CN105242540A - Network control system dynamic switching control method based on average residence time - Google Patents

Abstract

Provided is a network control system dynamic switching control method based on average residence time. The method based on the average residence time is characterized by converting the influence of uncertain and randomly-changing time delay and packet loss on dynamic property of a network control system into a kind of discrete switching event, and establishing a discrete switching control model of the network control system allowing to comprise an unstable subsystem; ensuring that the network-induced time delay is always smaller than one sampling period, and meanwhile, guaranteeing that a new control variable in the current sampling period can be acted on a controlled object by utilizing an improved initiative variable sampling hybrid node driving mechanism; and providing a sufficient condition for solving the existence of a modal dependence switching controller based on a multiple-Lyapunov function method, and providing a design method of the switching controller meeting indefinite subsystem switching sequence allowing the average residence time by adopting an optimal cycle algorithm, and ensuring the stability of the network control system under the condition of limited communication and incomplete information.

Description

Based on the network control system switching at runtime control method of average residence time

Technical field

The present invention relates to automation field, in particular to the network control system switching at runtime control method based on average residence time, the stochastic uncertainty that after the network control system sampling discretization that the time delay of uncertain change after introducing for network and packet loss cause, dynamic perfromance changed along with the sampling period, the switching Controlling model of the network control system comprising unstable subsystem is allowed based on average residence time method establishment, employing multiple-Lyapunov function method gives the adequate condition that the discretize network control system mode with time delay and packet loss relies on switch controller existence, and give the method for handover control meeting the indefinite subsystem transfer sequence allowing average residence time.

Background technology

Along with fusion and the development of 3C technology, traditional point-to-point Control system architecture can not meet requirement that is day by day complicated and Long-distance Control, and the advantages such as network control system is low with its cost, space distribution wide, connect flexibly, function is complicated are applied in Process Control System, robot, space operations, remote operation, intelligent grid and high-performing car operating system widely.Each network nodes such as controller, sensor and actuator complete control task based on the work of bus communication mechanism close coordination.The research of NCS has become a study hotspot of domestic and international control field.

While but being introduced in of network brings various advantage to control system, bring also to system and control theory and new have challenging problem, mainly contain: (1) multiple users share communication line and the irregular and different procotol of fluctuations in discharge will inevitably cause network inducement delay of different nature (2) network congestion, disconnecting and channel disturbance will inevitably cause the sequential entanglement of network packet and the loss of packet.The introducing of these factors changeable at random considerably increases the complicacy of system analysis and design, because: the moment that (1) data arrive is no longer permanent with well-regulated, causes the forfeiture of system steadiness; (2) data may occur to lose or mistake in the process of transmission, cause the forfeiture of data integrity; (3) uncertainty of data transmission period and the randomness of packet loss generation, the deterministic forfeiture of the system that result in.Therefore for the structure of network control system complexity, operation characteristic and existing uncertain network inducement, to set up new system model, analyze the behaviour of systems, reappraise and set up network Controlling model and control method be extremely important is also necessary.

Switched system is the very important hybrid system of a class, and typical switched system comprises one group of switching law how switched between (or discrete) subsystem and determinant system continuously, according to switching law switching at runtime between each subsystem.And network control system is said in some sense and can be referred to as a switched system, because it had both contained continuously dynamic controlled device subsystem, contained again the control device subsystem of Discrete Dynamic.Due to the existence of unpredictable time-delay and packet loss, the discrete controlled quentity controlled variable that actuator acts in controlled device can think uncertain event, between two discrete control events, the dynamic perfromance of controlled device is different, changes along with the time-varying control amount irregularly arrived.Therefore can think that network control system switches between continuous print subsystem according to the switch controller comprising switching law, control and coordinate whole system and normally run.Fig. 1 is the schematic diagram that network control system is converted into handover control system model.

Summary of the invention

The object of the invention is to the structure for network control system complexity, dynamically changeable operation characteristic and the existing uncertain network inducement causing system unstability, a kind of network control system switching at runtime control method based on average residence time is provided, in conjunction with average residence time method propose a kind of can better comprehensive description unpredictable time-delay and the packet loss discretize switched system modeling method that can comprise unstable subsystem that network control system dynamic property is affected, adopt mixed node driving mechanism and the time shaft gridding method of the active Variable sampling improved, provide based on multiple-Lyapunov function method the discretize network control system mode with time delay and packet loss being convenient to solve and rely on the adequate condition of switch controller existence and the satisfied method for designing allowing the switch controller of the indefinite subsystem transfer sequence of average residence time, ensure the stability that is limited and information imperfect situation lower network control system that communicates.

In order to achieve the above object, technical scheme of the present invention is as follows:

Based on the network control system switching at runtime control method of average residence time, comprise the following steps:

Step one, uncertain time-varying characteristics based on network inducement delay in closed loop network control system and packet loss, adopt the sensor node combination drive mechanism of initiatively Variable sampling, while ensureing that network inducement delay is always less than a sampling period, ensure that controlled quentity controlled variable new in the current sampling period can be applied in controlled device; Initiatively the concrete mechanism of Variable sampling is as follows:

(1) supposing that time shaft is divided into is hour layout of Δ l at equal intervals;

(2) if controlled device is a kth controlled quentity controlled variable to current being applied to, then sensor will trigger kth+1 sample event;

(3) tu is made _krepresent that kth is successfully sent to the time point of the controlled quentity controlled variable of actuator, and use network delay τ _krepresent that kth arrives the packet of actuator from sensor to controller

Time delay with the time delay of controller to actuator sum;

(4) in order to avoid not arriving actuator or data-bag lost due to the long data packet time and the network control system caused is in the system unstability problem that may cause under open loop situations, for a long time if the maximum permission sampling period is T _maxif, network transfer delay τ _kexceed T _max, then employing time type of drive is triggered next sample event by sensor;

(5) kth is set to be applied to the sampling time point of the controlled quentity controlled variable of controlled device as s _k, then next sampling time point s _k+1system of selection be:

$s_{k+1}=\left\{\begin{array}{cc}nl& {\mathrm{tu}}_k\∈\[(n-2)l,(n-1)l)\\ s_k+T_{\max }& {\mathrm{tu}}_k\≥s_k+T_{max}\end{array}\right.---\left(1\right)$

Wherein, T _maxallowed maximum sampling interval (T _max=N × Δ l, N is positive integer), n is positive integer and 0 < n < N;

(6) order set Γ={ t ₁, t ₂, t ₃... } represent and effectively reach time point, its implication is that controlled quentity controlled variable not only successfully arrives actuator and the time point be successfully applied in controlled device.Because when packet generation incorrect order, only have up-to-date packet just can be applied in controlled device.

(7) efficiently sampling time point i is introduced _m, namely successfully act on the packet of controlled device based on state variable sampling time point be called efficiently sampling time point, if I={i ₁, i ₂, i ₃... } and be the set of effective sampling points, under the effect of zero-order holder, feedback controller is at the time interval [t _k, t _k+1) in can be expressed as:

u(t)＝u(i _k)＝K(i _k)x(i _k)t _k≤t＜t _k+1(3)

Step 2, Time And Event mixed node driving mechanism based on the active Variable sampling of step one, set up the switching Controlling model of network control system:

If h _kfor the time span in a kth sampling period, then controlled device can be obtained based on above-mentioned improvement active Variable sampling technology state equation after discretize is:

x(i _k+1)＝Φ _kx(i _k)+Γ ₀(τ _k,h _k)u(i _k)+Γ ₁(τ _k,h _k)u(i _k-1)(4)

Wherein

${\mathrm{\&Phi;}}_k=e^{{\mathrm{Ah}}_k},{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)={\&Integral;}_0^{h_k-{\mathrm{\&tau;}}_k}{e}^{As}dsB,{\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)={\&Integral;}_{h_k-{\mathrm{\&tau;}}_k}^{h_k}{e}^{As}dsB---\left(5\right)$

Introduce new augmentation vector z (k)=[x (i _k) u (i _k-1)] ^t, then from the augmentation closed-loop system of the network control system below formula (4) and formula (5) can obtain:

z(k+1)＝Ψ _kz(k)(6)

Wherein

${\mathrm{\&Psi;}}_k=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_k+{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)K\left(i_k\right)& {\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)\\ K\left(i_k\right)& 0\end{array}\right]---\left(7\right)$

(1) d is defined _ktwo effective sampling points i _kand i _k+1between continual data package dropout number, then can obtain i _k+1-i _k=d _k+ 1.Suppose that maximum continual data package dropout number is d _max, then d _kspan be D={0,1 ..., d _max.Based on the rasterizing discrete method of time shaft, time-vary delay system τ _klimited discrete value will be changed into, the micro-Τ of value set=Δ l, 2 × Δ l ..., T _max(Δ l be above-mentioned time shaft is divided into little at equal intervals time layout, and T _max=Δ l × N).

