Robust control design through the attractive ellipsoid ... · Robust control design through the attractive ellipsoid technique for a class of linear stochastic models with multiplicative

IMA Journal of Mathematical Control and Informationdoi:10.1093/imamci/dns008

Robust control design through the attractive ellipsoid technique for a class oflinear stochastic models with multiplicative and additive noises

NORMA B. LOZADA-CASTILLO∗, HUSSAIN ALAZKI AND ALEXANDER S. POZNYAK

Department of Automatic Control, CINVESTAV-IPN, Av. Instituto Politecnico Nacional 2508,07300 Mexico D.F., Mexico

∗Corresponding author: [email protected] [email protected]@ctrl.cinvestav.mx

[Received on 28 July 2011; revised on 25 October 2011; accepted on 20 December 2011]

This paper concerns the robust ‘practical’ stabilization for a class of linear controlled stochastic differ-ential equations subject to both multiplicative and additive stochastic noises. Sufficient conditions of thestabilization are provided in two senses. In the first sense, it is proven that almost all trajectories of thestochastic model converge in a ‘mean-square sense’ to a bounded zone located in an ellipsoidal set, whilethe second one ensures the convergence to a zero zone in probability one. The considered control law isa linear state feedback. The stabilization problem is converted into the corresponding attractive averagedellipsoid ‘minimization’ under some constraints of bilinear matrix inequalities (BMIs) type. Some vari-ables permit to represent the BMIs problem in terms of linear matrix inequalities (LMIs) problem, whichare resolved in a straight manner, using the conventional LMI-MATLAB toolbox. Finally, the numericalsolutions of a benchmark example and a practical example are presented to show the efficiency of theproposed methodology.

Keywords: stochastic differential equations; linear matrix inequalities; attractive ellipsoid method.

1. Introduction

Some real-world phenomena present disturbances and perturbations of non-deterministic nature. Thedisturbances deal with additive noises, such as measurement noises in electrical systems (Gray et al.,1999), daily weather effects (Richardson, 1981), variations in population dynamics (Sykes, 1969),among others. On the other hand, some economic processes (e.g. El Karoui et al., 1998; Shreve, 2004),optoelectronic devices (Konstantatos et al., 2006) and many other physical systems are affected bymultiplicative disturbances of stochastic nature. Sometimes, these effects arise just in additive (multi-plicative) form, which reduce the mathematical analysis of the systems. However, in the case of someeconomic pricing options (Anteneodo & Riera, 2005), it is necessary to analyse the stability and thesynthesis of robust control strategies in dynamics affected by both additive and multiplicative stochasticnoises.

This class of models (stochastic differential equations) is governed by its behaviour properties whichcan be analysed by two main techniques: the quantitative techniques and the qualitative ones. ‘Quan-titative methods’ (Appleby & Flynn, 2006; El Bouhtouri & El Hadri, 2003) need an exact solution(closed-form expression) of the equations, which, for most of the systems, specially non-linear, is rarelypossible. These methods rely on general known properties of the systems, as well as boundary con-ditions, which are closely related to numerical approximations. The main advantage of these methodsis that they provide a specific information of the solution at any time. On the other hand, ‘qualitative

c© The author 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

(2013) 30, 1–19

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N. B. LOZADA-CASTILLO ET AL.

methods’ permit to investigate the general behaviour of the solutions, i.e. properties such as bounded-ness, periodicity, convergence, stability and so on, without the need of having an explicit form of thesolution. Examples of these methods are the Lyapunov’s second method, comparison theorems, amongothers (see, e.g. Arnold & Schmalfuss, 2001).

Concerning linear controlled stochastic differential equations (LCSDEs), one of the main qualita-tive properties is the, so-called, ‘robust practical stability’. Some quantitative methodologies have beenproposed to synthesize the control schemes. In Poznyak et al. (2002), a robust stochastic maximumprinciple is proven with the aid of deterministic min–max tools and the stochastic maximum principle.The well-known work of Zhou (1991) deals with an optimal stochastic control problem, where the diffu-sion coefficient also depends on the control, which can change its diffusion structural properties. In thisproblem, the maximum principle, dynamic programming and their connections are established within aunified framework of viscosity solution. The H∞ approach is considered in Ugrinovskii (1998), wherea state feedback control is proposed for a class of systems affected by uncertain multiplicative whitenoise perturbations, satisfying a certain variance constraint.

