Non-square Systems Research Articles

Distributed integral controllability for non-square processes: A comprehensive study and numerical analysis

Distributed control systems offer significant benefits such as enhanced flexibility, improved failure tolerance, and simplified system design compared to centralized systems. While the controllability of linear square systems under decentralized structures incorporating integral action is well-explored, challenges arise when dealing with non-square processes, which are prevalent in various control and optimization scenarios. Traditionally, non-square systems are manipulated into square configurations by adjusting inputs and outputs to apply decentralized integral controllability (DIC) analysis, potentially affecting the system’s reliability and controller flexibility. To address this issue, we propose a direct application of decentralized integral controllability to non-square processes, termed DIC-NSQ. We approach this problem by representing the system through a state space description, transforming it into standard singular perturbation form, and subsequently establishing its necessary as well as sufficient conditions using singular perturbation analysis. Through numerical examples, we demonstrate that DIC-NSQ effectively maintains offset-free tracking and ensures robust performance, even in the presence of actuator failures. This contribution marks an advancement in the field of distributed control systems, particularly for applications involving non-square processes.

On Krylov methods for large-scale CBCT reconstruction.

Krylov subspace methods are a powerful family of iterative solvers for linear systems of equations, which are commonly used for inverse problems due to their intrinsic regularization properties. Moreover, these methods are naturally suited to solve large-scale problems, as they only require matrix-vector products with the system matrix (and its adjoint) to compute approximate solutions, and they display a very fast convergence. Even if this class of methods has been widely researched and studied in the numerical linear algebra community, its use in applied medical physics and applied engineering is still very limited. e.g. in realistic large-scale computed tomography (CT) problems, and more specifically in cone beam CT (CBCT). This work attempts to breach this gap by providing a general framework for the most relevant Krylov subspace methods applied to 3D CT problems, including the most well-known Krylov solvers for non-square systems (CGLS, LSQR, LSMR), possibly in combination with Tikhonov regularization, and methods that incorporate total variation regularization. This is provided within an open source framework: the tomographic iterative GPU-based reconstruction toolbox, with the idea of promoting accessibility and reproducibility of the results for the algorithms presented. Finally, numerical results in synthetic and real-world 3D CT applications (medical CBCT andμ-CT datasets) are provided to showcase and compare the different Krylov subspace methods presented in the paper, as well as their suitability for different kinds of problems.

Tracking Control of Self-Restructuring Systems: A Low-Complexity Neuroadaptive PID Approach With Guaranteed Performance.

This article investigates the tracking control problem for a class of self-restructuring systems. Different from existing studies on systems with fixed structure, this work focuses on systems with varying structures, arising from, for instance, biological self-developing, unconsciously switching, or unexpected subsystem failure. As the resultant dynamic model is complicated and uncertain, any model-based control is too costly and seldom practical. Here, we explore a nonmodel-based low-complexity proportional-integral-derivative (PID) control. Unlike traditional PID with fixed gains, the proposed one is embedded with neural-network (NN)-based self-tuning adaptive gains, where the tuning strategy is analytically built upon system stability and performance specifications, such that transient behavior and steady-state performance are ensured. Both square and nonsquare systems are addressed by using the matrix decomposition technique. The benefits and feasibility of the proposed control method are also validated and confirmed by the simulations.

Emerging robust and data‐driven control methods for uncertain learning systems

Emerging robust and data‐driven control methods for uncertain learning systems

Plantwide control design using latent variables: An integration between control allocation and a measurement combination approach

This paper presents a plantwide control design methodology based on a novel structure which consists of a decentralized strategy complemented by a control allocation (CA) module and a measurement combination (MC) block. Taking into account a principal components analysis (PCA) selection approach, the CA and MC modules perform a dimensional reduction of the original input–output variable space in order to obtain sets of latent variables (or principal components) as control actions and controlled variables. The use of principal components in the controller design provides several interesting features given that: (i) the conditioning of the subsystem to be controlled can be improved, (ii) when performing combinations of variables, the CA and MC modules act as steady-state decouplers and thus an apparently diagonal process is obtained, which favors the reduction of the variables interaction and the pairing problem is automatically solved, and (iii) they allow to naturally handle nonsquare systems. The proposed design procedure is implemented through a multiobjective bilevel mixed-integer nonlinear programming (BMINLP) optimization problem. The leader problem is based on the minimization of three functional costs: 1- the well-known sum of squared deviations (SSD) index, 2- the number of selected manipulated variables (actuators), and 3- the number of selected measurements (sensors). The inner optimization minimizes the relative gain array number (RGAN). This provides a good trade-off between the degree of conditioning/controllability and the complexity/cost of the resulting system. This problem is efficiently solved through genetic algorithms and allows to perform: (i) the selection of the manipulated variables (actuators) and the measurements (sensors) to be used, (ii) the computation of the matrices that characterize the CA and MC modules, and (iii) the stability analysis of the multivariable control structure. The overall design procedure only requires steady-state models of the process. The Tennessee Eastman case study is considered for the simulation and performance evaluation of the proposed solutions.

