A combined feedforward-feedback controller design for nonlinear systems

2.1 FF controller design

By training an NN to function as an inverse model of the autonomous plant, direct inverse control (DIC) is achieved as a powerful method to control the nonlinear systems. Figure 1 shows the DIC generalized design [16,17].

Figure 1

Direct inverse control [2]. In the image, r ( t ) represents the input command, u ( t ) represents the control action, and y ( t ) represents the actual dynamic model response.

Figure 1 shows the training process for the MRWNN using an FF controller, with a focus on attaining optimum control actions to properly follow the intended reference signal. This figure provides a basic summary of how the network learns to control the system by optimizing its weights repeatedly. The training entails adjusting the MRWNN weights to reduce the integral squared error (ISE) criterion, as follows:

where

Here, N denotes the number of the time sample, and r(t) and y(t) 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 (i.e., the discrepancy between the actual system output and the command signal). Therefore, a controller created using this approach may result in a consistent disparity between the desired and the real outputs of the system [18]. Hence, to attain a desirable level of control precision, the FF controller must be integrated with the FB controller.

As shown in Figure 2, which illustrates the structure of the MRWNN, the FB connections from the output node to the wavelon layer are included to improve the approximation ability of the MRWNN [2]. Specifically, three network layers comprise the MRWNN: an input layer, a hidden layer (also known as the mother wavelet layer), and an output layer [2,14].

Figure 2

Architecture of the MRWNN [2].

The first layer (i.e., 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 an FF controller:

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 2. Each input node has a weight associated with it, a self-FB weight, and an FB weight from the output node. These input variables are used by the jth wavelon to determine the associated output, which is expressed as follows [19]:

where t _j and d _j are the translation and dilation parameters of the j th wavelon, N i represents the number of input variables in the first layer, v j i denotes the weight connecting the i th input node to the j th wavelon, x i represents the ith input variable, Ψ ( t − 1 ) indicates the preceding response from the j th wavelon, θ j denotes the parameter associated with the j th self-FB weight, y(t−1) represents the previous network output, and β j denotes the weight parameter connecting the output node to the j th  wavelon.

Choosing an appropriate wavelet activation function is as crucial as choosing the network architecture and training strategy [20]. Compared to other function types, the RASP1 function offered superior approximation performance in solving the control problem in the current work. Thus, the RASP1 function was employed to calculate the output of the j th wavelet using the following equation [21]:

The third layer consists of a single node that generates the ultimate output of the MRWNN structure, as expressed in the following equation:

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 jth wavelon and the output node, a _i represents the weight that connects the ith input node to the output node, and b represents a bias term of the output node. Based on the previously presented information, it is evident that the MRWNN structure has several adjustable weights, which may be encompassed in the set as follows:

To utilize the MRWNN structure as the FF controller, it is necessary to train the weights mentioned in Equation (7) by minimizing the ISE described in Equation (1).

2.2 FB controller design

To implement the FF-FB control system, the FB controller must be created after developing the appropriate FF controller, which was generated in the preceding step. The FB controller uses the error signal, also known as the control error, to determine the appropriate control action to force the system output to follow the command signal. The FB controller in this work was constructed as an H-infinity controller. To produce the FB controller’s output, denoted as u fb ( t ) , this controller must first receive input signals representing the control error e(t). The ultimate control action u ( t ) was produced by combining the FF controller output u ff ( t ) with the FB control signal.

In control theory, the H-infinity approach is employed to build controllers that achieve stability with assured performance and provide durability. In the late 1970s and early 1980s, Zames incorporated this approach into the domain of control theory [7].

H-infinity control is derived from the algebraic domain where the optimization takes place. H-infinity denotes the collection of analytic functions with matrix values that are limited to the open right half of the complex plane, specifically when the real portion is larger than zero. The H-infinity norm is defined as the largest singular value of a function across a certain space. The H-infinity technique is capable of minimizing the influence of disturbances on the closed-loop system.

The system dynamics in this section used a nonlinear approach, which included a linear description of a part of the state equation [21]. The linear representation is employed to precisely illustrate the nominal model of the system. Furthermore, to account for an extra disturbance in the nominal model, system nonlinearity and uncertainty were also included, described as follows:

where x ( t ) ∈ ℜ n represents a vector of system states, Δ d ( t ) ∈ ℜ m represents additive nonlinearity in this system, u c ( t ) ∈ ℜ l represents the control input, e ( t ) ∈ ℜ q represents the regulated output, y ( t ) ∈ ℜ p represents the output of the system, A ∈ ℜ n × n represents the system matrix, B 1 ∈ ℜ n × m represents the weight assigned to the perturbations, B 2 ∈ ℜ n × l represents the control matrix, C 1 ∈ ℜ q × n represents the weight matrix used for controlling the system state, D 12 ∈ ℜ q × l represents the management of the control inputs based on weight, and C 2 ∈ ℜ p × n represents the output weight matrix.

