bi   Ci   p(t )   0

0   g  pt     Ei  Di  mt      0

0 ui , i  1, 2 , Fi  i

(49)

PT

m   Ai  p    0   

ED

following nonlinear switched state space model comprised of two subsystems:

with:

CE

A1  6  I 3 , A2  7  I 3 , C1  5  I 3 , C2  6  I 3 , D1  diag 1,0.8,1.2, D2  I 3 ,

AC

0  5 0  5.5 0  0    B1   5 0 0 , B2   5.5 0 0 , E1  5  I 3 , E2  6  I 3 , F1  3  I 3 , F2  5  I 3 ,  0  5 0   0  5.5 0  and delay values   0.5 sec and   0.5 sec . The vector function g has nonlinear elements g j ( p) of Hill form given by:

g j  p 

p2 , j  1,2,3 . 1  p2

(50)

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ACCEPTED MANUSCRIPT This GRN model is exponentially stable with the average dwell time Tav  0.3171 in the absence of control signals, i.e., when u1,i , u2,i  0, 0, 0 T , i  1,2 . The respective Lyapunov functions and stability analysis are presented in [21]. By using Remark 4, the same Lyapunov functions given in [21] can be considered as storage functions, and the system is dissipative with zero supply and cross supply rate functions; hence the results of this paper can be implied for purposed GRN. Moreover,

1, it    2 ,

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let the switching law be given as:

t  2k  1T , 2kT 

t  2k  2T , 2k  1T 

(51)

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where k  1,2,3, and T  0.5 sec, which satisfies the average dwell time condition Tav  0.3171 . Fig.2 shows the simulation result for this switched system. The state initial conditions for all simulations are considered as m0  [0.5,0.7,0.8] and p0  [0.4,0.9,0.6] . Notice that, due to the shift of the equilibrium point to the origin, the states, which are shifted values of mRNA and protein

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concentrations given in (5), can be negative. As expected, the subsystems switch every 0.5 seconds.

Fig.2. State trajectory of shifted (a) mRNAs and (b) proteins of the unperturbed system.

By considering the following uncertainties:

A

T 1

 



 



T

, C1T  0.1m12 m22 , 0, 0, 0, 0, 0 , A2T , C2T  0, 0, 0, 0.1 p12 p22 , 0, 0 ,

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(52)

ACCEPTED MANUSCRIPT E1  0 

u

T 1,1

0  E  diag 0.1m1 p1 , 0, 0, 0.1m1 p1 , 0, 0 ,  2  F1   0 , u2T,1

  1.2 m , 0, 0, 0, 0, 1.2 p  , u T

T

1

3

T 1, 2

0   diag 0, 0.1m2 p2 , 0, 0, 0.1m2 p2 , 0 , (53) F2 

, u 2T, 2

  1.7 m , 0, 0, 1.7 p , 0, 0  T

T

1

1

.

(54)

T for the switched system (49), the switched system with the control signals u1,i , u2,i  0, 0, 0 ,

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i  1,2 becomes unstable as depicted in Fig.3.

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Fig.3. State trajectories of the switched GRN system with perturbations: (a) mRNAs and (b) proteins

robustifying terms:

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The perturbed GRN is in the form of (13) and, thus based on Theorem 2 and using 2-norm, the

1  1 2 2  2 2  m1 m2  1.2 m1  p3  1 sign( S s ) , 1  0.1 m1 p1  30 

(55)

u 2 m, p, t   

1  1 2 2  2 2  p1 p 2  1.7 m1  p1  2 sign( S s ) , 1  0.1 m2 p 2  50 

(56)

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CE

u1 m, p, t   

will stabilize the switched GRN (49) with the perturbations (52)-(54). Choosing the sliding surface matrix H  I 6 will make S s in Eq. (29) a diagonal 6 6 matrix which can be computed by MATLAB. The above robustifying control signals are shown in Fig.4 for 1 , 2  0 . The chattering of the sign function can be seen in this figure.

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Fig.4. The robustifying controllers (a) (55) and (b) (56).

The simulation results for the closed loop perturbed system with the robustifying controllers (55)

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and (56) where. 1 , 2  0 , are depicted in Fig.5. The results show that the states have the same behavior as the unperturbed ones in Fig.2, and that the robustifying controller has preserved the dissipative properties and the objectives of the closed-loop nominal system for the closed-loop

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perturbed system.

Fig.5. State trajectories of the closed-loop switched GRN system with perturbations: (a) mRNAs and (b) proteins with robustifying terms (55) and (56).

The chattering of control signals are due to the small deviations of trajectories from sliding surface caused by the perturbations. The diagonal elements of the sliding surface matrix are depicted in Fig.6.

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Fig.6. The sliding surface trajectories.

Conclusions

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5.

In this paper, a redesigned robust integral sliding mode control is developed for a class of nonlinear dissipative switched GRNs with affine subsystems. The derived variable structure

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controllers guarantee the stability and convergence of the perturbed switched GRN to the origin for norm bounded perturbations. The matched and unmatched perturbations are considered for the

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switched GRN. The integral sliding surface is considered such that the sliding mode is maintained from the initial time and the stability is preserved on this surface. The derived variable structure

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control in Theorem 2 preserves the sliding mode, and thus, the stability of the closed-loop perturbed

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nonlinear switched GRN is guaranteed with the same dissipativity functions as the unperturbed closed-loop ones. The designer can adjust the decay rate of states to the sliding surface if the state

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trajectory tends to leave this surface under the effect of perturbations. However, the larger value of decay rates results in the larger amplitude of the control signals. The simulation example show the effectiveness of the proposed sliding method in presence of the matched and unmatched perturbations for a perturbed delayed switched GRN.

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