Robust Control of Stochastic Structures Using Minimum Norm Quadratic Partial Eigenvalue Assignment Technique

The main objective of this research is to design a state-feedback controller for the rotary inverted pendulum module utilizing the linear quadratic regulator (LQR) technique. The controller maintains the pendulum in the inverted (upright) position and is robust enough to reject external disturbance to maintain its stability. The research work involves three major contributions: mathematical modeling, simulation, and real-time implementation. To design a controller, mathematical modeling has been done by employing the Newton-Euler, Lagrange method. The resulting model was nonlinear so linearization was required, which has been done around a working point. For the estimation of the controller parameters, MATLAB LQR function has been utilized. Simulation has been performed for the designed controller and it also has been implemented and tested over the real inverted pendulum. From the results, it is vivid that the designed controller keeps the inverted pendulum in an upright position and rejects the disturbances and falling under gravitational force by adjusting the rotation of the horizontal link.

Linear–quadratic regulator (LQR) is a control technique successfully implemented for several complex systems. However, LQR technique in certain cases fails due to disturbances mainly caused by the improper choice of weighting matrices Q and R. Methods using particle swarm optimisation (PSO) are used to find the weighting matrices of LQR and have been proven to be superior. The paper proposes an output feedback LQR proportional–integral–derivative (PID) controller, which is also compared with LQR and PSO-based LQR. The output feedback LQR–PID provides a better response, and the effectiveness of the proposed method is demonstrated through simulations, and experimental validations are conducted on various cases.

Rotary inverted pendulum (RIP) system is a nonlinear, non-minimum phase, unstable and underactuated system. Controlling such system can be a challenge and is considered a benchmark in control theory problem. Prior to designing a controller, equations that represent the behaviour of the RIP system must be developed as accurately as possible without compromising the complexity of the equations. Through Takagi-Sugeno (T-S) fuzzy modeling technique, the nonlinear system model is then transformed into several local linear time-invariant models which are then blended together to reproduce, or approximate, the nonlinear system model within local region. A parallel distributed compensation (PDC) based fuzzy controller using linear quadratic regulator (LQR) technique is designed to control the RIP system. The results show that the designed controller able to balance the RIP system.

Model predictive control (MPC) is a multi-objective control technique that can handle system constraints. However, the performance of an MPC controller highly relies on a proper prioritization weight for each objective, which highlights the need for a precise weight tuning technique. In this paper, we propose an analytical tuning technique by matching the MPC controller performance with the performance of a linear quadratic regulator (LQR) controller. The proposed methodology derives the transformation of a LQR weighting matrix with a fixed weighting factor using a discrete algebraic Riccati equation (DARE) and designs an MPC controller using the idea of a discrete time linear quadratic tracking problem (LQT) in the presence of constraints. The proposed methodology ensures optimal performance between unconstrained MPC and LQR controllers and provides a sub-optimal solution while the constraints are active during transient operations. The resulting MPC behaves as the discrete time LQR by selecting an appropriate weighting matrix in the MPC control problem and ensures the asymptotic stability of the system. In this paper, the effectiveness of the proposed technique is investigated in the application of a novel vehicle collision avoidance system that is designed in the form of linear inequality constraints within MPC. The simulation results confirm the potency of the proposed MPC control technique in performing a safe, feasible and collision-free path while respecting the inputs, states and collision avoidance constraints.

This study presents a pioneering method to address longitudinal vibrations in ship propeller shafts by integrating an intelligent semi-active control system. By leveraging Adaptive Neuro-Fuzzy Inference System (ANFIS) and Linear Quadratic Regulator (LQR) techniques, the proposed system offers a novel solution to a longstanding challenge in maritime engineering. The efficacy of the semi-active control system is assessed through two complementary analyses: frequency domain analyses and Percentage Reduction Index (PRI) values. The investigation reveals substantial PRI values, ranging from 24.68% to 46.87%, indicating notable enhancements in vibration control with the adoption of semi-active technology. Furthermore, frequency domain analyses demonstrate significant advantages, with a peak reduction of 68.21% observed at 750 r/min. To rigorously evaluate the statistical significance of these improvements, paired t-tests were conducted between the semi-active control system and passive strategies. The results confirm a statistically significant difference in vibration reduction (p < .05) between the two systems. Specifically, for Percentage Reduction Index Values, the t test yielded a p-value of .023, while for Frequency Domain Analyses, the t-statistic significantly exceeded the critical value (t ≈ 8.09, tcritical ≈ 2.571). Both data sets provide compelling evidence to reject the null hypothesis, indicating a significant difference in vibration reduction between the semi-active control system and passive strategies. The PRI values consistently demonstrate superior performance for the semi-active control system, highlighting its effectiveness in mitigating vibrations. These findings validate the effective ness of the system. These results show the practical implications of implementing semi-active control systems in maritime engineering, offering enhanced safety, fuel efficiency, and operational reliability in ship propeller shaft applications.

