CN102651641A - L2-L infinity filtering information processing method based on logarithmic quantization for network control system - Google Patents

Abstract

The invention relates to an L2-L infinity filtering information processing method based on logarithmic quantization for a network control system. The method comprises the following steps: (1) the sending end of the network control system samples, quantifies and encodes an input signal and then sends the input signal to the receiving end; (2) the receiving end decodes the received data and transmits the decoded data to a filter; and (3) the filter filters the received data and restores the original system information. Compared with the prior art, robust L2-L infinity filter processing with performance parameter Gamma of the network control system under disturbing influence is realized, influence factors brought by sampling, quantifying and packet loss in practice are given into full consideration in the design of the filtering information processing method, so that the L2-L infinity filtering information processing method provided by the invention more conforms to the practical application.

Description

Based on network control system L to quantification 2-L ∞The filtering information processing method

Technical field

The present invention relates to a kind of network control system, especially relate to based on network control system L to quantification ₂-L _∞The filtering information processing method.

Background technology

Traditional control system adopts point-to-point transmission mode, through some special circuits the signal that collects is sent to controller from transducer, again the control signal slave controller is sent to actuator, realizes the control to controlled device.Along with the fast development of industry, require production equipment can not limit to a factory building or very little zone in, traditional control system can not adapt to the needs of production, in this case, network control system rises gradually.It is how technological that network control system has merged control, communication, network etc.; Opening, interoperability, interchangeability integration, system reliability are high, maintainability reaches advantages such as low cost well but have, and have obtained in fields such as industrial automation control, intelligent transportation, Aero-Space and national defence using widely.

Although network control system has lot of advantages, yet, since signal through Network Transmission, brought some new between topic.Because inevitably there are some interference in some characteristics of existence, environment on every side and the system itself of communication network, like electromagnetic interference, the packet loss in the network, time delay, out of order, error code etc.Therefore, in network control system, how to become the problem that must solve in receiving terminal reduction primary signal.In the practical application, filter commonly used has: Kalman filter, robust H ₂Filter, robust H _∞Filter and robust L ₂-L _∞Filter etc.Kalman filter is a kind of full-fledged classical filter, and it can realize the filtering and the prediction of signal; The defective of its existence is can only be directed against under the condition that noise is a white Gaussian noise to carry out filtering, and therefore, its scope of application is restricted.For the filter of other kind, to the robust filter of time-lag system and network control system some researchs have been arranged, as: Mehrdad Sahebsara, people such as Yue Dong have designed some corresponding filters.

Suppose all in these present Filter Design that signal transmission (comprising that the signal that collects is sent to controller and the control signal slave controller is sent to actuator from transducer) is harmless transmission, and ignored the quantization factor in the network control system.As shown in Figure 1, on the one hand, because the signal continuous analog signal often in the control system; And will transmit and must these signals be transformed binary discrete digital signal through network; Therefore, network control system must be sampled to primary signal before the signal transmission, quantizes then, encodes; Then,, signal receiving end decodes after receiving signal.On the other hand, because the network bandwidth is limited, in order to reduce the congested of network; Guaranteeing under the enough prerequisites of signal message amount; Must quantize signal, and need reduce sampling rate and code length, thereby reduce required data packets for transmission quantity in the unit interval.Quantizing is exactly the block signal that actual signal is converted to a value in finite aggregate through quantizer.Because the existence that quantizes inevitably can bring quantization error, the existence meeting of quantization error causes some influences to the performance of system, and when serious even possibly produce limit cycle and chaos phenomenon, therefore, quantization error is to consider in Control System Design.From existing literature, in the existing filtering information treatment design process,, also there is not the designer that quantization factor is taken into account in order to simplify computing.

Summary of the invention

The object of the invention is exactly to provide based on the network control system L to quantification for the defective that overcomes above-mentioned prior art existence ₂-L _∞The filtering information processing method.

The object of the invention can be realized through following technical scheme:

Based on network control system L to quantification ₂-L _∞The filtering information processing method is characterized in that: may further comprise the steps:

1) the network control system transmitting terminal is sampled, is sent to receiving terminal after quantification and the encoding process the signal of input;

2) receiving terminal is decoded to the data that receive, and gives filter with decoded transfer of data;

3) filter carries out Filtering Processing to the data that receive, reduction original system information.

