introduction
Hydraulic technology is a relatively new engineering technology, primarily used in automated control systems requiring precise movements and rapid responses. With the development of atomic energy, space technology, and computer technology, hydraulic technology has also seen significant advancements and has permeated various industrial sectors. Currently, hydraulic technology is evolving towards higher pressure, higher speed, higher power, higher efficiency, lower noise, lower energy consumption, durability, and greater integration. Simultaneously, the control strategies of electro-hydraulic control systems are also continuously evolving. Over the years, hydraulic technology has progressed and improved through various novel control methods, from traditional PID control and adaptive control to variable structure control, robust control, and intelligent control.
The main problem studied in quadratic optimal control theory is to select an admissible control law based on the established mathematical model of the controlled object, so that the controlled object operates according to predetermined requirements and a given performance index reaches its minimum or maximum value. For linear control systems, if the integral of the quadratic functions of the state variables and control variables is taken as the performance index function, then this dynamic system optimization problem is called the optimal control problem of the quadratic performance index of the linear system, or simply the linear quadratic optimal control problem or linear quadratic problem. Linear quadratic problems occupy an important position in control theory. On the one hand, many control problems can be reduced to linear quadratic problems; on the other hand, it is theoretically relatively complete and mature. Because the optimal solution of a linear quadratic problem can be written as a unified analytical expression and the solution process can be standardized, and it can lead to a simple linear state feedback control law, it is easy to construct closed-loop optimal control, which is convenient for engineering implementation, and therefore it has been widely used in practical engineering.
1. Establishment of a mathematical model for a state-space-based electro-hydraulic servo control system
Research on electro-hydraulic servo control systems reveals that the main dynamic components include command and amplification devices, electro-hydraulic servo valves, hydraulic motors and controlled objects, and various measurement and feedback devices. The mathematical model of the electro-hydraulic control system is as follows:
2. Linear Quadratic Optimal Control Theory Equations
3. Simulation Analysis
The closed-loop step response curve can also be obtained, as shown in Figure 2. The figure shows that the system's settling time is 0.111 s. A comparison of Figures 1 and 2 reveals that the designed linear quadratic optimal control method for this electro-hydraulic servo control system exhibits better tracking characteristics and a faster response.
The magnitudes of the weighting matrices Q and R in the formula represent the designer's emphasis on both tracking error and control energy, and also reflect the relationship between the system's performance and the values of the weighting matrices.
The following simulations visually demonstrate the impact of weighting matrices Q and R on the system's dynamic performance. Assuming R=1, Figure 3 shows the step response curves for Q=10, 100, 1000, and 10000. Analysis of Figure 3 shows that as the Q value increases, the system's tracking performance improves. When Q is less than 100, the system response rise time is relatively long, and the tracking performance is poor, therefore it is not advisable. Assuming Q=1000, Figure 4 shows the step response curves for R=1, 10, 50, and 100. Analysis of Figure 4 shows that as the R value increases, the system's tracking performance deteriorates. When R is greater than 50, the system response rise time is relatively long, and the tracking performance is poor, therefore it is also not advisable.
The simulation results above show that when designing the optimal regulator for an electro-hydraulic servo control system, the dynamic performance of the system varies depending on the choice of the weighting matrices Q and R in the quadratic indices. When Q is fixed, increasing R continuously reduces the rise time and steady-state error, but increases the overshoot and transient response time. Similarly, when R is fixed, increasing Q continuously increases the rise time and steady-state error, but decreases the transient response time and overshoot. This analysis demonstrates that the selection of Q and R matrices influences the dynamic and static indices of the system according to certain rules. A comprehensive consideration should be given to the design requirements of the actual control system to select the appropriate weighting matrix.
4. Conclusion
This paper takes the electro-hydraulic servo system as the research object, analyzes and studies the quadratic optimal control theory, and applies it to the electro-hydraulic servo system of a tension leveling machine for experimental testing. MATLAB simulation results show that the designed system has fast control response speed, high control accuracy, and good dynamic characteristics. This paper comprehensively analyzes and considers the integrated characteristics of the linear quadratic optimal control system and achieves optimal control of the control system by adjusting the weighting matrices Q and R. When the control system parameters remain constant or change only slightly, the optimal quadratic control scheme is a relatively ideal control method for the electro-hydraulic servo control system.
5. Attachments
A=[0,1,0;-14196,0,1.78e-5;0,-88.44e8,-4.22]; B=[0;0;122.2e8];
Q1=diag([1000,0,0]);
R1=1
C=[1,0,0]；D=0;
[k1,P1]=lqr(A,B,Q1,R1)
disp('Theoptimalfeedbackgainmatrixk1is:')
k1
k11=k1(1);Ax1=AB*k1;Bx1=B*k11;Cx=C;Dx=D;
Q2=diag([1000,0,0]);
R2=10
[k2,P2]=lqr(A,B,Q2,R2)
disp('Theoptimalfeedbackgainmatrixk2is:')
k2
k22=k2(1);Ax2=AB*k2;Bx2=B*k22;Cx=C;Dx=D;
[k3,P3]=lqr(A,B,Q3,R3)
disp('Theoptimalfeedbackgainmatrixk3is:')
k3
k31=k3(1);Ax3=AB*k3;Bx3=B*k31;Cx=C;Dx=D;
Q1=diag([1000,0,0]);
R1=50;
C=[1,0,0]；D=0;
[k4,P4]=lqr(A,B,Q1,R1)
disp('Theoptimalfeedbackgainmatrixk4is:')
k4
k41=k4(1);
Ax4=AB*k4;Bx4=B*k41;Cx=C;Dx=D;
Q3=diag([1000,0,0]);R3=100
step(Ax1,Bx1,Cx,Dx)
holdon
step(Ax2,Bx2,Cx,Dx)
holdon
step(Ax3,Bx3,Cx,Dx)
holdon
step(Ax4,Bx4,Cx,Dx)
title('StepResponseofQuadraticOptimalControlSystem');
xlabel('Time-Sec');ylabel('Outputy=x1');
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