CN104049533A - Network control system controller design method based on hidden semi-markov model - Google Patents

Abstract

Disclosed is a network control system controller design method based on a hidden semi-markov model. The method uses the hidden semi-markov model to describe a network load state and then carries out modeling on a network control system in which time delay and packet loss exist and according to a robust control theory, a semi-markov jump linear system theory and a Lyapunov theory, the method uses interval expression of a state self-transfer probability to offer sufficient conditions for existence of an easily solvable state feedback robustness H <infinity> controller of the network control system and offer a design method for a control law so that multimode switching control of the network control system is realized.

Description

Controller Design for Networked Control Systems method based on hidden semi-Markov model

(1) technical field under

The present invention relates to a kind of method for designing of the network control system robust H ∞ controller based on hidden semi-Markov model, it is for the network control system with time delay and packet loss, utilize hidden semi-Markov model to be described and modeling, then utilize semi-Markov jump linear system theory and state to express from the interval of transition probability, provided the adequate condition of the network control system Robust State-Feedback H ∞ controller existence of being convenient to solve, and provided the method for designing of control law, and then realize the multi-modal switching controls of network control system, belong to automatic control technology field.

(2) background technology

The basic block diagram of network control system as shown in Figure 1, in network control system, by shared communication network exchange message, can realize resource sharing, save system cost and improve the objects such as system reliability by sensor, controller and actuator.But due to the introducing of communication network, inevitably have the problems such as network delay and data-bag lost in message transmitting procedure, these problems have been brought new challenge to the analysis and design of network control system.In network environment, due to a plurality of network nodes share same communication network and fluctuations in discharge irregular, be subject to network bandwidth limitations simultaneously, add in network owing to inevitably having network congestion and disconnecting, data will inevitably produce network delay and data-bag lost during by Internet Transmission.Network delay and data-bag lost are the subject matter facing in the research of network control system, they can reduce the performance of system, even cause system unstable, thereby it is very complicated that the analysis that makes system becomes, stability analysis and the controller design problem therefore with the network control system of packet loss cause people's concern day by day.

Network is a distributed stochastic system in essence, and its randomness mainly consists of many enchancement factors such as network load, network blockage, node competition, Route Selection.These enchancement factors cause network delay and packet loss to present randomness in various degree just.Conventionally many enchancement factors such as network load, network blockage, node competition, Route Selection are combined and are defined as a stochastic variable: " network state ".Visible, network state is an abstract concept, and its value is difficult to obtain by measuring directly.But the network performance index that network state can be surveyed by some reflects, in practice, our Adoption Network time delay and these two network service quality indexs of packet loss.We are defined as these network performance indexes one group of observed quantity of network state, and they are not only controlled by network state but also can reflect network state.As can be seen here, network state is hidden under network delay and packet loss and determines the random variation feature of network delay and packet loss, can indirectly obtain network state information by measuring network delay and packet loss again simultaneously.

Network traffics have self-similarity, and semi-Markov chain can be described self-similarity nature under certain condition, therefore can utilize semi-Markov chain to describe the variation of network state.The network state that transmission causes each time changes the once transfer of state generation that can be regarded as semi-Markov chain.Because network state is not directly observed, what can observe is time delay and the packet loss of reflection network state.In order to obtain the information of network state, need to analyze observation variable (being network delay and packet loss), thereby then network state is reconstructed and estimate to obtain the information of this semi-Markov chain.Accordingly, hidden semi-Markov model is a dual random process, and one of them is implicit semi-Markov chain, and another stochastic process is the statistical relationship between description state and observational variable.Thereby the relation between network state and network delay and packet loss is lucky and hidden semi-Markov model matches.Therefore, with hidden semi-Markov model, describe network control system and there is inborn superiority.We can form the semi-Markov chain state in hidden semi-Markov model by network state like this, with network delay and packet loss, form the observational variable in hidden semi-Markov model, thereby the probabilistic relation between network state and network delay and packet loss is modeled as to hidden semi-Markov model.

Under this technical background, the present invention provides a kind of network control system robust H ∞ controller design method based on hidden semi-Markov model, creatively proposed to utilize hidden semi-Markov model to describe offered load state, and for the network control system of considering time delay and packet loss simultaneously, utilize state from the interval of transition probability, to express the adequate condition that has provided the network control system Robust State-Feedback H ∞ controller existence of being convenient to solve, give the method for designing of control law, and realized the multi-modal switching controls of network control system.

(3) summary of the invention

1, goal of the invention

The object of the invention is: for the out of true of offered load state identification, propose a kind of method that can better describe offered load state, adopt hidden semi-Markov model to describe; Overcome the deficiency of existing control technology, for the network control system of considering these two large major issues of time delay and packet loss simultaneously, utilize state from the interval of transition probability, to express the adequate condition that has provided the network control system Robust State-Feedback H ∞ controller existence of being convenient to solve, and the method for designing of control law, realize the multi-modal switching controls of network control system.

The present invention is a kind of network control system robust H ∞ controller design method based on hidden semi-Markov model, its design philosophy is: for the network control system that has time delay and packet loss simultaneously, utilize hidden semi-Markov model to carry out modeling to the relation of network state and network delay and packet loss, estimation model parameter and offered load mode; Theoretical according to robust control theory, semi-Markov jump linear system theory and Liapunov again, utilize state from the interval of transition probability, to express the adequate condition that has provided the network control system Robust State-Feedback H ∞ controller existence of being convenient to solve, and provided the method for designing of control law, and then realize the multi-modal switching controls of network control system.

2, technical scheme

Below in conjunction with the step in FB(flow block) 2, specifically introduce the technical scheme of this method for designing.

The present invention is a kind of network control system robust H ∞ controller design method based on hidden semi-Markov model, and the method concrete steps are as follows:

The identification of first step parameter

Offered load is divided into: basic, normal, high, congested four mode, wherein basic, normal, high mode is distinguished corresponding different time delay distribution, the corresponding packet loss of congested mode.By offered load mode, form the semi-Markov chain state in hidden semi-Markov model, with network delay and packet loss, form the observational variable in hidden semi-Markov model, thereby the probabilistic relation between network state and network delay and packet loss is modeled as to hidden semi-Markov model.Consider the real-time demand of network control system, the present invention adopts discrete hidden semi-Markov model to set up random delay and the packet loss model of network control system.Compare with continuous hidden semi-Markov model, the parameter of discrete hidden semi-Markov model is relatively less, and training dataset is not had to excessive demand, parameter training process is faster, calculate consuming time lowlyer, take controller shorter operation time, more can meet the requirement of real-time of network control system.Thereby the present invention adopts discrete hidden semi-Markov model to set up the random delay model of network control system.