State equation after the network control system modeling in the described step 1) is following:

$\left\{\begin{array}{c}\stackrel{\·}{x}\left(t\right)=A_{1}x\left(t\right)+A_{2}x(t-\mathrm{\τ})+\mathrm{B\ω}\left(t\right)\\ y\left(t\right)=Q(\mathrm{Cx}\left(t\right)+\mathrm{D\ω}\left(t\right))\\ z\left(t\right)=\mathrm{Lx}\left(t\right)\end{array}\right.,t\∈[t_i,t_{i+1})---\left(1\right)$

Wherein, x (t) ∈ R ⁿBe the state vector of controlled device system,

I ∈ 0,1,2,3 ..., τ (t) is the time delay of network control system,

Y (t) ∈ R ^mBe the observation vector of system, A, B, C are for fitting dimension matrix, ω (t) ∈ R ^pBe to belong to L ₂[0, noise input ∞), z (t) is estimative signal;

This model has been considered the influence of factors such as the time delay that is produced in the Network Transmission, packet loss on the one hand, and there is a kind of L of satisfying in this system ₂[0, noise jamming ∞); What obtain at signal receiving end on the other hand is not complete system output information, but has comprised a kind of sampling, quantitative information of logarithm quantization error; In above-mentioned model, the packet loss in the network control system is used as time delay processing, time delay and packet loss are unified with τ (t) expression, and the disturbing factor in the network control system is modeled as a kind of L of satisfying ₂[0, signal ∞); At output, owing to need sample, quantize, encode before the Network Transmission, the output y of system (t) will pass through logarithm quantizer Q (), thereby can have quantization error; After receiving terminal receives this signal, in the filtering through conversion, to eliminate its influence.

Quantification treatment in the described step 1) is following:

Systematic observation vector Q (y after quantizer carries out obtaining quantizing after the quantification treatment to the data of sampling _iAnd quantization error e (t)) _q(Q (y _i(t))),

<math> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>&lt;;</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&lt;;</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein,

0＜ρ _i＜1, y _i(t) be the i dimension value of y (t), v _0i＞0, j=0, ± 1, ± 2....;

e _q(Q(y _i(t)))＝Q(y _i(t))-y _i(t)＝Δ(y _i(t))y _i(t) (3)

Δ (y wherein _i(t)) ∈ [δ _i, δ _i).

Described L ₂-L _∞The building process of filtering information processing method is following:

1) according to the modelling filter equation of network control system, confirm the parameters optimization that filter is required, wherein filter equation is following:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_f\left(t\right)=A_f{x}_f\left(t\right)+B_{f}y\left(t\right)\\ z_f\left(t\right)=C_f{x}_f\left(t\right)\end{array}\right.---\left(5\right)$

Definition

E (t)=z (t)-z _f(t), integrated (1) and formula (5) can get the state equation (6) of association system,

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{\tilde{x}}\left(t\right)={\tilde{A}}_1\tilde{x}\left(t\right)+{\tilde{A}}_2\tilde{x}(t-\mathrm{\&tau;})\\ +B_{1}Q(\stackrel{\&OverBar;}{C}\tilde{x}\left(t\right)+\mathrm{D\&omega;}\left(t\right))+B_2\mathrm{\&omega;}\left(t\right)\\ e\left(t\right)=z\left(t\right)-z_f\left(t\right)=\tilde{C}\tilde{x}\left(t\right)\end{array}\right.---\left(6\right)$

Wherein, ${\tilde{A}}_1=\left[\begin{array}{cc}{A}_1& 0\\ 0& A_f\end{array}\right],$ ${\tilde{A}}_2=\left[\begin{array}{cc}{A}_2& 0\\ 0& 0\end{array}\right],$ $B_1=\left[\begin{array}{c}0\\ B_f\end{array}\right],$ $B_2=\left[\begin{array}{c}B\\ 0\end{array}\right],$ $\tilde{C}=\left[\begin{array}{cc}L& -C_f\end{array}\right],$ $\stackrel{\&OverBar;}{C}=\left[\begin{array}{cc}C& 0\end{array}\right].$

And satisfy following two conditions:

(1) system stability of formula (6) expression;

(2) under zero initial conditions, for given performance parameter γ＞0, e (t) satisfies For non-zero ω (t) ∈ L arbitrarily ₂[t ₀, ∞), wherein

2) the definition Liapunov function is V=V ₁+ V ₂+ V ₃+ V ₄, wherein,

Can arrive as drawing a conclusion through computing:

The represented filter equation of formula (5) is the L with performance parameter γ ₂-L _∞Filter equation, if there is γ＞0, and matrix R＞0, X＞0,

