Interval Plant Research Articles

Multiobjective Robust PI Synthesis in Plants with Uncertain Poles

The use of multiobjective optimization procedures for control tuning has been constantly explored, presenting satisfactory results, and generating significant contributions to control engineering. In this way, the field of robust control can benefit significantly from the integration of multiobjective optimization design procedures for robust Proportional-Integral-Derivative (RPID) controller synthesis, enabling innovative solutions to complex control challenges. Due to this fact, this paper explores an approach to utilize multiobjective optimization algorithms in a simplified procedure, considering uncertain systems in the form of interval plants with uncertain poles, the Kharitonov theorem is used to obtain a region of stability as search space for the multiobjective algorithm, being the optimization process performed over a nominal system model. Two numerical cases and a flexible actuator model were used to demonstrate the procedure for RPI synthesis, resulting in robustly stable controllers with optimized performance over error and control signal effort measures.

Robust Tilt-Integral-Derivative Controllers for Fractional-Order Interval Systems

In this study, an innovative and sophisticated graphical tuning approach is postulated, aimed at the design of tilt-integral-derivative (TID) controllers that are specifically customized for fractional-order interval plants, whose numerators and denominators consist of fractional-order polynomials that are subjected to parametric uncertainties. By leveraging the powerful value set concept and the advanced D-composition technique, a comprehensive set of stabilizing TID controllers is obtained. The validity and effectiveness of the proposed methodology are demonstrated by some examples, which vividly illustrate its remarkable performance and potential.

Design of Robust PI Controllers for Interval Plants With Worst-Case Gain and Phase Margin Specifications in Presence of Multiple Crossover Frequencies

This article deals with the computation of robustly performing Proportional-Integral (PI) controllers for interval plants, where the performance measures are represented by the worst-case Gain Margin (GM) and Phase Margin (PM) specifications, in the event of multiple Phase Crossover Frequencies (PCFs) and/or Gain Crossover Frequencies (GCFs). The multiplicity of PCFs and GCFs poses a considerable complication in frequency-domain control design methods. The paper is a continuation of the authors’ previous work that applied the robust PI controller design approach to a Continuous Stirred Tank Reactor (CSTR). This preceding application represented the system with a single PCF and a single GCF, but the current article focuses on a case of multiple PCFs and GCFs. The determination of a robust performance region in the P-I plane is based on the stability/performance boundary locus method and the sixteen plant theorem. In the illustrative example, a robust performance region is obtained for an experimental oblique wing aircraft that is mathematically modeled as the unstable interval plant. The direct application of the method results in the (pseudo-)GM and (pseudo-)PM regions that “illogically” protrude from the stability region. Consequently, a deeper analysis of the selected points in the P-I plane shows that the calculated GM and PM boundary loci are related to the numerically correct values, but that the results may be misleading, especially for the loci outside the stability region, due to the multiplicity of the PCFs and GCFs. Nevertheless, the example eventually shows that the important parts of the GM and PM regions, i.e., the parts that have an impact on the final robust performance region, are valid. Thus, the method is applicable even to unstable interval plants and to the control loops with multiple PCFs and GCFs.

Robust PIDD2 Controller Design for Perturbed Load Frequency Control of an Interconnected Time-Delayed Power Systems

The extensive practice of open communication links in the power system network introduces unavoidable communication time delays (CTDs). These CTDs demean the performance of the load frequency control (LFC) system, and in the worst condition, the LFC system becomes unstable. Thus, this brief proposes a robust proportional integral derivative double derivative (PIDD2) controller design for the perturbed LFC of the interconnected time-delayed power system. In this brief, the worst case plant selection approach is applied to find the worst case plant model using Kharitonov’s stability theorem of the interval system. The proposed PIDD2 controller is designed for the selected worst case plant model of the original interval plant. Furthermore, the tuning of PIDD2 controller is carried out using an internal model control (IMC) approach. The notable feature of the presented IMC scheme is that the IMC filter coefficient ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> ) is determined in terms of maximum sensitivity ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M_{s}$ </tex-math></inline-formula> ) and CTD, which shows the tradeoff between robustness and performance. The proposed control scheme is validated on a two-area time-delayed power system model. The effectiveness and robustness of the proposed IMC-PIDD2 control approach are assessed under parametric uncertainties in system parameters as well as CTD, nonlinearities, and step load demand disturbances. Besides, delay margin (DM) is also computed in the sense of Walton and Marshall stability theorems for the proposed control system. The efficacy of the proposed IMC-PIDD2 control scheme is verified by comparing it with the recently reported control schemes.

