CN104102131B - The linear quadratic fault tolerant control method of the batch process that Infinite horizon optimizes - Google Patents

Abstract

The present invention proposes the linear quadratic fault tolerant control method of the batch process that a kind of Infinite horizon optimizes.The present invention passes through bonding state variable and output tracking error, establishes the Extended state space model of batch process, and then designs controller in infinite horizon.The method not only ensure that system has good tracking performance in the case of unknown disturbance and actuator failures, and the form that simultaneously also ensure that is simple and meets the needs of actual industrial process.

Description

The linear quadratic fault tolerant control method of the batch process that Infinite horizon optimizes

Technical field

The invention belongs to technical field of automation in industry, relate to the linear quadratic of the batch process that a kind of Infinite horizon optimizes Fault tolerant control method.

Background technology

Along with the high speed development of society, the requirement that high-quality batch is produced by people is more and more higher.This high request causes Producing and need to operate under conditions of more complicated, the probability of system jam also accordingly increases.In these faults In, actuator failures is modal a kind of fault.Owing to there is friction, dead band, the characteristic such as saturated, executor is in execution process In inevitably some faults, this causes it to be extremely difficult to specify or preferable position.If fault not by and Time detection and correct, production performance will necessarily deteriorate, and results even in the safety problem of equipment and personnel.It is therefore proposed that one Kind of new control method solves executor and breaks down in the process of implementation thus ensure that system control performance is the most necessary 's.

Summary of the invention

It is an object of the invention to for batch production process is likely encountered the problem that executor breaks down, it is proposed that one Plant the linear quadratic fault tolerant control method of the batch process that Infinite horizon optimizes.The method by bonding state variable and output with Track error, establishes the Extended state space model of batch process, and then designs controller in infinite horizon.The method is not only The system that ensure that has good tracking performance in the case of unknown disturbance and actuator failures, and the form that simultaneously also ensure that is simple also Meet the needs of actual industrial process.

The technical scheme is that and set up by data acquisition, model, predict the means such as mechanism, optimization, establish one Plant the linear quadratic fault tolerant control method of the batch process that Infinite horizon optimizes, utilize the method can be effectively improved system in the unknown Control performance in the case of disturbance and actuator failures.

The step of the inventive method includes:

Step (1). setting up the Extended state space model of controlled device, concrete grammar is:

A. the method utilizing Real-time data drive sets up local increment, and concrete grammar is: set up the reality of batch process Time runtime database, by data acquisition unit gather real-time process service data, will gather real-time process service data make Sample set for data-drivenWherein,Representing the input value of i-th group of technological parameter, y (i) represents i-th group The output valve of technological parameter, N represents sampling sum；Set up based on the real-time process service data set of this object based on The controlled local autoregressive moving-average model of the discrete differential equation form of young waiter in a wineshop or an inn's multiplication algorithm:

Wherein, y_LK () represents the output valve of the technological parameter of k moment local increment,Expression is obtained by identification The set of model parameter,Represent input and the set of output data, the u of the last time of the technological parameter of local increment (k-d-1) representing the control variable that k-d-1 etching processes parameter is corresponding, d+1 is the time lag of real process, and Τ is the transposition of matrix Symbol.

The identification means used are:

Wherein,It is two matrixes in identification with P,γ is forgetting factor,For unit square Battle array.

B. utilizing the coefficient obtained in a step, set up the difference equation model of batch process, its form is:

Δ y (k)+H Δ y (k-1)=F Δ u (k-d-1)

Wherein, Δ is difference operator, and F, H are that d is time lag item by debating the parameter that knowledge obtains in a step.

C. according to the difference equation in b step, setting up the state-space model of batch process, form is as follows:

$\left\{\begin{array}{c}\mathrm{\Δx}(k+1)=A_m\mathrm{\Δx}\left(k\right)+B_m\mathrm{\Δu}\left(k\right)\\ \mathrm{\Δy}\left(k\right)=C_m\mathrm{\Δx}\left(k\right)\end{array}\right.$

Wherein,

$\mathrm{\Δx}(k+1)=\left(\begin{array}{c}\mathrm{\Δy}(k+1)\\ \mathrm{\Δu}\left(k\right)\\ \mathrm{\Δu}(k-1)\\ \·\\ \·\\ \·\\ \mathrm{\Δu}(k-d+1)\end{array}\right),\mathrm{\Δx}\left(k\right)=\left(\begin{array}{c}\mathrm{\Δy}\left(k\right)\\ \mathrm{\Δu}(k-1)\\ \mathrm{\Δu}(k-2)\\ \·\\ \·\\ \·\\ \mathrm{\Δu}(k-d)\end{array}\right)$

C_m=(1 00 ... 0)

Wherein, A_mFor (d+1) × (d+1) rank matrix, B_mFor rank, (d+1) × 1 matrix, C_mIt it is 1 × (d+1) rank matrix.

D., the state-space model obtained in step c is converted to comprise the extension shape of state variable and output tracking error State space model, form is as follows:

Z (k+1)=Az (k)+B Δ u (k)=Az (k)+Bu (k)-Bu (k-1)

In formula,

$z(k+1)=\left[\begin{array}{c}\mathrm{\Δx}(k+1)\\ e(k+1)\end{array}\right],z\left(k\right)=\left[\begin{array}{c}\mathrm{\Δx}\left(k\right)\\ e\left(k\right)\end{array}\right]$

$A=\left(\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right),B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right]$

E (k)=r (k)-y (k)

Wherein, r (k) is the idea output in k moment, and e (k) is between k moment idea output and real output value Difference.

Step (2). the batch process linear quadratic fault-tolerant controller that the Infinite horizon of design controlled device optimizes, specifically side Method is:

A. choosing the object function of batch processed process, form is as follows:

$\begin{array}{c}J=\underset{k=k_0}{\overset{k_f-1}{\mathrm{\Σ}}}[z{\left(k\right)}^T\mathrm{Qz}\left(k\right)+\mathrm{\Δu}{\left(k\right)}^T\mathrm{R\Δu}\left(k\right)]+z{\left(k_f\right)}^T\mathrm{Qz}\left(k_f\right)\\ Q=\mathrm{diag}\{q_{j1},q_{j2},\·\·\·,q_{\mathrm{jp}+q-1},q_{\mathrm{je}}\}\end{array}$

Wherein, Q ＞ 0, R ＞ 0 is respectively the weighting matrix of process status, weighted input matrix, [k₀,k_f] for optimizing time domain； q_j1,q_j2,…q_jp+q+1For the weight coefficient of process status, q_jeFor the weight coefficient of output tracking error and take q_je=1.

B. utilize Pang Te lia king principle of minimum that the object function of a step is written as form:

$H_k=[z{\left(k\right)}^T\mathrm{Qz}\left(k\right)+\mathrm{\Δ}{u}^T\left(k\right)\mathrm{R\Δu}\left(k\right)]+p_{k+1}^T[\mathrm{Az}\left(k\right)+\mathrm{B\Δu}\left(k\right)]$

Wherein, p_k+1For Lagrange multiplier.

C. askAnd make it be equal to zero, can obtain

$\mathrm{\Δu}\left(k\right)=-\frac{1}{2}{R}^{-1}{B}^T{p}_{k+1}$

AssociatingCan obtain further

$\begin{array}{c}\mathrm{\Δu}\left(k\right)=-R^{-1}{B}^T{[I+H_{{k+1,k}_f}{\mathrm{BR}}^{-1}{B}^T]}^{-1}{H}_{{k+1,k}_f}\mathrm{Az}\left(k\right)\\ H_{{k,k}_f}=A^T[I+H_{{k+1,k}_f}{\mathrm{BR}}^{-1}{B}^T{]}^{-1}{H}_{{k+1,k}_f}A+Q\\ =A^T{H}_{{k+1,k}_f}A-A^T{H}_{{k+1,k}_f}B{(R+B^T{H}_{{k+1,k}_f}B)}^{-1}{B}^T{H}_{{k+1,k}_f}A+Q\\ H_{k_f,k_f}=Q\end{array}$

Wherein, R^-1Represent weighted input inverse of a matrix matrix.

D. k is made_fTend to the most infinite, can obtain

Δ u (k)=-R^-1B^Τ[I+K_∞BR^-1B^Τ]^-1K_∞Az(k)

K_∞=A^Τ[I+K_∞BR^-1B^Τ]^-1K_∞A+Q

=A^ΤK_∞A-A^ΤK_∞B(R+B^ΤK_∞B)^-1B^ΤK_∞A+Q

U (k)=u (k-1)+Δ u (k)

Wherein, K_∞For k_fWhen tending to the most infiniteValue.

E. controlled quentity controlled variable u (k) obtained in Step d is acted on controlled device.

F. at subsequent time, continue to solve new controlled quentity controlled variable u (k+1) according to the step of a to e, circulate successively.