The aforementioned methods can achieve excellent results in the presence of a certain minimalknowledge of the system. However, when there is a lack of information, these methods may not obtaindesirable results. The robust attractive ellipsoid methods (Bertsekas, 1994; Kurzhanskii & Valyi, 1997;Schweppe, 1973) come up with a qualitative approach which allows us to analyse and synthesize con-trol schemes for a class of uncertain systems. This method is based on the second Lyapunov methodand the concept of invariant sets. The solutions are expressed as an optimization problem restricted to‘bilinear matrix inequalities’ (BMIs) (see Safonov et al., 1994) and considering an effective numericalmethod for the design of the corresponding feedback control (Kurzhanskii & Valyi, 1997). The compre-hensive survey on invariant sets can be found in Blanchini & Miani (2007). In Polyak et al. (2006), theproblem of synthesis of a static state feedback controller for a linear time-invariant system, minimizingthe size of the corresponding invariant ellipsoid, was reduced to optimization of a linear function undersome set of ‘linear matrix inequality’ (LMI) application constraints (Nagapal et al., 1994; Abedor et al.,1996; Safonov et al., 1994).

In this article, we deal with the problem of robust practical stabilization (a zone-convergenceanalysis) of an LCSDE with multiplicative and additive noises, by means of attractive ellipsoid. Sincethe class of stochastic equations to analyse contains a Wiener process as an external noise, the Ito’sintegral is used as the main tool to obtain a stabilizing robust control law.

1.1 Main constraints and contribution

The main ‘constraints’ accepted in this paper are as follows:

- we consider the class of linear stochastic differential equations with diffusion terms linearlydependent on the state and control;

- as the control action, we use static linear feedback control.

The principal ‘contributions’ are as follows:

- A specific form of BMI (which by a special transformation is shown to be converted into an LMI)is obtained which provides the mean-square convergence into a bounded ellipsoidal set and, asa result, guarantees the robust ‘practical stability’ property of the designed controller for a wideclass of stochastic linear systems with controlled diffusion terms.

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ROBUST STABILIZATION OF LINEAR STOCHASTIC MODELS

- Sufficient conditions to exponential convergence to a zero zone.

- The numerical matrix optimization procedure, based on the ‘interior point method’ (Nesterov &Nemirovsky, 1994), which under the obtained LMI constraints provides the optimal numericalvalues of the control K guaranteeing a ‘minimal’ attractive ellipsoid.

1.2 Outline of the paper

In the next section, some necessary concepts, definitions and notation to present the formulation problemare introduced. Besides, the stochastic model description and the problem formulation are given. On theother hand, there are given some important propositions to present the main results which describe therobust attractive ellipsoid where all trajectories of the closed-loop system converge in mean square andwith the probability one. Then, in Section 3, the ideas for converting the BMI problem to LMI problemare shown. In Section 4, the numerical illustrative results for a benchmark example and actual exampleare presented in different cases. Finally, some conclusions are given.

2. System description and problem formulation

We consider Ω a non-empty set called ‘sample space’, F ⊆ 2Ω a σ -algebra called ‘event space’, Pa probability measure and t ∈ [0, T ] ⊆ R+. In what follows, (Ω,F, {Ft }t�0, P) denotes a filteredprobability space with {Wt }t�0 being an m-dimensional (Wt = W 1

t , . . . , W mt )� standard Brownian

motion, m ∈ N. We suppose the filtration satisfies the usual hypotheses.Consider the following stochastic differential equation:

dxt = (Axt + But + b)dt + ∑mi=1(Ci xt + Di ut + σi )dW i

t ,

xt , b, σi ∈ Rn ; A, Ci ∈ Rn×n ; B, Di ∈ Rn×k

and i ∈ {1, . . . , m}, xt0 ∈ R.

(2.1)

The variables of this equation have the following ‘physical’ interpretation:

- xt ∈ Rn is the vector of the states of the model (2.1) at time t � t0 � 0;

- ut ∈ Rk is the vector of the control action (at time t) which should be suggested by a designer tostabilize this system in the origin;

- W it (i ∈ {1, . . . , m}) is a standard scalar Brownian motion characterizing external random pertur-

bations to the vector-state dynamics satisfying (2.1); in engineering applications, it is referred asa ‘white noise’ external perturbation affecting the given model dynamics;

- the constant matrices A, Ci ∈ Rn×n , B, Di ∈ R

n×k and the constant scalars b and σi are as-sociated with the parameters of the considered models and are supposed below to be a prioriknown.

REMARK 2.1 Note that in most of the publications there are studied only the case Ci = 0, Di = 0(i = 1, . . . , m) corresponding to the, so-called, ‘additive noise’ presence which keeps the same ‘powersignal’ (σi = consti �= 0) independently of the closeness of the current state xt to the origin. Whenσi = 0 for all i and at least one of Ci �= 0 (but Di = 0 (i = 1, . . . , m)), we deal with the, so-called,‘multiplicative noise’ case which reflects the noise-decreasing (vanishing) property of some models

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been considered closer to the point x = 0. If at least one Di �= 0, the corresponding dynamics (2.1)has the controllable diffusion term reflecting the fact that in this situation the control action ut may alsoaffect the noise power.