(2) h _kfor a kth efficiently sampling cycle, be easy to obtain h _k=τ _k+ Δ l+T _maxd _k, as can be seen from formula, system matrix Φ _k, Γ ₀(τ _k, h _k), Γ ₁(τ _k, h _k) value by time delay τ _kwith continual data package dropout number d _k, therefore, augmented system (6) can be seen as the switched system that comprises time delay and packet loss information subsystem, wherein system matrix Ψ _kvalue from finite set below

Ω＝{Ψ ₁(τ _k＝Δl,d _k＝0),Ψ ₂(τ _k＝2×Δl,d _k＝0),...,Ψ _M(τ _k＝T _max,d _k＝d _max)}，

M＝N×(1+d _max).

(3) and then, we can be write as augmented system (6) form of switched system:

Wherein σ (l _k) ∈ Ι=1,2 ..., M}, M=N × (1+d _max) be called switching signal, as σ (l _k)=m, can obtain:

${\hat{A}}_{\mathrm{\&sigma;}\left(l_k\right)}={\hat{A}}_m=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_m+{\mathrm{\&Gamma;}}_{m,0}{K}_m& {\mathrm{\&Gamma;}}_{m,1}\\ K_m& 0\end{array}\right]---\left(9\right)$

(4) under normal circumstances, effectively sample all by the switching of a triggering subsystem each time, but when network condition is just in time the same, if time delay and packet loss are all τ _k=τ _a, d _k=d _a, therefore, can Ψ be obtained from formula (7) ₁=Ψ ₂, this illustrates that second time efficiently sampling does not have the switching between triggers system.Now need definition new variables l _kshow the real moment switching generation, so the switching instant point of system is (l ₁, l ₂..., l _m...);

Step 3, based on average residence time method analysis package containing the stable adequate condition of the switched system of unstable subsystem and switching signal average residence time needed for the condition that meets:

First the definition of average residence time is provided: for arbitrary l>=l ₀with arbitrary switching signal σ (k), l ₀≤ k < l, makes N _σ[l ₀, l) represent that σ (k) is at the time interval [l ₀, switching times l).If for N ₀>=0 and τ _a> 0, has N _σ[l ₀, l)≤N ₀+ (l-l ₀)/τ _aset up, then τ _abe called average residence time, N ₀represent boundary of trembling.For the purpose of simple also without loss of generality, we make N ₀=0,

Based on analysis above and definition, we can obtain lemma below, show that network control system switches Sufficient Conditions On Stability and the required average residence time met of switching signal of Controlling model (8).For the sake of simplicity, with (0,1,2 ..., l ...) and represent time point on time shaft (0, Δ l, 2 × Δ l ..., l × Δ l ...);

Lemma 1: consider switched system (8), and make | α _m| < 1, m ∈ Ι, if μ>=1 is given constant. there is positive definite function V _{σ (k)}: and Κ function β ₁, β ₂, and p (m), m ∈ Ι is that priori is known, then exist make:

β ₁(||x(l)||)≤V _m(l)≤β ₂(||x(l)||)(10)

$V_m\left(k\right)\&le;{\mathrm{\&mu;V}}_n\left(k\right),\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n---\left(12\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(13\right)$

So switched system (8) is Globally asymptotic, as long as the average residence time of switching signal meets:

${\mathrm{\&tau;}}_a>{\mathrm{\&tau;}}_a^{*}=-\frac{In\mathrm{\&mu;}}{In\mathrm{\&lambda;}}---\left(14\right)$

Wherein p (m), m ∈ Ι is the incidence of switched system subsystem m;

Prove: can obtain from (11)

ΔV _m(l)＝V _m(l+1)-V _m(l)≤α _mV _m(l)(15)

Then

$\begin{array}{c}{V}_m(l+1)\&le;(1+{\mathrm{\&alpha;}}_m)V_m\left(l\right)\\ \&le;{\left(1+{\mathrm{\&alpha;}}_m\right)}^2{V}_m\left(l-1\right)\\ .\\ .\\ .\\ \&le;{\left(1+{\mathrm{\&alpha;}}_m\right)}^{l+1-l_m}{V}_m\left(l_m\right)\end{array}---\left(16\right)$

Can obtain further

$V_m\left(l\right)\&le;{(1+{\mathrm{\&alpha;}}_m)}^{l-l_m}{V}_m\left(l_m\right)---\left(17\right)$

Again according to (12), the N in the definition of (17) and average residence time _σ(t ₀, t), obtain:

$\begin{array}{c}{V}_{\mathrm{\&sigma;}\left(l\right)}\left(l\right)\&le;{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{V}_{\mathrm{\&sigma;}\left(l_m\right)}\left(l_m\right)\\ \&le;{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{\mathrm{\&mu;V}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\left(l_m\right)\\ \&le;\mathrm{\&mu;}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\right)}^{l_m-l_{m-1}}{V}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\left(l_{m-1}\right)\\ .\\ .\\ .\\ \&le;{\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}\left(0,l\right)}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\right)}^{l_m-l_{m-1}}\mathrm{...}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(0\right)}\right)}^{l_1}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}\left(0,l\right)}\underset{m=1}{\overset{M}{\&Pi;}}{\left(1+{\mathrm{\&alpha;}}_m\right)}^{p\left(m\right)\&times;l}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ <{\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}\left(0,l\right)}{\mathrm{\&lambda;}}^l{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&mu;}}^{l/{\mathrm{\&tau;}}_a}{\mathrm{\&lambda;}}^l{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&lambda;}}^{\left(In\mathrm{\&mu;}/{\mathrm{\&tau;}}_{a}In\mathrm{\&lambda;}+1\right)\&times;l}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\end{array}---\left(18\right)$

According to Lyapunov stability theory, if average residence time meets (14), then switched system (8) is Globally asymptotic;

The adequate condition that the stability controller that step 4, the mode meeting average residence time providing closed loop network control system switching Controlling model (8) being convenient to solve based on multiple-Lyapunov function method rely on exists:

Adopt multiple-Lyapunov function method, namely each subsystem has the Lyapunov function of oneself, for subsystem m, namely as σ (l _kduring)=m ∈ Ι, its Lyapunov function is:

V _m(l)＝z ^T(l)P _mz(l)(19)

So can obtain from formula (11):

${\mathrm{\&Delta;V}}_m\left(l\right)-{\mathrm{\&alpha;}}_m{V}_m\left(l\right)=z^T\left(l\right)\&lsqb;{\hat{A}}_m^T{P}_m{\hat{A}}_m-P_m-{\mathrm{\&alpha;}}_m{P}_m\&rsqb;z\left(l\right)---\left(20\right)$

Can obtain from formula (12):

V _m(k)-μV _n(k)＝z ^T(k)[P _m-μP _n]z(k)(21)

So based on Lyapunov stability theory, if set up according to lemma 1 condition below:

${\hat{A}}_m^T{P}_m{\hat{A}}_m-P_m-{\mathrm{\&alpha;}}_m{P}_m<0---\left(22\right)$

P _m-μP _n≤0(23)

As long as then the average residence time of switching signal meets (14), closed loop network control system switches Controlling model (8) Globally asymptotic, therefore obtains following theorem:

Theorem 1: for given scalar | α _m| < 1, m ∈ Ι, if there is M symmetric positive definite matrix P in μ>=1 ₁, P ₂..., P _m, make $\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n:$

$\left[\begin{array}{cc}-P_m& \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{\hat{A}}_m^T{P}_m\\ \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{P}_m{\hat{A}}_m& -P_m\end{array}\right]<0,\&ForAll;m\&Element;I---\left(24\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(25\right)$

P _m≤μP _n(26)

When so meeting (14) for the average residence time of switching signal, closed loop network control system switches Controlling model (8) Globally asymptotic, and for subsystem m, its energy attenuation or escalating rate are

The mode that the switching signal that step 5, design are convenient to solve meets closed loop network control system switching Controlling model (8) of average residence time (14) relies on stability controller, realizes the point stabilization that is limited and information imperfect situation lower network control system that communicates;

In order to solve controller gain, first define matrix below:

$\begin{array}{c}\tilde{A}=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_m& {\mathrm{\&Phi;}}_{m,1}\\ 0& 0\end{array}\right],{\tilde{B}}_m=\left[\begin{array}{c}{\mathrm{\&Gamma;}}_{m,0}\\ I\end{array}\right],\\ {\tilde{K}}_m=\left[\begin{array}{cc}{K}_m& 0\end{array}\right],\mathrm{\&sigma;}\left(k\right)=m,m\&Element;I\end{array}---\left(27\right)$

Then closed loop network control system switching Controlling model can be write as form below:

