Control robustness refers to the ability of a control system to maintain satisfactory performance despite changes in the system or disturbances affecting it. There are two main aspects to control robustness: sensitivity and stability.

Sensitivity refers to the degree to which a control system is affected by changes in its inputs or parameters. A control system with high sensitivity will experience large variations in its output in response to small changes in the inputs or parameters. Conversely, a control system with low sensitivity will be able to maintain more stable output despite changes in its inputs or parameters.

Stability refers to the ability of a control system to remain within a desired range of operation, without oscillations or instability. A control system with good stability will be able to maintain its output within the desired range, even in the presence of disturbances or changes in the system. However, a control system with poor stability may experience oscillations or instability, leading to undesirable performance.

Achieving robust control requires a good understanding of the system dynamics, as well as careful selection of control strategies and tuning of control parameters. A robust control system will be able to maintain satisfactory performance even in the face of uncertainty, changes, or disturbances.

# Robustness Plots

Robustness plots, also known as Nichols plots or frequency response plots, are graphical representations of a control system’s sensitivity and stability. They are used to evaluate the robustness of a control system and to identify potential issues that may cause instability or poor performance.

A typical robustness plot consists of two graphs. The first graph shows the magnitude of the open-loop transfer function (OLTF) of the control system as a function of frequency. The second graph shows the phase angle of the OLTF as a function of frequency. The OLTF represents the transfer function of the system when there is no feedback loop.

The robustness plot is obtained by plotting the OLTF on a logarithmic scale, with frequency on the x-axis and magnitude or phase angle on the y-axis. The plot is then overlaid with lines that represent the limits of stability and sensitivity.

The stability limit is represented by a line that indicates the frequency at which the phase angle of the OLTF is equal to -180 degrees. If the OLTF crosses this line, the system is unstable. The sensitivity limit is represented by a line that indicates the frequency at which the magnitude of the OLTF is equal to 0 dB (unity gain). If the OLTF crosses this line, the system is highly sensitive to changes in the input signal.

By analyzing the robustness plot, engineers can identify potential issues with the control system and take corrective action. For example, if the OLTF crosses the stability limit, the system may require additional damping or gain adjustments to improve stability. If the OLTF crosses the sensitivity limit, the system may require additional filtering or other measures to reduce sensitivity to changes in the input signal.

Robustness plots are valuable tools for evaluating and optimizing control systems, particularly those that are complex or highly sensitive. They allow engineers to quickly identify potential issues and make informed decisions about how to improve system performance.

## Dynamic Parameters

Almost all control loops include dead time, which is a critical factor affecting control performance and robustness. The total loop dead time is the sum of small dead times and transportation lags in the loop.

The shape of the marginal stability curve is affected by changes in time constants, which can alter low-frequency stability. Larger lags and integration times have a more significant impact on the upper left section of the curve, while high-frequency components, such as the derivative mode settings of the controller, mostly affect the lower right portion.

System gain, which encompasses all gains in the loop, including process and controller gain, is plotted on the other axis of the robustness plot. Adjusting any of the gains will shift the marginal stability curve left or right, but the shape of the curve remains the same.

Increasing the loop gain moves the operating point of the loop closer to the marginal stability curve and toward instability. As the loop dynamics move down and to the right on the marginal stability curve, the frequency of sustained oscillation increases.

## Control Design with Nichols Plots

Designing a control loop with Nichols plot involves the following steps:

- Identify the process model: Determine the transfer function or process model for the system that you want to control. This can be done through experimental data or theoretical analysis.
- Sketch the Nichols plot: Plot the frequency response of the open-loop transfer function on a Nichols chart. The horizontal axis represents the magnitude of the frequency response in decibels, and the vertical axis represents the phase angle in degrees.
- Determine the stability margins: Analyze the Nichols plot to determine the stability margins of the system. The gain margin is the amount of gain that can be added before the system becomes unstable, and the phase margin is the amount of phase shift that can be added before the system becomes unstable. A stable system will have both a positive gain margin and a positive phase margin.
- Determine the required controller gain and phase shift: Using the Nichols plot, determine the required gain and phase shift of the controller to achieve the desired stability margins. This can be done by selecting a point on the Nichols plot that corresponds to the desired margins and finding the gain and phase shift values at that point.
- Choose a suitable controller type: Based on the required gain and phase shift, choose a suitable controller type, such as a proportional-integral-derivative (PID) controller or a lead-lag compensator.
- Tune the controller parameters: Once the controller type has been chosen, tune the controller parameters to achieve the desired response characteristics, such as settling time, overshoot, and steady-state error.
- Verify the design: Verify the design by simulating the closed-loop response of the system and testing it under various operating conditions. If necessary, adjust the controller parameters to improve the performance of the system.

# Conclusion

There is always a trade-off between loop sensitivity and stability in control loop design. When tuning a control loop, it is important to strive for a balance between these two objectives. Robustness plots can be useful graphical tools for this purpose, as they can illustrate the safety margin that the control system has before reaching instability.