﻿ 大时滞系统的无模型控制方法及应用
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (12): 1987-1993  DOI: 10.11990/jheu.201707002 0

### 引用本文

DONG Na, CHANG Jianfang, HAN Xueshuo, et al. Model-free control method and its application for large time-delay systems[J]. Journal of Harbin Engineering University, 2018, 39(12), 1987-1993. DOI: 10.11990/jheu.201707002.

### 文章历史

Model-free control method and its application for large time-delay systems
DONG Na , CHANG Jianfang , HAN Xueshuo , WU Aiguo
School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
Abstract: To address the large time delay characteristics of a system in actual application, a model-free adaptive control method with input rate constraint of time delay was designed to alleviate the impact of large time delay on the entire control process. A typical system with large time delay was simulated and compared. Then, the model-free control scheme was designed for a compressed refrigeration system to control the characteristics of its nonlinearity, large time delay, strong coupling, and difficulty in establishing accurate mathematical control model. Then, the control scheme was simulated and compared. The simulation results show that the new model-free adaptive control algorithm has excellent control performance and faster response speed, which improves the performance of the refrigeration system; thus, the effectiveness of this novel control method is validated.
Keywords: large time delay    nonlinear    model-free control    lag time constant    compression refrigeration system    dynamic response

1 无模型自适应控制算法

 $\left\{ \begin{array}{l} y\left( {k + 1} \right) - y\left( k \right) = \varphi \left( k \right)\left( {u\left( k \right) - u\left( {k - 1} \right)} \right)\\ \Delta y\left( {k + 1} \right) = \varphi \left( k \right)\Delta u\left( k \right) \end{array} \right.$ (1)

 $\begin{array}{*{20}{c}} {u\left( k \right) = u\left( {k - 1} \right) + }\\ {\frac{{\rho \varphi \left( k \right)}}{{\lambda + {{\left\| {\varphi \left( k \right)} \right\|}^2}}}\left[ {{y^ * }\left( {k + 1} \right) - y\left( k \right)} \right]} \end{array}$ (2)

$\hat \varphi \left( k \right) = \hat \varphi \left( 1 \right)$，若$\hat {\mathit{\Phi}} \left( k \right) \le \varepsilon$或Δu(k-1)≤ε，则：

 $\begin{array}{*{20}{c}} {\hat \varphi \left( k \right) = \hat \varphi \left( {k - 1} \right) + \frac{{\eta \Delta u\left( {k - 1} \right)}}{{\mu + {{\left\| {\Delta u\left( {k - 1} \right)} \right\|}^2}}} \cdot }\\ {\left[ {\Delta y\left( k \right) - \hat \varphi \left( {k - 1} \right)\Delta u\left( {k - 1} \right)} \right]} \end{array}$ (3)

2 大时滞系统的无模型控制算法 2.1 单输入单输出时滞系统的无模型自适应控制算法

 $\begin{array}{*{20}{c}} {\min J\left[ {u\left( k \right)} \right] = {{\left[ {{y^ * }\left( {k + 1} \right) - y\left( {k + 1} \right)} \right]}^2} + }\\ {\eta {{\left[ {\frac{{u\left( k \right) - u\left( {k - 1 - \tau } \right)}}{T}} \right]}^2} + }\\ {\lambda {{\left[ {u\left( k \right) - u\left( {k - 1} \right)} \right]}^2}} \end{array}$ (4)

 $\begin{array}{*{20}{c}} {\min J\left[ {\varphi \left( k \right)} \right] = \left[ {{y^ * }\left( k \right) - y\left( {k - 1} \right) - } \right.}\\ {{{\left. {\varphi \left( k \right)\left[ {u\left( {k - 1} \right) - u\left( {k - 2 - \tau } \right)} \right]} \right]}^2} + }\\ {\mu {{\left[ {\varphi \left( k \right) - \varphi \left( {k - 1} \right)} \right]}^2}} \end{array}$ (5)

