Carlos Osorio, MathWorks
了解如何在此MATLAB中构建一阶系统的Bode Plot®Tech Talk by Carlos Osorio.
We just saw how a function like Bode in MATLAB can quickly and easily create a frequency response plot directly from the dynamic equations of, or the input, output transfer functions of our system. The key as control engineers is not just to be able to create those plots. The important thing is having a good understanding of what those magnitude and phase traces are telling us about our system behavior and stability.
Bode Plots最初由Hendrik Bode博士开发,因此名称,而他在20世纪30年代在第二次世界大战之前工作的贝尔实验室。这家伙是一个聪明的控制工程师,他当时提出了使用渐近幅度和相位图的开创性理念,以便于频域中的稳定性分析和控制系统设计。
Remember that this was way before computers, so I guess engineers at the time were stuck with slide rulers, using them to manually calculate all those logarithms. The ideas behind the asymptotic approach are quite simple, but extremely powerful. And they will help us gain a better understanding of how those plots are actually built.
最简单的构造我可以从Laplace域中的1 / s对应于纯集成器。如果我们通过JW替换S,则我们的功能G成为负虚构轴上的向量。消极,因为我们正在将分子和分母乘以-1的平方根。
该矢量具有-90度的恒定相位角,幅度为1 / w。请注意,随着频率OMEGA从0到Infinity,我们的向量的大小将从无限远到0.在DB方面,分数的日志是分子的日志,在这种情况下,1,减去日志在这种情况下的分母,w。
We know that log of 1 is 0. So that part goes away and magnitude trace of the Bode plot becomes just a line, because we are plotting it on a horizontal axis that represents log of w. Notice that this line has a slope of -20 dBs per unit, which is a frequency decade in this case. The phase remains constant and -90 degrees, and is independent of the frequency.
Conversely, if we look at a pure differentiator, which corresponds to just s in the Laplace domain, because w is in the numerator now. In this case, the magnitude will be a line going up with a slope of +20 dBs per decade. And the phase will be a constant positive 90 degrees.
现在让我们继续前进我们的第一个结构艺术,就像一个单杆,带有TAU的时间常数。再一次,如果我们想查看频率响应,我们需要通过JW替换S.该矢量的幅度将是1的0,下降到0,减去分母幅度的幅度的20倍。
On first impression this looks like it is going to be very hard to draw. But if you think of that expression, on an asymptotic manner, and break the diagram in two parts-- when the frequency is well below the pole, in this case, below 1/tau, radiance per second, tau multiplied by w will become very small and the number 1 will dominate the expression.
请注意,这使得G变得接近1/1,这将是真实轴上的向量。这意味着相位将非常接近0,其幅度的日志也非常接近0。当频率高于杆的方式时,Tau * W将成为主导,在这种情况下,G变得靠近负,纯粹的虚数。这意味着相位将接近-90度,并且幅度的日志将接近一条直线,在-20度下滚动,然后过零,其中w等于1 / tau。
请注意,实际的Bode Plot从我们的渐近近似偏离。显然,我们将看到围绕1 / Tau的截止值的最大差异。从情节中,我们还可以看到相位角大约需要大约二十年来换90度。因此,如果您对角度稍微准确,我们可以假设在极值的值之前和之后每年45度下降45度。
使用相同的方法,我们可以看到单个0将导致类似的跟踪。只有在这种情况下,因为0位于分子中,相位将换+90度,并且磁铁到斜率将是每十年+ 20 dbs。
At this point, I would like to give you a feel for how all of this works in a more interactive fashion. What we are looking at here is the Bode diagram for a constant transfer function of 1. Our system G, over here, is equal to 1. This means log of 1, which is 0 dB magnitude and 0 degrees of phase, because it's a positive real number.
让我们看看我添加杆子时会发生什么。让我们靠近每秒1个弧度。我们可以看到该幅度绘图如何立即使用-20 dbs崩溃。并且相移到-90度。
如果我移动杆向右或向左,马king it faster or slower, all I am doing is shifting that kind of frequency. Let's erase that pole and bring in a 0. As expected, now we see a positive break in the magnitude and +90 in the phase. Notice that the magnitude of a pure 0 goes to infinity at high frequencies.
这是非常不良的行为,因为在其他坏事中,它将很可能在我们的系统中放大各种高频噪声。通常,如果您有纯差分器或0,则将始终伴随着在频率范围内的至少一个杆,以使增益树下降。
Because of superposition of the graphs, remember that multiplication becomes sums on a logarithmic scale.
The plus 20 dB slope of the 0 gets canceled by the minus 20 DB slope of the pole. Similarly with the phase, the +90 degrees of the 0 gets pulled down by -90 degrees of the pole. If I actually wanted some attenuation at the higher frequencies, all I need to do is add another pole close to the first one. And now the role of rate becomes minus 20 dBs per decade.
If you want a sharper drop but higher frequencies, just add another pole and, boom, -40 degrees per decade. Anyways I think you would agree that this interactive design tool is way better than slide rules and graph paper. I don't know about you, but I like to believe that Dr. Bode who, by the way, spent many years teaching controls just across the river at Harvard, would have truly loved MATLAB.
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