另一个测试。关闭循环

PEN

添加积分器。
PI控制的传递函数是
Gc（s）= Ki / s + Kp
请注意，积分器增益除以s。
乘以s得出导数。
除以s整合。
PI控制有两个问题：
1.除非使用前馈，否则积分器将结束并导致实际位置超调。
2.除非开环阻尼系数非常高，否则不可能将所有极点放置在稳定性好的区域。

PI控制产生具有两个实极和两个复极的闭环传递函数。当所需的特征方程将所有极点尽可能远离虚轴时，似乎会出现最佳结果。这通常意味着特征方程看起来像


（s +α）^ 2 *（（s +α）^ 2 +β^ 2）是期望的特征方程。
（K * Ki *ω^ 2 + K * Kp *ω^ 2 * s + 2 *ζ*ω* s ^ 3 +ω** 2 * s ^ 2 + s ^ 4）是实际特征方程

求解Ki，Kp，α和β。
（s +α）^ 2 *（（s +α）^ 2 +β^ 2）=（K * Ki *ω^ 2 + K * Kp *ω^ 2 * s + 2 *ζ*ω* s ^ 3 +ω** 2 * s ^ 2 + s ^ 4）


α是所有闭环极点的实部。这意味着误差将作为exp（-α* t）的函数衰减。你填写发现α=ζ*ω/ 2！这意味着最佳响应由液压和机械设计决定。

添加积分器会增加另一个闭环极点，所以现在有4个极点。这就像在停车灯处等候你前面的4辆车而不是3辆。

Adding the integrator.
The transfer function for PI control is
Gc(s)=Ki/s + Kp
Notice that the integrator gain is divided by s.
Multiplying by s differentiates.
Dividing by s integrates.
PI control has two problems:
1. the integrator will windup and cause the actual position to overshoot unless feed forwards are used.
2. It is impossible to place all the poles in the area of nice stability unless the open loop damping factor is very high.

PI control results in a closed loop transfer function with two real poles and two complex poles.  The best results seem to occur when the desired characteristic equation places all the poles as far away from the imaginary axis as possible. This usually means the characteristic equation should look like


(s+α)^2*((s+α)^2+β^2) is the desired characteristic equation.
(K*Ki*ω^2 + K*Kp*ω^2*s + 2*ζ*ω*s^3 + ω**2*s^2 + s^4) is the actual characteristic equation

Solve for Ki, Kp, α, and β.
(s+α)^2*((s+α)^2+β^2)= (K*Ki*ω^2 + K*Kp*ω^2*s + 2*ζ*ω*s^3 + ω**2*s^2 + s^4)


α is the real part of all the closed loop poles.  This means the errors will decay as a function of exp(-α*t). You fill find that α = ζ*ω/2!  This means that optimal response is determined by the hydraulic and mechanical design.

Adding the integrator adds another closed loop pole so now there are 4 poles.  This is like waiting at the stop light with 4 cars ahead of you instead of 3.

蜻蜓

蜻蜓
PI control has two problems:
1. the integrator will windup and cause the actual position to overshoot unless feed forwards are used.

请教：
1        因为没有前馈，误差大，所以i作用强烈，导致超调？

α is the real part of all the closed loop poles.  This means the errors will decay as a function of exp(-α*t). You fill find that α = ζ*ω/2!  This means that optimal response is determined by the hydraulic and mechanical design.

2. It is impossible to place all the poles in the area of nice stability unless the open loop damping factor is very high.


α=ζ*ω/ 2，误差将按exp（-α* t）的函数衰减。
从上面两个等式看，α是ζ*ω共同作用的结果，ω低，系统也不容易稳定。ω高，系统容易稳定。
通常容易理解的是：固有频率高，系统响应快； 阻尼系数高，系统稳定。
固有频率ω与稳定性的关系不太好理解。

积分有一个单独极点，好像与s在 （Ki/s）分母中有关。

PEN

低阻尼系数使系统难以稳定。
但是，如果使用二阶导数增益，阻尼系数可以为0并且仍然可以稳定。

你应该能够解释原因

A low damping factor makes the system hard to stabilize.
However, the damping factor can be 0 and still be stabilized if the second derivative gain is used.

You should be able to explain why.

蜻蜓

PI control has two problems:
1. the integrator will windup and cause the actual position to overshoot unless feed forwards are used.
2. It is impossible to place all the poles in the area of nice stability unless the open loop damping factor is very high.

if   want to use the benefits of integral, you need K2 even more. Without K2, you need to increase mechanical damping and consume energy. Those who look at the subject will understand. The reason is 34 #.

如果想要使用积分带来的好处，就更加需要K2了，不用K2就要增加机械阻尼，消耗能源。看这个主题的人会明白。理由在34#.。

1           Ga(s)= 5/(s*(s^2+2*s+5)
2            Gc(s)=Ki / s+Kp+Kd*s+K2*s^2
3            CLTF(s)=Gc(s)*Ga(s)  /  (1+Gc(s)*Ga(s))
将1, 2代入到3中，求出 CLTF(s) 的特征根方程。
没计算机不好算，pen老师，您讲解吧，

小满哥

您好，请教一个问题，
闭环传递函数CLTF（s）=（Kp + Kd * s）* 5 /（s ^ 3 + 2 * s ^ 2 +（1 + Kd）* 5 * s + 5 * Kp）
分子中的Kd乘以目标速度。
分母中的Kd乘以实际速度，然后从Kd *目标速度中减去。简化为Kd *（目标速度-实际速度）
这个实际速度和目标速度怎么理解呢

PEN

有三个位置，三个速度和三个加速度。
指令位置，速度和加速度由用户提供。
命令位置是要移动到的位置。 指令速度是移动的速度，而指令加速度决定了达到指令速度的速度。
运动控制器使用命令位置，命令速度和命令加速度每毫秒生成一次目标位置，速度和加速度。
实际位置，实际速度和实际加速度由反馈传感器得出。 闭环控制试图使实际位置，速度和加速度与目标位置，速度和加速度相同。

这是Delta Computer Systems和Rockwell使用的约定。 西门子使用相同的原理，但使用的名称略有不同

There are three positions, three velocities and three accelerations.
The command position, velocity and acceleration are provided by the user.
The command position is the position to move to.  The command velocity is the speed at which the move is made, and the command acceleration determines how fast the command velocity is reached.
The target position, velocity and acceleration are generated by the motion controller every millisecond using the command position, command velocity and command acceleration.  
The actual position, actual velocity and actual acceleration is derived by the feedback sensor.   The closed loop control tries to keep the actual position, velocity and acceleration the same as the target position, velocity and acceleration.

This is the convention used by Delta Computer Systems and Rockwell.  Siemens uses the same principle, but they use slightly different names.

mayseven

波士顿动力设计的机器人就是采用伺服液压驱动的，他们甚至可以完成后空翻动作，不知道他们采用的是哪种控制技术，传统的控制方法能否达到那种高度？

fawei1007

HELLO PEN：
　　　Thank you for sharing, which has helped me systematically understand many areas that I didn't understand before in theory. I see that the examples you mentioned are mostly the solution process under fixed vibration frequency. If it is a linear sweep frequency of 3-120Hz in about 20s, how should we solve it? Do you have any routines or literature on this? thank you.