# H2 Control of Markovian Jump Systems with Input Saturation and Incomplete Knowledge of Transition Probabilities

Authors: Bum Yong Park
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper proposes anℋ2 state-feedback controller for Markovian jump systems with input saturation and incomplete knowledge of transition probabilities. The proposed controller is developed using second-order matrix polynomials of an incomplete transition rate to derive less conservative stabilization conditions. The proposed controller not only guarantees ℋ2 performance but also rejects matched disturbances. The effectiveness of the proposed method is demonstrated using three numerical examples.


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## Body

## 1. Introduction

Over the last few decades, Markovian jump systems (MJSs) have been recognized as one of the most effective models for the representation of dynamic systems subjected to random and abrupt variations. Thus, numerous studies have been conducted to analyze and synthesize MJSs [1–7]. The findings of these studies have been applied in various practical systems, such as networked control systems [8], manufacturing systems [8], economic systems [9], power systems [10], and actuator saturation [11]. In particular, studies have focused on the analysis and synthesis of ideal MJSs having exact values of transition probabilities [12].However, such MJSs with exactly known transition probabilities have limited scope for application in practical systems because it is difficult to obtain complete knowledge of transition probabilities. Thus, recent studies on controller synthesis have focused on MJSs with incomplete knowledge of transition probabilities. Such studies have employed the free-connection weighting method and linear matrix inequalities (LMIs) [13–16].However, several practical systems suffer from input saturation because of the physical limitations of the control system [17–19]. It is well known that input saturation generally degrades control system performance and system stability [20].Thus, the control synthesis problem should be considered with input saturation in practical systems. In particular, the stochastic stabilization problem for MJSs subjected to actuator saturation was studied based on exactly known transition probabilities [21, 22]. Furthermore, the stabilization of saturated MJSs with incomplete knowledge of transition probabilities was studied using the free-connection weighting matrix approach [11]. In addition, the stabilization condition for MJSs in the presence of both partially unknown transition rates and input saturation was proposed [23].To the best of the author’s knowledge, intensive studies on theℋ2 control of MJSs with input saturation and incomplete knowledge of transition probabilities have not been conducted thus far. A previous study stabilized nonhomogeneous MJSs with input saturation [24]; however, the findings are not applicable to practical systems because disturbances were not considered.Thus, anℋ2 stabilization condition for MJSs with input saturation and incomplete knowledge of transition probabilities is proposed herein. The main contributions of this study are as follows:This is the first proposal to propose a stabilization condition to accomplish stochastic stability and guaranteeℋ2 performance for MJSs with input saturation and incomplete knowledge of transition probabilities.The proposed controller consists of two parts: a linear control part to guaranteeℋ2 performance and a nonlinear control part to reject the matched disturbances.Based on the proposed relaxation method using the second-order matrix polynomials of the incomplete transition rate, this paper presents less conservative stabilization conditions for estimating the domain of attraction.The effectiveness of the proposed controller is demonstrated using two numerical examples and a practical example.The remainder of this paper is organized as follows. Section2 provides a description of the system and some preliminary results. Section 3 introduces the proposed ℋ2 controller for MJSs with input saturation and incomplete knowledge of transition probabilities. Section 4 presents the simulations of three examples for verifying the proposed controller. Section 5 concludes the paper.Notation. The notations X≥Y and X>Y indicate that X−Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, ∗ is used as an ellipsis for terms that are induced by symmetry. Furthermore, HeX=X+XT stands for any matrix X, and E⋅ denotes the mathematical expectation. For any matrices Si and Sij,(1)Sii∈1,2,…,N=S11,S12,…,S1N,Siji,j∈1,2,…,N=S11S12⋯S1NS21S22⋱⋮⋮⋱⋱⋮SN1⋯⋯SNN.We also usexp to indicate the p-norm of x, i.e., xp≜x1p+⋯+xnp1/p,p≥1. λminX and λmaxX denote a minimum eigenvalue and a maximum eigenvalue of X, respectively. The notation ek indicates a unit vector with a single nonzero entry at the kth position, i.e., ek≜0…1︸kth…0T.


