Skip to main content
Engineering LibreTexts

18.3: Algebraic Constraints

  • Page ID
    24335
  • In general we would like to design feedback controllers to attenuate both noise and disturbances at the output. We have examined SISO and MIMO conditions that guarantee rejection of low frequency disturbances as well as similar conditions for the rejection of high frequency noise. However, one might wonder if we can

    1. minimize the influence of either noise or disturbances over all frequencies, and/or
    2. minimize the influence of both noise and disturbances at the same frequency.

    Let us begin this discussion by recalling the following:

    • \(S=(I+P K)^{-1}\) is the transfer function mapping disturbances to the output;
    • \(T=P K(I+P K)^{-1}\) is the transfer function mapping noise to the output.

    As we mentioned earlier, in a control design it is usually desirable to make both \(S\) and \(T\) small. However, because of algebraic constraints, both goals are not simultaneously achievable at the same frequency. These constraints are as follows.

    General Limitations

    \(S + T = I\) for all complex (Laplace domain) frequencies \(s\). This is easily verified, since

    \[\begin{aligned}
    S+T &=(I+P K)^{-1}+P K(I+P K)^{-1} \\
    &=(I+P K)(I+P K)^{-1} \\
    &=I
    \end{aligned}\nonumber\]

    This result implies that if \(\sigma_{\max }[S(j \omega)]\) is small in some frequency range, \(\sigma_{\max }[T(j \omega)] \sim 1\). The converse is also true.

    Fortunately, we rarely need to make both of these functions small in the same frequency region.

    Limitations Due to RHP Zeros and Poles

    Before we discuss these limitations, we quote the following fact from complex analysis:

    Let \(H(s)\) be a stable, causal, linear time-invariant continuous-time system. The maximum modulus principle implies that

    \[\sigma_{\max }[H(s)] \leq \sup _{\omega} \sigma \max [H(j \omega)]=\|H\|_{\infty} \quad \forall s \in \operatorname{RHP}\nonumber\]

    In other words, a stable function, which is analytic in the RHP, achieves its maximum value over the RHP when evaluated on the imaginary axis.

    Using this result, we can arrive at relationships between poles and zeros of the plant \(P\) located in the RHP and limitations on performance (e.g., disturbance and noise rejection).

    SISO Systems: Disturbance Rejection

    Consider the stable sensitivity function \(S=(1+P K)^{-1}\) for any stabilizing controller, \(K\); then,

    \[\begin{aligned}
    &S\left(z_{i}\right)=\left(1+P\left(z_{i}\right) K\left(z_{i}\right)\right)^{-1}=1 \quad \text { for all } \operatorname{RHP} \text { zeros } z_{i} \text { of } P\\
    &S\left(p_{i}\right)=\left(1+P\left(p_{i}\right) K\left(p_{i}\right)\right)^{-1}=0 \quad \text { for all } \operatorname{RHP} \text { poles } p_{i} \text { of } P
    \end{aligned}\nonumber\]

    Since the \(\mathcal{H}_{\infty}\) norm bounds the gain of a system over all frequencies,

    \[1=\left|S\left(z_{i}\right)\right| \leq\|S\|_{\infty}\nonumber\]

    This means that we cannot uniformly attenuate disturbances over the entire frequency range if there are zeros in the RHP.

    SISO Systems: Noise Rejection

    Since the transfer function relating a noise input to the output is \(T=P K(1+P K)^{-1}\), an argument for \(T\) similar to \(S\) can be made, but with the roles of poles and zeros interchanged. In this case, RHP poles of the plant restrict us from uniformly attenuating noise over the entire frequency range.

    MIMO Systems: Disturbance Rejection

    Suppose \(P\) has a transmission zero at \(\tilde{z} \in \mathrm{RHP}\) with left input zero direction \(\eta^{*}\). Then \(\eta^{*} P(\tilde{z}) K(\tilde{z})=0\), and thus

    \[\eta^{*}(I+P(\tilde{z}) K(\tilde{z}))^{-1}=\eta^{*}\nonumber\]

    Stated equivalently,

    \[\eta^{*} S(\tilde{z})=\eta^{*} \ \tag{18.10}\]

    Also, taking the conjugate transpose of both sides,

    \[S^{*}(\tilde{z}) \eta=\eta \ \tag{18.11}\]

    We then multiply the expressions in (18.10) and (18.11), obtaining

    \[\eta^{*} S(\tilde{z}) S^{*}(\tilde{z}) \eta=\eta^{*} \eta \nonumber\]

    which can be alternately written as

    \[\frac{\eta^{*} S(\tilde{z}) S^{*}(\tilde{z}) \eta}{\eta^{*} \eta}=1 \ \tag{18.12}\]

    Applying the maximum modulus principle (i.e., \(\max _{s \in \operatorname{RHP}} \sigma_{\max }[S(s)]\) occurs on the imaginary axis) and observing that the left hand side of (18.12) is less than or equal to \(\sigma_{\max }^{2}[S(\tilde{z})]\), we conclude that

    \[\|S\|_{\infty}^{2} \geq \frac{\eta^{*} S(\tilde{z}) S^{*}(\tilde{z}) \eta}{\eta^{*} \eta}=1\nonumber\]

    Thus, the conclusion regarding disturbance rejection for MIMO systems is the same as the conclusion we reached for SISO systems. Namely, RHP zeros make disturbance attenuation over all frequencies impossible.