 $\begin{array}{*{20}{c}} {u\left( k \right) = u\left( {k - 1} \right) + \frac{{\rho \varphi \left( k \right)}}{{\lambda + {{\left\| {\varphi \left( k \right)} \right\|}^2} + \frac{\eta }{{{T^2}}}}} \cdot }\\ {\left[ {{y^ * }\left( {k + 1} \right) - y\left( k \right)} \right] + \frac{{\left( {\eta /{T^2}} \right)}}{{\lambda + {{\left\| {\varphi \left( k \right)} \right\|}^2} + \frac{\eta }{{{T^2}}}}} \cdot }\\ {\left[ {u\left( {k - 1 - \tau } \right) - u\left( {k - 1} \right)} \right]} \end{array}$ (6)

φ(k)=φ(1)，若$\hat {\mathit{\Phi}} \left( k \right) \le \varepsilon$或Δu(k-1)≤ε，则：

 $\begin{array}{*{20}{c}} {\varphi \left( k \right) = \varphi \left( {k - 1} \right) + }\\ {\frac{{\xi \left[ {u\left( {k - 1} \right) - u\left( {k - 2 - \tau } \right)} \right]}}{{\mu + {{\left\| {u\left( {k - 1} \right) - u\left( {k - 2 - \tau } \right)} \right\|}^2}}} \cdot }\\ {\left\{ {\Delta y\left( k \right) - \varphi \left( {k - 1} \right)\left[ {u\left( {k - 1} \right) - u\left( {k - 2 - \tau } \right)} \right]} \right\}} \end{array}$ (7)

2.2 多输入多输出时滞系统的无模型自适应控制算法

 $\begin{array}{*{20}{c}} {\left[ \begin{array}{l} {y_1}\left( {k + 1} \right) - {y_1}\left( k \right)\\ {y_2}\left( {k + 1} \right) - {y_2}\left( k \right) \end{array} \right] = }\\ {\left[ {\begin{array}{*{20}{c}} {{\varphi _{11}}\left( k \right)}&{{\varphi _{12}}\left( k \right)}\\ {{\varphi _{21}}\left( k \right)}&{{\varphi _{22}}\left( k \right)} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_1}\left( k \right) - {u_1}\left( {k - 1} \right)}\\ {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \end{array}} \right]} \end{array}$ (8)

 $\begin{array}{*{20}{c}} {\min J\left[ {\mathit{\boldsymbol{U}}\left( k \right)} \right] = {{\left[ {y_1^ * \left( {k + 1} \right) - {y_1}\left( {k + 1} \right)} \right]}^2} + }\\ {{{\left[ {y_2^ * \left( {k + 1} \right) - {y_2}\left( {k + 1} \right)} \right]}^2} + }\\ {{\lambda _1}{{\left[ {{u_1}\left( k \right) - {u_1}\left( {k - 1} \right)} \right]}^2} + }\\ {{\lambda _2}{{\left[ {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \right]}^2}} \end{array}$ (9)

 $\begin{array}{*{20}{c}} {\min J\left[ {\mathit{\boldsymbol{U}}\left( k \right)} \right] = {{\left[ {y_1^ * \left( {k + 1} \right) - {y_1}\left( {k + 1} \right)} \right]}^2} + }\\ {{{\left[ {y_2^ * \left( {k + 1} \right) - {y_2}\left( {k + 1} \right)} \right]}^2} + {\lambda _1}\left[ {{u_1}\left( k \right) - } \right.}\\ {{{\left. {{u_1}\left( {k - 1} \right)} \right]}^2} + {\lambda _2}{{\left[ {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \right]}^2} + }\\ {{\eta _1}{{\left[ {\left( {{u_1}\left( k \right) - {u_1}\left( {k - 1 - \tau } \right)} \right)/T} \right]}^2} + }\\ {{\eta _2}{{\left[ {\left( {{u_2}\left( k \right) - {u_2}\left( {k - 1 - \tau } \right)} \right)/T} \right]}^2}} \end{array}$ (10)

