H-Infinity Performance

Performance as Generalized Disturbance Rejection

The modern approach to characterizing closed-loop performance objectives is to measure the size of certain closed-loop transfer function matrices using various matrix norms. Matrix norms provide a measure of how large output signals can get for certain classes of input signals. Optimizing these types of performance objectives over the set of stabilizing controllers is the main thrust of recent optimal control theory, such as L₁, H₂, H_∞, and optimal control. Hence, it is important to understand how many types of control objectives can be posed as a minimization of closed-loop transfer functions.

Consider a tracking problem, with disturbance rejection, measurement noise, and control input signal limitations, as shown in Generalized and Weighted Performance Block Diagram. K is some controller to be designed and G is the system you want to control.

Typical Closed-Loop Performance Objective

Plant K and controller G in a unit feedback configuration. The system also has a reference input to K, a control input measurement signal at the output of K, and external force disturbance input to the G, noise signal injected between the plant output and controller input. The tracking error is measured as the output of G minus the reference signal.

A reasonable, though not precise, design objective would be to design K to keep tracking errors and control input signal small for all reasonable reference commands, sensor noises, and external force disturbances.

Hence, a natural performance objective is the closed-loop gain from exogenous influences (reference commands, sensor noise, and external force disturbances) to regulated variables (tracking errors and control input signal). Specifically, let T denote the closed-loop mapping from the outside influences to the regulated variables:

You can assess performance by measuring the gain from outside influences to regulated variables. In other words, good performance is associated with T being small. Because the closed-loop system is a multiinput, multioutput (MIMO) dynamic system, there are two different aspects to the gain of T:

Spatial (vector disturbances and vector errors)
Temporal (dynamic relationship between input/output signals)

Hence the performance criterion must account for

Relative magnitude of outside influences
Frequency dependence of signals
Relative importance of the magnitudes of regulated variables

So if the performance objective is in the form of a matrix norm, it should actually be a weighted norm

∥W_LTW_R∥

where the weighting function matrices W_L and W_R are frequency dependent, to account for bandwidth constraints and spectral content of exogenous signals. A natural (mathematical) manner to characterize acceptable performance is in terms of the MIMO ∥· ∥_∞ (H_∞) norm. See Interpretation of H-Infinity Norm for an interpretation of the H_∞ norm and signals.

Interconnection with Typical MIMO Performance Objectives

The closed-loop performance objectives are formulated as weighted closed-loop transfer functions that are to be made small through feedback. A generic example, which includes many relevant terms, is shown in block diagram form in Generalized and Weighted Performance Block Diagram. In the diagram, G denotes the plant model and K is the feedback controller.

Generalized and Weighted Performance Block Diagram

Feedback control system with plant G and controller K, with weighting functions appended to all signals in the system.

The blocks in this figure might be scalar (SISO) and/or multivariable (MIMO), depending on the specific example. The mathematical objective of H_∞ control is to make the closed-loop MIMO transfer function T_ed satisfy ∥T_ed∥_∞ < 1. The weighting functions are used to scale the input/output transfer functions such that when ∥T_ed∥_∞ < 1, the relationship between $\tilde{d}$ and $\tilde{e}$ is suitable.

Performance requirements on the closed-loop system are transformed into the H_∞ framework with the help of weighting or scaling functions. Weights are selected to account for the relative magnitude of signals, their frequency dependence, and their relative importance. This is captured in the figure above, where the weights or scalings [W_cmd, W_dist,W_snois] are used to transform and scale the normalized input signals [d₁,d₂,d₃] into physical units defined as [d₁, d₂, d₃]. Similarly weights or scalings [W_act, W_perf₁,W_perf₂] transform and scale physical units into normalized output signals [e₁, e₂, e₃]. An interpretation of the signals, weighting functions, and models follows.