When offered load state and transition probability thereof are identified, first the delay data sequence gathering is averaged to quantification treatment, input using quantized result as hidden semi-Markov model, utilizes Baum-Welch Algorithm for Training hidden semi-Markov model, estimation model parameter.Then after the delay data of current time being quantized, form observed quantity, utilize the model and the improved Viterbi algorithm that have trained to carry out state recognition, thereby obtain offered load mode.

Second step is with analysis and the structure of the network control system model of time delay and packet loss

Consider network control system as shown in Figure 3, wherein between sensor and controller, by network, carry out message exchange, do not have network between controller and actuator, controller and actuator are positioned on same node.For easy analysis, do following hypothesis:

(1) sensor adopts clock to drive, and the sampling period is h, and controller and actuator adopt event-driven;

(2) when network delay is, become uncertain;

(3) node of sensor adopts single bag transmission, and between sensor and controller, has packet loss phenomenon;

(4) packet transmits without out of order situation and occurs in network.

The switching circuit closing when disconnected while regarding the network that has data-bag lost as one section, represents with on off state whether packet is lost, and when switch is closed, indicates without packet loss, and switch represents packet loss while disconnecting.If data-bag lost in transmitting procedure, sending node will transmit the packet of loss or the packet of retransmission of lost not again according to the communication network protocol adopting in system.In control system, the obliterated data of transmission can not reflect the time of day of controlled device again, so the node in network control system all adopts the not method of retransmission of lost packet conventionally.

Data are when transmission over networks, and network delay may be very long and surpassed time limit of regulation, and these data also can deliberately be abandoned.Because the data of long delay may can not meet the real-time demand of system, use the effect that these data produce may it would be better the original old data of utilization, thereby no longer there is value.Abandon in addition out-of-date data and also contribute to alleviate offered load, thereby utilize limited Internet resources to transmit up-to-date data, improve the performance of system.In order to meet the requirement of real-time of control system, adopt initiatively packet loss herein, when network delay is greater than a sampling period, think generation packet loss.

The present invention can On-line Estimation go out model parameter and offered load mode, so can design corresponding controller to different offered load mode.After On-line Estimation offered load mode, be switched in real time corresponding controller, thereby realize multi-modal switching controls, effectively improve the control performance of system, theory structure as shown in Figure 4.

According to above-mentioned analysis, can build the continuous model of network control system:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)+B_{w}w\left(t\right)\\ z\left(t\right)=\mathrm{Cx}\left(t\right)+\mathrm{Dw}\left(t\right)\end{array}\right.---\left(1\right)$

Wherein, x (t) ∈ R ⁿthe state of controlled device, u (t) ∈ R ^mcontrol inputs, w (t) ∈ R ^pthe external disturbance input of system, z (t) ∈ R ^qcontrolled output, A, B, B _w, C, D is the matrix of coefficients with corresponding dimension.

When packet is not lost by Internet Transmission, the discrete time model of the generalized controlled object that comprises network is:

$\left\{\begin{array}{c}{x}_{k+1}={\mathrm{\&Phi;x}}_k+{\mathrm{\&Gamma;}}_0\left({\mathrm{\&tau;}}_k\right)u_k+{\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right)u_{k-1}+{\mathrm{\&Gamma;}}_w{w}_k\\ z_k={\mathrm{Cx}}_k+{\mathrm{Dw}}_k\end{array}\right.---\left(2\right)$

Wherein, Φ=e ^ah, ${\mathrm{\&Gamma;}}_0\left({\mathrm{\&tau;}}_k\right)={\&Integral;}_0^{h-{\mathrm{\&tau;}}_k}{e}^{\mathrm{As}}\mathrm{dsB},$ ${\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right)={\&Integral;}_{h-{\mathrm{\&tau;}}_k}^h{e}^{\mathrm{As}}\mathrm{dsB},$ ${\mathrm{\&Gamma;}}_w={\&Integral;}_0^h{e}^{\mathrm{As}}\mathrm{ds}{B}_w.$

Due to random delay τ _kexistence, the represented network control system system of formula (2) is a Linear Time-Varying Discrete Systems.Order ${\mathrm{\&Gamma;}}_0\left({\mathrm{\&tau;}}_k\right)+{\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right)={\&Integral;}_0^h{e}^{\mathrm{As}}\mathrm{dsB}=H,$ So Linear Time-Varying Discrete Systems (2) can be rewritten as:

$\left\{\begin{array}{c}{x}_{k+1}={\mathrm{\&Phi;x}}_k+(H-{\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right))u_k+{\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right)u_{k-1}+{\mathrm{\&Gamma;}}_w{w}_k\\ z_k={\mathrm{Cx}}_k+{\mathrm{Dw}}_k\end{array}\right.---\left(3\right)$

We can be reduced to one by the indeterminate in Linear Time-Varying Discrete Systems model like this.

For different networking states, by network delay τ _kbe divided into:

τ wherein _ki(i=1,2 ..., N) be random delay, there is different span [a, b].Network delay can be described as:

Wherein, $\frac{a-b}{2}\&le;{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\&le;\frac{b-a}{2}.$

According to matrix theory, the indeterminate Γ in system model (3) ₁(τ _k) can be written as:

$\begin{array}{c}{\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right)={\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_{\mathrm{ki}}\right)\\ ={\&Integral;}_{h-{\mathrm{\&tau;}}_{\mathrm{ki}}}^h{e}^{\mathrm{As}}{d}_{s}B\\ ={\&Integral;}_{h-\frac{a+b}{2}-{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}{e}^{\mathrm{As}}{d}_{s}B}^h\\ ={\&Integral;}_{h-\frac{a+b}{2}}^h{e}^{\mathrm{As}}{d}_{s}B+{\&Integral;}_{h-\frac{a+b}{2}{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}}^{h-\frac{a+b}{2}}{e}^{\mathrm{As}}{d}_{s}B\\ ={\mathrm{\&Gamma;}}_{1i}+e^{A(h-\frac{a+b}{2})}{\&Integral;}_0^{-{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}}{e}^{\mathrm{As}}{d}_s(-B)\end{array}---\left(4\right)$

If

$\stackrel{\&OverBar;}{F}\left({\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\right)={\&Integral;}_0^{-{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}}{e}^{\mathrm{As}}{d}_s,\mathrm{\&delta;}=\underset{{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\&Element;[\frac{a-b}{2},\frac{b-a}{2}]}{\max }{\left|\right|\stackrel{\&OverBar;}{F}\left({\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\right)\left|\right|}_2=\underset{{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\&Element;[\frac{a-b}{2},\frac{b-a}{2}]}{\max }{\left|\right|{\&Integral;}_0^{-{\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}}{e}^{\mathrm{As}}{d}_s\left|\right|}_2={\left|\right|{\&Integral;}_0^{\frac{b-a}{2}}{e}^{\mathrm{As}}{d}_s\left|\right|}_2,E_i=-B,$ the matrix of coefficients of network control system can be illustrated as having probabilistic form:

