# Hysteretic controller

We can see that the hysteretic controller is a special case of other control techniques. For example, "sliding mode control" usually uses two state variables to determine one switching variable (switch ON or OFF). So the hysteretic controller is a special case of "1-dimensional" sliding mode. In general, there are many techniques under the name of "geometric control" that can be used to prove the stability of a general N-state system under a given switching rule. So I believe that you can apply some of these techniques to prove the stability of the hysteretic controller, although I have not tried to do this myself. The book "elements of power electronics" by Krein discusses that in chapter 17.

But I can talk more about one technique that I have used and in my opinion is the most general and elegant technique for non-linear systems. It is based on Lyapunov stability theory. You can use this technique to determine a switching rule to a general circuit with an arbitrary number of switches and state variables. It can be applied to the simple case of the hysteretic controller (i.e. 1 state variable, 1 switching variable) to verify if the system is stable and what are the conditions for stability. I have done this and verified that it is possible to prove the stability of hysteretic controllers, imposing very weak constraints (and, of course, no linearization needed). In a nutshell, to prove the system stable, you have to find a Lyapunov function for it.

What can expand is to go beyond a simple window comparator for hysteretic control.

#1) control bands, or switching limits can be variable and also part of a loop, especially if one wants to guarantee a nearly fixed frequency.

#2) using a latch or double latch after the comparator(s), one can define (remember) the state and define operations such as incorporating fixed Ton or Toff periods for additional time control... this permits the "voltage boost" scenario you previously said could not be done. This also prevents common "chaos" operation and noise susceptibility that others experience with simpler circuits.

#3) additional logic can assure multiphase topologies locked to a system clock and compete very well with typical POL buck regulators for high-end processors that require high di/dt response.

Time or state domain control systems such as this, can have great advantages over typical topologies. There really is no faster control method that provides a quicker load response without complete predictive processing, yet that can also be applied to hysteretic control.

But I can talk more about one technique that I have used and in my opinion is the most general and elegant technique for non-linear systems. It is based on Lyapunov stability theory. You can use this technique to determine a switching rule to a general circuit with an arbitrary number of switches and state variables. It can be applied to the simple case of the hysteretic controller (i.e. 1 state variable, 1 switching variable) to verify if the system is stable and what are the conditions for stability. I have done this and verified that it is possible to prove the stability of hysteretic controllers, imposing very weak constraints (and, of course, no linearization needed). In a nutshell, to prove the system stable, you have to find a Lyapunov function for it.

What can expand is to go beyond a simple window comparator for hysteretic control.

#1) control bands, or switching limits can be variable and also part of a loop, especially if one wants to guarantee a nearly fixed frequency.

#2) using a latch or double latch after the comparator(s), one can define (remember) the state and define operations such as incorporating fixed Ton or Toff periods for additional time control... this permits the "voltage boost" scenario you previously said could not be done. This also prevents common "chaos" operation and noise susceptibility that others experience with simpler circuits.

#3) additional logic can assure multiphase topologies locked to a system clock and compete very well with typical POL buck regulators for high-end processors that require high di/dt response.

Time or state domain control systems such as this, can have great advantages over typical topologies. There really is no faster control method that provides a quicker load response without complete predictive processing, yet that can also be applied to hysteretic control.

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