Additionally, C 1 T D 12 = 0 | and D 12 T D 12 = ρ I , where ρ is a matrix component, namely D 12 , which reflects the weight associated with regulating the control input.

Creating a control rule for the asymptotic tracking problem is necessary, such that the following holds [23]:

where e r ( t ) ∈ R n represents the tracking error vector, which is the difference between the system states x ( t ) and the reference model states x d ( t ) [23]:

The goal of the control challenge is to find the optimal controller, based on the system’s current state, to ensure that the infinite norm of the closed-loop transfer function ( T e Δ d ( s ) ) is below a given value of γ [24]. Thus, Equation (11) is required:

where T e Δ d ( s ) is the overall transfer function from Δ d (t) to e(t), and γ indicates that T e Δ d ( s ) exhibits linear development and is bounded by an upper limit that the controller always eliminates [25].

The condition in Equation (11) implies the following [22]:

where

The negative sign in Δ d indicates that the perturbation input is intended to optimize the cost function. Therefore, unlike the perturbation maximization activity, the controller would perform the minimization action.

Let the structure of the worst-case perturbation and recommended optimum controller be as follows [17,19]:

Substituting Equation (15) into Equation (8) results in the following:

By employing Assumption 1, the following result is obtained:

Therefore, the cost function is

Substituting Equations (14) and (15) into Equation (8) results in Equation (19):

Equation (18) is the performance index represented in the following form:

Therefore, Q i must be a matrix that is positive and definite.

The quadratic Lyapunov function is used to find the gain matrices for the worst-case perturbation, K Δ , and for the optimal controller, K [22].

The Lyapunov function defined in Equation (21) is also the time derivative:

where P i ∈ R n × n and Q i ∈ R n × n are real symmetric matrices, and V ̇ ( x ) and V ( x ) must be negative and positive definite functions, respectively:

To simplify the problem, the variables are expressed without considering the time parameter t. Subsequently, Equation (19) is inserted into Equation (23), resulting in the following:

By equalizing Equations (22) and (24), Equation (25) is obtained:

Equation (25) represents the Lyapunov equation for the closed-loop system in Equation (19). Usually, if the matrix Q i is positive-definite and the control input u c ( t ) causes the matrix [ A + B 1 K Δ + B 2 K ] ∈ R n × n to be Hurwitz, Equation (25) can be solved, and the positive-definite matrix P i represents its only solution [26].

The Lyapunov function value represents the reduction in cost from a certain beginning state x and time t [27]. As a result, we obtain Equation (26):

By taking the derivative of both sides with regard to time, we have Equation (27):

Based on Assumption 1, x ( ∞ ) = 0 . Therefore, V ̇ ( x ) becomes

By comparing Equations (22) and (27), it is observed that

The Lyapunov equation is then modified by substituting Equation (29) into Equation (25):

By differentiating Equation (30) with respect to K and setting the gradient ∇ K P i = 0 , the value of the controller gain matrix K can be determined as follows [17,20,23]:

Similarly, when Equation (30) is differentiated with respect to K Δ and the gradient ∇ K Δ P = 0 , the following is obtained [17,20,23]:

Inserting Equations (31) and (32) into Equation (30) gives Equation (33):

Equation (33) is referred to as the H-infinity algebraic Riccati equation (HIARE) [28]. Given that the HIARE is calculated based on the robustness requirements in Equation (11), the presence of a positive-definite solution P i is enough to meet this condition. Thus, the design issue is tackled in reverse order by first focusing on the HIARE and then using solution P to ascertain the optimum control rule [17,20,25,29].

2.3 MLP NN

The MLP is widely recognized as the predominant type of NN. MLPs belong to a large classification of structures referred to as FF NNs. The MLP is composed of at least three layers: an input layer, a hidden layer, and an output layer. The input data are subjected to processing, after which it is sent to the output layer via the hidden layer. Figure 3 shows the framework of the MLP. The sigmoid function, described in Equation (34), is often used as the activation function for hidden neurons [30]:

Figure 3

Structure of MLP [31].

Figure 3 illustrates that each layer of the MLP consists of many neurons or perceptrons. Each neuron in this set performs a calculation to produce an output determined by several different inputs. Usually, there is a specific quantity of nodes in each MLP layer [27].

The following equations are used to represent the MLP’s mathematical formulation [31]:

where z j represents the response of the jth hidden node, x i represents the ith input variable, and w i j denotes the magnitude of the linkage between the input node and the hidden node, w j . The connection weight between the hidden and output nodes represents the strength of the link between these two nodes, b j represents the bias, p ( · ) indicates a nonlinear activation function, and g ( · ) represents a function of linear activation.

