CN102063133A

CN102063133A - Adaptive control method of multiple-variable time lag process

Info

Publication number: CN102063133A
Application number: CN 201010558270
Authority: CN
Inventors: 贾英民; 苏海霞
Original assignee: Beihang University
Current assignee: Beihang University
Priority date: 2010-11-22
Filing date: 2010-11-22
Publication date: 2011-05-18

Abstract

The invention proposes an adaptive control method for a multivariable time-delay process, and the time-delay can be either an output time-delay or an input time-delay. The adaptive control method utilizes a model matching controller, and adopts gradient adaptive law to adjust the parameters of the controller, so as to ensure that all signals in the closed-loop system are bounded and the errors converge to zero. The method has high convergence precision and strong real-time performance, and has high application value.

Description

An adaptive control method for multivariable time-delay processes

技术领域technical field

本发明属于过程控制领域，具体涉及对多变量时滞过程设计自适应控制方案。The invention belongs to the field of process control, and in particular relates to designing an adaptive control scheme for a multivariable time-delay process.

背景技术Background technique

在很多的工业过程中都存在时间延迟，如蒸汽或流体在管道中的流动，网络系统中的信号传输，我们把这类含时间延迟的系统称为时滞系统。时滞的存在使得工业过程变得更加复杂和难以控制，降低控制的效果。所以对时滞系统的研究非常重要。另一方面，工业过程中总是有一些不确定性的因素，如一些控制参数的变化，建模过程中一些未知的动态，而自适应控制可以根据过程中这些因素的变化作相应的调整，达到比较理想的控制效果。所以对工业过程的自适应控制具有重要意义。There are time delays in many industrial processes, such as the flow of steam or fluid in pipelines, and signal transmission in network systems. We call such systems with time delays time-delay systems. The existence of time lag makes the industrial process more complex and difficult to control, reducing the effect of control. So the study of time-delay systems is very important. On the other hand, there are always some uncertain factors in the industrial process, such as the change of some control parameters, some unknown dynamics in the modeling process, and the adaptive control can make corresponding adjustments according to the changes of these factors in the process, To achieve a more ideal control effect. So the adaptive control of industrial process is of great significance.

目前对于单变量(一个输入一个输出)时滞过程的控制的研究比较多，所用的方法有著名的Smith预估器法，它可以消除闭环控制系统特征方程中的时滞，但是这种方法只适用于模型非常精确的过程，而且要求过程本身是稳定的。还有一种方法是将系统中的时滞先转化为扰动，然后用鲁棒控制的方法来解决。由于鲁棒控制是一种基于最优的方法，而其中所用的目标函数可以选择多个不同的函数，是一种需要多次尝试的方法，误差不可避免。而模型匹配控制技术是一种确定性的控制方法，先给定一个满足控制所要达到的目标的模型，设计一个控制器使得控制系统与这个模型完全匹配。对于控制者来说，这是一种理想的控制效果。At present, there are many studies on the control of single variable (one input and one output) time-delay process. The method used is the famous Smith predictor method, which can eliminate the time-delay in the characteristic equation of the closed-loop control system, but this method only It is suitable for the process with very accurate model, and requires the process itself to be stable. Another method is to convert the time delay in the system into a disturbance first, and then use a robust control method to solve it. Because robust control is an optimal method, and the objective function used in it can choose many different functions, it is a method that requires multiple attempts, and errors are inevitable. The model matching control technology is a deterministic control method. Firstly, a model that satisfies the goal to be achieved by the control is given, and a controller is designed to make the control system completely match the model. For controllers, this is an ideal control effect.

单变量时滞过程的模型匹配控制器已经设计出来，并且可以应用到自适应控制中，即对含参数变化的系统可以用设计的自适应控制方案。但是很多工业过程都有多个变量(多个输入多个输出)，如污泥焚烧流化床的控制，所以对含多个变量的时滞过程的模型匹配具有重要意义。Model matching controllers for univariate time-delay processes have been designed and can be applied to adaptive control, that is, the designed adaptive control scheme can be used for systems with parameter changes. However, many industrial processes have multiple variables (multiple inputs and multiple outputs), such as the control of sludge incineration fluidized bed, so the model matching of time-delay processes with multiple variables is of great significance.

本文提出一种基于模型匹配控制方法的对多变量时滞过程的自适应控制方案。This paper proposes an adaptive control scheme for multivariable time-delay processes based on model matching control method.

发明内容Contents of the invention

本发明的目的在于提出一种对多变量时滞过程的控制方案，是基于模型匹配技术而设计的，该方案实时性好精确度较高。The purpose of the present invention is to propose a control scheme for multi-variable time-delay process, which is designed based on model matching technology, and the scheme has good real-time performance and high precision.

