State Feedback Control for Vehicle Electro-Hydraulic Braking Systems Based on Adaptive Genetic Algorithm Optimization

. In traditional state feedback control, the difculty in determining the coefcient matrix is a signifcant factor that prevents achieving optimal control. To address this issue, this paper proposes the integration of adaptive genetic algorithms with state feedback control. Te efectiveness of the proposed algorithm is validated via an electro-hydraulic braking system. Firstly, a model of the electro-hydraulic braking system is introduced. Next, a state feedback controller optimized by parameter-adaptive genetic algorithm is designed. Additionally, a penalty term is introduced into the ftness function to suppress overshoots. Finally, simulations are conducted to compare the convergence speed of parameter-adaptive genetic algorithm with genetic algorithm, ant colony optimization, and particle swarm optimization. Furthermore, the performance of the proposed algorithm, the state feedback control, and the proportional-integral control are also compared. Te comparison results show that the proposed algorithm efectively accelerates the settling time of the electro-hydraulic braking system and suppresses the overshoots.


Introduction
Te electro-hydraulic brake system (EHB) is diferent from the traditional automobile braking system.EHB is an advanced mechatronic system by replacing local mechanical components with electronic components [1].EHB uses a comprehensive brake module to replace the pressure regulator and antilock braking system (ABS) and can adjust the braking pressure of four wheels independently [2,3].Compared with the traditional automobile braking system, EHB has the advantages of soft braking process, compact structure and fast response, which leads the development trend of automobile braking systems [4,5].
Te output pressure of an electro-hydraulic braking system (EHB) is primarily controlled by an electrical signal, typically voltage or current, regulated by an electronic control unit (ECU).When the driver presses the brake pedal, sensors on the pedal generate an electrical signal and transmit it to the EHB control unit.Te control unit then calculates and adjusts the pressure of the brake fuid based on the received signal, enabling precise control of the braking force on the wheels [6,7].But in fact, in the process of converting electrical signals into mechanical force, due to energy transfer losses, structural defects, and external environmental interference, the actual output pressure is far from the target pressure [8,9].
At present, there is a wide range of research focused on improving the performance of EHB.Among them, state feedback control is highly favoured by many scholars due to its applicability to multiinput multioutput systems and its simplicity.Reference [10] introduces an optimal feldoriented control using a linear quadratic regulator (LQR) for the in-wheel motor.An analytical method for determining Q and R matrices is examined.Reference [11] an analytical method is introduced to determine the coefcient matrix in the linear quadratic controller.Tis controller demonstrates an excellent disturbance rejection performance.In [12], a linear matrix inequality-based robust multiobjective LQR controller is designed for active trailer braking system with constraints on closed-loop pole locations and guarantee of robust stability.In [13], the lateral stability is achieved through independent motor torque control using LQR and PID.However, in the above research, the selection of coefcient matrices is based on empirical or experimental analysis methods.Tese approaches heavily rely on the designer's experience and it is highly difcult to obtain the optimal value.
Recently, some nature-inspired algorithms, such as the genetic algorithm (GA) and particle swarm optimization (PSO) have been applied to obtain the global optimal parameters of LQR controllers.In [14][15][16], PSO, adaptive particle swarm optimization (APSO), and ant colony optimization (ACO) algorithms are used to determine the coefcient matrix in the linear quadratic controller.In [17], a comparison was made between the GA-based LQR and the traditional LQR applied to a doubly-fed asynchronous generator system.Te results indicated that the GA-based LQR outperforms the traditional LQR in terms of stability and robustness.Reference [18] focuses on controlling a double inverted pendulum using pole placement and LQR control.To optimize LQR control parameters, GA and PSO are employed.Reference [19] addresses the design of LQR and PID controllers for an aircraft's pitch control system.GA is employed to optimize the parameters of both LQR and PID controllers.In the above research, most controllers use similar ftness functions, which aim to minimize the difference between the reference values and the actual values without incorporating additional constraints.In this situation, it may lead to excessive overshoots, as reported in [18,20,21].In addition, the issue of GA being susceptible to get trapped in local optima has not been efectively addressed.Terefore, to prevent extreme optimization and the risk of falling into local optima during the optimization process, it is essential to incorporate penalty terms into the ftness function and thus to improve the GA performance.
Considering the above issues, a parameter-adaptive genetic algorithm-based optimization control method has been proposed.Te following are the major contributions of this paper: (1) Combining GA with SFC, this approach uses the global optimization capabilities of GA to seek the optimal coefcient matrices within state feedback control.Tis innovative method addresses the challenge that the appropriate coefcient matrices are difcult to be determined in traditional SFC.
(2) In order to prevent extreme optimization scenarios, this paper modifes the ftness function by incorporating penalty terms designed to suppress overshoot.(3) Te adjustment of crossover and mutation probabilities based on individual ftness is employed to achieve parameter self-adaptation, efectively accelerating the convergence speed of the GA.
Te rest of this paper is organized as follows.Mathematical model of EHB is in Section 2. Ten the pressure tracking controller design is presented in Section 3. In Section 4, the simulation verifcation and result analysis are introduced, followed by conclusions.