Design of FOPID controller for higher order continuous interval system using improved approximation ensuring stability

This contribution deals with the design of a fractional-order proportional-integral-derivative (FOPID) controller through reduce-order modeling for continuous interval systems. First, a higher order interval plant (HOIP) is considered. The reduced-order interval plant (ROIP) for considered HOIP is derived by multipoint Padé approximation integrated with Routh table. Then, FOPID controller is designed for ROIP to satisfy the phase margin and gain cross over frequency. Thus obtained FOPID controller is implemented on HOIP also to validate the performance of designed FOPID on HOIP. A single-input-single-output (SISO) test system is taken up to elaborate the entire process of controller design. The outcomes affirm the validity of the designed FOPID controller. The designed FOPID controller produced stable results retaining the phase margin and gain cross-over frequency when implemented on HOIP. The results further proved that FOPID controller is working efficiently for ROIP and HOIP.

Sign function and ANN based pole placement for computing interval controls

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants.

Robust Stabilization of Interval Plants with Uncertain Time-Delay Using the Value Set Concept

This paper considers the robust stabilization problem for interval plants with parametric uncertainty and uncertain time-delay based on the value set characterization of closed-loop control systems and the zero exclusion principle. Using Kharitonov’s polynomials, it is possible to establish a sufficient condition to guarantee the robust stability property. This condition allows us to solve the control synthesis problem using conditions similar to those established in the loopshaping technique and to parameterize the controllers using stable polynomials constructed from classical orthogonal polynomials.

Optimal Robust Control for Unstable Delay System

Proportional-Integral-Derivative control system has been widely used in industrial applications. For uncertain and unstable systems, tuning controller parameters to satisfy the process requirements is very challenging. In general, the whole system’s performance strongly depends on the controller’s efficiency and hence the tuning process plays a key role in the system’s response. This paper presents a robust optimal Proportional-Integral-Derivative controller design methodology for the control of unstable delay system with parametric uncertainty using a combination of Kharitonov theorem and genetic algorithm optimization based approaches. In this study, the Generalized Kharitonov Theorem (GKT) for quasi-polynomials is employed for the purpose of designing a robust controller that can simultaneously stabilize a given unstable second-order interval plant family with time delay. Using a constructive procedure based on the Hermite-Biehler theorem, we obtain all the Proportional-Integral-Derivative gains that stabilize the uncertain and unstable second-order delay system. Genetic Algorithms (GAs) are utilized to optimize the three parameters of the PID controllers and the three parameters of the system which provide the best control that makes the system robust stable under uncertainties. Specifically, the method uses genetic algorithms to determine the optimum parameters by minimizing the integral of time-weighted absolute error ITAE, the Integral-Square-Error ISE, the integral of absolute error IAE and the integral of time-weighted Square-Error ITSE. The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example.

Sine Cosine Algorithm Assisted FOPID Controller Design for Interval Systems Using Reduced-Order Modeling Ensuring Stability

The focus of present research endeavor was to design a robust fractional-order proportional-integral-derivative (FOPID) controller with specified phase margin (PM) and gain cross over frequency (ωgc) through the reduced-order model for continuous interval systems. Currently, this investigation is two-fold: In the first part, a modified Routh approximation technique along with the matching Markov parameters (MPs) and time moments (TMs) are utilized to derive a stable reduced-order continuous interval plant (ROCIP) for a stable high-order continuous interval plant (HOCIP). Whereas in the second part, the FOPID controller is designed for ROCIP by considering PM and ωgc as the performance criteria. The FOPID controller parameters are tuned based on the frequency domain specifications using an advanced sine-cosine algorithm (SCA). SCA algorithm is used due to being simple in implementation and effective in performance. The proposed SCA-based FOPID controller is found to be robust and efficient. Thus, the designed FOPID controller is applied to HOCIP. The proposed controller design technique is elaborated by considering a single-input-single-output (SISO) test case. Validity and efficacy of the proposed technique is established based on the simulation results obtained. In addition, the designed FOPID controller retains the desired PM and ωgc when implemented on HOCIP. Further, the results proved the eminence of the proposed technique by showing that the designed controller is working effectively for ROCIP and HOCIP.

