# Networked recursive identification of nonlinear systems

### Background

Networked control systems have now been studied extensively for almost two decades. The focus has to a large extent been on the effect of quantization of control and feedback signals when plants are remotely feedback controlled. More recently there has been a renewed focus on the effects of delay on networked control systems, e.g. in wireless data flow control. Many networked controllers are model based, meaning that identification methods that can operate in a networked environment are needed as well. This has become more important with the standardization of the new 5G wireless systems. These systems are expected to enable a number of new use cases in the fields of industrial control and augmented reality with force feedback. Consequently, the study of identification methods that account for delay and quantization is of increasing importance.

The present research is performed on networked recursive identification, with a focus on nonlinear identification methods. The wish list for such algorithms, considering e.g. 5G wireless systems, include the following requirements

- Ability to identify networked delays, in combination with the plant dynamics.
- Ability to identify the plant dynamics, based on quantized control and feedback signals.
- Guaranteed convergence and stability properties.
- Low computational complexity.

The first and second requirements result since delay will be present as a part of the dynamics when networked identification and feedback control is performed over 5G wireless networks, at least for high bandwidth applications. The loop delay may become as low as <1 ms in so called ultra-reliable low latency communication (URLLC) and for critical machine type communication (C-MTC). Due to the high capacity of the 5G networks quantization effects are not as important as earlier, however they may still be present in case many users are served simultaneously. The latter is also a reason why a low computational complexity is preferrable. Further, since many users are expected, manual interaction may not be possible. Therefore the global convergence and stability properties of the applied algorithms become very important since it is only with such guarantees that the algorithms can be assumed to always deliver good performance.

### Recursive identification with non-uniform output quantization

The paper 11 presents an algorithm that is capable of recursive identification of the impulse response dynamics of a linear system, subject to simultaneous coarse, non-uniform quantization of the output signal and delay. The quantizer is assumed to be fix and known, with arbitrary quantization steps and switch points. The key idea used in the method is the replacement of the exact gradient with a smooth approximation. The algorithm is depicted in a networked identification setting in Fig. 1.

*Figure 1. The block diagram of the identified networked system.*

The model of the plant appears in Fig. 2.

*Figure 2: Block diagram of a dynamic system with output quantization. *

The algorithm has a very low computational complexity since it is of stochastic gradient type. A very important property of the scheme is that it is globally convergent to a perfect input-output setting of the identified impulse response dynamics, under relatively mild conditions. The algorithm hence meets all the above requirements. The paper 12 treats an IIR variant of the same algorithm, however then only local convergence has been proved. The performance of the algorithm was illustrated in a former research project on recursive identification of systems with output quantization.

An important remaining question concerns parametric convergence, i.e. the question if conditions that guarantee convergence to the true and unique parameter vector of the system can be found. This is not the same property as the input-output convergence discussed in 11. The papers 6 and 7 attempt to bring further clarity to this issue, by studying the conditions behind the convergence. The paper 7 focuses on the requirement of a monotone quantizer in the analysis of 11. The analysis defines a low order system with one single impulse response parameter being equal to 1 and a quantizer with three levels that is not monotone. The associated ODE analysis method pioneered by L. Ljung is then applied for convergence analysis. In the low order case it turns out that the average updating directions of the first order version of the algorithm of 11 can be analytically calculated. This opens up for a study of the convergence properties by means of numerical solution of the associated ODE. Fig. 3 and Fig. 4 illustrate the evolution of the associated ODE for different initial conditions, together with the evolution of the low adaptation gain estimates of the algorithm. The software package 10 was used to obtain parts of the results.

*Figure 3. Convergence to the unique true parameter vector.*

*Figure 4. Convergence to one of two mirrored parameter vectors, one being the true parameter vector.*

Both figures confirm that the associated ODE describes the asymptotic low gain evolution of the algorithm. As can be seen in Fig. 3., the associated ODE has a single fix point corresponding to the true parameter vector, despite the fact that the quantizer is not monotone. This results proves that:

- A monotone quantizer is not a necessary requirement for global parametric convergence.

Fig.4, on the other hand, illustrates a case where the associated ODE has fixed points equal to +/-1, i.e. global parametric convergence does not hold. Using similar techniques of simulation of associated ODEs, the paper 6 studies the effect of the input signals distribution and the coupling with the system gain and the quantizer. In summary, the paper 6 proves that at least the following additional restrictions need to be introduced in order to perform global parametric convergence proof:

- Input signals with discrete distributions need to excluded (in case the output signal is quantized).
- The input signal, the system gain and the quantizer must be such that there is signal energy in at least one switch point of the quantizer.

### Recursive identification of nonlinear state space models with delay

The paper 8 adresses a nonlinear identification problem, assuming the plant to be modeled by a general nonlinear differential equation model with a delayed output measurement. A 5G wireless networked setting where that model is useful is depicted in Fig. 5.

*Figure 5. The block diagram of the identified nonlinear networked system.*

An assumption of time invariance of the plant is needed in order to merge all delays of the loop into the output of the model. The dynamic model is formulated in state space form, with only one right hand side component defining the nonlinear ODE. A polynomial parameterization of the nonlinear right hand side component of the ODE is applied.

