Semi-active friction dampers (SAFDs), which can adaptively adjust the friction force, function as flexible semi-active control systems for the seismic protection of structures, including buildings exposed to strong ground accelerations. The effectiveness of SAFDs in dissipating seismic energy is highly dependent on the control strategy used to adjust the friction force of dampers. Due to the highly nonlinear and complex nature of buildings equipped with SAFDs, achieving optimal control is challenging and requires advanced adaptive control techniques. In this paper, we propose an intelligent control method based on deep reinforcement learning for the seismic control of buildings with SAFDs. We examine the potential of SAFDs to improve the seismic response of multi-story buildings and present a numerical case study to demonstrate the effectiveness of the proposed intelligent method. In this case study, the performance of the proposed intelligent control method is compared with two optimization methods and a reference controller: (i) friction dampers (FDs) using a constant friction force that is calibrated using a heuristic approach, (ii) FDs whose friction force is optimized through a Genetic Algorithm (GA), and (iii) SAFDs employing a rule-based approach for adaptive friction force adjustment. It was observed that the proposed DRL approach outperformed the alternatives, for example, achieving up to 63.7 % improvement over the rule-based controller. The core contribution of this study is the formulation of an MDP and reward-design framework that enables DRL to perform real-time friction control in SAFDs integrated into multi-story buildings.

Robust loss functions are crucial in deep neural network training for managing outliers and noisy data. This paper presents an adaptive, robust variant of the L2 loss function, leveraging the fractional calculus. The fractional derivative order α in the proposed Fractional L2 Loss (FL2L) function acts as a bridge, enabling a family of loss functions ranging from L2 to L1 losses. Specifically, as α increases the FL2L transitions from L2’s strict penalty on large residuals (associated with faster convergence), to L1’s penalizing less (robustness to outliers). Thus, α acts as an interpretable hyperparameter that adjusts the robustness level of the L2 loss function. Furthermore, we transform α to a dynamic parameter, allowing the FL2L to automatically adapt its loss landscape during gradient-based optimization, improving error minimization performance. Our experiments in linear regression, drone system identification, and battery cycle life prediction demonstrate the FL2L’s significant improvements.

In deep learning, robust loss functions are crucial for addressing challenges like outliers and noise. This paper introduces a novel family of adaptive robust loss functions, Fractional Loss Functions (FLFs), generated by deploying the fractional derivative operator into conventional ones. We demonstrate that adjusting the fractional derivative order a allows generating a diverse spectrum of FLFs while preserving the essential properties necessary for gradient-based learning. We show that tuning a gives the unique property to morph the loss landscape to reduce the influence of large residuals. Thus, a serves as an interpretable hyperparameter defining the robustness level of FLFs. However, determining a prior to training requires a manual exploration to pinpoint an FLF that aligns with the learning tasks. To overcome this issue, we reveal that FLFs can balance robustness against outliers while increasing penalization of inliers by tuning a. This inherent feature allows transforming a to an adaptive parameter as a trade-off that ensures balanced learning of a is feasible. Thus, FLFs can dynamically adapt their loss landscape, facilitating error minimization while providing robustness during training. We performed experiments across diverse tasks and showed that FLFs significantly enhanced performance. Our source code is available at https://github.com/mertcankurucu/Fractional-Loss-Functions.

Robust loss functions are crucial for training deep neural networks in the presence of label noise, yet existing approaches require extensive, dataset-specific hyperparameter tuning. In this work, we introduce Fractional Classification Loss (FCL), an adaptive robust loss that automatically calibrates its robustness to label noise during training. Built within the active-passive loss framework, FCL employs the fractional derivative of the Cross-Entropy (CE) loss as its active component and the Mean Absolute Error (MAE) as its passive loss component. With this formulation, we demonstrate that the fractional derivative order μ spans a family of loss functions that interpolate between MAE-like robustness and CE-like fast convergence. Furthermore, we integrate μ into the gradient-based optimization as a learnable parameter and automatically adjust it to optimize the trade-off between robustness and convergence speed. We reveal that FCL's unique property establishes a critical trade-off that enables the stable learning of μ: lower log penalties on difficult or mislabeled examples improve robustness but impose higher penalties on easy or clean data, reducing model confidence in them. Consequently, FCL can dynamically reshape its loss landscape to achieve effective classification performance under label noise. Extensive experiments on benchmark datasets show that FCL achieves state-of-the-art results without the need for manual hyperparameter tuning.

