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19.4: Robust Stability

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    24343
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    In this section we will show how we can analyze the stability of a feedback system when the plant is uncertain and is known to belong to a set of the form that we described earlier. We will start with the case of additive pertubations. Consider the unity feedback configuration in Figure 19.1. The open-loop transfer function is \(L(s)=\left(P_{0}(s)+W(s) \Delta(s)\right) K(s)\), and the

    Screen Shot 2020-08-05 at 12.13.17 AM.png

    Figure 19.5: Nyquist Plot

    nominal open-loop transfer function is \(L_{0}(s)=P_{0}(s) K(s)\). The nominal feedback system with the nominal open-loop transfer function \(L_{0}\) is stable, and we want to know whether the feedback system remains stable for all \(\Delta (s)\) satisfying \(|\Delta(j \omega)| \leq 1\) for all ! 2 R. We will assume that the nominal open-loop system is stable. This causes no loss of generality and the result holds in the general case. From the Nyquist criterion, we have that the Nyquist plot of \(L_{0}\) does not encircle the point \(-1\). For the perturbed system, we have that

    \[\begin{aligned}
    1+L(j \omega) &=1+P(j \omega) K(j \omega) \\
    &=1+\left(P_{0}(j \omega)+W(j \omega) \Delta(j \omega)\right) K(j \omega) \\
    &=1+L_{0}(j \omega)+W(j \omega) \Delta(j \omega) K(j \omega)
    \end{aligned}\nonumber\]

    From the Figure 19.6, it is clear that \(L(\mathrm{J} \omega)\) will not encircle the point \(-1\) if the following condition is satisfied,

    \[|W(j \omega) K(j \omega)|<\left|1+L_{0}(j \omega)\right|\nonumber\]

    which can be written as

    \[\left|\frac{W(j \omega) K(j \omega)}{1+L_{0}(j \omega)}\right|<1 \ \tag{19.6}\]

    A Small Gain Argument

    Next we will present a different derivation of the above result that does not rely on the Nyquist criterion, and will be the basis for the multivariable generalizations of the robust stability results. Since the nominal feedback system is stable, the zeros of 1 + L0(s) are all in the left half of the complex plane. Therefore, by the continuity of zeros, the perturbed system

    Screen Shot 2020-08-05 at 12.24.03 AM.png

    Figure 19.6: Nyquist Plot Illustration Robust Stability

    will be stable if and only if

    \[\left|1+\left(P_{0}(j \omega)+W(j \omega) \Delta(j \omega)\right) K(j \omega)\right|>0\nonumber\]

    for all \(\omega \in \mathbb{R},\|\Delta\|_{\infty} \leq 1\). By rearranging the terms, the perturbed system is stable if and only if

    \[\min _{|\Delta(j \omega)| \leq 1}\left|1+\frac{W(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)} \Delta(j \omega)\right|>0 \quad \text { for all } \omega \in \mathbb{R}\nonumber\]

    The following lemma will help us to transform this condition to the one given earlier.

    Lemma 19.1

    The following are equivalent

    1. \[\min _{|\Delta(j \omega)| \leq 1}\left|1+\frac{W(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)} \Delta(j \omega)\right|>0 \quad \text { for all } \omega \in \mathbb{R}\nonumber\]
    2. \[1-\left|\frac{W(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)}\right|>0 \quad \text { for all } \omega \in \mathbb{R}\nonumber\]
    Proof

    First we show that 2) implies 1), which is a consequence of the following inequalities

    \[\begin{aligned}
    \left|1+\frac{W(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)} \Delta(j \omega)\right| & \geq 1-\left|\frac{W(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)} \Delta(j \omega)\right| \\
    & \geq 1-\left|\frac{W(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)}\right|
    \end{aligned}\nonumber\]

    For the converse suppose 2) is violated, that is there exists \(\omega_{0}\) such that

    \[\left|\frac{W\left(j \omega_{0}\right) K\left(j \omega_{0}\right)}{1+P_{0}\left(j \omega_{0}\right) K\left(j \omega_{0}\right)}\right| \geq 1\nonumber\]

    Write

    \[\frac{W\left(j \omega_{0}\right) K\left(j \omega_{0}\right)}{1+P_{0}\left(j \omega_{0}\right) K\left(j \omega_{0}\right)}=a e^{j \phi}\nonumber\]

    and let \(\bar{\Delta}\left(j \omega_{0}\right)=\frac{1}{a} e^{-j \phi-j \pi}\). Clearly, \(\left|\bar{\Delta}\left(j \omega_{0}\right)\right| \leq 1\) and

    \[1+\frac{W\left(j \omega_{0}\right) K\left(j \omega_{0}\right)}{1+P_{0}\left(j \omega_{0}\right) K\left(j \omega_{0}\right)} \bar{\Delta}\left(j \omega_{0}\right)=0\nonumber\]

    Now select a real rational perturbation \(\bar{\Delta}(s)\) as

    \[\bar{\Delta}(s)=\pm \frac{1}{a} \frac{s-\alpha}{s+\alpha}\nonumber\]

    such that \(\pm \frac{j \omega_{0}-\alpha}{\omega_{0}+\alpha}=e^{-j \phi-j \pi}\).

    Screen Shot 2020-08-05 at 12.42.31 AM.png

    Figure 19.7: Representation of the actual plant in a servo loop via a multiplicative perturbation of the nominal plant.