 $\begin{array}{*{20}{c}} {\min \frac{{\partial J\left[ {U\left( k \right)} \right]}}{{\partial {u_1}\left( k \right)}} = 2\left[ {y_1^ * \left( {k + 1} \right) - {y_1}\left( k \right) - } \right.}\\ {{\varphi _{11}}\left( k \right)\left( {{u_1}\left( k \right) - {u_1}\left( {k - 1} \right)} \right) - {\varphi _{12}}\left( k \right)\left( {{u_2}\left( k \right) - } \right.}\\ {\left. {\left. {{u_2}\left( {k - 1} \right)} \right)} \right]\left( { - {\varphi _{11}}\left( k \right)} \right) + 2\left[ {y_2^ * \left( {k + 1} \right) - } \right.}\\ {{y_2}\left( k \right) - {\varphi _{21}}\left( k \right)\left( {{u_1}\left( k \right) - {u_1}\left( {k - 1} \right)} \right) - }\\ {\left. {{\varphi _{22}}\left( k \right)\left( {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \right)} \right]\left( { - {\varphi _{21}}\left( k \right)} \right) + }\\ {2{\lambda _1}\left[ {{u_1}\left( k \right) - {u_1}\left( {k - 1} \right)} \right] + }\\ {2{\eta _1}\left[ {\left( {{u_1}\left( k \right) - {u_1}\left( {k - 1 - \tau } \right)} \right)/T} \right]/{T^2}} \end{array}$ (11)
 $\begin{array}{*{20}{c}} {\min \frac{{\partial J\left[ {U\left( k \right)} \right]}}{{\partial {u_2}\left( k \right)}} = 2\left[ {y_1^ * \left( {k + 1} \right) - {y_1}\left( k \right) - } \right.}\\ {{\varphi _{11}}\left( k \right)\left( {{u_1}\left( k \right) - {u_1}\left( {k - 1} \right)} \right) - }\\ {\left. {{\varphi _{12}}\left( k \right)\left( {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \right)} \right]\left( { - {\varphi _{12}}\left( k \right)} \right) + }\\ {2\left[ {y_2^ * \left( {k + 1} \right) - {y_2}\left( k \right) - {\varphi _{21}}\left( k \right)\left( {{u_1}\left( k \right) - } \right.} \right.}\\ {\left. {\left. {{u_1}\left( {k - 1} \right)} \right) - {\varphi _{22}}\left( k \right)\left( {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \right)} \right] \cdot }\\ {\left( { - {\varphi _{22}}\left( k \right)} \right) + 2{\lambda _2}\left[ {{u_2}\left( k \right) - {u_2}\left( {k - 1} \right)} \right] + }\\ {2{\eta _2}\left[ {\left( {{u_2}\left( k \right) - {u_2}\left( {k - 1 - \tau } \right)} \right)/T} \right]/{T^2}} \end{array}$ (12)

 $\begin{array}{*{20}{c}} {\left[ {\varphi _{11}^2\left( k \right) + \varphi _{21}^2\left( k \right) + {\lambda _1} + \frac{{{\eta _1}}}{{{T^2}}}} \right]{u_1}\left( k \right) + }\\ {\left[ {{\varphi _{11}}\left( k \right){\varphi _{12}}\left( k \right) + {\varphi _{21}}\left( k \right){\varphi _{22}}\left( k \right)} \right]{u_2}\left( k \right) = }\\ {\frac{{{\eta _1}{u_1}\left( {k - 1 - \tau } \right)}}{{{T^2}}} + {\lambda _1}{u_1}\left( {k - 1} \right) + {\varphi _{21}}\left( k \right) \cdot }\\ {\left[ {y_2^ * \left( {k + 1} \right) - {y_2}\left( k \right) + {\varphi _{21}}\left( k \right){u_1}\left( {k - 1} \right) + } \right.}\\ {\left. {{\varphi _{22}}\left( k \right){u_2}\left( {k - 1} \right)} \right] + {\varphi _{11}}\left( k \right)\left[ {y_1^ * \left( {k + 1} \right) - } \right.}\\ {\left. {{y_1}\left( k \right) + {\varphi _{11}}\left( k \right){u_1}\left( {k - 1} \right) + {\varphi _{12}}\left( k \right){u_2}\left( {k - 1} \right)} \right]} \end{array}$ (13)
 $\begin{array}{*{20}{c}} {\left[ {{\varphi _{11}}\left( k \right){\varphi _{12}}\left( k \right) + {\varphi _{21}}\left( k \right){\varphi _{22}}\left( k \right)} \right]{u_1}\left( k \right) + }\\ {\left[ {\varphi _{12}^2\left( k \right) + \varphi _{22}^2\left( k \right) + {\lambda _2} + \frac{{{\eta _2}}}{{{T^2}}}} \right]{u_2}\left( k \right) = }\\ {\frac{{{\eta _2}{u_2}\left( {k - 1 - \tau } \right)}}{{{T^2}}} + {\lambda _2}{u_2}\left( {k - 1} \right) + {\varphi _{22}}\left( k \right) \cdot }\\ {\left[ {y_2^ * \left( {k + 1} \right) - {y_2}\left( k \right) + {\varphi _{21}}\left( k \right){u_1}\left( {k - 1} \right) + } \right.}\\ {\left. {{\varphi _{22}}\left( k \right){u_2}\left( {k - 1} \right)} \right] + {\varphi _{12}}\left( k \right)\left[ {y_1^ * \left( {k + 1} \right) - } \right.}\\ {\left. {{y_1}\left( k \right) + {\varphi _{11}}\left( k \right){u_1}\left( {k - 1} \right) + {\varphi _{12}}\left( k \right){u_2}\left( {k - 1} \right)} \right]} \end{array}$ (14)

