# Deployment of a Software to Simulate Control Systems in the State-Space

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## Abstract

**:**

## 1. Introduction

## 2. Easy JavaScript Simulations

## 3. Implementation of the First Plant (SISO System). Mathematical Model

#### 3.1. System Description

#### 3.2. Implementation of the Linearized Model

Listing 1. Fixed relations panel (SISO). |

h2ref_inc=h2ref-h20; |

Listing 2. Function to obtain the height of the first tank h1: height1_L (linear SISO). |

function height1_L (h1_inc,h2_inc,q1_inc) { |

if(h1_inc<-h10){ |

h1_inc=-h10; |

} |

if(h2_inc<-h20){ |

h2_inc=-h20; |

} |

//flow between both tanks |

q2_inc=((r1*h1_inc*(1/(2*Math.sqrt(h10-h20))))-(r1*h2_inc*(1/(2*Math.sqrt(h10-h20))))); |

return((1/A1)*(q1_inc-q2_inc)); //compute first derivative of H1 |

} |

Listing 3. Function to obtain the height of the second tank h2: height2_L (linear SISO). |

function height2_L (h1_inc,h2_inc){ |

if(h1_inc<-h10){ |

h1_inc=-h10; |

} |

if(h2_inc<-h20){ |

h2_inc=-h20; |

} |

//flow between both tanks |

q2_inc=((r1*h1_inc*(1/(2*Math.sqrt(h10-h20))))-(r1*h2_inc*(1/(2*Math.sqrt(h10-h20))))); |

// tank 2 output flow |

q3_inc=r2*(1/(2*Math.sqrt(h20)))*h2_inc; |

return((1/A2)*(q2_inc-q3_inc)); // compute first derivative of H2 |

} |

Listing 4. Code page: Linearized system (linear SISO). |

h1=h1_inc+h10; |

h2=h2_inc+h20; |

q1=q1_inc+q10; |

#### 3.3. Implementation of the Original Nonlinear Model

Listing 5. Function to obtain the height of the first tank h1: height1_NL (nonlinear SISO). |

function height1_NL (h1, h2, q1) { |

if (h1<0){ |

h1=0; |

} |

if(h2<0){ |

h2=0; |

} |

if (h1 < h2) { |

q2=-r1*Math.sqrt(h2-h1); |

else { |

q2=r1*Math.sqrt(h1-h2); |

} |

h1_inc=h1-h10; ; // compute the incremental value of h1 |

return (1/A1)*(q1-q2); |

} |

Listing 6. Function to obtain the height of the second tank h2: height2_L (nonlinear SISO). |

function height2_NL (h1, h2) { |

if (h1<0){ |

h1=0; |

} |

if(h2<0){ |

h2=0; |

} |

if (h1 < h2) { |

q2=-r1*Math.sqrt(h2-h1); |

else { |

q2=r1*Math.sqrt(h1-h2); |

} |

h2_inc=h2-h20; ; // compute the incremental value of h1 |

return (1/A2)*(q2-r2*Math.sqrt(h2)); |

} |

#### 3.4. Saturation

Listing 7. Code page: Saturation block (SISO). |

if(sat_upper){ |

if (q1_inc+q10 > value_sat_upper){ |

q1_inc=value_sat_upper-q10; |

} |

} |

if(sat_low){ |

if(q1_inc+q10 < value_sat_low){ |

q1_inc=value_sat_low-q10; |

} |

} |

## 4. Implementation of the First Plant (SISO System). Controllers

#### 4.1. Classical Control Theory

#### 4.1.1. Continuous-Time

Listing 8. Code page: Continuous-time PID controller (SISO system). |

error=h2ref_inc-h2_inc; |

derror=-h2_d; // value computed in function height2 |

q1_inc=kp*error+kint*ei+kd*derror; //control action |

#### 4.1.2. Discrete-Time

Listing 9. Code page: Discrete-time PID controller (SISO system). |

if ((Math.round(time/dt))% (Ts/dt) === 0){ |

//Derivative |

derror=(h2ref_inc-h2_inc)-error; //error[k]-error[k-1] |

//Proportional |

error=h2ref_inc-h2_inc; |

//Integral |

ei=error+ei; //error[k]+error[1:k-1] |

q1_inc=kp*error+ki*ei+kd*derror; |

} |

else{ |

error=error; |

derror=derror; |

ei=ei; |

q1_inc=q1_inc; |

h2_inc=h2_inc; |

} |

#### 4.2. Modern Control Theory

#### 4.2.1. Continuous-Time

Listing 10. Code page: Continuous-time control through observed state feedback (SISO system). |

