This paper presents the design and implementation of Adaptive Generalized Dynamic Inversion (AGDI) to track the position of a Linear Flexible Joint Cart (LFJC) system along with vibration suppression of the flexible joint. The proposed AGDI control law will be comprised of two control elements. The baseline (continuous) control law is based on principle of conventional GDI approach and is established by prescribing the constraint dynamics of controlled state variables that reflect the control objectives. The control law is realized by inverting the prescribed dynamics using dynamically scaled Moore-Penrose generalized inversion. To boost the robust attributes against system nonlinearities, parametric uncertainties and external perturbations, a discontinuous control law will be augmented which is based on the concept of sliding mode principle. In discontinuous control law, the sliding mode gain is made adaptive in order to achieve improved tracking performance and chattering reduction. The closed-loop stability of resultant control law is established by introducing a positive define Lyapunov candidate function such that semi-global asymptotic attitude tracking of LFJC system is guaranteed. Rigorous computer simulations followed by experimental investigation will be performed on Quanser's LFJC system to authenticate the feasibility of proposed control approach for its application to real world problems.

This paper presents a controller design for position tracking of a classroom equipment for mass-damper-spring quadratic systems called Linear Flexible Joint Cart (LFJC) system supplied by [

One of the most popular adaptive controllers is the Model Reference Adaptive Controller (MRAC) which belongs to the direct adaptive control method. MRAC has a structure where an additional control loop called adaptation loop is added to the normal feedback structure to adapt to changes in the system dynamic as well as to compensate disturbances. Since its first implementation in 1958, MRAC has evolved and some of recent works on the controller can be found in [

One type of adaptive controller under the family of indirect method is Model identification Adaptive Control (MIAC). The working principle of MIAC is to optimize the performance of a controller by identifying the system while it is running. This is very much similar to the concept of Self-tuning Regulators (STR) [

The two controllers discussed earlier used single model as reference, there exist a type of adaptive controller that uses large number of models as references called Multi-model Adaptive Control (MMAC). Not all reference models will be used all the time though. At an instance, only one model that is closest to the plant is chosen. Recent examples of MMAC implementation include a Dynamic Positioning System for Quadrotor Helicopters and Simulator for Solid Oxide Fuel Cell Gas Turbine Power Plants [

The usage of AI based technique in adaptive control is tempting as it generally provides a model free universal approximation. They were typically used together with classical controller to enhance their robustness, adaptivity to changes, and improve non-linear model approximation. Some examples are the

There is still an exhaustive list of types of adaptive controller yet to discussed in this paper like

In this paper, an inversion based AGDI control approach is applied for linear position control of LFJC system. A Moore-Penrose Generalized Inverse was used to parameterize the solution as the inversion could results in infinite number of solutions as given in [

An LFJC system as shown in

The differential equations of the system are given as:

The servo motor parameters in [

To track the linear position of LFJC system, a controller is designed to have two loops as shown in

By formulating constant time ordinary differential constraints, we begin our controller design:

To obtain the desired control action

From the system's dynamic model in

Now, we can obviously solve

However, there can exists infinite number of solutions for

Note that, the same control law is applied in both controller loops.

In order to make the controller more robust, a technique similar to SMC is employed to augment the controller. For that, a set of sliding surface vector function for both outer and inner controller loop is defined as in the following:

Therefore, the derivative of the vector functions are as follows:

Note that subscripts

Finally, based on the sliding vector functions in

The designed controller were made adaptive by introducing an update mechanism for the gains in

To guarantee stability of the controller, the error vectors

Substitute

Next, we intend to prove that the sliding mode dynamics

Let the positive definite Lyapunov energy function is defined as:

By substituting the

According to the Lyapunov's direct method, asymptotic stability stability of

The sliding mode dynamics in

It can be observed that from

It follows that

Then

To evaluate the performance of the designed controller, we perform numerical simulations with step and sine wave input. The same scenario is then repeated in the practical experiment on a real system. Two other controllers namely the GDI and Linear Quadratic Regulator (LQR) were also developed and tested in both simulations and experiments to provide comparison to the designed controller.

The LFJC system is simulated based on the dynamic model given in (

Then, sine-wave is used with the same frequency and amplitude as in the previous simulation and the results is as shown in

Controller | Square | Sine |
---|---|---|

LQR | 6.0151 | 5.8577 |

GDI | 6.8580 | 5.3998 |

AGDI | 5.9822 | 1.2352 |

Controller | Square | Sine |
---|---|---|

LQR | 4.6260 | 5.2164 |

GDI | 3.7727 | 4.7601 |

AGDI | 3.3902 | 0.4368 |

The input voltage generated in response to the set-point is shown in

The same scenario as in the simulation is tested on a real equipment where the experimental set-up is as shown in

The experimental results for the proposed controller are shown in

Controller | Square | Sine |
---|---|---|

LQR | 7.2477 | 7.1762 |

GDI | 4.7188 | 4.4185 |

AGDI | 4.8643 | 4.3898 |

Controller | Square | Sine |
---|---|---|

LQR | 12.0810 | 15.5544 |

GDI | 9.7485 | 15.1153 |

AGDI | 8.1561 | 13.3835 |

A two-loops adaptive controller based on GDI has been successfully implemented in tracking the linear position of LFJC system. The proposed AGDI control law contains two control elements. Laws of Control established using a baseline (continuous) control law designed to reflect control objectives. The baseline law derives from the conventional GDI approach and is based on the rules for GDI. By applying Moore-Penrose generalized inversion to the prescribed dynamics, the control law is established. The introduction of adaptation modulation gain has made the controller able to adapt to changes in the system. The superiority of the proposed controller over GDI and LQR was proven by the simulation and practical experimental results. However, there are still room for improvements especially in minimizing the oscillation produced by the input voltage due to the continuous movement of the cart.

This research work was funded by Institutional Fund Project under Grant No. (IFPHI-106-135-2020). Therefore, authors gratefully acknowledge technical and financial support from the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia.