Zamani et al.  proposed a stochastic optimization based tuning with a customized cost function consisting of various control objectives like maximum overshoot, rise time, settling time, steady state error, Integral of Absolute Error (IAE), squared control signal, inverse of phase margin and gain margin. Alomoush  optimized Integral of Time multiplied Absolute Error (ITAE) and Lee & Chang  optimized Integral of Square Error (ISE) as the integral performance index to find out the optimal set of controller parameters. An optimization based controller tuning by minimizing matrix norms as the cost functions has been proposed by Bouafoura & Braiek . Castillo et al.  proposed a tuningmethodology for FOPI controllers for first order systems from frequency domain specifications while also meeting few set of time domain specifications simultaneously. Bhambhani et al.  proposed a multi-objective optimization based FOPI controller tuningmethodology for Networked Control Systems (NCS) which simultaneously minimizes ITAE of the closed loop system and maximizes the jitter margin. Thus it can be seen that several time domain integral performance indices have been optimized by many contemporary researchers. Tavazoei in  has given a brief description of the finiteness of the integral performance indices for fractional order systems for step input and load disturbance excitation, which is required to be taken into account before the optimization. Caponetto et al.  investigated stabilization of FOPTD processes with FOPIDcontrollers. A similar stabilization problem with FOPD/FOPID controller for integer order integrating processes has been discussed by Hamamci & Koksal  and fractional order integrating processes by Hamamci . A PI D λ μ controller design for FO systems based on extended root-locus method has also been studied by Bayat & Ghartemani . Recently, Padula & Visioli  proposed empirical tuning rules for FOPIDcontrollers using IAE minimization criteria with constraints on maximum sensitivity for the FOPTD processes, which is rather a simplified approximation for higherorderprocesses with large modeling error.
Modern optimal control theory has a rich set of analytical tools to design control strategies satisfying desirable characteristics of the excursion of the system states according to the designer’s specifications in an optimal manner . The LQR is one such design methodology whereby quadratic performance indices involving the control signal and the state variables are minimized in an optimal fashion. Historically, in the area of industrial process control, PID controllers are tuned by minimizing a suitably chosen performance index for the control loop error function, which has yielded several thousands of tuning rules , in order to get an optimal PID setting. The tuning of PID controller uses the knowledge of the process model (mostly integer order models) like the gain (K), time-constant or lag (T) and time-delay (L). In spite of the huge advancements in the theoretical aspects of optimal control, successful integration of modern optimal control techniques in practical PID control problems was not there for decades due to several hidden heuristics in the design. For example, an effective choice of the weighting matrices (Q and R) in the optimal state feedback (LQR) design which is often impossible to know a priori, especially for the control of large industrial processes . There have been some previous efforts to merge the PID controller tuning problem with LQR theory as described in , , considering the error and integral of error as the state variables. The LQR technique has also been extended for tuning PID controllers for sluggish over-damped second orderprocesses in  by cancelling one of the real system poles with one of the zeros of PID controller. Thus, the approach, presented in  does not give the flexibility of tuning oscillatory processes by selecting the optimal controller gains via LQR for the three state variables i.e. error, its rate and integral. In the present paper, this concept is extended by simultaneously considering all the three state as the proportional, integral and derivative action of the controller and finding a synergism of the fractional calculus based enhancements of PID controllers  to circumvent the afore-said problems. The goal of the paper is to find out an answer to the following research questions – 1) how to optimally choose LQR weights, keeping in mind the final closed loop performance of a sluggish/oscillatory higherorder process in FO template, 2) how to handle the time delay terms, especially large delays in LQR formulation, while preserving both the stability and performance, 3) which time delay handling technique yields a better trade-off control design in terms of Pareto non-dominance for oscillatory and sluggish higherorderprocesses with varying level of lag and delay.
