As can be seen in Figure 17.1, \(H^2\) control with disturbance rejection leads to a change in the frequency response very narrowly around the given frequency. This is
undesirable if the frequency of the disturbance is uncertain.
The idea here is that we are interested in \(G_{zw}\), but we first multiply it by \(G_{w w_e}\) to emphasize certain frequencies. We then determine \(u\) based on \(G_{z w_e}\) (for example by \(H^2\) control design), but apply this \(u\) to
\(G_{zw}\) (i.e. after design we forget about the input-state-output exosystem).
Remark18.1. A typical input-state-output exosystem would be based on a
band-pass filter as in Section 6.3.2. Since we want to emphasize a certain frequency band, but not (almost) completely neglect the other frequencies, we would typically pick \(D_e=1\) (rather
than \(D_e=0\) which we would have for an actual band-pass filter). This results in
Here \(\omega _0\) is the center of the pass band, \(\zeta \) determines the bandwidth and \(\eta \) determines how much the pass band is emphasized over other frequencies: for the latter, note that the maximum of the modulus of the frequency
response is reached at \(\omega _0\) and equals \(1+\eta \) whereas the minimum (reached at both zero and infinite frequency) equals \(1\), so that \(\eta \) large corresponds to a large emphasis on the pass band and \(\eta \) close to zero
corresponds to a small emphasis on the pass band.
18.1 Examples
Example18.2. Consider similarly as in Example 17.3 the first order system
\[ T\dot {x}(t)+x(t)=K_1w_1(t)+K_2u(t), \]
where \(T,K_1,K_2>0\). As performance output \(z\) and measured output \(y\) consider
where \(\omega _e,\zeta ,\eta >0\). We then have that \(w_1\) is generated from \(w_{e,1}\) though a band-pass filter as in Remark 18.1, whereas \(w_2=w_{e,2}\).
In Figure 18.1a we give the Bode magnitude plot for \(G_{zw}\) (i.e., our original first order system) and for \(G_{z w_e}\) (i.e. the first order system composed with the
input-state-output exosystem). We see that the multiplication with the band-pass filter (plus one) gives a narrow additional peak around the chosen frequency \(\omega _e\). By choice of \(\zeta \) we could make this narrower or wider and by
choice of \(\eta \) we could make this peak higher or lower.
The \(H^2\) measurement control problem for the composed system is too difficult to solve by hand, so we solve it numerically. In Figure 18.1b we give the resulting Bode
magnitude plot when we attach this controller to the original system. In the figure we see that the magnitude is generally smaller than with no control and that it is especially smaller around the frequency \(\omega _0\) (due to the band-pass filter
structure in the input-state-output exosystem). Note that the controller has state space dimension 3 since it was based on the composed system which has state space dimension 3.
(a) Frequency response for first order system with and without the input-state-output exosystem.
(b) Frequency response for first order system: without control and with weighted \(H^2\) control.
Example18.3. Consider the undamped second order system
\begin{gather*}
A=\bbm {0&1\\-\omega _0^2&0},\quad B_1=\bbm {0&0\\K_1\omega _0^2&0},\quad B_2=\bbm {0\\K_2\omega _0^2}, \\ C_1=\bbm {0&1\\0&0},\quad D_{12}=\bbm {0\\\varepsilon },\quad C_2=\bbm
{0&1},\quad D_{21}=\bbm {0&\delta }.
\end{gather*}
We assume that \(w_1\) is concentrated around frequency \(\omega _e\), but that we have no such information about \(w_2\). This gives
where \(\omega _e,\zeta ,\eta >0\). We then have that \(w_1\) is generated from \(w_{e,1}\) though a band-pass filter as in Remark 18.1, whereas \(w_2=w_{e,2}\).
The \(H^2\) measurement control problem for the composed system is too difficult to solve by hand, so we solve it numerically. In Figure 18.2b we give the resulting Bode
magnitude diagram when we attach this controller to the original system.
(a) Frequency response for second order system (band pass) with and without the input-state-output exosystem.
(b) Frequency response for second order system (band pass): without control and with weighted \(H^2\) control.
18.2 Case study: control of a tape drive
For the tape drive problem, instead of disturbance rejection at the single frequency \(\omega _e\) which we did in Section 17.2, we can use a weight function (a second order band pass system plus a constant)
to emphasize frequencies around \(\omega _e\) (where the pass-band can be chosen). For comparison with the situation consider in Section 17.2, we keep the matrices from (17.4).
The \(H^2\) measurement control problem for the composed system is too difficult to solve by hand, so we solve it numerically. In Figures 18.3a and 18.3b we give the resulting Bode magnitude plot when we attach this controller to the original system.
(a) Frequency response from \(v_e\) to \(v_1\) for tape drive without control and with weighted \(H^2\) control.
(b) Frequency response from \(v_e\) to \(T\) for tape drive without control and with weighted \(H^2\) control.
