#
Robust H_{∞} Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions

^{1}

^{2}

^{*}

## Abstract

**:**

_{∞}finite-time control for discrete delayed nonlinear systems with Markovian jumps and external disturbances. It is usually assumed that the disturbance affects the system states and outputs with the same influence degree of 100%, which is not evident enough to reflect the situation where the disturbance affects these two parts by different influence degrees. To tackle this problem, a probabilistic distribution denoted by binomial sequences is introduced to describe the external disturbance. Throughout the paper, the definitions of the finite-time boundedness (FTB) and the H

_{∞}FTB are firstly given respectively. To extend the results further, a model which combines a linear dynamic system and a static nonlinear operator is referred to describe the system under discussion. Then by virtue of state feedback control method, some new sufficient criteria are derived which guarantee the FTB and H

_{∞}FTB performances for the considered system. Finally, an example is provided to demonstrate the effectiveness of the developed control laws.

## 1. Introduction

_{∞}finite-time control for Markovian jump systems. Further, subject to average dwell time and partially known transition probabilities, H

_{∞}filtering is studied to obtain the FTB performance for Markovian jump systems in [20].

_{∞}control concept [21]. However, to the best of our knowledge, in most of the literatures it is usually assumed that the disturbance is incorporated into both the system states and the outputs with the same influence degree of 100%, which actually is not accurate enough to reflect the real system due to the case where the disturbance affects the system states and outputs with different percentages of the total degrees, respectively. In particular, as is shown in Figure 1, the disturbance enters both the system states and system outputs. Here two cases are considered. In Case I, the disturbance affects the system states and system outputs by 100%. While in Case II, the disturbance enters the system states and system outputs by 100q% and 100(1−q)%, respectively, where q is called the distribution ratio (0 ≤ q ≤ 1), and 100q% and100(1−q)% refer to the disturbance influence degree. In this paper, this work will be carried out as one of the few attempts.

_{∞}finite-time control for discrete delayed nonlinear systems with Markovian jumps and disturbances of probabilistic distributions is addressed in this paper. Firstly, referring to the model in [22] which is the interconnection of a linear dynamic system and a static nonlinear operator, a new set of equations are established to describe the discrete-time delayed nonlinear system with Markovian jumps. By introducing the Bernoulli distribution and Binomial distribution sequences, the disturbance distributed into the system states and outputs by different influence degrees is incorporated into the model system. Then by employing the Lyapunov functions and state feedback control method, some new criteria are derived such that the robust H

_{∞}finite-time control performances are achieved for all possible Markovian jumps and disturbances of probabilistic distributions. Finally an example is provided to validate the developed control laws.

_{∞}finite-time controller design. In Section 4, an illustrative example is demonstrated to verify the effectiveness of the proposed control approaches. Finally, some conclusions are drawn in Section 5.

_{2}[0,∞) is the space of square integrable vector functions over [0, ∞). ℜ

^{n}denotes n dimensional Euclidean space, and ℜ

^{n×m}is the set of all n×m real matrices. I denotes identity matrix of appropriate orders. * denotes the symmetric parts. diag{…} stands for a block-diagonal matrix. ||x|| denotes the Euclid norm of vector x. The notation X > Y, where X and Y are matrices of the same dimensions, means that the matrix X−Y is positive definite. Pr{⋅} denotes the occurrence probability of event “⋅”. Pr{A|B} represents the occurrence probability of event A on condition B. E{⋅} stands for the mathematical expectation of event “⋅”. If X ∈ ℜ

^{p}and Y ∈ ℜ

^{q}, C(X;Y) denotes the space of all continuous functions mapping ℜ

^{p}→ ℜ

^{q}. N

_{0}represents the set of nonnegative integers.

