Robust and Intelligent Feedforward-Feedback Controller Design for Nonlinear Systems

All control activities can be classified into two categories: FB control and FF control. The FF and FB methods are often employed in control structures because of their uncomplicated setup and exceptional control capabilities. Nevertheless, each of these structures possesses disadvantages and constraints that could impact the precision of the entire control system [1-3]. The use of FF and FB control techniques is often the primary approach to addressing this issue and modeling the system using robust control, commonly referred to as the FF-FB control structure.

The FB control technique is distinguished by its simple design and good control performance. Nevertheless, in the event that the controlled system has a certain time delay, FB controller will not have an immediate impact on the system until a specified duration has passed. As a result, the delay in the FB controller reaction may negatively impact the overall control performance and lead to stability issues [4]. On the other hand, the FF controller is capable of anticipating changes in the reference signal and promptly applying the appropriate action to the controlled system [3]. Furthermore, the FF controller often does not need a FB signal, which means it does not create stability issues [5]. However, in order to create an effective FF controller, it is necessary to have a precise inverse model of the system. Obtaining such a model is challenging, particularly for complex nonlinear systems. In addition, the FF controller does not possess the capability to effectively manage unfavorable operating situations that may arise in the control system, such as changes in system dynamics parameters and unforeseen external disturbances.

Greater emphasis has been placed on using intelligent control approaches that can be directly implemented to address the complexity and nonlinearity of the systems. Specifically, artificial intelligence approaches may be integrated with traditional control strategies to create effective nonlinear control systems. Among several control methods, FB and FF strategies are often used owing to their easy setup and effective control performance.

Nevertheless, each of these structures has certain disadvantages and constraints that might potentially have a detrimental impact on the precision of the whole control system [3-5]. In order to address this issue, the FF and FB control techniques may be merged to provide a more robust control framework, often known as the FF-FB control scheme.

Neural networks perform this task with a high degree of accuracy as they do not have a linear structure of neurons [6]. In this way, the majority of systems rely on neural networks, which are fed by nonlinear data, for complex process control. Herein, studies [1, 2] discuss the network models that can be used to solve overall control problems by means of neural networks.

The WNN, a type of neural network, refers to SNNs as one of the recent trends in the academy. Due to the well-known advantage of this technique over the normal artificial neural network (ANN), it gives the opportunity to invert nonlinear models. Primarily, H-infinity control theory, being a mature and robust control theory tool, is widely [3] used to minimize the effect of disturbances and address nonlinearities in the context of model uncertainty. Precision and stability are the main aspects that should be elaborated on in control theory, as should how the FB controllers that could provide meticulous tracking over a long period of time will be designed and what the strategy for dealing with system uncertainties will be. Hence, the H-infinity FB controller is designed, which is usually the FF control scheme where a single control system can satisfy multiple control objectives such as disturbance rejection, performance optimization, and stabilization [5, 7, 8].

This paper focuses on the creation of an intelligent FF-FB system that uses the WNN as the FF and in the FB the H-infinity controller is involved. The main idea of the PSO is to work out the parameters of the FF and FB controllers.

The organizational flow of the remaining sections includes the following: Section 2 provides the outline and clarifies the PSO technique. Section 3 gives the details of designing the FF-FB control structure considered in this work. Section 5 presents the results of several evaluation tests and a comparative study to demonstrate the effectiveness of the FF-FB control structure. Finally, the conclusions from this work are given in Section 5.

Both the FF and FB control techniques have distinct advantages and disadvantages. Particularly, the FB control approach is specifically distinguished by utilizing an H-infinity controller. Nevertheless, in the presence of a specific time delay in the controlled system, the FB controller will not immediately impact the system until a given duration of time has passed. The FB controller is a witness to the response time delay in the system, which can threaten performance control and stability [1]. On the contrary, the FF controller has an advantage as it can be used for anticipating the shifts of the reference signal and taking the required action on the concerned system immediately, resulting in faster convergence [9]. Within this work, the structure of control, namely the FB and FF, which is depicted in Figure 1, is accomplished this way: it involves the FF technology and the FB control technique. This feature gives the stem cell more holistic control and makes it more powerful and active.

1.png

Figure 1. Schematic representation of the FF-FB controller in the form of a block diagram

The robust intelligent control law for the proposed controller is:

$u=u_{f f}+u_{f b}$     (6)

where,

$u_{f f}=f^{-1}(g(x))$     (7)

where, $f^{-1}$ is a nonlinear function representing the inverse system dynamics. ANNs can be trained to acquire the nonlinear function of Eq. (7) and $u_{f b}$ is the FB H-infinity control action.