Friction dampers are an economical and effective passive control solution for enhancing the seismic performance of multi-story buildings. Their performance strongly depends on their configuration (distribution or placement) in the building, and from a cost-effectiveness standpoint, the minimum feasible number of dampers should be installed. Therefore, addressing both of these aspects simultaneously is essential. However, the associated optimization problem is a large-scale, non-convex, combinatorial optimization problem. In this paper, we present a deep reinforcement learning-based solution for simultaneous optimization of the number and distribution of friction dampers in multi-story buildings for seismic protection. The presented solution is computationally efficient and is tested on a case study to demonstrate its performance.

Seismic protection of multi-story buildings using friction dampers (FDs) is a cheap and effective passive structural control solution. For optimization of system response, optimal distribution of FDs between floors is required, which is a challenging problem. In this paper, we propose novel intelligent computational methods based on reinforcement learning for the distribution of $n$ FDs for $m$ -story buildings. In order to demonstrate the effectiveness of the proposed methods, a case study of optimally distributing 16 FDs in a 3-story building is considered, and the results are compared with the optimal solution found from a statistical analysis based on a large number of earthquake accelerations.

It is known that analytical manipulation of the time-domain representation of fractional order transfer function is, hitherto, a challenging task. In this study, a controller design methodology based on the direct synthesis design method is proposed using bi-fractional order transfer function as a reference model and certain time-domain criteria. First, an analysis examines the effects of bi-fractional order transfer function parameters, that is, commensurate fractional order, damping ratio and natural frequency, on the system time-domain criteria. This examination has allowed us to set up a relation between damping ratio and commensurate fractional order. Next, a polynomial function fitting is established in order to express the damping ratio in terms of this commensurate order. Bi-fractional-order reference model is regenerated using the above-mentioned relationship. Furthermore, a control design algorithm that utilizes the newly derived bi-fractional order reference model is developed by considering control signal limitations. The proposed fractional order control design method is then compared with globally optimized fractional order proportional-integral-derivative (PID) controllers under the same circumstances and performance index. Finally, the implementation of the proposed controller design algorithm is done on a real-time active suspension system. The results are very satisfactory and incoherent with the simulations.

The design optimization of structures can be conducted in either the time domain or the frequency domain. The frequency domain approach is advantageous compared to its time domain counterpart, especially if the degree of freedom is large, the objectives and/or constraints are formulated in the frequency domain, or the structure is subject to random loading. In this paper, an attempt is undertaken to obtain feasible optimal solutions by implementing the Nevanlinna–Pick (NP) interpolation theory across multi-objective structural optimization problems in the frequency domain. The NP equations introduce a trade-off that originates from the interpolation theory for complex variables. According to the NP theory, a complex function cannot have an independent amplitude from its derivative at a certain frequency. Consequently, the frequency response of a physical system cannot be shaped arbitrarily at discrete frequencies. Our objectives within this paper include calculating the weight, natural frequency, fatigue life, frequency domain response, and its derivative. To illustrate our claims, sample parameter and topology optimization problems were formulated and solved, both with and without the NP constraints. It was found that the inclusion of NP constraints induced a considerable improvement in the optimal solutions, while also causing the convergence to the optimal solution to become smoother.

In recent years, the use of Variable-Order (VO) fractional operators in control system design has been gaining popularity due to their adaptive nature, enabled by their dynamically adjustable fractional derivative and integral orders. This paper presents an online tuning method for adjusting the VO fractional derivatives in fractional controllers. The method is formulated to strategically accelerate or decelerate the system response's rate of change to enhance reference tracking and disturbance rejection performance while preserving closed-loop stability. It uses the normalised acceleration of the system response, a metric that provides insights into the 'fastness' or 'slowness' of the system response. The effectiveness of the proposed online tuning method is validated through a case study on quadcopter position control. Our research includes both simulation and real-time testing of Variable-Order Fractional PD (VOFPD) controllers, which utilise our online tuning method to adjust their fractional derivatives in real-time. Stability analysis via the D-decomposition method confirms that the quadcopter's closed-loop stability is preserved. Comparative results show significant improvements in reference tracking and disturbance rejection in terms of time-domain criteria.