Vibration control based on a virtual controlled object (VCO) is a simple and practical model-free active vibration suppression strategy. To utilize VCO-based model-free control, controller tuning plays the critical role. This study presents a computationally efficient tuning strategy for the VCO-based model-free active vibration controller. A state equation for model-free controller design is derived based on an actuator model and the VCO. The parameters of the VCO are determined so as to achieve the vibration suppression in the predetermined controlled frequency band. Then, we introduce the reference controlled object (RCO). The control performance of the VCO-controller is evaluated through the vibration control simulation for the RCO. Iteratively executing the vibration suppression simulation for the RCO, the suitable design parameters of the VCO-controller are searched on the basis of the simultaneous perturbation stochastic approximation (SPSA) algorithm. The nonlinear parametrization technique enhances the tuning efficiency. The validity of the tuning scheme is demonstrated by applying it to the design problem of the VCO-based model-free linear quadratic regulator (LQR). The numerical simulations verify the applicability of the VCO-LQR tuned by the proposed approach to wide variety of controlled objects other than the RCO.

Friction stir welding (FSW) is a promising welding technology and is being extensively researched because it can produce high-quality joints. In the welding process, due to improper clamping, deformation of back support and other process variations, the axial force can vary significantly and produce welding defects. To solve this problem, a feedback controller of FSW axial force is desirable. However, time-delay, which widely exists within the FSW systems, may degrade the controller effect and make the system unstable. To regulate the axial force and compensate the time delay, this paper designs an axial force controller for FSW system with delay. At first, by utilizing a time delay compensation scheme, FSW systems with delayed control actions is transformed into standard differential equations without time delay terms. Then, an axial force controller is designed based on the linear-quadratic regulator (LQR) technique. Since the time delay is incorporated in the system model throughout the whole controller design process, the system performance and stability are assured and the control system is also simplified. Utilizing the LQR technique, aspects of system dynamic properties and simplicity are considered simultaneously. Experimental validations were carried out on a FSW platform. The experimental results demonstrate that the time delay is compensated effectively. The controller maintains the constant axial force and shows desirable dynamic behavior, even when the disturbance is encountered during the welding process.

The increasing penetration of renewable energy sources in power grids highlights the role of battery energy storage systems (BESSs) in enhancing the stability and quality of electricity. One of the significant applications of BESSs is to control and damp the frequency deviations in power grids. In this paper, we consider a two-area four-machine power grid system with large-scale battery integration. We model the battery energy storage with two dynamic models in d-q axis for charging and discharging modes and augment the battery model with the rest of the power grid system. For each of these modes, we design an optimum controller using linear quadratic regulator (LQR) techniques to damp the frequency deviations of all generators in the case study. Unlike the majority of existing studies merely focus on the active power of BESSs, we consider both active and reactive powers of BESSs. Next, we design a hybrid controller with a switching policy to select the battery's operating mode based on the cost functions of optimal controllers (LQRs). The hybrid controller selects the minimum cost function for each time interval and updates all states of the power system accordingly. The simulation results of the proposed methodology are provided in SIMULINK/MATLAB to illustrate this approach's effectiveness.

