For long time the optimization of controller parameters uses the well-known classical method such as the Ziegler-Nichols and the Cohen-Coon tuning techniques. Despite its effectiveness, these off-line tuning techniques can be time consuming especially for a case of complex nonlinear system. This paper attempts to show a great deal on how Metamodeling techniques can be utilized to tune the PID controller parameters quickly. Note that the plant use in this study is the cruise control system with 2 different models, which are the linear model and the nonlinear model. The difference between both models is that the disturbances were taken into consideration for the nonlinear model, but in the linear model the disturbances were assumed as zero. The Radial Basis Function Neural Network Metamodel is able to prove that it can minimize the time in tuning process as it is able to give a good approximation to the optimum controller parameters in both models of this system.

It is crucial to tune the controller parameters wisely for any control system in order to obtain the optimum results for its output. That is, the system output response should be as similar as possible due to the corresponding reference.

There were several techniques that commonly used to tune the controller parameters. For example, the Ziegler-Nichols tuning technique, the Cohen-Coon tuning technique, the tuning based on process response, the Damped Oscillation tuning method and so on. Each technique has its own pros and cons. But, for sure these conventional techniques consume lot of time as the tuning process need to be tuned again and again until the best response is met.

Usually, computer simulation is used to model the real complicated system to save cost and time. In spite of the advances in computer technology, the required time to simulate the actual model might still be long and thus it becomes impractical to rely exclusively on simulation for the purpose of design optimization. To overcome this timing issue, the Metamodeling techniques offer good approximation to the actual model due to the usage of simpler model, added on with less computation algorithms.

This paper illustrates how to optimize the PID controller parameters for both models (linear and nonlinear) of the cruise control system.

Metamodeling or sometimes called as “model of the model” has been used extensively in many fields to give simpler model of the input and output function that approximates the relationship between system performances and controller parameters of a system. The required significant set of data for each PID controller parameter that fit the actual set of data will give the best results of approximation.

Recently, as studied in [

In another example, in a test Metamodeling for optimization problem in Rashid [

The RBFs were first used to design Artificial Neural Networks in 1988 by Broomhead and Lowe [

In this study, it was used as the Metamodel for matching process between the input-output of the cruise control system. The architecture of the RBF-NN used in this study is illustrated in Figure

Radial Basis Function Neural Network.

The network consists of three layers: an input layer, a hidden layer and an output layer. Here,

A block diagram by Dorf and Bishop [

RBF NN offer several advantages compared to the Multilayer Perceptrons. RBF-NN has also been successfully applied to a large diversity of applications including interpolation [

This section presents the mathematical equation that represents both linear and nonlinear model of the cruise control system.

In this paper, all equations were modeled base on the block diagram given by Dorf and Bishop [

Cruise control system variables and values used for simulation.

Variables | Description | Values |
---|---|---|

Time constant (s) | 0.2 | |

Real time (s) | 1 | |

Mass of the vehicle (kg) | 1500 | |

Drive force minimum limit (N) | ||

Drive force maximum limit (N) | ||

Air drag force (N) | Variable | |

Gravitational force (N) | Variable | |

Constant related to engine block (no unit) | 743 | |

Constant related to air-drag force (Nm^{−1}s^{2}) | 1.19 | |

Constant related to gravitational force (ms^{−2}) | 9.81 | |

Variable related to wind disturbance | Variable | |

Variable related to desired speed | Variable | |

Variable related to gravity disturbance | Variable |

The following equations model the plant dynamics of nonlinear system:

Take a glance at (

MATLAB SIMULINK block diagram for the nonlinear model of cruise control system using PID controller.

MATLAB SIMULINK block diagram for the linear model of cruise control system using PID controller.

Comparison of metamodel and actual simulation output for nonlinear model.

Comparison of metamodel and actual simulation output for linear model.

Note that, the PID controller parameters used in this study are

The Taylor Series Expansion is utilized to check the system’s stability. It should be done before the tuning process started. Referring to the eigenvalues obtained, it shows that the system is indeed stable and hence the control of the system should be possible.

The tuning procedures for Metamodeling technique are listed below:

Define the input design space,

Obtain the ISE for speed parameter for all the design space defined in 1.

Create the target data set,

Fit the RBF NN using

Evaluate the RBF NN on a larger input space,

Find the minimum of the RBF NN output (estimated

Repeat step 1 to 6 should the controller parameter gains are not satisfactory.

In this case,

Initial and large data sets.

Initial and Large Data Sets | |||
---|---|---|---|

Model type | Nonlinear | Linear | |

Total number of data configuration | 90 | 285 | |

Total number of data configuration | 26691 | 26691 |

16 RBF centers are used. Centers are added one by one until the RBF NN reaches an error goal of 0.1.

The limitation of this study is the use of MATLAB on an INTEL PENTIUM M PC to simulate the whole process of optimization. For verification purpose, the actual simulation had been done on

The best controller gain that minimized the error or noted as ISE

Best gains for the cruise control system.

Process | Metamodel | Actual | ||
---|---|---|---|---|

Model type | Nonlinear | Linear | Nonlinear | Linear |

1.5 | 1.5 | 1.5 | 1.5 | |

0.23 | 0.22 | 0.3 | 0.09 | |

0.75 | 0.65 | 0.75 | 0.55 |

Simulated time for tuning process.

Process | Metamodel | Actual | ||
---|---|---|---|---|

Model type | Nonlinear | Linear | Nonlinear | Linear |

Time taken | 0.4172 min | 0.57318 min | 52.6874 min | 40.8492 min |

Using the optimized PID gains obtained by Metamodeling technique, the results of the system output response are sketched in Figures

Response of speed for nonlinear model.

Response of speed for linear model.

Tables

System response characteristics for nonlinear model.

Disturbances type | Characteristics | Actual | Metamodel |
---|---|---|---|

Wind force | 103s–203s | 103s–203s | |

110s–214s | 111s–219s | ||

%OS | −1.93%–1.93% | −1.58%–1.86% | |

Gravity force | 303s–403s | 303s–403s | |

317s–418s | 332s–434s | ||

%OS | −31.02%–30.99% | −31.11%–31.12% |

System response characteristics for linear model.

Characteristics | Actual | Metamodel |
---|---|---|

5s | 5s | |

32s | 32s | |

%OS | 26.9% | 32.92% |

As referred to Figures

Best PID controller parameter using actual simulation.

Best gains based on actual simulation | ||
---|---|---|

Model type | Nonlinear | Linear |

1.5 | 1.5 | |

0.3 | 0.09 | |

0.75 | 0.55 |

Time taken to optimize PID controller parameter using actual simulation.

Simulated time based on Actual Simulation | |
---|---|

Model type | |

Nonlinear | Linear |

52.6874 minutes | 40.8492 minutes |

Next, the comparison of the minimum errors between the actual process and Metamodel is done and implied in Table

Metamodel and Actual Result Comparison for PID Controller.

Model type | Nonlinear | Linear | ||
---|---|---|---|---|

Category | Metamodel | Actual | Metamodel | Actual |

min ( | 1651 | 1573 | 1063 | 1031 |

In this study, the initial data set,