Optimal design of robust resilient automatic voltage regulators

Uncertainties in the plant model parameters and perturbations in the controller gains imposed by implementation errors represent a challenge to ensure robust stability and controller non-fragility simultaneously. Optimal design of robust non-fragile proportional–integral–derivative (PID) controller is presented for an automatic voltage regulator (AVR). The PID design relies basically on Kharitonov theorem and optimization by future search algorithm (FSA). The proposed algorithm has low computational complexity and fast convergence rate because it utilizes both local and global search methods. Further, FSA can improve the exploration characteristic and prevent trapping in local optima by updating its random initial. The PID controller is optimized by FSA to cope with expected parametric uncertainties of the plant model and tolerate its gain perturbations such that robust stability and controller non-fragility are simultaneously met. An interval plant model is suggested to account for model uncertainties where only eight extreme plants derived by Kharitonov theorem are considered in design. FSA-based PID optimization is constrained by the stability conditions of Kharitonov’s plants derived using Routh–Hurwitz. A new figure-of-demerit (FoD) based performance index is suggested to enforce simultaneous minimization of the time domain specifications. The suggested objective function is represented by a weighted sum of FoD of nominal response and the sum of reciprocals of the perturbation radii of PID gains. The output results of the recommended design are compared to that of artificial bee colony (ABC) algorithm and teaching–learning based optimization (TLBO) algorithm, multi-objective extremal optimization (MOEO), and non-dominated sorting genetic algorithm II (NSGA II). The results can confirm better response of the suggested technique measured up to other techniques where robustness and non-fragility are simultaneously ensured.

Robust PI Control of Interval Plants With Gain and Phase Margin Specifications: Application to a Continuous Stirred Tank Reactor

The paper is focused on robust Proportional-Integral (PI) control of interval plants with gain and phase margin specifications and on the application of this approach to a Continuous Stirred Tank Reactor (CSTR). More specifically, the work aims at the determination of PI controller parameter regions, for which not only robust stability but also some level of robust performance of the closed-loop control system is guaranteed, and this robust performance is represented by the required gain and phase margin that has to be ensured for all potential members of the interval family of controlled plants, even for the worst case. The applied technique is based on the combination of the previously published generalization of stability boundary locus method (for specified gain and phase margin under the assumption of fixed-parameter plants) with the sixteen plant theorem. This extension enables the direct application of the method to design the robustly performing PI controllers for interval plants. The effectiveness of the improved method is demonstrated on a CSTR, modeled as the interval plant, for which the robust stability and robust performance regions are obtained.

Consensus of Higher Order Agents: Robustness and Heterogeneity

This paper explores the use of Kharitonov's Theorem on a class of linear multiagent systems. First, we study a network of the mth order (m ≥ 2) linear uncertain interval plants and provide conditions for achieving fullstate consensus, which relate the stability margins of each agent to the spectrum of the graph Laplacian. Then, a robustness analysis for such systems is presented when an edge weight in the underlying graph is perturbed. The same Kharitonov-based analysis proves useful in a related problem, where heterogeneous higher order linear models of agents are considered in a setup similar to pinning control, and conditions for consensus among the follower agents are derived. Numerous simulation examples validate the results.

Robust PID controller design for rigid uncertain spacecraft using Kharitonov theorem and vectored particle swarm optimization

This paper presents a robust Proportional Integral Derivative controller design methodology for three axis attitude control of a rigid spacecraft with parametric uncertainty using a combination of Kharitonov theorem and vectored particle swarm optimization based approaches. A controller is designed for each of the three axes using a systematic graphical approach. Here, a plot of the stability boundary loci in the integral gain versus proportional gain parameter plane, for the specified gain and phase margins for each of the Kharitonov interval plants is used to determine the region representing the set of all PID controllers that satisfy the desired performance and stability requirements. Vectored particle swarm optimization technique is used to determine the optimized proportional and integral gain values. The spacecraft attitude control system is simulated using Matlab-Simulink tool which shows that the designed controller provides stability, robustness, good reference pointing and disturbance rejection for perturbations within the specific bounds.

A comment on “Algorithm of robust stability region for interval plant with time delay using fractional order PIλDμ controller” [Commun Nonlinear Sci Numer Simulat 17 (2012) 979–991

In Ref. [1], it has been claimed that the value set of a fractional order interval plant, having one uncertain time delay is a polygon. This note illustrates that this claim is not true, and in fact the value set has a nonconvex shape. Therefore, Theorem 3.1 and the stabilization algorithm presented in the reference to design a PIλDμ controller, are not true.

