What is the H-infinity norm?

What is the H-infinity norm?

The phrase H∞ control comes from the name of the mathematical space over which the optimization takes place: H∞ is the Hardy space of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(s) > 0; the H∞ norm is the maximum singular value of the function over …

How do you find the norm of a transfer function?

The H2 norm of a continuous-time system with transfer function H(s) is given by: ‖ H ‖ 2 = 1 2 π ∫ − ∞ ∞ Trace [ H ( j ω ) H H ( j ω ) ] d ω . For a discrete-time system with transfer function H(z), the H2 norm is given by: ‖ H ‖ 2 = 1 2 π ∫ − π π Trace [ H ( e j ω ) H H ( e j ω ) ] d ω .

What is H norm?

Intitutively, this norm gives the maximum amplification of an input signal at the output that is caused by the system over all possible input signals. So, in this case the H∞ norm of this system is the worst-case effect of unwanted signals over error signals.

What is the H-infinity control problem?

The problem involved is the general H-infinity control problem, the so-called standard problem. It concerns the construction of a stabilizing controller with additional constraints on the maximum of the norm of the closed loop transfer function, taken over the values of the argument on the imaginary line.

What is H2 and H-infinity?

The H2 and H-infinity control theories involve suppressing the sensitivity matrix transfer function at the lower frequencies for high gain performance and suppressing the transmissivity at higher frequencies, i.e., loop shaping. the first design is an optimum design which imposed no limitations on control input.

What is H-infinity space?

The space H∞ is defined as the vector space of bounded holomorphic functions on the disk, with the norm.

What is norm in control system?

Norms for Signals and Systems. One way to describe the performance of a control system is in terms of the size of certain signals of interest. For example, the performance of a tracking system could be measured by the size of the error signal.

What are examples of norms?

Some examples of general class norms are: “be a good neighbor,” “respect others and yourself,” and “be kind.” Norms written at a specific level identify distinct behaviors, such as “raise hand before talking,” or “walk in the hallways,” and are usually only applicable in particular situations.

Which is an interpretation of the transfer function norm?

Both of these transfer function norms have input/output time-domain interpretations. If, starting from initial condition x (0) = 0, two signals d and e are related by For d, a unit intensity, white noise process, the steady-state variance of e is ∥ T ∥ 2.

Which is the best way to interpret the norm?

Because many different objectives are being lumped into one matrix and the associated cost is the norm of the matrix, it is important to use frequency-dependent weighting functions, so that different requirements can be meaningfully combined into a single cost function. Diagonal weights are most easily interpreted.

Is there a steady state interpretation of H-infinity?

As you would expect, a sinusoidal, steady-state interpretation of ∥ T ∥ ∞ is also possible: For any frequency , any vector of amplitudes , and any vector of phases , with ∥ a ∥ 2 ≤ 1, define a time signal Applying this input to the system T results in a steady-state response of the form

Which is the norm of a scalar signal T?

There are several ways of defining norms of a scalar signal e ( t) in the time domain. We will often use the 2-norm, ( L2 -norm), for mathematical convenience, which is defined as If this integral is finite, then the signal e is square integrable, denoted as e ∊ L 2.