本发明基于时滞补偿器和多项式放程组的解对多变量时滞系统设计模型匹配控制器，然后对参数未知的不确定系统设计自适应控制方案，使得闭环系统内所有信号有界，并且控制过程的输出尽可能的跟踪参考模型的输出。为达到上述目的，如图1所示，本发明的技术方案具体是这样实现的：The present invention designs a model matching controller for a multivariable time-delay system based on the solution of a time-delay compensator and a polynomial release group, and then designs an adaptive control scheme for an uncertain system with unknown parameters, so that all signals in the closed-loop system are bounded, and The output of the control process follows the output of the reference model as closely as possible. In order to achieve the above object, as shown in Figure 1, the technical solution of the present invention is specifically realized in this way:

1.将系统转化为对角线上增益不为零的形式。；1. Transform the system into a form in which the gain on the diagonal is not zero. ;

2.设计标称系统的模型匹配控制器；2. Design the model matching controller of the nominal system;

3.控制器参数化；3. Controller parameterization;

4.计算误差表达式；4. Calculate the error expression;

5.设计参数自适应律。5. Design parameter adaptive law.

本发明给出了一个系统化的多变量时滞过程自适应控制的方法，多变量时滞过程的模型也为一般化的模型，所以该发明可用于多种多变量时滞过程的自适应控制。The present invention provides a systematic method for adaptive control of multivariable time-delay processes, and the model of multivariable time-delay processes is also a generalized model, so the invention can be used for adaptive control of various multivariable time-delay processes .

附图说明Description of drawings

图1为自适应控制流程图；Fig. 1 is an adaptive control flow chart;

图2为标称系统的模型匹配控制器设计框图；Fig. 2 is a design block diagram of the model matching controller of the nominal system;

具体实施方式Detailed ways

参照附图并举实施例对本发明进一步详细说明。The present invention will be described in further detail with reference to the accompanying drawings and examples.

首先对多变量时滞过程建立一个统一的模型，过程有n个输入n个输出，从输入到输出的传递函数为Firstly, a unified model is established for the multivariable time-delay process. The process has n inputs and n outputs, and the transfer function from input to output is

$T T ((s the s)) = = [\begin{matrix} \frac{{g g}_{1111} {r r}_{1111} ((s the s))}{{p p}_{11} ((s the s))} {e e}^{- - {L L}_{1111} s the s} & . . . . . . & \frac{{g g}_{11 n no} {r r}_{11 n no} ((s the s))}{{p p}_{11} ((s the s))} {e e}^{- - {L L}_{11 n no} s the s} \\ . . . . . . & . . . . . . & . . . . . . \\ \frac{{g g}_{n no 11} {r r}_{n no 11} ((s the s))}{{p p}_{n no} ((s the s))} {e e}^{- - {L L}_{n no 11} s the s} & . . . . . . & \frac{{g g}_{nn n} {r r}_{nn n} ((s the s))}{{p p}_{n no} ((s the s))} {e e}^{- - {L L}_{nn n} s the s} \end{matrix}]$

其中从每个输入到每个输出都有时滞，记从第j个输入到第i个输出的时滞为L_ij。r_ij(s)和p_i(s)是首一多项式，它们的阶数分别为m_ij和n_i，记为

选定一个满足我们所要的控制效果的模型，称为参考模型。因为每一个输出仅由一个输入控制是理想的控制效果，所以模型选择为There is a time lag from each input to each output, and the time lag from the jth input to the ith output is L _ij . r _ij (s) and p _i (s) are the first polynomials, their orders are m _ij and ni _i respectively, denoted as

Select a model that satisfies the control effect we want, called the reference model. Since it is an ideal control effect that each output is controlled by only one input, the model is chosen as

${T T}_{r r} ((s the s)) = = diag diag ((\frac{{g g}_{r r 11} {r r}_{r r 11} ((s the s))}{{P P}_{r r 11} ((s the s))} {e e}^{- - {L L}_{r r 11} s the s},, . . . . . .,, \frac{{g g}_{rn rn} {r r}_{rn rn} ((s the s))}{{p p}_{rn rn} ((s the s))} {e e}^{- - {L L}_{rn rn} s the s}))$

我们的控制目标就是设计一个自适应控制方案使得过程的输出跟踪参考模型的输出。对了达到控制目标，对该模型有三个假设条件。Our control goal is to design an adaptive control scheme such that the output of the process tracks the output of the reference model. To achieve the control objectives, the model has three assumptions.