The present invention just optimizes time domain and extend to Infinite horizon, it is proposed that the batch process that a kind of Infinite horizon optimizes Linear quadratic fault tolerant control method.The method compensate for the deficiency of conventional linear linear quadratic control method, is effectively guaranteed system Good tracking performance in the case of unknown disturbance and actuator failures.

Detailed description of the invention

In injection moulding process as a example by the control of injection speed

In injection moulding process, the control of injection speed is a typical batch processed process, and regulating measure is for controlling proportioning valve The aperture of valve.

Step (1). setting up the Extended state space model of injection process, concrete grammar is:

A. setting up the real-time running data storehouse of injection process, gathering real-time process service data by data acquisition unit will The real-time process service data gathered is as the sample set of data-drivenWherein,Represent i-th group of proportioning valve The aperture of valve, y (i) represents the injection speed of i-th group of reality output；Real-time process service data collection with injection speed process It is combined into the controlled local autoregressive moving-average model of Foundation discrete differential based on least-squares algorithm equation form:

Wherein, y_LK () represents the real output value of k moment injection speed, θ represents the model parameter that obtained by identification Set,Representing input and the set of output data of the last time of injection process local increment, u (k-d-1) represents The aperture of proportioning valve valve in k-d-1 moment injection process, d+1 is the time lag of corresponding injection process, and Τ is the transposition symbol of matrix Number.

The identification means used are:

Wherein,It is two matrixes in identification with P,γ is forgetting factor,For unit square Battle array.

B. the form that injection process model conversion is difference equation that will obtain in a step:

Δ y (k)+H Δ y (k-1)=F Δ u (k-d-1)

Wherein, Δ is difference operator, and F, H are that d is time lag item by debating the parameter that knowledge obtains in a step.

C. choose state variable, according to the difference equation in b step, set up the state-space model of injection process, form As follows:

$\left\{\begin{array}{c}\mathrm{\Δx}(k+1)=A_m\mathrm{\Δx}\left(k\right)+B_m\mathrm{\Δu}\left(k\right)\\ \mathrm{\Δy}\left(k\right)=C_m\mathrm{\Δx}\left(k\right)\end{array}\right.$

Wherein,

$\mathrm{\Δx}(k+1)=\left(\begin{array}{c}\mathrm{\Δy}(k+1)\\ \mathrm{\Δu}\left(k\right)\\ \mathrm{\Δu}(k-1)\\ \·\\ \·\\ \·\\ \mathrm{\Δu}(k-d+1)\end{array}\right),\mathrm{\Δx}\left(k\right)=\left(\begin{array}{c}\mathrm{\Δy}\left(k\right)\\ \mathrm{\Δu}(k-1)\\ \mathrm{\Δu}(k-2)\\ \·\\ \·\\ \·\\ \mathrm{\Δu}(k-d)\end{array}\right)$

C_m=(1 00 ... 0)

Wherein, A_mFor (d+1) × (d+1) rank matrix, B_mFor rank, (d+1) × 1 matrix, C_mIt it is 1 × (d+1) rank matrix.

D. the state-space model of the injection process obtained in step c being converted to comprises state variable and output tracking misses The Extended state space model of difference, form is as follows:

Z (k+1)=Az (k)+B Δ u (k)=Az (k)+Bu (k)-Bu (k-1)

In formula,

$z(k+1)=\left[\begin{array}{c}\mathrm{\Δx}(k+1)\\ e(k+1)\end{array}\right],z\left(k\right)=\left[\begin{array}{c}\mathrm{\Δx}\left(k\right)\\ e\left(k\right)\end{array}\right]$

$A=\left(\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right),B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right]$

E (k)=r (k)-y (k)

E (k) is the difference between k moment ideal injection velocity amplitude and actual injection velocity amplitude.

Step (2). the batch process linear quadratic fault-tolerant controller that the Infinite horizon of design injection process optimizes, specifically side Method is:

A. choosing the object function of injection process, form is as follows:

$\begin{array}{c}J=\underset{k=k_0}{\overset{k_f-1}{\mathrm{\Σ}}}[z{\left(k\right)}^T\mathrm{Qz}\left(k\right)+\mathrm{\Δu}{\left(k\right)}^T\mathrm{R\Δu}\left(k\right)]+z{\left(k_f\right)}^T{Q}_{f}z\left(k_f\right)\\ Q=\mathrm{diag}\{q_{j1},q_{j2},\·\·\·,q_{\mathrm{jp}+q-1},q_{\mathrm{je}}\}\end{array}$

Wherein, Q ＞ 0, R ＞ 0 weighting matrix of injection process state, weighted input matrix, [k respectively₀,k_f] for injecting The optimization time domain of journey；q_j1,q_j2,…q_jp+q+1For the weight coefficient of injection speed process status, q_jePower for output tracking error Weigh coefficient and take q_je=1.

