1 Introduction

One of the important topics in control engineering is how to reduce higher order systems [1, 2], where the objective is to design a controller that has a lower order, while preserving effectiveness when used with higher order systems [1].

Higher order systems can be found in multifarious industries such as oil, cement, chemical, pharmaceutical, aircraft system, atomic nuclear plant, flexible robot manipulator [3], quadrotor with a variable degree of freedom (DOF) [4], fuel injector and spark timing of automobiles, etc. [5]. Nonlinear systems are usually linearized at different operating points, and appear to be higher order systems [6, 7]. In such a case, higher order system can be obtained using the first principle method or the finite-element model [8, 9]. Due to poor-conditioning, time and memory limitation, most of these models may not be suitable for many applications such as improving the design of control systems or analyzing them, especially in large-scale systems [10, 11]. Due to the complexity and high cost, it is difficult to design a precise control of large-scale control systems. Therefore, many researches proposed suitable controllers based on model order reduction, the reduction methods in most of these researches are: Routh approximation, Pade approximation, minimizing and rounding the model by reducing the flow of errors, the principal component analysis method, and the model reduction using impulse/step error minimization, in all these approaches, the higher order system is approximated to a lower order one by using model reduction, through basic step-to-step method for element analysis [12] and balanced-truncation method [13]. Recently, several reduction methods have been suggested based the optimization algorithms such as in [14–17]. These algorithms consist of finding a solution to problems that may minimize or maximize costs. Depending on model reduction methods, many controllers can be simply designed to control the higher order system such as PID [18, 19], Internal Model Controller (IMC) [20], Robust Dead-beat Control (RDBC) [21, 22].

The RDBC is one of the common controllers that has been designed to control the lower and higher linear systems. This controller has been successfully implemented on first and second order systems. It was shown from the result of the tested system that the settling time increases when the order of the system is increased, in addition to the difficulty in obtaining the controller parameters [22]. So the main challenges in this paper are to simplify the calculation of the RDBC parameters as well as achieving the predefined specifications and hence reduce the settling time of the overall control system. Therefore in this paper, the model order reduction method, called Mixed Reduction Method (MRM) is first optimized by Flower Pollination Algorithm (FPA), then the hybrid neuro-deadbeat control scheme is proposed to control different higher order systems, where the RDBC is combined with a simple adaptive modified radial basis neural network, to enhance the robustness and the performance of the RDBC in tracking any set point. FPA is used here because this algorithm is adaptable, simple and efficient, it is established to balance between exploitation and exploration by a variable called switching probability.

The rest of this paper is organized as follows: Section 2, presents the model order MRM. Section 3 explains the suggested optimized MRM by using flower pollination algorithm. Section 4 illustrates the suggested RDBC design, while Section 5 shows the proposed improved RDBC by using simple modified radial neural network. Section 6 addresses the simulation results of the proposed controller for different test examples. Finally, summary and conclusions are presented in Section 7.

2 Model order by Mixed Reduction Method (MRM)

One of the techniques utilized for order reduction is the MRM, which is used in numerical simulations to reduce the complexity of the mathematical models [18]. Due to the complexity and high dimensionality, most modern mathematical models are challenging, with regard to numerical simulations. In this method, the approximation of the original model is calculated by subtracting the space dimensions of the state accompanying the degrees of freedom of the model. This allows for a reduced model to be evaluated with less precision, yet in much less time [18]. To get the kth reduced order models, two steps must be applied:

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The algorithm of modified cluster center

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

By cross multiplying Eq.(1) and comparing terms of similar powers of (s) for both sides:

The unknown numerator coefficients d₀, d₁,…., d_q can be found by solving the above equations, and the reduced order numerator can be written as [18]:

In this paper, steps 1 and 2 are used to reduce the order of the higher order equations to second order equations that has the following form:

3 Flower Pollination Algorithm (FPA)

The FPA was proposed by Yang in 2012 [24]. It is inspired by the process of flowers pollination in zoology where flowers' pollens are spread to another plant for reproduction. Sometimes the flowers pollinate with the aid of biotical using several animals or insects. Others pollinate via water or wind. Pollination [24] can be accomplished by cross-pollination or self-pollination, depending on whether it involves flowers from different plants or flowers from the same plant. This method was developed to tackle optimization problems of global domain with multiple objectives and multiple varied principles. This technique requires the participation of several individuals called pollinators. Self (Biotic) pollination depends on local pollination operations while the global pollination operations are used with cross-pollination. The technique supposes that a solution is a single pollen. Pollination technique can be employed to motivate members' reproduction, which includes vector estimation and candidate solution. The size of the step in the next epoch is based on the pollination intensity. The Levy flights equation is used to simulate the pollinator movement for a global pollination [25]. To evaluate the pollination, the candidate vectors are selected randomly from a uniform distribution. If the selected new member is optimal, it is re-trained, and updated the worst members, however, one ignored. This iterative process is continued until the optimum member is obtained. Furthermore, this algorithm has more points to be investigated, such as non-homogeneity, multiple dimensionality and complexity of time [26]. The FPA depends on the following four rules [27]:

