CN106054667A - Coking furnace pressure system stable switching controller design method - Google Patents

Abstract

The present invention discloses a coking furnace pressure system stable switching controller design method. The coking furnace pressure system stable switching controller design method is established through the means such as data collection, model establishment, optimization and the like to effectively solve the instability problem caused by the system switching and ensure that the system has good control effect in the condition of the stable system.

Description

A kind of coking furnace furnace pressure system stability switch controller method for designing

Technical field

The invention belongs to technical field of automation, relate to a kind of coking furnace furnace pressure system stability switch controller design Method.

Background technology

During actual industrial control, due to real process object exist much not well known complicated physics or Chemical characteristic, produces interference to system control process.A lot of real systems are all the switchings of use system, such as, ecological, Industrial engineering, Chemical Engineering, economics etc..It is known that system stability is the basis of industrial process, control system is set Count extremely important.Switched system is a class hybrid system, carries out layout by subsystem and rule and realizes switching between them, can To ensure that it is stable.For the dynamic characteristic of coking heater furnace pressure, system mode feedback has good control Effect, solves traditional control method and switches the unstability difficult problem caused.

Summary of the invention

It is an object of the invention to that this problem unstable occurs for the models switching process of coking furnace furnace pressure object, By means such as data acquisition, model foundation, optimizations, it is provided that a kind of coking furnace furnace pressure system stability switch controller sets Meter method.The method inputoutput data by gatherer process object, uses switching signal to optimize handoff procedure and shakiness occurs Fixed.The method has higher accuracy, can well improve the dynamic characteristic of process object.

The technical scheme is that by means such as data acquisition, model foundation, optimizations, establish a kind of coking furnace stove Thorax pressure stability switch controller method for designing, utilizes the method can effectively solve the unstability difficulty that system switching causes Topic, it is ensured that there is on the premise of system stability good control effect.

The step of the inventive method includes:

Step 1, setting up the state-space model of coking furnace furnace pressure, concrete grammar is:

1.1 inputoutput datas first gathering coking furnace furnace pressure, utilize these data to set up coking furnace furnace pressure State-space model, form is as follows:

Y (t)=C_σ(t)x(t)

Wherein, x (t) is coking furnace system mode, and y (t) is coking furnace output pressure, u_σ(t)T () is for controlling baffle opening, σ (t) For switching signal represent from [0, ∞) to finite aggregate S={1,2 ..., the mapping of N}, A_σ(t)For Metzler matrix, each σ (t) ∈ S is had, B_σ(t)≥0,C_σ(t)≥0。

Step 2, the state feedback controller of design coking furnace furnace pressure, comprise the concrete steps that:

2.1 design switching signal σ (t) and 0≤t₁≤t₂, meet following condition:

N_σ(t₂,t₁)≤N₀+(t₂-t₁)/τ^*

Wherein, N_σ(t₂,t₁) it is that switched system is at (t₁,t₂Switching times in), τ^*> 0 it is the average residence of switching signal Time (ADT), N₀≥0。

2.2 design A_p+B_pK_pC_p, p ∈ SMetzler matrix, K_pFor gain, switching sequence is 0≤t₀<t₁<t₂< ..., t ∈ [t_k,t_k+1), k ∈ N is at t ∈ [t_k,t_k+1) design liapunov function and its derivative be:

V_i=x^Tv⁽ⁱ⁾,i∈S

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)=x^T{A}_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}+u_{\mathrm{\&sigma;}\left(t_k\right)}^T{B}_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)},t\&Element;\&lsqb;t_k,t_{k+1})$

2.3 design constants ρ > 0, λ > 1,And vector v^(p)∈Rⁿ, v^(q)∈Rⁿ, z^(p)∈R^sIt is made to meet following condition:

$(A_p^T+{\mathrm{\&rho;I}}_n)v^{\left(p\right)}+z^{\left(p\right)}<0$

v^(p)＞ 0

v^(p)≤λv^(q)

Wherein,It is given non-vanishing vector, and meets at ADT

2.4 design heating-furnace gun pressure Force systemEquilibrium point in suitable switching signal σ It is globally consistent Exponential Stability (GUES) under (t), to arbitrary initial state x (t₀), design constant α and β make its system mode ring Should meet following condition:

$\left|\right|x\left(t\right)\left|\right|\&le;{\mathrm{\&alpha;e}}^{-\mathrm{\&beta;}(t-t_0)}\left|\right|x\left(t_0\right)\left|\right|$

$\mathrm{\&alpha;}=\frac{\sqrt{n}\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)}{\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)}{\mathrm{\&alpha;}}^{\&prime;}$

Wherein,

Above formula can be converted into by 2.5 by step 2.4:

$x^T\left(t\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t\right){v_i}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&GreaterEqual;\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t\right)\&GreaterEqual;\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)\left|\right|x\left(t\right)\left|\right|$

$x^T\left(t_0\right)v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t_0\right){v_i}^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\&le;\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t_0\right)\&le;\sqrt{n}\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)\left|\right|x\left(t\right)\left|\right|$

2.6 can be such as lower inequality according to step 2.4 and 2.5:

$x^T\left(t\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;{\mathrm{\&alpha;}}^{\&prime;}{e}^{-\mathrm{\&beta;}(t-t_0)}{x}^T\left(t_0\right)v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}$

In conjunction with step 2.3, can be converted into further:

$\begin{array}{c}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;e^{N_{\mathrm{\&sigma;}}\ln \mathrm{\&lambda;}}{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_0\left)\right)\\ \&le;e^{(N_0+\frac{t-t}{{\mathrm{\&tau;}}^{*}})\ln \mathrm{\&lambda;}}{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_0\right)}\left(x\right(t_0\left)\right)\\ \&le;{\mathrm{\&alpha;}}^{\&prime;}{e}^{-\mathrm{\&beta;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_0\left)\right)\end{array}$

2.7 according to last inequality of step 2.3, can obtain following form:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;{\mathrm{\&lambda;e}}^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_{k-1}\right)}\left(x\right(t_k\left)\right)\&le;...\&le;{\mathrm{\&lambda;}}^k{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_0\right)}\left(x\right(t_0\left)\right)$

It is further converted to:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;{\mathrm{\&lambda;e}}^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_{k-1}\right)}\left(x\right(t_k\left)\right)$

$x^T\left(t_k\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;{\mathrm{\&lambda;x}}^T\left(t_k\right)v^{\left(\mathrm{\&sigma;}\right(t_{k-1}\left)\right)}$

Can be obtained by 2.3:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;e^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_k\left)\right),t\&Element;\&lsqb;t_k,t_{k+1})$

Obtain with upper inequality derivation:

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;-{\mathrm{\&rho;x}}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;-{\mathrm{\&rho;V}}_{\mathrm{\&sigma;}\left(t_k\right)}$

Can obtain in conjunction with 2.3:

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)=x^T(A_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}+z^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)})$

2.8 combining step 2.2 are to step 2.7 available coking furnace furnace pressure STATE FEEDBACK CONTROL amount, and form is such as Under:

$u_p\left(t\right)=K_{p}x\left(t\right)=\frac{1}{{\stackrel{\&OverBar;}{v}}^{\left(p\right)T}{B}_p^T{v}^{\left(p\right)}}{\stackrel{\&OverBar;}{v}}^{\left(p\right)}{z}^{\left(p\right)T}x\left(t\right)$

The present invention proposes a kind of coking furnace furnace pressure system stability switch controller method for designing.The method establishes The state-space model of system, carrys out design point feedback controller by the liapunov function of design system, it is ensured that be System switching is stable.

Detailed description of the invention

With coking furnace furnace pressure as practical object, with the aperture of damper for input, with coking furnace furnace pressure it is Output, sets up the model of coking furnace furnace pressure.

The step of the inventive method includes:

Step 1, setting up the state-space model of coking furnace furnace pressure, concrete grammar is:

1.1 inputoutput datas first gathering coking furnace furnace pressure, utilize these data to set up coking furnace furnace pressure State-space model, form is as follows:

$\stackrel{\&CenterDot;}{x}\left(t\right)=A_{\mathrm{\&sigma;}\left(t\right)}x\left(t\right)+B_{\mathrm{\&sigma;}\left(t\right)}{u}_{\mathrm{\&sigma;}\left(t\right)}\left(t\right)$

Y (t)=C_σ(t)x(t)