U, V satisfy following LMI (7) (8),

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>&amp;Psi;</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>RA</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>RA</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>XA</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>XA</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mi>RB</mi> </mtd> </mtr> <mtr> <mtd> <mi>XB</mi> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>UB</mi> <mi>f</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>D</mi> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> </mtable> </mfenced> <msup> <mi>&amp;Delta;</mi> <mi>T</mi> </msup> <mi>&amp;Delta;D</mi> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mi>B</mi> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <msup> <mi>D</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Delta;</mi> <mi>T</mi> </msup> <mi>&amp;Delta;D</mi> <mo>-</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&lt;;</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow></math>

${\mathrm{\&Xi;}}_2=\left[\begin{array}{cc}{\mathrm{\&Xi;}}_{11}& {\mathrm{\&Xi;}}_{12}\\ *& {\mathrm{\&gamma;}}^{2}I\end{array}\right]>0---\left(8\right)$

Wherein,

${\mathrm{\&Psi;}}_{11}=\left[\begin{array}{cc}{A}_1^{T}R+{\mathrm{RA}}_1+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}& {A_1}^{T}X+{\tilde{A}}_f^T+{\mathrm{RA}}_1+C^T{\tilde{B}}_f^T+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}\\ *& {A_1}^{T}X+{\mathrm{XA}}_1+{\tilde{B}}_{f}C+C^T{\tilde{B}}_f^T+{\tilde{B}}_f{\tilde{B}}_f^T+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}\end{array}\right]$ ， ${\mathrm{\&Xi;}}_{11}=\left[\begin{array}{cc}R& R\\ R& X\end{array}\right],$ ${\mathrm{\&Xi;}}_{12}=\left[\begin{array}{c}{L}^T-{\tilde{C}}_f\\ L^T\end{array}\right];$

Find the solution LMI (7) and (8) and can get L ₂-L _∞The parameter of filter:

Wherein, UV ^T=I-XR ^-1

For simple process, do not considered the influence of bandwidth and the error that quantification brought in the transmission in the existing filtering information processing method.Yet; For signal can transmit effectively; Reduce data amount transmitted, reduce factors such as network congestion, packet loss, quantification is a requisite link in the network control system, does not consider that these factors can possibly cause very big influence to systematic function; Possibly cause system to disperse for feedback control system, even produce limit cycle and chaos phenomenon.The present invention has realized the robust L with performance parameter γ of network control system under interference effect ₂-L _∞The design of filtering information processing method.Under the situation of considering Network Packet Loss, time delay; Set up a kind of continuity model, and adopted the logarithm quantizer, quantization error has been transformed to the uncertainty relevant with system mode handles; Draw one and satisfied the filtering information processing method of disturbing rejection condition; Through Lyapunov method and LMI method, can solve each parameter of filtering information processing method, and prove the validity of algorithm through an example.The sampling because the present invention has taken into full account in the filtering information treatment design in the reality, quantification, packet loss with influence factor, thereby more realistic operating position.

Description of drawings

Fig. 1 is existing network control system;

Fig. 2 is a flow chart of the present invention;

Fig. 3 is the result of logarithm quantizer of the present invention.

Embodiment

Below in conjunction with accompanying drawing and specific embodiment the present invention is elaborated.

Embodiment

As shown in Figure 2, based on network control system L to quantification ₂-L _∞The filtering information processing method may further comprise the steps:

1) the network control system transmitting terminal is sampled, is sent to receiving terminal after quantification and the encoding process the signal of input;

2) receiving terminal is decoded to the data that receive, and gives filter with decoded transfer of data;

3) filter carries out Filtering Processing to the data that receive, reduction original system information.

State equation after the network control system modeling in the described step 1) is following:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=A_{1}x\left(t\right)+A_{2}x(t-\mathrm{\&tau;})+\mathrm{B\&omega;}\left(t\right)\\ y\left(t\right)=Q(\mathrm{Cx}\left(t\right)+\mathrm{D\&omega;}\left(t\right))\\ z\left(t\right)=\mathrm{Lx}\left(t\right)\end{array}\right.,t\&Element;[t_i,t_{i+1})---\left(1\right)$

Wherein, x (t) ∈ R ⁿBe the state vector of controlled device system, i ∈ 0,1,2,3 ...,

τ (t) is the time delay of network control system,

Y (t) ∈ R ^mBe the observation vector of system, A, B, C are for fitting dimension matrix, ω (t) ∈ R ^pBe to belong to L ₂[0, noise input ∞), z (t) is estimative signal.

