Modeling Uncertainty

Dealing with and understanding the effects of uncertainty are important tasks for the control engineer. Reducing the effects of some forms of uncertainty (initial conditions, low-frequency disturbances) without catastrophically increasing the effects of other dominant forms (sensor noise, model uncertainty) is the primary job of the feedback control system.

Closed-loop stability is the way to deal with the (always present) uncertainty in initial conditions or arbitrarily small disturbances.

High-gain feedback in low-frequency ranges is a way to deal with the effects of unknown biases and disturbances acting on the process output. In this case, you are forced to use roll-off filters in high-frequency ranges to deal with high-frequency sensor noise in a feedback system.

Finally, notions such as gain and phase margins (and their generalizations) help quantify the sensitivity of stability and performance in the face of model uncertainty, which is the imprecise knowledge of how the control input directly affects the feedback variables.

At the heart of robust control is the concept of an uncertain LTI system. Model uncertainty arises when system gains or other parameters are not precisely known, or can vary over a given range. Examples of real parameter uncertainties include uncertain pole and zero locations and uncertain gains. You can also have unstructured uncertainties, by which is meant complex parameter variations satisfying given magnitude bounds.

With Robust Control Toolbox™ software you can create uncertain LTI models as MATLAB® objects specifically designed for robust control applications. You can build models of complex systems by combining models of subsystems using addition, multiplication, and division, as well as with Control System Toolbox™ commands like `feedback` and `lft`.

Robust Control Toolbox software has built-in features allowing you to specify model uncertainty simply and naturally. The primary building blocks, called uncertain elements (or uncertain Control Design Blocks) are uncertain real parameters and uncertain linear, time-invariant objects. These can be used to create coarse and simple or detailed and complex descriptions of the model uncertainty present within your process models.

Once formulated, high-level system robustness tools can help you analyze the potential degradation of stability and performance of the closed-loop system brought on by the system model uncertainty.

Summary of Robustness Analysis Tools

Function

Description

`ureal`

Create uncertain real parameter.

`ultidyn`

Create uncertain, linear, time-invariant dynamics.

`umargin`

Model uncertain gain and phase in a feedback loop.

`uss`

Create uncertain state-space object from uncertain state-space matrices.

`ufrd`

Create uncertain frequency response object.

`loopsens`

Compute all relevant open and closed-loop quantities for a MIMO feedback connection.

`diskmargin`

Compute loop-at-a-time as well as MIMO gain and phase margins for a multiloop system, including the simultaneous gain/phase margins.

`robgain`

Robustness performance of uncertain systems.

`robstab`

Compute the robust stability margin of a nominally stable uncertain system.

`wcgain`

Compute the worst-case gain of a nominally stable uncertain system.

`wcdiskmargin`

Compute worst-case (over uncertainty) loop-at-a-time disk-based gain and phase margins.