3.1 FF controller design

By training a neural network to function as an inverse model of the plant, direct inverse control (DIC) is achieved as a powerful method to control nonlinear systems. Figure 2 depicts the DIC's generalized design [13, 14].

2.png

Figure 2. Direct inverse control

Figure 2 represents the training process of the WNN as an FF controller, emphasizing the goal of achieving optimal control actions to track the desired reference signal accurately. It provides a conceptual overview of how the network learns to control the system through iterative optimization of its weights.

The training entails adjusting the WNN weights in order to reduce the integral squared error (ISE) criterion, as given below:

$J=0.5 * \sum_{t=1}^N(e(t))^2$     (8)

where,

$e(t)=r(t)-y(t)$     (9)

In Eq. (8), N denotes the number of time samples, whereas r(t) and y(t) in Eq. (9) correspond to the reference signal and the plant output, respectively. Nevertheless, a drawback of this control method is that the inverse modeling stage does not effectively minimize the output error, which refers to the discrepancy between the actual system output and the command signal. Therefore, the controller created using this approach may result in a consistent disparity between the desired and real outputs of the system [15]. Hence, to attain a desirable level of control precision, the FF controller is integrated with the FB controller, which will be further discussed in the subsequent section.

3.2 WNN structure

The structure of the proposed WNN for representing the FF controller is shown in Figure 3.

As depicted in Figure 3, the WNN consists of three layers, which are explained in the following [16, 17]:

The first layer, which is the input layer, is responsible for directly passing the input variables to the next layer without any modification. In this work, the input variables must have the following format to exploit the WNN as a FF controller.

$\begin{gathered}y(t+1), y(t), \ldots, y(t-n+1), \\ u(t-1), \ldots, u(t-m), r(t)\end{gathered}$     (10)

The mother wavelet, or the wavelet layer, is the second layer. Each node in this layer, referred to as a wavelon, receives three input variables, as shown in Figure 3, every input node possesses an associated weight, including a self-FB weight and a FB weight from the output node. These input variables are used by the $j^{t h}$ wavelon to determine the associated output, which is expressed as follows:

$\begin{gathered}z_j=d_j \\ \left(\sum_{i=1}^{N i} v_{j i} x_i+\psi_j(k-1) \cdot \theta_j+y(k-1) \cdot \beta_j\right)-t_j\end{gathered}$     (11)

where, the variables $d_j$ and $t_j$ represent the dilation and translation parameters, respectively. $N_i$ represents the node number in the input layer. $v_{j i}$ denotes the weight connecting the $i^{t h}$ input node and the $j^{\text {th }}$ wavelon. $x_i$ represents the i-th input variable. $\Psi(k-1)$ represents the network memory, which stores past information from the $j^{\text {th }}$ wavelon. $\theta_j$ represents the $j^{\text {th }}$ adjustable connection at the self-FB, which determines the rate of information storage. $y(k-1)$ denotes the output of the preceding network. $\beta_j$ denotes tghe weight parameter connecting output node to the $j^{\text {th }}$ wavelon.

3.png

Figure 3. Architecture of the WNN

The importance of selecting a suitable wavelet activation function is now well acknowledged, since it is considered equally crucial as picking the network design and the training approach [18]. In order to tackle this problem, a series of tests were carried out utilizing various wavelet functions. The RASP1 function exhibited superior approximation performance compared to other function types. Thus, the RASP1 function was utilized for calculating the output of the $j^{t h}$ wavelet using the following equation [19]:

$\Psi_j\left(z_j\right)=\frac{z_j}{\left(z_j^2+1\right)^2}$     (12)

The third layer consists of a single node that generates the ultimate output of the WNN structure utilizing the subsequent formula:

$y=\sum_{j=1}^{N w} c_j \psi_j\left(z_j\right)+\sum_{i=1}^{N i} a_i x_i+b$     (13)

where, $N_w$ is the number of wavelon layer nodes, $N_i$  represents the total number of nodes in the input layer, c_j denotes the weight connection between the j^th wavelon and output node, $a_i$ represents the weight that connects the i^th input node to the output node, and finally, b represents a bias term for the output node. Based on the previously presented information, it is evident that the WNN structure has several adjustable weights, which may be encompassed under the set given below: 

$S=\left[v_{j i} d_j t_j c_j \theta_j \beta_j a_i b\right]$     (14)

To utilize the WNN structure as the FF controller, it is necessary to train the weights mentioned in Eq. (14) by minimizing the ISE described in Eq. (8).