φ(k)是泛模型的特征参量。一方面，希望实际输出与泛模型输出之间的偏差平方和最小，使得泛模型近似于实际模型；另一方面，采用泛模型代替非线性系统时，相邻采样时刻的Δφ(k)不能变化太大，故文中使用下式(15)所表示的目标函数：

 $\begin{array}{*{20}{c}} {\min J\left[ {\varphi \left( k \right)} \right] = {{\left[ {y_1^ * \left( k \right) - {y_1}\left( k \right)} \right]}^2} + }\\ {{{\left[ {y_2^ * \left( k \right) - {y_2}\left( k \right)} \right]}^2} + {\mu _{11}}{{\left[ {{\varphi _{11}}\left( k \right) - {\varphi _{11}}\left( {k - 1} \right)} \right]}^2} + }\\ {{\mu _{12}}{{\left[ {{\varphi _{12}}\left( k \right) - {\varphi _{12}}\left( {k - 1} \right)} \right]}^2} + {\mu _{21}}\left[ {{\varphi _{21}}\left( k \right) - } \right.}\\ {{{\left. {{\varphi _{21}}\left( {k - 1} \right)} \right]}^2} + {\mu _{22}}{{\left[ {{\varphi _{22}}\left( k \right) - {\varphi _{22}}\left( {k - 1} \right)} \right]}^2}} \end{array}$ (15)

 $\begin{array}{*{20}{c}} {\min J\left[ {\varphi \left( k \right)} \right] = \left[ {y_1^ * \left( k \right) - {y_1}\left( {k - 1} \right) - } \right.}\\ {{\varphi _{11}}\left( {k - 1} \right)\left[ {{u_1}\left( {k - 1} \right) - {u_1}\left( {k - 2 - \tau } \right)} \right] - }\\ {{{\left. {{\varphi _{12}}\left( {k - 1} \right)\left[ {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right]} \right]}^2} + }\\ {\left[ {y_2^ * \left( k \right) - {y_2}\left( {k - 1} \right) - {\varphi _{21}}\left( {k - 1} \right)\left[ {{u_1}\left( {k - 1} \right) - } \right.} \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right] - {\varphi _{22}}\left( {k - 1} \right)\left[ {{u_2}\left( {k - 1} \right) - } \right.}\\ {{{\left. {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right]} \right]}^2} + {\mu _{11}}{{\left[ {{\varphi _u}\left( k \right) - {\varphi _{11}}\left( {k - 1} \right)} \right]}^2} + }\\ {{\mu _{12}}{{\left[ {{\varphi _{12}}\left( k \right) - {\varphi _{12}}\left( {k - 1} \right)} \right]}^2} + {\mu _{21}}\left[ {{\varphi _{21}}\left( k \right) - {\varphi _{21}}\left( {k - } \right.} \right.}\\ {{{\left. {\left. 1 \right)} \right]}^2} + {\mu _{22}}{{\left[ {{\varphi _{22}}\left( k \right) - {\varphi _{22}}\left( {k - 1} \right)} \right]}^2}} \end{array}$ (16)