q1_inc=(h2ref_inc*Ks) - ((Kc1*Xe1) + (Kc2*Xe2)); |

#### 4.2.2. Discrete-Time

Listing 11. Code page: Discrete-time control through observed state feedback (SISO system). |

if ((Math.round(time/dt3))% (Ts3/dt3) === 0){ |

Xe1_1=(h2_inc*Ko1)+(q1_inc*H1)+(Xe1*r11)+(Xe2*r12); |

Xe2_1=(h2_inc*Ko2)+(q1_inc*H2)+(Xe1*r21)+(Xe2*r22); |

q1_inc=(h2ref_inc*Ks)-((Kc1*Xe1)+(Kc2*Xe2)); |

Xe1=Xe1_1; |

Xe2=Xe2_1; |

} |

else{ |

Xe1=Xe1_1; |

Xe2=Xe2_1; |

q1_inc=q1_inc; |

h2_inc=h2_inc; |

} |

## 5. Implementation of the Second Plant (MIMO System): Mathematical Model

#### 5.1. System Description

#### 5.2. Implementation of the Linearized Model

Listing 12. Function to obtain the height of fluid in the first tank h1: height1_L (linear MIMO). |

function height1_L (h1_inc, gamma_inc, V_inc) { |

if(h1_inc<-h10){ |

h1_inc=-h10; |

} |

qe1_inc=-Kp*V0*gamma_inc+Kp*(1-gamma0)*V_inc; // input flow |

qs1_inc=B1*Math.sqrt(g/(2*h10))*h1_inc; // output flow |

return((1/A1)*(qe1_inc-qs1_inc)); |

} |

Listing 13. Function to obtain the height of fluid in the second tank h2: height2_L (linear MIMO). |

function height2_L (h2_inc, gamma_inc, V_inc) { |

if(h2_inc<-h20){ |

h2_inc=-h20; |

} |

qe2_inc=-Kp*V0*gamma_inc+Kp*gamma0*V_inc; |

qs2_inc=B2*Math.sqrt(g/(2*h20))*h2_inc; |

return((1/A2)*(qe2_inc-qs2_inc)); |

} |

Listing 14. Page code: Linearized system (linear MIMO). |

h1=h1_inc+h10; |

h2=h2_inc+h20; |

gamma=gamma_inc+gamma0; |

V=V_inc+V0; |

#### 5.3. Implementation of the Original Nonlinear Model

Listing 15. Function to obtain the height of fluid in the first tank h1: height1_NL (nonlinear MIMO). |

function height1_NL (h1, gamma, V) { |

if (h1<0){ |

h1=0; |

} |

qe1=Kp*V*(1-gamma); |

qs1=B1*Math.sqrt(2*g*h1); |

h1_d=(1/A1)*(qe1-qs1); //save the first derivative value of h1 |

h1_inc=h1-h10; // compute the incremental value of h1 |

return((1/A1)*(qe1-qs1)); |

} |

Listing 16. Function to obtain the height of of fluid in the second tank h2: height2_NL (nonlinear MIMO). |