$z(k+1)=({\tilde{A}}_m+{\tilde{B}}_m{\tilde{K}}_m)z\left(k\right)---\left(28\right)$

The LMI that solving state feedback modalities relies on stability controller gain so can be provided by theorem below, specific as follows:

Theorem 2: for given scalar | α _m| < 1, m ∈ Ι, if there is M symmetric positive definite matrix G in μ>=1 respectively _m, V _m, m ∈ Ι, and M matrix R _m, m ∈ Ι, makes $\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n:$

$\left[\begin{array}{cccc}-G_m& 0& \frac{1}{\sqrt{\left(1+{\mathrm{\&alpha;}}_m\right)}}{G}_m^T{\mathrm{\&Phi;}}_m^T+R_m^T{\mathrm{\&Gamma;}}_{m,0}^T& \frac{1}{\sqrt{\left(1+{\mathrm{\&alpha;}}_m\right)}}{R}_m^T\\ *& -V_m& \frac{1}{\sqrt{\left(1+{\mathrm{\&alpha;}}_m\right)}}{V}_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -G_m& 0\\ *& *& *& -V_m\end{array}\right]<0,\&ForAll;m\&Element;I---\left(29\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(30\right)$

G _m-μG _n≤0(31)

V _m-μV _n≤0(32)

When so meeting (14) for the average residence time of switching signal, closed loop network control system switches Controlling model (8) Globally asymptotic, and the mode obtained relies on the gain of feedback controller is:

$K_m=R_m{G_m}^{-1},m\&Element;I---\left(33\right)$

Step 6, subsystem energy attenuation rate the determination of span:

Switched system (8) both may comprise stable subsystem, also unstable subsystem may be comprised, therefore, on the one hand in order to ensure the stability of whole system, the attenuation rate of stabistor system is wanted could cut down the energy increased in unstable subsystem to a certain extent greatly; On the other hand, the attenuation rate of stabistor system can not be too large, because the increase of system attenuation rate means needs, exigent controller is calmed, and this brings difficulty will to solving of controller gain.The span design optimization problem below of the subsystem energy attenuation rate so mentioned in theorem 1, adopts LMI to solve:

Maximumε

Subjectto

$\begin{array}{c}\left[\begin{array}{cccc}-G_m& 0& {\mathrm{\&epsiv;G}}_m^T{\mathrm{\&Phi;}}_m^T+{\mathrm{\&epsiv;R}}_m^T{\mathrm{\&Gamma;}}_{m,0}^T& {\mathrm{\&epsiv;R}}_m^T\\ *& -V_m& {\mathrm{\&epsiv;V}}_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -G_m& 0\\ *& *& *& -V_m\end{array}\right]<0,\&ForAll;m\&Element;I\\ G_m>0,\\ V_m>0,\\ G_m\&le;{\mathrm{\&mu;G}}_n,\\ V_m\&le;{\mathrm{\&mu;V}}_n,\&ForAll;m,n\&Element;I,m\&NotEqual;n\\ \mathrm{\&epsiv;}>1\end{array}---\left(34\right)$

Wherein $\mathrm{\&epsiv;}=\frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}},m\&Element;I.$

In order to solve with MatlabLMI tool box, above formula is changed into canonical form below:

Minimizeω

Subjectto

$\begin{array}{c}\left[\begin{array}{cccc}-G_m& 0& G_m^T{\mathrm{\&Phi;}}_m^T+R_m^T{\mathrm{\&Gamma;}}_{m,0}^T& R_m^T\\ *& -V_m& V_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -M_m& 0\\ *& *& *& -N_m\end{array}\right]<0,\&ForAll;m\&Element;I,\\ G_m\&le;{\mathrm{\&mu;G}}_n,\\ V_m\&le;{\mathrm{\&mu;V}}_n,\&ForAll;m,n\&Element;I,m\&NotEqual;n\\ 0<\mathrm{\&omega;}<1,\\ G_m,V_m,M_m,N_m>0,\\ Y_m<{\mathrm{\&omega;G}}_m,M_m<{\mathrm{\&omega;Y}}_m\\ Z_m<{\mathrm{\&omega;V}}_m,N_m<{\mathrm{\&omega;Z}}_m\end{array}---\left(35\right)$

Step 7, the mode adopting LMI tool box to solve the network control system with random delay and packet loss in matlab rely on the gain of feedback controller and the border of subsystem energy attenuation rate, complete the design of network control system controller, ensure the validity of stability controller within the scope of required subsystem energy attenuation.

Controller performance simulating, verifying:

The mode adopting LMI tool box to solve the network control system with random delay and packet loss for given simulation example in matlab relies on the gain of feedback controller and the border of subsystem energy attenuation rate, result is implanted to access control performance in Networked controller.In matlab programmed process, in order to ensure under the restriction not having subsystem switching characteristic sequence that mode relies on the validity of feedback controller, namely the switching sequence of subsystem is unknowable, and we are to G _m≤ μ G _nand V _m≤ μ V _nhave employed a round-robin algorithm, i.e. forn=1:M, ifn ≠ mG _m≤ μ G _nv _m≤ μ V _n, ensure that the validity of controller under switching sequence indefinite.

Advantage of the present invention:

(1) the present invention is directed to the structure of network control system complexity, dynamically changeable operation characteristic and the existing uncertain network inducement causing system unstability, in conjunction with average residence time method propose a kind of can better comprehensive description unpredictable time-delay and the packet loss discretize switched system modeling method that can comprise unstable subsystem that network control system dynamic property is affected, and go out to be convenient to the switch controller with the discretize network control system of time delay and packet loss of the indefinite subsystem transfer sequence of the satisfied permission average residence time solved based on multiple-Lyapunov function method design.

(2) the present invention is based on the uncertain time-varying characteristics of network inducement delay and packet loss in closed loop network control system, propose the sensor node combination drive mechanism of the active Variable sampling of improvement, while guarantee network inducement delay is always less than a sampling period, ensures that controlled quentity controlled variable new in the current sampling period can be applied in controlled device, establishes and comprise time delay and packet loss information subsystem network control system switching Controlling model.

(3) the present invention adopts average residence time method to analyze the stability of switched system, and this method can carry out the design of stability analysis and stability controller to the switched system comprising unstable subsystem.Present invention employs multiple-Lyapunov function method simultaneously, avoid the conservative property adopting announcement effects method to ask for fixed gain, be more convenient to solve the discretize network control system mode with time delay and packet loss based on multiple-Lyapunov function method and rely on the adequate condition of switch controller existence and the gain of controller.

(4) the present invention have employed a round-robin algorithm when solving controller gain LMI MATRIX INEQUALITIES, can ensure under the restriction not having subsystem switching characteristic sequence that mode relies on the validity of feedback controller, namely the switching sequence of subsystem is in unknowable situation, and the network control system mode of our design relies on switch controller and still can realize the point stabilization that is limited and information imperfect situation lower network control system that communicates.

(5) gridding method is incorporated in the Stabilizing Controller Design of network control system by the present invention, the time delay of uncertain change is converted into discrete finite value, thus in conjunction with discrete continual data package dropout value, time-varying system is converted into the limited discrete-time switched systems of switching law, and ensure that the feasibility of the LMI (linearmatrixinequalities, LMIs) solving mode dependent status feedback control gain and the solvability of convex optimization problem.

Accompanying drawing explanation

Fig. 1 is that the switching Controlling model of network control system in prior art transforms schematic diagram.

Fig. 2 is the active Variable sampling method sampling process schematic diagram that the present invention improves.

Fig. 3 is the data packet transmission example of the active Variable sampling method that the present invention improves.

Fig. 4 is the state rail τ of the network control system with time delay and packet loss _max=0.15ms.

Fig. 5 has the state trajectory τ of the network control system of time delay and packet loss _max=0.45ms.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described in detail.

Based on the network control system switching at runtime control method of average residence time, comprise the following steps:

Step one, uncertain time-varying characteristics based on network inducement delay in closed loop network control system and packet loss, adopt the sensor node combination drive mechanism of initiatively Variable sampling, ensure that network inducement delay ensures that controlled quentity controlled variable new in the current sampling period can be applied in controlled device while being always less than a sampling period.