假设1记

R(s)是渐近稳定的，即最小相位的.Hypothesis 1

R(s) is asymptotically stable, i.e. minimum phase.

假设2参考模型中每行的相对阶都大于或等于控制模型中每行的相对阶。Assumption 2 The relative order of each row in the reference model is greater than or equal to the relative order of each row in the control model.

假设3常数矩阵

是非奇异的，其中d_i是一个常数，使得是非零的有限向量。Tⁱ(s)是T(s)的第i行.Assume 3 constant matrices

is nonsingular, where d _i is a constant such that is a nonzero finite vector. T ⁱ (s) is the ith row of T(s).

1将系统转化为对角线上增益不为零的形式。因为B^*是非奇异的，总可以找到一个常数矩阵使得系统自右乘之后转化为b_ii＝g_ii≠0，所以首先将系统转化为b_ii＝g_ii≠0的形式。1 transforms the system into a form in which the gain on the diagonal is non-zero. Because B ^* is non-singular, a constant matrix can always be found to transform the system into b _ii = g _ii ≠ 0 after right multiplication, so first transform the system into the form of b _ii = g _ii ≠ 0.

2标称系统模型匹配控制器设计2 Nominal system model matching controller design

首先是时滞补偿器设计，控制系统每行总最小的时滞为L_il，有下面的表达式The first is the design of the time-delay compensator. The total minimum time-lag of each line of the control system is L _il , which has the following expression

$\frac{{r r}_{i i}^{* *} ((s the s)) {p p}_{i i} ((s the s)) - - {g g}_{ij ij} {r r}_{ij ij} ((s the s)) {p p}_{i i}^{* *} ((s the s))}{{r r}_{i i}^{* *} ((s the s)) {p p}_{i i} ((s the s))} = = {Σ Σ}_{k k = = 11}^{{m m}_{i i} + + {n no}_{i i}} \frac{{β β}_{ij ij}^{k k} {e e}^{(({L L}_{ij ij} - - {L L}_{il il})) {z z}_{i i}^{k k}}}{s the s - - {z z}_{i i}^{k k}} + + 11 - - {b b}_{ij ij},, j j &NotEqual; &NotEqual; l l$

定义多项式φ_ij(s)满足下面的方程Define the polynomial φ _ij (s) to satisfy the following equation

$\frac{{r r}_{i i}^{* *} ((s the s)) {p p}_{i i} ((s the s)) - - {φ φ}_{ij ij} ((s the s))}{{r r}_{i i}^{* *} ((s the s)) {p p}_{i i} ((s the s))} = = {Σ Σ}_{k k = = 11}^{{m m}_{i i} + + {n no}_{i i}} \frac{{β β}_{ij ij}^{k k} {e e}^{{L L}_{ij ij} {z z}_{i i}^{k k}}}{s the s - - {z z}_{i i}^{k k}} + + 11 - - {b b}_{ij ij},, j j &NotEqual; &NotEqual; l l$

选择阶数为m_ij的多项式

(j≠l)，可以写为Choose a polynomial of order m _ij

(j≠l), can be written as

$- - \frac{{g g}_{ij ij} {\overset{&OverBar; &OverBar;}{r r}}_{ij ij} ((s the s))}{{r r}_{i i}^{* *} ((s the s))} = = {Σ Σ}_{k k = = 11}^{{m m}_{i i}} \frac{{\overset{&OverBar; &OverBar;}{β β}}_{ij ij}^{k k}}{s the s - - {z z}_{i i}^{k k}} - - {b b}_{ij ij}$

定义多项式

满足下面的方程define polynomial

satisfy the following equation

$- - \frac{{\overset{&OverBar; &OverBar;}{φ φ}}_{ij ij} ((s the s))}{{r r}_{i i}^{* *} ((s the s))} = = {Σ Σ}_{k k = = 11}^{{m m}_{i i}} \frac{{\overset{&OverBar; &OverBar;}{β β}}_{ij ij}^{k k} {e e}^{(({L L}_{ij ij} - - {L L}_{il il})) {z z}_{i i}^{k k}}}{s the s - - {z z}_{i i}^{k k}} - - {b b}_{ij ij}$

然后是多项式方程组求解。选择一个阶数为n_i-m_i-1的多项式τ_i(s)，则下面的多项式方程组有解。Then there is the solution of polynomial equations. Choose a polynomial τ _i (s) whose order is n _i -m _i -1, then the following polynomial equations have solutions.

k_ij(s)p_i(s)+g_ijh_i(s)r_ij(s)＝b_ijτ_i(s)r^* _i(s)p_i(s)-τ_i(s)φ_ij(s)k _ij (s)p _i (s)+g _ij h _i (s)r _ij (s)＝b _ij τ _i (s)r ^* _i (s)p _i (s)-τ _i (s)φ _ij (s)

