Abstract
In traditional state feedback control, the difficulty in determining the coefficient matrix is a significant factor that prevents achieving optimal control. To address this issue, this paper proposes the integration of adaptive genetic algorithms with state feedback control. The effectiveness 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 fitness 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. The comparison results show that the proposed algorithm effectively accelerates the settling time of the electro-hydraulic braking system and suppresses the overshoots.
1. Introduction
The electro-hydraulic brake system (EHB) is different 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].
The 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. The control unit then calculates and adjusts the pressure of the brake fluid 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 field-oriented control using a linear quadratic regulator (LQR) for the in-wheel motor. An analytical method for determining and matrices is examined. Reference [11] an analytical method is introduced to determine the coefficient matrix in the linear quadratic controller. This 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 coefficient matrices is based on empirical or experimental analysis methods. These approaches heavily rely on the designer’s experience and it is highly difficult 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 [14–16], PSO, adaptive particle swarm optimization (APSO), and ant colony optimization (ACO) algorithms are used to determine the coefficient 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. The 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 fitness 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 effectively addressed. Therefore, 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 fitness function and thus to improve the GA performance.
Considering the above issues, a parameter-adaptive genetic algorithm-based optimization control method has been proposed. The 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 coefficient matrices within state feedback control. This innovative method addresses the challenge that the appropriate coefficient matrices are difficult to be determined in traditional SFC.(2)In order to prevent extreme optimization scenarios, this paper modifies the fitness function by incorporating penalty terms designed to suppress overshoot.(3)The adjustment of crossover and mutation probabilities based on individual fitness is employed to achieve parameter self-adaptation, effectively accelerating the convergence speed of the GA.
The rest of this paper is organized as follows. Mathematical model of EHB is in Section 2. Then the pressure tracking controller design is presented in Section 3. In Section 4, the simulation verification and result analysis are introduced, followed by conclusions.
2. Electro-Hydraulic Braking System Model
The simplified diagram of single wheel electric-hydraulic braking is shown in Figure 1. The brake valve used in the system is proportional reducing valve. The electronic control unit of the EHB calculates the brake pressure required for the wheel based on information such as brake pedal travel, road adhesion coefficient, and vehicle speed. Then 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.
The mathematical model of the EHB can be represented as follows [9]:
Select status variable and the state space expression can be written as
The brake system model given by equations (2)–(6) can be described in the standard form of linear state equations as follows:where
3. Pressure Tracking Controller Design
The performance objective of an effective braking system is to follow the desired pressure target rapidly and precisely. Therefore, the control problem for the system can be defined by the following performance function:where is the weight coefficient matrix with ; is the control weight coefficient with ; is the error values of the state variables; and are the start time and the end time, respectively. The control problem currently is: find an optimal control to make the performance functional take a minimum.
According to Pontryagin’s minimum principle, it can be concluded that the optimal control iswhere and satisfy the following equations:
The values of and have a significant impact on the dynamic performance of the system. When the diagonal elements of the takes the same value, it indicates that all state variables of the system are equally important. The larger the value, the more important the corresponding state variable. On the other hand, the matrix is closely related to the control inputs. The larger the values of the diagonal elements, the greater the limitations imposed on the corresponding control inputs. This may lead to overshooting, but it can reduce the control effort.
After obtaining the control input via solving the equations (10)–(12), it appears that a solution to the optimal control problem has been obtained. However, since the and in equation (10) are manually specified, different choices of and lead to different values of , consequently resulting in varying cost values . To address this issue and determine the values of and that minimize , 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.
This paper adopts binary encoding method. The length of binary code is related to the accuracy of solving the problem. Suppose an individual code is expressed as
The corresponding decoding method iswhere and represent the range of decimal values for parameters, and 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. The first fifty represent matrix , with a value range of 1–1000, and the last ten represent matrix with a value range of 1–10.
In GA, The fitness 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 specific problem. Here, we first initialize the definition of the objective function as
For the purpose of facilitating comparisons with other algorithms, we have modified the “roulette” selection as shown in formula (17). After the modification, individuals with smaller fitness values will have a larger selection probability.
In traditional GA, the probabilities of crossover and mutation are fixed. Choosing values that are too large can result in nonconvergence, while values that are too small can lead to slow convergence. Therefore, 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 fitness, individual fitness, and minimum fitness, as expressed below.where is the crossover probability, is the mutation probability, is the individual fitness, is the minimum fitness, is the average fitness, and are adjustment parameters.
In the above, we initially define 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. This overshooting can have adverse effects on safety. To effectively suppress overshooting, a penalty term is introduced into the objective function.where , , , . is the penalty coefficient, which will be determined in Section 4.
Figure 3 is the flow diagram of the control system proposed in this paper, which is composed of two parts. The 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 and 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. Then, the fitness is calculated through the formula (19). When overshoots occur, that is, , the fitness will increase significantly, thereby reducing the probability of selecting individuals with overshoots, and further suppressing the generation of overfitting. The 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 fitness 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. The individual with the highest fitness in the last generation is the optimal individual, and the decoded and are the optimal weight coefficients.
