Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis

In this paper, a robust fractional‐order PID (FOPID) controller design method for fractional‐order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D‐decomposition method. Then, the optimal fractional orders λ and μ of FOPID controller are achieved at the biggest three‐dimensional PSR, which means the best robustness. Given the optimal λ and μ, the other FOPID controller parameters k_p, k_i, k_d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI). The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm.     

Control quality enhancement using fractional PIλDμ controller

The proportional–integral–derivative (PID) controllers have remained, by far, the most commonly and practically used in all industrial feedback control applications; therefore, there is a continuous effort to improve the system control quality performances. More recently Podlubny has proposed the fractional PI^λD^μ controller, a generalisation of the classical PID controller, involving an integration action of order λ and differentiation action of order μ. Since then, many researchers have been interested in the use and tuning of this type of controller. In this article, a new conception method of this fractional PI^λD^μ controller is considered. The basic ideas of this new tuning method are based, in the first place, on the classical Ziegler–Nichols tuning method for setting the parameters of the fractional PI^λD^μ controller for λ = μ = 1, which means setting the parameters of the classical PID controller, and on the minimum integral squared error criterion by using the Hall–Sartorius method for setting the fractional integration action order λ and the fractional differentiation action order μ. Illustrative examples and simulation results are presented to show the control quality enhancement of this proposed fractional PI^λD^μ controller conception method compared to the PID controller conception using Ziegler–Nichols tuning method.     

Optimal fractional order PIλDμ controller for stabilization of cart‐inverted pendulum system: Experimental results

The cart‐inverted pendulum is a non‐minimum phase system having right half s‐plane pole and zero in close vicinity to each other. Linear time invariant (LTI) classical controllers cannot achieve satisfactory loop robustness for such systems. Therefore, in the present work the fractional order PI^λD^μ (FOPID) controller is addressed for robust stabilization of the system, since fractional order controller design allows more degrees of freedom compared to its integer order counterparts by virtue of its two parameters λ and μ. The controller parameters are tuned by three evolutionary optimization techniques. In order to select the controller parameters optimally, a novel non‐linear fitness function using integral time square error (ITSE), settling‐time, and rise time is proposed here. The control algorithm is implemented successfully in real‐time. Moreover, stability analysis of the system compensated with a fractional order controller is presented using Riemann surface. Robustness of the physical cart‐inverted pendulum system towards multiplicative gain variations and plant parameter variations is verified. In this regard, it is shown that the fractional order controller provides satisfactory robust performance in both simulation and real‐time system.     

Robustness analysis and design of fractional order Iλ Dμ controllers using the small gain theorem

ABSTRACT

In this paper, a simple method is proposed to tune the parameters of Fractional Integral-Fractional Derivative (FIFE) I^λ D^μ controllers based on the Bode diagram. The proposed technique provides a practical approach for tuning FIFE controllers to compensate stable plants. Using the small gain theorem and based on the sensitivity functions analysis, it is proved that by applying the designed FIFE controller the robustness of the compensated system in the presence of plant uncertainties is improved in comparison to the PI controller in a similar structure. Moreover, the closed-loop phase margin and gain crossover frequency are adjustable by tuning the free controller parameters. Simulation results are presented to demonstrate the simplicity of application and effectiveness of the tuned controller.     

自适应模糊控制技术在铝塑板剪切生产线中的实现

飞剪控制系统是一个非线性、强耦合、时变的复杂系统,用普通的PID难以达到理想的控制效果.设计出一种基于模糊控制原理的自适应PID控制系统,根据偏差和偏差变化率来实时调整kp,kl,kd的值.仿真和应用结果表明,这种模糊自适应PID比常规PID控制器在飞剪控制中具有更好的控制性能.     

Computer package for microscopic spin Hamiltonian analysis of the 3d4 and 3d6 (spin S = 2) ions at orthorhombic and tetragonal symmetry sites

A FORTRAN package based on the microscopic spin Hamiltonian expressions derived by computer algebra (ALTRAN) for the high spin (S = 2) 3d⁴ and 3d⁶ ions with an orbital singlet ground state at orthorhombic (point groups: C_2v, D₂, D_2h) and tetragonal (point groups: C_4v, D₄, D_4h, D₂d) symmetry sites is presented. The spin—orbit (λ) and the spin—spin (ρ) coupling contributions up to the fourth-order perturbation theory are taken into account within the ⁵D approximation. The package enables efficient numerical calculations of the Zeeman electronic (Ze) parameters and the zero-field splitting (ZFS) parameters for the S = 2 3d⁴ and 3d⁶ ions. The following terms are included: λ and λ² for the Ze parameters, λ², λ³ λ⁴, ρ, ρ² and λ²ρ for the second-rank ZFS ones, and λ⁴, ρ² and λ²ρ for the fourth-rank ZFS ones. The program is applicable to all possible energy level schemes with a ground orbital singlet arising from the ⁵D multiplet due to orthorhombic or tetragonal symmetry crystal fields. The input parameters are λ, ρ, the energy levels Δ_j (j = 1, 2, 3, 4) and the mixing coefficient s, which can be obtained from other spectroscopic data. The ZFS parameters output in the extended Stevens notation B_k^q and b_k^q, as well as the conventional notation (D, E; a, F, K), is provided.     

Positivity properties in pencils of matrices†

If A and B are positive definite self-adjoint matrices, then all the matrices P_i, occurring in the expansion of an exterior power Λ^k(A+ λB) = P ₀ + P ₁λ + … + P _kλ^k are positive definite.     

>H∞-optimal solutions of robust regulator problem for linear MIMO systems

The robust regulator problem for linear MIMO systems in the frequency domain is considered. The stability robustness of the overall system is analysed with respect to additive perturbations in the mathematical model of the plant, and a controller that has been designed to solve the regulator problem for the nominal plant P_o(s). It is shown that in the special case where the reference signal (or the disturbance) has a pole at s =jω_k, the robustness can be upper-bounded by the minimum singular value of P_o(jω_k). This upper bound can be achieved in the case of a stable plant and step reference and disturbance. A design procedure for obtaining >H∞ -optimal solutions is given.     

Algorithms for constrained <Emphasis Type="Italic">k</Emphasis>-nearest neighbor queries over moving object trajectories

An important query for spatio-temporal databases is to find nearest trajectories of moving objects. Existing work on this topic focuses on the closest trajectories in the whole data space. In this paper, we introduce and solve constrained k-nearest neighbor (CkNN) queries and historical continuous CkNN (HCCkNN) queries on R-tree-like structures storing historical information about moving object trajectories. Given a trajectory set D, a query object (point or trajectory) q, a temporal extent T, and a constrained region CR, (i) a CkNN query over trajectories retrieves from D within T, the k (≥ 1) trajectories that lie closest to q and intersect (or are enclosed by) CR; and (ii) an HCCkNN query on trajectories retrieves the constrained k nearest neighbors (CkNNs) of q at any time instance of T. We propose a suite of algorithms for processing CkNN queries and HCCkNN queries respectively, with different properties and advantages. In particular, we thoroughly investigate two types of CkNN queries, i.e., CkNN_P and CkNN_T, which are defined with respect to stationary query points and moving query trajectories, respectively; and two types of HCCkNN queries, namely, HCCkNN_P and HCCkNN_T, which are continuous counterparts of CkNN_P and CkNN_T, respectively. Our methods utilize an existing data-partitioning index for trajectory data (i.e., TB-tree) to achieve low I/O and CPU cost. Extensive experiments with both real and synthetic datasets demonstrate the performance of the proposed algorithms in terms of efficiency and scalability.     

2D Bilinear programming for robust PID/DD controller design

A new design method of PID structured controllers to achieve robust performance is developed. Both robust stabilization and performance conditions are losslessly expressed by bilinear constraints in the proportional‐double derivative variable ( k _P, k _DD) and the integral‐derivative variable ( k _I, k _D). Therefore, the considered control design can be efficiently solved by alternating optimization between ( k _P, k _DD) and ( k _I, k _D), which is a 2D computationally tractable program. The proposed method works equally efficiently whenever even higher order differential or integral terms are included in PID control to improve its robustness and performance. Numerical examples are provided to show the viability of the proposed development. Copyright © 2016 John Wiley & Sons, Ltd.     

PI‐PD controller design for time delay systems via the weighted geometrical center method

In this study, a PI‐PD controller tuning method is presented using the weighted geometrical center method, which is based on the calculation of the weighted geometric center of the stability region obtained by the stability boundary locus method. The proposed method for tuning of PI‐PD controller parameters (k_d,k_f,k_p and k_i ) is performed in three steps. In the first step, the (k_d,k_f) parameter region for the inner loop with PD controller is obtained, and then the weighted geometric center of this region is calculated. In the second step, the inner PD loop is reduced to a single block using the numerical values of (k_d,k_f) that are obtained in the first step. Then, the (k_p,k_i) values of the external loop with PI controller are determined by the same procedure. This tuning method has some advantages over other tuning methods in terms of simplicity and robustness. The simulation examples show that a PI‐PD controller designed using the proposed method provides good performance results when compared to other tuning methods presented in the literature.     