3.3 FB controller design

The next section delineates the design methodology of the FB controller for a system with a structure of [20, 21]:

$\begin{gathered}\dot{x}(t)=A x(t)+B_1 \Delta_a(t, x, u)+B_2 u(t) ; x\left(t_o\right)=x_o \\ h(t)=C_1 x(t)+D_{12} u(t) \\ y(t)=C_2 x(t)\end{gathered}$     (15)

where,

$x(t) \in \Re^n$ is a vector of system states.

$\Delta_a(t, x, u) \in \Re^m$ is additive nonlinearity in this system.

$u(t) \in \Re^l$ is the input of control.

$e(t) \in \Re^q$ is vector of controlled outputs.

$y(t) \in \Re^p$ is the output vector of the system.

$A \in \Re^{n \times n}$ is the matrix of the system.

$B_1 \in \Re^{n \times m}$ represents the weight matrix of perturbation.

$B_2 \in \Re^{n \times l}$ is the matrix of control.

$C_1 \in \Re^{q \times n}$ is a system state weight matrix.

$D_{12} \in \Re^{q \times l}$ is a matrix with weights for regulating the inputs to the controller.

$C_2 \in \mathfrak{R}^{p \times n}$ is the weight matrix of the output.

The schematic representation of the dynamics of a nonlinear system in the controlled formulation, as described in Eq. (15), is depicted in Figure 4.

4.png

Figure 4. A block diagram illustrating the controllable form of a nonlinear system

The typical arrangement of the all-encompassing state FB H-infinity control is illustrated in Figure 5.

5.png

Figure 5. The revised default setup of the control problem where $M_e$ represents the plant matrix [21]

To successfully execute a comprehensive state FB H-infinity control, it is necessary to have access to all the states of the system for FB, which indicates that:

$C_2=I . D_{11}, D_{21}$ and $D_{22}$ equals zero.

Thus, the plant matrix M is transformed into:

$M_e=\left[\begin{array}{ccc}A & B_1 & B_2 \\ C_1 & 0 & D_{12} \\ C_2 & 0 & 0\end{array}\right]$     (16)

The goal is to maintain the internal stability of the system so that ( $T_{h_e \Delta_a}$ ) stays within acceptable bounds, namely below a predefined threshold value of $\gamma$ [20]:

$\left\|T_{h_e \Delta_a}(j w)\right\|_{\infty} \leq \gamma$     (17)

where, $\gamma$ indicates that $\Delta_a\left(t, e, u_e\right)$ demonstrates linear development and has an upper bound, which the controller consistently handles [22].

The condition in Eq. (17) implies that [21]:

$\min _{u_e} \max _{\Delta_a} J\left(u_e, \Delta_a\right)<\infty$     (18)

and the function cost is [23]:

$J_{p s o}\left(\gamma, \rho, C_1\right)=\int_0^{\infty} e^2(t) d t$     (19)

The perturbation $\Delta_a$ aims to optimize the cost function $J\left(u_e, \Delta_a\right)$, whereas the control signal $u_e$ aims to reduce it.

The Lyapunov quadratic function is employed to derive the optimum controller and the worst-case perturbation gain matrices values, denoted as $K_e$ and $K_{\Delta}$, respectively [21].

The Lyapunov function and its derivative are defined as [24]:

$V(e)=e^T P_e e$     (20)

$\dot{V}(e)=-e^T Q_e e$     (21)

where, the matrices $P_e$ and $Q_e$, which are real symmetric matrices of size $\Re^{\omega \times \omega}$, define the definiteness of the functions $V(e)$ and $\dot{V}(e) . P_e$ must be positively definite, whereas $Q_e$ must be negatively definite.

The structure of the optimum control and the worst-case perturbation is as follows [25]:

$\Delta_a\left(t, e, u_e\right)=K_{\Delta} x(t) ; K_{\Delta} \in \Re^{m \times \omega}$     (22)

$u_e(t)=K_e(t)$     (23)

Evaluating Eq. (22) and Eq. (23) in Eq. (15), yields:

$\dot{x}=\left(A+B_1 K_{\Delta}+B_2 K_e\right) x$     (24)

Using assumption 3, the following equation will be obtained:

$h_e^T h_e=X^T\left(C_1^T C_1+K_e^T K_e\right) x$     (25)

Therefore,

$J\left(u_e, \Delta_a\right)=\int_0^{\infty}\left[e^T\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right) e\right] d t$     (26)

Now let:

$Q=\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right)$     (27)

The matrix Q must be positive definite. The optimum control law assumes that the system described in Eq. (27) is stable.

