Chaotic Van Der Pol Oscillator Control Algorithm Comparison

1. Introduction

The concept of chaos theory can be illustrated through the butterfly effect: the flapping of a butterfly’s wings can create a small gust of air that ultimately leads to a storm on the other side of the world. Chaos theory refers to systems that are bounded, recurrent, and highly sensitive to initial conditions (see Figure 1). For example, weather patterns are chaotic systems that can be influenced by small changes in initial conditions such as temperature, pressure, and wind patterns. This sensitivity to initial conditions makes predicting the weather challenging, even with the help of advanced computers.

In 2000, Boccalettia reviewed the major ideas involved in the control of chaos and proposed two methods: the Ott–Grebogi–Yorke (OGY) method and the adaptive method both seeking to bring a trajectory to a small neighborhood of a desired location, seeking stabilized desired chaotic orbits embedded in a chaotic attractor including a review of relevant experimental applications of the techniques [2]. Ott, C. Grebogi and J. A. Yorke observed that the infinite number of unstable periodic orbits typically embedded in a chaotic attractor could be taken advantage of for the purpose of achieving control by means of applying only very small perturbations [3]. Song, et. al, proposed combining feedback control with OGY method [4], and this trend to utilize feedforward followed by application of feedback might be considered canonical. Back in 1992, Pyragas had already offered continuous control of chaos by self-controlling feedback [4].

Adaptive methods as offered by Slotine and Li [5] strictly rely on a feedforward implementation augmented by feedback adaption of the feedforward dynamics, but also utilize elements of classical feedback adaption (e.g. the M.I.T. rule [6]) to adapt classical feedback control gains and adapt the desired trajectory to eliminate tracking errors. Following kinetic improvements to Slotine’s approach by Fossen [7,8] and subsequent researchers [9] the approach was experimentally validated in [10]. In 2017, Cooper and Heidlaulf [11] proposed extracting the feedforward elements of Slotine’s approach for controlling chaos of the famous van der Pol oscillator [12] which has grown to become a benchmark system for representing chaotic systems like relaxation oscillators [13], frequency demultiplication [14], heartbeats [15] and nonlinear electric oscillators in general [16].

Cooper and Heidlauf’s techniques were initially paired with linear feedback control (linear quadratic optimal control), but the combination of feedback and feedforward proved ineffective. Smeresky and Rizzo offered 2-norm optimal feedback using pseudoinverse in 2020 [17], and the combination proved effective for controlling highly nonlinear, coupled (non-chaotic) Euler’s moment equations. The combination of feedforward and optimal feedback learning was formulated for five-degree of freedom oceanic vehicles in [18] and labeled deterministic artificial intelligence. Shortly afterwards, Zhai directly compared stochastic artificial intelligence methods: neural networks and physics-informed deep learning in [19,20].

Originally discovered by Gauss in the 1800s, the work lay unused until Plackett rediscovered his work and applied it to signal processing [21]. Several methods have emerged since extending and resolving weakness of the recursive least squares method. Several of these such as the addition of exponential forgetting and posterior residuals their follow-on techniques are discussed by Astrom and Wittenmark and in their textbook on adaptive control [22]. However, these works often fail to consider the cost required to determine the control input dependence of the system. In order to excite the system so that the response can be observed, and a model fit to it, resources such as fuel or electric power must be expended. In addition, the convergence rate per unit cost may be deciding factor when choosing an approach. A controller must understand the system it is controlling before it is able to drive it to a desired state. A secondary challenge is the inherent non-linearity of many systems present in real world applications. Such systems cannot be written using simple linear models with respect to the system states and require different analysis techniques. An overview of existing analysis methods can be found in [23]. However, as mentioned before these techniques do not attempt to minimize the cost of system identification. It is demonstrated in this work that an excitation controller based on optimal learning techniques and self-awareness statements provides a significantly higher accuracy per unit cost than other excitation signals. Optimal learning and self-awareness are components of deterministic artificial intelligence, which seeks to create intelligent systems that derive results through conditions restricted to obey physical models rather than purely stochastically through bulk data.

In this manuscript, we focus on the Van der Pol chaotic oscillator. Scientists have used this chaotic system for multiple applications, such as modeling action potentials and countering electromagnetic pulse attacks. Adaptive control theory offers a means of stabilizing the Van der Pol system through synchronization of a drive and response system.

Cooper and Heidlauf showed that the Van der Pol system can be forced to asymptotically follow a desired trajectory in [11]. They accomplished this by using a Van der Pol oscillator, the driver, to force another Van der Pol oscillator, the responder. We refer to this technique as feedforward Van der Pol.