## 2. Problem Formulations and Preliminaries

^{n}is the system state, u(k) ∈ ℜ

^{m}is the control input, w(k) ∈ ℜ

^{s}is the external disturbance which belongs to l

_{2}[0, ∞). ξ ∈ ℜ

^{L}is the input of the nonlinear function ϕ, ϕ ∈ C(ℜ

^{L}; ℜ

^{L}) is the nonlinear function satisfying ϕ (0) = 0, L ∈ N

_{0}is the number of nonlinear functions. A(r

_{k}) ∈ ℜ

^{n×n}, A

_{d}(r

_{k}) ∈ ℜ

^{n×n}, B

_{p}(r

_{k}) ∈ ℜ

^{n×L}, B

_{u}(r

_{k}) ∈ ℜ

^{m}, B

_{w}(r

_{k}) ∈ ℜ

^{n×s}, C

_{q}(r

_{k}) ∈ ℜ

^{L×n}, C

_{qd}(r

_{k}) ∈ ℜ

^{L×n}, D

_{p}(r

_{k}) ∈ ℜ

^{L×L}, D

_{u}(r

_{k}) ∈ ℜ

^{L×m}, and D

_{w}(r

_{k}) ∈ ℜ

^{L×s}are mode-dependent matrices where r(k) denotes the discrete-time Markov chain taking values from a finite set V = {1,2, ⋯, s} with the mode-to-mode transition probabilities as follows:

_{ij}≤ 1, $0\le {\mu}_{ij}\le 1,{\displaystyle {\sum}_{j=1}^{s}{\mu}_{ij}=1}\phantom{\rule{0.2em}{0ex}}\forall i\in V$.

_{k}) as Q

_{i}∀r

_{k}= i, i ∈ V, i.e., A(r

_{k}) is denoted by A

_{i}, A

_{d}(r

_{k}) by A

_{di}, and so on.

_{i}∈ ℜ

^{m×n}, i ∈ V.

**Remark 1.**We choose linear state feedback control here because it is a quite classical and effective method to stabilize the system. If nonlinear feedback is applied, better control performances may be achieved although, the implementation may become a little more complex or difficult than that of linear state feedback. Furthermore, once the state feedback is successfully applied to the desired issue, based on which Luenburger-like state estimator which is of nonlinear type can be constructed to investigate the current issue further. To this regard, we made the choice of linear state feedback for its important role in further study.

**Definition 1.**(FTB): Given 0≤c

_{1}≤β, c

_{2}≥0, R

_{i}> 0, N ∈ N

_{0}and the time delay τ, if

_{1},c

_{2},β,R

_{i},N).

_{zi}∈ ℜ

^{l×n}and D

_{zwi}∈ ℜ

^{l×s}are both mode-dependent matrices.

**Definition 2.**(H

_{∞}FTB): With the FTB control performance defined in

**Definition 1**achieved, if the following index holds under zero initial conditions:

_{∞}, FTB for any nonzero w(k), where γ > 0 is called the disturbance attenuation rate.

**Assumption 1.**[22]: We assume the nonlinear functions in (1) are monotonically non-decreasing and globally Lipschitz, i.e., the following relation holds:

_{1}, ε

_{2}∈ ℜ, ε

_{1}≠ ε

_{2}, l = 1,…,L, h

_{l}>0.

_{0}(k). According to (4) and (6), the disturbance enters the system states and the outputs respectively. Here we consider two main cases as follows, which is also shown in Figure 1:

_{0}(k) = 1} = E{α

_{0}(k)} = b.

_{1}= 1,2,…,v, where v denotes the total trial numbers, b

_{1}denotes E{q}. Hence E{α

_{1}(k)} = vb

_{1}.

**Remark 2.**The Markovian process denoted by the Markov chain r

_{k}is independent of the Bernoulli distribution α

_{0}(k).

**Remark 3.**For the system under discussion, there are two kinds of noises (disturbances), i.e., the process noise which enters the system states and the measurement noise which enters the system outputs. Usually, these two kinds of noises are taken as mutually uncorrelated white noises when dealing with control problems such as state estimation. However, in practical engineering, colored noises may occur which makes it difficult for the controlled system to guarantee this assumption, especially for a discrete time system sampled from a continuous time system where the process noise is correlated to the measurement noise [23]. Motivated by this kind of application or the like, in this paper we propose the idea of distributed disturbance subject to certain probabilistic distribution shown Figure 1 to make an alternative research.