    A similar set of results can be obtained for the case of multiplicative perturbations. In particular, a robust stability of the configuration in Figure 19.7 can be guaranteed if the system is stable for the nominal plant \(P_{0}\) and

    \[\left|\frac{W(j \omega) P_{0}(j \omega) K(j \omega)}{1+P_{0}(j \omega) K(j \omega)}\right|<1 . \quad \text { for all } \omega \in \mathbb{R} \ \tag{19.7}\]

    Example 19.3 Stabilizing a Bean

    We are interested in deriving a controller that stabilizes the beam in Figure 19.8 and tracks a step input (with good properties). The rigid body model from torque input to the tip deflection is given by

    \[P_{0}(s)=\frac{6.28}{s^{2}}\nonumber\]

    Screen Shot 2020-08-05 at 12.31.01 AM.png

    Figure 19.8: Flexible Beam.

    Consider the controller

    \[K_{0}(s)=\frac{500(s+10)}{s+100}\nonumber\]

    The loop gain is given by

    \[P_{0}(s) K_{0}(s)=\frac{3140(s+10)}{s^{2}(s+100)}\nonumber\]

    and is shown in Figure 19.9. The closed loop poles are located at -49.0, -28.6, -22.4, and the nominal Sensitivity function is given by

    \[S_{0}(s)=\frac{1}{1+P_{0}(s) K_{0}(s)}=\frac{s^{2}(s+100)}{s^{3}+100 s^{2}+3140 s+31400}\nonumber\]

    and is shown in Figure 19.10. It is evident from this that the system has good disturbance rejection and tracking properties. The closed loop step response is show in Figure 19.11

    While this controller seems to be an excellent design, it turns out that it performs quite poorly in practice. The bandwidth of this controller (which was never constrained) is large enough to excite the flexible modes of the beam, which were not taken into account in the model. A more complicated model of the beam is given by

    \[P_{1}(s)=\underbrace{\frac{6.28}{s^{2}}}_{\text {nominal plant }}+\underbrace{\frac{12.56}{s^{2}+0.707 s+28}}_{\text {flexible mode }}\nonumber\]

    If \(K_{0}\) is connected to this plant, then the closed loop poles are -1.24, 0.29, 0.06, -0.06, which implies instability.

    Instead of using the new model to redesign the controller, we would like to use the nominal model \(P_{0}\), and account for the flexible modes as unmodeled dynamics with a certain frequency concentration. There are several advantages in this. For

    Screen Shot 2020-08-05 at 12.33.44 AM.png

    Figure 19.9: Open-loop Bode Plot

    one, the design is based on a simpler nominal model and hence may result in a simpler controller. This approach also allows us to accomodate additional flexible modes without increasing the complexity of the description. And finally, it enables us to tradeoff performance for robustness.

    Consider the set of plants:

    \[\Omega=\left\{P=P_{0}(1+\Delta) ;|\Delta(j \omega)| \leq l(\omega), \Delta \text { is stable }\right\}\nonumber\]

    where

    \[l(\omega) \leq 2\left|\frac{\omega^{2}}{28-\omega^{2}+0.707 j \omega}\right|\nonumber\]

    Screen Shot 2020-08-05 at 12.35.48 AM.png

    Figure 19.10: Nominal Sensitivity

    This set includes the model \(P_{1}\). The stability Robustness Condition is given by:

    \[|T(j \omega)|<\frac{1}{l(\omega)}\nonumber\]

    Where \(T\) is the nominal closed loop map with any controller \(K\). First, consider the stability analysis of the initial controller \(K_{0}(s)\). Figure 19.12 shows both the frequency response for \(\left|T_{0}(j \omega)\right|\) and \([l(\omega)]^{-1}\). It is evident that the Stability robustness condition is violated since

    \[\left|T_{0}(j \omega)\right| \nless \frac{1}{\ell(\omega)}, \quad 3 \leq \omega \leq 70 \mathrm{rad} / \mathrm{sec}\nonumber\]

    Screen Shot 2020-08-05 at 12.38.26 AM.png

    Figure 19.11: Step Response

    Screen Shot 2020-08-05 at 12.39.15 AM.png

    Figure 19.12:\(\left|T_{0}(j \omega)\right| \text { and }[l(\omega)]^{-1}\)

    Let's try a new design with a different controller

    \[K_{1}(s)=\frac{\left(5 \times 10^{-4}\right)(s+0.01)}{s+0.1}\nonumber\]

    The new loop-gain is

    \[P_{0}(s) K_{1}(s)=\frac{\left(3.14 \times 10^{-3}\right)(s+0.01)}{s^{2}(s+0.1)}\nonumber\]

    which is shown in the Figure 19.13 We first check the robustness condition with the new controller. \(T_{1}\) is given by

    \[T_{1}(s)=\frac{P_{0}(s) K_{1}(s)}{1+P_{0}(s) K_{1}(s)}\nonumber\]

    Figure 19.14 depicts both \(\left|T_{1}(j \omega)\right|\) and \([\ell(\omega)]^{-1}\). It is clear that the condition is satisfied. Figure 19.15 shows the new nominal step response of the system. Observe that the response is much slower than the one derived by the controller \(K_{0}\). This is essentially due to the limited bandwidth of the new controller, which was necessary to prevent instability.


    This page titled 19.4: Robust Stability is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Mohammed Dahleh, Munther A. Dahleh, and George Verghese (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.