1. Introduction

The control of electric motor systems, particularly in precision position control, has been a subject of extensive research due to its widespread applications in industrial automation, robotics, and servo systems ^[1,2]. Ensuring accurate motor position tracking in the presence of model uncertainties, nonlinearities, and external disturbances remains a challenging task. Various control strategies have been developed to address these challenges, with cascade controllers emerging as one of the most effective solutions for improving system robustness and performance ^[3-6].

Cascade control structures, where a primary controller, typically for position tracking, is augmented with a secondary controller, often for velocity or current control, offer advantages in terms of design flexibility. Commonly, proportional (P) and proportional-integral (PI) controllers are used in cascade configurations due to their simplicity and effectiveness in linear systems ^[5,6]. However, these conventional controllers can not maintain nominal performance under nonlinear dynamics and time-varying disturbances, leading to the need for more advanced robust control methods. The most common approach used to enhance robustness in nominal controllers is the DOBC ^[7,8]. Among the various DOBC techniques, Reduced-Order Proportional-Integral Observer (ROPIO), stands out as it integrates the simplicity of a PI framework with the robustness of a disturbance observer, enabling it to handle uncertainties and compensate for unmodeled dynamics ^[9,10]. However, the performance of ROPIO-based controllers can degrade significantly when subjected to complex, time-varying disturbances, such as sinusoidal inputs with unknown frequency.

To address the limitations of linear robust control methods, advanced nonlinear control techniques have garnered increasing attention ^[11,12]. Radial Basis Function Neural Networks (RBF-NN), in particular, have been widely used in control systems for their effective real-time approximation capabilities, making them well-suited for compensating complex, nonlinear disturbances and uncertainties while continuously adapting based on system feedback ^[13-15]. The versatility of RBF-NNs has made them applicable across a wide range of systems ^[16-20]. To highlight a few recent works, ^[16] integrates an RBF-NN into a maglev vehicle control system, enabling adaptation to real-time control trends, which enhances robustness against nonlinearities and improves control precision. The RBF-NN-based adaptive robust controller for nonlinear bilateral teleoperation manipulators in ^[17] employs RBF-NN to estimate and compensate for nonlinear environmental uncertainties, resulting in increased system robustness and enhanced stability despite time delays and external disturbances. In ^[18] RBF-NN was used to optimize control parameters in real-time to compensate for time-varying parameters in permanent magnet synchronous motors, achieving superior decoupling performance and reducing torque and current fluctuations. Similarly, ^[19] implements RBF-NN in a robust back-stepping control strategy for a snake robot’s head, approximating external disturbances and compensating for nonlinearities, leading to improved performance in dynamic environments. Another recent work presented in ^[20] introduces an RBF-NN compensator integrated with a deadbeat active disturbance rejection control system for electric drives. This compensator, trained online to suppress harmonic disturbances without prior knowledge of harmonic frequencies, significantly enhances current tracking and overall motor performance.

Motivated by these works, this paper considers two RBF NN-based robust cascade controllers for improved reference tracking performance considering nonlinear friction and unknown dynamic uncertainty. In the context of this study, the term ‘uncertainty’ is used to refer to the combined effect of model parameter variations, nonlinearities in the closed-loop system as well as external disturbance, while 'dynamic uncertainty' specifically refers to the elements with time-varying dynamics. The friction model considered in the study is the dynamic LuGre model ^[21], acting on the system along with an input channel sinusoidal disturbance with unknown amplitude and frequency. A comprehensive comparison is made between the proposed scheme and conventional cascade controllers including an ROPIO-based robust cascade controller. The effectiveness of the approach is validated through simulations on an uncertain DC motor position control system. The simulation results show that while the ROPIO-based cascade controller provides robustness against nonlinear friction and model uncertainties, its performance deteriorates when subjected to time-varying disturbances. The adaptive RBF-NN-based cascade controller, despite offering robust compensation through online learning, can be limited under extreme variations in system dynamics. In contrast, the proposed supervisory RBF-NN-based cascade controller provides a more robust solution against dynamic disturbances and model uncertainties. Thus, the purpose of this paper is to propose a robust cascade controller enhanced with an RBF-NN that enables nominal performance recovery in the presence of nonlinear friction and dynamic external disturbances, whose combined effects often prove too challenging for ROPIOs to address.

The contributions of the paper can be summarized as follows:

Development of a supervisory RBF-NN-based cascade control scheme for nonlinear friction compensation under dynamic uncertainty.

