H 2 Control for the Continuous-Time Markovian Jump Linear Uncertain Systems with Partly Known Transition Rates and Input Quantization

For a class of continuous-timeMarkovian jump linear uncertain systems with partly known transition rates and input quantization, theH 2 state-feedback control design is considered.The elements in the transition ratesmatrix include completely known, boundary known, and completely unknown ones. First, anH 2 cost index for Markovian jump linear uncertain systems is introduced; then by introducing a new matrix inequality condition, sufficient conditions are formulated in terms of linear matrix inequalities (LMIs) for theH 2 control of the Markovian jump linear uncertain systems. Less conservativeness is achieved than the result obtained with the existing technique. Finally, a numerical example is given to verify the validity of the theoretical results.


Introduction
Recently, much attention has been devoted to the study of the stochastic stability for the Markovian jump systems, and many important results have been published [1][2][3][4].This is because the Markovian jump systems have been widely employed to model many practical systems, such as manufacturing systems, the power systems, and the economic systems in which they may experience abrupt changes in their structures and parameters [5,6].It is worth noticing that these results require that the transition probabilities/rates must be known a priori.However, in many practical engineering applications, the likelihood for obtaining the perfect information on all transition probabilities/rates elements is questionable, and the cost might be expensive in some cases.Therefore, the study of the stabilization of the Markovian jump systems with partly known transition probabilities/rates becomes interesting, and some well-known results have been published.The idea for the stochastic stability of the Markovian jump linear uncertain systems with partly known transition probabilities/rates is developed in Zhang et al. [7].It is then applied to the  ∞ control design in Zhang and Boukas [8].In those papers, the feature of the information about the transition probabilities/rates matrix considered includes two kinds of elements: completely known and completely unknown ones.As a matter of fact, the transition probabilities matrix might involve completely known, completely unknown, and boundary known elements.In Shen and Yang [9], an  2 state-feedback controller design method is proposed for the continuous-time Markovian jump linear uncertain systems with the three kinds of transition rates matrix elements.In order to yield the design condition for analysis and synthesis, a matrix inequality is introduced to present LMIs conditions.In this paper, less conservative design conditions will be formulated by introducing a new relaxed matrix inequality condition.
On the other hand, in many modern engineering practices, all kinds of information processing devices, such as analog-to-digital and digital-to-analog converters, have been widely used.By the utilization of such information processing Mathematical Problems in Engineering devices, some advantages have been brought, for example, lower cost, reduced weight and power, simple installation, and maintenance.However, some new phenomena have also been induced, which might cause server deterioration of system performance or even lead to system instability.Signal quantization is one of the important aspects that should be fully considered in such cases, which always exists in computer-based control systems.Nowadays, many well-known results have been published on quantized feedback control.For example, the feedback stabilization problem is considered by utilizing dynamic quantizers [10][11][12][13] and static quantizers [14][15][16][17].In addition, the filter design [18] and the  ∞ control design [19] are also investigated.Specially, quantization errors have adverse effects on the network control systems which can often be modeled as Markovian jump systems.In Xiao et al. [20], the stabilization problem for single-input discrete Markovian jump linear uncertain systems via mode-dependent quantized state-feedback is addressed, but the transition rates are assumed to be completely known.
To the best of our knowledge, no result has been presented for the control design of the continuous-time Markovian jump linear uncertain systems with partly known transition rates and input signal quantization.In this paper, the  2 control for a class of continuous-time Markovian jump linear uncertain systems with respect to partly known transition rates and input signal quantization is addressed.The structure of the controller consists of two parts: the nonlinear part is provided to eliminate the effect of input quantization, and the linear part is obtained by solving LMIs for achieving the  2 performance against unknown transition rates and model uncertainties.In comparison with the design utilizing the LMIs technique in Shen and Yang [9], the design method has less conservativeness by introducing a relaxed inequality condition.
The rest of this paper is organized as follows.The problem statement and preliminaries are presented in Section 2. The main results are given in Section 3. In Section 4, a numerical example is presented to illustrate the effectiveness of the results, and the conclusions are drawn in Section 5.
Notations.Throughout this paper, the following notations are used.  denotes the -dimensional Euclidean space;   denotes the transpose of matrix ;  and 0 represent the identity matrix and a zero matrix in appropriate dimensions, respectively; E{⋅} denotes the mathematical expectation operator;  >  ( ≥ ), where  and  are symmetric matrices, means that  −  is positive definite (positive semi-definite); ||  denotes the -norm of the vector ; that is, The notation | ⋅ |, in particular denotes the absolute value of a scalar, the standard Euclidean norm of a vector, and the induced norm of a matrix, respectively.In symmetric block matrices, an * is used to represent a term that is induced by symmetry.Finally, the symbol () is used to represent  +   .