In this study, an online tuning strategy for the fractional derivative order term of the variable‐order fractional proportional–integral–derivative (PID) controller is proposed for processes with dead time. The classical step response is divided into regions, and meta‐rules are developed for each region in order to improve the control performance. To achieve the goals of the meta‐rules, a set of equations that are the functions of absolute error and model parameters are proposed to manipulate the fractional order derivative during the process. These equations can handle the changes in model parameters since the coefficients of these equations are functions of model parameters. On both simulation studies and experimental results on the active suspension system, we show that the proposed method improves the time domain performance criteria both in relation to reference tracking and load disturbance rejection. Moreover, the robustness of the proposed method has also been tested and analyzed for the dead time variation within the process.

In this study, a reduced inverse integer-order controller design methodology for single fractional-order pole model is proposed. First, the higher integer-order equivalent of this model is found using the Oustaloup approximation to fractional operator. Then, the poles and zeros of the higher integer-order model are determined in terms of fractional-order system model parameters employing characteristic equation format and root-locus idea. The order of the higher integer-order model is reduced using dominant pole–zero couples. The proximity of the pole–zero couples to each other as well as their proximity to the origin is key dominance requirement for order reduction. The controller is obtained inverting the reduced order integer system model in terms of fractional-order system model parameters. The proposed controller is compared with integer and fractional proportional–integral–derivative (PID) controllers using the experiments carried out on two-tank liquid-level process. The real-time experiment outcomes demonstrate that the proposed controller has superior performance over integer and fractional proportional–integral–derivative controllers.

In this paper, a novel fractional-order Proportional-Integral Derivative (PID) controller design depending on optimal selection of frequency domain specifications is proposed for time delay systems. The frequency domain specification sets, namely, (i) phase margin and gain crossover frequency and (ii) phase margin and gain margin are determined so that the reference model is optimal according to three time domain performance indices, i.e. Integral Square Error (ISE), Integral Time Square Error (ITSE) and Integral Absolute Error (IAE). Here, the delayed Bode's ideal transfer function is employed in the reference model. Moreover, the stability region of the reference model is given via a theorem. In simulation studies, the proposed methodology is compared with other two different methods using the same frequency domain specifications. It is observed that the proposed optimal fractional-order PID controllers outperform according to the mentioned performance indices, and they also possess considerably acceptable performance in terms of other time domain specifications such as overshoot, settling time, etc.

In this study, a robust fractional-order controller design methodology for a type of fractional-order or integer-order model with dead time is proposed using phase and gain margin specifications. The delayed Bode's ideal transfer function is used as a reference model to design the controller analytically. The delay term in delayed Bode's ideal transfer function provides the exact determination of these frequency domain specifications when the system owns a dead time. The analytical robust controller design problem is transformed to solving four nonlinear equations with four unknown variables, two of which are the desired specifications; namely, phase and gain margins. The remaining two are the phase and gain cross-over frequencies. Next, some conditions are set based on the desired specifications so that nonlinear equations provide a unique solution. The proposed method is compared with the other existing robust controller methods based on the same frequency domain specifications. The simulation results reveal that the proposed method outperforms the other methods and also gives closer outcomes to the desired specifications. (c) 2022 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

In this study, a novel design method for half-cycle and modified posicast controller structures is proposed for a class of the fractional order systems. In this method, all required design variable values, namely, the input step magnitudes and their application times are obtained as functions of fractional system parameters. Moreover, empirical formulas are obtained for the overshoot values of the compensated system with half-cycle and modified posicast controllers designed utilizing this method. The proposed design methodology has been tested via simulations and ball balancing real-time system. It is observed that the derived formulas are in coherence with outcomes of the simulation and real-time application. Furthermore, the performance of modified posicast controller designed using proposed method is much better than other posicast control method.

Abstract: Fractional order PI controllers based on two different analytical design methods are applied to a magnetic levitation system in this paper. The controller parameters are specified in order to fulfill specific frequency criteria. The first design method utilizes a unity feedback reference model whose forward path includes Bode’s ideal loop transfer function. The second method uses the reference model that has been obtained via delayed Bode’s ideal loop transfer function. The achievement of these two controllers are contrasted with each other on the magnetic levitation system using various criteria.

In this study, an optimal fractional-order controller is proposed for a type of fractional-order model utilising the direct synthesis method. In that respect, the fractional counterpart of the second-order integer transfer function is selected as a closed-loop reference transfer function. The stability region of the fractional-order closed-loop reference transfer function is given via a theorem and related lemmas. Considering that a unity feedback loop is used, the parameters of the fractional-order closed-loop reference transfer function are specified based on the integral square error performance index within the specified stability region using a genetic algorithm. The time-domain characteristics of the optimal fractional-order closed-loop reference transfer function are compared with those of the optimal second-order integer closed-loop reference transfer function. The fractional- and integer-order controllers that are designed based on optimal closed-loop reference transfer functions are implemented on a real-time system. The performance of the fractional-order controller outperforms that of the integer counterpart on the integral square error criterion. Moreover, the simulation and practical results are consistent with each other.