其中多项式k_ij(s)，h_i(s)的阶数分别为

设计的模型匹配控制器为[u₁，u₂，…，u_n]^T＝(B^*)^-1[c₁，c₂，…，c_n]^T Among them, the orders of polynomials k _ij (s), h _i (s) are respectively

The designed model matching controller is [u ₁ , u ₂ ,…, u _n ] ^T = (B ^* ) ^-1 [c ₁ , c ₂ ,…, c _n ] ^T

其中in

${c c}_{i i} = = {Σ Σ}_{j j = = 11}^{n no} \frac{{k k}_{ij ij} ((s the s))}{{τ τ}_{i i} ((s the s)) {r r}_{i i}^{* *} ((s the s))} {u u}_{j j} ((s the s)) {e e}^{- - {L L}_{ij ij} s the s} + + \frac{{h h}_{i i} ((s the s))}{{τ τ}_{i i} ((s the s)) {r r}_{i i}^{* *} ((s the s))} {y the y}_{i i} ((s the s)) + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} \frac{{g g}_{ij ij} {\overset{&OverBar; &OverBar;}{r r}}_{ij ij} ((s the s))}{{r r}_{i i}^{* *} ((s the s))} {u u}_{j j} ((s the s))$

$+ + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {Σ Σ}_{k k = = 11}^{{m m}_{i i} + + {n no}_{i i}} {β β}_{ij ij}^{k k} {e e}^{- - σ σ {z z}_{i i}^{k k}} {u u}_{j j} ((s the s)) {e e}^{σs σs} dσ dσ + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {Σ Σ}_{k k = = 11}^{{m m}_{i i} + + {n no}_{i i}} {β β}_{ij ij}^{k k} {e e}^{- - σ σ {z z}_{i i}^{k k}} {u u}_{j j} ((s the s)) {e e}^{σs σs} dσ dσ$

$- - {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} \frac{{\overset{&OverBar; &OverBar;}{φ φ}}_{ij ij} ((s the s))}{{r r}_{i i}^{* *} ((s the s))} {u u}_{j j} ((s the s)) {e e}^{- - (({L L}_{ij ij} - - {L L}_{il il})) s the s} + + \frac{{g g}_{ri the ri} {r r}_{ri the ri} {p p}^{* *}_{i i} ((s the s))}{{p p}_{ri the ri} {r r}^{* *}_{i i} ((s the s))} {&upsi; &upsi;}_{i i} ((s the s)) {e e}^{- - (({L L}_{ri the ri} - - {L L}_{il il})) s the s}$

3控制器参数化3 Controller parameterization

上面的控制器结构设计出来之后，要设计自适应控制方案来使得控制器可以在实际的过程中参数变化时也有很好的控制效果。首先要将控制器参数化。控制器可以写为After the above controller structure is designed, it is necessary to design an adaptive control scheme so that the controller can have a good control effect when the parameters change in the actual process. The first step is to parameterize the controller. The controller can be written as

${u u}_{i i} = = \frac{11}{{b b}_{ii i}} {{{Σ Σ}_{j j = = 11}^{n no} \frac{{k k}_{ij ij} ((p p))}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {u u}_{j j} ((t t - - {L L}_{ij ij})) + + \frac{{h h}_{i i} ((p p))}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {y the y}_{i i} ((t t)) + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} \frac{{g g}_{ij ij} {\overset{&OverBar; &OverBar;}{r r}}_{ij ij} ((p p))}{{r r}_{i i}^{* *} ((p p))} {u u}_{j j} ((t t))$

$+ + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {Σ Σ}_{k k = = 11}^{{m m}_{i i} + + {n no}_{i i}} {β β}_{ij ij}^{k k} {e e}^{- - σ σ {z z}_{i i}^{k k}} {u u}_{j j} ((t t + + σ σ)) d d σ σ + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {Σ Σ}_{k k = = 11}^{{m m}_{i i}} {\overset{&OverBar; &OverBar;}{β β}}_{ij ij}^{k k} {e e}^{- - σ σ {z z}_{i i}^{k k}} {u u}_{j j} ((t t + + σ σ)) dσ dσ$

