# Stability Criterion of Linear Stochastic Systems Subject to MixedH2/Passivity Performance

Authors: Cheung-Chieh Ku; Chih-Peng Lu; Min-Ta Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

TheH2 control scheme and passivity theory are applied to investigate the stability criterion of continuous-time linear stochastic system subject to mixed performance. Based on the stochastic differential equation, the stochastic behaviors can be described as multiplicative noise terms. For the considered system, the H2 control scheme is applied to deal with the problem on minimizing output energy. And the asymptotical stability of the system can be guaranteed under desired initial conditions. Besides, the passivity theory is employed to constrain the effect of external disturbance on the system. Moreover, the Itô formula and Lyapunov function are used to derive the sufficient conditions which are converted into linear matrix inequality (LMI) form for applying convex optimization algorithm. Via solving the sufficient conditions, the state feedback controller can be established such that the asymptotical stability and mixed performance of the system are achieved in the mean square. Finally, the synchronous generator system is used to verify the effectiveness and applicability of the proposed design method.


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## Body

## 1. Introduction

In real circumstance, the external disturbance usually causes kinds of the poor control performance and instability of the dynamic systems. For achieving the good control performance of the disturbed systems, many control schemes [1–3] have been proposed to constrain the effect of external disturbance on the systems. In [3–6], the H∞ control scheme is proposed to achieve the attenuation γ based on the relation between state and disturbance. On the other hand, the passivity theory was applied to deal with the disturbance attenuation performance of the systems in [7, 8]. With setting the power supply function [9], passivity theory can be converted into various constrains including H∞ control scheme, H2 control scheme, positive real theory, and several passive constraints. Thus, the passivity theory becomes the powerful and general tool to discuss the disturbance attenuation performance of the systems.Recently, the mixed control method [10, 11] via combining the H2 and H∞ control schemes has been proposed to solve the control problems subject to required performances. Generally, the H2 control scheme is applied to minimize the output energy of the system and to guarantee the stability of the system under initial conditions. The attenuation performance is discussed via H∞ control scheme with given attenuation γ for the worst case of external disturbance input. In [12], the mixed H2/H∞ performance of uncertain spacecraft systems is achieved via designed PID tracking controller. Based on mixed H2/H∞ performance control method, the stability and performance of robotic manipulator have been discussed in [13]. In addition to mixed H2/H∞ performance control scheme, the passivity theory was also applied to discuss the mixed performance of the system via combining the H2 control scheme or H∞ control scheme. Referring to [14, 15], passivity theory is used to analyze the stability of system and H2 control scheme is used to constrain the concerned signal resources. However, most of the literatures discussing mixed performance control problem are focused on deterministic systems. Few papers [16] have been proposed to study the mixed performance of stochastic systems. Since the stochastic behavior usually exists in the practical operating conditions, the investigation for mixed performance control problem of stochastic system is worth being discussed.Stochastic systems have received much attention since stochastic differential equation is proposed with modeling approach [17–21]. Through the modeling approach, the stochastic behaviors can be described via the multiplicative noise term structured by multiplying state and noise. The noise is considered as Brownian motion [18, 19] to imitate natural random variation. For analyzing the stability of stochastic systems, the Itô formula [18] and Lyapunov function are applied to derive the sufficient conditions. During the analyzing process, the mean square calculation is the most useful tool for solving the stability and stabilization problems of stochastic system in the sense of mean square. According to the modeling approach and Itô formula, many stability criteria of the deterministic system have been extended to stochastic cases.For the above motivation and illustration, the mixedH2/passivity performance controller design for linear stochastic systems is discussed in this paper. The H2 control scheme applied in this paper is to minimize the output energy of the system. Moreover, the asymptotical stability of the system can be guaranteed under initial conditions via H2 control scheme. For constraining the effect of external disturbance on the system, the passivity theory is used with general power supply function which decides the attenuation performance index. Based on the Itô formula and Lyapunov function, the sufficient conditions are derived such that the asymptotical stability and mixed performance of the system are guaranteed in the sense of mean square. For applying convex optimization algorithm, the sufficient conditions are converted into LMI form [22]. The main contribution of this paper is to propose a mixed H2/passivity performance controller design method for the linear stochastic systems. Through the setting power supply function, the choice of attenuation performance is more flexible and general than previous mixed H2/H∞ performance methods. Finally, the synchronous generator system with multiplicative noise is applied to demonstrate the proposed design method.This paper is organized as follows: In Section2, the linear stochastic systems and mixed performance control problem are illustrated. Through the Itô formula and Lyapunov function, the sufficient conditions are derived in Section 3. A numerical simulation is proposed in Section 4. Finally, the conclusion is in Section 5.