LQR based dominant pole placement tuning results for the three kinds of second order plants having the transfer functions (51), (53) and (55) are shown in section 4, with the obtained PID controllers (52), (54) and (56) respectively as the first stage of proposed PID controller tuningmethodology. Now, one pole and two complex zeros of the PID controllers in the complex s -plane are replaced by the corresponding fractional order poles/zeros - while keeping the similar order of the poles and zeros with an extra tuning knob. This extra flexibility induced by the equivalent integer order PID controller zeros at a certain point in -plane can be thought of as to get replaced by the corresponding fractional order zeros of a FOPID controller at the same position in the - plane, suggesting preservation of the desired control action for specified closed loop damping and frequency. Using first two steps, presented in the tuning algorithm, the FOPID controller order is gradually decreased up to 0.9 which give the approximated sub-optimal PID controller that was initially tuned via LQR as in section 4. The damping and frequency of the dominant closed loop poles are calculated and presented in Table 1. The equivalent PID controller gains for the FOPIDcontrollers are also reported in Table 1.
this paper with a Linear Quadratic Regulator (LQR) based technique that minimizes the change in trajectories of the state variables and the control signal. A class of fractional order systems having single non-integer order element which show highly sluggish and oscillatory open loop responses have been tuned with an LQR based FOPID controller. The proposed controller design methodology is compared with the existing time domain optimal tuning techniques with respect to change in the trajectory of state variables, tracking performance for change in set-point, magnitude of control signal and also the capability of load disturbance suppression. A real coded genetic algorithm (GA) has been used for the optimal choice of weighting matrices while designing the quadratic regulator by minimizing the time domain integral performance index. Credible simulation studies have been presented to justify the proposition.
It is fact that the single objective optimization with (12) has been carried out only with set-point changes and additionally we have shown the load-disturbance characteristics of such optimum set-point tracking based FO fuzzy PID controllers. Since, modulating the maximum magnitude of sensitivity function to control the load disturbance characteristics like that in  are difficult and more mathematically involved for highly nonlinear controllers as in our case, we restricted our study on various performance comparison with the family of FO fuzzy PID controllers for optimum set-point based tuning only. The variation in control signal or manipulated variable has also been taken into consideration for unit set-point change only in the optimization based controller tuning process. The summary of the comparative performances of the family of fuzzy FOPIDcontrollers are presented in Table 6 for three classes of oscillatory fractional orderprocesses with various levels of relative dead-time. The proposed family of FO hybrid fuzzy PID controller structure is believed to dominate future process control industries over present day’s fuzzy PID controllers, if the hardware implementation issues can be circumvented for both the fuzzy  and fractional differ- integral modules -. In addition to the recommended structure, it is interesting to see the achievable design trade-offs for different controller structure which requires multi- objective formulation of the controller tuning problem using different conflicting objectives (13) and (14).
control policies to take network delay and dropout into consideration -. In  the NCS is viewed as an asynchronous dynamical system with the network existing only between the sensor and the controller. The stability analysis of such systems is then done using Bilinear Matrix Inequalities in a Lyapunov framework. Other approaches have analyzed NCS using the concept of switched systems - and lifted sampling period . The focus of these papers is mainly the stability aspect of the NCS in the presence of variable delay and packet dropout. However for process control applications, various other performance criterions like set point tracking, load disturbance rejection etc. are also of prime importance and are not addressed adequately by these literatures. Other literatures like - have tried to address these issues of process control in NCS applications by incorporating the concept of jitter margin for variable time delays and formulating various PID tuning rules for the same. However, in -, - the NCS models do not account for packet drops and are limited to only variable delays which are unrealistic in a real time network. In the present paper both variable delays and packet drops encountered in a practical NCS application is incorporated in the tuningmethodology and the major issues of process control application, like set-point tracking, load disturbance rejection etc. are addressed with the help of evolutionary algorithms. The tuningmethodology is applicable to both PID and Fractional Order PID controllers and is illustrated by credible simulation studies.
In this paper we show that, in a distributed higher-order process language, locality of chan- nels can be enforced by a typing system with subtyping. The essential idea is to control the input capability of channels, guaranteeing at any one time this capability resides at exactly one location. As discussed in Section 3, ensuring locality in higherorderprocesses is much more difficult than in systems which only allows name passing. However, using our typing system we only have to static type-check each local configuration to guarantee the required global invari- ance, namely locality of channels.