$- - {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} \frac{{\overset{&OverBar; &OverBar;}{φ φ}}_{ij ij} ((p p))}{{r r}_{i i}^{* *} ((p p))} {u u}_{j j} ((t t - - (({L L}_{ij ij} - - {L L}_{il il})))) - - {Σ Σ}_{j j &NotEqual; &NotEqual; i i}^{n no} {b b}_{ij ij} {u u}_{j j} ((t t)) + + \frac{{g g}_{ri the ri} {r r}_{ri the ri} ((p p)) {p p}^{* *}_{i i} ((p p))}{{p p}_{ri the ri} ((p p)) {r r}^{* *}_{i i} ((p p))} {&upsi; &upsi;}_{i i} ((t t - - (({L L}_{ri the ri} - - {L L}_{il il}))))}}$

将多项式h_i(s)，k_ij(s)和

分别写为The polynomials h _i (s), k _ij (s) and

respectively written as

${h h}_{i i} ((s the s)) = = {h h}_{{n no}_{i i} - - 11}^{i i} {s the s}^{{n no}_{i i} - - 11} + + {h h}_{{n no}_{i i} - - 22}^{i i} {s the s}^{{n no}_{i i} - - 22} + + . . . . . . + + {h h}_{00}^{i i},,$ ${k k}_{ij ij} ((s the s)) = = {k k}_{{n no}_{i i} - - 22}^{ij ij} {s the s}^{{n no}_{i i} - - 22} + + {k k}_{{n no}_{i i} - - 33}^{ij ij} {s the s}^{{n no}_{i i} - - 33} + + . . . . . . + + {k k}_{00}^{ij ij},,$

${\overset{&OverBar; &OverBar;}{φ φ}}_{ij ij} ((s the s)) = = {q q}_{{m m}_{i i}}^{ij ij} {s the s}^{{m m}_{i i}} + + {q q}_{{m m}_{i i} - - 11}^{ij ij} {s the s}^{{m m}_{i i} - - 11} + + . . . . . . + + {q q}_{00}^{ij ij},, j j &NotEqual; &NotEqual; l l,,$

定义参数向量Define parameter vector

${θ θ}_{i i} = = \frac{11}{{g g}_{ii i}} [[{k k}_{{n no}_{i i} - - 22}^{i i 11},, . . . . . .,, {k k}_{00}^{i i 11},, . . . . . .,, {k k}_{{n no}_{i i} - - 22}^{in in},, . . . . . .,, {k k}_{00}^{in in},, {h h}_{{n no}_{i i} - - 11}^{i i},, . . . . . .,, {h h}_{00}^{i i},, {g g}_{i i 11},, . . . . . .,, {g g}_{i i,, ((l l - - 11))},, {g g}_{i i,, ((l l - - 11))},, . . . . . .,, {g g}_{in in},, {q q}_{{m m}_{i i}}^{i i 11},,$

$. . . . . .,, {q q}_{00}^{i i 11},, . . . . . .,, {q q}_{{m m}_{i i}}^{i i,, ((l l - - 11))},, . . . . . .,, {q q}_{00}^{i i,, ((l l - - 11))},, {q q}_{{m m}_{i i}}^{i i,, ((l l + + 11))},, . . . . . .,, {q q}_{00}^{i i,, ((l l + + 11))},, . . . . . .,, {q q}_{{m m}_{i i}}^{in in},, . . . . . .,, {q q}_{00}^{in in},, {b b}_{i i 11},, . . . . . .,,$

${{b b}_{i i,, i i - - 11},, {b b}_{i i,, i i + + 11},, . . . . . .,, {b b}_{in in},, 11]]}^{T T}$

${λ λ}_{ij ij} = = {Σ Σ}_{k k = = 11}^{{m m}_{i i} + + {n no}_{i i}} {β β}_{ij ij}^{k k} {e e}^{- - σ σ {z z}_{i i}^{k k}},, i i,, j j = = 1,2 1,2,, . . . . . .,, n no,,$

${ρ ρ}_{ij ij} = = {Σ Σ}_{k k = = 11}^{{m m}_{i i}} {\overset{&OverBar; &OverBar;}{β β}}_{ij ij}^{k k} {e e}^{- - σ σ {z z}_{i i}^{k k}},, j j &NotEqual; &NotEqual; l l$

信号向量signal vector

${ω ω}_{i i} ((t t)) = = [[\frac{{p p}^{{n no}_{i i} - - 22}}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {u u}_{11} ((t t - - {L L}_{i i 11})),, . . . . . .,, \frac{11}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {u u}_{11} ((t t - - {L L}_{i i 11})),, . . . . . .,, \frac{{p p}^{{n no}_{i i} - - 22}}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {u u}_{n no} ((t t - - {L L}_{in in})),, . . . . . .,,$