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 individual is represented by a 60 bit binary number. After decoding, it is transformed into coefficient matrices (5 × 5) and (1 × 1), which are then brought into the simulation models of LQR and EHB to obtain the response curve. The fitness of the individual is calculated using the fitness calculation formula (19). Due to the addition of the penalty term, the overshoot part will be amplified ten times, resulting in an increase in fitness and subsequently a decrease in the probability of selection and inheritance to the next generation. Over multiple generations of evolution, individuals carrying the gene for overshoot will gradually disappear from the population, effectively preventing the occurrence of overshoot.
4. 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. The 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 for PAGA derived from multiple experimental trials.
To determine the appropriate penalty coefficient, we compared the effects of different penalty coefficients on suppressing overshoot by using the after 20 iterations as the evaluation criterion. The formula for calculating the is as follows:where and are the time of the start overshoot and end overshoot.
Figure 5 shows the after 20 iterations for different values of . It can be observed that when the value of exceeds 10, the is suppressed to nearly zero. Therefore, this study selects = 10. So, the .
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 fitness convergence curve of four algorithms after 10 experiments with the same fitness 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 finds 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 different minimum values. The 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 briefly encountered local optima, whereas in PAGA, this problem was effectively resolved. This 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 find the global optimum solution. The simulation results demonstrate that the proposed parameter-adaptive GA effectively avoids local optima and accelerates convergence.
Figure 7 shows the response curve during the auto-tuning 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. This results in extreme operating conditions in the result. However, in Figure 7(b), with the introduction of a penalty term, this risky condition is effectively suppressed.
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After 40 iterations, the coefficient matrices with and without penalty term are shown below. It can be observed that the overall values of are higher than , while is lower than . This indicates that a larger places more emphasis on state values, and a smaller results in less emphasis on control values.
To evaluate the stability of the proposed algorithm, we conducted tests under three different operating conditions. Additionally, a comparison was made with the traditional SFC (, SFC optimized by PAGA with (PAGA-SFC1), SFC optimized by PAGA with (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.
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4.1. 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 significantly faster rise times and steady-state times. PAGA-SFC2 effectively 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 and 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.
4.2. 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. The 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 significantly different. 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 effectively suppresses pressure overshoot due to the introduction of the penalty term. This difference can be mainly explained by the variations in the weighting matrices associated with the control signals. The is greater than and . Therefore, 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 different algorithms. Similarly, PAGA-SFC1 and PAGA-SFC2 exhibit significantly smaller fluctuations in their ranges.
4.3. 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 . The braking pressure output by the EHB is recorded in Figure 10(a). It can be observed that, compared to the two previous operating conditions, all algorithms show some steady-state error under the varying load pressure condition. However, PAGA-SFC1 and PAGA-SFC2 show significantly faster response speed and higher accuracy than the other two. This behavior can be explained by the weighting matrix , where larger values indicate more constraints on the corresponding variables, and is related to the pressure error. In , = 1000; in , = 250.1; while in , = 100. This 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.
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5. Conclusions
The selection of the coefficient matrices and in traditional LQR control significantly influences 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 fitness. Furthermore, to prevent overshooting, a penalty factor is incorporated into the fitness function to suppress the occurrence of overshoot. The 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 effect 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.
Abbreviations
| : | Electro-hydraulic brake |
| : | Antilock braking system |
| : | Genetic algorithm |
| : | Parameter adaptive genetic algorithm |
| : | Particle swarm optimization |
| : | Ant colony optimization |
| : | Adaptive particle swarm optimization |
| : | State feedback control |
| : | Linear quadratic regulator |
| : | SFC optimized by PAGA with |
| : | SFC optimized by PAGA with |
| : | Total overshoot |
| : | Minimum fitness |
| : | Iteration times to minimum fitness |
| : | Size of the population |
| : | The number of iterations |
| : | Mutation probability |
| : | Crossover probability |
| : | Adjustment parameters |
| : | Learning factor 1 |
| : | Learning factor 2 |
| : | Inertia weight |
| : | Pheromone evaporation factor |
| : | Transition probability |
| : | Fitness function with no penalty |
| : | Fitness function with a penalty |
| : | |
| : | Amplifier output voltage () |
| : | Outlet pressure of pressure () |
| : | Valve spool end face area () |
| : | Equivalent mass () |
| : | Armature displacement () |
| : | Comprehensive damping coefficient () |
| : | Spring stiffness of pressure reducing valve () |
| : | Hydrodynamic stiffness coefficient () |
| : | Proportional valve flow gain () |
| : | Proportional valve flow pressure gain () |
| : | Piston area () |
| : | Piston displacement () |
| : | The volume of hydraulic cylinder control chamber () |
| : | Bulk elastic modulus of liquid () |
| : | The total mass of the piston and load () |
| : | Viscous damping coefficient () |
| : | Load spring stiffness (). |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was funded by the Guangzhou Science and Technology Plan Project (202201020173 and 202201010475), the National Natural Science Foundation of China (No. 52205090), and the College-Innovation-Foundation-of Guangdong (No. 2022KQNCX058).