Robustness validation considering dynamic uncertainties through comparative simulation against a robust ROPIO-based controller and an adaptive RBF-NN-based scheme.

The paper is organized as follows: Section 2 introduces the system model used in the study and develops classical cascade controllers, including the robust ROPIO-based cascade controller. The latter part of this section focuses on the development of the adaptive and supervisory RBF-NN controllers. The robustness validation of the proposed scheme is detailed in Section 3. Finally, Section 4 concludes the paper.

2. System Description and Controller Designs

2.2 Conventional Cascade Controllers

Using the system model (2) and assuming $\psi = 0$, a nominal cascade controller can be designed to track a constant reference position signal $x_{r}$. To design the controller the outer-loop tracking error is defined as $e_{1}= x_{r}- x_{1}$. The outer-loop tracking error dynamics, given as $\dot{e}_{1}= -x_{2}$, indicates that if $x_{2}$ tracks an inner-loop reference defined as $x_{2}^{*}= k_{1}e_{1}$, with $k_{1}>0$, then $e_{1}\to 0$ as $t\to\infty$ with $\dot{e}_{1}= - k_{1}e_{1}$. Hence, the outer-loop gain is designed based on a desired time constant $\tau_{1}= k_{1}^{-1}$. To ensure $x_{2}\to x_{2}^{*}$, the inner-loop tracking error is defined as $e_{2}= x_{2}^{*}- x_{2}$, and the inner-loop control objective can be achieved through the nominal control law

(5)

$u_{1}^{*}=\dfrac{1}{b}\left(ax_{2}+\dot{x}^{*_{2}}+ k_{2}e_{2}\right)$

where the inner-loop dynamics is restricted as $k_{2}\ge 5k_{1}$.

It is obvious that $u_{1}^{*}$ would not ensure $e_{1}\to 0$, as $\psi ne 0$. A more suitable controller that would ensure zero steady-state error in the presence of a constant uncertainty is a controller with an integral term such as a proportional-integral (PI) controller. In the cascade framework, this can be incorporated by taking the inner-loop control law as

(6)

$\dot{\eta}= e_{2},\: u_{2}^{*}= k_{p}e_{2}+ k_{i}\eta$

where $k_{p}$ and $k_{i}$ are the inner-loop proportional and integral gains, respectively. The inner-loop gains are designed for the extended inner-loop system

(7)

$\dot{\psi}=\begin{bmatrix}-a &0 \\-1&0\end{bmatrix}\psi +\begin{bmatrix}b \\0\end{bmatrix}u =\overline{A}\psi +\overline{B}u$

where $\psi =\begin{bmatrix}x_{2}&\eta\end{bmatrix}^{T}$. Since system (7) is controllable, the closed-loop eigenvalues of (6)-(7) can be placed at arbitrary location $\begin{bmatrix}-\lambda_{1},\: & -\lambda_{1}\end{bmatrix}$ for $\lambda_{1}>0$. Similar to $u_{1}^{*}$, the inner-loop dynamics is restricted as $\lambda_{1}\ge 5k_{1}$.

Fig. 1. ROPIO-based robust cascade controller closed-loop system

2.3 ROPIO-based Robust Cascade Controller

Although the P-PI cascade controller offers zero steady-state error in the presence of constant uncertainty, $u_{2}^{*}$ will not maintain $e_{1}\to 0$ under the dynamic $\psi$. However, by augmenting the cascade controller with an ROPIO, as depicted in Fig 1, the dynamic uncertainty could be compensated through a disturbance estimation.

To design the ROPIO the reduced-order system model (2) is rewritten as

(8a)

$\dot{x}_{1}= x_{2}$

(8b)

$\dot{x}_{2}= -ax_{2}+ b(u + d)$

(8c)

$\dot{d}= 0$.

Remark 1 : Although the ROPIO is designed considering a constant disturbance $d$ as the additional system state, through the use of high-gain, its effects can be compensated as shown in the simulation results presented in Section 3.