B. utilize Pang Te lia king principle of minimum that the object function of a step is written as form:

$H_k=[z{\left(k\right)}^T\mathrm{Qz}\left(k\right)+\mathrm{\Δ}{u}^T\left(k\right)\mathrm{R\Δu}\left(k\right)]+p_{k+1}^T[\mathrm{Az}\left(k\right)+\mathrm{B\Δu}\left(k\right)]$

Wherein, p_k+1For Lagrange multiplier.

C. askAnd make it be equal to zero, can obtain

$\mathrm{\Δu}\left(k\right)=-\frac{1}{2}{R}^{-1}{B}^T{p}_{k+1}$

Associating $p_k=2H_{{k,k}_f}z\left(k\right)$ Can obtain

$\begin{array}{c}\mathrm{\Δu}\left(k\right)=-R^{-1}{B}^T{[I+H_{{k+1,k}_f}{\mathrm{BR}}^{-1}{B}^T]}^{-1}{H}_{{k+1,k}_f}\mathrm{Az}\left(k\right)\\ H_{{k,k}_f}=A^T[I+H_{{k+1,k}_f}{\mathrm{BR}}^{-1}{B}^T{]}^{-1}{H}_{{k+1,k}_f}A+Q\\ =A^T{H}_{{k+1,k}_f}A-A^T{H}_{{k+1,k}_f}B{(R+B^T{H}_{{k+1,k}_f}B)}^{-1}{B}^T{H}_{{k+1,k}_f}A+Q\\ H_{k_f,k_f}=Q\end{array}$

Wherein, R^-1Represent weighted input inverse of a matrix matrix.

D. k is made_fTend to the most infinite, can obtain

Δ u (k)=-R^-1B^T[I+K_∞BR^-1B^T]^-1K_∞Az(k)

K_∞=A^T[I+K_∞BR^-1B^T]^-1K_∞A+Q

=A^TK_∞A-A^TK_∞B(R+B^TK_∞B)^-1B^TK_∞A+Q

U (k)=u (k-1)+Δ u (k)

Wherein, K_∞For k_fWhen tending to the most infiniteValue.

E. proportioning valve valve opening u (k) obtained in Step d is acted on injection machine.

F. at subsequent time, continue to solve aperture u (k+1) of new proportioning valve valve according to the step of a to e, and successively Circulation.

Claims (1)

1. the linear quadratic fault tolerant control method of the batch process that Infinite horizon optimizes, it is characterised in that the concrete steps of the method It is:

Step (1). setting up the Extended state space model of injection process, concrete grammar is:

A. setting up the real-time running data storehouse of injection process, gathering real-time process service data by data acquisition unit will gather Real-time process service data as the sample set of data-drivenWherein,Represent i-th group of proportioning valve valve Aperture, y (i) represents the injection speed of i-th group of reality output；It is combined into the real-time process service data collection of injection speed process The controlled local autoregressive moving-average model of Foundation discrete differential based on least-squares algorithm equation form:

Wherein, y_LK () represents the real output value of k moment injection speed, θ represents the set of the model parameter obtained by identification,Representing input and the set of output data of the last time of injection process local increment, u (k-d-1) represents k-d-1 The aperture of proportioning valve valve in moment injection process, d+1 is the time lag of corresponding injection process, and T is the transposition symbol of matrix；

The identification means used are:

Wherein,It is two matrixes in identification with P,γ is forgetting factor,For unit matrix；

B. the form that injection process model conversion is difference equation that will obtain in a step:

Δ y (k)+H Δ y (k-1)=F Δ u (k-d-1)

Wherein, Δ is difference operator, and F, H are that d is time lag item by debating the parameter that knowledge obtains in a step；

C. choosing state variable, according to the difference equation in b step, set up the state-space model of injection process, form is as follows:

$\left\{\begin{array}{c}\mathrm{\Δ}x(k+1)=A_m\mathrm{\Δ}x\left(k\right)+B_m\mathrm{\Δ}u\left(k\right)\\ \mathrm{\Δ}y\left(k\right)=C_m\mathrm{\Δ}x\left(k\right)\end{array}\right.$

