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1、南京农业大学工学院,第三讲 控制系统根轨迹法 分析与设计,常用的基本命令,rlocus(sys),r,k=rlocus(sys)rlocus(sys,k),r,k=rlocus(sys,k)r,k=rlocfind(sys)k,poles=rlocfind(sys,p),例3-1 自动焊接头控制,自动焊接头需要进行精确定位控制，其控制系统结构图如图3-1所示。图中，K1为放大器增益，K2为微分反馈增益。,图3-1 自动焊接头控制系统,例3-1 自动焊接头控制,本例设计要求为用根轨迹法选择参数K1与K2，使系统满足如下性能指标：,系统对斜坡输入响应的稳态误差 斜坡幅值的35%；,系统主导极点的阻
2、尼比 0.707；,系统阶跃响应的调节时间 3s(=2%)。,要求用根轨迹法选择参数K1与K2,使系统满足性能指标,解：由图3-1知，系统开环传递函数,显然，该系统为型系统，在斜坡输入作用下，存在稳态误差。系统的误差信号：,要求用根轨迹法选择参数K1与K2,使系统满足性能指标,令R(s)=R/s2，则稳态误差：,根据系统对稳态误差的性能指标要求，K1与K2的选取，应满足如下要求：,上式表明，为了获得较小的稳态误差，应该选择小的K2值。,要求系统对斜坡输入响应的稳态误差 斜坡幅值的35%,根据系统对主导极点的阻尼比要求，系统的闭环极点应位于s平面上=0.707的45o斜线之间；再由对系统的调节时
3、间的指标要求可知，主导极点实部的绝对值应满足：,因此有1.5。于是，满足设计指标要求的闭环极点，应全部位于图3-2所示的扇形区域内。,要求用根轨迹法选择参数K1与K2,使系统主导极点的阻尼比 0.707,首先考虑参数=K1的选择。令=0，则变化时的根轨迹方程为：,要求用根轨迹法选择参数K1与K2,阶跃响应的调节时间 3s(=2%),事实上,对于闭环特征方程：s2+2s+s+=0可作以下表达：,令从0到，其根轨迹如图3-3(a)所示。相应的matlab文本为:,变化时的根轨迹方程为：,要求用根轨迹法选择参数K1与K2,阶跃响应的调节时间 3s(=2%),num=1;den=1 2 0;G=tf(
4、num,den);rlocus(G)grid on;,利用模值条件，在图3-3(a)中试取K1=20，其对应的闭环极点为-1j4.36。于是参数=20K2。,3-3(a)为可变参数,要求用根轨迹法选择参数K1与K2,阶跃响应的调节时间 3s(=2%),r,k=rlocus(G,20)r=-1.0000+4.3589i-1.0000-4.3589i,当两参数一起变化时,成为一簇根轨迹,即,其次，考虑参数的选择。在闭环特征方程D(s)=0中，代入=20，则变化时的根轨迹方程为,D(s)=s2+2s+s+,要求用根轨迹法选择参数K1与K2,阶跃响应的调节时间 3s(=2%),要求用根轨迹法选择参数K
5、1与K2,阶跃响应的调节时间 3s(=2%),令从0变化到，其根轨迹图如图3-3(b)所示，,3-3(b)为可变参数,要求用根轨迹法选择参数K1与K2,阶跃响应的调节时间 3s(=2%),当取模值条件=4.3=20K2，即K2=0.215时，就得到了满足阻尼比=0.707的闭环主导极 点，其实部绝对值=3.15，由其决定的调节时间：,分离点坐标d=-4.47。,相应的稳态误差值,因而K1=20，K2=0.215的设计值，满足全部设计指标要求。系统的单位阶跃响应和单位斜坡响应分别如图3-4和图3-5所示。,要求系统对斜坡输入响应的稳态误差 斜坡幅值的35%,要求用根轨迹法选择参数K1与K2,阶跃
6、响应的调节时间 3s(=2%),系统单位阶跃响应的Matlab文本：,要求用根轨迹法选择参数K1与K2,阶跃响应的调节时间 3s(=2%),图3-4 系统的单位阶跃响应,要求系统对斜坡输入响应的稳态误差 斜坡幅值的35%,系统单位斜坡响应的Matlab文本：,要求系统对斜坡输入响应的稳态误差 斜坡幅值的35%,图3-6 系统的单位斜坡响应,设计要求：当r(t)=Rt时，要求e=0.2R,其它要求不变。,ss,Automobile velocity control design,Automated Highway System,Automobile velocity control design
7、,Zero steady-state error to a step inputSteady-state error due to a ramp input less than 25%of the input magnitudePercent overshoot less than 5%to a step inputSettling time less than 1.5 seconds to a step input(using a 2%criterion to establish settling time),The design specifications are:,DS1:Zero s
8、teady-state error to a step input,We find that we need a type one system to guarantee a zero steady-state error to a step input,DS2:Steady-state error due to a ramp input less than 25%of the input magnitude,To meet DS2 we need to have:,DS3:Percent overshoot less than 5%to a step input,Allow us to de
9、fine a target damping ratio:,Automobile velocity control design,DS4:Settling time less than 1.5 seconds to a step input(using a 2%criterion to establish settling time),From the settling time specification,we have:,Automobile velocity control design,Consider a PI controller:,The closed-loop transfer
10、function is:,Consider the stability of the system,we should producethe Routh sheet:,Automobile velocity control design,Automobile velocity control design,So we yield:,The open-loop transfer function as follow:,Automobile velocity control design,From the open-loop transfer function,we find that this
11、system has three poles and one zero,so there aretwo branches of the loci to go to infinity along two asymptotes at and centered at:,If we have,then the two branches of the loci will bend into the desired regions,that is,Automobile velocity control design,or,It follows from DS2 that:,Therefore,the in
12、tegral gain must satisfy:,Now we have the following equations:,Suppose we choose:,Thus,the closed-loop characteristic equation is:,Automobile velocity control design,Automobile velocity control design,Then,we can draw the locus as varies.,Automobile velocity control design,The closed-loop transfer function is:,Automobile velocity control design,