Problem Statement and Preliminaries
Consider a class of the continuous-time Markovian jump linear uncertain systems in the following probability space (Ω, F, P): where () ∈   is the system state and () ∈   is the control input.{  ,  ≥ 0} is a continuous-time Markovian process with right continuous trajectories taking values in the finite set S = {1, 2, . . ., N}.It governs the switching among the different system modes with the following mode transition probabilities: where ℎ > 0, lim ℎ → 0 ((ℎ)/ℎ) = 0, and   ≥ 0 (,  ∈ S;  ̸ = ) denote the switching rate from mode  at time  to mode  at time  + ℎ and that   = − ∑ N =1, ̸ =    for each  ∈ S. In general, the Markovian process transition rates matrix Λ is defined by: In this paper, the transition rates of the jumping process are assumed to be partly available; that is, some elements in matrix Λ have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use.For instance, for the system (1) with four operation modes, the transition rates matrix might be described by Furthermore, let S   = { :   ≤   ≤   }; then one can obtain S   = S  1 ∪ S  2 .If S   ̸ = Ø, it can be described as Similarly, if S   ̸ = Ø, let us denote that where in the th row of the matrix Λ.K   2 denotes the  2 th completely unknown element with the index K   2 in the th row of the matrix Λ.    and    represent the number of elements in S   and S   , respectively.For example, considering the transition rates matrix (4), one can easily check that Remark 1.When the lower and upper bounds of the elements in S  2 are equal, the transition rates matrix is reduced to the considered case in Zhang et al. [7].It is obvious that the solving method there can only treat  ∈ S  2 as the completely unknown case which can result in some conservativeness.
For convenience, the notations   = (  ),   = (  ), Δ  = Δ(  ), and Δ  = Δ(  ) are used for each possible value   = ,  ∈ S, where   and   are known constant matrices with appropriate dimensions.Then, the system (1) can be described by The following assumptions are assumed to be valid.Assumption 1.The pair (  ,   ) is controllable.Assumption 2. Consider the following: where   ,   ,   , and   are known constant matrices with appropriate dimensions,   () and Ξ  () are time-varying uncertain matrices satisfying      ≤  and Ξ  Ξ   ≤ , and the parameter   satisfies 0 ≤   < 1.
In addition, the quantizer (⋅) is defined by an operator function round(⋅) that rounds towards the nearest integer; that is, where (>0) is called a quantizing level of the quantizer.In computer-based control systems, the value of  depends on the sampling accuracy and is known a priori.(⋅) is the uniform quantizer with the fixed level .Define   = (()) − (), since each component of   is bounded by the half of the quantizing level ; thus, we have The objective of this paper is to design the state-feedback control law  () =    +   ,   =  (  ) , when   = , (10) such that the resulting closed-loop system is stochastically stable and obtains as small value of the  2 cost index for the Markovian jump linear uncertain systems given in the following as possible where (  ) and (  ) are positive definite matrices.The nonlinear part of the controller   is designed against the effect of signal quantization, and the linear part    is proposed to deal with model uncertainties and to unknown transition rates and achieve optimal  2 performance.

Main Results
Theorem 8.For the system (1) subject to Assumptions 1 and 2, suppose that there exist the symmetric positive definite matrices   , general matrices   , and positive scalars   ,   , and  such that where   ] , Then, the controller designed as Proof.Take the Lyapunov function candidate  =     ; then, along the system trajectory of plant (1), the weak infinitesimal operator I   [⋅] of the process {(),   ,  ≥ 0} for plant (8) at the point {, , } is given by Kushner [23] as follows: For Case 1, pre-and postmultiplying  −1  in the first inequality in (27) and using Lemma 7 in the second inequality in (21), one can get that Let   =  −1  ,   =    −1  , and   =  −1  ; then, we have Applying the Schur complement formula, one can get (16).For Case 2, pre-and postmultiplying  −1  in the first inequality in (28) and applying Lemma 7 to the second inequality in (28), one can obtain that Let   =  −1  and   =    −1  ; then, one can see that Thus, the LMIs in ( 17) are derived by using the Schur complement formula.
From the above proof, one can see that It follows from Kushner [23] that Since E[(, , ) |  0 ] ≥ 0, by some simple calculation, one can achieve that Therefore, the minimum cost can be obtained by minimizing .Thus, the proof is achieved.
Remark 9.The merit of the proposed results lies in that the transition rates of the jumping process are assumed to be more general, which means that some elements in the transition rates matrix have been exactly known, some ones have been merely known with lower and upper bounds, and others may have no information to use.Dealing with the unknown transition rates, a less conservative method is used.At the same time, the impact of the input signal quantization on the system is also considered.Finally, the controller design conditions are presented in the framework of LMIs.
In order to make comparison with the design method using the LMIs technique in Shen and Yang [9], we present the conditions designed by the utilization of Lemma 2 there.Proposition 10.For the system (1) subject to Assumptions 1 and 2, suppose that there exist symmetric positive definite matrices   , general matrices   , and positive scalars   ,   , and  such that  (22).The details are omitted here for space limitation.

Numerical Example
An example is presented to illustrate the effectiveness of the proposed method.
Consider the MJLSs with four operation modes as follows: The considered transition rates matrix is given as follows: where −1.
The switching mode, the control input, and the response curves of the system states are presented in Figures 1, 2, and 3, respectively.Among them, Figure 1 shows a possible system modes evolution which meets the transition rates given in this example.As shown in Figure 1, the system has 4 modes and is in different modes at a different time.Figure 2 shows the curve of the control input ().With this controller, Figure 3 depicts the state response curves of the closed-loop system.It can be seen that the considered continuous-time Markovian jump linear uncertain system is stochastically stable in spite of mismatched uncertainty, the input signal quantization, and the partly known transition rates covering the completely known, the boundary unknown, and the completely unknown elements in the transition rates matrix.performance against model uncertainties and unknown transition rates.In comparison with the existing result in the literature, less conservativeness has been obtained by introducing new relaxed inequality conditions.Finally, a numerical example is given to show the effectiveness of the proposed design method.

2 Figure 1 :
Figure 1: Evolution of the system mode.

Figure 2 :
Figure 2: The curve of the control input ().

𝐻 2 Figure 3 :
Figure 3: The response curve of the state .

Table 1 :
Comparison of optimal .