Data-driven load disturbance rejection

Question

1. What are the main differences between the proposed work and the works in Reference 14 and 15?

2. What is the role of the control designer?

3. What is the objective of the prediction error framework?

4. What is the prediction error for the linear predictor in the absence of noise?

Accepted Answer

The main differences between the proposed work and the works in Reference 14 and 15 are as follows. Firstly, the proposed work is not restricted to linearly parametrized controllers, allowing for the identification of the controller's poles using two different one-step ahead predictors. Secondly, the proposed work introduces the use of two different optimization algorithms: one that minimizes the norm of the one-step ahead prediction error and another that minimizes the correlation between the one-step ahead error and an external signal. These differences set the proposed work apart from the previous works and contribute to its unique approach in data-driven disturbance rejection with model matching.

Accepted Answer

The role of the control designer is to choose the parameter vector of the controller in order to obtain good performance for the closed-loop system. This involves selecting the appropriate values for the controller parameters to ensure that the closed-loop system meets the desired performance criteria. The control designer must consider factors such as stability, responsiveness, and robustness when designing the controller. By optimizing the controller parameters, the control designer can achieve the desired system behavior and ensure that the closed-loop system operates effectively and efficiently.

Accepted Answer

The objective of the prediction error framework is to identify the controller parameters using a dataset from an open-loop or closed-loop experiment. It involves a thought exercise where the real dataset is assumed to come from a closed-loop experiment with an ideal controller. By pretending that the virtual experiment is noise-free and a regulation setup, the virtual error and virtual control action can be determined. This allows for the identification of the dynamical relation between the input/output signals, leading to the data-driven identification of the controller's parameters. The controller structure includes an unknown portion that must be identified, while some fixed portion may be known or required, such as an integral action. The prediction error signal is minimized using different approaches, such as minimizing the 2norm of the error or the correlation with an instrumental variable, to estimate the controller's parameters.

Accepted Answer

In the absence of noise, the prediction error obtained from the linear predictor is given by EQUATION regardless if the data are collected in open-loop or closed-loop. This prediction error is linear on which simplifies the analysis and design of identification algorithms. By replacing (30) in (33), the linear prediction error may be written as EQUATION. Using these two equalities, we can calculate the noiseless prediction error for the linear predictor and propose a solution for the parameters estimate. Lemma 1 states that in the absence of noise, the prediction error obtained from the linear predictor is given by EQUATION regardless if the data are collected in open-loop or closed-loop. The proof involves considering both open-loop and closed-loop cases, and removing the noise-dependent term from the equations to obtain the noiseless prediction error. Theorem 1 further confirms that when Assumption 1 holds and the linear predictor is calculated from noiseless data, the ideal parameters map to the global minimum of EQUATION. This is because the error evaluates to zero, and so does the cost, which is quadratic. Therefore, the noiseless prediction error for the linear predictor is EQUATION, and the ideal parameters are estimated by EQUATION, which is the minimum of the quadratic function (72).

Accepted Answer

The nonlinear predictor equation is given by EQUATION, which makes the prediction error simply EQUATION. This predictor is non-linear with respect to the parameters, as some parameters appear in its denominator. The two proposed predictors are equivalent when all parameters are on the numerator of the controller, i.e., (, ) = 1. When the controller's denominator is fixed, the transfer operator becomes linearly parametrized, and solutions for identifying the controller have been proposed before. In the absence of noise, the prediction error from the non-linear predictor (35) is given by EQUATION, regardless of whether the data are collected open-loop or closed-loop.

Accepted Answer

Minimizing the 2-norm of the error can benefit controller parameter identification by utilizing well-known least squares and prediction error methods. This approach has been employed with data-driven methods such as VRFT and VDFT. By solving an optimization problem that minimizes the 2-norm of a filtered version of the error signal, the controller parameters can be estimated effectively. The filtered error signal, denoted as (, ), is obtained by filtering the prediction error through the filter (, ). This additional degree of freedom allows for modification of the objective function, enhancing the accuracy of parameter estimation. Users can choose between linear or non-linear predictors, resulting in two distinct algorithms for estimating controller parameters. The choice of predictor type depends on the specific requirements and characteristics of the system being analyzed. Overall, minimizing the 2-norm of the error provides a robust and efficient method for controller parameter identification, leading to improved system performance and stability.