Wherein, x (t) is coking furnace system mode, and y (t) is coking furnace output pressure, u_σ(t)T () is for controlling baffle opening, σ (t) For switching signal represent from [0, ∞) to finite aggregate S={1,2 ..., the mapping of N}, A_σ(t)For Metzler matrix, each σ (t) ∈ S is had, B_σ(t)≥0,C_σ(t)≥0。

Step 2, the state feedback controller of design coking furnace furnace pressure, comprise the concrete steps that:

2.1 design switching signal σ (t) and 0≤t₁≤t₂, meet following condition:

N_σ(t₂,t₁)≤N₀+(t₂-t₁)/τ^*

Wherein, N_σ(t₂,t₁) it is that switched system is at (t₁,t₂Switching times in), τ^*> 0 it is the average residence of switching signal Time (ADT), N₀≥0。

2.2 design A_p+B_pK_pC_p, p ∈ SMetzler matrix, K_pFor gain, switching sequence is 0≤t₀<t₁<t₂< ..., t ∈ [t_k,t_k+1), k ∈ N is at t ∈ [t_k,t_k+1) design liapunov function and its derivative be:

V_i=x^Tv⁽ⁱ⁾,i∈S

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)=x^T{A}_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}+u_{\mathrm{\&sigma;}\left(t_k\right)}^T{B}_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)},t\&Element;\&lsqb;t_k,t_{k+1})$

2.3 design constants ρ > 0, λ > 1,And vector v^(p)∈Rⁿ, v^(q)∈Rⁿ, z^(p)∈R^sIt is made to meet following condition:

$(A_p^T+{\mathrm{\&rho;I}}_n)v^{\left(p\right)}+z^{\left(p\right)}<0$

v^(p)＞ 0

v^(p)≤λv^(q)

Wherein,It is given non-vanishing vector, and meets at ADT

2.4 design heating-furnace gun pressure Force systemEquilibrium point in suitable switching signal σ It is globally consistent Exponential Stability (GUES) under (t), to arbitrary initial state x (t₀), design constant α and β make its system mode ring Should meet following condition:

$\left|\right|x\left(t\right)\left|\right|\&le;{\mathrm{\&alpha;e}}^{-\mathrm{\&beta;}(t-t_0)}\left|\right|x\left(t_0\right)\left|\right|$

$\mathrm{\&alpha;}=\frac{\sqrt{n}\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)}{\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)}{\mathrm{\&alpha;}}^{\&prime;}$

Wherein,

Above formula can be converted into by 2.5 by step 2.4:

$x^T\left(t\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t\right){v_i}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&GreaterEqual;\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t\right)\&GreaterEqual;\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)\left|\right|x\left(t\right)\left|\right|$

$x^T\left(t_0\right)v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t_0\right){v_i}^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\&le;\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t_0\right)\&le;\sqrt{n}\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)\left|\right|x\left(t\right)\left|\right|$

2.6 can be such as lower inequality according to step 2.4 and 2.5:

$x^T\left(t\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;{\mathrm{\&alpha;}}^{\&prime;}{e}^{-\mathrm{\&beta;}(t-t_0)}{x}^T\left(t_0\right)v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}$

In conjunction with step 2.3, can be converted into further:

$\begin{array}{c}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;e^{N_{\mathrm{\&sigma;}}\ln \mathrm{\&lambda;}}{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_0\left)\right)\\ \&le;e^{(N_0+\frac{t-t}{{\mathrm{\&tau;}}^{*}})\ln \mathrm{\&lambda;}}{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_0\right)}\left(x\right(t_0\left)\right)\\ \&le;{\mathrm{\&alpha;}}^{\&prime;}{e}^{-\mathrm{\&beta;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_0\left)\right)\end{array}$

2.7 according to last inequality of step 2.3, can obtain following form:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;{\mathrm{\&lambda;e}}^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_{k-1}\right)}\left(x\right(t_k\left)\right)\&le;...\&le;{\mathrm{\&lambda;}}^k{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_0\right)}\left(x\right(t_0\left)\right)$

It is further converted to:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;{\mathrm{\&lambda;e}}^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_{k-1}\right)}\left(x\right(t_k\left)\right)$