Electro-Hydraulic Braking System Model
Te simplifed diagram of single wheel electric-hydraulic braking is shown in Figure 1.Te brake valve used in the system is proportional reducing valve.Te electronic control unit of the EHB calculates the brake pressure required for the wheel based on information such as brake pedal travel, road adhesion coefcient, and vehicle speed.Ten the electronic control unit controls the opening amount of the proportional reducing valve, allowing hydraulic oil to enter the brake wheel cylinder, and then brake the wheel.
Te mathematical model of the EHB can be represented as follows [9]: (1) x p , x 5 � p c and the state space expression can be written as 2 International Journal of Intelligent Systems Te brake system model given by equations ( 2)-( 6) can be described in the standard form of linear state equations as follows: where

Pressure Tracking Controller Design
Te performance objective of an efective braking system is to follow the desired pressure target rapidly and precisely.Terefore, the control problem for the system can be defned by the following performance function: where Q is the weight coefcient matrix with Q � diag q 1 q 2 q 3 q 4 q 5   ; R is the control weight coefcient with R � [r]; e � x r − x is the error values of the state variables; t 0 and t f are the start time and the end time, respectively.Te control problem currently is: fnd an optimal control u to make the performance functional J take a minimum.
According to Pontryagin's minimum principle, it can be concluded that the optimal control u(t) is where P and g satisfy the following equations: Te values of Q and R have a signifcant impact on the dynamic performance of the system.When the diagonal elements of the Q takes the same value, it indicates that all state variables of the system are equally important.Te larger the value, the more important the corresponding state variable.On the other hand, the R matrix is closely related to the control inputs.Te larger the values of the diagonal elements, the greater the limitations imposed on the corresponding control inputs.Tis may lead to overshooting, but it can reduce the control efort.
After obtaining the control input u(t) via solving the equations ( 10)-( 12), it appears that a solution to the optimal control problem has been obtained.However, since the Q International Journal of Intelligent Systems and R in equation ( 10) are manually specifed, diferent choices of Q and R lead to diferent values of u(t), consequently resulting in varying cost values J.To address this issue and determine the values of Q and R that minimize J, this paper employs the GA.
GA is an optimization algorithm inspired by the natural process of evolution.It simulates the principles of biological evolution to search for optimal solutions.Figure 2 illustrates the optimization process of the GA.
Tis paper adopts binary encoding method.Te length of binary code is related to the accuracy of solving the problem.Suppose an individual code is expressed as Te corresponding decoding method is where U max and U min represent the range of decimal values for parameters, and l is the number of bits for binary encoding.
In this paper, the individual code length is 60, and its genotype is composed of 60 binary numbers.Te frst ffty represent matrix Q, with a value range of 1-1000, and the last ten represent matrix R with a value range of 1-10.
In GA, Te ftness function is used to assess the performance or quality of each individual solution in the problem space, quantifying the degree to which an individual is superior or inferior in solving a specifc problem.Here, we frst initialize the defnition of the objective function as For the purpose of facilitating comparisons with other algorithms, we have modifed the "roulette" selection as shown in formula (17).After the modifcation, individuals with smaller ftness values will have a larger selection probability.
In traditional GA, the probabilities of crossover and mutation are fxed.Choosing values that are too large can result in nonconvergence, while values that are too small can lead to slow convergence.Terefore, this paper proposes a PAGA (Parameter adaptive genetic algorithm) in which the crossover and mutation probabilities for each individual are determined by the population's average ftness, individual ftness, and minimum ftness, as expressed below.
where P c is the crossover probability, P m is the mutation probability, F i(i�1...M) is the individual ftness, F min is the minimum ftness, F ave is the average ftness, and k i(i�1...4) are adjustment parameters.
In the above, we initially defne the objective function as (15), which means the sum of errors between various state variables.Because it does not impose any restrictions on the overoptimization situations, it may lead to considerable overshooting.Tis overshooting can have adverse efects on safety.To efectively suppress overshooting, a penalty term is introduced into the objective function. where . a is the penalty coefcient, which will be determined in Section 4. Figure 3 is the fow diagram of the control system proposed in this paper, which is composed of two parts.Te upper part is the PAGA, and the lower part is the LQR and EHB model.At the beginning, the PAGA randomly generates an initial population consisting of 20 individuals.Each individual decodes the corresponding Q and R according to the formula (14), and then substitutes them into the simulation model of LQR and EHB one by one to get the actual response curve.Ten, the ftness is calculated through the formula (19).When overshoots occur, that is, e pc < 0, the ftness will increase signifcantly, thereby reducing the probability of selecting individuals with overshoots, and further suppressing the generation of overftting.Te next step will judge whether the number of evolutions reaches the set value, if not, then the next generation will be produced according to the individual ftness in the way of "roulette" selection (16), and it is possible that crossover and mutation will happen according to the set probability.Cycle the above process until the number of evolutions reaches the set end value.Te individual with the highest ftness in the last generation is the optimal individual, and the decoded Q and R are the optimal weight coefcients.
To better illustrate the mechanism of how the penalty term suppresses overshoot, we provide a detailed explanation combined with the following Figure 4. We initial that the population size is 20, and the initial state of each International Journal of Intelligent Systems individual is represented by a 60 bit binary number.After decoding, it is transformed into coefcient matrices Q (5 × 5) and R (1 × 1), which are then brought into the simulation models of LQR and EHB to obtain the response curve.Te ftness of the individual is calculated using the ftness calculation formula (19).Due to the addition of the penalty term, the overshoot part will be amplifed ten times, resulting in an increase in ftness and subsequently a decrease in the probability of selection and inheritance to the next generation.Over multiple generations of evolution,   Generate initial population randomly  International Journal of Intelligent Systems individuals carrying the gene for overshoot will gradually disappear from the population, efectively preventing the occurrence of overshoot.