2.4 Robust intelligent controller design

Both the FF and FB control techniques include distinct advantages and disadvantages. In this work, the FB control approach utilized an H-infinity controller. However, 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. Thus, the delay in the reaction of the FB controller may compromise the overall control performance and lead to stability issues [32]. On the other hand, the FF controller can anticipate changes in the reference signal and directly apply the appropriate action to the controlled system [33]. Furthermore, due to the absence of an FB signal requirement, the FF controller typically does not induce stability issues [34]. The FF-FB control structure, shown in Figure 4, was achieved by integrating the FF and the FB control techniques, resulting in a more robust and effective control system.

Figure 4

Schematic representation of the FF-FB controller.

The robust intelligent control law for the proposed controller is given as follows:

where f − 1 in Equation (38) is a nonlinear representation of the inverse dynamics of the system, and y ( t ) is the nonlinear system dynamics. ANNs can be taught to acquire the nonlinear function defined by Equation (38).

2.5 PSO method

The PSO algorithm employs particles as individuals within the population. Each particle navigates through a multidimensional search space with a velocity that is continuously adjusted based on the particle’s personal experience and the experiences of its neighboring particles or the entire swarm. This technique has been implemented in various domains, such as optimization problems in engineering, machine learning for training NNs, and robotics for path planning [35]. In particular, the implementation of the PSO algorithm is carried out in the following manner:

1. The individual solutions (called particles) constitute the population size (n).

2. The particles begin with a stochastic initialization and subsequently navigate through a search space to minimize an objective function.

3. The objective function is minimized to estimate the parameters.

4. The genotype’s fitness is calculated from the objective function of the particle, indicating the position of ( X p best ) (the best personal position) and ( X g best ) (the global best position). In such positions, these particles perform the required calculations.

5. The particles are attracted toward their appropriate X p best positions and the general ( X g best ) positions in a scenario that favors the particles landing in better spaces [36].

The velocity of the ith particle, denoted as v i , is computed using the following equation:

where for the ith particle in the kth iteration, ( x i ) represents the position; ( p best i ) is the past best position; ( g best ) is the past global best position of the particles; and the acceleration coefficients ( c m 1 ) and ( c m 2 ) represent the cognitive and the social scaling characteristics, respectively.

In addition, r m 1 and r m 2 are two arbitrary integers between 0 and 1, and the constriction coefficient (χ) is defined as follows [37]:

where ϕ = c m 1 + c m 2  and  ϕ > 4 . Consequently, this coefficient serves to prevent explosions and guarantee convergence. The ith particle’s new position is then expressed as follows [38]:

The velocity in the standard PSO is calculated as follows [35]:

By multiplying Equation (43) by w, where w ≥ 0 , which is defined as the inertia weight factor, the velocity equation becomes

To this end, previous experimental studies on PSO with the inertia weight have shown that a relatively large w has more global search ability, while a relatively small w results in a faster convergence [35].

When the maximum number of iterations is achieved, or a suitable cost is obtained, the PSO operation ends. After several iterations, the optimal costs remain unchanged, suggesting that there are no more optimal possibilities available [35,36].

In most applications, the weights of the MLP networks are first assigned small random values. However, a disadvantage of this strategy is that it might result in local minima inside the search area. WNNs differ from MLP networks in that the wavelet functions in WNNs define their properties using dilation and translation parameters. The wavelet represents a waveform characterized using certain dilation and translation parameters. These variables control the effectiveness of the wavelet over a limited period. As a result, the random initialization of dilation and translation parameters might lead to the production of inefficient wavelons, which in turn has a detrimental impact on the convergence rate of the network.

The goal of training the WNN structure to optimize its parameters is to reduce the difference between the system’s output and the model. Multiple changeable parameters need to be optimized. The parameters can be represented using the following settings:

For the WNN structure to achieve optimal performance, the parameters in Equation (45) must be optimized using a suitable optimization approach. This work utilized the particle swarm algorithm optimizer to determine these parameters.

The FB controller optimization was employed offline to tackle the optimization problem of the design procedure. The goal was to determine the optimal value of ρ and the optimal values of the elements in the matrix C 1 such that the infinity norm of ∥ T e Δ d ( j w ) ∞ ∥ is less than or equal to the optimal value of γ . The selection of the optimization cost function is as follows [29]:

Figure 5 illustrates the block diagram of the suggested control method. Additionally, the technique for designing the controller is summarized in the following steps:

1. Using the PSO algorithm, the optimal values of γ, ρ , and C 1 were found by minimizing the cost function specified in Equation (33).

2. The HIARE equation was then solved for the variable P i , utilizing the calculated optimal values.

3. Subsequently, matrix P i was employed in Equation (32) to derive the values of the gain matrix K .

4. Equation (31) yielded the optimal control rule, which was then applied to the initial system model.

Figure 5

Proposed controller’s block diagram using PSO.