1. (i)Biotic and Levy flights can be used for carrying pollinators, for global pollination, and Cross-pollination.
2. (ii)Self -pollination and abiotic-pollination are used for local pollination.
3. (iii)Pollinators, like insects, can increase flower firmness. This role is equivalent to cloning probability, which refers to the similarity of two implicated flowers.
4. (iv)The switching between local and global pollination can be controlled by interaction probability P_n ∈ [0, 1].

To formulate the algorithm update procedure, the assumptions above are required to be translated into computable steps. Such as, the flower pollen gametes are moved by birds or insects for the global pollination, the birds or insects can often move and fly for a long distance. Rule (i) and the flower firmness can be illustrated mathematically as in Eq. (6) below [28]:

Since insects may fly over a long range with different length of steps, the Levy flight can be used to simulate this case. Let L > 0 from Levy equation:

Here, $\Gamma \beta$ is the standard gamma function, and this equation is proper for large steps S > 0. Next, to simulate the local pollination, both rule (ii) and rule (iii) can be illustrated as:

3.1 Optimized Mixed Reduction Method (OMRM) by using FPA

Each flower (or pollen) in FPA refers to a solution in control problem. Likewise, when employing FPA to improve the MRM for reducing the higher order system, every individual agrees to a candidate solution which is referred to as higher order factor. The first step of FPA is the initialization of the population. The initial population consists of N individuals. Each member presents a candidate solution in on N-dimensional space of solutions. The population parameters initialization and the fitness function are represented as:

(9)$\text{Pop}=\left[n_1,n_2,m_1,\text{and}{m}_2\right]$

where n₁ is $X_1^t,n_2$ is $X_2^t,m_1$ is $X_3^t,m_2$ is $X_4^t$, and t is the iteration index. The Fitness function is chosen as:

(10)$\text{Fitness}\text{function}={\displaystyle \sum _{i=1}^N\left(y_n(i)-y_r(i)\right)}$

where: $y_h(i)\text{and}{y}_r(i)$ are the original higher order system and the reduced model by MRM (which optimize by flower pollination parameters) respectively. The implementation steps of the optimized MRM by FPA are summarized in Fig. 2.

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The flow chart of FPA for optimizing the MRM

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

4 Robust deadbeat controller design

Deadbeat control was originally introduced as a linear state-feedback control for discrete-time systems. It is the simplest of all performance methods, and it is the most common method applied today [29]. The overall control system provides good parametric robustness in a steady state; the control error is practically zero [30] which is the main goal of the control system to reach the desired value within specified settling time [31]. A deadbeat response should have zero steady state error (e_s.s), minimum rise time (t_r), controllable settling time (t_s), overshoot (M_p) from 0 to 2%, undershoot <2%. Also have robustness against external disturbance and parameter uncertainty [31]. However, uncertainties on the input gain remain problematic [32].

4.1 Deadbeat controller design for higher order system

Fig. 3 shows the basic structure of the deadbeat controller.

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Block diagram for the basic robust deadbeat control structure

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

The main control law of this structure is:

(11)$u(s)=u_o(s)+u_c(s)$

with

(12)$u_0(s)=H_2(s)y(s)=K_{a}y(s)$

(13)$u_c(s)=K{K}_C\frac{\left(S^2+Xs+Y\right)}{S}{e}_h(s)$

where u_o(s) is an output feedback control, while u_c(s) is the portion of the control signal generated by cascade PID with series gain K, which is introduced in order to compensate the higher-order plants, to deal with uncertainties, and to be able to perform the desired specifications such as settling time, error steady state and maximum overshoot.

The error signal e_h(s) is defined by;

(14)$e_h(s)=r(s)-H_1(s)y(s)$

where $H_1(s)$ is an additional derivative controller, its structure depends on the order of G_c(s)G_p(s) transfer function as shown in Table 1.

Table 1.