 $\begin{array}{*{20}{c}} {\min \frac{{\partial J\left[ {\varphi \left( k \right)} \right]}}{{\partial {\varphi _{11}}\left( k \right)}} = - 2\left[ {y_1^ * \left( k \right) - {y_1}\left( {k - 1} \right) - } \right.}\\ {\left( {{\varphi _{11}}\left( k \right) - \Delta {\varphi _{11}}\left( k \right)} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - \left( {{\varphi _{12}}\left( k \right) - \Delta {\varphi _{12}}\left( k \right)} \right) \cdot }\\ {\left. {\left( {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right)} \right]\left[ {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right] + 2{\mu _{11}}\left[ {{\varphi _{11}}\left( k \right) - {\varphi _{11}}\left( {k - 1} \right)} \right]} \end{array}$ (17)
 $\begin{array}{*{20}{c}} {\min \frac{{\partial J\left[ {\varphi \left( k \right)} \right]}}{{\partial {\varphi _{12}}\left( k \right)}} = - 2\left[ {y_1^ * \left( k \right) - {y_1}\left( {k - 1} \right) - } \right.}\\ {\left( {{\varphi _{11}}\left( k \right) - \Delta {\varphi _{11}}\left( k \right)} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - \left( {{\varphi _{12}}\left( k \right) - \Delta {\varphi _{12}}\left( k \right)} \right) \cdot }\\ {\left. {\left( {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right)} \right]\left[ {{u_2}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right] + 2{\mu _{12}}\left[ {{\varphi _{12}}\left( k \right) - {\varphi _{12}}\left( {k - 1} \right)} \right]} \end{array}$ (18)
 $\begin{array}{*{20}{c}} {\min \frac{{\partial J\left[ {\varphi \left( k \right)} \right]}}{{\partial {\varphi _{21}}\left( k \right)}} = - 2\left[ {y_2^ * \left( k \right) - {y_2}\left( {k - 1} \right) - } \right.}\\ {\left( {{\varphi _{21}}\left( k \right) - \Delta {\varphi _{21}}\left( k \right)} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - \left( {{\varphi _{22}}\left( k \right) - \Delta {\varphi _{22}}\left( k \right)} \right) \cdot }\\ {\left. {\left( {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right)} \right]\left[ {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right] + 2{\mu _{21}}\left[ {{\varphi _{21}}\left( k \right) - {\varphi _{21}}\left( {k - 1} \right)} \right]} \end{array}$ (19)
 $\begin{array}{*{20}{c}} {\min \frac{{\partial J\left[ {\varphi \left( k \right)} \right]}}{{\partial {\varphi _{22}}\left( k \right)}} = - 2\left[ {y_2^ * \left( k \right) - {y_2}\left( {k - 1} \right) - } \right.}\\ {\left( {{\varphi _{21}}\left( k \right) - \Delta {\varphi _{21}}\left( k \right)} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - \left( {{\varphi _{22}}\left( k \right) - \Delta {\varphi _{22}}\left( k \right)} \right) \cdot }\\ {\left. {\left( {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right)} \right]\left[ {{u_2}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right] + 2{\mu _{22}}\left[ {{\varphi _{22}}\left( k \right) - {\varphi _{22}}\left( {k - 1} \right)} \right]} \end{array}$ (20)