## 2. System Description and Preliminaries

Consider the following continuous-time MJS with input saturation and a matched disturbance:(2)x˙t=Artxt+Brtsatut+dt,(3)zt=Crtxt,where xt∈ℜn is the state, ut∈ℜm is the control input, dt∈ℜm is the matched disturbance, and zt∈ℜq is the controlled output. The matched disturbance dt is assumed to be ekTdt<ɛ. Here, rt,t≥0 is a continuous-time Markov jumping process in a finite set D=1,2,3,…,N with mode transition probabilities:(4)Prt+δt=j|rt=i=πijδt+oδt,ifi≠j,1+πijδt+oδt,ifi=j,where δt>0, limδt⟶0oδt/δt=0, and πij is the transition rate from mode i to j at time t+δt. For rt=i∈D, to simplify the notation, Art=Ai, Brt=Bi, Drt=Di, and Crt=Ci. Further, sat⋅ denotes a saturation operator, which is defined as(5)satui≜ui,ui<μ,μ,ui≥μ,−μ,ui≤μ,where μ>ɛ is the saturation level. Furthermore, the transition rate matrix Π belongs to(6)SΠ≜πiji,j∈D|0≤πij,fori≠j,πii=−∑j=1,i≠jNπij.In view of the aforementioned relations,πij accords with the following relationships, for all i,j∈D:(7)νijπij≥0,∑j=1Nπij=0,−νijπijπij+πii≥0,where(8)νij=1,i≠j,−1,i=j.For future convenience, two sets are defined with respect to the measurability of the transition rate fori,j∈D:(9)Di+≜j|πijis known fori,Di−≜j|πijis unknown fori.The following lemma and definitions are introduced as preliminaries required to prove the theorems presented in the subsequent sections.Lemma 1 (see [25]).
Letu,v∈ℜm,(10)u=u1u2…umT,v=v1v2…vmT.Assume thatekTv≤μ for all k∈1,m, and then(11)satu∈CoEsu+Es−v|,s∈1,2m,where Es denotes a diagonal matrix with all possible combinations of 1 and 0 diagonal entries, Es−≜I−Es, and Co is the convex hull.Definition 1 (see [21]).
A setS∈ℜn is called the domain of attraction in the mean square sense of (2), if for any initial mode r0∈D and initial state x0∈S, the state xt of (2) satisfies(12)limT⟶∞∫0TE∥xτ∥2dτ|x0,r0<xT0Ψx0,where Ψ>0.