Γ ₁(τ _k)＝Γ ₁(τ _ki)＝Γ _1i+D _iF(τ′ _ki)E _i (5)

D in above formula _i, E _ifor constant matrices, F (τ ' _ki) value with network delay τ ' _kivariation and change, and satisfy condition

F ^T(τ′ _ki)F(τ′ _ki)≤I (6)

By formula (5) substitution system equation formula (3), variable being changed to of formula (3):

$\left\{\begin{array}{c}{x}_{k+1}={\mathrm{\&Phi;x}}_k+(H-{\mathrm{\&Gamma;}}_{1i}-D_{i}G\left({\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\right)E_i)u_k+({\mathrm{\&Gamma;}}_{1i}+D_{i}F\left({\mathrm{\&tau;}}_{\mathrm{ki}}^{\&prime;}\right)E_i)u_{k-1}+{\mathrm{\&Gamma;}}_w{w}_k\\ z_k={\mathrm{Cx}}_k+{\mathrm{Dw}}_k\end{array}\right.---\left(7\right)$

Hereinafter for mark convenient by F (τ ' _ki) be abbreviated as F _i.

If augmentation is vectorial ${\tilde{x}}_k={\left[\begin{array}{cc}{x}_k^T& u_{k-1}^T\end{array}\right]}^T,$ As system mode q _k=i (i=1,2 ... 3), time, formula (7) can turn to:

$\left\{\begin{array}{c}{\tilde{x}}_{k+1}={\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_i{\tilde{x}}_k+{\widetilde{\mathrm{\&Gamma;}}}_{0i}{u}_k+{\widetilde{\mathrm{\&Gamma;}}}_w{w}_k\\ z_k=\tilde{C}{\tilde{x}}_k+\tilde{D}{w}_k\end{array}\right.---\left(8\right)$

Wherein ${\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_i=\left[\begin{array}{cc}\mathrm{\&Phi;}& {\mathrm{\&Gamma;}}_{1i}+D_i{F}_i{E}_i\\ 0& 0\end{array}\right],{\widetilde{\mathrm{\&Gamma;}}}_{0i}=\left[\begin{array}{c}H-{\mathrm{\&Gamma;}}_{1i}-D_i{F}_i{E}_i\\ I\end{array}\right],{\widetilde{\mathrm{\&Gamma;}}}_w=\left[\begin{array}{c}{\mathrm{\&Gamma;}}_w\\ 0\end{array}\right],\tilde{C}=\left[\begin{array}{cc}C& 0\end{array}\right],\tilde{D}=D.$ If controller adopts state feedback control law the state equation of closed loop network control system can be described as:

$\left\{\begin{array}{c}{\tilde{x}}_{k+1}={\widetilde{\mathrm{\&Phi;}}}_{1i}{\tilde{x}}_k+{\widetilde{\mathrm{\&Gamma;}}}_w{w}_k\\ z_k=\tilde{C}{\tilde{x}}_k+\tilde{D}{w}_k\end{array}\right.---\left(9\right)$

Wherein ${\widetilde{\mathrm{\&Phi;}}}_{1i}={\widehat{\mathrm{\&Phi;}}}_{1i}+{\widehat{\mathrm{\&Gamma;}}}_{0i}{K}_i+{\hat{D}}_i{F}_i({\hat{E}}_i-E_i{K}_i),$ ${\widehat{\mathrm{\&Phi;}}}_{1i}=\left[\begin{array}{cc}\mathrm{\&Phi;}& {\mathrm{\&Gamma;}}_{1i}\\ 0& 0\end{array}\right],{\hat{D}}_i=\left[\begin{array}{c}{D}_i\\ 0\end{array}\right],{\hat{E}}_i=\left[\begin{array}{cc}0& E_i\end{array}\right],{\widehat{\mathrm{\&Gamma;}}}_{0i}=\left[\begin{array}{c}H-{\mathrm{\&Gamma;}}_{1i}\\ I\end{array}\right],$ ${\widetilde{\mathrm{\&Gamma;}}}_w=\left[\begin{array}{c}{\mathrm{\&Gamma;}}_w\\ 0\end{array}\right],\tilde{C}=\left[\begin{array}{cc}C& 0\end{array}\right],\tilde{D}=D.$

When packet occurs to lose by Internet Transmission, actuator keeps the value of previous moment, i.e. u _k=u _k-1, the state equation of network control system is:

$\left\{\begin{array}{c}{\tilde{x}}_{k+1}={\widetilde{\mathrm{\&Phi;}}}_2{\tilde{x}}_k+{\widetilde{\mathrm{\&Gamma;}}}_w{w}_k\\ z_k=\tilde{C}{\tilde{x}}_k+\tilde{D}{w}_k\end{array}\right.---\left(10\right)$

Wherein ${\widetilde{\mathrm{\&Phi;}}}_2=\left[\begin{array}{cc}\mathrm{\&Phi;}& \mathrm{\&Gamma;}\\ 0& I\end{array}\right],\mathrm{\&Gamma;}={\&Integral;}_0^h{e}^{\mathrm{As}}\mathrm{dsB},{\widetilde{\mathrm{\&Gamma;}}}_w=\left[\begin{array}{c}{\mathrm{\&Gamma;}}_w\\ 0\end{array}\right],\tilde{C}=\left[\begin{array}{cc}C& 0\end{array}\right],\tilde{D}=D.$

Like this, when considering network delay and packet loss, network control system can be described with following discrete time semi-Markov jump linear system simultaneously:

$\left\{\begin{array}{c}{\tilde{x}}_{k+1}={\widetilde{\mathrm{\&Phi;}}}_{{\mathrm{\&sigma;}}_k,{\mathrm{\&theta;}}_k}{\tilde{x}}_k+{\widetilde{\mathrm{\&Gamma;}}}_w{w}_k\\ z_k=\tilde{C}{\tilde{x}}_k+\tilde{D}{w}_k\end{array}\right.{\mathrm{\&sigma;}}_k=\left\{\mathrm{1,2}\right\},{\mathrm{\&theta;}}_k=\{\mathrm{1,2},...N\}---\left(11\right)$

Work as σ _k=1, represent that network control system does not exist packet drop, θ _k=1,2 ... N} represents that network control system is respectively in basic, normal, high load.Work as σ _k=2 o'clock, represent network control system generation packet loss, in congestion load mode.

The 3rd step has the network control system robust H of time delay and packet loss _∞the design of controller

This step is theoretical according to robust control theory and Liapunov, design point feedback robust H _∞controller, thus realize the multi-modal switching controls of network control system.But because analysis and design method is comparatively complicated, so this step is divided three small steps,---proofs to three theorems---carry out.