## 2. System Description and Definition Statement

In this paper, the linear stochastic system with external disturbance is described as follows:(1a)dxt=Axt+But+Evtdt+A-xt+B-utdβt,(1b)yt=C1xt+D1vt,(1c)zt=C2xt+D2ut,
where xt∈Rn is the state vector, yt∈Rm is the measured output vector, zt∈Rp is the controlled output vector, ut∈Rq is the control input vector, vt∈Rm is the disturbance input vector, and βt is a scalar of Brownian motion which satisfies the independent increment properties [18], such as Edβt=0 and Edβtdβt=1. E· denotes the expected value of ·. The A, A-, B, B-, C1, C2, D1, D2, and E are constant matrices with compatible dimensions.For the stabilization problem of (1a), (1b), and (1c), the following controller is proposed in this paper:
(2)ut=Fxt,
where F is feedback gain with compatible dimensions. Substituting (2) into (1a), (1b), and (1c), the following closed-loop system is formulated:
(3a)dxt=Afxt+Evtdt+A-fxtdβt,(3b)yt=C1xt+D1vt,(3c)zt=C2fxt,
where Af=A+BF, A-f=A-+B-F, and C2f=C2+D2F.For considering mixed performances, theH2 control scheme and passivity theory are applied. By setting the power supply function, the passivity theory includes several performance constraints [9]. In this paper, the general power supply function is considered for providing flexible choice of performance index to constrain the external disturbance input. Thus, the general passive constraint is introduced as in the following definition.Definition 1.
The closed-loop system (3a), (3b), and (3c) with external disturbance input vt and measured output yt is called passive if there exist known matrices S1, S2, and S3 such that
(4)E2∫0tpyTtS1vtdt>E∫0tpyTtS2ytdt+∫0tpvTtS3vtdt,
where tp>0.Remark 2.
According to the matricesS1, S2, and S3 defined by [9], the passivity theory is more general and flexible than H∞ control scheme for constraining the effect of external disturbance on the system. Besides, both of state-dependent noise and input-dependent noise are considered for the stochastic system in this paper. Thus, the proposed mixed H2/passivity performance control design method is more general and flexible than method in [16] for disturbance attenuation performance of stochastic systems.In addition, theH2 control scheme is employed to guarantee the stability of the closed-loop system (3a), (3b), and (3c) under given initial condition. And, the output energy can also be minimized via achieving the H2 performance. Thus, the H2 performance is defined in the following definition.Definition 3.
The controller (2) is an H2 performance measurement if the following inequality is satisfied:
(5)E∫0∞zTtztdt<α.
This definition can also be called as minimizing problem for output with α.Through the above definitions, the mixed performance of (3a), (3b), and (3c) is discussed to minimize output energy with constraining effect of external disturbance. Based on the above illustration, the sufficient conditions are derived for stability criterion of closed-loop system (3a), (3b), and (3c) in the next section.


