Tags: H Infinity Control
Reference: ^{(Bibel & Malyevac, 1992)}
Author(s): Bibel, J. E., & Malyevac, D. S.
Year: 1992

Properties of feedback control

From the figure 1, we have:

\begin{align*} y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\
e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\
u(s) &= S(s)K(s) r(s) - S(s)K(s) d(s) - S(s)K(s) n(s) \end{align*}

With the following definitions

\(L(s) = G(s)K(s)\) is the loop transfer matrix
\(S(s) = [I+G(s)K(s)]^{-1}\) is the Sensitivity function matrix
\(T(s) = [I+G(s)K(s)]^{-1}G(s)K(s)\) is the Transmissibility function matrix

\[ S(s) + T(s) = 1 \]

Command following: \(S=0\) and \(T=1\) => large gains
Disturbance rejection: \(S=0\) => large gains
Sensor noise attenuation: \(T\) small where the noise is concentrated
Control Sensitivity minimization: \(K S\) small
Robustness to modeling errors: \(T\) small in the frequency range of the expected model undertainties

SISO tradeoff

We want \(S\) small for command following and disturbance rejection. We want \(T\) small to remain insensitive to sensor noise and modeling errors and to reduce control sensitivity.

However we cannot keep both \(S\) and \(T\) small as \(S(s)+T(s)=1\).

We must determine some tradeoff between the sensitivity and the complementary sensitivity functions.

Usually, reference signals and disturbances occur at low frequencies, while noise and modeling errors are concentrated at high frequencies. The tradeoff, in a SISO sense, is to make \(|S(j\omega)|\) small as low frequencies and \(|T(j\omega)|\) small at high frequencies.

\(H_\infty\) and weighting functions

\(\mathcal{H}_\infty\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller’s performance and robustness characteristics.

Figure 2: \(\mathcal{H}_\infty\) control framework

New design framework (figure 2): \(P(s)\) is the generalized plant transfer function matrix:

\(w\): exogenous inputs
\(z\): regulated performance output
\(u\): control inputs
\(y\): measured output variables

The plant \(P\) has two inputs and two outputs, it can be decomposed into four sub-transfer function matrices: \[P = \begin{bmatrix}P_{11} & P_{12} \ P_{21} & P_{22} \end{bmatrix}\]

Lower Linear Fractional Transformation

The transformation from the input \(w\) to the output \(z\), \(T_{zw}\) is called the Lower Linear Fractional Transformation \(F_l (P, K)\).

\[T_{zw} = F_l (P, K) = P_{11} + P_{12}K (I-P_{22})^{-1} P_{21}\]

The \(H_\infty\) control problem is to find a controller that minimizes \(\| T_{zw} \|_\infty\) over the space of all realizable controllers \(K(s)\) that stabilize the closed-loop system.

Weights for inputs/outputs signals

Since \(S\) and \(T\) cannot be minimized together at all frequency, weights are introduced to shape the solutions. Not only can \(S\) and \(T\) be weighted, but other regulated performance variables and inputs (figure 3).

Figure 3: Input and Output weights in \(\mathcal{H}_\infty\) framework

The weights on the input and output variables are selected to reflect the spatial and frequency dependence of the respective signals and performance specifications.

These inputs and output weighting functions are defined as rational, stable and minimum-phase transfer function (no poles or zero in the right half plane).

General Guidelines for Weight Selection: \(W_S\)

\(W_S\) is selected to reflect the desired performance characteristics. The sensitivity function \(S\) should have low gain at low frequency for good tracking performance and high gain at high frequencies to limit overshoot. We have to select \(W_S\) such that \({W_S}^-1\) reflects the desired shape of \(S\).

Low frequency gain: set to the inverse of the desired steady state tracking error
High frequency gain: set to limit overshoot (\(0.1\) to \(0.5\) is a good compromise between overshoot and response speed)
Crossover frequency: chosen to limit the maximum closed-loop time constant (\(\omega_c \approx 1/\tau\))

General Guidelines for Weight Selection: \(W_T\)

We want \(T\) near unity for good tracking of reference and near zero for noise suppresion.

A high pass weight is usualy used on \(T\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:

The crossover frequency is chosen to limit the closed-loop bandwidth
The high frequency gain is set high to proide sensor noise rejection and high frequency gain attenuation

When using both \(W_S\) and \(W_T\), it is important to make sure that the magnitude of theise weights at the crossover frequency is less that one to not violate \(S+T=1\).

Unmodeled dynamics weighting function

Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an unmodeled dynamics uncertainty block that may be added from the plant input to the plant output (figure 4).

The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure 5.

Figure 5: Example of unmodeled dynamics weight

Inputs and Output weighting function

It is possible to weight the control input and actuator rate. This is used to prevent actuator saturation and limit amplification of sensor noise signals on the control input signal.

Typically actuator input weights are constant over frequency and set at the inverse of the saturation limit.

Order of the weighting functions

The order of the optimal controller is equal to the order of the nominal plant model plus the order of the weights. The complexity of the controller is increase as the order of the weights increases.

The order of the weights should be kept reasonably low to reduce the order of th resulting optimal compensator and avoid potential convergence problems in the DK interactions.

Bibliography

Bibel, J. E., & Malyevac, D. S., Guidelines for the selection of weighting functions for h-infinity control (1992). ↩

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