On the stability radius of control systems with interval plants and first‐order controllers

This paper considers the stability radius problem of control systems with interval plants and first‐order controllers. The nominal plant P₀(s) used for the controller design is the centre of the interval plant. Under the condition that a lead controller C(s) stabilizes P₀(s), the stability radius of the closed‐loop system is determined in terms of the eigenvalues of four frequency‐independent matrices of the form H⁻¹_{β, i}H_{γ, i}, where both H_{β, i} and H_{γ, i} are Hurwitz‐like matrices. Copyright © 2000 John Wiley & Sons, Ltd.     

Controller reduction via stable factorization and balancing

A controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper. For a certain right coprime factorization of an LQG designed controller K(s) = N(s)D^-1(s), we approximate using a balancing technique the pair [D(s), N(s)]^T by a low-order pair [D₁(s), N₁(s)]^T defining a factorization of the reduced-order controller K ₁(s) = N₁(s)D₁ ^-1(s). We show that reducing the controller order in this way is motivated in a natural way, which leads to the expectation of both good stability properties and good accuracy of approximation of closed-loop behaviour. This is also demonstrated in some examples.     

Necessary and sufficient conditions on Q ( = C( I + PC)) for stabilization of the linear feedback system S(P, C)

We find the following necessary and sufficient conditions for Q (:=C(I+PC)⁻¹) to -stabilize the standard linear time-invariant unity feedback system S(P, C) where P has the l.c.f. (D_pl, N_pl) and the r.c.f. (N_pr, D_pr); and is a principal ideal domain. (i) Q must have elements in (ii) (resp. (iii)) Q must factorize in with D_pr, (resp. D_pl) as a left (resp. right) factor and (iv) (I – QP) must factor in with D_pr, as a left factor.     

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

This paper presents an algorithm for the Quillen–Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k [ x₀, . . . ,x_n ] / I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .     

<Emphasis Type="Italic">k</Emphasis>-Nearest-Neighbor Clustering and Percolation Theory

Let P be a realization of a homogeneous Poisson point process in ℝ ^d with density 1. We prove that there exists a constant k _d , 1<k _d <∞, such that the k-nearest neighborhood graph of P has an infinite connected component with probability 1 when k≥k _d . In particular, we prove that k ₂≤213. Our analysis establishes and exploits a close connection between the k-nearest neighborhood graphs of a Poisson point set and classical percolation theory. We give simulation results which suggest k ₂=3. We also obtain similar results for finite random point sets. Part of the work was done while S.-H. Teng was at Xerox Palo Alto Research Center and MIT. The work of F.F. Yao was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 1165/04E].     

Tuning of PIλDμ controllers based on sensitivity constraint

A graphical tuning method for fractional-order PID (PI^λD^μ) controllers is proposed based on the sensitivity function constraint of the closed-loop, which provides the information on robustness to plant uncertainties. The stabilizing regions in integral-derivative plane of the controller are first identified using a graphical stability criterion applicable to fractional-delay systems. Then, via Leibniz Sector Formula, the stabilizing region is optimized with respect to the two fractional orders of the controller to expect bigger stabilizing regions. Finally, the sensitivity function constraint of the closed-loop is mapped into stabilizing region by means of the explicit algebraic equations which can be solved efficiently. Numerical examples of a second-order integrating delay process are followed in each design procedure to show the effectiveness of the method.     

Two-step compensation of nonlinear systems

We study the problem of stabilization of nonlinear plants. We show that given a nonlinear plant P, if there exists a (nonlinear) compensator F, possibly unstable, which stabilizes P, then, with P₁: = P(I − F(− P))⁻¹, any C defined by C:= F + Q(I − P₁Q)⁻¹ for some finite-gain stable Q will stabilize P.     

New upper and lower bounds for the eigenvalues of the Sturm-Liouville problem

We derive new inequalities for the eigenvaluesλ _k of the Sturm-Liouville problem?u″+qu=λu, u(0)=u(π)=0 under the usual hypothesis thatq has mean value zero. The estimates give upper and lower bounds of the form |λ _k ?k ²|≤p _1,m k ^?m +P _2,m k ^2m ,k ²≥3‖q‖ _m ,m=1, 2 where ‖q‖ _m is the norm ofq in a Sobolev spaceH ^m (0, π) and theP's are homogeneous polynomials of degree at most 3 in ‖q‖ _m . Such bounds are used in the analysis of the inverse Sturm-Liouville problem.