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 哈尔滨工程大学学报  2018, Vol. 39 Issue (6): 1046-1051  DOI: 10.11990/jheu.201612044 0

### 引用本文

CHEN Zengqiang, WU Xia, SUN Mingwei, et al. Stability of active disturbance rejection generalized predictive control based on frequency domain[J]. Journal of Harbin Engineering University, 2018, 39(6), 1046-1051. DOI: 10.11990/jheu.201612044.

### 文章历史

1. 南开大学 计算机与控制工程学院, 天津 300350;
2. 天津市智能机器人重点实验室, 天津 300350

Stability of active disturbance rejection generalized predictive control based on frequency domain
CHEN Zengqiang1,2, WU Xia1, SUN Mingwei1, SUN Qinglin1
1. College of computer and control engineering, Nankai University, Tianjin 300350, China;
2. Key Lab of Intelligent Robotics of Tianjin, Tianjin 300350, China
Abstract: Generalized predictive control (GPC) is an advanced control algorithm that has been applied successfully in the process industry, but it has a large online computation burden. To overcome the drawback of GPC, a novel active disturbance rejection generalized predictive control (ADRC-GPC) algorithm was proposed. First, the principle of the ADRC-GPC was introduced. Second, the closed-loop feedback structure of the new algorithm was deduced based on a first-order nonlinear uncertain system, and its frequency domain characteristic was analyzed to verify the stability of the proposed method. Finally, the principles and laws for selecting the parameters of a controller were summarized by utilizing the Bode diagram. Results show that the proposed method could be used to obtain the general solutions of the Diophantine equations offline in comparison with traditional GPC. In addition, identifying the parameters of the controlled objects is unnecessary, so the online calculation burden can be reduced greatly. Compared with the β-GPC method, the proposed scheme has better dynamic performance and higher control precision.
Key words: generalized predictive control(GPC)    active disturbance rejection control(ADRC)    active disturbance rejection generalized predictive control    closed-loop feedback structure    frequency domain characteristics    stability

1 自抗扰广义预测控制算法的设计 1.1 自抗扰广义预测控制的结构图

1.2 自抗扰广义预测控制的原理

 $\dot y = g\left( {y, \dot y, t} \right) + w + b\tilde u$ (1)

 ${{\dot x}_1} = u$ (2)

 $A({z^{ - 1}})y\left( k \right) = B({z^{ - 1}})u\left( {k - 1} \right) + \zeta \left( k \right)/\Delta$ (3)

 $\left\{ \begin{array}{l} A({z^{ - 1}}) = 1 + {a_1}{z^{ - 1}} + \cdots + {a_{{n_a}}}{z^{ - n}}\\ B({z^{ - 1}}) = {b_0} + {b_1}{z^{ - 1}} + \cdots + {b_{{n_b}}}{z^{ - n}} \end{array} \right.$ (4)

 ${G_0}\left( s \right) = \frac{1}{s}$ (5)

 ${G_0}({z^{ - 1}}) = (1 - {z^{ - 1}}){\rm Z}\left[ {\frac{{{G_0}\left( s \right)}}{s}} \right] = {z^{ - 1}}\frac{T}{{1 - {z^{ - 1}}}}$ (6)

 $y\left( k \right) = {G_0}({z^{ - 1}})u\left( k \right)$ (7)

 ${G_0}({z^{ - 1}}) = {z^{ - 1}}\frac{{B({z^{ - 1}})}}{{A({z^{ - 1}})}}$ (8)

 $A({z^{ - 1}}) = 1 - {z^{ - 1}}, B({z^{ - 1}}) = T$ (9)

 $\left\{ \begin{array}{l} 1 = {E_j}({z^{ - 1}})A({z^{ - 1}})\Delta + {z^{ - j}}{F_j}({z^{ - 1}})\\ {E_j}({z^{ - 1}})B({z^{ - 1}}) = {G_j}({z^{ - 1}}) + {z^{ - j}}{H_j}({z^{ - 1}}) \end{array} \right.$ (10)

 $\begin{array}{l} {e_j} = j, \;\;f_1^j = j + 1, \;\;f_2^j = - j\\ \;\;\;\;{g_j} = jT, \;\;\;{H_j}({z^{ - 1}}) = 0 \end{array}$ (11)

 $\begin{array}{l} J = \sum\limits_{j = 1}^N {{{\left( {y\left( {k + j} \right) - w\left( {k + j} \right)} \right)}^2} + } \\ \;\;\;\;\;\;\;\sum\limits_{j = 1}^{{N_u}} {\lambda {{(\Delta u\left( {k + j - 1} \right))}^2}} \end{array}$ (12)

 $\begin{array}{l} \mathit{\boldsymbol{W}} = {\left[ {w\left( {k + 1} \right), \cdots , w\left( {k + N} \right)} \right]^{\rm{T}}} = \\ \;\;\;\;\;\;\;\;{\mathit{\boldsymbol{F}}_\alpha }y\left( k \right) + {{\mathit{\boldsymbol{\bar F}}}_\alpha }{y_r}\left( k \right) \end{array}$ (13)