$\frac{11}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {u u}_{n no} ((t t - - {L L}_{in in})),, \frac{{p p}^{{n no}_{i i} - - 11}}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {y the y}_{i i} ((t t)),, . . . . . .,, \frac{11}{{τ τ}_{i i} ((p p)) {r r}_{i i}^{* *} ((p p))} {y the y}_{i i} ((t t)),, \frac{{\overset{&OverBar; &OverBar;}{r r}}_{i i 11} ((p p))}{{r r}_{i i}^{* *} ((p p))} {u u}_{11} ((t t)),, . . . . . .,,$

$\frac{{\overset{&OverBar; &OverBar;}{r r}}_{i i,, ((l l - - 11))} ((p p))}{{r r}_{i i}^{* *} ((p p))} {u u}_{l l - - 11} ((t t)),, \frac{{\overset{&OverBar; &OverBar;}{r r}}_{i i,, ((l l + + 11))} ((p p))}{{r r}_{i i}^{* *} ((p p))} {u u}_{l l + + 11} ((t t)),, . . . . . . \frac{{\overset{&OverBar; &OverBar;}{r r}}_{in in} ((p p))}{{r r}_{i i}^{* *} ((p p))} {u u}_{n no} ((t t)),, - - \frac{{p p}^{{m m}_{i i}}}{{r r}_{i i}^{* *} ((p p))} {u u}_{11} ((t t - - (({L L}_{i i 11} - - {L L}_{il il})))),, . . . . . .,,$

$- - \frac{11}{{r r}_{i i}^{* *} ((p p))} {u u}_{11} ((t t - - (({L L}_{i i 11} - - {L L}_{il il})))),, . . . . . .,, - - \frac{{p p}^{{m m}_{i i}}}{{r r}_{i i}^{* *} ((p p))} {u u}_{l l - - 11} ((t t - - (({L L}_{i i,, ((l l - - 11))} - - {L L}_{il il})))),, . . . . . .,,$

$- - \frac{11}{{r r}_{i i}^{* *} ((p p))} {u u}_{l l - - 11} ((t t - - (({L L}_{i i,, ((l l - - 11))} - - {L L}_{il il})))),, . . . . . .,, - - \frac{{p p}^{{m m}_{i i}}}{{r r}_{i i}^{* *} ((p p))} {u u}_{l l + + 11} ((t t - - (({L L}_{i i,, ((l l + + 11))} - - {L L}_{il il})))),, . . . . . .,,$

$- - \frac{11}{{r r}_{i i}^{* *} ((p p))} {u u}_{l l + + 11} ((t t - - (({L L}_{i i ((l l + + 11))} - - {L L}_{il il})))),, . . . . . .,, - - \frac{{p p}^{{m m}_{i i}}}{{r r}_{i i}^{* *} ((p p))} {u u}_{n no} ((t t - - (({L L}_{in in} - - {L L}_{il il})))),, . . . . . .,,$

$- - \frac{11}{{r r}_{i i}^{* *} ((p p))} {u u}_{n no} ((t t - - (({L L}_{in in} - - {L L}_{il il})))),, - - {u u}_{11} ((t t)),, . . . . . .,, - - {u u}_{i i - - 11} ((t t)),, - - {u u}_{i i + + 11} ((t t)) . . . . . .,, - - {u u}_{n no} ((t t)),, {\overset{&OverBar; &OverBar;}{&upsi; &upsi;}}_{i i} ((t t - - {L L}_{il il})) {]]}^{T T}$

则控制器的参数化表达式为Then the parameterized expression of the controller is

${u u}_{i i} ((t t)) = = {\overset{^^}{θ θ}}_{i i}^{T T} ((t t)) {ω ω}_{i i} ((t t)) + + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {\overset{^^}{λ λ}}_{ij ij} ((t t,, σ σ)) {u u}_{j j} ((t t + + σ σ)) dσ dσ + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {\overset{^^}{ρ ρ}}_{ij ij} ((t t,, σ σ)) {u u}_{j j} ((t t + + σ σ)) dσ dσ$

其中

分别是θ_i，λ_ij，ρ_ij的估计。in

are estimates of θ _i , λ _ij , and ρ _{ij ,} respectively.