In order to get the most efficient cost function, Eq. (27) will be replaced with Eq. (18), resulting in:

$V(x)=-x^T\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right) x$     (28)

$x^T\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right) x=-\frac{d}{d t} x^T P x$     (29)

By performing the process of integration on both sides of Eq. (27) throughout the interval, we have:

$\begin{gathered}\int_0^{\infty} x^T\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right) x d t=-\int_0^{\infty} \frac{d}{d t} x^T P x d t\end{gathered}$     (30)

$J\left(u_e, \Delta_a\right)=-x(\infty)^T P x(\infty)-\left(-x(0)^T P x(0)\right)$     (31)

In order for the system described by Eq. (19) to remain stable under the control law, it is necessary that $x(\infty)=0$.

Hence, the most efficient cost function is:

$J^*=-x(0)^T P x(0)$     (32)

The matrix P, is positive definite, represents the evaluation of the Lyapunov equation given below:

$\left(A+B_1 K_{\Delta}+B_2 K_e\right) P^T+P\left(A+B_1 K_{\Delta}+B_2 K_e\right)=-Q$     (33)

$\begin{gathered}\left(A+B_1 K_{\Delta}+B_2 K_e\right) P^T+\left(A+B_1 K_{\Delta}+B_2 K_e\right)=-\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right)\end{gathered}$     (34)

$\begin{gathered}\left(A+B_1 K_{\Delta}+B_2 K_e\right) P^T+P\left(A+B_1 K 6_{\Delta}+B_2 K_e\right)\left(C_{1_a}^T C_{1_a}+\rho K_e^T K_e-\gamma^2 K_{\Delta}^T K_{\Delta}\right)=0\end{gathered}$     (35)

To determine the most efficient control law, it is necessary to derive Eq. (35) $K_e$ and make $\partial P / \partial K c i j=0$, and we get:

$K_e=-\frac{1}{\rho} B_{2_a}^T P_e$     (36)

As a result,

$u_e *=K_e x=-\frac{1}{\rho} B_{2_a}^T P_e x$     (37)

$\Delta_a\left(t, e, u_e\right)=K_{\Delta} x(t)=\frac{1}{\gamma^2} B_{1_a}^T P_e$     (38)

Now, Eq. (39) is solved, which is known as the H-infinity algebraic Riccati equation (HIARE) [26]:

$\begin{gathered}P_e A_a+A_a^T P_e+C_{1_a}^T C_{1_a}-P_e\left(\frac{1}{\rho} B_{2_a} B_{2_a}^T-\frac{1}{\gamma^2} B_{1_a} B_{1_a}^T\right) P_e=0_\omega\end{gathered}$     (39)

The purpose of training the WNN structure is to optimize the WNN parameters by reducing the discrepancy between the reference signal and the system's actual output. In particular, multiple changeable parameters in the FF controller need to be optimized. The parameters can be represented using the following settings:

$S=\left[v_{j i} d_j t_j c_j \theta_j \beta_j a_i b\right]$     (40)

For the WNN structure to achieve optimal performance, the parameters in Eq. (40) must be optimized using a suitable optimization approach. This work utilizes the particle swarm algorithm optimizer to determine these parameters.

The FB controller optimization is employed offline to tackle the optimization problem of the design procedure. The goal is to find the optimal value of $\rho$ and the optimal values of the elements in matrix $C_1$, such that the infinity norm of $\left\|T_{e \Delta_d}(j w)\right\|_{\infty}$ is less than or equal to the optimal value of $\gamma$. The selection of the optimization cost function is as follows:

$J_{p s o}\left(\gamma, \rho, C_1\right)=\int_0^{\infty} e^2(t) d t$     (41)

In this work, a FF-FB control structure was utilized to enhance the system's ability to handle nonlinear systems. The utilization of the WNN in the FF path represents an intelligent control strategy, where WNNs are known for their capability to model complex nonlinear systems. The nonlinear H-infinity controller in the FB loop suggests an emphasis on robustness and tracking. The PSO algorithm was utilized for optimizing the weights of the WNN and the parameters of the H-infinity controller. The efficacy of the FF-FB control system was demonstrated by simulation experiments conducted on a nonlinear leg movement model. Specifically, the system was evaluated based on control accuracy, the ability to track different reference signals, and its robustness against external disturbances. The overall conclusion is that the proposed FF-FB control system, incorporating the intelligent WNN in the FF path and the nonlinear H-infinity controller in the FB path, is effective in achieving the desired control performance for the nonlinear leg movement system. In particular, the control system was shown to be robust and accurate in tracking the reference signals. The comparative study has clearly highlighted the superiority of the FF (WNN)-FB(H-infinity) control structure over the MLP-Hinfinity controller. For future work, the proposed control approach will be implemented in real-time to adaptively control nonlinear control systems.