Smeresky and Rizzo proposed an improvement on feedforward Van der Pol in [17] using deterministic artificial intelligence (DAI). The DAI method combines nonlinear adaptive control and physics-based controls methods. It is based on self-awareness statements enhanced by optimal feedback, which is reparametrized into a standard regression form. Online estimation methods such as recursive least squares estimate the system parameters for this optimal feedback.

This manuscript compares the performance of DAI for non-linear adaptive control of a Van der Pol oscillator when using three different online estimation methods: a Kalman filter, recursive least squares with exponential forgetting (RLS-EF), and least mean squares with a normalized gradient (LMS-NG).

2. Materials and Methods

The Van der Pol chaotic oscillator is defined by Equation (1):

$$F\left(t\right)=\alpha x+\mu \left(1-x^2\right)\dot{x}+\beta \ddot{x}$$

(1)

where $\alpha$, $\mu$, and $\beta$ define the system parameters, $\theta$, and $F\left(t\right)$ is the forcing function. With DAI, the system parameters are estimated at each time step. Figure 3 is a Simulink model that represents the system.

, drives the Van der Pol oscillator. State information is passed into the forcing function block, to be used by DAI to estimate the system parameters.

Online estimation methods analyzed in this paper require two inputs: the regressors ${\mathsf{\Phi }}_{true}$ observed from the system and $F\left(t\right)$ from the previous time step, $F\left(t-1\right)$. We define the regressors as:

$${\mathsf{\Phi }}_{true}=\left[\begin{array}{c}x\\ \left(1-x^2\right)\dot{x}\\ \ddot{x}\end{array}\right]$$

(2)

Given system inputs ${\mathsf{\Phi }}_{true}$ and $F\left(t-1\right)$, the estimator performs regression to calculate the estimated system parameters, $\widehat{\theta }$. These are used in conjunction with self-awareness statements to calculate $F\left(t\right)$.

Self-awareness statements are formulated using desired trajectory $⌊\begin{array}{ccc}{x}_d& {\dot{x}}_d& {\ddot{x}}_d\end{array}⌋$:

$${\mathsf{\Phi }}_d=\left[\begin{array}{c}{x}_d\\ \left(1-x^2\right){\dot{x}}_d\\ {\ddot{x}}_d\end{array}\right]$$

(3)

and represent the known physics of the Van der Pol system. All together, we have Equation (4):

$$F\left(t\right)=\left[{\mathsf{\Phi }}_d\right]\left[\widehat{\theta }\right]=\left[\begin{array}{c}{x}_d\\ \left(1-x^2\right){\dot{x}}_d\\ {\ddot{x}}_d\end{array}\right]⌊\begin{array}{ccc}{x}_d& {\dot{x}}_d& {\ddot{x}}_d\end{array}⌋$$

(4)

See Figure 4 and Figure 5 for the Simulink architectures of the forcing function with feedforward only and feedforward with DAI.

3. Results

We used the Runge-Kutta MATLAB 2022b solver in Simulink with a step size of 0.01 and a simulation time of 100 seconds to run the model in Figure 3 given different forcing functions:

• Unforced
• Uniform noise
• Sine wave
• Feedforward
• Deterministic artificial intelligence (DAI) – Kalman estimator
• Deterministic artificial intelligence (DAI) – RLS-EF estimator
• Deterministic artificial intelligence (DAI) – LMS-NG estimator

For the deterministic artificial intelligence methods, we initialized $\widehat{\theta }$ ($\alpha$, $\mu$, and $\beta$) as 0.1; the ground truth value was $\theta =⌊\begin{array}{ccc}5& 1& 1\end{array}⌋$. Despite the inaccuracy of the initial estimates, all deterministic artificial intelligence methods succeeded in iteratively estimating the truth values, with $\widehat{\theta }$ converging to $\theta$.

Table 1. Mean absolute error (MAE) for the entire trajectory.

Method	$\mathbf{M}\mathbf{A}\mathbf{E}$$\mathit{x}$	Method	$\mathbf{MAE}\dot{\mathit{x}}$
1	3.3234	1	3.2031
2	3.3334	2	3.2110
3	3.9843	3	3.9275
4	0.2091	4	0.2284
5	0.4975	5	0.3927
6	0.2041	6	0.2237
7	0.3089	7	0.2877

Table 1. Mean absolute error (MAE) for the entire trajectory.

Method	$\mathbf{M}\mathbf{A}\mathbf{E}$$\mathit{x}$	Method	$\mathbf{MAE}\dot{\mathit{x}}$
1	3.3234	1	3.2031
2	3.3334	2	3.2110
3	3.9843	3	3.9275
4	0.2091	4	0.2284
5	0.4975	5	0.3927
6	0.2041	6	0.2237
7	0.3089	7	0.2877

Table 2. Mean absolute error (MAE) when excluding the first 10% of the trajectory.