4计算误差表达式4 Calculate the error expression

定义误差为Define the error as

e_i(t)＝y_i(t)-y_ri(t)(i＝1，2，…，n)e _i (t) = y _i (t) - y _ri (t) (i = 1, 2, ..., n)

计算得到误差的表达式为The expression to calculate the error is

${e e}_{i i} ((t t)) = = {b b}_{ii i} \frac{{r r}_{i i}^{* *} ((p p))}{{p p}_{i i}^{* *} ((p p))} {q q}^{- - {L L}_{il il}} {{{\overset{~ ~}{θ θ}}_{i i}^{T T} ((t t)) {ω ω}_{i i} ((t t)) + + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {\overset{~ ~}{λ λ}}_{ij ij} ((t t,, σ σ)) {u u}_{j j} ((t t + + σ σ)) dσ dσ + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {\overset{~ ~}{ρ ρ}}_{ij ij} ((t t,, σ σ)) {u u}_{j j} ((t t + + σ σ)) dσ dσ}}$

其中为

是时滞算子，即

which is

is the delay operator, that is,

为了设计自适应律来估计控制器中的参数，需要利用增广误差。定义In order to design an adaptive law to estimate the parameters in the controller, the augmented error needs to be exploited. definition

${η η}_{i i} ((t t)) = = {{{\overset{~ ~}{θ θ}}_{i i}^{T T} ((t t)) {\overset{&OverBar; &OverBar;}{ω ω}}_{i i} ((t t)) + + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {\overset{~ ~}{λ λ}}_{ij ij} ((t t,, σ σ)) {\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t + + σ σ)) dσ dσ + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {\overset{~ ~}{ρ ρ}}_{ij ij} ((t t,, σ σ)) {\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t + + σ σ)) dσ dσ}}$

$- - \frac{{r r}_{i i}^{* *} ((p p))}{{p p}_{i i}^{* *} ((p p))} {q q}^{- - {L L}_{il il}} {{{\overset{~ ~}{θ θ}}_{i i}^{T T} ((t t)) {ω ω}_{i i} ((t t)) + + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {\overset{~ ~}{λ λ}}_{ij ij} ((t t,, σ σ)) {u u}_{j j} ((t t + + σ σ)) dσ dσ$

$+ + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{n no} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {\overset{~ ~}{ρ ρ}}_{ij ij} ((t t,, σ σ)) {u u}_{j j} ((t t + + σ σ)) dσ dσ}}$

其中in

${\overset{&OverBar; &OverBar;}{ω ω}}_{i i} ((t t)) = = \frac{{r r}_{i i}^{* *} ((p p))}{{p p}_{i i}^{* *} ((p p))} {q q}^{- - {L L}_{il il}} {ω ω}_{i i} ((t t)),,$ ${\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t)) = = \frac{{r r}_{i i}^{* *} ((p p))}{{p p}_{i i}^{* *} ((p p))} {q q}^{- - {L L}_{ij ij}} {u u}_{j j} ((t t)),, j j = = 1,2 1,2 . . . . . .,, n no$

定义增广误差为Define the augmented error as

${ϵ ϵ}_{i i} ((t t)) = = {e e}_{i i} ((t t)) + + {\overset{^^}{g g}}_{ii i} ((t t)) {η η}_{i i} ((t t))$

$= = {g g}_{ii i} {{{\overset{~ ~}{θ θ}}_{i i}^{T T} ((t t)) {\overset{&OverBar; &OverBar;}{ω ω}}_{i i} ((t t)) + + {Σ Σ}_{j j = = 11}^{n no} {&Integral; &Integral;}_{- - {L L}_{ij ij}}^{- - (({L L}_{ij ij} - - {L L}_{il il}))} {\overset{~ ~}{λ λ}}_{ij ij} ((t t,, σ σ)) {\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t + + σ σ)) dσ dσ + + {Σ Σ}_{j j &NotEqual; &NotEqual; l l}^{N N} {&Integral; &Integral;}_{- - (({L L}_{ij ij} - - {L L}_{il il}))}^{00} {\overset{~ ~}{ρ ρ}}_{ij ij} ((t t,, σ σ)) {\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t + + σ σ)) dσ dσ}} + + {\overset{~ ~}{g g}}_{ii i} ((t t)) {η η}_{i i} ((t t))$

其中

是g_ii的估计。in

is an estimate of g _ii .

5根据误差表达式选择自适应律5 Select the adaptive law according to the error expression

定义一个信号向量define a signal vector

${Ω Ω}_{i i} ((t t)) = = [[{\overset{&OverBar; &OverBar;}{ω ω}}_{i i} ((t t)),, \underset{- - {L L}_{i i 11} \leq \leq σ σ \leq \leq - - (({L L}_{i i 11} - - {L L}_{il il}))}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{11} ((t t + + σ σ)),, . . . . . . \underset{- - {L L}_{i i,, ((l l - - 11))} \leq \leq σ σ \leq \leq - - (({L L}_{i i,, ((l l - - 11))} - - {L L}_{il il}))}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{l l - - 11} ((t t + + σ σ)),, \underset{- - {L L}_{il il} \leq \leq σ σ \leq \leq 00}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{l l} ((t t + + σ σ)),,$