Due to the complicacy of network control system itself, different procotols, the type of drive that each node is different, different sampling patterns and Internet Transmission situation (length, packet loss, packet reordering etc. of time delay) all can have influence on the model of network control system, thus have influence on the design effect of controller.In sampled-data control system, usually all the sampling period is a constant constant and equal interval sampling, but because network inducement delay and packet loss are random, uncertain, so usually cause the complicacy of network control system model, especially the augmented system dimension for multi-time Delay can be very high, and computation and analysis gets up all quite difficult.Therefore mode sensor being taked to Time And Event combination drive is considered, when the controlled quentity controlled variable through Internet Transmission is sent to actuator from controller, also being successfully applied to controlled device triggers next sample event, namely sensor adopts event driven manner, simultaneously in order to ensure newly to controlled quentity controlled variable can act in controlled device, next sample event is not when new controlled quentity controlled variable is once arriving triggering, but wait for that its effect chronomere's (will explain below) triggers next sample event again, like this can the better control performance that must ensure system.If but sensor only adopts event driven manner, then long delay and continual data package dropout will cause system to cause unstability owing to being in open loop situations for a long time, therefore maximum permission sampling interval will be set, if the sampling period exceedes maximum permission sampling interval, sensor then adopts the time to drive, and initiatively the concrete mechanism of Variable sampling is as follows:

(1) supposing that time shaft is divided into is hour layout of Δ l at equal intervals;

(2) if controlled device is a kth controlled quentity controlled variable to current being applied to, then sensor will trigger kth+1 sample event;

(3) tu is made _krepresent that kth is successfully sent to the time point of the controlled quentity controlled variable of actuator, and use network delay τ _krepresent that kth arrives the time delay of packet from sensor to controller of actuator with the time delay of controller to actuator sum;

(4) in order to avoid not arriving actuator or data-bag lost due to the long data packet time and the network control system caused is in the system unstability problem that may cause under open loop situations, for a long time if the maximum permission sampling period is T _maxif, network transfer delay τ _kexceed T _max, then employing time type of drive is triggered next sample event by sensor;

(5) kth is set to be applied to the sampling time point of the controlled quentity controlled variable of controlled device as s _k, then next sampling time point s _k+1system of selection be:

$s_{k+1}=\left\{\begin{array}{cc}nl& {\mathrm{tu}}_k\&Element;\&lsqb;(n-2)l,(n-1)l)\\ s_k+T_{\max }& {\mathrm{tu}}_k\&GreaterEqual;s_k+T_{max}\end{array}\right.---\left(1\right)$

Wherein, T _maxallowed maximum sampling interval (T _max=N × Δ l, N is positive integer), n is positive integer and 0 < n < N.

With reference to Fig. 2, Fig. 2 illustrates the sampling process of the active Variable sampling of improvement, the active Variable sampling method improved is explained as follows: if a kth controlled quentity controlled variable successfully arrives the time point of actuator at the time interval [(n-2) l, (n-1) l), the time point that so kth+1 sample event triggers not is (n-1) l, but nl.Under this approach, the new subsystem produced will be subject to the effect of new controlled quentity controlled variable, and namely feedback information can be applied in controlled device in time, more easily can obtain the solution of the stability controller making system stability like this.

(6) order set Γ={ t ₁, t ₂, t ₃... } represent and effectively reach time point, its implication is that controlled quentity controlled variable not only successfully arrives actuator and the time point be successfully applied in controlled device.Because when packet generation incorrect order, only have up-to-date packet just can be applied in controlled device, Fig. 3 illustrates Packet Generation and arrival actuator and the situation being applied to controlled device, we can see the 1st, 2,4,6 data from sensor sample have successfully been used control controlled device, and the data of the 3rd sampling are initiatively abandoned due to long time delay, and the 5th data wrap in transmitting procedure and are not used to control owing to there is packet loss.Therefore we can obtain:

t ₁＝tu ₁,t ₂＝tu ₂,t ₃＝tu ₄,t ₄＝tu ₆(2)

(7) efficiently sampling time point i is introduced _m, namely successfully act on the packet of controlled device

Based on state variable sampling time point be called efficiently sampling time point, if I={i ₁, i ₂, i ₃... } and be the set of effective sampling points, under the effect of zero-order holder, feedback controller is at the time interval [t _k, t _k+1) in can be expressed as:

u(t)＝u(i _k)＝K(i _k)x(i _k)t _k≤t＜t _k+1(3)

Step 2, Time And Event mixed node driving mechanism based on the active Variable sampling of step one, set up the switching Controlling model of network control system:

If h _kfor the time span in a kth sampling period, then controlled device can be obtained based on above-mentioned improvement active Variable sampling technology state equation after discretize is:

x(i _k+1)＝Φ _kx(i _k)+Γ ₀(τ _k,h _k)u(i _k)+Γ ₁(τ _k,h _k)u(i _k-1)(4)

Wherein

${\mathrm{\&Phi;}}_k=e^{{\mathrm{Ah}}_k},{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)={\&Integral;}_0^{h_k-{\mathrm{\&tau;}}_k}{e}^{As}dsB,{\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)={\&Integral;}_{h_k-{\mathrm{\&tau;}}_k}^{h_k}{e}^{As}dsB---\left(5\right)$

Introduce new augmentation vector z (k)=[x (i _k) u (i _k-1)] ^t, then from the augmentation closed-loop system of the network control system below formula (4) and formula (5) can obtain:

z(k+1)＝Ψ _kz(k)(6)

Wherein

${\mathrm{\&Psi;}}_k=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_k+{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)K\left(i_k\right)& {\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)\\ K\left(i_k\right)& 0\end{array}\right]---\left(7\right)$

(1) d is defined _ktwo effective sampling points i _kand i _k+1between continual data package dropout number, then can obtain

To i _k+1-i _k=d _k+ 1.Suppose that maximum continual data package dropout number is d _max, then d _kspan be D={0,1 ..., d _max.Based on the rasterizing discrete method of time shaft, time-vary delay system τ _klimited discrete value will be changed into, the micro-Τ of value set=Δ l, 2 × Δ l ..., T _max(Δ l be above-mentioned time shaft is divided into little at equal intervals time layout, and T _max=Δ l × N).

(2) h _kfor a kth efficiently sampling cycle, be easy to obtain h _k=τ _k+ Δ l+T _maxd _k, as can be seen from formula, system matrix Φ _k, Γ ₀(τ _k, h _k), Γ ₁(τ _k, h _k) value by time delay τ _kwith continual data package dropout number d _k, therefore, augmented system (6) can be seen as the switched system that comprises time delay and packet loss information subsystem, wherein system matrix Ψ _kvalue Ω={ Ψ from finite set below ₁(τ _k=Δ l, d _k=0), Ψ ₂(τ _k=2 × Δ l, d _k=0) ..., Ψ _m(τ _k=T _max, d _k=d _max), M=N × (1+d _max).

(3) and then, we can be write as augmented system (6) form of switched system:

Wherein σ (l _k) ∈ Ι=1,2 ..., M}, M=N × (1+d _max) be called switching signal, as σ (l _k)=m, can obtain:

${\hat{A}}_{\mathrm{\&sigma;}\left(l_k\right)}={\hat{A}}_m=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_m+{\mathrm{\&Gamma;}}_{m,0}{K}_m& {\mathrm{\&Gamma;}}_{m,1}\\ K_m& 0\end{array}\right]---\left(9\right)$

(4) under normal circumstances, effectively sample all by the switching of a triggering subsystem each time, but when network condition is just in time the same, such as, shown in Fig. 2 the 1st time efficiently sampling cycle is network-like the same with the 2nd efficiently sampling cycle, and time delay and packet loss are all τ _k=1, d _k=0, therefore, can Ψ be obtained from formula (7) ₁=Ψ ₂, this illustrates that second time efficiently sampling does not have the switching between triggers system.Now need definition new variables l _kshow the real moment switching generation, so the switching instant point of system is (l ₁, l ₂..., l _m...).

Step 3, based on average residence time method analysis package containing the stable adequate condition of the switched system of unstable subsystem and switching signal average residence time needed for the condition that meets:

First the definition of average residence time is provided: for arbitrary l>=l ₀with arbitrary switching signal σ (k), l ₀≤ k < l, makes N _σ[l ₀, l) represent that σ (k) is at the time interval [l ₀, switching times l).If for N ₀>=0 and τ _a> 0, has N _σ[l ₀, l)≤N ₀+ (l-l ₀)/τ _aset up, then τ _abe called average residence time, N ₀represent boundary of trembling.For the purpose of simple also without loss of generality, we make N ₀=0.

Based on analysis above and definition, we can obtain lemma below, show that network control system switches Sufficient Conditions On Stability and the required average residence time met of switching signal of Controlling model (8).For the sake of simplicity, with (0,1,2 ..., l ...) expression time point (0, Δ l, 2 × Δ l ..., l × Δ l ...).

Lemma 1: consider switched system (8), and make | α _m| < 1, m ∈ Ι, if μ>=1 is given constant. there is positive definite function V _{σ (k)}: and Κ function β ₁, β ₂, and p (m), m ∈ Ι is that priori is known, then exist make:

β ₁(||x(l)||)≤V _m(l)≤β ₂(||x(l)||)(10)

$V_m\left(k\right)\&le;{\mathrm{\&mu;V}}_n\left(k\right),\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n---\left(12\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(13\right)$

So switched system (8) is Globally asymptotic, as long as the average residence time of switching signal meets:

${\mathrm{\&tau;}}_a>{\mathrm{\&tau;}}_a^{*}=-\frac{In\mathrm{\&mu;}}{In\mathrm{\&lambda;}}---\left(14\right)$

Wherein p (m), m ∈ Ι is the incidence of switched system subsystem m.