Accepted Answer

When Assumption 1 holds and using the linear predictor, the ideal parameters can be estimated using the equation ( ). The parameters are calculated as (, )() and u () = (, ) u(). The linear predictor offers the advantage of having an analytic solution to the optimization problem (38), given by ( 50). This method is effective in identifying the optimal controller when the noise level is low, resulting in small bias and variance. However, when the data are collected with noise () 0, it becomes a classical errors-in-variables identification problem, leading to biased estimates if the noise is not negligible. In such cases, alternative solutions should be explored.

Accepted Answer

Filter design can be improved by considering the violation of Assumption 1 and using non-linear predictors. Theorem 4 suggests that a filter can be designed to force the minimum of the prediction error cost to match the minimum of the disturbance response cost. This can be achieved by using the equation (63) and approximating it with equation (65) when (, ) and d () are similar. The quantities involved in (65) can be estimated from data. However, an iterative optimization procedure is still needed due to the non-convexity of the problem. This approach allows for compensating for the violation of Assumption 1 and optimizing the filter design in the presence of noise.

Accepted Answer

In closed-loop experiments, the prediction error is represented by EQUATION, where (61) comes from (9). After removing the noise-dependent term from (61), the prediction error becomes the same as the one in (56). This error is crucial in determining the optimal parameters for the system.

Accepted Answer

To minimize the correlation between input and error, we can use the 2-norm of the cross-correlation between the one-step ahead prediction error and the experiment's excitation input. This involves calculating the cross-correlation function between the prediction error and another signal driving the experiment. A finite length approximation of the cross-correlation function is used, and the estimation of the controller's parameters is done by minimizing the 2-norm of this approximate correlation. The estimate is given by a specific equation, which depends on the data collected during an open-loop or closed-loop experiment.

Accepted Answer

When Assumption 1 doesn't hold, the filter design can be optimized by bringing the minima of the disturbance response and correlation costs closer together. This can be achieved by designing the filter to make the minimum of the correlation cost (72) close to the minimum of the disturbance response cost (12). Theoretical theorems and proofs provide guidance on achieving this optimization. For example, Theorem 7 suggests a filter that forces the correlation cost to be the same as the disturbance response cost for a given disturbance signal. Similarly, Theorem 10 proposes a filter for the non-linear predictor case. By comparing the cost functions and solving for the filter, it is possible to approximate the filter that makes the two functions the same. Remark 9 and 10 further discuss the approximation of the filter and the availability of necessary quantities for optimization. However, due to the non-convex nature of the cost function, an iterative optimization process is still required.

Accepted Answer

The frequency-domain version of disturbance response cost can be compared by equating equations (53) and (79) and solving for the filter. This comparison shows that the two costs are the same if the filter is given by equation (78). Remark 7 highlights that the filter (78) can be approximated by equation ( ) when ( , ) d ( ). Remark 8 explains that while all quantities in equation (80) are available or estimable from data, the filter depends on parameters even after approximation. To address this, an iterative optimization approach can be used, where a least squares problem is solved at each iteration. Initially, the filter's denominator is set to 1 in the first iteration, and then updated after each iteration. In the noiseless case, the squared cross power spectrum can be expressed as equations ( ) for open-loop data and ( ) for closed-loop data. These equalities can simplify the filter when using the disturbance to be rejected as the probing signal, i.e., () = () in open-loop or () = () in closed-loop.