function height2_NL (h2,gamma,V) { |

if (h2<0){ |

h2=0; |

} |

qe2=Kp*V*gamma; |

qs2=B2*Math.sqrt(2*g*h2); |

h2_d=(1/A2)*(qe2-qs2); //save the first derivative value of h2 |

h2_inc=h2-h20; // compute the incremental value of h2 |

return((1/A2)*(qe2-qs2)); |

} |

#### 5.4. Saturation

Listing 17. Code page: Saturation block (MIMO). |

// valve constant (gamma) saturation |

if (gamma_inc+gamma0 > value_gamma_sat_upper){ |

gamma_inc=value_gamma_sat_upper-gamma0; |

} |

if (gamma_inc+gamma0 < value_gamma_sat_low){ |

gamma_inc=value_gamma_sat_low-gamma0; |

} |

// voltage applied to pump (V) saturation |

if (sat_V_upper){ |

if (V_inc+V0 > value_V_sat_upper){ |

V_inc=value_V_sat_upper-V0; |

} |

} |

if(sat_V_low){ |

if(V_inc+V0 < value_V_sat_low){ |

V_inc=value_V_sat_low-V0; |

} |

} |

## 6. Implementation of the Second Plant (MIMO System): Controllers

#### 6.1. Classical Control Theory

#### 6.1.1. Continuous-Time

Listing 18. Code page: Continuous-time PID controller (MIMO system). |

error1=h1ref_inc-h1_inc; |

error2=h2ref_inc-h2_inc; |

derror1=-h1_d; // first derivative value of h1 |

derror2=-h2_d; // first derivative value of h2 |

V_inc=kp1*error1+ki1*ei1+kd1*derror1; |

gamma_inc=kp2*error2+ki2*ei2+kd2*derror2; |

#### 6.1.2. Discrete-Time

Listing 19. Code page: Discrete-time PID controller (MIMO system). |

if ((Math.round(time/dt))% (Ts/dt) === 0){ |

//Derivative |

derror1=(h1ref_inc-h1_inc)-error1; |

derror2=(h2ref_inc-h2_inc)-error2; |

//Proportional |

error1=h1ref_inc-h1_inc; |

error2=h2ref_inc-h2_inc; |

//Integral |

ei1=error1+ei1; |

ei2=error2+ei2; |

V_inc=kp1*error1+ki1*ei1+kd1*derror1; |

gamma_inc=kp2*error2+ki2*ei2+kd2*derror2; |

} |

else{ |

V_inc=V_inc; |

gamma_inc=gamma_inc; |

derror1=derror1; |

derror2=derror2; |

error1=error1; |

error2=error2; |

ei1=ei1; |

ei2=ei2; |

} |

#### 6.2. Modern Control Theory

#### 6.2.1. Continuous-Time

Listing 20. Code page: Observer state feedback control (MIMO system). |

V_inc=(Ks11*h1ref_inc)+(Ks12*h2ref_inc)-(Kc11*Xe1+Kc12*Xe2); |

gamma_inc=Ks21*h1ref_inc+Ks22*h2ref_inc-Kc21*Xe1+Kc22*Xe2; |

#### 6.2.2. Discrete-Time

Listing 21. Code page: Discrete observer state feedback control (MIMO system). |

if ((Math.round(time/dt3))% (Ts3/dt3) === 0){ |

V_inc=(Ks11*h1ref_inc)+(Ks12*h2ref_inc)-(Kc11*Xe1+Kc12*Xe2); |

gamma_inc=(Ks21*h1ref_inc)+(Ks22*h2ref_inc)-(Kc21*Xe1+Kc22*Xe2); |

//first derivative value of the first estimated state |

Xe1_1=(H11*V_inc+H12*gamma_inc)+(r11*Xe1)+(r12*Xe2)+(Ko11*h1_inc+Ko12*h2_inc); |

//first derivative value of the second estimated state |

Xe2_1=(H21*V_inc+H22*gamma_inc)+(r21*Xe1)+(r22*Xe2)+(Ko21*h1_inc+Ko22*h2_inc); |

// save these values |

Xe1=Xe1_1; |

Xe2=Xe2_1; |

} |

else{ |

Xe1=Xe1_1; |

Xe2=Xe2_1; |

gamma_inc=gamma_inc; |

V_inc=V_inc; |

h1_inc=h1_inc; |

h2_inc=h2_inc; |

} |

## 7. Results of the Simulation

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Parameters and Variables

#### Appendix A.1. SISO System

Parameter | Description |
---|---|

$A1$,$A2$ | Cross section of each tank. |

$r1$,$r2$ | Discharge coefficients. |

Variable | Incremental Variable | Description |
---|---|---|

$q1$ | $q1\_inc$ | Input flow. |

$q2$ | $q2\_inc$ | Flow from the first to the second tank. |

$q3$ | $q3\_inc$ | Output flow of the second tank. |

$h1$ | $h1\_inc$ | Fluid level of the first tank. |

$h2$ | $h2\_inc$ | Fluid level of the second tank. |

Parameter | Description |
---|---|

$q10$ | Value of q1 at the operating point. |

$q20$ | Value of q2 at the operating point. |

$q30$ | Value of q3 at the operating point. |

$h10$ | Value of h1 at the operating point. |