Prove: can obtain from (11)

ΔV _m(l)＝V _m(l+1)-V _m(l)≤α _mV _m(l)(15)

Then

$\begin{array}{c}{V}_m(l+1)\&le;(1+{\mathrm{\&alpha;}}_m)V_m\left(l\right)\\ \&le;{\left(1+{\mathrm{\&alpha;}}_m\right)}^2{V}_m\left(l-1\right)\\ .\\ .\\ .\\ \&le;{\left(1+{\mathrm{\&alpha;}}_m\right)}^{l+1-l_m}{V}_m\left(l_m\right)\end{array}---\left(16\right)$

Can obtain further:

$V_m\left(l\right)\&le;{(1+{\mathrm{\&alpha;}}_m)}^{l-l_m}{V}_m\left(l_m\right)---\left(17\right)$

Again according to (12), the N in the definition of (17) and average residence time _σ(t ₀, t), obtain:

$\begin{array}{c}{V}_{\mathrm{\&sigma;}\left(l\right)}\left(l\right)\&le;{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{V}_{\mathrm{\&sigma;}\left(l_m\right)}\left(l_m\right)\\ \&le;{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{\mathrm{\&mu;V}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\left(l_m\right)\\ \&le;\mathrm{\&mu;}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\right)}^{l_m-l_{m-1}}{V}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\left(l_{m-1}\right)\\ .\\ .\\ .\\ \&le;{\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}\left(0,l\right)}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)}\right)}^{l-l_m}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\right)}^{l_m-l_{m-1}}\mathrm{...}{\left(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(0\right)}\right)}^{l_1}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}\left(0,l\right)}\underset{m=1}{\overset{M}{\&Pi;}}{\left(1+{\mathrm{\&alpha;}}_m\right)}^{p\left(m\right)\&times;l}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ <{\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}\left(0,l\right)}{\mathrm{\&lambda;}}^l{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&mu;}}^{l/{\mathrm{\&tau;}}_a}{\mathrm{\&lambda;}}^l{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&lambda;}}^{\left(In\mathrm{\&mu;}/{\mathrm{\&tau;}}_{a}In\mathrm{\&lambda;}+1\right)\&times;l}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\end{array}---\left(18\right)$

According to Lyapunov stability theory, if average residence time meets (14), then switched system (8) is Globally asymptotic.

The adequate condition that the stability controller that step 4, the mode meeting average residence time providing closed loop network control system switching Controlling model (8) being convenient to solve based on multiple-Lyapunov function method rely on exists:

The Lyapunov function chosen, because we will design the controller of mode dependence, namely each subsystem has the controller of oneself, therefore we adopt multiple-Lyapunov function method, namely each subsystem has the Lyapunov function of oneself, for subsystem m, namely as σ (l _kduring)=m ∈ Ι, its Lyapunov function is:

V _m(l)＝z ^T(l)P _mz(l)(19)

So can obtain from formula:

${\mathrm{\&Delta;V}}_m\left(l\right)-{\mathrm{\&alpha;}}_m{V}_m\left(l\right)=z^T\left(l\right)\&lsqb;{\hat{A}}_m^T{P}_m{\hat{A}}_m-P_m-{\mathrm{\&alpha;}}_m{P}_m\&rsqb;z\left(l\right)---\left(20\right)$

Can obtain from formula:

V _m(k)-μV _n(k)＝z ^T(k)[P _m-μP _n]z(k)(21)

So based on Lyapunov stability theory, if set up according to lemma 1 condition below:

${\hat{A}}_m^T{P}_m{\hat{A}}_m-P_m-{\mathrm{\&alpha;}}_m{P}_m<0---\left(22\right)$

P _m-μP _n≤0(23)

As long as then the average residence time of switching signal meets (14), closed loop network control system switches Controlling model (8) Globally asymptotic, therefore obtains following theorem:

Theorem 1: for given scalar | α _m| < 1, m ∈ Ι, if there is M symmetric positive definite matrix P in μ>=1 ₁, P ₂..., P _m, make $\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n:$

$\left[\begin{array}{cc}-P_m& \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{\hat{A}}_m^T{P}_m\\ \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{P}_m{\hat{A}}_m& -P_m\end{array}\right]<0,\&ForAll;m\&Element;I---\left(24\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(25\right)$

P _m≤μP _n(26)

When so meeting (14) for the average residence time of switching signal, closed loop network control system switches Controlling model (8) Globally asymptotic, and for subsystem m, its energy attenuation or escalating rate are

The mode that the switching signal that step 5, design are convenient to solve meets closed loop network control system switching Controlling model (8) of average residence time relies on stability controller, realizes the point stabilization that is limited and information imperfect situation lower network control system that communicates.

In order to solve controller gain, first define matrix below:

$\begin{array}{c}\tilde{A}=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_m& {\mathrm{\&Gamma;}}_{m,1}\\ 0& 0\end{array}\right],{\tilde{B}}_m=\left[\begin{array}{c}{\mathrm{\&Gamma;}}_{m,0}\\ I\end{array}\right],\\ {\tilde{K}}_m=\left[\begin{array}{cc}{K}_m& 0\end{array}\right],\mathrm{\&sigma;}\left(k\right)=m,m\&Element;I\end{array}---\left(27\right)$

Then closed loop network control system switching Controlling model can be write as form below:

$z(k+1)=({\tilde{A}}_m+{\tilde{B}}_m{\tilde{K}}_m)z\left(k\right)---\left(28\right)$

The LMI that solving state feedback modalities relies on stability controller gain so can be provided by theorem below, specific as follows:

Theorem 2: for given scalar | α _m| < 1, m ∈ Ι, if there is M symmetric positive definite matrix G in μ>=1 respectively _m, V _m, m ∈ Ι, and M matrix R _m, m ∈ Ι, makes $\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n:$

$\left[\begin{array}{cccc}-G_m& 0& \frac{1}{\sqrt{\left(1+{\mathrm{\&alpha;}}_m\right)}}{G}_m^T{\mathrm{\&Phi;}}_m^T+R_m^T{\mathrm{\&Gamma;}}_{m,0}^T& \frac{1}{\sqrt{\left(1+{\mathrm{\&alpha;}}_m\right)}}{R}_m^T\\ *& -V_m& \frac{1}{\sqrt{\left(1+{\mathrm{\&alpha;}}_m\right)}}{V}_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -G_m& 0\\ *& *& *& -V_m\end{array}\right]<0,\&ForAll;m\&Element;I---\left(29\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(30\right)$

G _m-μG _n≤0(31)

V _m-μV _n≤0(32)

When so meeting (14) for the average residence time of switching signal, closed loop network control system switches Controlling model (8) Globally asymptotic, and the mode obtained relies on the gain of feedback controller is:

$K_m=R_m{G_m}^{-1},m\&Element;I---\left(33\right)$

Step 6, subsystem energy attenuation rate the determination of span:

Switched system both may comprise stable subsystem, also unstable subsystem may be comprised, therefore, on the one hand in order to ensure the stability of whole system, the attenuation rate of stabistor system is wanted could cut down the energy increased in unstable subsystem to a certain extent greatly; On the other hand, the attenuation rate of stabistor system can not be too large, because the increase of system attenuation rate means needs, exigent controller is calmed, and this brings difficulty will to solving of controller gain.How does the span of the subsystem energy attenuation rate so mentioned in theorem 1 solve? optimization problem below can be designed, adopt LMI to solve:

Maximumε

Subjectto

$\begin{array}{c}\left[\begin{array}{cccc}-G_m& 0& {\mathrm{\&epsiv;G}}_m^T{\mathrm{\&Phi;}}_m^T+{\mathrm{\&epsiv;R}}_m^T{\mathrm{\&Gamma;}}_{m,0}^T& {\mathrm{\&epsiv;R}}_m^T\\ *& -V_m& {\mathrm{\&epsiv;V}}_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -G_m& 0\\ *& *& *& -V_m\end{array}\right]<0,\&ForAll;m\&Element;I\\ G_m>0,\\ V_m>0,\\ G_m\&le;{\mathrm{\&mu;G}}_n,\\ V_m\&le;{\mathrm{\&mu;V}}_n,\&ForAll;m,n\&Element;I,m\&NotEqual;n\\ \mathrm{\&epsiv;}>1\end{array}---\left(34\right)$

Wherein $\mathrm{\&epsiv;}=\frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}},m\&Element;I.$

In order to solve with MatlabLMI tool box, above formula is changed into canonical form below:

Minimizeω

Subjectto

$\begin{array}{c}\left[\begin{array}{cccc}-G_m& 0& G_m^T{\mathrm{\&Phi;}}_m^T+R_m^T{\mathrm{\&Gamma;}}_{m,0}^T& R_m^T\\ *& -V_m& V_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -M_m& 0\\ *& *& *& -N_m\end{array}\right]<0,\&ForAll;m\&Element;I,\\ G_m\&le;{\mathrm{\&mu;G}}_n,\\ V_m\&le;{\mathrm{\&mu;V}}_n,\&ForAll;m,n\&Element;I,m\&NotEqual;n\\ 0<\mathrm{\&omega;}<1,\\ G_m,V_m,M_m,N_m>0,\\ Y_m<{\mathrm{\&omega;G}}_m,M_m<{\mathrm{\&omega;Y}}_m\\ Z_m<{\mathrm{\&omega;V}}_m,N_m<{\mathrm{\&omega;Z}}_m\end{array}---\left(35\right)$

Step 7, the mode adopting LMI tool box to solve the network control system with random delay and packet loss in matlab rely on the gain of feedback controller and the border of subsystem energy attenuation rate, complete the design of network control system controller, ensure the validity of stability controller within the scope of required subsystem energy attenuation.