Simulation Verification
In order to verify the performance of the proposed algorithm, this paper establishes a simulation model for the EHB, and compares the performance of the GA, PAGA, PSO, and ACO algorithms.Te relevant parameters of the EHB and the GA are listed in Tables 1 [2] and 2. In this simulation, typical values were chosen for the parameters of all optimization algorithms, while the adjustment parameters k i(i�1...4) for PAGA derived from multiple experimental trials.
To determine the appropriate penalty coefcient, we compared the efects of diferent penalty coefcients on suppressing overshoot by using the TO after 20 iterations as the evaluation criterion.Te formula for calculating the TO is as follows: where t s and t e are the time of the start overshoot and end overshoot.
Figure 5 shows the TO after 20 iterations for diferent values of a.It can be observed that when the value of a exceeds 10, the TO is suppressed to nearly zero.Terefore, this study selects a � 10.So, the λ � −10, e pc < 0 1, e pc ≥ 0  .
To highlight the advantages of PAGA in terms of optimization speed and the ability to overcome local optima, three additional algorithms are introduced, PSO, ACO, and GA for parameters turning process.Figure 6 is the average ftness convergence curve of four algorithms after 10 experiments with the same ftness function (15).Table 3 presents the statistical data for four algorithms across ten runs, where MF stands for "Minimum Fitness" and TItMF is the abbreviation for "Iteration Times to Minimum Fitness." By comparing Figure 6 and Table 3, it can be observed that, under the same number of iterations, the PAGA converges much faster.PAGA fnds the optimal solution at the 19th iteration, whereas PSO converges at the 36th iteration, ACO converges at the 26th iteration, and GA converges at the 20th iteration.In addition, although all three algorithms converge after 40 iterations, but they end with diferent minimum values.Te minimum value found by PSO is 9.6762; ACO reaches a minimum value of 9.4348, while GA obtains a minimum value of 9.3583.In addition, the GA briefy encountered local optima, whereas in PAGA, this problem was efectively resolved.Tis suggests that during the parameter tuning process, PSO, ACO, and GA might have become trapped in local optima, while only PAGA managed to avoid local optima and fnd the global optimum solution.Te simulation results demonstrate that the proposed parameter-adaptive GA efectively avoids local optima and accelerates convergence.
Figure 7 shows the response curve during the autotuning procedure with PAGA.With the increase in the iteration, the response curve tends to converge more closely to the reference value.In Figure 7(a) (with no penalty term), the algorithm tends to minimize tracking errors as much as possible without imposing constraints on the overshooting behaviour.Tis results in extreme operating conditions in the result.However, in Figure 7(b), with the introduction of a penalty term, this risky condition is efectively suppressed.
After 40 iterations, the coefcient matrices with and without penalty term are shown below.It can be observed that the overall values of Q npen are higher than Q pen , while   Te bold values are those yielded with the proposed method, while the unbold ones in that row are those yielded with compared methods.
To evaluate the stability of the proposed algorithm, we conducted tests under three diferent operating conditions.Additionally, a comparison was made with the traditional SFC (Q 0 � diag([100 100 100 100 100], R 0 � diag [2]), SFC optimized by PAGA with F npen (PAGA-SFC1), SFC optimized by PAGA with F pen (PAGA-SFC2), and the PID controller optimized by GA (GA-PID).Figures 7-9 depict the responses of the EHB under no load, constant load, and variable load conditions, respectively.