Method	$\mathbf{M}\mathbf{A}\mathbf{E}$$\mathit{x}$	Method	$\mathbf{MAE}\dot{\mathit{x}}$
1	3.3218	1	3.1891
2	3.3262	2	3.1941
3	4.0150	3	3.9487
4	0.0642	4	0.0743
5	0.1220	5	0.1379
6	0.0629	6	0.0728
7	0.0329	7	0.0599

Table 2. Mean absolute error (MAE) when excluding the first 10% of the trajectory.

Method	$\mathbf{M}\mathbf{A}\mathbf{E}$$\mathit{x}$	Method	$\mathbf{MAE}\dot{\mathit{x}}$
1	3.3218	1	3.1891
2	3.3262	2	3.1941
3	4.0150	3	3.9487
4	0.0642	4	0.0743
5	0.1220	5	0.1379
6	0.0629	6	0.0728
7	0.0329	7	0.0599

We analyzed the error between the trajectory of the responder Van der Pol oscillator and the desired trajectory, a circle of radius 5. The first three methods did not employ feedforward or deterministic artificial intelligence feedback. The unforced and uniform noise forced oscillators both exhibited the expected chaotic trajectory of a Van der Pol oscillator, visualized at the center of Figure 6. The sine wave forced the responder oscillator to deviate from the typical Van der Pol trajectory, but the resulting trajectory was chaotic.

The method 4, feedforward using the drive-response methodology in [1], successfully forced the responder oscillator to follow the target trajectory. Methods 5-7 built upon method 4, adding deterministic artificial intelligence feedback as described in [25] to estimate and learn the system parameters.

4. Discussion

A Van der Pol oscillator subtends toward a limit cycle, an invariant set determined by the initial conditions. This occurs when both sides of Equation (1) are equal; or:

$$\alpha x+\mu \left(1-x^2\right)\dot{x}+\beta \ddot{x}-F\left(t\right)=0$$

(5)

In other words, the oscillator will, after some time, converge to a fixed trajectory. The goal of [1] was to command a Van der Pol oscillator such that, no matter the initial conditions, it will asymptotically approach the same limit cycle. Smeresky and Rizzo went a step further in [25] by showing that deterministic artificial intelligence (DAI) can improve the ability of the forcing function, $F\left(t\right)$, to force the responder oscillator to follow a desired trajectory.

We build upon the results of [25] by comparing three different DAI system parameter estimators: a Kalman filter, recursive least squares with exponential forgetting (RLS-EF), and least mean squares with a normalized gradient (LMS-NG).

For all methods with feedforward (methods 4-7), the responder Van der Pol oscillator has significant error in the beginning that decreases as the oscillator stabilizes to the desired trajectory. This error approaches zero.

Over the length of the entire simulation, 10,000 time steps for 100 seconds at a sample time of 0.01, the DAI estimator that had the lowest mean absolute error (MAE) was RLS-EF, with a 2.41% and 2.01% reduction in MAE for $x$ and $\dot{x}$, respectively, compared to feedforward alone. See the first two results columns of Table 3 for percent improvement. We found that DAI with the Kalman filter and LMS-NG had worse performance than feedforward alone. These results suggested that the Kalman filter and LMS-NG estimators should be avoided in favor of RLS-EF.

However, we noticed that DAI using the LMS-NG estimator exhibited decreased MAE as the system approached the limit cycle. See the last two results columns of Table 3 for percent improvement after t=1000. DAI with LMS-NG had 48.7% and 19.3% less MAE for $x$ and $\dot{x}$, respectively, compared to feedforward alone, and exhibited a 47.7% and 17.7% reduction in MAE for $x$ and $\dot{x}$ compared to DAI with RLS-EF.

Our results show that the RLS-EF algorithm for parameter estimation is optimal for the first 1,000 time steps of the simulation, after which LMS-NG exhibits superior performance. One algorithm may be preferable over the other depending on the intended application. For instance, RLS-EF having relatively minimal error when the system is far from the limit cycle could indicate that the algorithm is better-suited for systems where frequent external disturbances are expected.

As a next step for future research, both techniques could be used together, with the algorithm using RLS-EF for the first t time steps to estimate system parameters $\theta$ and then switching over to LMS-NG once the system has found stability. Another research opportunity would be adding logic for the model to switch between different estimation methods when an external disturbance to the system is introduced. Such efforts may build off the research of Sands [26] for countering the effects of electromagnetic pulses.