Consider below two ‘practical’ motivating example models studied in earlier publications.

EXAMPLE 2.1 Consider a gene regulatory network, which is a collection of DNA segments that interactwith each other through their messenger RNA (mRNA) and some protein expression products. Theconditions that govern the genetic expression of some proteins permit to model this class of systems asstochastic models. Under certain conditions, the variations may produce genetic disorders, leading tomedical diseases. Controlling the mRNA is a gene therapy alternative, which allows us to modify themRNA contents. This action helps to prevent diseases or to produce certain types of proteins.

The model of a gene regulatory network can be represented by a linear stochastic model of the form(see Ozbudak et al., 2002; El Samad & Khammash, 2004; Chunguang et al., 2006)

dxt = Axt dt + Ξ xt dWt

xt ∈ R2, A ∈ R2×2, Ξ ∈ R2×2,(2.2)

where

- x1,t is the number of mRNA molecules;

- x2,t is the number of protein molecules. The states are defined in terms of an equilibrium point;

-

A =[−γR −K2

K Po −γPo

],

where γR and γPo represent the decay rates of mRNA and protein, respectively, K Po is the tran-scription rate and K R is the translation rate, whose form is set to be K R = K2 − K1 Po, whereK1, K2 ∈ R, and Po the end proteins product, according to El Samad & Khammash (2004).

- Ξ represents the constant matrix which corrupts the parameters γR, γPo , K Po and K R .

EXAMPLE 2.2 (The reinsurance–dividend management; Taksar & Zhou, 1998) Consider the following{Ft }t�0-adapted R-valued random processes

dy(t) = [a(t)μα − δα − c(t)]dt − a(t)σα dW (t),

y(0) = y0,

where

- y(t) is the value of the liquid assets of a company at time t ,

- c(t) is the dividend rate paid out to the shareholder at time t ,

- μα is a difference between premium rate and expected payment on claims per unit time (‘safetyloading’),

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- δα is the rate of the debt repayment,

- [1 − a(t)] is the reinsurance fraction,

- σα := √λα E{η2} (λα is the intensity of Poisson process, η is the size of claim).

Here, the controls are

u1(t) := a(t) ∈ [0, 1] and u2(t) := c(t) ∈ [0, c+].

Certainly, this example corresponds the situation with a controllable diffusion term.

Below, we will consider the linear feedback

ut := K xt , K ∈ Rk×n, (2.3)

and study the capacity of such feedback to stabilize the considered system in some probabilistic sense.Substituting (2.3) into (2.1) leads to

dxt = [(A + BK )xt + b]dt +m∑

i=1

[(Ci + Di K )xt + σi ]dW it . (2.4)

Let P be a positive definite matrix in Rn×n satisfying

0 < β In×n � P � α In×n .

Define also the quadratic function

V (x) := x� Px

for which at the trajectories x = xt of the system (2.4) there exists (under the accepted assumptions) themathematical expectation

V (t) := E{V (xt )} (2.5)

(which is, obviously, differentiable if we apply the Ito’s calculus to dV (xt )). Now we are ready toformulate the first main result.

Introduce the extended vector

zt := (x�

t b� σ�1 · · · σ�

m

)� ∈ Rn(m+2). (2.6)

LEMMA 2.1 Under the accepted assumptions for any non-negative α, ε and κ , the following represen-tation holds:

d

dtV (t) = E{z�

t W zt } − αV (t) + β, (2.7)

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where

W :=

⎡⎢⎢⎢⎢⎢⎣W11 W12 W13 · · · W1,m+2W21 W22 0 · · · 0W31 0 W33 · · · 0

......

.... . .

...Wm+2,1 0 0 · · · Wm+2,m+2

⎤⎥⎥⎥⎥⎥⎦ , (2.8)

W11 :=[

P(

A + BK + α

2In×n

)+

(A + BK + α

2In×n

)�P

]+

m∑i=1

[(Ci + Di K )� P(Ci + Di K )] ∈ Rn×n,

W12 = W �21 := P, W22 := −ε In×n, W j, j := P − κ In×n,

W j1 = W �1 j := P(C j−2 + D j−2 K ), j ∈ {3, . . . , m + 2},

and

β := ε‖b‖2 + κ

m∑i=1

‖σi‖2. (2.9)

Proof. From the multidimensional Ito’s formula (see, e.g. Poznyak, 2009), we derive

dV (xt ) = ∂V

∂t(xt )dt +

[∂V

∂x(xt )

]�dxt

+1

2tr

⎧⎨⎩[

∂2V

∂x2(xt )

]� m∑i=1

[(Ci + Di K )xt + σi ][(Ci + Di K )xt + σi ]�⎫⎬⎭ dt.