Lemma 1 (Schur mends lemma): to given symmetric matrix $J_{11}=\left(\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}\right),$ J wherein ₁₁∈ R ^{r * r}, below three conditions be of equal value:

(1)J＜0；

$\left(2\right)J_{11}<0,J_{22}-J_{12}^T{J}_{11}^{-1}{J}_{12}<0;$

$\left(3\right)J_{22}<0,J_{11}-J_{12}{J}_{22}^{-1}{J}_{12}^T<0;$

Lemma 2: establish W, M, N, F is the real matrix with suitable dimension, wherein F meets F ^tf≤I, W is symmetrical matrix, so W+MFN+N ^tf ^tm ^t< 0, and and if only if exists constant ε > 0, makes W+ ε MM ^t+ ε ^-1n ^tn < 0.

Theorem 1: for given constant γ, if there is one group of positive definite symmetric matrices P _i(i ∈ Q), meets following LMI (LMI), and the semi-Markov jump linear system that formula (11) is described is random stable and have a H _∞performance γ.

${\mathrm{\&Omega;}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\mathrm{\&Xi;}}_i\left(d\right)& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)<0,i=\mathrm{1,2}...N,1\&le;d\&le;D---\left(12\right)$

${\mathrm{\&Theta;}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\mathrm{\&Pi;}}_i\left(d\right)& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_2^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)<0,i=\mathrm{1,2}...N,1\&le;d\&le;D---\left(13\right)$

Wherein, $Q_i=(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i,{\mathrm{\&Xi;}}_i\left(d\right)={\widetilde{\mathrm{\&Phi;}}}_{1i}^T[(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i]{\widetilde{\mathrm{\&Phi;}}}_{1i}-P_i,$ ${\mathrm{\&Pi;}}_i\left(d\right)={\widetilde{\mathrm{\&Phi;}}}_2^T[(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i]{\widetilde{\mathrm{\&Phi;}}}_2-P_i,$ λ _ii(d) be state i residence time while being d from transition probability, P _ijit is the state transition probability that utilizes Baum_Welch algorithm identified.

Proof: select Lyapunov function q wherein _kfor offered load mode, and P _qk(be P _i) be symmetric positive definite matrix.

When there is not packet loss by Internet Transmission in packet, σ in corresponding (11) _k=1.

$\begin{array}{c}E\left\{V({\tilde{x}}_{k+1},q_{k+1})\right|{\tilde{x}}_k,q_k=i\}-V({\tilde{x}}_k,q_k=i)\\ ={\tilde{x}}_{k+1}^T[(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;1}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i]{\tilde{x}}_{k+1}-{\tilde{x}}_k^T{P}_i{\tilde{x}}_k\end{array}$

$={\tilde{x}}_k^T\left\{{\widetilde{\mathrm{\&Phi;}}}_{1i}^T\right[(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i]{\widetilde{\mathrm{\&Phi;}}}_{1i}-P_i\}{\tilde{x}}_k---\left(14\right)$

$\begin{array}{c}={\tilde{x}}_k^T{\mathrm{\&Xi;}}_i\left(d\right){\tilde{x}}_k\\ \&le;\max \left\{{\mathrm{\&lambda;}}_{\max }{\mathrm{\&Xi;}}_i\left(d\right)\right\}{\tilde{x}}_k^T{\tilde{x}}_k\end{array}$

When packet is during by Internet Transmission generation packet loss, σ in corresponding (11) _k=2.

$\begin{array}{c}E\left\{V({\tilde{x}}_{k+1},q_{k+1})\right|{\tilde{x}}_k,q_k=i\}-V({\tilde{x}}_k,q_k=i)\\ ={\tilde{x}}_{k+1}^T[(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i]{\tilde{x}}_{k+1}-{\tilde{x}}_k^T{P}_i{\tilde{x}}_k\end{array}$

$={\tilde{x}}_k^T\left\{{\widetilde{\mathrm{\&Phi;}}}_2^T\right[(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i]{\widetilde{\mathrm{\&Phi;}}}_2-P_i\}{\tilde{x}}_k---\left(15\right)$

$\begin{array}{c}={\tilde{x}}_k^T{\mathrm{\&Pi;}}_i\left(d\right){\tilde{x}}_k\\ \&le;\max \left\{{\mathrm{\&lambda;}}_{\max }{\mathrm{\&Pi;}}_i\left(d\right)\right\}{\tilde{x}}_k^T{\tilde{x}}_k\end{array}$

So, have

$\begin{array}{c}E\left\{V({\tilde{x}}_{k+1},q_{k+1})\right\}-V({\tilde{x}}_0,q_0)\\ =E\{\underset{m=0}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&delta;}}_m{\tilde{x}}_m^T{\mathrm{\&Xi;}}_i\left(d\right){\tilde{x}}_m+\underset{m=0}{\overset{k}{\mathrm{\&Sigma;}}}(1-{\mathrm{\&delta;}}_m){\tilde{x}}_m^T{\mathrm{\&Pi;}}_i\left(d\right){\tilde{x}}_m\}\end{array}$

$\&le;\max \left\{{\mathrm{\&lambda;}}_{\max }{\mathrm{\&Xi;}}_i\left(d\right)\right\}E\left\{\underset{m=0}{\overset{k}{\mathrm{\&Sigma;}}}{\mathrm{\&delta;}}_m{\tilde{x}}_m^T{\tilde{x}}_m\right\}+\max \left\{{\mathrm{\&lambda;}}_{\max }{\mathrm{\&Pi;}}_i\left(d\right)\right\}E\left\{\underset{m=0}{\overset{k}{\mathrm{\&Sigma;}}}(1-{\mathrm{\&delta;}}_m){\tilde{x}}_m^T{\tilde{x}}_m\right\}---\left(16\right)$

$\&le;\mathrm{\&beta;E}\left\{\underset{m=0}{\overset{k}{\mathrm{\&Sigma;}}}{\tilde{x}}_m^T{\tilde{x}}_m\right\}$

δ wherein _m∈ { 0,1}, β=max{ λ _maxΞ _i(d), λ _max∏ _i(d) }.

If Ξ _i(d) < 0, ∏ _i(d) < 0, and β < 0, and

$\begin{array}{c}E\left\{\underset{k=0}{\overset{m}{\mathrm{\&Sigma;}}}{\tilde{x}}_m^T{\tilde{x}}_m\right\}\&le;{\mathrm{\&beta;}}^{-1}E\left\{V({\tilde{x}}_{k+1},q_{k+1})\right\}-{\mathrm{\&beta;}}^{-1}V({\tilde{x}}_0,q_0)\\ <-{\mathrm{\&beta;}}^{-1}V({\tilde{x}}_0,q_0)\end{array}---\left(17\right)$

As k → ∞, can obtain

$\begin{array}{c}E\left\{\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\tilde{x}}_m^T{\tilde{x}}_m\right\}<-{\mathrm{\&beta;}}^{-1}V({\tilde{x}}_0,q_0)\\ <\&infin;\end{array}---\left(18\right)$

Therefore work as Ξ _i(d) < 0, ∏ _i(d), during < 0, system is that robust is random stable.