## 3. Mixed Controller Design for Considered System

Applying the Lyapunov function and Itô formula, the sufficient conditions are derived to analyze the stability of closed-loop system (3a), (3b), and (3c). Furthermore, the sufficient conditions are applied to find the feedback gain F to achieve the mixed H2/passivity performance of (3a), (3b), and (3c). For using convex optimization algorithm, the conditions are converted into LMI problems for finding the feasible solutions.Theorem 4.
Given matricesS1, S2, and S3, the closed-loop system (3a), (3b), and (3c) is asymptotically stable and satisfies the mixed H2/passivity performance in the sense of mean square, if there exist positive scalar α, positive definite matrix P, and feedback gain matrix F such that one has the following:minimizeα subject to(6)C1TS2C1+AfTP+PAf+A-fTPA-f*-S1TC1+D1TS2C1+ETPS3-D1TS1-S1D1+D1TS2D1<0,(7)AfTP+PAf+A-fTPA-f+C2fTC2f<0,(8)xT0Px0<α.Proof.
Choose the following Lyapunov function:(9)Vxt=xTtPxt.
And the derivation of the Vxt along with trajectories of (3a), (3b), and (3c) can be obtained by Itô formula [18], such as
(10)dVxt=LVxtdt+2xTtA-fxtdβt,
where
(11)LVxt=xTtAfTP+PAf+A-fTPA-fxt+vTtETPxt+xTtPEvt.
Arranging (11), one has
(12)LVxt=xtvtTAfTP+PAf+A-fTPA-f*ETP0xtvt.
Let us take expectation of (10); one has the following equation with the independent increment property as Extdβt=0 and Evtdβt=0:
(13)EdVxt=ELVxtdt.
In case as nonzero external disturbancevt≠0, the following cost function can be defined with zero initial conditions:
(14)Γx,v,t=E∫0tpyTtS2yt+vTtS3vtllllllllllllllll-yTt2yTtS1vtdt∫0tpyTtS2yt+vTtS3vt=E∫0tpyTtS2yt+vTtS3vt-2yTtllllllllllllllll·S1vt+LVxtyTtS2yt+vTtS3vtdt-Vxtp∫0tp2yTtS1vt+LVxtyTtS2yt+vTtS3vt≤E∫0tpΨx,v,tdt,
where
(15)Ψx,v,t=yTtS2yt+vTtS3vt-2yTtS1vt+LVxt.
Substituting (3b) and (12) into (15), one has the following:
(16)Ψx,v,t=xtvtTΛxtvt,
where(17)Λ=C1TS2C1+AfTP+PAf+A-fTPA-f*-S1TC1+D1TS2C1+ETPS3-D1TS1-S1D1+D1TS2D1.Obviously, (17) is equal to the left-hand side of inequality (6). Thus, if the condition of this theorem is held then one can find the Λ<0 from (17). Since the Λ<0, the Ψ(x,v,t)<0 is found from (16). Due to Ψ(x,v,t)<0, Γx,v,k<0 can be inferred from (14). And then, the following inequalities can be found since Γx,v,k<0:
(18a)E∫0tpyTtS2yt+vTtS3vt-2yTtS1vtdt<0
or
(18b)E2∫0tpyTtS1vtdt>E∫0tpyTtS2ytdt+∫0tpvTtS3vtdt.
Because (18b) is equivalent to (4), the closed-loop system (3a), (3b), and (3c) is passive with given S1, S2, and S3. Next, the asymptotical stability of (3a), (3b), and (3c) is proven. Assuming the disturbance input is zero vt=0, (12) can be rewritten as follows:
(19)LVxt=xTtAfTP+PAf+A-fTPA-fxt.
Furthermore, the following relation can be derived with (3c):
(20)LVxt=xTtAfTP+PAf+A-fTPA-fxt+zTtzt-zTtzt=xTtAfTP+PAf+A-fTPA-f+C2fTC2fxt-zTtzt<Ξ,
where Ξ=xTtAfTP+PAf+A-fTPA-f+C2fTC2fxt.
One can find that if condition (7) is held then Ξ<0 can easily be found. Based on the Ξ<0, the LVxt<0 can also be obtained from (20). Since LVxt<0 and vt=0, the following inequality can be obtained from (13):
(21)EdVxt=ELVxtdt<0.
According to EdVxt<0 and referring to [17], the closed-loop system (3a), (3b), and (3c) is asymptotically stable in the sense of mean square with zero external disturbance.
Integrating both side of (13) from 0 to tg, one has the following equation with nonzero initial condition:
(22)ExTtgPxtg-ExT0Px0=E∫0tgLVxtdt=E∫0tgΞ-zTtztdt.
Also, (22) can be rewritten as follows:
(23)ExTtgPxtg-ExT0Px0+E∫0tgzTtztdt=E∫0tgΞdt.
Since condition (7) is held, the Ξ<0 can easily be found. And then, the following inequalities can also be obtained:
(24a)ExTtgPxtg-ExT0Px0+E∫0tgzTtztdt<0,(24b)ExTtgPxtg-ExT0Px0<-E∫0tgzTtztdt.
Due to the fact that the closed-loop system (3a), (3b), and (3c) is asymptotically stable in the mean square, one can find that the xtg→0 as tg→+∞. Thus, the xTtgPxtg→0 can also be found with tg→+∞. And (24b) can be rewritten as in the following inequality:
(25)E∫0+∞zTtztdt<ExT0Px0.
Obviously, the xT0Px0 is the upper bound of output energy from (25). If condition (8) is satisfied, one has
(26)xT0Px0<α.
From (25) and (26), the following relation can directly be found:
(27)E∫0+∞zTtztdt<α.
Since (27) is equivalent to (5), one can find that if conditions (7) and (8) are held, then the H2 performance of closed-loop system is achieved. The proof of this theorem is completed.In Theorem4, the stability problem of closed-loop system (3a), (3b), and (3c) is discussed via finding the feasible solutions. However, the conditions of Theorem 4 belong to BMI (bilinear matrix inequality) problems which cannot directly be calculated by convex optimization algorithm. For this reason, the conditions of Theorem 4 are converted into LMI problems in the next theorem.Theorem 5.
Given matricesS1, S2, and S3, the closed-loop system (3a), (3b), and (3c) is asymptotically stable and satisfies mixed H2/passivity performances in the sense of mean square if there exist positive scalar α, positive definite matrix P, and feedback gain matrix F such that one has the following:minimizeα subject to
(28)XTAT+KTBT+AX+BK***-S1TC1X+ETS3-D1TS1-S1TD1**A-X+B-K0-X*S2C1XS2D10-I<0,(29)XTAT+KTBT+AX+BK**A-X+B-K-X*C2X+D2K0-I<0,(30)-αxT0x0-X<0,where X=P-1 and K=FX. I denotes the identity matrix with approximate dimension.Proof.
Multiplying the both sides of (6) by P-1*0I, the following inequality can be obtained:(31)P-1C1TS2C1P-1+P-1AfT+AfP-1+P-1A-fTPA-fP-1*-S1TC1P-1+D1TS2C1P-1+ETS3-D1TS1-S1TD1+D1TS2D1<0.
Applying Schur complement to (31), one has
(32)P-1AfT+P-1AfT***-S1TC1P-1+ETS3-D1TS1-S1TD1**A-fP-10-P-1*S2C1P-1S2D10-I<0.
According to Af=A+BF and A-f=A-+B-F, one has the following inequality from (32) with setting X=P-1 and K=FX:
(33)XTAT+KTBT+AX+BK***-S1TC1X+ETS3-D1TS1-S1TD1**A-X+B-K0-X*S2C1XS2D10-I<0.
Thus, (28) is obtained from (33). And, one can find that if (28) is satisfied, condition (6) can be held. Based on the above procedure, condition (34) can easily be obtained. And, if (29) is held, condition (7) can be satisfied. Thus, the proof of (29) is omitted here.
Based on (8), one has the following inequality:
(34)xT0Px0-α<0.
Using Schur complement for (34), the following inequality can be found with X=P-1:
(35)-αxT0x0-X<0.
Since (35) is equivalent to (30), one knows that if condition (8) is held then (30) can also be satisfied. On the other hand, if the conditions in this theorem are held then the conditions in Theorem 4 are satisfied. And the asymptotical stability and mixed H2/passivity performance of the closed-loop system (3a), (3b), and (3c) are achieved in the sense of mean square. Thus, the proof of the theorem is complete.Applying the convex optimization algorithm, the feasible solutions of Theorem5 can directly be found to establish controller (2). Based on the proposed stability criterion, the closed-loop system (3a), (3b), and (3c) driven by designed controller is asymptotically stable and satisfies mixed H2/passivity performance in the sense of mean square. In the next section, the synchronous generator system is utilized to verify the effectiveness and application of the proposed design method.