Replacing dxt from (2.4) and taking into account that

∂V

∂t(xt ) ≡ 0,

∂V

∂x(xt ) = 2x� P,

∂2V

∂x2(xt ) = 2P,

we obtain in the integral form

V (xt ) − V(xt0

) =t∫

t0

2x�s P[(A + BK )xs + b]ds +

∫ t

t0

m∑i=1

2x�s P[(Ci + Di K )xs + σi ]dW i

s

+∫ t

t0

m∑i=1

tr{P[(Ci + Di K )xs + σi ][(Ci + Di K )xs + σi ]�}ds.

Taking mathematical expectation of both parts and in view of the local properties of Ito’s integral, weconclude that

E

{∫ t

t02x�

s P[(Ci + Di K )xs + σi ]dW is

}= 0

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and

E{V (xt )} − E{

V(xt0

)} = E

{∫ t

t02x�

s P[(A + BK )xs + b]ds

}+

m∑i=1

∫ t

t0tr{E{P[(Ci + Di K )xs + σi ][(Ci + Di K )xs + σi ]

�}}ds.

By the property tr{AC} = tr{C A}, the last equality can be represented as

E{V (xt )} − E{

V(xt0

)} =t∫

t0

2E{x�s P[(A + BK )xs + b]}ds

+m∑

i=1

E∫ t

t0tr{[(Ci + Di K )xs + σi ]

� P[(Ci + Di K )xs + σi ]}ds.

Calculating the derivative of both sides of the last identity, we obtain

d

dtE{V (xt )} = E{2x�

t P[(A + BK )xt + b]}

+m∑

i=1

tr{E{[(Ci + Di K )xt + σi ]� P[(Ci + Di K )xt + σi ]}}.

Using the definition (2.5) and the property 2x� Ax = x�(A + A�)x , we are able to rewrite the lastequation as

d

dtV (t) =E{x�

t [P(A + BK ) + (A + BK )� P+]xt } + E

{x�

t

m∑i=1

[(Ci + Di K )� P(Ci + Di K )]xt

}

+ E{x�t Pb} + E

{x�

t

m∑i=1

[(Ci + Di K )� Pσi ]

}+ E{bPxt }

+ E

{m∑

i=1

[σ�i P(Ci + Di K )]xt

}+

m∑i=1

σ�i Pσi , (2.10)

and adding and subtracting the terms [±αV (xt )], [±ε‖b‖2] and[ ± ∑m

i=1 κ‖σi‖2], one can represent

(2.10) as the ordinary differential equation (2.7). �The following result is the consequence of the above lemma.

COROLLARY 2.1 For any matrices P, K and constants α, β and κ such that W < 0 (2.8), the followingdifferential inequality holds:

d

dtV (t) � −αV (t) + β (2.11)

so that

V (t) � e−α(tn−t0)V (t0) + β

α(1 + e−α(tn−t0)) = β

α+ O(e−α(t−t0)). (2.12)

Proof. It follows directly from (2.7) if we take into account that W < 0. �

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By the previous corollary, we obtain the result that describes the sufficient conditions of the statestabilization.

THEOREM 2.1 If there exists a real positive number τ < β and the collection {K , P, α, β, τ } satisfyingthe following matrix inequality:

Υ := W − τα

βI � P I � 0 (2.13)

with

I = (In×n On×n · · · On×n

) ∈ Rn×n(m+2),

then the best parameters minimizing the convergence attractive ellipsoid are

{K ∗, P∗, α∗, β∗, τ ∗} = Arg supα>0,β>0,τ>0,Υ <0

tr

{α

β − τP

}.

Proof.From (2.11), (2.8) and (2.6), define

A0 := W ∈ Rn(m+2)×n(m+2),

A1 :=

⎡⎢⎢⎢⎣−α

β P On×n · · · On×n

On×n On×n · · · On×n...

.... . .

...On×n On×n · · · On×n

⎤⎥⎥⎥⎦ ∈ Rn(m+2)×n(m+2)

and

fi (zt ) := zt Ai zt , i = 1, 2,

where

xt = In×nzt .

By S-procedure (see 12.3.2, Poznyak, 2008), the inequality

f1(zt ) < −1

implies that

f0(zt ) < 0

if only if there exists τ � 0 such that

A0 − τ A1 < 0

given by (2.13). Since the attractive ellipsoid is given by

lim supt→∞

V (t) = lim supt→∞

E{x�t Pxt },

then the maximum of P (in fact, its trace) implies a smaller dynamics xt . �

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The previous result deals with the convergence to zero zone (ellipsoid zone) in mean-square sense.The next corollary shows that the trajectories of the closed-loop system also converge to the sameellipsoid with probability one.