By defining the state $x_{b}=\begin{bmatrix}x_{2}&d\end{bmatrix}^{T}$ the ROPIO can be constructed as

(9)

$\dot{\hat{x}}_{b}=\begin{bmatrix}-a & b \\ 0&0\end{bmatrix}\hat{x}_{b}+\begin{bmatrix}b \\ 0\end{bmatrix}u + L\left(x_{2}-\hat{x}_{2}\right)$

where $L$ is the observer gain matrix and the disturbance estimation $\hat{d}=\begin{bmatrix}0 &1\end{bmatrix}\hat{x}_{b}$. The estimation error dynamics resulting from (9) is summarized as

(10)

$\dot{\widetilde{x}}_{b}=\left(\begin{bmatrix}-a & b \\ 0&0\end{bmatrix}- L\begin{bmatrix}1&0\end{bmatrix}\right)\widetilde{x}_{b}= A_{o}\widetilde{x}_{b}$.

where $\widetilde{x}_{b}= x_{b}-\hat{x}_{b}$. The observer gains are designed to ensure the ROPIO dynamics is significantly faster than the inner-loop controller. To avoid using the derivative term we define $x_{c}=\hat{x}_{b}- L x_{1}$, equation (9) can be rewritten as

(11)

$\dot{x}_{c}=\left(\begin{bmatrix}-a & b \\ 0&0\end{bmatrix}- L\begin{bmatrix}1&0\end{bmatrix}\right)\hat{x}_{b}+\begin{bmatrix}b \\ 0\end{bmatrix}u$.

Hence, the disturbance estimation is obtained as

(12)

$\hat{d}=\begin{bmatrix}0 &1\end{bmatrix}\hat{x}_{b}=\begin{bmatrix}0 &1\end{bmatrix}\left(x_{c}+ L x_{1}\right)$.

Using the disturbance estimation, the P-PI cascade controller can be enhanced as

(13)

$u_{3}= u_{2}^{*}-\hat{d}= k_{p}e_{2}+ k_{i}\eta -\hat{d}$.

Remark 2 : Although the integral term of (6) can offer nominal control performance in addition to removing steady-state errors caused by constant disturbances, the cumulative uncertainty resulting from model parameter variation, nonlinear friction and external dynamic disturbance cannot be addressed by simply using a PI controller. In this context, the disturbance estimation, $\hat{d}$, produced by the ROPIO designed in this section will serve as the robust component of the cascade control framework to address these effects.

Fig. 2. Adaptive RBF NN-based cascade controller closed-loop system

2.4 Adaptive RBF NN-based Robust Cascade Controller

While it is possible to compensate for the nonlinear friction and dynamic disturbance in $\psi$ using a high-gain ROPIO, as noted in Remarks 1 and 2, this approach can potentially lead to closed-loop instability ^[22] and induce vibrations in the system due to the aggressive control actions triggered by rapid disturbance estimation. To address the nonlinear uncertainties in the closed-loop system without encountering the issues associated with high-gain DOBs, an adaptive RBF neural network can be employed ^[14]. The proposed adaptive RBF-NN-based cascade controller is depicted in Fig 2. Unlike the previous result the proposed controller utilizes the error $e_{2}= x_{2}^{*}- x_{2}$ in the input vector $\xi$ via the cascade control approach as depicted in Fig 2.

To approximate the combined uncertainty $\psi$ in (2), an RBF NN is constructed as

(14)

$h_{j}(\xi)=\exp\left(-\dfrac{\left . ∥\xi - c_{ij}\right .∥^{2}}{2b_{j}^{2}}\right)$

(15)

$\psi = W^{T}h(\xi)+\varepsilon$

where $\xi$ denotes the input vector, $i$ denotes the input neural net number in the input layer, $j$ denotes the hidden node number in the hidden layer, $h =\begin{bmatrix}h_{1},\: h_{2},\: \cdots ,\: h_{n}\end{bmatrix}^{T}$ denotes the output of the hidden layer, $W$ represents the weight values and $\varepsilon$ is the estimation error and is bounded as $\varepsilon\le\varepsilon_{N}$ ^[14]. For an arbitrary vector $v$, $∥ v∥^{2}= v^{T}v$.

Choosing the input vector as $\xi =\begin{bmatrix}e_{1}& e_{2}\end{bmatrix}^{T}$, the nonlinear uncertainty estimation is produced as

(16)

$\hat{\psi}=\hat{W}^{T}h(\xi)$

where $\hat{W}$ is an adaptive weight. The optimal weight value is obtained as

(17)

$W^{*}=\arg\min_{W\in\Omega}[{sup}|\hat{\psi}-\psi |]$.

Let the modeling error be defined as $\omega =\hat{\psi}\left(\xi | W^{*}\right)-\psi(\xi)$ and $\widetilde{\psi}=\psi -\hat{\psi}$. Using the adaptive RBF NN estimation, the nominal control law $u^{*_{1}}$ can be enhanced as

(18)

$u_{4}=\dfrac{1}{b}\left(ax_{2}+\dot{x}^{*_{2}}+ k_{2}e_{2}-\hat{\psi}\right)$.