$\underset{- - {L L}_{i i,, ((l l + + 11))} \leq \leq σ σ \leq \leq - - (({L L}_{i i,, ((l l + + 11))} - - {L L}_{il il}))}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{l l + + 11} ((t t + + σ σ)),, . . . . . . \underset{- - {L L}_{in in} \leq \leq σ σ \leq \leq - - (({L L}_{in in} - - {L L}_{il il}))}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{n no} ((t t + + σ σ)),, \underset{- - (({L L}_{i i,, ((l l - - 11))} - - {L L}_{il il})) \leq \leq σ σ \leq \leq 00}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{11} ((t t + + σ σ)),, . . . . . .$

$\underset{- - {L L}_{i i,, ((l l - - 11))} \leq \leq σ σ \leq \leq 00}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{l l - - 11} ((t t + + σ σ)),, \underset{- - (({L L}_{i i,, ((l l + + 11))} - - {L L}_{il il})) \leq \leq σ σ \leq \leq 00}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{l l + + 11} ((t t + + σ σ)),, . . . . . . \underset{- - (({L L}_{in in} - - {L L}_{il il})) \leq \leq σ σ \leq \leq 00}{sup sup} {\overset{&OverBar; &OverBar;}{u u}}_{n no} ((t t + + σ σ)),, {η η}_{i i} ((t t)) {]]}^{T T}$

选择自适应律Choose an Adaptive Law

${\overset{. .}{\overset{^^}{θ θ}}}_{i i} ((t t)) = = - - {τ τ}_{{θ θ}_{i i}} \frac{{\overset{&OverBar; &OverBar;}{ω ω}}_{i i} ((t t)) {ϵ ϵ}_{i i} ((t t))}{11 + + {| | | | {Ω Ω}_{i i} ((t t)) | | | |}^{22}},, i i = = 1,2 1,2,, . . . . . .,, n no$

${\overset{. .}{\overset{^^}{g g}}}_{ii i} ((t t)) = = - - {τ τ}_{{g g}_{ii i}} \frac{{η η}_{i i} ((t t)) {ϵ ϵ}_{i i} ((t t))}{11 + + {| | | | {Ω Ω}_{i i} ((t t)) | | | |}^{22}},, i i = = 1,2 1,2,, . . . . . .,, n no$

${\overset{. .}{\overset{^^}{λ λ}}}_{ij ij} ((t t,, σ σ)) = = - - {τ τ}_{{λ λ}_{ij ij}} \frac{{\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t + + σ σ))}{11 + + {| | | | {Ω Ω}_{i i} ((t t)) | | | |}^{22}} {ϵ ϵ}_{i i} ((t t)),, - - {L L}_{ij ij} \leq \leq σ σ \leq \leq - - (({L L}_{ij ij} - - {L L}_{il il})),, j j = = 1,2 1,2,, . . . . . .,, n no$

${\overset{. .}{\overset{^^}{ρ ρ}}}_{ij ij} ((t t,, σ σ)) = = - - {τ τ}_{{ρ ρ}_{ij ij}} \frac{{\overset{&OverBar; &OverBar;}{u u}}_{j j} ((t t + + σ σ))}{11 + + {| | | | {Ω Ω}_{i i} ((t t)) | | | |}^{22}} {ϵ ϵ}_{i i} ((t t)),, - - (({L L}_{ij ij} - - {L L}_{il il})) \leq \leq σ σ \leq \leq 00,, j j &NotEqual; &NotEqual; l l$

其中

是可调参数。通过调整这些参数，来得到控制器中的参数，从而来控制系统。由此自适应控制方案设计完成，即它是由控制器和自适应律所组成。in

is an adjustable parameter. By adjusting these parameters, the parameters in the controller are obtained to control the system. Thus the design of the adaptive control scheme is completed, that is, it is composed of the controller and the adaptive law.

Claims

1. an adaptive control method of a multivariable time-delay process, characterized in that:

Skew compensator design;

Solving polynomial equations;

Nominal system model matching controller design;

Controller parameterization;

Choose an adaptive law based on the error expression.

2. The method according to claim 1, characterized in that we design a time lag compensator for compensating multiple time lags in a multivariable time lag process.

3. The method according to claim 1, characterized in that solving polynomial equations to obtain their solutions.

4. The method according to claim 1, characterized in that a model matching controller of the nominal system is designed, which can make the transfer function of the closed-loop system equal to the transfer function of the reference model.

5. The method according to claim 1, characterized in that, parameterizing the obtained model matching controller to obtain a parameterized expression.

6. The method according to claim 1, characterized in that, using the parameterized expression of the controller, the parameterized expression of the error is calculated, so as to select the adaptive law of the unknown parameter.