## 3. Main Results

This section considers the design problem of theℋ2 state-feedback controller.A controller is proposed for system (2) as follows:(13)ut=Krtxt+u¯rt,xt,where Krt is the linear controller part to guarantee the ℋ2 performance and u¯rt,xt is the nonlinear controller part to reject the matched disturbance Brtdt. Thus, the proposed controller is designed to stochastically stabilize and minimize the upper bound of the following linear quadratic cost:(14)Jt=E∫t∞xTτQrtxτdτ,where Qrt=Qi≥0. Here, for rt=i∈D, Krt=Ki and u¯rt,xt=u¯ixt. Using system (2) and the proposed controller (13), the resultant closed-loop system is expressed as follows:(15)x˙t=Aixt+BisatKixt+u¯ixt+dt.Theorem 1.
Consider system (15) with input saturation and incomplete knowledge of the transition rate. For i,j∈D, s∈1,2m, and k∈1,m, suppose that there exist symmetric matrices P¯i and Rij, matrices K¯i, H¯i, Λij, Yij, Si0, and Sij, and a scalar γ such that(16)P¯i>0,(17)Λij+ΛijT>0,Yij+YijT>0,(18)RijP¯i∗P¯j>0,i≠j,(19)Γ¯siΓjij∈Di−∗Γjlij,l∈Di−<0,(20)γxT0x0P¯0>0,(21)P¯iH¯iTekekTH¯iμ−ɛ2>0,(22)1xT0x0P¯i>0,wherei∈Di+(23)Γ¯si=Ω¯si+Πi+ETHeSi0E+∑j∈Di+νijπijETHeΛijE−∑j∈Di+νijπijπij+πiiETHeYijE,Γji=12ETGij+ETSi0+Πi+Sij+νijETΛij−νijπiiETYij,Γjji=HeSij−νijHeYij,Γjli=Sil+Sij,i∈Di−(24)Γ¯si=Ω¯si+Πi+ETHeSi0E+∑j∈Di+νijπijETHeΛijE−∑j∈Di+νijπij2ETHeYijE,Γji=12ETGij+ETSi0+Πi+ETSij+νijETΛij−∑j∈Di+cijνijπijETYij,Γjji=HeSij−2νijHeYij,Γjli=Sil+Sij,i≠l,Sil+Sij−νijYij,i=l,Πi+=∑j∈D+πij,Ω¯si≜HeAiP¯i+BiEsK¯i+BiEs−H¯i+∑j∈Di+πijGijP¯iP¯i−Qi−1,Gij≜κijRij+1−κijP¯i,κij=1,i≠j,0,i=j,E=I0∈ℜn×2n.Then, the set∩i=1NΩPi is contained in the domain of attraction, and the proposed system (15) is stochastically stable with the ℋ2 cost in (14) guaranteed by γ. Furthermore, the proposed controller is constructed as ut=Kixt+u¯ixt for mode i, where Ki=K¯iP¯i−1 and each component of u¯ixt is defined as(25)u¯ixtk=−ɛsgnekTBiTPixt.Proof.
Consider the following mode-dependent control inputut and the auxiliary input vt:(26)ut=Krtxt+u¯ixt,vt=Hrtxt+u¯ixt,where vt is used to handle the input saturation in Lemma 1. For the representation method (10) in Lemma 1, the following condition should be satisfied:(27)ekTHrtxt+ekTu¯ixt≤μ.
From the definition ofu¯ixt in (25), the left side of (27) can be derived as follows:(28)ekTHrtxt+ekTu¯ixt≤ekTHrtxt+ekTu¯ixt=ekTHrtxt+ɛ.
Then, the sufficient condition for (27) is given as follows:(29)ekTHrtxt≤μ−ɛ.
Therefore, the representation method in Lemma1 can be used if xt∈LHrt for k∈1,m, where(30)LHrt=xt∈ℜn|ekTHrtxt≤μ−ɛ.
To establish a set invariance condition [25], the ellipsoid ΩPrt≜xt∈ℜn|xtTPrtxt≤1 is in the linear region LHrt, that is, for k∈1,m,(31)xTtPixt>xTtHiTek1μ−ε2ekTHixt,or equivalently,(32)PiHiTekekTHiμ−ɛ2>0.
Then, multiplying both sides of the above equation bydiagPi−1,I yields (20), where P¯i=Pi−1 and H¯i=HiP¯i.