The structure of the $H_1(s)$ equation

Gc(s)Gp(s) order	2	3 or 4	5
H1(s) equation	H1(s) = 1	H1(s) = 1 + Kbs	H1(s) = 1 + Kbs + Kcs2

The deadbeat controller parameters can be determined by comparing the equivalent characteristic equation of closed-loop controlled system with the following desired deadbeat characteristic equation:

(15)$$s^{n_p}+{\alpha }_1{\omega }_n{s}^{n_p-1}+{\alpha }_2{\omega }_n^2{s}^{n_p-2}+\cdots +{\omega }_n^{n_p}$$

The ${\alpha }_i$ coefficients and the deadbeat settling time t_s of the step response corresponding to different G_c(s)G_p(s) order are illustrated in Table 2.

Table 2.

Coefficient of the desired deadbeat characteristic equation [21, 33].

Gc(s)Gp(s) order	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$	${\alpha }_4$	ts
2nd	1.82	–	–	–	4.82
3rd	1.90	2.2			4.04
4th	2.20	3.50	2.8		4.81
5th	2.70	4.90	5.40	3.40	5.43

Depending on desired settling time, the normalized frequency $\left({\omega }_n\right)$ is described as follows:

(16)${\omega }_n=\frac{t_s}{\mu t_{ds}}$

where μ is a selected positive constant between $0.8\le \mu \le 0.95$.

4.2 Design the Robust deadbeat based on reduction model

The block diagram for the suggested simple, robust deadbeat controller based on reduction model is shown in Fig. 4 below.

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Block diagram of the suggested robust deadbeat controller based on reduction model structure

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

where G_r(s) is the reduction model (Eq. (5)) which parameters values (n₁, n₂, m₁, and m₂) are determine by the improved OMRM method. While, the G_c is only the PI controller multiplied by the series gain K:

(17)$G_c=\frac{K\left(K_{c}s+K_{c}x\right)}{s}=\frac{K\left[K_c\left(s+x\right)\right]}{s}$

with $x=\frac{1}{T_i}$

$T_i$ is a suitable selected integral time, while $H_1\text{and}{H}_2$ as shown in Fig. 3, are given by:

(18)$H_1(s)=\left(1+K_{b}s\right)$

(19)$H_2(s)=K_a$

Since K_b is a derivative gain, then it can be written as;

(20)$K_b=K_c{T}_d=K_c\frac{T_i}{4}=\frac{4K_c}{T_i}$

And, since K_a is just a proportional gain, it can be determined from K_b as follows:

(21)$K_a=\frac{K_p}{T_d}=\frac{4K_p}{T_i}$

So, the transfer function of closed loop controlled system will be:

(22)$\frac{Y(s)}{R(s)}=\frac{G_c(s)G_r(s)}{1+G_r(s)H_2(s)+G_c(s)G_r(s)H_1(s)}$

or

(23)$\text{C}\text{.L}\text{.T}\text{.F}=\frac{2.394xKaKd+2.394KaKds}{s^2+\left(2.657+2.394KaKd\right)s+2.394Ka+2.394xKaKd}$

The desired characteristic equation which will compare with the characteristic equation of the closed-loop controlled system is:

(24)$s^3+{\alpha }_1{\omega }_n{s}^2+{\alpha }_2{\omega }_n^{2}s+{\omega }_n^3$

In this case, the procedure for design the robust deadbeat includes the following steps:

1. The α₁ and α₂ are taken from Table 2, while ω_n is selected according to specified t_s (Eq. (16)) then the desired characteristic equation is determined.
2. K_b is chosen to be less than one.
3. K_a is calculated according to Eq. (21)
4. After comparing the overall closed loop equation with desired characteristic equation, the K_c is determined as below:

(25)$K_c=\frac{{\omega }_n^3}{b_{2}x-{\omega }_n^3{K}_b{b}_1}$

5 The proposed hybrid modified RBF neuro-deadbeat controller design

Instead of explicitly choosing a value for the series gain K of the designed deadbeat controller, an adaptive self-tune function K(t) is designed based on the modified feed-forward radial basis function (RBF) neural network.

In general, the basic structure of the RBF neural network consists of three layers (input layer, hidden layer and output layer). The input layer is just buffered units, which pass the signal without altering it. The hidden layers are nonlinear activation functions (Gauss basis function). The output layers have linear units which add the signals fed to them [34].

In this paper, the suggested modified radial basis function (MRBF) neural network is shown in Fig. 5, this network consists of input layer with single node (neuron) g(t), hidden layer with three node h_j (j = 1, 2, 3) and single output layer in addition to nonlinear function f(t), which is used to improve the performance of RBF network.

