2.1. Continuous-Time Markovian Jump Linear Systems
Consider the following continuous-time SMJS described by
(1)Ex˙(t)=A(η(t))x(t)+B(η(t))u(t)+F(η(t))w(t)y(t)=C(η(t))x(t)+D(η(t))w(t),
where x(t)∈ℝn is the state vector, u(t)∈ℝm is the control input, w(t)∈ℝp is the disturbance input, and y(t)∈ℝq is the control output. Matrix E may be singular assumed rank(E)=r≤n. Parameter η(t) is the continuous-time Markov processes with right continuous trajectories taking values in a finite set 𝕊={1,2,…,N} with transition probabilities:
(2)Pr(η(t+Δt)=j∣η(t)=i)={λijΔt+o(Δt)i≠j1+λiiΔt+o(Δt)i=j,
where Δt>0, limΔt→0(o(Δt)/Δt)=0, and the transition probability rate satisfies λij≥0, for i,j∈𝕊,i≠j, and
(3)λii=-∑j=1,j≠iNλij.

For notational simplicity, in the sequel, for each possible η(t)=i, i∈𝕊, the matrix Aη(t) will be denoted by Ai, and so on.

Assumption 1.
The transition probabilities for system (1) are assumed to be unknown but vary between two known bounds satisfying the following:
(4)0≤λ_ij≤λij≤λ¯ij ∀i,j∈𝕊,
where i≠j.

From Assumption 1, when we only know that λ_i=minj≠i∈𝕊λ_ij and λ¯i=minj≠i∈𝕊λ¯ij, we have λ_i≤λij≤λ¯i, which is the same as in [25]. Thus, we may say that Assumption 1 is more natural.

Definition 2 (see [<xref ref-type="bibr" rid="B27">26</xref>]).
(
1
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The nominal system (1) is said to be regular if det(sE-Ai) is not identically zero for every i∈𝕊.

(
2
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The nominal system (1) is said to be impulse free if deg(det(sE-Ai))=rank(E) for every i∈𝕊.

(
3
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The nominal system in (1) is said to be stochastically stable if, when u(t)=0 and w(t)=0, there exists a constant M(x0,η0) such that
(5)ℰ{∫0∞∥x(t)∥2dt∣x0,η0}≤M(x0,η0).

(
4
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The nominal system in (1) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.

Definition 3.
Given γ>0, the nominal system in (1) is said to be stochastically admissible with γ-disturbance attenuation if it is stochastically admissible and satisfying the following such that
(6)ℰ{∫0∞∥y(t)∥2dt}<γ2𝔼{∫0∞∥w(t)∥2dt}
holds for zero-initial condition and any nonzero w(t)∈ℒ2[0,∞).

In this paper, the H∞ controller such that the resulting closed-loop system is stochastically admissible with (6) is
(7)u(t)=Kix(t),
where Ki is to be determined.

Lemma 4.
Given a scalar γ>0, the unforced system (1) is stochastically admissible with H∞ performance if there exist matrices Pi such that
(8)ETPi=PiTE≥0,[AiTPi+PiTAi+∑j=1NλijETPjPiTFiCiT*-γ2IDiT**-I]<0.

2.2. Discrete-Time Markovian Jump Linear Systems
Consider the following discrete-time SMJS described by
(9)Ex(k+1)=A(θ(k))x(t)+B(θ(k))u(k)+F(θ(k))w(k)y(k)=C(θ(k))x(k)+D(θ(k))w(k),
where x(k)∈ℝn is the state vector, u(k)∈ℝm is the control input, w(k)∈ℝp is the disturbance input, and y(k)∈ℝq is the control output. Matrix E may be singular assumed rank(E)=r≤n. Parameter θ(k) is a discrete-time, discrete-state Markovian chain taking values in a finite set 𝕊 with transition probabilities:
(10)Pr(θ(k+1)=j∣θ(k)=i)=πij,
where πij≥0, for i,j∈𝕊, and
(11)∑j=1Nπij=1.

Assumption 5.
The transition probabilities for (10) are assumed to be unknown but vary between two known bounds satisfying the following:
(12)0≤π_ij≤πij≤π¯ij ∀i,j∈𝕊.

From [24], we know that ∑j=1Nπ_ij and ∑j=1Nπ¯ij may not equal 1. It is not necessary to exactly know the transition probabilities but only the bounds of πij, which are π_ij and π¯ij.

Definition 6 (see [<xref ref-type="bibr" rid="B27">26</xref>]).
(
1
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The nominal system in (9) is said to be regular if det(sE-Ai) is not identically zero for every i∈𝕊.

(
2
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The nominal system in (9) is said to be causal if deg(det(sE-Ai))=rank(E) for every i∈𝕊.

(
3
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The nominal system in (9) is said to be stochastically stable if, when u(k)=0 and w(k)=0, there exists a constant M(x0,θ0) such that
(13)ℰ{∑k=0N∥x(k)∥2∣x0,θ0}≤M(x0,θ0).

(
4
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The nominal system in (9) is said to be stochastically admissible if it is regular, causal, and stochastically stable.

Definition 7.
Given γ>0, the nominal system in (9) is said to be stochastically admissible with γ-disturbance attenuation such that
(14)ℰ{∑k=0N∥y(k)∥2}<γ2𝔼{∑k=0N∥w(k)∥2}
holds for zero-initial condition and any nonzero w(k)∈ℒ2[0,∞).

The corresponding H∞ controller for system (9) is
(15)u(k)=Kix(k),
where Ki is to be determined.

Lemma 8.
Given a scalar γ>0, the unforced system (9) is stochastically admissible with H∞ performance if there exist matrices Pi=PiT such that
(16)ETPiE≥0(17)[AiTP^iAi-ETPiEAiTP^iFiCiT*-γ2I+FiTP^iFiDiT**-I]<0,
where P^i=∑j=1NπijPj.

Lemma 9 (see[<xref ref-type="bibr" rid="B28">27</xref>]).
Given any real matrices X, Y, and Z with appropriate dimensions and such that Y>0 and is symmetric, then, one has
(18)XTYX+XTZ+ZTX+ZTY-1Z≥0.