In general, all the hybridized evolutionary optimization algorithms use the routine “first diversification and then intensification” approach. In other words, these hybridized methods all begin with a global search mode using a highly random initial search population, and then switch to an intense local search mode at some stage. The population initialization is still a crucial point in the hybridized evolutionary optimization algorithms since it can affect the convergence speed and the quality of the final solution. In this work, we introduce a new approach by creating a paradigm shift that reverses the “diversification” and then “intensification” routines. Here, instead of starting with a random initial population, we first find a unique starting point by conducting an initial exhaustive search based on the coordinate exhaustive search local optimization algorithm only for a single-step iteration in order to collect a rough but some meaningful knowledge about the nature of the problem. Thus our main assertion is that this approach will ameliorate the convergence rate of any evolutionary optimization algorithm. In this work, we illustrate how one can use this unique starting point in the initialization of two evolutionary optimization algorithms including but not limited to the Big Bang-Big Crunch optimization and the Particle Swarm Optimization. The experiments performed on a commonly used benchmark test suite, which consists of mainly rotated and shifted functions, show that the proposed initialization procedure leads to a great improvement for the above-mentioned two evolutionary optimization algorithms.

In this study, model predictive control (MPC) and inverse optimal control (IOC) approaches are merged with each other and a new control strategy is evolved. The key feature in this strategy is to solve the IOC problem repeatedly for each receding horizon of the model predictive control approach. From another perspective, MPC structure is inserted to IOC problem and thus, IOC problem is solved repeatedly using different initial conditions at the beginning of each receding horizon. In the solution phase of IOC, the parameters of the candidate control Lyapunov function matrix are estimated using the global evolutionary Big Bang-Big Crunch (BB-BC) optimization algorithm in an on-line manner. Thus, the proposed control structure solves the optimal control problem in classical MPC approach to the search of an appropriate candidate control Lyapunov function matrix for each control horizon. The comparison of the proposed method with the other related control methods are performed on the ball and beam system via simulations and real-time applications.

In general, all of the hybridized evolutionary optimization algorithms use “first diversification and then intensification” routine approach. In other words, these hybridized methods all begin with a global search mode using a highly random initial search population and then switch to intense local search mode at some stage. The population initialization is still a crucial point in the hybridized evolutionary optimization algorithms since it can affect the speed of convergence and the quality of the final solution. In this study, we introduce a new approach by creating a paradigm shift that reverses the “diversification” and then “intensification” routines. Here, instead of starting from a random initial population, we firstly find a unique starting point by conducting an initial exhaustive search based on the coordinate exhaustive search local optimization algorithm only for single step iteration in order to collect a rough but some meaningful knowledge about the nature of the problem. Thus, our main assertion is that this approach will ameliorate convergence rate of any evolutionary optimization algorithms. In this study, we illustrate how one can use this unique starting point in the initialization of two evolutionary optimization algorithms, including but not limited to Big Bang-Big Crunch optimization and Particle Swarm Optimization. Experiments on a commonly used benchmark test suite, which consist of mainly rotated and shifted functions, show that the proposed initialization procedure leads to great improvement for the above-mentioned two evolutionary optimization algorithms.

In this paper, a fractional order PID controller cascaded with a fractional filter is proposed for higher order processes. In this analytical design methodology, one or two reduced fractional orders plus time delay models are used to represent higher order system transfer functions. The controller parameters are determined so as to meet certain frequency domain specifications. A unity feedback reference model is employed where Bode’s ideal loop transfer function plus time delay of the fractional order model is placed in the forward path. The addition of this time delay provides the exact determination of frequency domain specifications if the system either intrinsically owns a time delay or a time delay is injected by its reduced order model. The proposed methodology is compared with two other related methodologies and it has been observed that the proposed controller performs much better than the others. Moreover, some empirical formulas for time domain characteristics of the reference model are numerically derived in terms of certain frequency domain specifications and time delay of the fractional reduced order model. The accuracy of these formulas is tested by simulations. The iso-damping, noise attenuation and load disturbance suppression performances of the proposed controller are also considered and compared with those of other related controllers.