COROLLARY 2.2 Under the conditions of Theorem 2.1, we may guarantee that

P

{ω: lim

t→∞ d

(xt , S

(0,

α

β − τP

))= 0

}= 1, (2.14)

whered(x, S) := inf

y∈S‖x − y‖,

S(0, P) := {x ∈ Rn : x� Px � 1}.Proof. Let {tk}k=0,1,2,... be any monotonically increasing sequence from R+

(tk < tk+1, tk →

k→∞ ∞).

Then, by the Borel–Cantelli lemma∞∑

k=0

χ

{[V

(xtk

) − β

α

]+� ε > 0

}a.s.< ∞, (2.15)

χ{A} ={

1 if the event A occurs,0 if not,

[z]+ :={

z if z � 0,0 if z < 0,

if∞∑

k=0

P

{[V

(xtk

) − β

α

]+� ε > 0

}< ∞.

But, by the generalized Chebyshev inequality (Poznyak, 2009) and in view of (2.12), it follows that∞∑

k=0

P

{[V

(xtk

) − β

α

]+� ε

}� ε−1

∞∑k=0

E

{[V

(xtk

) − β

α

]+

}

� ε−1∞∑

k=0

E

{[e−α(tk−t0)

[V

(xt0

) + β

α

]]+

}

= ε−1[

V(xt0

) + β

α

] ∞∑k=0

e−α(tk−t0) < ∞.

The convergence of the series (2.15) with probability one for any small ε > 0 and any sequence{tk}k=0,1,2,... exactly means that for almost all ω ∈ Ω and any sequence {tk}k=0,1,2,..., there exists a

random (but finite) number k+(ω)a.s.< ∞ such that for all k � k+(ω) we have χ

{[V(xtk

) − βα

]+ �

ε > 0} a.s.= 0. If this is true for any sequence {tk}k=0,1,2,..., it means that

[V (xt ) − β

α

]+

a.s.→ 0, whereast → ∞. Corollary is proven. �COROLLARY 2.3 If β = 0, i.e. in (2.9) ‖b‖ = ‖σi‖ = 0 (i = 1, . . . , m), and τ = 0 in Theorem 2.1, wehave global stability with probability one of the origin x = 0 of the closed-loop system (2.4) containingthe diffusion term depending linearly on state and control (2.3).

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3. LMI representation of the optimization problem

As it follows from (2.4), if one wishes to ‘maximize’ P by the matrix K and other scalar parameters insome ‘matrix’ sense, then the matrix gain K should be found as the solution of

α

β − τtrP → sup

K ,ε,κ,α,τ : Υ <0. (3.1)

From Theorem 2.1, we define Λ = −Υ > 0

Λ =

⎡⎢⎢⎢⎢⎢⎣−W(2) Λ12 Λ13 · · · Λ1,m+2Λ21 Λ22 0 · · · 0Λ31 0 Λ3,3 · · · 0...

......

. . ....

Λm+2,1 0 0 · · · Λm+2,m+2

⎤⎥⎥⎥⎥⎥⎦ ,

W(2) :=[

P

(A + α

2

(1 + τ

β

)n×n

+ BK

)+

(A + α

2

(1 + τ

β

)In×n + BK

)�P

]

+[

m∑i=1

(Ci + Di K )� P(Ci + Di K )

],

Λ12 = Λ�21 := −W12, Λ22 := −W22, Λ j, j := −W j, j ,

Λ j1 = Λ�1 j := −W1 j , j ∈ {3, . . . , m + 2}.

Note that the matrix Λ is not linear with respect to our variables even for fixed scalar parameters.The principal problem is represented in the expression (Ci + Di K )� P(Ci + Di K ). Let Q1,i be a non-negative matrix such that (Ci +Di K )� P((Ci +Di K )�)� � Q1,i for i ∈ {1, . . . , m}. By the applicationof the Schur’s complement, we have[

Q1,i (Ci + Di K )�(Ci + Di K ) P−1

]� 0. (3.2)

Multiplying (3.2) by two positive matrices, the equivalent inequality is[In×n 0

0 P

] [Q1,i (Ci + Di K )�

(Ci + Di K ) P−1

] [In×n 0

0 P

]� 0,[

Q1,i C�i P + (Di K )� P

PCi + P Di K P

]=

θ1,i =P Di K

[Q1,i C�

i P + θ�1,i

PCi + θ1,i P

]� 0

(3.3)

for i ∈ {1, . . . , m}. With similar arguments, we analyse P(Ci + Di K ). For P BK , consider θ2 = P BK .Thus,

W :=

⎡⎢⎢⎢⎢⎢⎣W(3) W12 W13 · · · W1m+2

W21 W22 0 · · · 0W31 0 W33 · · · 0

......