$h20$ | Value of h2 at the operating point. |

Parameter | Description |
---|---|

sat_upper | Boolean variable to activate the upper saturation. |

sat_lower | Boolean variable to activate the lower saturation. |

value_sat_upper | Upper limit value. |

value_sat_lower | Lower limit value. |

Parameter | Description |
---|---|

$kp$ | Proportional constant. |

$kd$ | Derivative constant. |

$kint$ | Integral constant. |

$error$ | Error signal. |

$derror$ | Derivative of the error signal. |

$ei$ | Integral of the error signal. |

Parameter | Description |
---|---|

$[B1;B2]$ | Input matrix (state space representation continuous-time). |

$[H1;H2]$ | Input matrix (state space representation discrete-time). |

$[r11r12;r21r22]$ | Proportional constant. |

$[Xe1;Xe2]$ | Estimated state (in discrete-time: $Xei\left[k\right]$). |

$[Xe{1}_{1};Xe{2}_{1}]$ | Estimated state value in the next sampling instant $Xei[k+1]$. |

$[Ko1;Ko2]$ | Observer matrix. |

$\left[Kc1Kc2\right]$ | Feedback state matrix. |

$Ks$ | Input gain. |

#### Appendix A.2. MIMO System

Parameter | Description |
---|---|

$A1$,$A2$ | Cross section of each tank. |

$B1$,$B2$ | Cross section of their respective outlet pipes |

$Kp$ | Constant gain of the pump |

Variable | Incremental Variable | Description |
---|---|---|

V | $V\_inc$ | Voltage applied to pump. |

$gamma$ | $gamma\_inc$ | Valve opening. |

$qe1$ | $qe1\_inc$ | Input flow to tank 1. |

$qs1$ | $qs1\_inc$ | Output flow from the tank 1. |

$qe2$ | $qe2\_inc$ | Input flow to tank 2. |

$qs2$ | $qs2\_inc$ | Output flow from the tank 2. |

$h1$ | $h1\_inc$ | Height of the fluid in the tank 1. |

$h2$ | $h2\_inc$ | Height of the fluid in the tank 2. |

Parameter | Description |
---|---|

$V0$ | Value of V at the operating point. |

$gamma0$ | Value of gamma at the operating point. |

$qe10$ | Value of qe1 at the operating point. |

$qs10$ | Value of qs1 at the operating point. |

$qe20$ | Value of qe2 at the operating point. |

$qs20$ | Value of qs2 at the operating point. |

$h10$ | Value of h1 at the operating point. |

$h20$ | Value of h2 at the operating point. |

Parameter | Description |
---|---|

$sat\_V\_upper$ | Boolean variable to activate the upper saturation of the voltage applied to pump. |

$sat\_V\_lower$ | Boolean variable to activate the lower saturation of the voltage applied to pump. |

$value\_V\_sat\_upper$ | Upper limit value of the voltage applied to pump. |

$value\_V\_sat\_lower$ | Lower limit value of the voltage applied to pump. |

$value\_gamma\_sat\_upper$ | Upper limit value of the Valve opening. |

$value\_gamma\_sat\_lower$ | Lower limit value of the Valve opening. |

Parameter | Description |
---|---|

$kp1$ | Proportional constant of the first PID. |

$kd1$ | Derivative constant of the first PID. |

$ki1$ | Integral constant of the first PID. |

$kp2$ | Proportional constant of the second PID. |

$kd2$ | Derivative constant of the second PID. |

$ki2$ | Integral constant of the second PID |

$error1$ | Error signal (first PID). |

$derror1$ | Derivative of the error signal (first PID). |

$ei1$ | Integral of the error signal (first PID). |

$error2$ | Error signal (second PID). |

$derror2$ | Derivative of the error signal (second PID). |

$ei2$ | Integral of the error signal (second PID). |

Parameter | Description |
---|---|

$[B11B12;B21B22]$ | Input matrix (state space representation continuous-time). |

$[H11H12;H21H22]$ | Input matrix (state space representation discrete-time). |

$[r11r12;r21r22]$ | Proportional constant. |

$[Xe1;Xe2]$ | Estimated state (in discrete-time: $Xei\left[k\right]$). |

$[Xe{1}_{1};Xe{2}_{1}]$ | Estimated state value in the next sampling instant $Xei[k+1]$. |