If definition $Q_i=(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{\underset{j\&NotEqual;i}{j=1}}{\overset{N}{\mathrm{\&Sigma;}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i,$ When there is not packet loss by Internet Transmission in packet,

$z_k^T{z}_k-{\mathrm{\&gamma;}}^2{w}_k^T{w}_k+E\left\{V({\tilde{x}}_{k+1},q_{k+1})\right|{\tilde{x}}_k,q_k=i\}-V({\tilde{x}}_k,q_k=i)$

$=\left(\begin{array}{cc}{\tilde{x}}_k^T& w_k^T\end{array}\right)\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\mathrm{\&Xi;}}_i\left(d\right)& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)\left(\begin{array}{c}{\tilde{x}}_k\\ w_k\end{array}\right)---\left(19\right)$

$={\hat{x}}_k^T{\mathrm{\&Omega;}}_i\left(d\right){\hat{x}}_k$

When packet is during by Internet Transmission generation packet loss,

$z_k^T{z}_k-{\mathrm{\&gamma;}}^2{w}_k^T{w}_k+E\left\{V({\tilde{x}}_{k+1},q_{k+1})\right|{\tilde{x}}_k,q_k=i\}-V({\tilde{x}}_k,q_k=i)$

$=\left(\begin{array}{cc}{\tilde{x}}_k^T& w_k^T\end{array}\right)\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\mathrm{\&Pi;}}_i\left(d\right)& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_2^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)\left(\begin{array}{c}{\tilde{x}}_k\\ w_k\end{array}\right)---\left(20\right)$

$={\hat{x}}_k^T{\mathrm{\&Theta;}}_i\left(d\right){\hat{x}}_k$

Wherein ${\hat{x}}_k={\left[\begin{array}{cc}{\tilde{x}}_k^T& w_k^T\end{array}\right]}^T,{\mathrm{\&Omega;}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\mathrm{\&Xi;}}_i\left(d\right)& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right),$

${\mathrm{\&Theta;}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\mathrm{\&Pi;}}_i\left(d\right)& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_2^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{Q}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right).$

Due to Ω _i(d) < 0 is Ξ _i(d) adequate condition of < 0, and Θ _i(d) < 0 is ∏ _i(d) adequate condition of < 0.So, if Ω _i(d) < 0, Θ _i(d) < 0, and system is that robust is random stable.

Further, have $E\left\{\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{z}_k^T{z}_k\right\}<{\mathrm{\&gamma;}}^{2}E\left\{\underset{k=0}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{w}_k^T{w}_k\right\}+E\left\{V\left(0\right)\right\}-E\left\{V(\&infin;)\right\}.$ Under zero initial condition, || z|| ₂≤ γ || w|| ₂.Therefore when system meets formula (12) and (13), system is that robust is stable and have a H at random _∞performance γ.

Although it is stable and have a H at random that theorem 1 has provided system (11) _∞the adequate condition of performance γ, but due to λ _ii(d) randomness, cannot directly verify the feasibility of theorem.Thereby utilize state to express from the interval of transition probability, by time become unsolvable linear inequality and be converted to the form that can separate, and further derive and be convenient to solve the method for designing of controller according to theorem 1.

Theorem 2: for given constant γ, if there is one group of positive definite symmetric matrices P _i(i ∈ Q), meets following LMI (LMI), and the semi-Markov jump linear system that formula (11) is described is random stable and have a H _∞performance γ.

${\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\tilde{Q}}_{1i}-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)<0,i=\mathrm{1,2}...N,1\&le;d\&le;D---\left(21\right)$

${\underset{\&OverBar;}{\mathrm{\&Omega;}}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Phi;}}}_{1i}-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)<0,i=\mathrm{1,2}...N,1\&le;d\&le;D---\left(22\right)$

${\stackrel{\&OverBar;}{\mathrm{\&Theta;}}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_2^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Phi;}}}_2-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_2^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)<0,i=\mathrm{1,2}...N,1\&le;d\&le;D---\left(23\right)$

${\underset{\&OverBar;}{\mathrm{\&Theta;}}}_i\left(d\right)=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_2^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Phi;}}}_2-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_2^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)<0,i=\mathrm{1,2}...N,1\&le;d\&le;D---\left(24\right)$

Wherein, ${\stackrel{\&OverBar;}{Q}}_i=(1-\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)P_i,$ ${\underset{\&OverBar;}{Q}}_i=(1-p_i\left(1\right)p_{\mathrm{ii}})\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+p_i\left(1\right)p_{\mathrm{ij}}{P}_i.$

Proof: from theorem 1, when there being one group of positive definite symmetric matrices P _i(i ∈ Q) meets formula (12) and (13), and the semi-Markov jump linear system that formula (11) is described is random stable and have a H _∞performance γ.The λ becoming in the time of in formula (12) _ii(d) can utilize mathematics conversion as follows:

${\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)={\&PartialD;}_1{\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}}+{\&PartialD;}_2{\underset{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}}---\left(25\right)$

Wherein, ${\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}}=\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right),$ λ _ii＝p _i(1)p _ii， ${\&PartialD;}_1+{\&PartialD;}_2=1$ And ${\&PartialD;}_1,{\&PartialD;}_2>0.$

So have ${\stackrel{\&OverBar;}{Q}}_i=(1-\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)P_i,$ ${\underset{\&OverBar;}{Q}}_i=(1-p_i\left(1\right)p_{\mathrm{ii}})\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+p_i\left(1\right)p_{\mathrm{ij}}{P}_i.$ So, can obtain ${\&PartialD;}_1{\stackrel{\&OverBar;}{Q}}_i+{\&PartialD;}_2{\underset{\&OverBar;}{Q}}_i={\&PartialD;}_1\left[(1-\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)P_i)\right]+{\&PartialD;}_2[(1-p_i\left(1\right)p_{\mathrm{ij}})\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+p_i\left(1\right)p_u{P}_i]$

$={\&PartialD;}_1[(1-{\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}})\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\stackrel{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}}{P}_i]+{\&PartialD;}_2[(1-{\underset{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}})\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\underset{\&OverBar;}{\mathrm{\&lambda;}}}_{\mathrm{ii}}{P}_i]---\left(26\right)$

$\begin{array}{c}=(1-{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right))\underset{j\&NotEqual;i}{\overset{N}{\underset{j=1}{\mathrm{\&Sigma;}}}}{p}_{\mathrm{ij}}{P}_j+{\mathrm{\&lambda;}}_{\mathrm{ii}}\left(d\right)P_i\\ =Q_i\end{array}$