Let us chooseVxt=xTtPrtxt as a Lyapunov function, where Prt is a positive definite matrix. Then, from the weak infinitesimal operator ∇ of the Markov process, ∇Vxt is given by(33)∇V=limδt⟶01δtEVxt+δt,rt+δt=j|xt,rt=i−Vxt,rt=i=dVxt,idt+∑j=1NπijVxt,j=2xTtPix˙t+xTt∑j=1NπijPjxt.
According to the convex property and condition (21), there exist variables ηs such that(34)satut=∑s=12mηsEsut+Es−vt=∑s=12mηsEsKrtxt+Es−Hrtxt+u¯ixt=∑s=12mηsEsKrtxt+Es−Hrtxt+u¯ixt,where ∑s=12mηs=1.
Then,∇Vxt can be rewritten as(35)∇Vxt=2xTtPiAi+∑s=12mηsPiBiEsKi+Es−Hixt+2xTtPiBi∑s=12mηsu¯ixt+dt+xTt∑j=1NπijPjxt=∑s=12mηs2xTtPiAi+PiBiEsKi+Es−Hi+2xTtPiBiu¯ixt+dt+xTt∑j=1NπijPjxt.
Furthermore, from (25) and ekTdt<ɛ, we have(36)2xTtPiBiu¯ixt+dt≤0.
Thus, if the following condition holds, fori∈D and s∈1,2m,(37)HePiAi+PiBiEsKi+PiBiEs−Hi+Qi+∑j=1NπijPj<0,then, from (36) and (35), ∇Vxt can be expressed as the following relation:(38)∇Vxt≤2∑s=12mηsxTtPiAi+PiBiEsKi+Es−Hi+xTt∑j=1NπijPjxt<−xTtQixt.
Using the generalized Dynkin’s formula [26], the above relation allows(39)EVt−V0=E∫0t∇Vτdτ|x0,r0<−E∫0txTτQrτxτdτ|x0,r0≤−mini∈DλminQiE∫0txτ2dτ|x0,r0,which leads to(40)mini∈DλminQiE∫0txτ2dτ|x0,r0<V0−EVt≤V0,because the following equation is valid:(41)E∫0txτ2dτ|x0,r0<V0mini∈DλminQi.
From (41), it is allowed that(42)limT⟶∞E∫0txτ2dτ|x0,r0≤xT0Ψx0,where(43)Ψ=maxr0∈DλmaxPr0mini∈DλminQi>0.
Furthermore, from (39), we have(44)J0<V0=xT0Pr0x0,which guarantees the ℋ2 cost through (20), indicating that xT0Pr0x0<γ using the Schur complement.
Subsequently, by pre- and postmultiplying (37) with Pi−1, we have(45)HeAsi+P¯iQiP¯i+∑j=1NπijP¯iPjP¯i<0,where Asi=AiP¯i+BiEsK¯i+BiEs−H¯i, P¯i=Pi−1, and K¯i=KiP¯i.
Note that fori=j, P¯iPjP¯i=P¯i, and for i≠j, (18) leads to P¯iPjP¯i≤Rij. Equation (45) holds because of the following condition:(46)HeAsi+P¯iQiP¯i+∑j=1NπijGij<0,where Gij≜κijRij+1−κijP¯i.
Applying the Schur complement to (46) yields(47)HeAsi+∑j=1NπijGijP¯iP¯i−Qi−1<0.
To derive the LMI conditions, (47) can be written as follows:(48)Ωsi≜Ω¯si+∑j∈Di−πijETGijE<0.
In addition, according to condition (7), the following equations can be derived from (19):(49)Ci1≜HeΠi++∑j∈Di−πijETSi0+∑j∈Di−πijSijE=0,(50)Ci2≜∑j=1NνijπijETHeΛijE≥0,(51)Ci3≜−∑j=1Nνijπijπij+πiiETHeYijE≥0.
Then, the positive semidefinite matrixLi is constructed using (49)–(51) in the following form:(52)Li≜Ci1+Ci2+Ci3=L¯i+∑j∈Di−πijHeLjiE+∑j∈Di−∑l∈Di−l>jπijπilETHeLjliE+∑l∈Di−l=jπij2ETHeLjjiE≥0,wherei∈Di+(53)L¯i=Πi+ETHeSi0E+∑j∈Di+πijETHeνijΛijE−∑j∈Di+πij2ETHeνijYijE−∑j∈Di+πijπiiETHeνijYijE,Lji=ETSi0+Πi+Sij+ETνijΛij−ETνijπiiYij,Ljli=Sil+Sij,Ljji=Sil−νijYij,i∈Di−(54)L¯i=Πi+ETHeSi0E+∑j∈Di+πijETHeνijΛijE−∑j∈Di+πij2ETHeνijYijE,Lji=ETSi0+Πi+Sij+ETνijΛij−∑j∈Di+cijπijETνijYij,Ljli=Sil+Sij,i≠l,Sil+Sij−νijYij,i=l,Ljji=Sij−2νijYij,where(55)cij=1,i=j,0,i≠j.
Based on theS-procedure, if Ωsi<0 whenever Li≥0, the following sufficient condition is formulated:(56)Li+Ωsi<0,which can be converted to the following LMI condition:(57)IπijEj∈Di−TΓ¯siΓjij∈Di−∗Γjlj,l∈Di−IπijEj∈Di−<0,where(58)Γ¯si=Ω¯si+L¯i,Γji=12ETGij+Lji,Γjli=Ljli.
Then, (57) holds because of the LMI conditions (16)–(19).