Controller performance simulating, verifying:

The mode adopting LMI tool box to solve the network control system with random delay and packet loss for given simulation example in matlab relies on the gain of feedback controller and the border of subsystem energy attenuation rate, result is implanted to access control performance in Networked controller.In matlab programmed process, in order to ensure under the restriction not having subsystem switching characteristic sequence that mode relies on the validity of feedback controller, namely the switching sequence of subsystem is unknowable, and we are to G _m≤ μ G _nand V _m≤ μ V _nhave employed a round-robin algorithm, i.e. forn=1:M, ifn ≠ mG _m≤ μ G _nv _m≤ μ V _n, ensure that the validity of controller under switching sequence indefinite.

Adopt the technical program to implement and the illustrating of operating process under different network condition to embodiment below, and verify superiority of the present invention.

For certain continuity second order controlled device, its state-space expression is as follows respectively:

Controlled device:

$\begin{array}{c}\frac{dx}{dt}=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]x+\left[\begin{array}{c}0\\ 1\end{array}\right]u\\ y=\left[\begin{array}{cc}1& 0\end{array}\right]x\end{array}---\left(36\right)$

A. network condition one: Δ l=0.05ms, τ _max=0.15ms, d _max=2

Mode dependent status feedback switch controller specific design step is as follows:

(1) based on improvement of the present invention initiatively Variable sampling method and time shaft gridding method, by time-vary delay system discretize.According to network condition one, time shaft is divided into 0.05 _mslittle time, i.e. Δ l=0.05ms, maximum allowable delay τ _max=0.15ms, then possible network inducement delay is τ ₁=0.05ms, τ ₂=0.1ms, τ ₃=0.15ms, for the sake of simplicity, supposes maximum continual data package dropout number d _max=2, this shows d _i={ 0,1,2}

(2) determine that the value in a kth sampling period is h _k=τ _k+ T _maxd _k, like this, through type can obtain system matrix Φ in the kth sampling period _k, Γ ₀(τ _k, h _k), Γ ₁(τ _k, h _k) value, through the permutation and combination of the possible value of time delay and number of dropped packets, the switching signal that can obtain system always has 9 kinds.

(3) the mode dependent status feedback switch controller of MatlabLMI tool box solving system is applied based on theorem 2; The span of subsystem attenuation rate is asked for: 1 < ε < 1.2865 based on LMI optimization problem (35) application MatlabLMIGEVP solver.Table 1 indicates the mode dependent status feedback switch controller that subsystem is tried to achieve in attenuation rate span, has 9 kinds of switching laws.

Table 1 switching law and switch controller gain (μ=1.2, τ _max=0.15ms)

(4) following, we calculate and verify whether the average residence time of system quickly meets the demands.We represent with η (m) number of times that subsystem m is switched in whole system Dynamic simulation process, and in this example, we have done 100 efficiently samplings, and total working time is 59.8ms, and so we can according to formula calculate subsystems operation ratio, the result obtained is as follows:

p(1)＝0.0669,p(2)＝0.0753,p(3)＝0.1003

p(4)＝0.1012,p(5)＝0.1204,p(6)＝0.1413(37)

p(7)＝0.0936,p(8)＝0.1003,p(9)＝0.2007

In this example, through type (30) we can calculate therefore we can select λ=0.7 to meet formula (30), so we can obtain on the other hand, in this example, because we have done 100 efficiently samplings, so that is switch for 100 times the time interval [0,59.8] inner generation at most, so minimum average residence time ADT can be calculated be meet formula (14).The average residence time demonstrating subsystem like this meets the demands.

(5) access control device performance in Matlab, we are put in network control system by solving the mode dependent status feedback switch controller obtained, and arrange network environment τ _max=0.15ms, d _max=2, register system state trajectory as shown in Figure 4, can see that the mode dependent status feedback switch controller that the present invention designs has good inhibiting effect to uncertain network inducing delay and packet loss, make controlled device still can have good performance for stability under the interference of the network is guided factor, thus describe validity of the present invention.

B. network condition two: Δ l=0.15ms, τ _max=0.45ms, d _max=2

In this example, we increase maximum permission propagation delay time, and specific design method is as follows:

(1) based on improvement of the present invention initiatively Variable sampling method and time shaft gridding method, by time-vary delay system discretize.According to network condition one, time shaft is divided into 0.15 _mslittle time, i.e. Δ l=0.15ms, maximum allowable delay τ _max=0.45ms, then possible network inducement delay is τ ₁=0.15ms, τ ₂=0.3ms, τ ₃=0.45ms, for the sake of simplicity, supposes maximum continual data package dropout number d _max=2, this shows d _i={ 0,1,2}

(2) determine that the value in a kth sampling period is h _k=τ _k+ T _maxd _k, like this, through type (5) can obtain system matrix Φ in the kth sampling period _k, Γ ₀(τ _k, h _k), Γ ₁(τ _k, h _k) value, through the permutation and combination of the possible value of time delay and number of dropped packets, the switching signal that can obtain system always has 9 kinds.

(3) the mode dependent status feedback switch controller of MatlabLMI tool box solving system is applied based on theorem 2; The span of subsystem attenuation rate is asked for: 1 < ε < 1.2248 based on LMI optimization problem (35) application MatlabLMIGEVP solver.Table 2 indicates the mode dependent status feedback switch controller that subsystem is tried to achieve in attenuation rate span, has 9 kinds of switching laws.

Table 2 switching law and switch controller gain (μ=1.2, τ _max=0.45ms)

(4) following, we calculate and verify whether the average residence time of system quickly meets the demands.We still represent with η (m) number of times that subsystem m is switched in whole system Dynamic simulation process, and in this example, we have also been made 100 efficiently samplings, and total working time is 177.15ms, and so we can according to formula calculate subsystems operation ratio, the result obtained is as follows:

p(1)＝0.0610,p(2)＝0.0610,p(3)＝0.0847

p(4)＝0.1583,p(5)＝0.1321,p(6)＝0.1761(38)

p(7)＝0.1185,p(8)＝0.1270,p(9)＝0.0813

In this example, through type (30) we can calculate therefore we can select λ=0.8 to meet formula (30), so we can obtain on the other hand, in this example, because we have done 100 efficiently samplings, so minimum average residence time ADT can be calculated be meet formula (14).The average residence time demonstrating subsystem like this meets the demands.

(5) access control device performance in Matlab, we are put in network control system by solving the mode dependent status feedback switch controller obtained, and arrange network environment τ _max=0.45ms, d _max=2, register system state trajectory as shown in Figure 4, can be seen that the mode dependent status feedback switch controller that the present invention designs still has good inhibiting effect when uncertain network inducing delay increases, thus describe validity of the present invention.

Along with the increase at point interval such as maximum allowable delay and time shaft from these two examples, the subsystem attenuation rate interval that mode can be found to rely on the feasible solution of feedback control gain reduces.In other words, network condition have impact on the performance for stability of system, when network condition is deteriorated, if required subsystem attenuation rate is too high, will can not find the feasible solution of controller gain.In addition, we can also see in example B be greater than in example A this shows that, after time delay increases, subsystem residence time increases, this realistic situation, because long delay will cause the long efficiently sampling cycle, this must cause long average residence time.