And then, have

${\&PartialD;}_1{\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}_i+{\&PartialD;}_2{\underset{\&OverBar;}{\mathrm{\&Omega;}}}_i={\&PartialD;}_1\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Phi;}}}_{1i}-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{\stackrel{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)+{\&PartialD;}_2\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Phi;}}}_i-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T{\underset{\&OverBar;}{Q}}_i{\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)$

$\begin{array}{c}=\left(\begin{array}{cc}{\tilde{C}}^T\tilde{C}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T({\&PartialD;}_1{\stackrel{\&OverBar;}{Q}}_i+{\&PartialD;}_2{\underset{\&OverBar;}{Q}}_i){\widetilde{\mathrm{\&Phi;}}}_{1i}-P_i& {\tilde{C}}^T\tilde{D}+{\widetilde{\mathrm{\&Phi;}}}_{1i}^T({\&PartialD;}_1{\stackrel{\&OverBar;}{Q}}_i+{\&PartialD;}_2{\underset{\&OverBar;}{Q}}_t){\widetilde{\mathrm{\&Gamma;}}}_w\\ *& {\tilde{D}}^T\tilde{D}+{\widetilde{\mathrm{\&Gamma;}}}_w^T({\&PartialD;}_1{\stackrel{\&OverBar;}{Q}}_i+{\&PartialD;}_2{\underset{\&OverBar;}{Q}}_i){\widetilde{\mathrm{\&Gamma;}}}_w-{\mathrm{\&gamma;}}^{2}I\end{array}\right)\\ ={\mathrm{\&Omega;}}_i\left(d\right)\end{array}---\left(27\right)$

So when system meets with time, meet Ω _i(d) < 0.With should system meeting with time, meet therefore when system meets formula (21)-(24), system is stable and have a H at random _∞performance γ, theorem 2 must be demonstrate,proved.

In theorem 2, provided system at random stable and meet H _∞the adequate condition of performance γ, but can not solve controller gain, thereby need the method for solving of further derivation controller gain, the i.e. parametric expressions of controller.

Theorem 3: for given constant γ, if there is one group of positive definite symmetric matrices X _iand the matrix Y of corresponding dimension _i, positive scalar ε, wherein i ∈ Q, sets up following LMI, and semi-Markov jump system (11) is the random stable H that has of robust _∞performance γ, wherein state feedback controller gain matrix is

$\left(\begin{array}{ccccc}-X_i& 0& {\stackrel{\&OverBar;}{\mathrm{\&Delta;}}}_i^T& X_i{\tilde{C}}^T& {({\hat{E}}_i{X}_i-E_i{Y}_i)}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_i^T& {\tilde{D}}^T& 0\\ *& *& -\underset{\&OverBar;}{Z}+\mathrm{\&epsiv;}{\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i^T{\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i& 0& 0\\ *& *& *& -I& 0\\ *& *& *& *& -\mathrm{\&epsiv;I}\end{array}\right)<0---\left(28\right)$

$\left(\begin{array}{ccccc}-X_i& 0& {\underset{\&OverBar;}{\mathrm{\&Delta;}}}_i^T& X_i{\tilde{C}}^T& {({\hat{E}}_i{X}_i-E_i{Y}_i)}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\underset{\&OverBar;}{W}}_i^T& {\tilde{D}}^T& 0\\ *& *& -\underset{\&OverBar;}{Z}+\mathrm{\&epsiv;}{\underset{\&OverBar;}{\mathrm{\&Pi;}}}_i^T{\underset{\&OverBar;}{\mathrm{\&Pi;}}}_i& 0& 0\\ *& *& *& -I& 0\\ *& *& *& *& -\mathrm{\&epsiv;I}\end{array}\right)<0---\left(29\right)$

$\left(\begin{array}{cccc}-X_i& 0& {\stackrel{\&OverBar;}{U}}_{2i}^T& X_i{\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_{2i}^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)<0---\left(30\right)$

$\left(\begin{array}{cccc}-X_i& 0& {\underset{\&OverBar;}{U}}_{2i}^T& X_i{\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\underset{\&OverBar;}{W}}_{2i}^T& {\tilde{D}}^T\\ *& *& -\underset{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)<0---\left(31\right)$

Wherein,

${\stackrel{\&OverBar;}{\mathrm{\&Delta;}}}_i^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{({\widehat{\mathrm{\&Phi;}}}_{1i}{X}_i+{\widehat{\mathrm{\&Gamma;}}}_{0i}{Y}_i)}^T,$

${\stackrel{\&OverBar;}{W}}_i^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\widetilde{\mathrm{\&Gamma;}}}_w^T,\underset{\&OverBar;}{Z}=\mathrm{diag}\{X_1,...X_{i-1},X_{i-2},...X_N,X_i\},$

${\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\hat{D}}_i^T,\stackrel{\&OverBar;}{Z}=\mathrm{diag}\{X_i,...X_{i-1},X_{i-2},...X_N,X_i\},$

${\underset{\&OverBar;}{\mathrm{\&Delta;}}}_i^T=[\sqrt{p_{i1}}...\sqrt{p_{\mathrm{ij}}}...\sqrt{p_{\mathrm{iN}}}\sqrt{p_i\left(1\right)p_{\mathrm{ii}}}]{({\widehat{\mathrm{\&Phi;}}}_{1i}{X}_i+{\widehat{\mathrm{\&Gamma;}}}_{0i}{Y}_i)}^T,{\underset{\&OverBar;}{W}}_i^T=[\sqrt{p_{i1}}...\sqrt{p_{\mathrm{ij}}}...\sqrt{p_{\mathrm{iN}}}\sqrt{p_i\left(1\right)p_{\mathrm{ii}}}]{\widetilde{\mathrm{\&Gamma;}}}_w^T,$

${\underset{\&OverBar;}{\mathrm{\&Pi;}}}_i=[\sqrt{p_{i1}}...\sqrt{p_{\mathrm{ij}}}...\sqrt{p_{\mathrm{iN}}}\sqrt{p_i\left(1\right)p_{\mathrm{ii}}}]{\hat{D}}_i^T,{\underset{\&OverBar;}{U}}_{2i}^T=[\sqrt{p_{i1}}...\sqrt{p_{\mathrm{ij}}}...\sqrt{p_{\mathrm{iN}}}\sqrt{p_i\left(1\right)p_{\mathrm{ii}}}]X_i{\widetilde{\mathrm{\&Phi;}}}_2^T,$

${\stackrel{\&OverBar;}{U}}_{2i}^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]X_i{\widetilde{\mathrm{\&Phi;}}}_2^T,$

${\stackrel{\&OverBar;}{W}}_{2i}^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\widetilde{\mathrm{\&Gamma;}}}_w^T,$

${\underset{\&OverBar;}{W}}_{2i}^T=[\sqrt{p_{i1}}...\sqrt{p_{\mathrm{ij}}}...\sqrt{p_{\mathrm{iN}}}\sqrt{p_i\left(1\right)p_{\mathrm{ii}}}]{\widetilde{\mathrm{\&Gamma;}}}_w^T,$ 1≤j≤N and j ≠ i, P _ijand P _i(d) be respectively state transition probability and the residence time probability that utilizes the Baum_Welch algorithm identified of hidden semi-Markov model.