As shown in Figure 3, the quantification treatment in the described step 1) is following:

Systematic observation vector Q (y after the logarithm quantizer carries out obtaining quantizing after the quantification treatment to the data of sampling _iAnd quantization error e (t)) _q(Q (y _i(t))),

<math> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>&lt;;</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&lt;;</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein,

0＜ρ _i＜1, y _i(t) be the i dimension value of y (t), v _0i＞0, j=0, ± 1, ± 2....;

e _q(Q(y _i(t)))＝Q(y _i(t))-y _i(t)＝Δ(y _i(t))y _i(t) (3)

Δ (y wherein _i(t)) ∈ [δ _i, δ _i).

The building process of described filtering information processing method is following:

1) according to the modelling filter equation of network control system, confirm the parameters optimization that filter is required, wherein filter equation is following:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_f\left(t\right)=A_f{x}_f\left(t\right)+B_{f}y\left(t\right)\\ z_f\left(t\right)=C_f{x}_f\left(t\right)\end{array}\right.---\left(5\right)$

Definition

E (t)=z (t)-z _f(t), integrated (1) and formula (5) can get the state equation (6) of association system,

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{\tilde{x}}\left(t\right)={\tilde{A}}_1\tilde{x}\left(t\right)+{\tilde{A}}_2\tilde{x}(t-\mathrm{\&tau;})\\ +B_{1}Q(\stackrel{\&OverBar;}{C}\tilde{x}\left(t\right)+\mathrm{D\&omega;}\left(t\right))+B_2\mathrm{\&omega;}\left(t\right)\\ e\left(t\right)=z\left(t\right)-z_f\left(t\right)=\tilde{C}\tilde{x}\left(t\right)\end{array}\right.---\left(6\right)$

Wherein, ${\tilde{A}}_1=\left[\begin{array}{cc}{A}_1& 0\\ 0& A_f\end{array}\right],$ ${\tilde{A}}_2=\left[\begin{array}{cc}{A}_2& 0\\ 0& 0\end{array}\right],$ $B_1=\left[\begin{array}{c}0\\ B_f\end{array}\right],$ $B_2=\left[\begin{array}{c}B\\ 0\end{array}\right],$ $\tilde{C}=\left[\begin{array}{cc}L& -C_f\end{array}\right],$ $\stackrel{\&OverBar;}{C}=\left[\begin{array}{cc}C& 0\end{array}\right].$

And satisfy following two conditions:

(1) system stability of formula (6) expression;

(2) under zero initial conditions, for given performance parameter γ＞0, e (t) satisfies

For non-zero ω (t) ∈ L arbitrarily ₂[t ₀, ∞), wherein

2) the definition Liapunov function is V=V ₁+ V ₂+ V ₃+ V ₄, wherein,

Can arrive as drawing a conclusion through computing:

The represented filter equation of formula (5) is the L with performance parameter γ ₂-L _∞Filter equation, if there is γ＞0, and matrix R＞0, X＞0,

U, V satisfy following LMI (7) (8),

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>&amp;Psi;</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>RA</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>RA</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>XA</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>XA</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mi>RB</mi> </mtd> </mtr> <mtr> <mtd> <mi>XB</mi> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>UB</mi> <mi>f</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>D</mi> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> </mtable> </mfenced> <msup> <mi>&amp;Delta;</mi> <mi>T</mi> </msup> <mi>&amp;Delta;D</mi> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mi>B</mi> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <msup> <mi>D</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Delta;</mi> <mi>T</mi> </msup> <mi>&amp;Delta;D</mi> <mo>-</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&lt;;</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow></math>

${\mathrm{\&Xi;}}_2=\left[\begin{array}{cc}{\mathrm{\&Xi;}}_{11}& {\mathrm{\&Xi;}}_{12}\\ *& {\mathrm{\&gamma;}}^{2}I\end{array}\right]>0---\left(8\right)$

Wherein,

${\mathrm{\&Psi;}}_{11}=\left[\begin{array}{cc}{A}_1^{T}R+{\mathrm{RA}}_1+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}& {A_1}^{T}X+{\tilde{A}}_f^T+{\mathrm{RA}}_1+C^T{\tilde{B}}_f^T+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}\\ *& {A_1}^{T}X+{\mathrm{XA}}_1+{\tilde{B}}_{f}C+C^T{\tilde{B}}_f^T+{\tilde{B}}_f{\tilde{B}}_f^T+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}\end{array}\right]$ ， ${\mathrm{\&Xi;}}_{11}=\left[\begin{array}{cc}R& R\\ R& X\end{array}\right],$ ${\mathrm{\&Xi;}}_{12}=\left[\begin{array}{c}{L}^T-{\tilde{C}}_f\\ L^T\end{array}\right];$

Find the solution the parameter that LMI (7) and (8) can get filter:

Wherein, UV ^T=I-XR ^-1

With following case verification Design of Filter:

Consider the equation of state parameter network control system

d = 0.1.