Robust stabilization of multilinear interval plants by Takagi–Sugeno fuzzy controllers

• Stabilization problem of multilinear interval plants is studied. • A T–S fuzzy system is expressed in the form of an interval system. • T–S fuzzy system is considered as a controller for multilinear interval plant. • Sufficient conditions for robust stability of fuzzy controller are derived. • The results are applied to a two-mass-spring-damper system. This study investigates the stabilization problem of interval and multilinear interval plants. For this purpose, a Takagi–Sugeno fuzzy state model, expressed in the form of an interval system, is considered as a robust stabilizing controller for the plants with interval and multilinear interval models. Based on the Zero Exclusion Condition, sufficient conditions that ensure robust stability of the presented fuzzy controller are derived as well. A two-mass-spring-damper system with multilinear uncertain parameters is studied to demonstrate the application and the effectiveness of the design process.

Impulse energy approximation of higher-order interval systems using Kharitonov’s polynomials

This research work proposes a method to obtain a stable reduced-order interval model from its stable higher-order interval plant. The reduced-order interval numerator and denominator polynomials are determined by using Kharitonov’s polynomials and a general form of the Routh approximation method. The proposed reduction algorithm retains stability and full impulse response energy of the higher-order interval system in its reduced-order interval model. In addition to this, the proposed method has useful features like matching of time moments and mathematical simplicity. Moreover, a few numerical examples in the literature are taken into consideration and simulated through MATLAB to illustrate the effectiveness of the proposed method.

Comment on “Design of PID controllers for interval plants with time delay”

In [1] three theorems are presented which relate to the problem of stabilizing PID controller design for interval time-delay plants. We present a counterexample for these results.

Robust power system stabiliser design via interval arithmetic

In this paper, an interval arithmetic (IA) approach is proposed to rigorously address load uncertainties associated with the design of PSS. The proposed approach characterises the robust stabilising three-term/parameter PSSs computed for a single-machine infinite-bus system. A robust PSS can properly function over a wide range of operating conditions. Image-set and interval plant models are developed to express all uncertainties in the model parameters imposed by continuous variation in generation and load patterns. For robust PSS design purpose, an interval characteristic polynomial of the closed loop system is generated. Interval Routh-Hurwitz (RH) array is considered to get a fast but conservative calculation of the stability region in the controller-parameter plane using IA computation. Degenerate interval (RH) array for an image-set polynomial of the plant is thereafter developed where the region of robust stability is exactly computed. Simulation results are provided to confirm the effectiveness of the proposed approach in computing robust PSSs.

Robust fuzzy stabilisation of interval plants

ABSTRACTThe paper deals with the problem of stabilisation of interval systems. To this end, by using Takagi–Sugeno fuzzy mechanism, a Mamdani-type PID-like fuzzy controller is modified and extended to develop a new PID-like Takagi–Sugeno fuzzy stabilising controller for the plant described by an interval system. Indeed, a PID-like Takagi–Sugeno fuzzy controller and an interval plant are considered in the forward path of a unity feedback system, and parameters in Takagi–Sugeno fuzzy controller are determined so that the stability of the closed-loop system is assured. The closed-loop system has a multilinear uncertainty structure. Therefore, based on the Zero Exclusion Condition for multilinear uncertain systems, a new theorem presenting sufficient conditions for the Takagi–Sugeno fuzzy controller to be robust stability guaranteed is also derived. An example is given to illustrate the application and the effectiveness of the proposed controller.

Application of polynomial control to design a robust oscillation-damping controller in a multimachine power system

This paper addresses the application of a static Var compensator (SVC) to improve the damping of interarea oscillations. Optimal location and size of SVC are defined using bifurcation and modal analysis to satisfy its primary application. Furthermore, the best-input signal for damping controller is selected using Hankel singular values and right half plane-zeros. The proposed approach is aimed to design a robust PI controller based on interval plants and Kharitonov׳s theorem. The objective here is to determine the stability region to attain robust stability, the desired phase margin, gain margin, and bandwidth. The intersection of the resulting stability regions yields the set of kp–ki parameters. In addition, optimal multiobjective design of PI controller using particle swarm optimization (PSO) algorithm is presented. The effectiveness of the suggested controllers in damping of local and interarea oscillation modes of a multimachine power system, over a wide range of loading conditions and system configurations, is confirmed through eigenvalue analysis and nonlinear time domain simulation.