## 3. Main Results

**Theorem 1.**Given 0 ≤ c

_{1}≤ β, c

_{2}≥ 0, R

_{i}> 0, and N ∈ N

_{0}, system (9) is said to be FTB with respect to (c

_{1}, c

_{2}, β, R

_{i}, N) provided there exist σ

_{1}

^{−}

^{1}> 0, σ

_{2}> 0, σ

_{3}

^{−}

^{1}> 0, α ≥1, symmetric positive definite matrices P

_{i}, Γ, Q

_{i}, a set of diagonal positive definite matrices Λ

_{i}, and matrices K

_{i}such that the following LMIs hold:

**Proof.**Take the following Lyapunov functional:

_{i}> 0, Γ > 0.

**Remark 4.**Recently, the Differential/Difference LMIs (D/DLMIs) are applied to FTS of linear systems and deterministic hybrid systems [24,25], where necessary and sufficient conditions are derived. However, in this paper, due to the introduction of Markov jumps, P

_{i}is a mode-dependent matrix which swifts between different values with the time instant going on. Since the number of modes is finite, P

_{i}takes values from a finite set. If Difference Lyapunov functional is adopted, P will be derived by recursive algorithm rather than chosen from a finite set, which makes it hard to introduce the Markov jumps. Therefore in our opinions DLMIs cannot be applied to our current work directly. However, future work will concentrate on FTB issue of the nonlinear system without Markov jumps based on Difference Lyapunov functional such that less conservative criteria can be derived and output feedback control is also accessible.

_{i}>0, α ≥ 1, and d > 0.

_{k}

_{+1}= j)|r

_{k}= i}, then

**Assumption 1**, the inequality (8) can be rewritten as follows:

_{il}> 0, l = 1,⋯,L.

_{i}= {λ

_{i}

_{1}, ⋯, λ

_{iL}}, H = {h

_{1},⋯,h

_{L}}.

^{T}(k) x

^{T}(k−1) ⋯ x

^{T}(k−τ+1)x

^{T}(k−τ)ϕ

^{T}(ξ(k ))w

^{T}(k)]

^{T}.

_{i}< 0, we have

**Definition 1**,

**Definition 1**, system (9) is FTB with respect to {c

_{1}, c

_{2}, β, R

_{i}, N}. Thus the proof is completed. □

**Theorem 2.**Given 0 ≤ c

_{1}≤ ß,c

_{2}≥ 0, R

_{i}> 0, N ∈ N

_{0}, and d >0, if there exist σ

_{1}

^{−}

^{1}> 0, σ

_{3}

^{−1}> 0, α ≥ 1, symmetric positive definite matrices P

_{i}, Γ, a set of diagonal positive definite matrices Λ

_{i}, and matrices K

_{i}such that the following LMIs hold:

_{∞}performances, $\gamma =\sqrt{d}$is called the disturbance attenuation rate.

**Proof.**According to the Schur Complement [26], ${\overline{G}}_{i}<0$is equivalent to

**Theorem 1**, let Q

_{i}= I, then G

_{i}becomes the principle minor of the left side of (40). Thus G

_{i}< 0 is derived according to (40). Together with the conditions (37)–(39), it can be concluded that system (9) is FTB based on

**Theorem 1**. On the other hand, consider the following function:

**Definition 2**, it concludes that system (9) has H

_{∞}performances with the disturbance attenuation rate $\gamma =\sqrt{d}$. Thus the proof is completed. □

**Theorem 3**. Given 0 ≤ c

_{1}≤ ß, c

_{2}≥ 0, R

_{i}> 0, N ∈ N

_{0}, and d > 0, if there exist σ

_{1}> 0, σ

_{3}> 0, α ≥ 1, symmetric positive definite matrices X

_{i}, Y, a set of diagonal positive definite matrices S

_{i}, and matrices W

_{i}such that the following LMIs hold:

_{i}= W

_{i}X

_{i}

^{−1}.

**Proof.**

_{i}< 0 in (49).

**Remark 5.**If τ = 0 or A

_{di}= C

_{qdi}= 0, system (9) becomes a non-delayed system denoted as follows

_{∞}FTB controller can still be designed by virtue of the following corollary.