$x^T\left(t_k\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;{\mathrm{\&lambda;x}}^T\left(t_k\right)v^{\left(\mathrm{\&sigma;}\right(t_{k-1}\left)\right)}$

Can be obtained by 2.3:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;e^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_k\left)\right),t\&Element;\&lsqb;t_k,t_{k+1})$

Obtain with upper inequality derivation:

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;-{\mathrm{\&rho;x}}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;-{\mathrm{\&rho;V}}_{\mathrm{\&sigma;}\left(t_k\right)}$

Can obtain in conjunction with 2.3:

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)=x^T(A_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}+z^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)})$

2.8 combining step 2.2 are to step 2.7 available coking furnace furnace pressure STATE FEEDBACK CONTROL amount, and form is such as Under:

$u_p\left(t\right)=K_{p}x\left(t\right)=\frac{1}{{\stackrel{\&OverBar;}{v}}^{\left(p\right)T}{B}_p^T{v}^{\left(p\right)}}{\stackrel{\&OverBar;}{v}}^{\left(p\right)}{z}^{\left(p\right)T}x\left(t\right)$

Claims (1)

1. a coking furnace furnace pressure system stability switch controller method for designing, it is characterised in that the method includes following step Rapid:

Step 1, set up the state-space model of coking furnace furnace pressure, specifically:

First step 1.1 gathers the inputoutput data of coking furnace furnace pressure, utilizes these data to set up coking furnace furnace pressure State-space model, form is as follows:

$\stackrel{\&CenterDot;}{x}\left(t\right)=A_{\mathrm{\&sigma;}\left(t\right)}x\left(t\right)+B_{\mathrm{\&sigma;}\left(t\right)}{u}_{\mathrm{\&sigma;}\left(t\right)}\left(t\right)$

Y (t)=C_σ(t)x(t)

Wherein, x (t) is coking furnace system mode, and y (t) is coking furnace output pressure, u_σ(t)T (), for controlling baffle opening, σ (t) is for cutting Change signal represent from [0, ∞) to finite aggregate S={1,2 ..., the mapping of N}, A_σ(t)For Metzler matrix, each σ (t) ∈ S is had, B_σ(t)≥0,C_σ(t)≥0；

Step 2, the state feedback controller of design coking furnace furnace pressure, specifically:

Step 2.1 designs switching signal σ (t) and 0≤t₁≤t₂, meet following condition:

N_σ(t₂,t₁)≤N₀+(t₂-t₁)/τ^*

Wherein, N_σ(t₂,t₁) it is that switched system is at (t₁,t₂Switching times in), τ^*> 0 it is the average residence time of switching signal, N₀≥0；

Step 2.2 designs A_p+B_pK_pC_p, p ∈ SMetzler matrix, K_pFor gain, switching sequence is 0≤t₀<t₁<t₂< ..., t ∈ [t_k,t_k+1), k ∈ N is at t ∈ [t_k,t_k+1) design liapunov function and its derivative be:

V_i=x^Tv⁽ⁱ⁾,i∈S

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)=x^T{A}_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}+u_{\mathrm{\&sigma;}\left(t_k\right)}^T{B}_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)},t\&Element;\&lsqb;t_k,t_{k+1})$

Step 2.3 design constant ρ > 0, λ > 1,And vector v^(p)∈Rⁿ, v^(q)∈Rⁿ, z^(p)∈R^sIt is made to meet following condition:

$(A_p^T+{\mathrm{\&rho;I}}_n)v^{\left(p\right)}+z^{\left(p\right)}<0$

v^(p)＞ 0

v^(p)≤λv^(q)

Wherein,It is given non-vanishing vector, and meets at average residence time

Step 2.4 designs heating-furnace gun pressure Force systemEquilibrium point in suitable switching signal σ It is globally consistent Exponential Stability under (t), to arbitrary initial state x (t₀), design constant α and β make the response of its system mode meet Following condition:

$\left|\right|x\left(t\right)\left|\right|\&le;{\mathrm{\&alpha;e}}^{-\mathrm{\&beta;}(t-t_0)}\left|\right|x\left(t_0\right)\left|\right|$

$\mathrm{\&alpha;}=\frac{\sqrt{n}\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}\right)}{\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\right)}{\mathrm{\&alpha;}}^{\&prime;}$

Wherein,

Above formula can be converted into by step 2.5 by step 2.4:

$x^T\left(t\right)v^{\left(\mathrm{\&sigma;}\left(t_k\right)\right)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t\right){v_i}^{\left(\mathrm{\&sigma;}\left(t_k\right)\right)}\&GreaterEqual;\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\left(t_k\right)\right)}\right)\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t\right)\&GreaterEqual;\underset{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\left(t_k\right)\right)}\right)\left|\right|x\left(t\right)\left|\right|$

$x^T\left(t_0\right)v^{\left(\mathrm{\&sigma;}\left(t_0\right)\right)}=\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t_0\right){v_i}^{\left(\mathrm{\&sigma;}\left(t_0\right)\right)}\&le;\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\left(t_0\right)\right)}\right)\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{x}_i\left(t_0\right)\&le;\stackrel{\&OverBar;}{\mathrm{\&mu;}}\left(v^{\left(\mathrm{\&sigma;}\left(t_0\right)\right)}\right)\left|\right|x\left(t\right)\left|\right|$

Step 2.6 can be such as lower inequality according to step 2.4 and 2.5:

$x^T\left(t\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;{\mathrm{\&alpha;}}^{\&prime;}{e}^{-\mathrm{\&beta;}(t-t_0)}{x}^T\left(t_0\right)v^{\left(\mathrm{\&sigma;}\right(t_0\left)\right)}$

In conjunction with step 2.3, can be converted into further:

$\begin{array}{c}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;e^{N_{\mathrm{\&sigma;}}ln\mathrm{\&lambda;}}{e}^{-\mathrm{\&rho;}\left(t-t_0\right)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_0\left)\right)\\ \&le;e^{\left(N_0+\frac{t-t}{{\mathrm{\&tau;}}^{*}}\right)ln\mathrm{\&lambda;}}{e}^{-\mathrm{\&rho;}\left(t-t_0\right)}{V}_{\mathrm{\&sigma;}\left(t_0\right)}\left(x\right(t_0\left)\right)\\ \&le;{\mathrm{\&alpha;}}^{\&prime;}{e}^{-\mathrm{\&beta;}\left(t-t_0\right)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_0\left)\right)\end{array}$

Step 2.7, according to last inequality of step 2.3, can obtain following form:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;{\mathrm{\&lambda;e}}^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_{k-1}\right)}\left(x\right(t_k\left)\right)\&le;...\&le;{\mathrm{\&lambda;}}^k{e}^{-\mathrm{\&rho;}(t-t_0)}{V}_{\mathrm{\&sigma;}\left(t_0\right)}\left(x\right(t_0\left)\right)$

It is further converted to:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;{\mathrm{\&lambda;e}}^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_{k-1}\right)}\left(x\right(t_k\left)\right)$

$x^T\left(t_k\right)v^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;{\mathrm{\&lambda;x}}^T\left(t_k\right)v^{\left(\mathrm{\&sigma;}\right(t_{k-1}\left)\right)}$

Can be obtained by 2.3:

$V_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;e^{-\mathrm{\&rho;}(t-t_k)}{V}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t_k\left)\right),t\&Element;\&lsqb;t_k,t_{k+1})$

Obtain with upper inequality derivation:

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)\&le;-{\mathrm{\&rho;x}}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}\&le;-{\mathrm{\&rho;V}}_{\mathrm{\&sigma;}\left(t_k\right)}$

Can obtain in conjunction with 2.3:

${\stackrel{\&CenterDot;}{V}}_{\mathrm{\&sigma;}\left(t_k\right)}\left(x\right(t\left)\right)=x^T(A_{\mathrm{\&sigma;}\left(t_k\right)}^T{v}^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)}+z^{\left(\mathrm{\&sigma;}\right(t_k\left)\right)})$

Step 2.8 combining step 2.2 is to step 2.7 available coking furnace furnace pressure STATE FEEDBACK CONTROL amount, and form is such as Under:

$u_p\left(t\right)=K_{p}x\left(t\right)=\frac{1}{{\stackrel{\&OverBar;}{v}}^{\left(p\right)T}{B}_p^T{v}^{\left(p\right)}}{\stackrel{\&OverBar;}{v}}^{\left(p\right)}{z}^{\left(p\right)T}x\left(t\right).$