Claims (2)

1., based on the network control system switching at runtime control method of average residence time, comprise the following steps:

Step one, uncertain time-varying characteristics based on network inducement delay in closed loop network control system and packet loss, adopt the sensor node combination drive mechanism of initiatively Variable sampling, ensure that network inducement delay ensures that controlled quentity controlled variable new in the current sampling period can be applied in controlled device while being always less than a sampling period;

Step 2, Time And Event mixed node driving mechanism based on the active Variable sampling of step one, set up the switching Controlling model of network control system:

If h _kfor the time span in a kth sampling period, then controlled device can be obtained based on above-mentioned improvement active Variable sampling technology state equation after discretize is:

$x\left(i_{k+1}\right)={\mathrm{\&Phi;}}_{k}x\left(i_k\right)+{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)u\left(i_k\right)+{\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)u\left(i_{k-1}\right)---\left(4\right)$

Wherein

${\mathrm{\&Phi;}}_k=e^{{\mathrm{Ah}}_k},{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)={\&Integral;}_0^{h_k-{\mathrm{\&tau;}}_k}{e}^{As}dsB,{\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)={\&Integral;}_{h_k-{\mathrm{\&tau;}}_k}^{h_k}{e}^{As}dsB---\left(5\right)$

Introduce new augmentation vector z (k)=[x (i _k) u (i _k-1)] ^t, then from the augmentation closed-loop system of the network control system below formula (4) and formula (5) can obtain:

z(k+1)＝Ψ _kz(k)(6)

Wherein

${\mathrm{\&Psi;}}_k=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_k+{\mathrm{\&Gamma;}}_0({\mathrm{\&tau;}}_k,h_k)K\left(i_k\right)& {\mathrm{\&Gamma;}}_1({\mathrm{\&tau;}}_k,h_k)\\ K\left(i_k\right)& 0\end{array}\right]---\left(7\right)$

(1) d is defined _ktwo effective sampling points i _kand i _k+1between continual data package dropout number, then can obtain i _k+1-i _k=d _k+ 1.Suppose that maximum continual data package dropout number is d _max, then d _kspan be D={0,1 ..., d _max.Based on the rasterizing discrete method of time shaft, time-vary delay system τ _klimited discrete value will be changed into, the micro-Τ of value set=Δ l, 2 × Δ l ..., T _max(Δ l be above-mentioned time shaft is divided into little at equal intervals time layout, and T _max=Δ l × N);

(2) h _kfor a kth efficiently sampling cycle, be easy to obtain h _k=τ _k+ Δ l+T _maxd _k, as can be seen from formula, system matrix Φ _k, Γ ₀(τ _k, h _k), Γ ₁(τ _k, h _k) value by time delay τ _kwith continual data package dropout number d _k, therefore, augmented system (6) can be seen as the switched system that comprises time delay and packet loss information subsystem, wherein system matrix Ψ _kvalue from finite set below

Ω＝{Ψ ₁(τ _k＝Δl,d _k＝0),Ψ ₂(τ _k＝2×Δl,d _k＝0),...,Ψ _M(τ _k＝T _max,d _k＝d _max)}，

M＝N×(1+d _max).

(3) and then, we can be write as augmented system (6) form of switched system:

Wherein σ (l _k) ∈ Ι=1,2 ..., M}, M=N × (1+d _max) be called switching signal, as σ (l _k)=m, can obtain:

${\hat{A}}_{\mathrm{\&sigma;}\left(l_k\right)}={\hat{A}}_m=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_m+{\mathrm{\&Gamma;}}_{m,0}{K}_m& {\mathrm{\&Gamma;}}_{m,1}\\ K_m& 0\end{array}\right]---\left(9\right)$

(4) under normal circumstances, effectively sample all by the switching of a triggering subsystem each time, but when network condition is just in time the same, if time delay and packet loss are all τ _k=τ _a, d _k=d _a, therefore, can Ψ be obtained from formula (7) ₁=Ψ ₂, this illustrates that second time efficiently sampling does not have the switching between triggers system.Now need definition new variables l _kshow the real moment switching generation, so the switching instant point of system is (l ₁, l ₂..., l _m...);

Step 3, based on average residence time method analysis package containing the stable adequate condition of the switched system of unstable subsystem and switching signal average residence time needed for the condition that meets:

First the definition of average residence time is provided: for arbitrary l>=l ₀with arbitrary switching signal σ (k), l ₀≤ k < l, makes N _σ[l ₀, l) represent that σ (k) is at the time interval [l ₀, switching times l).If for N ₀>=0 and τ _a> 0, has N _σ[l ₀, l)≤N ₀+ (l-l ₀)/τ _aset up, then τ _abe called average residence time, N ₀represent boundary of trembling; For the purpose of simple also without loss of generality, we make N ₀=0;

Based on analysis above and definition, we can obtain lemma below, show that network control system switches Sufficient Conditions On Stability and the required average residence time met of switching signal of Controlling model (8), for the sake of simplicity, with (0,1,2 ..., l, ...) represent time point (0 on time shaft, Δ l, 2 × Δ l ..., l × Δ l ...);

Lemma 1: consider switched system (8), and make | α _m| < 1, m ∈ Ι, if μ>=1 is given constant. there is positive definite function and Κ function β ₁, β ₂, and p (m), m ∈ Ι is that priori is known, then exist make:

β ₁(||x(l)||)≤V _m(l)≤β ₂(||x(l)||)(10)

$V_m\left(k\right)\&le;{\mathrm{\&mu;V}}_n\left(k\right),\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n---\left(12\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(13\right)$

So switched system (8) is Globally asymptotic, as long as the average residence time of switching signal meets:

${\mathrm{\&tau;}}_a>{\mathrm{\&tau;}}_a^{*}=-\frac{In\mathrm{\&mu;}}{In\mathrm{\&lambda;}}---\left(14\right)$

Wherein p (m), m ∈ Ι is the incidence of switched system subsystem m.

Prove: can obtain from (11)

ΔV _m(l)＝V _m(l+1)-V _m(l)≤α _mV _m(l)(15)

Then

$\begin{array}{c}{V}_m(l+1)\&le;(1+{\mathrm{\&alpha;}}_m)V_m\left(l\right)\\ \&le;{(1+{\mathrm{\&alpha;}}_m)}^2{V}_m(l-1)\\ .\\ .\\ .\\ \&le;{(1+{\mathrm{\&alpha;}}_m)}^{l+1-l_m}{V}_m\left(l_m\right)\end{array}---\left(16\right)$

Can obtain further:

$V_m\left(l\right)\&le;{(1+{\mathrm{\&alpha;}}_m)}^{l-l_m}{V}_m\left(l_m\right)---\left(17\right)$

Again according to (12), the N in the definition of (17) and average residence time _σ(t ₀, t), obtain:

$\begin{array}{c}{V}_{\mathrm{\&sigma;}\left(l\right)}\&le;{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)})}^{l-l_m}{V}_{\mathrm{\&sigma;}\left(l_m\right)}\left(l_m\right)\\ \&le;{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)})}^{l-l_m}{\mathrm{\&mu;V}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\left(l_m\right)\\ \&le;\mathrm{\&mu;}{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)})}^{l-l_m}{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)})}^{l_m-l_{m-1}}{V}_{\mathrm{\&sigma;}\left(l_{m-1}\right)}\left(l_{m-1}\right)\\ .\\ .\\ .\\ \&le;{\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}(0,l)}{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_m\right)})}^{l-l_m}{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(l_{m-1}\right)})}^{l_m-l_{m-1}}\mathrm{...}{(1+{\mathrm{\&alpha;}}_{\mathrm{\&sigma;}\left(0\right)})}^{l_1}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}(0,l)}\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)\&times;l}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ <{\mathrm{\&mu;}}^{N_{\mathrm{\&sigma;}}(0,l)}{\mathrm{\&lambda;}}^l{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&mu;}}^{l/{\mathrm{\&tau;}}_a}{\mathrm{\&lambda;}}^l{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\\ ={\mathrm{\&lambda;}}^{(In\mathrm{\&mu;}/{\mathrm{\&tau;}}_{a}In\mathrm{\&lambda;}+1)\&times;1}{V}_{\mathrm{\&sigma;}\left(0\right)}\left(0\right)\end{array}---\left(18\right)$

According to Lyapunov stability theory, if average residence time meets (14), then switched system (8) is Globally asymptotic;

The adequate condition that the stability controller that step 4, the mode meeting average residence time providing closed loop network control system switching Controlling model (8) being convenient to solve based on multiple-Lyapunov function method rely on exists:

Adopt multiple-Lyapunov function method, namely each subsystem has the Lyapunov function of oneself, for subsystem m, namely as σ (l _kduring)=m ∈ Ι, its Lyapunov function is:

V _m(l)＝z ^T(l)P _mz(l)(19)

So can obtain from formula (11):

${\mathrm{\&Delta;V}}_m\left(l\right)-{\mathrm{\&alpha;}}_m{V}_m\left(l\right)=z^T\left(l\right)\&lsqb;{\hat{A}}_m^T{P}_m{\hat{A}}_m-P_m-{\mathrm{\&alpha;}}_m{P}_m\&rsqb;z\left(l\right)---\left(20\right)$