Applying (18) to (2b) yields the tracking error dynamics

(19)

$\dot{e}_{2}= -k_{2}e_{2}-\widetilde{\psi}$$= -k_{2}e_{2}+\left(\hat{W}- W^{*}\right)^{T}h(\xi)+\omega$.

Next choose the Lyapunov function as

(20)

$V =\dfrac{1}{2}e_{2}^{2}+\dfrac{1}{2\gamma}\left(\hat{W}- W^{*}\right)^{T}\left(\hat{W}- W^{*}\right)$

where $\gamma$ is a positive constant. Taking the derivative of the function yields

(21)

$\dot{V}= e_{2}\dot{e}_{2}+\dfrac{1}{\gamma}\left(\hat{W}- W^{*}\right)^{T}\dot{\hat{W}}$$= -k_{2}e_{2}^{2}+ e_{2}\omega +\dfrac{1}{\gamma}(\hat{W}- W^{*})\left[\gamma e_{2}h(\xi)+\dot{\hat{W}}\right]$.

Constructing the RBF-NN weight adaptive update law as

(22)

$\dot{\hat{W}}= -\gamma e_{2}h(\xi)$

makes the derivative of the Lyapunov function

(23)

$\dot{V}= -k_{2}e_{2}^{2}+ e_{2}\omega$.

Hence if the approximation error $\omega$ can be made to have a small value using the adaptive RBF-NN, $\dot{V}\le 0$ can be obtained.

Fig. 3. Conventional RBF supervisory control system ^[15]

2.5 Supervisory RBF NN-based Cascade Controller

The RBF NN supervisory controller in this section is a design method combining a switching control technique to improve the control performance of the existing one as shown in Fig 3 ^[15]. To improve the conventional supervisory controller in ^[14] we consider a switching controller for the following error equation from (8b)

(24)

$\dot{e}_{2 =}- u -\delta$

where $e_{2}= x_{2}^{*}- x_{2}$ and $\delta = -ax_{2}+(b- 1)u + bd -\dot{x}_{2}^{*}$.

Consider the following control input

(25)

$u = k_{2}e_{2}+ u_{s}$, $u_{s}= M{sgn}(e_{2})$

with positive gains $k_{2}$ and $M$. By using (25) time derivative of the Lyapunov function $V = 0.5 e_{2}^{2}$ yields

(26)

\begin{align*}\dot{V}= e_{2}\dot{e}_{2}= e_{2}\left(-k_{2}e_{2}- u_{s}-\delta\right)\\\le -2k_{2}V + ｜｜\delta ｜｜ ｜｜ e_{2}｜｜ - e_{2}u_{s}\\ \le -2 k_{2}V -\overline{\gamma}｜｜ e_{2}｜｜ \\\le -\overline{\gamma}\sqrt{2 V}.\end{align*}

when $M -｜\delta｜>\overline{\gamma}> 0$. This implies $e_{2}\to 0$ within a finite time.

In Fig 3 the supervisory control input is given by

(27)

$u = u_{p}+ u_{n}$, $u_{p}= k_{2}e_{2}$,

(28)

$u_{n}= h_{1}w_{1}+ h_{2}w_{2}+\cdots + h_{m}w_{m}=: w^{T}h(y_{d})$

and the parameter $w$ is trained to minimize the following cost function

(29)

$J =\dfrac{1}{2}\left(u_{n}- u\right)^{2}$.

The switching control input (25) can be used to help training the supervisory control input (27)-(28) because the following property can be assumed on the two controllers in the steady state

(30)

$w^{T}h\approx M{sgn}(e_{2})$.

Motivated by (30) the proposed RBF NN supervisory controller is represented by

(31)

$u_{5}= u_{p}+ u_{ns}$, $u_{ns}= w^{T}h(x_{2}^{*})+\mu\tan h(\sigma u_{p})$

where $u_{p}= k_{2}e_{2}$ and the switching gain $\mu$ is a positive constant $0 <\mu\le M$. The function $\tan h(\sigma u_{p})$ is used to avoid the use of a discontinuous function ${sgn}(u_{p})$ with a positive constant $\sigma$ ^[15].

By using the Gradient Descent Method, the parameter update is modified by using the cost function (29) with (31) instead of (28) as follows.

(32)

$\triangle w_{j}= -\eta\dfrac{\partial J}{\partial w_{j}}= -\eta(u_{ns}- u_{5})h_{j}(x_{2}^{*})$

where the learning rate $\eta$ has a value between 0 and 1 (see Fig 4). Unlike in the previous section the input vector $y_{d}= x_{2}^{*}$ in Fig 4 instead of $\xi$ in Fig 2.