**Corollary 1.**Given 0 ≤ c

_{1}≤ ß,c

_{2}≥ 0, R

_{i}> 0, N ∈ N

_{0}, and d > 0, if there exist σ

_{1}> 0, α ≥ 1, symmetric positive definite matrices X

_{i}, a set of diagonal positive definite matrices S

_{i}, and matrices W

_{i}such that the following LMIs hold:

_{i}= W

_{i}X

_{i}

^{−1}.

## 4. Numerical Example

**Example.**Consider system (9) with the following parameters:

**Theorem 3**, we can design the feedback controller as follows:

^{T}(k)R

_{i}x(k)} (i =1,2) in Figure 5 doesn’t exceed the prescribed level ß = 10. The trajectories of system states are depicted in Figures 6 and 7, respectively. And the system outputs perform as shown in Figures 8 and 9. From Figures 6 and 7, it also conclude that the FTB performance is achieved as the trajectories converge to zeros asymptotically in the finite time interval [0, 20].

_{1}values, d

_{min}becomes larger with the increase of b, which demonstrates that with the same distribution ratios, a larger disturbance attenuation rate is required for a lower occurrence probability of disturbance distribution. On the other hand, Table 2 presents that with the same b values, larger values of b

_{1}demands higher d

_{min}, which means with the same occurrence probability of disturbance distribution, if the disturbance affects the system states more than the outputs, the disturbance attenuation performances deserve more strict criteria. In a word, H

_{∞}performances demands higher levels if the disturbance affects the system states more than the outputs and is distributed with a lower occurrence probability.

_{1}= Null), d

_{min}is larger than the counterparts in the case of distributed disturbances. And Table 2 shows the same conclusion.

_{1}any more. Actually whatever b

_{1}takes over the interval [0, 1], the simulation results for d

_{min}make no difference. Therefore, the expression “Null” is used to denote the value of b

_{1}in the case of b = 1.

## 5. Conclusions

_{∞}FTB issues for discrete delayed nonlinear systems with Markovian jumps and disturbances of probabilistic distributions. Concepts of FTB and H

_{∞}FTB are proposed for the discussed system. Then new criteria are derived which guarantee the FTB and H

_{∞}FTB performances for discrete delayed nonlinear systems with Markovian jumps and disturbances of probabilistic distributions. And finally a numerical example is provided to validate the designed controller. The system we considered here contains no uncertainties and linear feedback control is used. Thus in our future work, output feedback control method will be considered for the FTB and H

_{∞}FTB of discrete delayed nonlinear uncertain systems with Markovian jumps and disturbances of probabilistic distributions.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Table 1.**Relations between the disturbance distribution and the attenuation rate: Part I (α = 1.1364).

b | b_{1} | d_{min} |
---|---|---|

0.2 | 0.4 | 1.1400 |

0.4 | 0.4 | 1.6900 |

0.6 | 0.4 | 2.2900 |

0.8 | 0.4 | 2.9300 |

1 | Null | 3.6200 |

**Table 2.**Relations between the disturbance distribution and the attenuation rate: Part II (α = 1.1338).

b | b_{1} | d_{min} |
---|---|---|

0.8 | 0.2 | 2.4400 |

0.8 | 0.4 | 2.5700 |

0.8 | 0.6 | 2.7300 |

0.8 | 0.8 | 2.9300 |

1 | Null | 3.1500 |

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**MDPI and ACS Style**

Chen, H.; Liu, M.; Zhang, S.
Robust H_{∞} Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions. *Entropy* **2015**, *17*, 346-367.
https://doi.org/10.3390/e17010346

**AMA Style**

Chen H, Liu M, Zhang S.
Robust H_{∞} Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions. *Entropy*. 2015; 17(1):346-367.
https://doi.org/10.3390/e17010346

**Chicago/Turabian Style**

Chen, Haiyang, Meiqin Liu, and Senlin Zhang.
2015. "Robust H_{∞} Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions" *Entropy* 17, no. 1: 346-367.
https://doi.org/10.3390/e17010346