## 4. Numerical Examples

In this section, theℋ2 performance is investigated through numerical examples to verify the effectiveness of the proposed method.


### 4.1. Example 1

Consider an MJS with four modes (N=4), whose system matrices are(59)A1=0.35−7.301.480.81,A2=0.89−3.111.480.21,A3=−0.11−0.852.31−0.10,A4=−0.17−1.481.59−0.27,B1=0.571.23,B2=0.78−0.49,B3=1.340.39,B4=−0.381.07,C1=0.0−0.1,C2=0.10.0,C3=0.00.1,C4=0.10.0,Π=−1.30.2π13π14π21π220.30.30.6π32−1.5π340.4π42π43π44,Q1=Q2=Q3=Q4=100010,ɛ=0.1,μ=1,where π13, π14, π21, π22, π32, π34, π42, π43, and π44 are the unknown transition rates. The following sets can be obtained using the transition rate matrix Π:(60)D1+=1,2,D2+=3,4,D3+=1,3,D4+=1,D1−=3,4,D2−=1,2,D3−=2,4,D4−=2,3,4.Considering the initial conditionx0=0.2−0.15T, the state trajectories of the closed-loop system shown in Figure 1 are stochastically stable with incomplete knowledge of transition rates under the input saturation and matched disturbances. Here, we set r0=2 and dt=0.1sin2t2−0.7.Figure 1
State trajectories for Example 1.According to Theorem1, the ℋ2 performance γ=0.2803, and the proposed controller gains are obtained as follows:(61)K1=3.2384×105−2.13426,K2=−1.5913×106−7.8766×105,K3=−8.9047×105−8.0083×105,K4=−9.0151×106−1.5631×107,P1=4.3829−4.5290−4.52901.8561,P2=47.96860.40460.40488.256,P3=6.33421.19451.194516.543,P4=480.52196.07396.073151.11.Figure2 presents the domain of attraction for the proposed controller. As shown in the figure, the state trajectory of the closed-loop system (15) converges to the origin as time progresses, as long as the initial state is in ∩i=14ΩPi.Figure 2
Domain of attraction for Example 1.


### 4.2. Example 2

Consider the following multiinput system with three different modes (N=3) [24]:(62)A1=−2.50.30.81−30.200.5−2,A2=−2.51.20.3−0.55−10.251.25,A3=21.5−0.42.230.71.10.9−2,B1=0.70700010001,B2=0.70700010000.707,B3=0.70700010001,Π=−3π12π13π21π221π310.3π33,Q1=Q2=Q3=100010001,ɛ=0.01,μ=1,where π12, π13, π21, π22, π31, and π33 are the unknown transition rates. The following sets can be obtained using the transition rate matrix Π:(63)D1+=1,D2+=3,D3+=2,D1−=2,3,D2−=1,2,D3−=1,3.According to Theorem1, the ℋ2 performance γ=0.0803, and the proposed controller gains are obtained as follows:(64)K1=−94.941−12.248−2.4019−8.6349−61.833−6.8275−1.7987−6.8828−76.698,K2=−24.110−3.7947−0.69917−2.7040−13.943−2.0031−0.69626−2.8494−26.380,K3=−93.208−25.60718.331−18.085−67.048−14.08412.952−14.079−76.181,P1=1.8198×10−12.3490×10−23.0898×10−32.3490×10−21.6739×10−11.8759×10−23.0898×10−31.8759×10−22.0758×10−1,P2=3.3589×10−15.3394×10−29.2638×10−35.3394×10−22.7157×10−14.0095×10−29.2638×10−34.0095×10−23.6452×10−1,P3=2.5519×10−17.0131×10−2−5.1063×10−27.0131×10−22.5930×10−15.4768×10−2−5.1063×10−25.4768×10−22.9602×10−1.Figure3 shows the state trajectories and the mode evolution obtained using the aforementioned controller gains. Figure 4 shows the saturated control input, where x0=0.5−0.3−0.4T and r0=3. Here, we set dt=0.01sint2+0.1. Figures 5 and 6 show the domains of attraction on the x1t−x2t and x2t−x3t planes, respectively. As shown in the figures, the state trajectory of the closed-loop system (15) converges to the origin as time progresses, as long as the initial state is in ∩i=13ΩPi. These figures show that the proposed controller stabilizes the MJS with input saturation and incomplete knowledge of the transition rates under the matched disturbance.Figure 3
State trajectories for Example 2.Figure 4
Control input for Example 2.Figure 5
Domain of attraction (x1t−x2t planes) for Example 2.Figure 6
Domain of attraction (x2t−x3t planes) for Example 2.