Constant Load Condition.
When the load pressure is set to 1 MPa, all pressure tracking controllers are able to achieve zero steady-state error.As shown in Figure 8(a), the SFC shows a slower rise time compared to the other controllers and takes a longer time to reach the target value.Although GA-PI has a faster rise time than SFC, the pressure values show larger variations, and the settling time is also longer.However, PAGA-SFC1 and PAGA-SFC2 demonstrate signifcantly faster rise times and steady-state times.PAGA-SFC2 efectively suppresses overshoot due to the introduction of the penalty term, but it slightly increases the rise time.Figures 8(b) and 8(c) present the state values x v and x p for SFC, PAGA-SFC1, PAGA-SFC2, and GA-PI.In this test, both reference values are set to zero.With the help of the penalty term, PAGA-SFC1 and PAGA-SFC2 exhibit smaller oscillation amplitudes and smoother curves compared to the other two algorithms.

Sudden Load Condition.
In this condition, the load pressure is initially set to 1 MPa.At 0.075 s and 0.2 s, 0.1 MPa sudden load pressure disturbances are introduced to test the disturbance rejection performance.Te variations in pressure and system state responses are shown in Figure 8.
As shown in Figure 9(a), when a sudden load pressure disturbance occurs, all three SFC controllers undergo an increase or decrease in pressure, with amplitudes smaller than that of GA-PI.However, the time it takes for each of them to return to the original state is signifcantly diferent.Among them, PAGA-SFC1 and PAGA-SFC2 exhibit much faster recovery speeds.PAGA-SFC1 shows a slight overshoot after the disturbance ends, while GA-SFC2 efectively suppresses pressure overshoot due to the introduction of the penalty term.Tis diference can be mainly explained by the variations in the weighting matrices R associated with the control signals.Te R 0 is greater than R 1 and R 2 .Terefore, SFC imposes more constraints on the control signals, limiting the amount of energy given to the system.As a result, GA-SFC1 and SFC2 demonstrate better disturbance rejection characteristics compared to GA-PI, while SFC performance in this regard is less satisfactory.Figures 9(b) and 9(c) show the variations in the state variables under diferent algorithms.Similarly, PAGA-SFC1 and PAGA-SFC2 exhibit signifcantly smaller fuctuations in their ranges.

Variable Load Condition.
In this simulation experiment, the load pressure is set to vary from 0.8 to 1.2 MPa according to the expression 0.2 × sin(50t) + 1. Te braking pressure output by the EHB is recorded in Figure 10        International Journal of Intelligent Systems conditions, all algorithms show some steady-state error under the varying load pressure condition.However, PAGA-SFC1 and PAGA-SFC2 show signifcantly faster response speed and higher accuracy than the other two.Tis behavior can be explained by the weighting matrix Q, where larger values indicate more constraints on the corresponding variables, and q 5 is related to the pressure error.In Q 1 , q 5 � 1000; in Q 2 , q 5 � 250.1; while in Q 1 , q 5 � 100.Tis means that PAGA-SFC1 and PAGA-SFC2 are more concerned about velocity reference tracking compared to SFC.As a result, GA-SFC1 and GA-SFC2 demonstrate better performance in following the varying load pressure.

Conclusions
Te selection of the coefcient matrices Q and R in traditional LQR control signifcantly infuences control performance.To address this issue, this paper proposes an approach that combines the genetic algorithm with LQR for automatic parameter tuning.In addition, to accelerate the convergence speed of the genetic algorithm, an adaptive genetic algorithm is introduced, which adjusts crossover and mutation probabilities based on individual ftness.Furthermore, to prevent overshooting, a penalty factor is incorporated into the ftness function to suppress the occurrence of overshoot.Te proposed algorithm, compared to PSO and ACO, has a faster convergence speed and is less likely to fall into local optima.When compared to traditional LQR and GA-PI, it has a faster response speed and stability time.However, there are also shortcomings in this paper.Only a single brake system model was digitally simulated without considering the impact on the control efect when the control object is the entire vehicle.Subsequent work will attempt to change the control object to the entire vehicle and verify the performance of the proposed control method under more external factors.

Figure 5 :
Figure 5: TO with diferent a.

Table 1 :
Parameters of the electro-hydraulic braking system.

Table 3 :
Algorithm performance comparison table.
(a).It can be observed that, compared to the two previous operating