Proof: formula (28) use Schur complement fixed is managed:

$\left(\begin{array}{cccc}-P_i& 0& {\stackrel{\&OverBar;}{V}}_i^T& {\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_i^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)<0---\left(32\right)$

Wherein $\stackrel{\&OverBar;}{Z}=\mathrm{diag}\{X_1,...X_{i-1},X_{i+1},...X_N,X_i\},X_i=P_i^{-1};$

${\stackrel{\&OverBar;}{V}}_i^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{({\widehat{\mathrm{\&Phi;}}}_{1i}+{\widehat{\mathrm{\&Gamma;}}}_{0i}{K}_i+{\hat{D}}_i{F}_i({\hat{E}}_i-E_i{K}_i))}^T;$

${\stackrel{\&OverBar;}{W}}_i^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\widetilde{\mathrm{\&Gamma;}}}_w^T.$

Use again diag{X _i, θ } and (θ=diag (I wherein ₁, I ₂... I _n), I _iunit matrix for corresponding dimension) difference left and right multiplier (32), can obtain:

$\left(\begin{array}{cccc}-X_i& 0& {\stackrel{\&OverBar;}{U}}_i^T& X_i{\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_i^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)<0---\left(33\right)$

Wherein ${\stackrel{\&OverBar;}{U}}_i^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{({\widehat{\mathrm{\&Phi;}}}_{1i}{X}_i+{\widehat{\mathrm{\&Gamma;}}}_{0i}{K}_i{X}_i+{\hat{D}}_i{F}_i({\hat{E}}_i-E_i{K}_i)X_i)}^T.$

Formula (33) is equivalent to

$\left(\begin{array}{cccc}-X_i& 0& {\stackrel{\&OverBar;}{\mathrm{\&Theta;}}}_i^T& X_i{\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_i^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)+{\left[\begin{array}{cccc}0& 0& {\mathrm{\&Pi;}}_i& 0\end{array}\right]}^T{F}_i\left[\begin{array}{cccc}{\hat{E}}_i{X}_i-E_i{K}_i{X}_i& 0& 0& 0\end{array}\right]+{\left[\begin{array}{cccc}{\hat{E}}_i{X}_i-E_i{K}_i{X}_i& 0& 0& 0\end{array}\right]}^T{F}_i^T\left[\begin{array}{cccc}0& 0& {\mathrm{\&Pi;}}_i& 0\end{array}\right]<0---\left(34\right)$

Wherein ${\stackrel{\&OverBar;}{\mathrm{\&Theta;}}}_i^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{({\widehat{\mathrm{\&Phi;}}}_{1i}{X}_i+{\widehat{\mathrm{\&Gamma;}}}_{0i}{K}_i{X}_i)}^T;$

${\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\hat{D}}_i^T.$

According to lemma 2, there is constant ε > 0, make

$\left(\begin{array}{cccc}-X_i& 0& {\stackrel{\&OverBar;}{\mathrm{\&Theta;}}}_i^T& X_i{\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_i^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)+\mathrm{\&epsiv;}{\left[\begin{array}{cccc}0& 0& {\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i& 0\end{array}\right]}^T\left[\begin{array}{cccc}0& 0& {\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i& 0\end{array}\right]+{\mathrm{\&epsiv;}}^{-1}{\left[\begin{array}{cccc}{\hat{E}}_i{X}_i-E_i{K}_i{X}_i& 0& 0& 0\end{array}\right]}^T\left[\begin{array}{cccc}{\hat{E}}_i{X}_i-E_i{K}_i{X}_i& 0& 0& 0\end{array}\right]<0---\left(35\right)$

Utilize Schur complement fixed reason, can obtain

$\left(\begin{array}{ccccc}-X_i& 0& {\stackrel{\&OverBar;}{\mathrm{\&Theta;}}}_i^T& X_i{\tilde{C}}^T& {({\hat{E}}_i{X}_i-E_i{K}_i{X}_i)}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_i^T& {\tilde{D}}^T& 0\\ *& *& -Z+\mathrm{\&epsiv;}{\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i^T{\stackrel{\&OverBar;}{\mathrm{\&Pi;}}}_i& 0& 0\\ *& *& *& -I& 0\\ *& *& *& *& -\mathrm{\&epsiv;I}\end{array}\right)<0---\left(36\right)$

Make Y _i=K _ix _i, just can obtain formula (28).

In like manner, by formula (22), can obtain formula (29).

Formula (23) is used to Schur complement fixed reason, can obtain:

$\left(\begin{array}{cccc}-\stackrel{\&CenterDot;}{P_i}& 0& {\stackrel{\&OverBar;}{V}}_{2i}^T& {\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_{2i}^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)<0---\left(37\right)$

Wherein $\stackrel{\&OverBar;}{Z}=\mathrm{diag}\{X_1,...X_{i-1},X_{i-2},...X_N,X_i\},X_i=P_i^{-1},$

${\stackrel{\&OverBar;}{V}}_{2i}^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\widetilde{\mathrm{\&Phi;}}}_2^T,$

${\stackrel{\&OverBar;}{W}}_{2i}^T=[\sqrt{p_i\left(1\right)p_{i1}}...\sqrt{p_i\left(1\right)p_{\mathrm{ij}}}...\sqrt{p_i\left(1\right)p_{\mathrm{iN}}}\sqrt{\underset{d=2}{\overset{D}{\mathrm{\&Sigma;}}}{p}_i\left(d\right)}]{\widetilde{\mathrm{\&Gamma;}}}_w^T.$

Use again diag{X _i, θ } and (θ=diag (I wherein ₁, I ₂... I _n), I _iunit matrix for corresponding dimension) difference left and right multiplier (37), can obtain:

$\left(\begin{array}{cccc}-X_i& 0& X_i{\stackrel{\&OverBar;}{V}}_{2i}^T& X_i{\tilde{C}}^T\\ *& -{\mathrm{\&gamma;}}^{2}I& {\stackrel{\&OverBar;}{W}}_{2i}^T& {\tilde{D}}^T\\ *& *& -\stackrel{\&OverBar;}{Z}& 0\\ *& *& *& -I\end{array}\right)<0---\left(38\right)$

Make Y _i=K _ix _i, just can obtain formula (30).