The designing of fuzzy PID controller have three input terms: error, integral error, derivative error, a rule base three inputs, such as fuzzy proportional controller reduce error response and disturbance, Derivative action helps to predict the error and the proportional –derivative controller uses the derivative action to improve closed loop stability. If there is a sustained error in steady state, integral action is necessary. The integral action will increase the control signal if there is a small positive error, no matter how small the error is: the integral action will decrease if error is negative. The input to the Self-tuning Fuzzy PID Controller is speed error "e (t)" and Change-in-speed error "de (t)". The input shown in figure is described by:
In this paper, we present a framework for maximum likelihood estimation of GIN AR(p) processes based on a recursive representation of the transition proba- bilities. Using the resulting likelihood, we derive the score function and the Fisher information matrix for the model, which form the basis for conditional maximum likelihood estimation and inference. As in FM, we go on to represent all elements of the Fisher information matrix in terms of time t conditional moments of model components. Using the IN AR(2)-P speci…cation, we investigate the asymptotic gain of implementing the ML method over the commonly used CLS method by calculating the ARE ratio between the two estimators. Our results con…rm that the proposed MLE is asymptotically more e¢ cient than the CLSE and the ef- …ciency gain is most substantial for persistent processes. A Monte Carlo study suggests that there are often small sample gains in terms of bias and MSE to be had.
This fact is strengthened by the results presented in Table I and Figure 13. As already mentioned, most of the original stock return series cannot indeed be considered as Normal (see Table I). The same is also observed regarding the optimized portfolio returns. The Jarque-Bera, Anderson-Darling and Liliefors tests highly reject the Gaussian hypothesis, whilst the Kolmogorov-Smirnov test overall result is more contrasted. The two former tests mostly evaluate diﬀerences in the tails of the distributions, while the Lilliefors is especially sensitive to distribution gaps located at the node. For the Kolmogorov-Smirnov test, only the size of the largest diﬀerence counts. Diﬀerences between empirical probability den- sity functions and the theoretical Gaussian-benchmark ones are then mainly diﬀerences in the skewness and the kurtosis, small diﬀerences in the tails and in the centers of distri- butions, more likely than large diﬀerences somewhere specific. It is fair to mention here that most of the series - original stock and various optimized portfolio returns - clearly diﬀer from the ideal Gaussian hypothesis of the Markowitz’ model. Our negative result concerning the impact of higher moments on the utility-based optimal choice is not due to a hypothetical (almost-)Gaussianity of the underlying series. What could happen now if we were to intensify the original values of the skewness and the kurtosis (keeping the two first moments unchanged)? In Figure 13, we select and represent, from the set of all directionally optimized portfolios, those that are optimal for one of the various Quartic 35 Despite the widespread approach in the investor preference literature being to scale the initial agent
ABSTRACT : PID controllers are most popular controllers because of simplicity of implementation and broad applicability. In order to obtain the desired control performance correct tuning of PID controller is very important. There are many tuning algorithms available for tuning the PID controller. Most of the tuningprocesses are implemented manually. These processes are difficult and time consuming. Soft computing techniques have been widely used to tune the parameters of PID. In this paper parameters of PID controller are tuned using two sets of soft computing techniques which are Differential Evolution (DE) and Hybrid Differential Evolution (HDE). The optimal PID control parameters are applied for a composition control system. The performance of two techniques is evaluated by setting its objective function with Integral Square Error (ISE), Integral Absolute Error (IAE), Integral of Time multiplied by Absolute Error (ITAE). This paper also compares performance of tuned PID controller using DE and HDE methods with Ziegler-Nichols method.
The parallel structure has an important advantage because it allows substantial reduction of rule base size. This attractive feature will be further exploited in this paper. In addition, we also aim to extend the capability of the PD+I fuzzy logic controller by including the self tuning features so that plants with time varying nonlinear dynamics can be handled. First proposed self tuning fuzzy logic controller (STPD+I_31) uses rule base structure similar to the original PD+I fuzzy logic controller. Second proposed controller STPD+I_9 is much more efficient than STPD+I_31 because additional constraint is imposed on the size of rule bases. This constraint allows only minimum number of rules in the rule bases. As a result, there are only 3 fuzzy labels for the main PD controller and 2 fuzzy labels for the two auxiliary controllers. Consequently, STPD+I_9 works on 9 rules in total. To the best of our knowledge, STPD+I_9 is the first working 9 rules adaptive fuzzy logic controller that can be applied to the highly complex robot tracking control problems. How good can this seemingly simple fuzzy logic controller perform? To this end, two classical nonlinear controllers are used as the benchmark for comparison: one is based on parameter based adaptive control and another on sliding mode control theory. They are compared on their tracking performance in controlling a two-links revolute robot with different speed settings. It is well known that the two-links revolute robot is a highly coupled nonlinear system  because the inertia loading, the coupling between joints and the gravity effects are highly sensitive to position and velocity conditions. During high speed motions, the inertia loading terms can change drastically. As such, it serves as a good test bed for the comparison of the newly proposed controllers and the benchmark controllers. Extensive simulation results are included to support the performance analysis.