.... . .

...

Wm+2,1 0 0 · · · Wm+2,m+2

⎤⎥⎥⎥⎥⎥⎦ (3.4)

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and

W(3) :=[

P

(A + α

2

(1 + τ

β

)In×n

)+

(A + α

2

(1 + τ

β

)In×n

)�P

]

+θ2 + θ�2 +

m∑i=1

Q1,i ,

W12 = W �21 := P, W22 := −ε In×n, W j, j := P − κ In×n,

W j1 = W �1 j := PC j + θ1, j , j ∈ {3, . . . , m + 2}.

For solving these LMIs (3.2), (3.3) and (3.4), first we fix the scalar parameters α, κ, β, τ and ε and solveour problem with respect to the matrix variables which satisfy the LMI constraints. And second, for thefound matrix variables θ1,i , θ2 and Q1,i for i ∈ {1, . . . , m}, we solve our problem with respect to thescalar parameters α, κ, β, τ and ε.

Finally, we find the solution α∗, κ∗, β∗, τ ∗, ε∗, θ∗1,i , θ

∗2 and Q∗

1,i for i ∈ {1, . . . , m}. We find P∗ andK ∗ from (3.3). This problem can be solved using the MATLAB, LMI toolbox, SeDuMi or Yalmip.

4. Numerical example

Below, we present some numerical simulations to confirm the effectiveness of the proposed stabilizationstrategy. Consider a stochastic system of the form (2.4) and a proportional control law as in (2.3). Theblock diagram of the control approach is depicted in Fig.1. We consider two examples: benchmark andgene regulatory network.

4.1 Benchmark example

For the numerical illustrative example, we consider i = 1, m = 1 and the following stochastic differen-tial equation of the second order:

dxt = [Axt + But ]dt + (Cxt + Dut + σ)dWt

FIG. 1. The formal formulation problem.

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with A, C ∈ R2×2, B, D, σ ∈ R2×1 and

u(t) = K xt , K ∈ R1×2,

so that

dxt = [A + BK ]xt dt + [(C + DK )xt + σ ]dWt . (4.1)

By (4.1), for the matrices (3.3), (3.4) we have

A =[

0.4 −0.81.8 −2.1

], B =

[0.51

],

C =[−1.9 −1.8

0.8 0.5

], D =

[0.11

], σ =

[0.30.3

].

We also have

W :=(

W(3) C� P + θ�1

PC + θ1 P − κ I2×2

),

W(3) :=[

P

(A + α

2

(1 + τ

β

)In×n

)+

(A + α

2

(1 + τ

β

)In×n

)�P

]+ θ2 + θ�

2 + Q1, (4.2)

where Q1 � 0 satisfies the additional inequality (3.3)[Q1 C� P + θ�

1PC + θ1 P

]� 0,

θ1 = P DK , θ2 = P BK .

Consider now three case studies:

- the first one is free of any restrictions (τ = 0);- in the second one, we try to obtain a smaller ellipsoidal region using the parameter τ as an addi-

tional tool for the optimization process;- in the last case, we show the state convergence when β = 0, i.e. when uncertainties are absent, and

we guarantee the convergence to the origin with probability one.

When τ = 0, in (4.2),

W(3) :=[

P(

A + α

2In×n

)+

(A + α

2In×n

)�P

]+ θ2 + θ�

2 + Q1.

The numerical implementation of the noisy signal in Simulink is given in terms of a Gaussian noisesignal generator Wt , using the following approximation:

�Wt

�t≈ Wt − Wt−h

h,

where 0 < h 1. Figures 2 and 3 show the state trajectories within the corresponding ellipsoid.

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FIG. 2. The trajectories of state.

FIG. 3. The trajectory of state in the corresponding ellipsoid.

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The numerical solutions of the optimization problem (3.1) are

P∗ =[

0.2384 −0.0000−0.0000 0.2384

], K ∗ = [−5.2662 − 2.6492],

β+ = 0.2622, α∗ = 0.5233, κ∗ = 0.26.

When the restricting parameter τ is positive, we obtain the above problem formulation as in (4.2) and(3.3). The corresponding state trajectories are depicted in Figs 4 and 5.

In this case,

P∗ =[

0.0259 −0.0000−0.0000 0.0259

], K + = [−434.3058 − 268.8806],

τ ∗ = 0.001, β∗ = 0.0285, α∗ = 13.4588, κ∗ = 0.1.