$[Ko11Ko12;Ko21Ko22]$ | Observer matrix. |

$[Kc11Kc12;Kc21Kc22]$ | Feedback state matrix. |

$[Ks11Ks12;Ks21Ks22]$ | Steady gain matrix. |

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**Figure 10.**The evolution of the outputs considering the equations of the original (nonlinear) system.

**Figure 11.**The evolution of the control actions considering the equations of the original (nonlinear) system.

State | Derivative |
---|---|

$\frac{dh1\_inc\left(t\right)}{dt}$ | $height1\_L(h1\_inc,h2\_inc,q1\_inc)$ |

$\frac{dh2\_inc\left(t\right)}{dt}$ | $height2\_L(h1\_inc,h2\_inc)$ |

State | Derivative |
---|---|

$\frac{dh1\_inc\left(t\right)}{dt}$ | $height1\_NL(h1,h2,q1)$ |

$\frac{dh2\_inc\left(t\right)}{dt}$ | $height2\_NL(h1,h2)$ |

State | Derivative |
---|---|

$\frac{dei}{dtime}$ | error |

State | Derivative |
---|---|

$\frac{dXe1}{dtime}$ | $B1\ast q1\_inc+r11\ast Xe1+r12\ast Xe2+Ko1\ast h2\_inc$ |

$\frac{dXe2}{dtime}$ | $B2\ast q1\_inc+r21\ast Xe1+r22\ast Xe2+Ko2\ast h2\_inc$ |

State | Derivative |
---|---|

$\frac{dh1\_inc}{dtime}$ | $height1\_L(h1\_inc,gamma\_inc,V\_inc)$ |

$\frac{dh2\_inc}{dtime}$ | $height2\_L(h2\_inc,gamma\_inc,V\_inc)$ |

State | Derivative |
---|---|

$\frac{dh1}{dtime}$ | $height1\_NL(h1,gamma,V)$ |

$\frac{dh2}{dtime}$ | $height2\_NL(h2,gamma,V)$ |

State | Derivative |
---|---|

$\frac{dei\_1}{dtime}$ | $error1$ |

$\frac{dei\_2}{dtime}$ | $error2$ |

State | Derivative |
---|---|

$\frac{dXe1}{dtime}$ | $(B11\ast V\_inc+B12\ast gamma\_inc)+(r11\ast Xe1)+(r12\ast Xe2)+(Ko11\ast h1\_inc+Ko12\ast h2\_inc)$ |

$\frac{dXe2}{dtime}$ | $(B21\ast V\_inc+B22\ast gamma\_inc)+(r21\ast Xe1)+(r22\ast Xe2)+(Ko21\ast h1\_inc+Ko22\ast h2\_inc)$ |

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## Share and Cite

**MDPI and ACS Style**

Flores, M.; Payá, L.; Valiente, D.; Gallego, J.; Reinoso, O.
Deployment of a Software to Simulate Control Systems in the State-Space. *Electronics* **2019**, *8*, 1205.
https://doi.org/10.3390/electronics8111205

**AMA Style**

Flores M, Payá L, Valiente D, Gallego J, Reinoso O.
Deployment of a Software to Simulate Control Systems in the State-Space. *Electronics*. 2019; 8(11):1205.
https://doi.org/10.3390/electronics8111205

**Chicago/Turabian Style**

Flores, María, Luis Payá, David Valiente, Julio Gallego, and Oscar Reinoso.
2019. "Deployment of a Software to Simulate Control Systems in the State-Space" *Electronics* 8, no. 11: 1205.
https://doi.org/10.3390/electronics8111205