Identification of Closed Loop Systems

Often it is necessary to identify a system that must operate in a closed-loop fashion under some type of feedback control. This may be due to safety reasons, an unstable plant that requires control, or the expense required to take a plant offline for test. In these cases it is necessary to perform closed-loop identification.

There are three basic approaches to closed-loop identification. These approaches are direct, indirect, and joint input-output. In this article we outline each approach and the system identification techniques that may be used to implement them.

Direct

The first method of interest is the Direct Approach. In this method we measure the output of the system y(t) and the input to the plant u(t), ignoring any feedback and the reference signal, to obtain the model. This is illustrated in Figure 1. This has the advantage of requiring no knowledge about the feedback in the system and becomes an open-loop identification problem.

The suggested system identification model structures when using this method are ARX, ARMAX and state-space models. Optimal accuracy occurs if the chosen model structure contains the true system (including the noise properties) and the main drawback to the method is that a poor noise model can introduce bias into the model. This bias will be small when any or all of the following hold

· The noise model is representative of the actual noise
· The feedback contribution to the input spectrum is small
· The signal to noise ratio is high

Spectral analysis will not provide correct results in the closed-loop case when using the direct approach so avoid non-parametric methods of identification such as impulse response and bode response estimation.

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Figure 1 Direct Approach to Closed-Loop System Identification.

Indirect

The second method of interest in closed-loop identification is the Indirect Approach as shown in Figure 2. In this method we identify the closed loop system (Gcl) using measurements of the reference input r(t) and the output y(t) and retrieve the plant model making use of a known regulator structure. The transfer function for the open loop plant G, with regulator H, can be retrieved from



The advantages in using the indirect approach are that any method will work in determining the closed-loop transfer function Gcl and the need for a accurate representation of the noise model is alleviated. The main disadvantage is that any error in H (including deviations due to saturations or anti-windup logic) will be imposed directly into G resulting in bias errors.

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Figure 2 Indirect Approach to Closed-Loop System Identification.

Joint Input-Output

The last method is the Joint Input-Output Approach. As shown in Figure 3, we consider the plant input u(t) and the system output y(t) as outputs of the system. The inputs to the system are the reference signal r(t) and the noise signal v(t).

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Figure 3 Joint Input-Output Approach to Closed-Loop System Identification.

This identification method results in a multidimensional system of the form

Where the system matrix A is comprised of two models, the closed-loop model Gcl and the model relating u(t) to r(t), Gru. The plant model, G, is then estimated from the relation

This approach is advantageous because the regulator structure is not needed nor is an accurate noise model necessary. It suffers from the disadvantages of requiring additional acquisition hardware (sensors) and requires acquiring a greater quantity of data.

When using the indirect and joint input-output methods, the reference signal r(t) should be as informative as possible. This means it should provide good spectral coverage of the domain of interest. This may be done by adjustments to the system set points (or adjustments to the regulator) as much as allowed by the system being identified.

Conclusion

It is often necessary to perform identification under closed-loop conditions to increase safety or reduce the costs of the modeling. The three approaches outlined in this article provide accurate estimations of plant dynamics under feedback control using simple measurements. Using the LabVIEW System Identification Toolkit provides the necessary identification algorithms to aid in these closed-loop identification problems.

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