Fig. 4. Supervisory RBF NN-based cascade controller closed-loop system

In the next section we verify the performance of the proposed controller through comparative simulations under dynamic uncertainties.

3. Performance Validation

The performance of the proposed RBF NN-based cascade controllers is validated through a comparative simulation against the nominal P-P cascade, P-PI cascade, and ROPIO-based robust cascade controller. The simulation has been constructed considering nominal plant model parameters $a = 8.389$ and $b = 1.7028$ (see Table 2 in the next page). Using the nominal parameter values, the cascade controller and ROPIO gains were designed choosing the outer-loop time constant $\tau_{1}= 0.05[{s}]$. The two RBF NN models considered in the simulation were constructed with $11$ hidden nodes. The controller and obs

erver gains as well as the RBF NN parameters used in the simulation are summarized in Table 1.

The reference signal used in the simulation was the variable step trajectory constructed as

(33)

The uncertainties considered in the simulation are detailed in Table 2. These include the parameters of the dynamic friction model (3), input channel sinusoidal disturbance model represented by (4), as well as model parameter perturbations.

Fig. 5. Reference tracking performance comparison between ROPIO-based cascade controller and non-robust cascade controllers (top) and the tracking error (bottom) comparison

The reference tracking performance comparison obtained from the simulation is presented in Fig 5 and Fig 6. From the simulation the following observations can be made: the P-P cascade controller does not offer zero steady-state error under the nonlinear dynamic friction. This challenge is resolved by the P-PI cascade controller owing to the integral term in the inner-loop.

Fig. 6. Reference tracking performance comparison between ROPIO-based cascade controller and RBF NN-based cascade controllers (top) and the tracking error (bottom) comparison

However, the P-PI cascade controller loses zero steady-state error once the sinusoidal disturbance is introduced into the closed-loop system. Furthermore, as can be noticed in the reference tracking error comparison (bottom plot of Fig 6), the ROPIO compensated cascade controller suffers from a sinusoidal error owing to the disturbance.

In Fig 6 a relatively less effective compensation can be obtained from the adaptive RBF NN-based P-P cascade controller, although its performance may be improved by integrating it with a PI inner-loop controller. The proposed supervisory RBF NN-based cascade controller on the other hand offers a superior compensation of the nonlinear friction as well as the dynamic uncertainties despite having a P-P cascade structure as the main controller as shown in the reference tracking error comparison in Fig 6 (bottom plot).

Fig. 7. Nominal performance recovery comparison between ROPIO-based cascade controller and RBF-NN-based robust cascade controllers

Furthermore, the nominal performance recovery of the ROPIO-based and RBF NN-based cascade controllers are presented in Fig 7. This shows the supervisory cascade controller maintains nominal performance at the sinusoidal disturbance introduction at $t = 6[{s}]$. In addition, as a quantitative measure of the performance, the norms of the errors($Vert e Vert$) from Fig 7 are compared in Table 3 during the time interval ($6\le t\le 7$). This further demonstrates the effectiveness of the supervisory scheme against dynamic uncertainties.

The control effort comparison of the controllers is presented in Fig 8 and Fig 9. Throughout the simulations a control input saturation of $\overline{u}=\pm 24.0[{V}]$ was imposed. From the control efforts comparison it can be seen that the improved performance of the RBF NN-based schemes was not obtained at the cost of large magnitude control input.

Fig. 8. Control effort comparison between ROPIO-based cascade controller and non-robust cascade controllers

Fig. 9. Control effort comparison between the ROPIO-based cascade controller and RBF NN-based cascade controllers

4. Conclusion

A supervisory RBF NN-based cascade control scheme has been proposed for enhanced nonlinear friction compensation in the presence of dynamic uncertainties. The scheme was compared with both a robust ROPIO-based cascade controller and an adaptive RBF NN-based cascade controller. Comparative simulations, conducted on a DC motor position control objective, highlighted the limitations of conventional robust linear controllers in addressing dynamic disturbances and demonstrated the superior performance of RBF NN-based controllers. In particular, the supervisory RBF-NN controller outperformed the adaptive RBF NN in terms of disturbance rejection and robustness. The proposed scheme can be applied to position control of various kinds of mechanical systems as well as electric motors. To test its feasibility of real-world applications future work will focus on experimental validation of the proposed scheme with a laboratory motor system.