### 4.3. Example 3

Consider the following inverted pendulum system controlled using a DC motor [27]:(65)x˙1t=x2t,x˙2t=glsinx1t+NKmml2x3t,Lax˙3t=KbNx2t−Rrtx3t+satut,where x1t is the angle of the inverted pendulum, x2t is the angular velocity, x3t is the input current, ut is the control input voltage, g is the acceleration of gravity, m and l are the mass and length of the inverted pendulum, respectively, Kb is the back-EMF constant, Km is the motor torque constant, and N is the gear ratio. Here, Rrt is the resistance in the DC motor, which is defined as(66)Rrt=Ra,ifrt=1,Rb,ifrt=2.LetLa=1, g=9.8m/s2, l=1m, m=1kg, N=10, Km=0.1Nm/A, Kb=0.1Vs/rad, Ra=1Ω, and Rb=0.5Ω.Using the aforementioned parameters, system (65) can be linearized as the following MJS with two modes:(67)x˙t=Artxt+Brtsatut+dt,zt=Crtxt,where(68)xt=x1tx2tx3tT,A1=0109.80101−1,A2=0109.80101−0.5,B1=B2=001,C1=0.100,C2=0.200,Π=−0.61270.6127π21π22,dt=0.01e−0.5tsin10t2,ɛ=0.01,μ=12,where π21 and π22 are the unknown transition rates. Here, it is assumed that the matched disturbance dt exists. Based on Theorem 1, the ℋ2 performance γ=0.1399, and the proposed controller gains are obtained as follows:(69)K1=−2.6199×107−8.6656×106−1.9736×106,K2=−1.25×107−4.1531×107−1.0244×107,P1=1.0289×1023.2962×1012.93233.2962×1011.0739×1019.6989×10−12.93239.6989×10−12.2090×10−1,P2=0.0698×1042.1904×1021.3991×1012.1904×1026.9178×1014.64831.3991×1014.64831.1465.Based on the aforementioned control gains, Figure7 shows the state trajectories for x0=−0.10.20T and the mode evolution rt. As shown in the figure, the state trajectories of the closed-loop systems with the proposed controller converge to zero as time progresses.Figure 7
State trajectories for Example 3.


## 4.1. Example 1

Consider an MJS with four modes (N=4), whose system matrices are(59)A1=0.35−7.301.480.81,A2=0.89−3.111.480.21,A3=−0.11−0.852.31−0.10,A4=−0.17−1.481.59−0.27,B1=0.571.23,B2=0.78−0.49,B3=1.340.39,B4=−0.381.07,C1=0.0−0.1,C2=0.10.0,C3=0.00.1,C4=0.10.0,Π=−1.30.2π13π14π21π220.30.30.6π32−1.5π340.4π42π43π44,Q1=Q2=Q3=Q4=100010,ɛ=0.1,μ=1,where π13, π14, π21, π22, π32, π34, π42, π43, and π44 are the unknown transition rates. The following sets can be obtained using the transition rate matrix Π:(60)D1+=1,2,D2+=3,4,D3+=1,3,D4+=1,D1−=3,4,D2−=1,2,D3−=2,4,D4−=2,3,4.Considering the initial conditionx0=0.2−0.15T, the state trajectories of the closed-loop system shown in Figure 1 are stochastically stable with incomplete knowledge of transition rates under the input saturation and matched disturbances. Here, we set r0=2 and dt=0.1sin2t2−0.7.Figure 1
State trajectories for Example 1.According to Theorem1, the ℋ2 performance γ=0.2803, and the proposed controller gains are obtained as follows:(61)K1=3.2384×105−2.13426,K2=−1.5913×106−7.8766×105,K3=−8.9047×105−8.0083×105,K4=−9.0151×106−1.5631×107,P1=4.3829−4.5290−4.52901.8561,P2=47.96860.40460.40488.256,P3=6.33421.19451.194516.543,P4=480.52196.07396.073151.11.Figure2 presents the domain of attraction for the proposed controller. As shown in the figure, the state trajectory of the closed-loop system (15) converges to the origin as time progresses, as long as the initial state is in ∩i=14ΩPi.Figure 2
Domain of attraction for Example 1.