## 4. Simulation

Referring to [8], the dynamic equations of the synchronous generator system are given as follows:
(36)x˙1t=2πf0x2t-0.1vt,x˙2t=-DHx2tllllllllllll+ω0Hxd-xd′Vs2xdΣxdΣ′xd-xd′Vs2xdΣxdΣ′Pm-VsxdΣ′x3t+Eq0′sinx1t+δ0llllllllllllllllllllllll+xd-xd′Vs2xdΣxdΣ′lllllllllll×sinx1t+δ0cos⁡x1t+δ0,x˙3t=-xdΣTD0xdΣ′x3t+Eq0′llllllllllll+xd-xd′TD0xdΣ′Vscos⁡x1t+δ0+kAxadTD0Rfu0+ut,
where x1t is the angular position of the rotor of the generator with respect to synchronously rotating reference, which is selected here to be the infinite bus; x2t is the angular velocity of the rotor, x3t is the electromotive force the q-axis of the generator, ut is control input, and vt is a zero mean white noise with unit variance. And, the parameters of the synchronous generator system (36) are given as f0=50 Hz, D=0.8, H=8s, ω0=1 p.u., Pm=0.79 p.u., Vs=1 p.u., xdΣ′=1.1808 p.u., xdΣ=2.3108 p.u., Eq0′=1.2723 p.u., δ0=60∘, xd=1.5 p.u., xd′=0.3 p.u., TD0=3 s, kA=10, Rf=0.0045 p.u., u0=7.2942×10-4 p.u., and xad=1.3 p.u. Base on the linearization technique applied in [8], the linear equation of (36) can be obtained with equilibrium point as xept=000T. Considering the stochastic behavior, the constant matrices A- and B- of the multiplicative noise terms are assumed as follows:
(37)A-=100000000,B-=96.2963.
Based on the stochastic modeling approach, one has the following dynamic linear stochastic equation to characterize the local trajectories of (36) with added multiplicative noise term around equilibrium point:
(38)dxt=Axt+But+Evtdt+A-xt+B-utdβt,
where A=0314.15930-0.1009-0.1-0.0957-0.31210-0.6937, B=00962.963, and E=0.100T. Moreover, the measured output yt and controlled output zt for (38) are given as in the following equations:
(39)yt=C1xt+D1vt,zt=C1xt+D2ut,
where C1=C2=100 and D1=D2=1. With given S1=1, S2=1, and S3=0.98 and initial condition x(0)=30∘0.10T, the following feasible solutions can be obtained via using LMI Toolbox of MATLAB to solve the sufficient conditions in Theorem 5:
(40)P=1.932513.6451-0.013913.6451160.5379-0.2368-0.0139-0.23680.0005,F=2.351537.9586-0.0863,α=3.5641.
With obtained feedback gain matrix F, the linear controller can be designed as follows:
(41)ut=Fxt.In order to emphasize the advantages of the proposed design method, the approaches proposed by [6, 16] are also applied to control the synchronous generator system (36) with added multiplicative noise term (37). Studying [16], the mixed H2/H∞ performance controller design method has been proposed for state-dependent noised systems. Applying [16], the following controller can be obtained via setting the γ2=1.01 performance index for H∞ control scheme and the same initial condition x(0)=30∘0.10T:
(42)ut=Fxt,
where F=1.987827.8338-0.0771.Besides, the state feedbackH∞ control for stochastic systems has been developed in [6] without considering the H2 performance. And, the following controller can be established by [6] with γ2=1.01 performance index for H∞ control scheme:
(43)ut=Fxt,
where F=0.03843.2799-0.0151.Applying controller (41) to the synchronous generator system (36) with added multiplicative noise term, the simulation results are presented by Figures 1–3 with initial condition as x(0)=30∘0.10T. From the figures, system (36) driven by (41) is asymptotically stable in the mean square. In order to check the achievement of mixed H2/passivity performance, the ratio value of the following functions can be obtained by introducing simulation responses:(44)E2∫0tpyTtS1vtdtE∫0tpyTtS2ytdt+∫0tpvTtS3vtdt=1.002,(45)∫0tpzTtzt=0.2293.
Since the ratio value of (44) is bigger than one, one can find that Definition 1 is satisfied with given matrices as S1=1, S2=1, and S3=0.98. Hence, the passivity of considered system is achieved by designed controller (41). Besides, (41) shows that the output energy of system is smaller than α=3.5641. Thus, the inequality of Definition 3 can be satisfied.Figure 1
Responses of statex1(t).Figure 2
Responses of statex2(t).Figure 3
Responses of statex3(t).On the other hand, the simulated responses of (36) driven by (42) and (43) are also shown in Figures 1–3. From these responses, the overshoot of (43) is the biggest among the others because the H2 performance is not considered. Thus, one can find that the control performance of (43) is the worst case in this simulation. In addition, the requirements for attenuation performance and H2 performance of (36) driven by (42) are achieved. However, the overshoot and setting time of (41) are smaller than ones of (42). Therefore, in this simulation, the state responses of the proposed design method are better than that driven by the approaches in [6, 16]. Based on the results, the asymptotical stability and mixed H2/passivity performance of the stochastic system (36) are achieved by designed controller (41).


## 5. Conclusions

In this paper, theH2 control scheme was applied to minimize the output energy and to stabilize the considered system with initial condition. In addition, the disturbance attenuation performance of the system was discussed via the passivity theory with general power supply function. Through the H2 control scheme and passivity theory, the stability criterion for the linear stochastic system has been proposed subject to mixed H2/passivity performance. Furthermore, the Lyapunov function and Itô formula were employed to derive the sufficient conditions which have been converted into LMI problems. Thus, the feasible solutions can directly be found via convex optimization algorithm for establishing the controller such that the asymptotical stability and mixed H2/passivity performance of linear stochastic system are achieved in the sense of mean square. Through the simulation results, the proposed design method is useful and effective for stabilizing linear stochastic systems subject to mixed H2/passivity performance.



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