The last case, β = 0, is illustrated in Fig. 6

K ∗ = [2.9832 − 20.8990], α∗ = 98.

4.2 A gene regulation system

Consider a model of a gene regulatory network as in Example 2.1, which is affected by the presence ofstochastic noise. Using the approach presented in El Samad & Khammash (2004), our aim is actually tostabilize (2.2) to the equilibrium point

(R∗, P∗o ) =

(K1γPo

K2 K Po + γRγPo

,K1 K Po

K2 K Po + γRγPo

),

FIG. 4. The trajectory of state in the corresponding ellipsoid, in the case τ �= 0.

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FIG. 5. The trajectory of state in the corresponding ellipsoid, in the case τ �= 0.

FIG. 6. The trajectory of state in probability one.

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with the coordinate change given by

x(t) =[

x1(t)x2(t)

]=

[R − R∗Po − P∗

o

],

where R and Po are the actual system states. As a consequence, if we wish to stabilize x�(t) :=[x1, x2]� to the origin, it means that we wish to show that the state x(t) will converge to the equilibriumpoint (R∗, P∗

o ). In terms of the new variables, we have[dx1(t)dx2(t)

]=

[−γR −K2K Po −γPo

] [x1(t)x2(t)

]dt. (4.3)

When the parameters −γR, −K2, K Po and −γPo are perturbed by a stochastic Gaussian noise {Wt }t�0,we can finally rewrite (4.3) as[

dx1(t)dx2(t)

]=


] [x1(t)x2(t)

]dt +

[δ1 δ2δ3 δ4

] [x1(t)x2(t)

]dWt .

On the other hand, if we consider the additive disturbances (such as measurement noise or environmentalvariations), the above equation becomes[

dx1(t)dx2(t)

]=


] [x1(t)x2(t)

]dt +

[[δ1 δ2δ3 δ4

] [x1(t)x2(t)

]+

[σ1σ2

]]dWt .

Taking u(t) = K x(t) implies that[dx1(t)dx2(t)

]=

([−γR −K2K Po −γPo

]+

[K1 00 K2

])[x1(t)x2(t)

]dt

+[([−δ1 −δ2

δ3 −δ4

]+

[K1 00 K2

])[x1(t)x2(t)

]+

[σ1σ2

]]dWt .

(4.4)

Denote

A =[−γR −K2

K Po −γPo

], C =

[−δ1 −δ2δ3 −δ4

], σ =

[σ1σ2

]and for (4.4), consider the following matrix values:

A =[−0.35 0.001

0.8 −0.0049

], C =

[−0.8 −0.10.2 −0.5

], σ =

[0.30.3

].

Following the same procedure as in the first numerical example (see (4.2)), we obtain the followingresults:

P∗o =

[0.0115 −0.0000

−0.0000 0.0115

], K ∗ =

[−24.2925 00 −24.6376

],

τ∗ = 0.01, β∗ = 0.0127, α∗ = 0.0152, κ = 0.28.

The corresponding trajectories of x(t) are presented below in Figs 7–9.

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FIG. 7. The trajectory of state x1.

FIG. 8. The trajectory of state x2.

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FIG. 9. The convergence trajectory of the states into the ellipsoid.

5. Conclusions

This paper proposes the implemented algorithm for the designing of a robust stabilizing control for theclass of LCSDEs with uncertainties. As a result of the suggested numerical optimization procedure,we construct an attractive ellipsoid of a ‘minimal size’ and calculate numerically the corresponding‘optimal’ gain matrix. The computational algorithm proposed in this work makes it possible to generatethe ‘best’ proportional feedback control law. Two examples illustrate the effectiveness of the suggestedapproach.

REFERENCES

ABEDOR, J., NAGAPAL, K. & POOLA, K. (1996) A linear matrix inequality approach to peak-to-peak gainminimization. Int. J. Robust Nonlinear Control, 6, 899–927.

ANTENEODO, C. & RIERA, R. (2005) Additive-multiplicative stochastic models of financial mean-reverting pro-cesses. Phys. Rev., 72, 1–7.

APPLEBY, J. A. D. & FLYNN, A. (2006) Stabilization of Volterra equations by noise. J. Appl. Math. Stoch. Anal.,2006, 1–29.

ARNOLD, L. & SCHMALFUSS, B. (2001) Lyapunov’s second method for random dynamical systems. J. Differ.Equ., 177, 235–265.

BERTSEKAS, D. (1994) Infinite time reachability of state-space regions by using feedback control. IEEE Trans.Autom. Control, 17, 604–613.

