11. H2 and H-Infinity Control

1. Review on Sensitivity Functions

First, look at the linear system with disturbance,

Give the transfer function of this system:

\[ \begin{aligned} Y(s) &= T(s)(y^\circ(s)-N(s)) + S(s)D_y(s)+G(s)S(s)D_u(s) \\ U(s) &= K(s)(y^\circ(s)-N(s) - D_y(s)) + S(s)D_u(s) \end{aligned} \]

And give the sensitivity functions:

Description	Formulation
Complementary sensitivity function	\(T(s) = \frac{L(s)}{1+L(s)}\)
Sensitivity function	\(S(s) = \frac{1}{1+L(s)}\)
Control sensitivity function	\(K(s) = R(s)S(s)\)

Where \(L(s) = R(s)G(s)\).

We have the Stability margin: \(\varphi_m \geq \bar \varphi_m\), \(g_m \geq \bar g_m\),

\(\Rightarrow\) \(M_T = ||T||_\infty \leq \bar M_T\), \(M_S = ||S||_\infty \leq \bar M_S\)

And the noise and disturbance distributes like figure below:

And we look at the bode diagram of the noise and disturbance distribution,

2. Additive Uncertainty

Given a nominal model without considering any disturbance and noise, we have the following schematics:

And we designed an \(R(s)\) to make this closed loop nominal model A.S.. We want to research the stability of this nominal model at \(G(s) \neq \bar G(s)\).

We can represent the uncertainty part with nominal part and the error on system:

\[ G(s) = \bar G(s) + \Delta G_a(s) \]

Thus, we can change the system schematics:

Assume that:

\(\Delta G_a (s)\) is A.S. (\(\Rightarrow P_G=P_{\bar G}\))
There have no zero-pole cancellation between \(R(s)\) and \(G(s)\) in the positive real part.

Now, we can know the TF of \(\eta \to \phi\): \(-\frac{R(s)}{1+R(s)\bar G(s)}\)

And we can simplify the control schematics:

There have: \(\frac{R(s)}{1+R(s)\bar G(s)} = R(s)\bar S(s) = \bar K(s)\), is A.S.

To make the system above stable, we need to meet the condition: \(||\Delta G_a(s)\bar K(s)||_\infty < 1\), we can draw the Nyquist diagram to explain this condition is a sufficient condition to A.S.:

For the infinity norm, there have: \(||\Delta G_a(s)\bar K(s)||_\infty = \sup_\omega |\Delta G_a(s)\bar K(s)|\). And within the Nyquist stability theorem and the given condition, \(\Delta G_a(s)\bar K(s)\) will have no contact to \((-1+j0)\) point, the system is A.S.. But this is not a necessary condition for the stability, because even if \(||\Delta G_a(s)\bar K(s)||_\infty \geq 1\), the system could still be stable.

To get the condition above, we can get \(|\bar K(j\omega)| < \frac{1}{|\Delta G_a(j\omega)|}, \forall \omega\) \(\Rightarrow\) \(|\bar K(j\omega)|\) is small when \(|\Delta G_a(j\omega)|\) is big (typically in low frequency). And because \(|\bar K(j\omega)| = |R(s)\bar||S(s)|\), then \(M_S\) is small.

3. Multiplicative Uncertainty

Given the TF,

\[ G(s) = \bar G(s)(1+\Delta G_m(s)) \]

And the schematic below:

Assume that \(\Delta G_m(s)\) is A.S. \((P_G=P_{\bar G})\).

The TF from \(\eta \to \phi\): \(-\frac{R(s)\bar G(s)}{1+R(s)\bar G(s)} = -\bar T(s)\). The system schematic can be changed to:

The condition of stability is: \(||\bar T \Delta G_m||_\infty < 1\).

And to satisfy the condition, we can let (\(|\bar T| < \frac{1}{|\Delta G_m|}, \forall \omega\)) \(\Rightarrow\) (\(M_T < \frac{1}{|\Delta G_m|}, \forall \omega\)), \(M_T\) should be small.

Example

\[ \begin{aligned} \bar G(s) &= \frac{1}{s} \\ G(s) &= \frac{1}{s} e^{\tau s},\quad (\tau > 0) \end{aligned} \]

\[ G(s) = \bar G(s)(1+\Delta G_m(s)) \Rightarrow \Delta G_m(s) = e^{\tau s} - 1 \]

We can draw the bode plot:

And we continue with additive uncertainty, which gives:

\[ G(s) = \bar G(s) +\Delta G_a(s) \Rightarrow \Delta G_a(s) = \frac{1}{s}(e^{\tau s} - 1) \]

Within the \(H_\infty\) norm, we have \(||\bar K \Delta G_a||_\infty < 1\), we should design a \(\bar K\) shown in the bode plot:

Example

\[ \left\{\begin{aligned} G(s) &= \frac{\bar K + \Delta K}{s + a} \\ \bar G(s) &= \frac{\bar K}{s + a} \end{aligned}\right., a > 0 \]

Given the bode plot for \(|S|\) and \(|T|\),

\(\omega_{BS}\) is the frequency where \(|S|_{dB}\) crosses \(-3dB\) line from below , and
\(\omega_{BT}\) is the frequency where \(|T|_{dB}\) crosses \(-3dB\) line from above.