 $\begin{array}{*{20}{c}} {{\varphi _{11}}\left( k \right) = {\varphi _{11}}\left( {k - 1} \right) + }\\ {\frac{{{u_1}\left( {k - 1} \right) - {u_1}\left( {k - 2 - \tau } \right)}}{{{\mu _{11}} + {{\left\| {{u_1}\left( {k - 1} \right) - {u_1}\left( {k - 2 - \tau } \right)} \right\|}^2}}}\left[ {y_1^ * - } \right.}\\ {{y_1}\left( {k - 1} \right) - {\varphi _{11}}\left( {k - 1} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - {\varphi _{12}}\left( {k - 1} \right)\left( {{u_2}\left( {k - 1} \right) - } \right.}\\ {\left. {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right)} \right]} \end{array}$ (21)
 $\begin{array}{*{20}{c}} {{\varphi _{12}}\left( k \right) = {\varphi _{12}}\left( {k - 1} \right) + }\\ {\frac{{{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)}}{{{\mu _{12}} + {{\left\| {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right\|}^2}}}\left[ {y_1^ * - } \right.}\\ {{y_1}\left( {k - 1} \right) - {\varphi _{11}}\left( {k - 1} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - {\varphi _{12}}\left( {k - 1} \right)\left( {{u_2}\left( {k - 1} \right) - } \right.}\\ {\left. {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right)} \right]} \end{array}$ (22)
 $\begin{array}{*{20}{c}} {{\varphi _{21}}\left( k \right) = {\varphi _{21}}\left( {k - 1} \right) + }\\ {\frac{{{u_1}\left( {k - 1} \right) - {u_1}\left( {k - 2 - \tau } \right)}}{{{\mu _{21}} + {{\left\| {{u_1}\left( {k - 1} \right) - {u_1}\left( {k - 2 - \tau } \right)} \right\|}^2}}}\left[ {y_2^ * - } \right.}\\ {{y_2}\left( {k - 1} \right) - {\varphi _{21}}\left( {k - 1} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - {\varphi _{22}}\left( {k - 1} \right)\left( {{u_2}\left( {k - 1} \right) - } \right.}\\ {\left. {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right)} \right]} \end{array}$ (23)
 $\begin{array}{*{20}{c}} {{\varphi _{22}}\left( k \right) = {\varphi _{22}}\left( {k - 1} \right) + }\\ {\frac{{{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)}}{{{\mu _{22}} + {{\left\| {{u_2}\left( {k - 1} \right) - {u_2}\left( {k - 2 - \tau } \right)} \right\|}^2}}}\left[ {y_2^ * - } \right.}\\ {{y_2}\left( {k - 1} \right) - {\varphi _{21}}\left( {k - 1} \right)\left( {{u_1}\left( {k - 1} \right) - } \right.}\\ {\left. {{u_1}\left( {k - 2 - \tau } \right)} \right) - {\varphi _{22}}\left( {k - 1} \right)\left( {{u_2}\left( {k - 1} \right) - } \right.}\\ {\left. {\left. {{u_2}\left( {k - 2 - \tau } \right)} \right)} \right]} \end{array}$ (24)

3 仿真研究

3.1 一阶大时滞系统仿真

 $G\left( s \right) = \frac{1}{{40s + 1}}{{\rm{e}}^{ - {\rm{20s}}}}$ (25)

 Download: 图 1 一阶大时滞系统阶跃响应曲线 Fig. 1 The step response curves of the first-order large-delay system
3.2 多输入多输出大时滞系统仿真

 Download: 图 2 基本MFAC算法阶跃响应曲线 Fig. 2 The step response curves of basic MFAC algorithm
 Download: 图 3 本文算法阶跃响应曲线 Fig. 3 Step response curve of this algorithm
 $\mathit{\boldsymbol{G}}\left( s \right) = \left[ {\begin{array}{*{20}{c}} {\frac{{12.8{{\rm{e}}^{ - s}}}}{{16.7s + 1}}}&{\frac{{ - 18.9{{\rm{e}}^{ - 3s}}}}{{21s + 1}}}\\ {\frac{{6.6{{\rm{e}}^{ - 7s}}}}{{10.9s + 1}}}&{\frac{{ - 19.4{{\rm{e}}^{ - 3s}}}}{{14.4s + 1}}} \end{array}} \right]$ (26)
4 新型无模型自适应控制方法在制冷系统中的应用 4.1 制冷系统模型辨识

 $\left[ {\begin{array}{*{20}{c}} {\Delta {T_s}}\\ {\Delta {T_e}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\frac{{ - 0.825}}{{38s + 1}}{{\rm{e}}^{ - 9s}}}&{\frac{{1.84}}{{35s + 1}}{{\rm{e}}^{ - 26s}}}\\ {\frac{{0.778}}{{35s + 1}}{{\rm{e}}^{ - 10s}}}&{\frac{{ - 0.752}}{{41s + 1}}{{\rm{e}}^{ - 28s}}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\Delta {v_a}}\\ {\Delta f} \end{array}} \right]$ (27)

4.2 制冷系统控制方案

 Download: 图 5 压缩式制冷系统无模型控制方案 Fig. 5 Model-free control scheme for compression refrigeration syste
4.3 制冷系统仿真实验