Wherein,

$\mathrm{\Δ}x(k+1)=\left(\begin{array}{c}\mathrm{\Δ}y(k+1)\\ \mathrm{\Δ}u\left(k\right)\\ \mathrm{\Δ}u(k-1)\\ \begin{array}{c}.\\ .\\ .\end{array}\\ \mathrm{\Δ}u(k-d+1)\end{array}\right),\mathrm{\Δ}x\left(k\right)=\left(\begin{array}{c}\mathrm{\Δ}y\left(k\right)\\ \mathrm{\Δ}u(k-1)\\ \mathrm{\Δ}u(k-2)\\ \begin{array}{c}.\\ .\\ .\end{array}\\ \mathrm{\Δ}u(k-d)\end{array}\right)$

C_m=(1 00 ... 0)

Wherein, A_mFor (d+1) × (d+1) rank matrix, B_mFor rank, (d+1) × 1 matrix, C_mIt it is 1 × (d+1) rank matrix；

D., the state-space model of the injection process obtained in step c is converted to comprise state variable and output tracking error Extended state space model, form is as follows:

Z (k+1)=Az (k)+B Δ u (k)=Az (k)+Bu (k)-Bu (k-1)

In formula,

$z(k+1)=\left[\begin{array}{c}\mathrm{\Δ}x(k+1)\\ e(k+1)\end{array}\right],z\left(k\right)=\left[\begin{array}{c}\mathrm{\Δ}x\left(k\right)\\ e\left(k\right)\end{array}\right]$

$A=\left(\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right),B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right]$

E (k)=r (k)-y (k)

E (k) is the difference between k moment ideal injection velocity amplitude and actual injection velocity amplitude；

Step (2). the batch process linear quadratic fault-tolerant controller that the Infinite horizon of design injection process optimizes, concrete grammar It is:

A. choosing the object function of injection process, form is as follows:

$J=\underset{k=k_0}{\overset{k_f-1}{\Σ}}\[z{\left(k\right)}^{T}Qz\left(k\right)+\mathrm{\Δ}u{\left(k\right)}^{T}R\mathrm{\Δ}u\left(k\right)\]+z{\left(k_f\right)}^T{Q}_{f}z\left(k_f\right)$

Q=diag{q_j1,q_j2,…,q_jp+q-1,q_je}

Wherein, Q ＞ 0, R ＞ 0 weighting matrix of injection process state, weighted input matrix, [k respectively₀,k_f] it is injection process Optimize time domain；q_j1,q_j2,…q_jp+q+1For the weight coefficient of injection speed process status, q_jeWeight system for output tracking error Count and take q_je=1；

B. utilize Pang Te lia king principle of minimum that the object function of a step is written as form:

$H_k=\[z{\left(k\right)}^{T}Qz\left(k\right)+{\mathrm{\Δu}}^T\left(k\right)R\mathrm{\Δ}u\left(k\right)\]+p_{k+1}^T\[Az\left(k\right)+B\mathrm{\Δ}u\left(k\right)\]$

Wherein, p_k+1For Lagrange multiplier；

C. askAnd make it be equal to zero, can obtain

$\mathrm{\Δ}u\left(k\right)=-\frac{1}{2}{R}^{-1}{B}^T{p}_{k+1}$

AssociatingCan obtain

$\mathrm{\Δ}u\left(k\right)=-R^{-1}{B}^T{\[I+H_{k+1,k_f}{\mathrm{BR}}^{-1}{B}^T\]}^{-1}{H}_{k+1,k_f}Az\left(k\right)$

$\begin{array}{c}{H}_{k,k_f}=A^T{\[I+H_{k+1,k_f}{\mathrm{BR}}^{-1}{B}^T\]}^{-1}{H}_{k+1,k_f}A+Q\\ =A^T{H}_{k+1,k_f}A-A^T{H}_{k+1,k_f}B{\left(R+B^T{H}_{k+1,k_f}B\right)}^{-1}{B}^T{H}_{k+1,k_f}A+Q\end{array}$

$H_{k_f,k_f}=Q$

Wherein, R^-1Represent weighted input inverse of a matrix matrix；

D. k is made_fTend to the most infinite, can obtain

Δ u (k)=-R^-1B^T[I+K_∞BR^-1B^T]^-1K_∞Az(k)

K_∞=A^T[I+K_∞BR^-1B^T]^-1K_∞A+Q

=A^TK_∞A-A^TK_∞B(R+B^TK_∞B)^-1B^TK_∞A+Q

U (k)=u (k-1)+Δ u (k)

Wherein, K_∞For k_fWhen tending to the most infiniteValue；

E. proportioning valve valve opening u (k) obtained in Step d is acted on injection machine；

F. at subsequent time, continue to solve aperture u (k+1) of new proportioning valve valve according to the step of a to e, and circulate successively.