Signal	Meaning
d₁ ${\tilde{d}}_{1}$	Normalized reference command Typical reference command in physical units
d₂ ${\tilde{d}}_{2}$	Normalized exogenous disturbances Typical exogenous disturbances in physical units
d₃ ${\tilde{d}}_{3}$	Normalized sensor noise Typical sensor noise in physical units
e₁ ${\tilde{e}}_{1}$	Weighted control signals Actual control signals in physical units
e₂ ${\tilde{e}}_{2}$	Weighted tracking errors Actual tracking errors in physical units
e₃ ${\tilde{e}}_{3}$	Weighted plant errors Actual plant errors in physical units

Signal

Meaning

d₁

${\tilde{d}}_{1}$

Normalized reference command

Typical reference command in physical units

d₂

${\tilde{d}}_{2}$

Normalized exogenous disturbances

Typical exogenous disturbances in physical units

d₃

${\tilde{d}}_{3}$

Normalized sensor noise

Typical sensor noise in physical units

e₁

${\tilde{e}}_{1}$

Weighted control signals

Actual control signals in physical units

e₂

${\tilde{e}}_{2}$

Weighted tracking errors

Actual tracking errors in physical units

e₃

${\tilde{e}}_{3}$

Weighted plant errors

Actual plant errors in physical units

W_cmd

W_cmd is included in H_∞ control problems that require tracking of a reference command. W_cmd shapes the normalized reference command signals (magnitude and frequency) into the actual (or typical) reference signals that you expect to occur. It describes the magnitude and the frequency dependence of the reference commands generated by the normalized reference signal. Normally W_cmd is flat at low frequency and rolls off at high frequency. For example, in a flight control problem, fighter pilots generate stick input reference commands up to a bandwidth of about 2 Hz. Suppose that the stick has a maximum travel of three inches. Pilot commands could be modeled as normalized signals passed through a first-order filter:

$W_{c m d} = \frac{3}{\frac{1}{2 \cdot 2 π} s + 1} .$

W_model

W_model represents a desired ideal model for the closed-looped system and is often included in problem formulations with tracking requirements. Inclusion of an ideal model for tracking is often called a model matching problem, i.e., the objective of the closed-loop system is to match the defined model. For good command tracking response, you might want the closed-loop system to respond like a well-damped second-order system. The ideal model would then be

$W_{m o d e l} = \frac{ω^{2}}{s^{2} + 2 ζ ω + ω^{2}}$

for specific desired natural frequency ω and desired damping ratio ζ. Unit conversions might be necessary to ensure exact correlation between the ideal model and the closed-loop system. In the fighter pilot example, suppose that roll-rate is being commanded and 10º/second response is desired for each inch of stick motion. Then, in these units, the appropriate model is:

$W_{m o d e l} = 10 \frac{ω^{2}}{s^{2} + 2 ζ ω + ω^{2}} .$

W_dist

W_dist shapes the frequency content and magnitude of the exogenous disturbances affecting the plant. For example, consider an electron microscope as the plant. The dominant performance objective is to mechanically isolate the microscope from outside mechanical disturbances, such as ground excitations, sound (pressure) waves, and air currents. You can capture the spectrum and relative magnitudes of these disturbances with the transfer function weighting matrix W_dist.

W_perf1

W_perf₁ weights the difference between the response of the closed-loop system and the ideal model W_model. Often you might want accurate matching of the ideal model at low frequency and require less accurate matching at higher frequency, in which case W_perf1 is flat at low frequency, rolls off at first or second order, and flattens out at a small, nonzero value at high frequency. The inverse of the weight is related to the allowable size of tracking errors, when dealing with the reference commands and disturbances described by W_cmd and W_dist.

W_perf2

W_perf2 penalizes variables internal to the process G, such as actuator states that are internal to G or other variables that are not part of the tracking objective.

W_act

W_act is used to shape the penalty on control signal use. W_act is a frequency varying weighting function used to penalize limits on the deflection/position, deflection rate/velocity, etc., response of the control signals, when dealing with the tracking and disturbance rejection objectives defined above. Each control signal is usually penalized independently.

W_snois

W_snois represents frequency domain models of sensor noise. Each sensor measurement feedback to the controller has some noise, which is often higher in one frequency range than another. The W_snois weight tries to capture this information, derived from laboratory experiments or based on manufacturer measurements, in the control problem. For example, medium grade accelerometers have substantial noise at low frequency and high frequency. Therefore the corresponding W_snois weight would be larger at low and high frequency and have a smaller magnitude in the mid-frequency range. Displacement or rotation measurement is often quite accurate at low frequency and in steady state, but responds poorly as frequency increases. The weighting function for this sensor would be small at low frequency, gradually increase in magnitude as a first- or second-order system, and level out at high frequency.