## 4.2. Example 2

Consider the following multiinput system with three different modes (N=3) [24]:(62)A1=−2.50.30.81−30.200.5−2,A2=−2.51.20.3−0.55−10.251.25,A3=21.5−0.42.230.71.10.9−2,B1=0.70700010001,B2=0.70700010000.707,B3=0.70700010001,Π=−3π12π13π21π221π310.3π33,Q1=Q2=Q3=100010001,ɛ=0.01,μ=1,where π12, π13, π21, π22, π31, and π33 are the unknown transition rates. The following sets can be obtained using the transition rate matrix Π:(63)D1+=1,D2+=3,D3+=2,D1−=2,3,D2−=1,2,D3−=1,3.According to Theorem1, the ℋ2 performance γ=0.0803, and the proposed controller gains are obtained as follows:(64)K1=−94.941−12.248−2.4019−8.6349−61.833−6.8275−1.7987−6.8828−76.698,K2=−24.110−3.7947−0.69917−2.7040−13.943−2.0031−0.69626−2.8494−26.380,K3=−93.208−25.60718.331−18.085−67.048−14.08412.952−14.079−76.181,P1=1.8198×10−12.3490×10−23.0898×10−32.3490×10−21.6739×10−11.8759×10−23.0898×10−31.8759×10−22.0758×10−1,P2=3.3589×10−15.3394×10−29.2638×10−35.3394×10−22.7157×10−14.0095×10−29.2638×10−34.0095×10−23.6452×10−1,P3=2.5519×10−17.0131×10−2−5.1063×10−27.0131×10−22.5930×10−15.4768×10−2−5.1063×10−25.4768×10−22.9602×10−1.Figure3 shows the state trajectories and the mode evolution obtained using the aforementioned controller gains. Figure 4 shows the saturated control input, where x0=0.5−0.3−0.4T and r0=3. Here, we set dt=0.01sint2+0.1. Figures 5 and 6 show the domains of attraction on the x1t−x2t and x2t−x3t planes, respectively. As shown in the figures, the state trajectory of the closed-loop system (15) converges to the origin as time progresses, as long as the initial state is in ∩i=13ΩPi. These figures show that the proposed controller stabilizes the MJS with input saturation and incomplete knowledge of the transition rates under the matched disturbance.Figure 3
State trajectories for Example 2.Figure 4
Control input for Example 2.Figure 5
Domain of attraction (x1t−x2t planes) for Example 2.Figure 6
Domain of attraction (x2t−x3t planes) for Example 2.


## 4.3. Example 3

Consider the following inverted pendulum system controlled using a DC motor [27]:(65)x˙1t=x2t,x˙2t=glsinx1t+NKmml2x3t,Lax˙3t=KbNx2t−Rrtx3t+satut,where x1t is the angle of the inverted pendulum, x2t is the angular velocity, x3t is the input current, ut is the control input voltage, g is the acceleration of gravity, m and l are the mass and length of the inverted pendulum, respectively, Kb is the back-EMF constant, Km is the motor torque constant, and N is the gear ratio. Here, Rrt is the resistance in the DC motor, which is defined as(66)Rrt=Ra,ifrt=1,Rb,ifrt=2.LetLa=1, g=9.8m/s2, l=1m, m=1kg, N=10, Km=0.1Nm/A, Kb=0.1Vs/rad, Ra=1Ω, and Rb=0.5Ω.Using the aforementioned parameters, system (65) can be linearized as the following MJS with two modes:(67)x˙t=Artxt+Brtsatut+dt,zt=Crtxt,where(68)xt=x1tx2tx3tT,A1=0109.80101−1,A2=0109.80101−0.5,B1=B2=001,C1=0.100,C2=0.200,Π=−0.61270.6127π21π22,dt=0.01e−0.5tsin10t2,ɛ=0.01,μ=12,where π21 and π22 are the unknown transition rates. Here, it is assumed that the matched disturbance dt exists. Based on Theorem 1, the ℋ2 performance γ=0.1399, and the proposed controller gains are obtained as follows:(69)K1=−2.6199×107−8.6656×106−1.9736×106,K2=−1.25×107−4.1531×107−1.0244×107,P1=1.0289×1023.2962×1012.93233.2962×1011.0739×1019.6989×10−12.93239.6989×10−12.2090×10−1,P2=0.0698×1042.1904×1021.3991×1012.1904×1026.9178×1014.64831.3991×1014.64831.1465.Based on the aforementioned control gains, Figure7 shows the state trajectories for x0=−0.10.20T and the mode evolution rt. As shown in the figure, the state trajectories of the closed-loop systems with the proposed controller converge to zero as time progresses.Figure 7
State trajectories for Example 3.


## 5. Conclusion

This paper proposed anℋ2 mode-dependent state-feedback controller for MJSs with input saturation and an incomplete knowledge of transition probabilities. Specifically, an invaluable relaxation method was developed into the second-order matrix polynomials of the unknown transition rate using all possible slack variables for the incomplete transition rates to obtain less conservative stabilization conditions. Consequently, the proposed controller guaranteed ℋ2 performance and removed the matched disturbances. The effectiveness of the proposed controller was demonstrated using three examples.



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*Source: 8868234-2020-12-10.xml*