Can obtain from formula (12):

V _m(k)-μV _n(k)＝z ^T(k)[P _m-μP _n]z(k)(21)

So based on Lyapunov stability theory, if set up according to lemma 1 condition below:

${\hat{A}}_m^T{P}_m{\hat{A}}_m-P_m-{\mathrm{\&alpha;}}_m{P}_m<0---\left(22\right)$

P _m-μP _n≤0(23)

As long as then the average residence time of switching signal meets (14), closed loop network control system switches Controlling model (8) Globally asymptotic, therefore obtains following theorem:

Theorem 1: for given scalar | α _m| < 1, m ∈ Ι, if there is M symmetric positive definite matrix P in μ>=1 ₁, P ₂..., P _m, make $\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n:$

$\left[\begin{array}{cc}-P_m& \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{\hat{A}}_m^T{P}_m\\ \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{P}_m{\hat{A}}_m& -P_m\end{array}\right]<0,\&ForAll;m\&Element;I---\left(24\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(25\right)$

P _m≤μP _n(26)

When so meeting (14) for the average residence time of switching signal, closed loop network control system switches Controlling model (8) Globally asymptotic, and for subsystem m, its energy attenuation or escalating rate are

The mode that the switching signal that step 5, design are convenient to solve meets closed loop network control system switching Controlling model (8) of average residence time (14) relies on stability controller, realizes the point stabilization that is limited and information imperfect situation lower network control system that communicates;

In order to solve controller gain, first define matrix below:

$\begin{array}{c}{\tilde{A}}_m=\left[\begin{array}{cc}{\mathrm{\&Phi;}}_m& {\mathrm{\&Gamma;}}_{m,1}\\ 0& 0\end{array}\right],{\tilde{B}}_m=\left[\begin{array}{c}{\mathrm{\&Gamma;}}_{m,0}\\ I\end{array}\right],\\ {\tilde{K}}_m=\left[\begin{array}{cc}{K}_m& 0\end{array}\right],\mathrm{\&sigma;}\left(k\right)=m,m\&Element;I\end{array}---\left(27\right)$

Then closed loop network control system switching Controlling model can be write as form below:

$z(k+1)=({\tilde{A}}_m+{\tilde{B}}_m{\tilde{K}}_m)z\left(k\right)---\left(28\right)$

The LMI that solving state feedback modalities relies on stability controller gain so can be provided by theorem below, specific as follows:

Theorem 2: for given scalar | α _m| < 1, m ∈ Ι, if there is M symmetric positive definite matrix G in μ>=1 respectively _m, V _m, m ∈ Ι, and M matrix R _m, m ∈ Ι, makes $\&ForAll;\mathrm{\&sigma;}\left(k\right)=m\&Element;I,\&ForAll;\mathrm{\&sigma;}(k-1)=n\&Element;I,m\&NotEqual;n:$

$\left[\begin{array}{cccc}-G_m& 0& \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{G}_m^T{\mathrm{\&Phi;}}_m^T+R_m^T{\mathrm{\&Gamma;}}_{m,0}^T& \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{R}_m^T\\ *& -V_m& \frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}}{V}_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -G_m& 0\\ *& *& *& -V_m\end{array}\right]<0,\&ForAll;m\&Element;I---\left(29\right)$

$\underset{m=1}{\overset{M}{\&Pi;}}{(1+{\mathrm{\&alpha;}}_m)}^{p\left(m\right)}<\mathrm{\&lambda;}<1---\left(30\right)$

G _m-μG _n≤0(31)

V _m-μV _n≤0(32)

When so meeting (14) for the average residence time of switching signal, closed loop network control system switches Controlling model (8) Globally asymptotic, and the mode obtained relies on the gain of feedback controller is:

K _m＝R _mG _m ^-1,m∈Ι(33)

Step 6, subsystem energy attenuation rate the determination of span:

Switched system (8) both may comprise stable subsystem, also unstable subsystem may be comprised, therefore, on the one hand in order to ensure the stability of whole system, the attenuation rate of stabistor system is wanted could cut down the energy increased in unstable subsystem to a certain extent greatly; On the other hand, the attenuation rate of stabistor system can not be too large, because the increase of system attenuation rate means needs, exigent controller is calmed, and this brings difficulty will to solving of controller gain.The span design optimization problem below of the subsystem energy attenuation rate so mentioned in theorem 1, adopts LMI to solve:

Maximumε

Subjectto

$\left[\begin{array}{cccc}-G_m& 0& {\mathrm{\&epsiv;G}}_m^T{\mathrm{\&Phi;}}_m^T+{\mathrm{\&epsiv;R}}_m^T{\mathrm{\&Gamma;}}_{m,0}^T& {\mathrm{\&epsiv;R}}_m^T\\ *& -V_m& {\mathrm{\&epsiv;V}}_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -G_m& 0\\ *& *& *& -V_m\end{array}\right]<0,\&ForAll;m\&Element;I$

(34)

G _m＞0,

V _m＞0,

G _m≤μG _n,

$V_m\&le;{\mathrm{\&mu;V}}_n,\&ForAll;m,n\&Element;I,m\&NotEqual;n$

ε＞1

Wherein $\mathrm{\&epsiv;}=\frac{1}{\sqrt{(1+{\mathrm{\&alpha;}}_m)}},m\&Element;I.$

In order to solve with MatlabLMI tool box, above formula is changed into canonical form below:

Minimizeω

Subjectto

$\left[\begin{array}{cccc}-G_m& 0& G_m^T{\mathrm{\&Phi;}}_m^T+R_m^T{\mathrm{\&Gamma;}}_{m,0}^T& R_m^T\\ *& -V_m& V_m^T{\mathrm{\&Gamma;}}_{m,1}^T& 0\\ *& *& -M_m& 0\\ *& *& *& -N_m\end{array}\right]<0,\&ForAll;m\&Element;I,$

G _m≤μG _n,

$V_m\&le;{\mathrm{\&mu;V}}_n,\&ForAll;m,n\&Element;I,m\&NotEqual;n$

0＜ω＜1,(35)

G _m,V _m,M _m,N _m＞0,

Y _m＜ωG _m,M _m＜ωY _m

Z _m＜ωV _m,N _m＜ωZ _m

Step 7, the mode adopting LMI tool box to solve the network control system with random delay and packet loss in matlab rely on the gain of feedback controller and the border of subsystem energy attenuation rate, complete the design of network control system controller, ensure the validity of stability controller within the scope of required subsystem energy attenuation.

2. the network control system switching at runtime control method based on average residence time according to claim 1, is characterized in that, described in step one, initiatively the concrete mechanism of Variable sampling is as follows:

(1) supposing that time shaft is divided into is hour layout of Δ l at equal intervals;

(2) if controlled device is a kth controlled quentity controlled variable to current being applied to, then sensor will trigger kth+1 sample event;

(3) tu is made _krepresent that kth is successfully sent to the time point of the controlled quentity controlled variable of actuator, and use network delay τ _krepresent that kth arrives the time delay of packet from sensor to controller of actuator with the time delay of controller to actuator sum;

(4) in order to avoid not arriving actuator or data-bag lost due to the long data packet time and the network control system caused is in the system unstability problem that may cause under open loop situations, for a long time if the maximum permission sampling period is T _maxif, network transfer delay τ _kexceed T _max, then employing time type of drive is triggered next sample event by sensor;

(5) kth is set to be applied to the sampling time point of the controlled quentity controlled variable of controlled device as s _k, then next sampling time point s _k+1system of selection be:

$s_{k+1}=\left\{\begin{array}{cc}nl& {\mathrm{tu}}_k\&Element;\&lsqb;(n-2)l,(n-1)l)\\ s_k+T_{\max }& {\mathrm{tu}}_k\&GreaterEqual;s_k+T_{max}\end{array}\right.---\left(1\right)$

Wherein, T _maxallowed maximum sampling interval (T _max=N × Δ l, N is positive integer), n is positive integer and 0 < n < N;

(6) order set Γ={ t ₁, t ₂, t ₃... } represent and effectively reach time point, its implication is that controlled quentity controlled variable not only successfully arrives actuator and the time point be successfully applied in controlled device.Because when packet generation incorrect order, only have up-to-date packet just can be applied in controlled device.

(7) efficiently sampling time point i is introduced _m, namely successfully act on the packet of controlled device based on state variable sampling time point be called efficiently sampling time point, if I={i ₁, i ₂, i ₃... } and be the set of effective sampling points, under the effect of zero-order holder, feedback controller is at the time interval [t _k, t _k+1) in can be expressed as:

u(t)＝u(i _k)＝K(i _k)x(i _k)t _k≤t＜t _k+1(3)。