H_sens

H_sens represents a model of the sensor dynamics or an external antialiasing filter. The transfer functions used to describe H_sens are based on physical characteristics of the individual components. These models might also be lumped into the plant model G.

This generic block diagram has tremendous flexibility and many control performance objectives can be formulated in the H_∞ framework using this block diagram description.

Robustness in the H-Infinity Framework

Performance and robustness tradeoffs in control design were discussed in the context of multivariable loop shaping in Tradeoff Between Performance and Robustness. In the H_∞ control design framework, you can include robustness objectives as additional disturbance to error transfer functions — disturbances to be kept small. Consider the following figure of a closed-loop feedback system with additive and multiplicative uncertainty models.

Plant G and controller K in unit negative feedback configuration. A multiplicative uncertainty ΔM(s) takes the output of K as its input w1 and produces output z1, which is added to the input of G. An additive uncertainty ΔA(s) takes the input of G as its input w2 and produces output z2, which is added to the plant output to produce system output y.

The transfer function matrices are defined as:

$\begin{array}{l} T F {(s)}_{z_{1} \to w_{1}} = T_{I} (s) = K G {(I + G K)}^{- 1} \\ T F {(s)}_{z_{2} \to w_{2}} = K S_{O} (s) = K {(I + G K)}^{- 1} \end{array}$

where T_I(s) denotes the input complementary sensitivity function and S_O(s) denotes the output sensitivity function. Bounds on the size of the transfer function matrices from z₁ to w₁and z₂ to w₂ ensure that the closed-loop system is robust to multiplicative uncertainty, Δ_M(s), at the plant input, and additive uncertainty, Δ_A(s), around the plant G(s). In the H_∞ control problem formulation, the robustness objectives enter the synthesis procedure as additional input/output signals to be kept small. The interconnection with the uncertainty blocks removed follows.

Control structure with additional outputs w1 and w2 measuring the output of K and input of G, respectively, and additional inputs z1 and z2 injected at the input and output of G, respectively.

The H_∞ control robustness objective is now in the same format as the performance objectives, that is, to minimize the H_∞ norm of the transfer matrix from z, [z₁,z₂], to w, [w₁,w₂].

Weighting or scaling matrices are often introduced to shape the frequency and magnitude content of the sensitivity and complementary sensitivity transfer function matrices. Let W_M correspond to the multiplicative uncertainty and W_A correspond to the additive uncertainty model. Δ_M(s) and Δ_A(s) are assumed to be a norm bounded by 1, i.e., |Δ_M(s)|<1 and |Δ_A(s)|<1. Hence as a function of frequency, |W_M(jω)| and |W_A(jω)| are the respective sizes of the largest anticipated additive and multiplicative plant perturbations.

The multiplicative weighting or scaling W_M represents a percentage error in the model and is often small in magnitude at low frequency, between 0.05 and 0.20 (5% to 20% modeling error), and growing larger in magnitude at high frequency, 2 to 5 ((200% to 500% modeling error). The weight will transition by crossing a magnitude value of 1, which corresponds to 100% uncertainty in the model, at a frequency at least twice the bandwidth of the closed-loop system. A typical multiplicative weight is

$W_{M} = 0.10 \frac{\frac{1}{5} s + 1}{\frac{1}{200} s + 1} .$

By contrast, the additive weight or scaling W_A represents an absolute error that is often small at low frequency and large in magnitude at high frequency. The magnitude of this weight depends directly on the magnitude of the plant model, G(s).

Numeric Considerations

Do not choose weighting functions with poles very close to s = 0 (z = 1 for discrete-time systems). For instance, although it might seem sensible to choose W_cmd = 1/s to enforce zero steady-state error, doing so introduces an unstable pole that cannot be stabilized, causing synthesis to fail. Instead, choose W_cmd = 1/(s + δ). The value δ must be small but not very small compared to system dynamics. For instance, for best numeric results, if your target crossover frequency is around 1 rad/s, choose δ = 0.0001 or 0.001. Similarly, in discrete time, choose sample times such that system and weighting dynamics are not more than a decade or two below the Nyquist frequency.