H-infinity methods in control theory

H∞ (i.e. "H-infinity") methods are used in <a href="/facts/Control_theory/aIuERpn6">control theory</a> to synthesize controllers to achieve stabilization with guaranteed performance. To use H∞ methods, a control designer expresses the control problem as a <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a> problem and then finds the controller that solves this optimization. H∞ techniques have the advantage over classical control techniques in that H∞ techniques are readily applicable to problems involving multivariate systems with cross-coupling between channels; disadvantages of H∞ techniques include the level of mathematical understanding needed to apply them successfully and the need for a reasonably good model of the system to be controlled. It is important to keep in mind that the resulting controller is only optimal with respect to the prescribed cost function and does not necessarily represent the best controller in terms of the usual performance measures used to evaluate controllers such as settling time, energy expended, etc. Also, non-linear constraints such as saturation are generally not well-handled. These methods were introduced into control theory in the late 1970s-early 1980s
by <a href="/facts/George_Zames/zz3KtycK">George Zames</a> (sensitivity minimization), J. William Helton (broadband matching),
and <a href="/facts/Allen_Tannenbaum/TRlHSN0n">Allen Tannenbaum</a> (gain margin optimization).
The phrase H∞ control comes from the name of the mathematical space over which the optimization takes place: H∞ is the <a href="/facts/Hardy_space/aN45dgx7">Hardy space</a> of <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>-valued functions that are <a href="/facts/Analytic_function/JhL3z8FP">analytic</a> and bounded in the open right-half of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> defined by Re(s) > 0; the H∞ norm is the <a href="/facts/Supremum/oXCKVopz">supremum</a> <a href="/facts/Singular_value_decomposition/8t3v6wk3">singular value</a> of the matrix over that space. In the case of a scalar-valued function, the elements of the Hardy space that extend continuously to the boundary and are continuous at infinity is the <a href="/facts/Disk_algebra/9QVWbISL">disk algebra</a>. For a matrix-valued function, the norm can be interpreted as a maximum gain in any direction and at any frequency; for <a href="/facts/Single-Input_and_Single-Output/Bk9Vyswk">SISO</a> systems, this is effectively the maximum magnitude of the frequency response. 
H∞ techniques can be used to minimize the closed loop impact of a perturbation: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance. Simultaneously optimizing robust performance and robust stabilization is difficult. One method that comes close to achieving this is <a href="/facts/H-infinity_loop-shaping/MWYmuUHf">H∞ loop-shaping</a>, which allows the control designer to apply classical loop-shaping concepts to the multivariable frequency response to get good robust performance, and then optimizes the response near the system bandwidth to achieve good robust stabilization.
Commercial software is available to support H∞ controller synthesis.

H-infinity methods in control theory open-in-new

H-infinity methods in control theory