tion is larger as shown in Table II and Fig. 8. The networked PI control system using fixed and the GSM could maintain the system performance much better. As shown in Table II and Fig. 8(a), all control schemes can satisfy the performance re- quirements by resulting in because RTT delay and its variation are relatively low. With longer RTT delay and more delay variation, the requirements cannot be satisfied as shown in Table II, and Fig. 8(b) and (c). Nevertheless, the fixed and the GSM schemes can still maintain the system performance satis- factorily. The performance is much better than the system with the nominal since the PI controller gains in this case are adapted to be more suitable for the network traffic condi- tions. However, as shown in Table II and Fig. 8(d), both schemes cannot perfectly maintain the system performance to meet the specifications with very long RTT delay and very high variation, but can still reasonably stabilize the networked control system. In addition, the PI controllers using the pre-computed optimal and the GSM with respect to real-time RTT delay character- istics have similar performance. The only exception in these ex- amples is that when RTT delays are very long with high vari- ation (e.g., RTT delays from ADAC Lab to www.ku.ac.th). In actual IP networks, the probability density of RTT delay may vary, and sometimes the variation can be large. Therefore, the fixed optimal gain approach may no longer be suitable in some situations such as IP network congestion. Using the GSM to dy- namically adjust the PI controller gains could provide an accept- able and more flexible control in a real IP environment.
Abdulameer, Sulaiman, Aras and Saleem (2016) discussed using the tuning methods: Ziegler- Nichols, Chen-Hrones-Reswick and explained the advantages and disadvantages of each formula of the two methods. The DC motor they used as a plant had a second order transfer function with 0.897 damping ratio and 3.89 rad/s natural frequency. The maximum percentage overshoot they got using the studied techniques was 13.1 to 26.3 % while the settling time was between 0.378 and 1.35 seconds . Issa, Hassanien and Abdelbaset (2017) used the Egyptian Vulture Optimization Algorithm to tune PID controllers by minimizing the ISE error function. They showed that using their proposed approach enhanced the performance of the controlled process than the Ziegler-Nichols method . Alzuabi (2018) studied the application of the Bacterial Foraging optimization method to tune a PID controller used with a DC motor. His simulation results illustrated the enhancement of the control system response. He didn't compare with other tuning techniques used in the literature to control the PID controller . Zhang, Zhang and Dong (2019) presented a tuning technique for the PID controller based on the Mind Evolutionary Algorithm. They applied their tuning technique to control five different processes using an ITAE error criterion and compared their results with two other tuning techniques .
Femtocells have gained attention in the research community, which is evident from several recently launched projects (e.g., HOMESNET , BeFEMTO , FREEDOM ). First publications were devoted to residential scenarios with standalone femtocells. For these scenarios, Claussen et al. propose in  a self-tuning algorithm to adjust transmit power for uplink and down- link in an Universal mobile telecommunication system (UMTS) femtocell to mitigate interference to macrocells and ensure a constant femtocell radius, regardless of the position of the latter within the macrocell area. In , the authors present a self-tuning algorithm for pilot power in an UMTS femtocell to improve coverage and minimize the total number of HO attempts. For the same purpose,  presents a self-tuning algorithm for selecting femto- cell pilot power and antenna pattern, while  presents an adaptive algorithm for selecting the hysteresis margin based on user position. More recent studies have con- sidered networked femtocell environments, among which is the enterprise scenario . In these scenarios, most eﬀorts have been paid to the design of advanced RRM algorithms to manage inter-cell interference in orthogo- nal frequency division multiple access (OFDMA) schemes . L´opez et al.  propose an integer linear program- ming model to dynamically assign modulation and coding scheme, radio bearer and transmit power to users, while minimizing the total cell transmit power and meeting user throughput demands. Similarly, several distributed admis- sion control and scheduling schemes have been inspired in self-organizing principles taken from cognitive net- works , machine learning [32,33] and game theory . More related to the study presented here, deal- ing with self-optimization,  proposes a decentralized algorithm for tuning pilot power in UMTS femtocells to balance cell load and minimize total pilot transmit power in an oﬃce scenario. In , the problem of power and frequency planning in mobile wireless interoperability for microwave access (WiMAX) enterprise femtocells is for- mulated as a mixed integer programming model, whose goal can be either to maximize the sum of transmit power, given that the overall connection quality impairment is kept within acceptable limits, or to maximize network Shannon capacity.