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The structure of the proposed modified RBF neural network

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

The function of the single node input layer (g(t)) is given by:

The nonlinear function of the hidden layer node is [35]:

The f(t) is designed as [36]:

As mentioned previously, the proposed MRBF network is used to generate the K(t) for the deadbeat controller instead of using series gain K. Accordingly, combining the deadbeat controller and the MRBF, the control law of Eq. (11) will be modified to the following form:

6 Simulation results

Firstly, the efficiency of the reduction methods (MRM method and the improved OMRM) was evaluated by calculating the Integral Square Error (ISE) between the output response of the original model and the reduced models by using Matlab software version R2015a. Table 3 shows three higher order tested examples whose reduction models by MRM, and OMRM are given in Table 4. The simulation results for these examples tested by unit step input signal are illustrated by Fig. 6. It can be seen from these results and the values of ISE in Table 4 that the OMRM result in more accurate reduction model, which is given a very close behavior to the performance of the original higher order model either in transient or steady state response than the MRM. This is due to the use of the FPA, which gives optimum values for the reduction model by MRM.

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Comparison of step responses for the original model and the reduced model of the three tested examples

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

The above results also shows that if the order of the original system is very high (Ex.3), there will be an error in transient response of the obtained reduction model by MRM, but this error is less in the model obtained by the OMRM.

In other words, the suggested RDBC, RDBC-MRBF are designed to control the higher order system of Table 3 depending on the reduced model of OMRM, with t_s = 4.82 s, μ = 0.95, t_ds = 2.5368 s and hence ω_n = 2 rad/s for all examples, and the final parameters for both controllers are given in Table 5, expect K value is not used with RDBC-MRBF. While the parameters of the designed MRBF neural network are explained by Table 6.

The robustness, stability assurance, and the efficiency of the proposed RDBC, and RDBC-MRBF are illustrated by simulation results in Fig. 7 and Table 7 for step input signal, these figures show output responses, and control efforts of the original three tested systems. These results show that the performance of both controllers (RDBC, and RDBC-MRBF) are efficient, especially RDBC-MRBF because it makes the original higher order model flow the reference input signal with small settling time, small peak time, and small overshoot as compared with RDBC with same steady state control signal (u_s.s), as illustrated in Table 7. The efficiency of the RDBC-MRBF is due to making the constant gain K update by the suggested MRBF, which gives the appropriate value to improve the efficiency of the RDBC.

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Simulation results for original higher order examples controlled by (RDBC, and RDBC-MRBF) with step reference input, (a) Output response of Ex.1, (b) control signal of Ex.1, (c) Output response of Ex.2, (d) control signal of Ex.2; (e) Output response of Ex.3, (f) control signal of Ex.3

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

By using the same designed parameter values of Table 5 and unit step input, the performance efficiency of the RDBC, and RDBC-MRBF is also evaluated by testing the original higher order examples with 20% uncertainty in their parameters, the simulation results in this case are illustrated in Table 8, which shows the efficiency of both RDBC and RDBC-MRBF specially RDBC-MRBF because the higher order models flow the reference signal with small M_p, t_p , t_s, e_s.s, ISE with this controller.

To check the ability of the designed RDBC, and RDBC-MRBF to force the higher order tested examples to follow an arbitrary reference input signal at same designed parameter values given by Table 5 are used to test the performance of Ex.3 (eighth order model) for an arbitrary input signal. The simulation results of this case for output response, control signal, error signal, and the update K(t) are shown in Fig. 8. Again, the simulation results show the efficiency of both controllers, especially RDBC-MRBF, which make the higher order model flow the arbitrary input signal very fast with nearly zero steady state error and no overshoot, in addition to smooth control signal with adaptive parameters for the K(t).

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Simulation results for original 8th order model (Ex. 3) controlled by (RDBC, and RDBC-MRBF) with arbitrary input signal, (a) Output response, (b) control signal, (c) error signal, (d) nonlinear function K(t) response

Citation: International Review of Applied Sciences and Engineering IRASE 11, 3; 10.1556/1848.2020.00090

7 Summary and conclusions

The RBF neural network has a simple design with promising performance, it shows tolerance to different noises that come with inputs and data real-time capture capability. The characteristics of this network make it very suitable for adaptive control framework. Therefore, in this paper, a simple modified RBF neural network is proposed for enhancing the performance of the suggested RDBC, which is used to control higher order linear systems based on a reduced second order model. The reduced model is produced by a suggested optimized mixed reduction method.

The simulation results for three higher order systems have shown that the proposed hybrid controller scheme effectively controlled these systems, even with arbitrary input or there is 20% uncertainty in system parameters. The hybrid controller scheme is simple; therefore, it can be used to control the single input-single output (SISO) nonlinear systems based on a linearized model such as inverted pendulum, mass-spring, dc motor, and single robotic arm and so on.