BLANCHINI, F. & MIANI, S. (2007) Set Theoretic Methods in Control. Systems & Control: Foundations & Appli-cations. Boston, MA: Birkhauser.

CHUNGUANG, L., CHEN, L. & AIHARA, K. (2006) Stability of genetic networks with SUM regulatory logic:Lur’e system and LMI approach. IEEE Trans. Circuits Syst. I Regul. Pap., 53, 2451–2458.

18


³n y Estudios A


on April 30, 2013

http://imam


ownloaded from



EL BOUHTOURI, A. & EL HADRI, K. (2003) Robust stabilization of jump linear systems with multiplicative noise.IMA J. Math. Control Inf., 20, 1–19.

EL KAROUI, N., JEANBLANC-PICQUE, M. & SHREVE, S. E. (1998) Robustness of the Black and Scholesformula. Math. Financ., 8, 93–126.

EL SAMAD, H. & KHAMMASH, M. (2004) Stochastic stability and its application to the analysis of gene regulatorynetworks. IEEE Conf. Decis. Control, 3, 3001–3006.

GRAY, W. S., GONZALES, O. R. & DOGAN, M. (1999) Stochastic perturbation models of electromagnetic dis-turbances in closed-loop computer controlled flight systems. Proceedings of the 18th Digital Avionics SystemsConference, St Louis, MO, vol. 2, IEEE Conference Publications, pp. 1–10.

KONSTANTATOS, G., HOWARD, I., FISCHER, A., HOOGLAND, S., CLIFFORD, J., KLEM, E., LEVINA, L. &SARGENT, E. H. (2006) Ultrasensitive solution-cast quantum dot photodetectors. Nat. Pub. Group, 442,180–183.

KURZHANSKII, A. & VALYI, I. (1997) Ellipsoidal Calculus for Estimation and Control. Boston, MA: Birkhauser.NAGAPAL, K., ABEDOR, J. & POOLA, K. (1994) An LMI approach to peak-to-peak gain minimization: filtering

and control. Proceedings of the American Control Conference, Baltimore, MD, vol. 1, Conference Publica-tions, pp. 742–746.

NESTEROV, Y. E. & NEMIROVSKY, A. (1994) Interior-Point Polynomial Methods in Convex Programming. Studiesin Applied Mathematics, vol. 13. Philadelphia, PA: SIAM.

OZBUDAK, E. M., THATTAI, M., KURSTER, I., GROSSMAN, A. D. & VAN OUDERNARDEN, A. (2002) Regula-tion of noise in the expression of a single gene. Nat. Genet., 31, 69–73.

POLYAK, B., NAZIN, A. V., TOPUNOV, M. V. & NAZIN, S. A. (2006) Rejection of bounded disturbances viainvariant ellipsoids technique. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego,CA, Conference Publications, pp. 1429–1434.

POZNYAK, A., DUNCAN, T., PASIK-DUNCAN, B. & BOLTYANSKY, V. (2002) Robust stochastic maximum prin-ciple for multi-model worst case optimization. Int. J. Control, 75, 1032–1048.

POZNYAK, A. S. (2008) Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Tech-niques, vol. 1. London: Elsevier.

POZNYAK, A. S. (2009) Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques,vol. 2. London: Elsevier.

RICHARDSON, C. W. (1981) Stochastic simulation of daily precipitation, temperature, and solar radiation. WaterResour. Res., 17, 182–190.

SAFONOV, M. G., GOH, K. C. & LY, J. H. (1994) Control system synthesis via bilinear matrix inequalities.American Control Conference, Baltimore, MD, (June-1 July), vol. 1, Conference Publications, pp. 45–49.

SCHWEPPE, F. (1973) Uncertain Dynamic Systems. Englewood Cliffs, NJ: Prentice-Hall.SHREVE, S. E. (2004) Stochastic Calculus for Finance. II. Continuous-Time Models. New York: Springer.SYKES, Z. M. (1969) Some stochastic versions of the matrix model for population dynamics. J. Am. Stat. Assoc.

Pub. Info., 64, 111–130.TAKSAR, M. I. & ZHOU, X. Y. (1998) Optimal risk and dividend control for a company with a debt liability. Insur

Math. Econ., 22, 105–122.UGRINOVSKII, V. A. (1998) Robust H-infinity control in the presence of stochastic uncertainty. Int. J. Control, 71,

219–237.ZHOU, X. Y. (1991) A unified treatment of maximum principle and dynamic programming in stochastic controls.

Stoch. Int. J. Probab. Stoch. Process. (formerly Stochastics and Stochastics Reports), 36, 137–161.

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Robust control design through the attractive ellipsoid ... · Robust control design through the attractive ellipsoid technique for a class of linear stochastic models with multiplicative