And the phase margin for this system have: \(\varphi_m < 90^\circ\), and \(\omega_{BS} < \omega_c < \omega_{BT}\).

4. Design Specifications in Terms of the Sensitivity Function

For the design specifications, we need to concern about:

the shape of \(S(s)\):
1. small at low frequency, \(\leftarrow\) for compensating the disturbances, small or null error when tracking a constant \(y^\circ\). \(\leftarrow\) \(M_L\) is large, integrator in \(L(s)\).
2. \(\simeq 1\) at high frequency
3. small pick of resonance
Minimum frequency \(\omega_B\),
\(M_S \leq \bar M_S\), where \(\bar M_S\) is the robustness of the stability.

These specifications defines a "desired sensitivity function" \(S_{des}(s)\), and its bode plot:

We introduce the sensitivity shaping function: \(W_S(s) = S^{-1}_{des}(s)\).

4.1. H-infinity Control Approach

We want to find a regulator \(R(s)\) that satisfying:

\[ \begin{aligned} &|S(j\omega)| < \frac{1}{|W_S(j\omega)|}, \forall \omega \\ \Rightarrow& ||SW_S||_\infty < 1 \end{aligned} \]

A possible (standard) choice for \(W_S\):

\[ W_S(s) = \frac{s/M + \omega_B}{s+A\omega_B} \]

We can draw the bode plot:

We want \(|\frac{1}{A}|\) to be high, \(A \ll 1\) \(\Rightarrow\) desired attenuation at low frequency
\(M\) relates to \(M_S\)

5. Design Specifications in Terms of the Complementary Sensitivity Function

For the design specifications, we need to concern about:

shape of \(T(s)\)
1. \(\simeq 1\) at low frequency
2. small at high frequency
3. small pick of resonance
Max frequency \(\omega_{BT}\)
\(M_T\) is small enough

We can draw the desired bode plot for \(T(s)\),

Give the complementary sensitivity shaping function: \(W_T(s) = T_{des}^{-1}(s)\).

\[ \begin{aligned} &|T(j\omega)| < \frac{1}{W_T(j\omega)}, \forall \omega \\ \Rightarrow& ||TW_T||_\infty < 1 \end{aligned} \]

A possible (standard) choice for \(W_T\) is:

\[ W(s) = \frac{s + \omega_{BT}/M}{As + \omega_{BT}} \]

And the bode diagram:

\(A\) is desired attenuation at high frequency.

We can do the same for the control sensitivity function: \(W_K(s) = K_{des}^{-1}(s)\).

\[ \begin{aligned} &K(j\omega) < \frac{1}{|W_K(j\omega)|}, \forall \omega \\ \Rightarrow& ||KW_K||_\infty < 1 \end{aligned} \]

6. General H-infinity Control Approach

To design a \(H_\infty\) controller, our goal is to design a \(R(s)\) such that:

\[ \begin{aligned} ||W_SS||_\infty &< 1 \\ ||W_TT||_\infty &< 1 \\ ||W_KK||_\infty &< 1 \end{aligned} \]

To go for more general formulation of \(H_\infty\) control, we draw the system schematics:

Where \(z = \begin{bmatrix} z_S & z_T & z_K \end{bmatrix}^T\) is the performance variables. \(w\) is external signals, and we have: \(z = \underbrace{\begin{bmatrix} W_SS & W_TT & W_KK \end{bmatrix}^T}_{G_{ZW}} w\)

Now, our goal is to design \(R(s)\) to minimize \(||G_{ZW}||_\infty\), so that if you obtain that \(||G_{ZW}||_\infty < \gamma\), \(\Rightarrow\) \(||W_SS||_\infty < \gamma\), \(||W_KK||_\infty < \gamma\), \(||W_TT||_\infty < \gamma\).

6.1 H-2 Control

For \(H_2\) control, we want to design a \(R(s)\) for minimizing:

\[ ||G_{ZW}||_2^2 = \frac{1}{2\pi} \int_{-\infty}^{\infty} |G_{ZW}(j\omega)|^2 d\omega \]

Give the exogenous signals:

\[ W = \begin{bmatrix} du \\ dy \\ y^\circ \\ n \end{bmatrix} \]

and \(u\) is the control input, \(v\) is the measured variable.

Design \(R(s)\) such that the \(H_2/H_\infty\) norm of the transfer matrix \(G_{ZW}\) between \(W\) and \(Z\) is minimized.