Definition

Satisfy UV ^T=I-XR ^-1. utilizing matlab to find the solution LMI (7) (8) availability can parameter γ _Opt=0.7163, L ₂-L _∞Parameter may be selected to be

Claims (4)

1. based on network control system L to quantification ₂-L _∞The filtering information processing method is characterized in that: may further comprise the steps:

1) the network control system transmitting terminal is sampled, is sent to receiving terminal after quantification and the encoding process the signal of input;

2) receiving terminal is decoded to the data that receive, and gives filter with decoded transfer of data;

3) filter carries out Filtering Processing to the data that receive, reduction original system information.

2. according to claim 1 based on network control system L to quantification ₂-L _∞The filtering information processing method is characterized in that, the state equation after the network control system modeling in the described step 1) is following:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=A_{1}x\left(t\right)+A_{2}x(t-\mathrm{\&tau;})+\mathrm{B\&omega;}\left(t\right)\\ y\left(t\right)=Q(\mathrm{Cx}\left(t\right)+\mathrm{D\&omega;}\left(t\right))\\ z\left(t\right)=\mathrm{Lx}\left(t\right)\end{array}\right.,t\&Element;[t_i,t_{i+1})---\left(1\right)$

Wherein, x (t) ∈ R ⁿBe the state vector of controlled device system,

I ∈ 0,1,2,3 ..., τ (t) is the time delay of network control system, Y (t) ∈ R ^mBe the observation vector of system, A, B, C are for fitting dimension matrix, ω (t) ∈ R ^pBe to belong to L ₂[0, noise input ∞), z (t) is estimative signal;

This model has been considered the influence of factors such as the time delay that is produced in the Network Transmission, packet loss on the one hand, and there is a kind of L of satisfying in this system ₂[0, noise jamming ∞); What obtain at signal receiving end on the other hand is not complete system output information, but has comprised a kind of sampling, quantitative information of logarithm quantization error; In above-mentioned model, the packet loss in the network control system is used as time delay processing, time delay and packet loss are unified with τ (t) expression, and the disturbing factor in the network control system is modeled as a kind of L of satisfying ₂[0, signal ∞); At output, owing to need sample, quantize, encode before the Network Transmission, the output y of system (t) will pass through logarithm quantizer Q (), thereby can have quantization error; After receiving terminal receives this signal, in the filtering through conversion, to eliminate its influence.

3. according to claim 1 based on network control system L to quantification ₂-L _∞The filtering information processing method is characterized in that, the quantification treatment in the described step 1) is following:

Systematic observation vector Q (y after quantizer carries out obtaining quantizing after the quantification treatment to the data of sampling _iAnd quantization error e (t)) _q(Q (y _i(t))),

<math> <mrow> <mi>Q</mi> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> <mo>&lt;;</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&amp;le;</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <msubsup> <mi>&amp;rho;</mi> <mi>i</mi> <mi>j</mi> </msubsup> <msub> <mi>&amp;upsi;</mi> <mrow> <mn>0</mn> <mi>i</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mi>Q</mi> <mrow> <mo>(</mo> <mo>-</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mi>if</mi> <msub> <mi>y</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&lt;;</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow></math>

Wherein,

0＜ρ _i＜1, y _i(t) be the i dimension value of y (t), v _0i＞0, j=0, ± 1, ± 2....;

e _q(Q(y _i(t)))＝Q(y _i(t))-y _i(t)＝Δ(y _i(t))y _i(t) (3)

Δ (y wherein _i(t)) ∈ [δ _i, δ _i).