Accepted Answer

In noiseless data, the prediction error becomes zero at the ideal parameters, regardless of open-loop or closed-loop experiment. This results in a correlation of zero between the input and prediction error, making c (, d) = 0 a global minimum of the cost (72). However, the cost (72) is non-convex, requiring iterative optimization with an initial guess close to the minimum parameters.

Accepted Answer

Linear feedback controllers can be tuned for disturbance rejection by employing the proposed methodology, which involves simulation case studies. The system simulated is a stable one with unitary static gain, and the control objective is to reject step disturbances in steady-state. This can be achieved by including an integrator in the controller structure. Within the data-driven framework, the process model is unknown, but it is presented for example purposes. The theory suggests that a broad range of controllers can be tuned using these methods. For the examples, controllers from the PID family are chosen due to their simplicity, power, and widespread use. A PIDF controller has an integrator and four parameters to be identified, while a simpler PI controller has the same fixed part. The complete controller must be causal for implementability. The ideal controller, obtained from the reference model, is a PIDF controller. If the available controller structure is PIDF, Assumption 1 holds, and the estimated controller is optimal. However, if the structure is not PIDF, the methods yield a controller that may not be optimal. A filter can be employed to estimate a controller closer to the optimum. These three cases are explored in the following sections.

Accepted Answer

In the matching case, noise affects controller identification by introducing errors in the signals collected in open-loop and closed-loop. This creates an errors-in-variables identification problem. When noise is present, a filter (, ) = d () is employed to reduce high frequency noise. Controller identification is performed using different methods, such as minimizing the norm of the prediction error or minimizing the correlation cost. The presence of noise makes the non-linear predictor less sensitive to noise and achieves better performance compared to the linear predictor. The correlation method is also less sensitive to noise and achieves smaller costs. Overall, noise impacts the accuracy of controller identification and the choice of method to minimize the cost function.

Accepted Answer

Mismatching controller structure leads to different disturbance responses and performance optimization. In real-life scenarios, the process model is not identified, causing performance loss. The algorithm minimizes prediction error's norm to identify controllers, achieving similar performance with linear and non-linear predictors. However, PI controllers perform worse than PIDF controllers due to the inability to achieve the reference model exactly. Open-loop data results in better performance than closed-loop data, but the difference is minimal. The best performance is obtained by minimizing the norm of the prediction error for open-loop data and using the correlation approach for closed-loop data.

Accepted Answer

The main idea of the presented system identification framework is to obtain virtual signals from experiments in open-loop or closed-loop, and then estimate the parameters of a restricted order controller. This approach aims to solve an optimization problem that does not depend on a process model. The framework proposes the use of two different one-step ahead predictors: one linear and another non-linear. Additionally, two different approaches are proposed to estimate the parameters of the controller, with the first minimizing the quadratic norm of the prediction error and the second minimizing the correlation between the prediction error and an external signal. When restricted order controllers are employed, an extra filter is proposed to reduce the difference between the obtained and prescribed load disturbance sensitivities. Simulations have shown that the proposed methods achieve very good performance for both full order and reduced order controllers.

Data-driven load disturbance rejection

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Most frequently asked questions

1. What are the main differences between the proposed work and the works in Reference 14 and 15?

2. What is the role of the control designer?

3. What is the objective of the prediction error framework?

4. What is the prediction error for the linear predictor in the absence of noise?

5. What is the nonlinear predictor equation?

6. How can minimizing the 2-norm of the error benefit controller parameter identification?

7. What are the ideal parameters estimated using the linear predictor?

8. How can filter design be improved?

9. What is the prediction error in closed-loop experiments?

10. How to minimize correlation between input and error?

11. How can filter design be optimized when Assumption 1 doesn't hold?

12. How can the frequency-domain version of disturbance response cost be compared?

13. What happens with non-linear predictor in noiseless data?

14. How can linear feedback controllers be tuned for disturbance rejection?

15. How does noise affect controller identification in matching case?

16. What is the impact of mismatching controller structure?

17. What is the main idea of the presented system identification framework?

Citations

The data-driven approach to classical control theory

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