Based on two successfully and widely used control techniques in many industrial applications under normal (fault-free) operation conditions, the Gain-Scheduled Proportional-Integral-Derivative (GS-PID) control and Model Reference Adaptive Control (MRAC) strategies have been extended, implemented, and experimentally tested on a quadrotor helicopter Unmanned Aerial Vehicle (UAV) test- bed available at Concordia University, for the purpose of investigation of these two typical and different control techniques as two useful Fault-Tolerant Control (FTC) approaches. Controllers are designed and implemented in order to track the desired trajectory of the helicopter in both normal and faulty scenarios of the flight. A Linear Quadratic Regulator (LQR) with integral action controller is also used to control the pitch and roll motion of the quadrotor helicopter. Square trajectory, together with specified autonomous and safe taking-off and landing path, is considered as the testing trajectory and the experimental flight testing results with both GS-PID and MRAC are presented and compared with tracking performance under partial loss of control power due to fault/damage in the propeller of the quadrotor UAV. The performance of both controllers showed to be good. Although GS-PID is easier for development and implementation, MRAC showed to be more robust to faults and noises, and is friendly to be applied to the quadrotor UAV.

The electric motor is a device that converts electrical power into mechanical power. The DC motor has some advantages compared to the AC motor on its easier way to control the speed or position and wide adjustable range. However, it still has some issues, such as uncertainty and disturbance; most existing controllers still provide bad system performance or have major drawbacks in solving these issues. The alternative solution for a control method with simple, easy to design, and can be implemented in a multi-input multi-output system is the integral state feedback control technique, such as in Linear Quadratic Regulator (LQR). The weakness of LQR control is that there is no exact method to determine the best value of matrix Q and R used to calculate the state feedback gain; trial or manual tuning method is mostly used. Therefore, this paper proposed an improved LQR control using Particle Swarm Optimization (PSO) method and the Kalman filter estimator. The PSO was used to find optimal Q and R-value, while the Kalman filter reduced the number of sensors used to measure the system state. Both performance and energy consumption were compared using the Integral of Absolute Error (IAE) and total energy consumption. The simulation results showed that the proposed method is superior in performance and energy compared to manually-tuned LQR. However, based on the hardware implementation, although the manually-tuned LQR was found to have the smallest energy consumption, the proposed method has the smallest IAE, which can reduce the IAE by 11.28% with only 1% higher energy compared to manually-tuned LQR.

Linear quadratic regulator (LQR) and proportional-integral-derivative (PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions. The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.

Chaos of power electronic systems and electric motor drives were introduced a few decades ago and follow with numerous of research literature. However, chaos of permanent magnet synchronous motors, in particular, seems not receive enough deserved interest. Considering this fact, this paper designs a linear quadratic regulator (LQR) to suppress chaotic behaviors of PMSMs. The stability and robustness properties of the proposed LQR are fully presented. It should be noted that this is the first time the robustness of LQR for chaos suppression of PMSMs are discussed. Due to its robustness properties, the proposed LQR scheme not only can eliminate the chaos but also provide better performances than conventional methods. Simulations results under various scenarios prove the superiority of the proposed control technique over conventional controllers.

DC motors are widely utilized in industrial applications for their reliability and efficiency. To optimize their performance, it is crucial to employ control systems supported by mathematical modeling to predict motor responses under varying conditions. This study investigates the first- and second-order models of DC motors and examines the impact of internal disturbances (noise) on system performance. The output responses of two DC motors, the 42BLFX02 and Maxon EC-I 40 (70W), are compared under both undisturbed and noisy conditions using simulations. The results reveal that the second-order model offers a more stable response and better aligns with the desired target compared to the first-order model. Furthermore, the application of the Linear Quadratic Regulator (LQR) control method significantly enhances the speed and accuracy of reaching the motor set point. However, when noise is introduced, the LQR method fails to maintain stability, and the motor's output starts to mirror the disturbance pattern. These findings highlight that while LQR is effective under ideal conditions, its performance diminishes when exposed to disturbances. Therefore, additional strategies are necessary to ensure stability and optimal performance in real-world conditions, particularly in environments with significant noise or disturbances.

An adaptive method with weight matrix as a function of the state to design the rotatory flexible system control law

This article is a continuation of our previously published work that presented a comparison of nine attitude quaternion-based controllers of the quadrotor in simulation environment. In this article, the best three controllers were implemented into the real quadrotor. Namely proportional derivative (PD), linear quadratic regulator (LQR) and backstepping quaternion-based control techniques were evaluated. As a suitable test stand was not available on the basis of literature analysis, the article also outlines the requirements and the development of a new innovative test stand. In order to provide a comprehensive overview, the hardware and software that was used is also presented in the article. The main contribution of this article is a performance comparison of the controllers, which was based on absolute quaternion (positioning) error and energy consumption.