4. according to claim 1 based on network control system L to quantification ₂-L _∞The filtering information processing method is characterized in that, described L ₂-L _∞The building process of filtering information processing method is following:

1) according to the modelling filter equation of network control system, confirm the parameters optimization that filter is required, wherein filter equation is following:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_f\left(t\right)=A_f{x}_f\left(t\right)+B_{f}y\left(t\right)\\ z_f\left(t\right)=C_f{x}_f\left(t\right)\end{array}\right.---\left(5\right)$

Definition

E (t)=z (t)-z _f(t), integrated (1) and formula (5) can get the state equation (6) of association system,

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{\tilde{x}}\left(t\right)={\tilde{A}}_1\tilde{x}\left(t\right)+{\tilde{A}}_2\tilde{x}(t-\mathrm{\&tau;})\\ +B_{1}Q(\stackrel{\&OverBar;}{C}\tilde{x}\left(t\right)+\mathrm{D\&omega;}\left(t\right))+B_2\mathrm{\&omega;}\left(t\right)\\ e\left(t\right)=z\left(t\right)-z_f\left(t\right)=\tilde{C}\tilde{x}\left(t\right)\end{array}\right.---\left(6\right)$

Wherein, ${\tilde{A}}_1=\left[\begin{array}{cc}{A}_1& 0\\ 0& A_f\end{array}\right],$ ${\tilde{A}}_2=\left[\begin{array}{cc}{A}_2& 0\\ 0& 0\end{array}\right],$ $B_1=\left[\begin{array}{c}0\\ B_f\end{array}\right],$ $B_2=\left[\begin{array}{c}B\\ 0\end{array}\right],$ $\tilde{C}=\left[\begin{array}{cc}L& -C_f\end{array}\right],$ $\stackrel{\&OverBar;}{C}=\left[\begin{array}{cc}C& 0\end{array}\right].$

And satisfy following two conditions:

(1) system stability of formula (6) expression;

(2) under zero initial conditions, for given performance parameter γ＞0, e (t) satisfies

For non-zero ω (t) ∈ L arbitrarily ₂[t ₀, ∞), wherein

2) the definition Liapunov function is V=V ₁+ V ₂+ V ₃+ V ₄, wherein,

Can arrive as drawing a conclusion through computing:

The represented filter equation of formula (5) is the L with performance parameter γ ₂-L _∞Filter equation, if there is γ＞0, and matrix R＞0, X＞0,

U, V satisfy following LMI (7) (8),

<math> <mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>&amp;Psi;</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>RA</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>RA</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>XA</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>XA</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mi>RB</mi> </mtd> </mtr> <mtr> <mtd> <mi>XB</mi> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>UB</mi> <mi>f</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mi>D</mi> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> </mtable> </mfenced> <msup> <mi>&amp;Delta;</mi> <mi>T</mi> </msup> <mi>&amp;Delta;D</mi> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>3</mn> </msub> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> <mtd> <msubsup> <mi>A</mi> <mn>1</mn> <mi>T</mi> </msubsup> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mtd> <mtd> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>4</mn> </msub> </mtd> <mtd> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mi>B</mi> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <mover> <mi>&amp;tau;</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <msup> <mi>D</mi> <mi>T</mi> </msup> <msup> <mi>&amp;Delta;</mi> <mi>T</mi> </msup> <mi>&amp;Delta;D</mi> <mo>-</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>&lt;;</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow></math>

${\mathrm{\&Xi;}}_2=\left[\begin{array}{cc}{\mathrm{\&Xi;}}_{11}& {\mathrm{\&Xi;}}_{12}\\ *& {\mathrm{\&gamma;}}^{2}I\end{array}\right]>0---\left(8\right)$

Wherein,

${\mathrm{\&Psi;}}_{11}=\left[\begin{array}{cc}{A}_1^{T}R+{\mathrm{RA}}_1+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}& {A_1}^{T}X+{\tilde{A}}_f^T+{\mathrm{RA}}_1+C^T{\tilde{B}}_f^T+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}\\ *& {A_1}^{T}X+{\mathrm{XA}}_1+{\tilde{B}}_{f}C+C^T{\tilde{B}}_f^T+{\tilde{B}}_f{\tilde{B}}_f^T+2{\tilde{P}}_2+(1+\stackrel{\&OverBar;}{\mathrm{\&tau;}})C^T{\mathrm{\&Delta;}}^T\mathrm{\&Delta;C}\end{array}\right]$ ， ${\mathrm{\&Xi;}}_{11}=\left[\begin{array}{cc}R& R\\ R& X\end{array}\right],$ ${\mathrm{\&Xi;}}_{12}=\left[\begin{array}{c}{L}^T-{\tilde{C}}_f\\ L^T\end{array}\right];$

Find the solution LMI (7) and (8) and can get L ₂-L _∞The parameter of filter:

Wherein, UV ^T=I-XR ^-1

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