# Closed-Loop Systems Model Estimation Methods (System Identification Toolkit)

Systems in many real-world applications contain feedback. Feedback is a process in which the output signal of a plant is passed, or fed back, to the input to regulate the next output. Systems without feedback are open-loop systems. Systems with feedback are closed-loop systems.

In an open-loop system, the stimulus signal and the output noise do not correlate with each other. In a closed-loop system, the stimulus signal correlates to the output noise. Though you can apply many open-loop model estimation methods to closed-loop data, not all open-loop model estimation methods handle the correlation between the stimulus signal and output noise well.

## Feedback in Systems

Feedback is common in control systems. With feedback, the system output corresponds to a reference input. Feedback also reduces the effect of input disturbances. One example of a closed-loop system is a system that regulates room temperature, as shown in the following figure. In this example, the reference input is the temperature *T*
_{set} at which you want the room to stay. The thermostat senses the actual temperature, *T*
_{actual}, of the room. Based on the difference between *T*
_{actual }and *T*
_{set}, the thermostat activates the heater or the air conditioner. The thermostat returns *T*
_{actual} as the feedback to compare again with *T*
_{set}. Then the thermostat uses the difference between *T*
_{actual }and _{
T
set to regulate the temperature at the next moment.}

You must verify if feedback exists before choosing a model estimation method because not all open-loop model estimation methods work correctly with closed-loop data.

Note You must know whether the data you collect is from an open-loop system or a closed-loop system according to the real-world system configuration. If you do not have such information, you can determine if feedback exists by using the SI Detect Feedback VI or by obtaining the impulse responses of a plant. You can use the Least Squares instances of the SI Estimate Impulse Response VI to estimate the impulse response of a plant. |

The following figure shows a comparison of the impulse responses of the plant in a closed-loop system and an open-loop system.

The values outside the** upper limit** and **lower limit** range at the negative lag, which appears between –10 and 0 on the x-axis, are considered significant values. Significant values in the impulse response at negative lags imply feedback in data. As shown in the following figure, significant values exist in the **Closed-loop data** plot. Therefore, feedback exists in the closed-loop system. No significant impulse response values exist in the **Open-loop data** plot. Thus, feedback does not exist in the open-loop system.

## Understanding Closed-Loop Model Estimation Methods

Closed-loop model estimation methods use data from a closed-loop system to build a model for a plant that a controller regulates. The following figure shows a system that consists of a plant and a controller. In this system, *G*
_{0 }is the plant, *F*
_{y} is the controller, *H* is the stochastic part of the plant, *u* is the stimulus signal, *y* is the response signal, *r* is the reference signal that is an external signal, and *e* is the output noise. In control engineering, this system is known as a feedback-path closed-loop system, which is a typical closed-loop system.

In some cases, the controller comes before the plant in a closed-loop system. This system is known as a feedforward-path closed-loop system, as shown in the following figure.

Depending on the amount of prior knowledge you have about the feedback, the controller, and the reference signal of a system, you can categorize closed-loop model estimation approaches into the following three groups:

- Direct identification—Uses the stimulus and response signals to identify the plant model as if the plant is in an open-loop system. You can apply the direct identification approach to compute all types of models except state-space models by using the LabVIEW System Identification Toolkit.
- Indirect identification—Identifies a closed-loop system by using the reference signal and the response signal and then determines the plant model based on the known controller of the closed-loop system. You can apply the indirect identification approach to compute transfer function models.
- Joint input-output identification—Considers the stimulus signal and the response signal as outputs of a cascaded system. The reference signal and the noise jointly perturb the system, and the plant model is identified from this joint input-output system. You can apply the joint input-output identification approach to compute transfer function models.

You can choose a suitable model identification approach according to the information you have about the closed-loop system. The following table summarizes the information you must have to use each identification approach.

Stimulus Signal | Response Signal | Reference Signal | Controller Information | |
---|---|---|---|---|

Direct |
X | X | — | — |

Indirect |
— | X | X | X |

Joint Input-Output |
X | X | X | — |

With the LabVIEW System Identification Toolkit, you can choose to use the direct, indirect, or joint input-output identification approaches for different types of closed-loop systems. The direct identification approach supports single-input single-output (SISO), multiple-input single-output (MISO), and multiple-input multiple-output (MIMO) systems. The indirect and joint input-output identification approaches support SISO systems only.