Csdrhrt

I have often read that RHP zeros and poles set limits on the maximum achievable performance of LTI systems.

However, what does that exactly mean and how can you compute these performance limits?

For example, take these three transfer functions:

$$
begin{align}
G_1(s) &= frac{1}{(s + 1)(s - 2)} quadtext{(RHP pole)} \
G_2(s) &= frac{(s - 3)}{(s + 1)(s + 2)} quadtext{(RHP zero)} \
G_3(s) &= frac{(s - 3)}{(s + 1)(s + 2)(s - 5)} quadtext{(RHP pole and zero)}
end{align}
$$

What would be the limits of performance for these transfer functions? And are there differences between RHP zeros and RHP poles in terms of achievable performance?

Although this is not a rigorous proof, it should at least demonstrate the limitation that RHP poles and zeros put on the bandwidth in combination with the peak of the sensitivity.

If you have a system with only one RHP pole or only one RHP zero then, although bad practice, you can always cancel the remaining poles, zeros and gain in the controller. In order ensure that the controller has a proper transfer function one can always add a high bandwidth low-pass filter of sufficiently high order. This low-pass filter shouldn't effect the closed loop much, since this is mainly dominated around the frequency range where the magnitude of the open loop (system times controller) crosses the 0 dB line. Furthermore by using time scaling the RHP pole or zero can always be normalized to $-1$.

In the case of only one RHP pole the considered system can be generalized to

$$
G(s) = frac{1}{s - 1}.
$$

By using a controller of the form

$$
C(s) = frac{a,s + b}{s}
$$

then the sensitivity transfer function would look like

$$
S(s) = frac{s(s - 1)}{s^2 + (a - 1)s + b}.
$$

Using $b = omega^2$ and $a=2,zeta,omega+1$ gives a more standard form

$$
S(s) = frac{s(s - 1)}{s^2 + 2,zeta,omega,s + omega^2},
$$

where $omega$ can be used as a measure of the bandwidth. As expected for a sensitivity transfer function at really low frequencies the assymptote of $S(s)$ has a positive slope, namely $+1$, and at really high frequencies the assymptote of $S(s)$ has a slope of zero and a magnitude of 0 dB.

When $omega < 1$ then the slope of the assymptote of $S(s)$ after a frequency of $omega$ will decrease by two to $-1$ and eventually increase to the final assymptote with slope zero after a frequency of $1$. So, before the assymptote goes to the 0 dB line, the slope is negative which means that the magnitude of $S(s)$ during that interval is above 0 dB. The further $omega$ lies below one, the higher the magnitude of $S(s)$ will go above 0 dB.

When $omega > 1$ then the slope of the assymptote of $S(s)$ after a frequency of $1$ will increase by one to $+2$ and eventually decrease to the final assymptote with slope zero after a frequency of $omega$. So, before the assymptote goes to the 0 dB line, the slope is always positive which means that there shouldn't be a magnitude of $S(s)$ which goes significantly above 0 dB.

The two cases above (and the case when $omega=1$) are also illustrated in the figure below which uses $zeta = tfrac{1}{2}sqrt{2}$:

In the case of only one RHP zero the considered system can be generalized to

$$
G(s) = frac{s - 1}{s + p}.
$$

The pole at $p>0$ is just added to make the system proper. Now by using a controller of the form

$$
C(s) = frac{a(s + p)}{s^2 + b,s}
$$

then the sensitivity transfer function would look like

$$
S(s) = frac{s (s + b)}{s^2 + (a + b)s - a}.
$$

Using $a = -omega^2$ and $b=omega(2,zeta + omega)$ again gives a more standard form

$$
S(s) = frac{s (s + omega(2,zeta + omega))}{s^2 + 2,zeta,omega,s + omega^2},
$$

where $omega$ can again be used as a measure of the bandwidth. As expected for a sensitivity transfer function at really low frequencies the assymptote of $S(s)$ has a positive slope, namely $+1$, and at really high frequencies the assymptote of $S(s)$ has a slope of zero and a magnitude of 0 dB. The transition for the magnitude of the peak of the sensitivity now does not lie near $omega=1$ but roughly at $omega=2,zeta$.

When $omega < 2,zeta$ then the zero of $S(s)$ is of the same magnitude as the bandwidth (assuming a normal value for $zeta$). This means that the slope of the assymptote of $S(s)$ a little after a frequency of $omega$ will eventually decrease by one to zero (decrease by two and increase by one). The damping coefficient can influence this a little, namely the zero might lie a little ahead or behind $omega$, but for realistic values for $zeta$ the assymptote does not change much.

When $omega > 2,zeta$ then the zero of $S(s)$ scales with the square of the bandwidth. So between the frequencies $omega$ and roughly $omega^2$ the assymptote of $S(s)$ will be $-1$ and thus its magnitude of $S(s)$ will have a significant portion above 0 dB.

The two cases above (and the case when $omega=1$) are also illustrated in the figure below which uses $zeta = tfrac{1}{2}sqrt{2}$:

For a RHP pole the maximal magnitude of the sensitivity quickly goes up when the bandwidth is chosen below the break frequency of the pole. The opposite is true for a RHP zero, so when the bandwidth is chosen above the break frequency of the zero. So it is possible to place the bandwidth anywhere you want if you have a single RHP pole or zero, but you will have poor performance. Namely a large maximal magnitude of the sensitivity transfer function means large amplification of disturbances that act of the system. So often one would want to keep the magnitude of the sensitivity below roughly 6 dB.