In like manner, by formula (24), obtain formula (29).

Therefore when system meets formula (25-29), system is that robust is stable and have a H at random _∞performance γ.

The 4th step controller performance check

Whether this step meets design requirement the selection of check controller parameter, by means of conventional numerical evaluation and Control System Imitation instrument Matlab, carries out.In actual applications, first utilize Baum-Welch algorithm to realize the identification to the estimation of offered load state and transition probability thereof, then design robust H _∞controller, makes network control system in the situation that considering time delay, packet loss and random perturbation, meet robust stable and have a H ∞ performance γ at random.

The 5th step design finishes

Whole design process emphasis has been considered the estimation of offered load state and the identification of transition probability thereof and has been had time delay and the multi-modal switching controls of the network control system of packet loss.First in the above-mentioned first step, offered load state and transition probability thereof are identified; In second step, analyze and set up the network control system model with time delay and packet loss; In the 3rd step, designed the network control system robust H ∞ controller with time delay and packet loss, to reach the object that makes system stability; After above steps, design finishes.

3, advantage and effect

The present invention is a kind of network control system robust H ∞ controller design method based on hidden semi-Markov model, for realizing the multi-modal switching controls of network control system.The advantage of the method comprises two aspects: one, and utilize hidden semi-Markov model to describe network control system, solved because of the prior known problem that causes being difficult to use in real network control system of hypothesis offered load state-transition matrix.Its two, considered time delay and packet loss simultaneously, in the time of CONTROLLER DESIGN, considered parameter uncertainty, thereby made controller there is robustness.

(4) accompanying drawing explanation

Fig. 1: typical network control system basic block diagram

Fig. 2: a kind of network control system robust H ∞ controller design method schematic flow sheet based on hidden semi-Markov model of the present invention

Fig. 3: the network control system structural drawing with time delay and packet loss

Fig. 4: the multi-modal switching controls structural drawing of network control system based on hidden semi-Markov model

Fig. 5: State-output curve

Fig. 6: disturbance input curve

Fig. 7: the controlled curve of output of network control system

The horizontal ordinate of Fig. 5-7 represents the sampling period; The ordinate of Fig. 5 represents the state of system, dimensionless; The ordinate of Fig. 6 represents the amplitude of disturbance input, dimensionless; The ordinate of Fig. 7 represents the controlled output of system, dimensionless.

(5) embodiment

Design object of the present invention comprises two aspects: one, realize the identification to the estimation of offered load state and transition probability thereof; Its two, for the network control system with time delay and packet loss, design robust H ∞ controller, makes network control system meet robust stable at random.Wherein the specific targets of target two are that the horizontal γ of Disturbance Rejection of system is less than setting value 1.5.In concrete enforcement, the emulation of the estimation of offered load state and the identification of transition probability and closed-loop control system and check all realize by means of Matlab.

The first step: parameter identification

Offered load is divided into: low (L), in (M), high (H), congested (C) four mode, utilize Baum-Welch Algorithm for Training, obtain model parameter (being state transition probability) and be:

$P=\left(\begin{array}{cccc}0.6598& 0.1639& 0.0787& 0.0976\\ 0.8472& 0.1179& 0.0349& 0.0000\\ 0.5496& 0.3120& 0.1274& 0.0110\\ 0.8230& 0.1681& 0.0058& 0.0030\end{array}\right)---\left(39\right)$

Second step: with the structure of the network control system model of time delay and packet loss

In system, the model of controlled device is as shown in (1), wherein $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],B_w=\left[\begin{array}{c}0.1\\ 0.2\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=0.1.$

Set up departments system sampling period be h=0.5s, discrete time model is suc as formula shown in (2):

$x_{k+1}={\mathrm{\&Phi;x}}_k+{\mathrm{\&Gamma;}}_0\left({\mathrm{\&tau;}}_k\right)u_k+{\mathrm{\&Gamma;}}_1\left({\mathrm{\&tau;}}_k\right)u_{k-1}+{\mathrm{\&Gamma;}}_w{w}_k$

The 3rd step: there is the design of the network control system robust H ∞ controller of time delay and packet loss

According to theorem 3, utilize the state transition probability P obtain above, and apply the LMI tool box solving state feedback controller of MATLAB, obtain controller gain and be:

K ₁＝(-0.9739 -1.6795 -0.1645)

K ₂＝(-0.8750 -1.6583 -0.3896)

K ₃＝(-1.2105 -2.2559 -0.8242)

The 4th step: tracking performance check

Whether this step meets design requirement checking system tracking performance, by means of conventional numerical evaluation and Control System Imitation instrument Matlab, carries out.

Even can see and have network delay and data-bag lost by Fig. 5, at the state of the effect lower network control system of designed controller, can reach robust stable at random.By Fig. 6-7, can be found out, according to the state feedback controller of this paper method design, make network control system meet robust stable at random, and through calculating || z (k) || ₂/ || w (k) || ₂=0.6582, be less than horizontal γ=1.5 of Disturbance Rejection of setting.Therefore, network control system meets H ∞ disturbance decay indices, thereby has illustrated that method proposed by the invention is effective.

The 5th step: design finishes

Whole design process emphasis has been considered the estimation of offered load state and the identification of transition probability thereof and has been had time delay and the multi-modal switching controls of the network control system of packet loss.Around these two emphasis, the estimation of offered load state and the identification of transition probability thereof first in the above-mentioned first step, have been carried out; In second step, built the network control system model with time delay and packet loss; In the 3rd step, for the network control system with time delay and packet loss, design robust H ∞ state feedback controller, made network control system meet robust stable at random; After above steps, design finishes.

Claims (3)

1. the Controller Design for Networked Control Systems method based on hidden semi-Markov model, it is characterized in that: utilize hidden semi-Markov model to describe offered load state, again for the network control system with time delay and packet loss, according to robust control theory, semi-Markov jump linear system is theoretical and Liapunov is theoretical, utilize state from the interval of transition probability, to express the adequate condition that provides the network control system Robust State-Feedback H ∞ controller existence of being convenient to solve, and provide the method for designing of control law, and then realize the multi-modal switching controls of network control system.

2. the Controller Design for Networked Control Systems method based on hidden semi-Markov model according to claim 1, is characterized in that: utilize hidden semi-Markov model to describe offered load state, realize the estimation of offered load state and the identification of transition probability thereof.

3. the Controller Design for Networked Control Systems method based on hidden semi-Markov model according to claim 1, it is characterized in that: when network control system being carried out to the design of modeling and control device, utilized state to express from the interval of transition probability and provided the adequate condition of the controller existence of being convenient to solve and the method for designing of control law; And considered the major issue in time delay and these two network control systems of packet loss simultaneously.