Matched and unmatched uncertainties

Question

In control theory, according to the definition of matched uncertainty, the uncertainty enters the system through the same channel as the control.

If we consider the system: $$\dot x = F(x)+G(x)u+Δ(x, t),$$

where $F$ and $G$ are known functions comprising the nominal system and $Δ$ is an uncertain function known only to lie within some bounds. For example, we may know a function $ρ(x)$ so that $| Δ(x, t) |≤ ρ(x)$.

If we require that the uncertainty $Δ$ is of the form: $$Δ(x, t) = G(x) · \barΔ(x, t)$$

for some uncertain function $\barΔ$, this form is called the matching condition because it allows the system to be written as: $$\dot x = F(x)+G(x)[u+ \barΔ(x, t)]$$

where now the uncertainty $\barΔ$ is matched with the control $u$, that is, it enters the system through the same channel as the control.

What are the physical examples for matched and unmatched uncertainties?

(Levine, William S., ed. The control handbook. CRC press, 1996.)

Answer 1

One answer can be:

Magnetic suspension systems, in general, suffer from two principal components of uncertainty. The first of these are the parameter variations, where the electromagnetic coil characteristics change due to resistance heating, and the coefficients of the – curve drift with temperature. The second important source of uncertainty is the external interaction forces on the suspended body, which are typically unknown. Robustness to parameter variations relaxes the accuracy constraints on the force–current–airgap characterization of the electromagnets, while resistance to external forces determines the dynamic stiffness of the suspension. If the dynamical equations for the voltage–current characteristics of the power supply are not modeled in the suspension equations, then the parameter variations and the external disturbance force terms appear in the same level of differentiation as the system input, which happens to be the coil current in this case. Uncertainties that occur in the same order of differentiation as the control inputs are termed as matched uncertainties.

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The matching conditions are typically restrictive for arbitrary nonlinear systems and, if the voltage–current dynamics of the power supply are modeled in the suspension equations, then the uncertainty terms appear in levels of differentiation other than that of the voltage input to the system and are, therefore, termed as unmatched uncertainties. Modeling of the voltage– current characteristics and consequent compensation of the resulting unmatched uncertainties is essential to an efficient and cost-effective design of the magnetic suspension system. By ignoring the dynamics of the voltage–current characteristics, the controller may demand arbitrarily high current slewing rates which, in turn, lead to an oversized power supply system. Therefore, compensation of unmatched uncertainties in nonlinear control system design is an important step in the context of magnetic suspension systems.

Mittal, Samir, and Chia-Hsiang Menq. "Precision motion control of a magnetic suspension actuator using a robust nonlinear compensation scheme." Mechatronics, IEEE/ASME Transactions on 2.4 (1997): 268-280.