 Download: 图 6 仿真1的过热度和蒸发温度阶跃响应曲线 Fig. 6 The superheat and the evaporation temperature step response curves in simulation 1
 Download: 图 7 仿真2的过热度和蒸发温度阶跃响应曲线 Fig. 7 The superheat and the evaporation temperature step response curves in simulation 2

 Download: 图 8 制冷系统无模型控制阶跃响应 Fig. 8 Model-free control step response of refrigeration system
5 结论

1) 选取典型的单输入单输出和双输入双输出大时滞系统进行了仿真实验。仿真结果表明，本文所设计的控制算法具有较小的超调，能在更短的时间内取得稳定的跟踪结果，输出稳定且动态性能良好。

2) 将无模型自适应控制算法应用到压缩式制冷系统控制器设计中，采用制冷系统二阶模型进行了控制方案的仿真验证。仿真结果表明，本文设计算法控制下的过热度和蒸发温度能够对于给定信号实现快速稳定的跟踪，并且对于变化的给定信号也具有良好的跟踪能力，缓解了系统的耦合问题，验证了控制算法的有效性。

 [1] WANG Xiaofeng, LI Xing, WANG Jianhui, et al. Data-driven model-free adaptive sliding mode control for the multi degree-of-freedom robotic exoskeleton[J]. Information sciences, 2016, 327: 246-257. DOI:10.1016/j.ins.2015.08.025 (0) [2] GIGLIO S, MAGGIORI M, STROEBEL J. No-Bubble condition:model-free tests in housing markets[J]. Econometrica, 2016, 84(3): 1047-1091. DOI:10.3982/ECTA13447 (0) [3] KALDMÄE A, CALIFANO C, MOOG C H. Integrability for nonlinear time-delay systems[J]. IEEE transactions on automatic control, 2016, 61(7): 1912-1917. DOI:10.1109/TAC.2015.2482003 (0) [4] 金尚泰, 侯忠生. 一类非线性大滞后系统的改进无模型自适应控制[J]. 控制理论与应用, 2008, 25(4): 623-626. JIN Shangtai, HOU Zhongsheng. An improved model-free adaptive control for a class of nonlinear large-lag systems[J]. Control theory & applications, 2008, 25(4): 623-626. (0) [5] 程保华.制冷系统动态建模及优化控制方法研究[D].天津: 天津大学, 2013. CHENG Baohua. Dynamic modeling of refrigeration sys-tem and research on optimization control method[D]. Tianjin: Tianjin University, 2013. (0) [6] OH J S, BINNS M, PARK S, et al. Improving the energy efficiency of industrial refrigeration systems[J]. Energy, 2016, 112: 826-835. DOI:10.1016/j.energy.2016.06.119 (0) [7] CHEN Jianyong, JARALL S, HAVTUN H, et al. A review on versatile ejector applications in refrigeration systems[J]. Renewable and sustainable energy reviews, 2015, 49: 67-90. DOI:10.1016/j.rser.2015.04.073 (0) [8] HU Kaiyong, ZHU Jialing, ZHANG Wei, et al. Effects of evaporator superheat on system operation stability of an organic Rankine cycle[J]. Applied thermal engineering, 2016, 111: 793-801. (0) [9] ZHU Yuanming, HOU Zhongsheng. Data-driven MFAC for a class of discrete-time nonlinear systems with RBFNN[J]. IEEE transactions on neural networks and learning systems, 2014, 25(5): 1013-1020. DOI:10.1109/TNNLS.2013.2291792 (0) [10] 陈琛, 何小阳. 一种大时滞系统的无模型自适应控制改进算法[J]. 计算技术与自动化, 2012, 31(3): 10-13. CHEN Chen, HE Xiaoyang. An improved algorithm of model-free adaptive control for large time-delay system[J]. Computing technology and automation, 2012, 31(3): 10-13. DOI:10.3969/j.issn.1003-6199.2012.03.003 (0) [11] LIU Tao, ZHANG Weidong, GU Danying. Analytical multiloop PI/PID controller design for two-by-two processes with time delays[J]. Industrial & engineering chemistry research, 2005, 44(6): 1832-1841. (0)