A closed loop control system encounters different combina- tions of plant and controllers while handling real world prob- lems. They include the integer or fractional order of either plant or controller or both. In practice, the plant models have been obtained as integer order models and it is natural to consider the fractional nature of the controller. The efforts of control engi- neers and scientists lead to the development of fractional order controller (PI λ /PI λ D μ ) tuning rules. Podlubny  had proposed the fractional order PI and PID controllers that demonstrated better response with integrator and differentiator rose to the fractional powers λ and μ respectively. There are several other PI λ and PI λ D μ controllers in the literature [2–6] whose tuning rules are developed using evolutionary algorithms by minimiz- ing objective functions. Some heuristics based controllers are also reported.
Cautionary note & related works. In general, a projection search algorithm can be evaluated according to two orthogonal axes, namely its time and data complexity (i.e. how many iterations and measurements do we need to obtain a projection?) or the informativeness of its outputs (which relates to the data complexity of an attack exploiting the projections obtained). Hence, it is first worth recalling that (standard) dimentionality reductions for unprotected imple- mentations indeed optimize informativeness, whereas existing solutions to detect POIs in masked implementations focus on the complexity issue (because of the more challenging nature of the problem). In this context, we note that compar- ing different projection’s informativeness (e.g. PCA, LDA and the recent works in [11, 17]), in the context of unprotected implementations, is of limited inter- est anyway. Indeed, it has been shown in  that the objective functions of LDA (which improves over PCA in terms of informativeness) and [11, 17] are essentially equivalent in this case, meaning that LDA, these works and our new projections all have similar informativeness as well (up to statistical artifacts). Nevertheless, and for completeness, we show empirical evidence of the gain they provide over PCA and its impact in the DPA contest v2 . As for compar- isons in the case of higher-order leakages and masked implementations, the main issue is that none of the previous dimensionality reductions generalizes to such contexts 2 . In fact, and as witnessed by the previous state-of-the-art, existing
2. Consideration of higherorder harmonics Let us consider an open loop stable second order plus time delay (SOPTD) system with equal time constants (1). Consider a symmetric relay feedback system. From the Fourier series analysis, it can be easily shown  that a relay consists of many sinusoidal waves of odd multiples of fundamental frequency ‘ω’ and with the amplitude 4h/(nπ) (n=1, 3, 5, ..). For a SOPTD system, the output wave is also a sinusoidal wave. Here, y(t) is the output response.
Abstract – Position control of the manipulator has been noted for its difficulty as the result of the so-called dynamic stability problem, parameter uncertainty and dynamic coupling. This work focused on position control of the first four degree of freedom (DOF) of IT-Robot manipulator using Fractional Order PID (FOPID) controller that can conquer this difficulty. All procedure description to model of the manipulator and control has been detailed and simulated using MATLAB R2015a/Simulink; from the mechanical model generation in SimMechanic where, the manipulator joint is moved using DC motor. Parameters of FOPID controller are optimized using GA. The controller effectiveness is analyzed for set point tracking. By simulation results, it was observed that FOPID controller give better response with minimum error than PID controller for the position control of the IT-robot manipulator.
Another tool widely exploited to solve optimal control problems is dy- namic programming . Dynamic programming (DP) is a very different approach to solve optimal control problems than the ones presented pre- viously. The methodology was developed in the fifties and sixties of the 20th century, most prominently by Richard Bellman  who also coined the term dynamic programming. Interestingly, dynamic programming is easi- est to apply to systems with discrete state and control spaces. When DP is applied to discrete time systems with continuous state spaces, some ap- proximations have to be made, usually by discretization. Generally, this discretization leads to exponential growth of computational cost with re- spect to the dimension of the state space, what Bellman called “the curse of dimensionality”. It is the only but major drawback of DP and limits its practical applicability. On the positive side, DP can easily deal with all kinds of hybrid systems or non-differentiable dynamics, and it even allows us to treat stochastic optimal control with recourse, or minimax games, without much additional effort.