 $\mathit{\boldsymbol{U}} = {({\mathit{\boldsymbol{G}}^{\rm{T}}}\mathit{\boldsymbol{G}} + \lambda \mathit{\boldsymbol{I}})^{ - 1}}{\mathit{\boldsymbol{G}}^{\rm{T}}}\left[ {\mathit{\boldsymbol{W}} - \mathit{\boldsymbol{Fy}}\left( k \right)} \right]$ (14)

U的第一个元素作为Δu(k)，则针对式(2)所示的积分器串联型系统所施加的控制律为

 $u\left( k \right) = u\left( {k - 1} \right) + \Delta u\left( k \right)$ (15)
2 自抗扰广义预测控制闭环离散形式

 $T({z^{ - 1}})\Delta u\left( k \right) = R{y_r}\left( k \right) - S({z^{ - 1}})y\left( k \right)$ (16)

 $y\left( k \right) = \frac{{{G_0}({z^{ - 1}})D({z^{ - 1}})}}{{1 + G({z^{ - 1}})H({z^{ - 1}})}}{y_r}\left( k \right)$ (17)

 $G\left( s \right) = \frac{{{b_0}{{(s + {\omega _o})}^2}}}{{T{b_0}{s^3} + (2{\omega _o}{b_0}T + 1){s^2} + (2{\omega _o}{b_0} + {\omega _o}^2{b_0})s}}$ (18)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;G({z^{ - 1}}) = \\ {\rm Z}\left[ {\frac{{1 - {{\rm{e}}^{ - \tau s}}}}{s} \cdot \frac{{{b_0}{{(s + {\omega _o})}^2}}}{{T{b_0}{s^3} + (2{\omega _o}{b_0}T + 1){s^2} + (2{\omega _o}{b_0} + {\omega _o}^2{b_0})s}}} \right] \end{array}$ (19)

 $1 + G({z^{ - 1}})H({z^{ - 1}}) = 0$ (20)

 $G({z^{ - 1}})H({z^{ - 1}}) = \frac{{{z^{ - 1}}B({z^{ - 1}})S({z^{ - 1}})}}{{A({z^{ - 1}})T({z^{ - 1}})\mathit{\Delta }}}$ (21)

 ${G_0}\left( s \right) = \frac{2}{{2s + 1}}$ (22)

3.1 观测器带宽ωo对系统性能的影响

ωo的值分别取2、5、8、12、15、20、30、40时，取N=10, α=0.2, λ=0.005, Nu=1，开环系统的Bode图如图 5所示。

3.2 N改变对系统性能的影响

N分别取值为5、8、10、15、20、30、40时，取ωo=10, α=0.2, λ=0.005, Nu=1。开环系统的Bode图曲线如图 6所示，相应的相角裕度和截止频率的值见表 1所示。

3.3 Nu改变对系统性能的影响

Nu的值分别取1、3、5、6、7、8、9时，其他可调参数不变，即N=10, ωo=1, λ=0.005, α=0.2。开环系统的Bode图如图 7所示。

3.4 控制加权常数λ的变化对系统性能的影响

λ的取值分别为1、0.5、0.1、0.05、0.01、0.005、0.001、0时, 为更明显看出λ的变化对系统的性能的影响，取N=10, α=0.2, ωo=10, Nu=2，采样周期T=0.1，开环系统的Bode图如图 8所示。

λ是用来限制控制增量Δu的剧烈变化，以防止对被控对象造成过大冲击。从图中可以看出，λ对系统的影响效果最明显。λ的减小使得系统的截止频率升高，响应速度加快，但是系统的相角裕度减小，稳定性降低，系统逐渐的开始出现超调，增大λ会使得相角裕度增加，可以实现无超调的控制，但是也会使得控制作用减弱。因此实际选择时λ取值较小。

3.5 柔化因子α的变化对系统性能的影响

α为柔化因子。从图 9可以看出，α的增加降低了系统的截止频率，减缓了系统的响应速度，对系统的动态性能有很大影响。但是却提高了系统的稳定性。因此，在实际进行参数选择时，若确定了预测时域N的值，为了保证闭环系统的稳定性，α应充分接近1。

3.6 实例验证

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;G({z^{ - 1}})H({z^{ - 1}}) = \\ \frac{{1.241{z^{ - 1}} - 1.89{z^{ - 2}} + 0.8858{z^{ - 3}} - 0.1316{z^{ - 4}}}}{{1 - 2.7{z^{ - 1}} + 2.529{z^{ - 2}} - 0.9574{z^{ - 3}} + 0.1287{z^{ - 4}}}} \end{array}$ (23)

 Download: 图 10 一阶系统的奈式曲线 Fig. 10 Nyquist diagram of a first-order linear system

 Download: 图 11 两种算法的阶跃响应曲线 Fig. 11 Step response of these two GPC methods

N=10, Nu=1, λ=0, α=0.1, β=0.36, T=0.1。

4 结论