Based on the model an approximate recursive prediction error identification algorithm is then derived. In order to model the effect of delay, multiple models are used for each integer delay defined by one sampling period. The fractional component of the delay is obtained by interpolation between multiple models, adjacent to the running delay estimate of the complete model. The discretization in time is handled by an Euler discretization method.

A case where a first order nonlinear system is identified is depicted in Fig. 6 – Fig. 8. In this case the delay of the networked system is 1.22 s. As can be seen the identified delay is very close to the true one, a fact that also holds for the remaining parameters, c.f. 8. The software of 9 was used to generate the results.

*Figure 6. The input signal, the output signal of the plant, and the delayed output signal of the plant processed by the identification algorithm of 8.*

*Figure 7. The parameter estimates obtained by the identification algorithm of 8.*

*Figure 8. The input signal, the delayed output signal processed by the identification algorithm, and the simulated delayed output signal obtained from the identified model with delay. Note that the two output signals are now well aligned.*

Due to the importance of the algorithm of 8 for rapid detection of delay attacks on feedback control systems, the convergence properties were analysed in 1. Averaging analysis using the method with associated ODEs was applied to prove:

- Under standard conditions and the assumption that the system is in the model set, the true parameter vector is in the set of global stationary points of the algoithm.

It was also discussed in 1 why the standard way of proving local convergence fails, instead numerical examples in 1, 8 and Fig. 7 illustrate that stability of the true parameter vector usually holds.

The SW package 5 updates the algorithm of 8 and 9 with an unknown output equation that is jointly identified with the ODE and delay. The output model is divided into a lower, mid, and upper interval of the first state of the ODE. The mid interval fixes the input-output static gain of the total model by using a linear model with fixed slope and free bias. The lower and upper intervals are parametrized as multipolynomial models in the input and states of the ODE, constrained so that continuity of the output model is enforced at the interval boundaries. Fig. 9 illustrates an example of the performance of the new algorithm.

*Figure 9. The parameters identified by the new algorithm of 5 while converging to the true parameter vector. See sections 9-13 of 5 for details on model and parameters.*

Recently, the algorithm of 8 has been applied for detection of disguised delay attacks on networked feedback loops with promising results. As shown in the paper 4, the algorithm can detect a delay change of a feedback loop disguised in jitter long before the attack becomes evident by visual inspection. In addition, delay attacks on automotive cruise control feedback loops can be rapidly detected, despite the fact that that the attacks are completely disguised by application of the attack in the feedback path, see Fig. 10, Fig. 11, and the papers 2 and 3 for details.

*Figure 10. The control signal, disturbance and feedback signal used for delay attack detection of the algorithm of 1 and 8. *

*Figure 11. The identified delay and nonlinear dynamic parameters of the algorithm of 1 and 8, while used for delay attack detection on the automotive cruise control system of 2. The attack appears at time t = 6000 s.*

### References

1. T. Wigren, "Convergence in delayed recursive identification of nonlinear systems", to appear in European Control Conference, Stockholm, Sweden, June 25-28, 2024.

2. T. Wigren and A. Teixeira, "Delay attack and detection in feedback linearized control systems", to appear in European Control Conference, Stockholm, Sweden, June 25-28, 2024.

3. T. Wigren and A. Teixeira, "Feedback path delay attacks and detection", Proc. IEEE CDC , Singapore, pp. 3864-3871, December 13-15, 2023.

4. T. Wigren and A. Teixeira, "On-line identification of delay attacks in networked servo control", Proc. IFAC World Congress, pp. 1041-1047, Yokohama, Japan, July 9-14, 2023.

5. T. Wigren, "MATLAB software for nonlinear and delayed recursive identification - revision 2", Technical Reports from the Department of Information Technology, 2022-002, Uppsala University, Uppsala, Sweden, January, 2022.

6. S. Yasini and T. Wigren, "Convergence in networked recursive identification with output quantization", SYSID 2018, Stockholm, Sweden, pp. 915-920, July 9-11, 2018.

7. S. Yasini and T. Wigren, "Counterexamples to parametric convergence in recursive networked identification", ACC 2018, Milwaukee, USA, pp. 258-264, June 27-29, 2018.

8. T. Wigren, "Networked and delayed recursive identification of nonlinear systems", IEEE CDC 2017, Melbourne, Australia, pp. 5851-5858, Dec. 12-15, 2017.

9. T. Wigren, "MATLAB Software for Nonlinear and Delayed Recursive Identification - Revision 1", Technical Reports from the Department of Information Technology, 2017-007, Uppsala University, Uppsala, Sweden, April, 2017. Available: http://www.it.uu.se/research/publications/reports/2017-007/RecursiveNonlinearNetworkedIdentificationSW.zip

10. T. Wigren, "MATLAB software for recursive identification of systems with output quantization - Revision 1 ", Technical Reports from the department of Information Technology 2007-015, Uppsala University, Uppsala, Sweden, April, 2007. Available: http://www.it.uu.se/research/publications/reports/2007-015/QRISRev1.zip

11. T. Wigren, "Adaptive filtering using quantized output measurements", IEEE Trans. Signal Processing , vol. 46, no. 12, pp. 3423-3426, 1998.

12. T. Wigren, "Approximate gradients, convergence and positive realness in recursive identification of a class of nonlinear systems", Int. J. Adaptive Contr. Signal Processing, vol. 9, no. 4, pp. 325-354, 1995.