Robust Adaptive Learning-Based Path Tracking Control of Autonomous Vehicles Under Uncertain Driving Environments

This paper investigates the path tracking control problem of autonomous vehicles subject to modelling uncertainties and external disturbances. The problem is approached by employing a 2-degree of freedom vehicle model, which is reformulated into a newly defined parametric form with the system uncertainties being lumped into an unknown parametric vector. On top of the parametric system representation, a novel robust adaptive learning control (RALC) approach is then developed, which estimates the system uncertainties through iterative learning while treating the external disturbances by adopting a robust term. It is shown that the proposed approach is able to improve the lateral tracking performance gradually through learning from previous control experiences, despite only partial knowledge of the vehicle dynamics being available. It is noteworthy that a novel technique targeting at the non-square input distribution matrix is employed so as to deal with the under-actuation property of the vehicle dynamics, which extends the adaptive learning control theory from square systems to non-square systems. Moreover, the convergence properties of the RALC algorithm are analysed under the framework of Lyapunov-like theory by virtue of the composite energy function and the <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> -norm. The effectiveness of the proposed control scheme is verified by representative simulation examples and comparisons with existing methods.

A control philosophy for complex Non-square chemical process

All the complex non-square chemical processes may have thousands of measurement and control loops and these loops should be paired properly. A term control philosophy decides the structural decisions of the overall plant instead of the term plantwide control which is not meant for tuning each of these loops. Operational safety require direct inclusion of control system into associated safety to handle disturbances and failure in critical devices of a chemical process. Though most of the process units are non-linear and multivariable in nature, their operations are based on linear controllers to provide overall stability and safe operations. As the design of an appropriate control structure for complex non-square chemical processes are difficult, it is very challenging to control these processes. Non-square systems offers safe operational procedures and ensures plant safety. The proposed method of control configuration selection for non-square Multi Input Multi Output (MIMO) system is an extension of square MIMO system. An appropriate selection of control configuration for non-square chemical processes have been addressed in this paper. The proposed method of dynamic loop interactions is based on the computation and comparison of areas under responses for all possible combinations of manipulated and controlled variables. The control configurations that emerged from subsequent analyses are able to provide adequate inherent safety to the overall process in a closed-loop sense. This paper assesses the interaction among the loops having unequal number of input and output variables. This procedure is illustrated via simulation of examples taken for both the under-defined and over-defined non–square MIMO system. Also the quantitative measure of process safety index is obtained for the process.

Hybridization of Manta-Ray Foraging Optimization Algorithm with Pseudo Parameter-Based Genetic Algorithm for Dealing Optimization Problems and Unit Commitment Problem

The manta ray foraging optimization algorithm (MRFO) is one of the promised meta-heuristic optimization algorithms. However, it can stick to a local minimum, consuming iterations without reaching the optimum solution. So, this paper proposes a hybridization between MRFO, and the genetic algorithm (GA) based on a pseudo parameter; where the GA can help MRFO to escape from falling into the local minimum. It is called a pseudo genetic algorithm with manta-ray foraging optimization (PGA-MRFO). The proposed algorithm is not a classical hybridization between MRFO and GA, wherein the classical hybridization consumes time in the search process as each algorithm is applied to all system variables. In addition, the classical hybridization results in an extended search algorithm, especially in systems with many variables. The PGA-MRFO hybridizes the pseudo-parameter-based GA and the MRFO algorithm to produce a more efficient algorithm that combines the advantages of both algorithms without getting stuck in a local minimum or taking a long time in the calculations. The pseudo parameter enables the GA to be applied to a specific number of variables and not to all system variables leading to reduce the computation time and burden. Also, the proposed algorithm used an approximation for the gradient of the objective function, which leads to dispensing derivatives calculations. Besides, PGA-MRFO depends on the pseudo inverse of non-square matrices, which saves calculations time; where the dependence on the pseudo inverse gives the algorithm more flexibility to deal with square and non-square systems. The proposed algorithm will be tested on the test functions that the main MRFO failed to find their optimum solution to prove its capability and efficiency. In addition, it will be applied to solve the unit commitment (UC) problem as one of the vital power system problems to show the validity of the proposed algorithm in practical applications. Finally, several analyses will be applied to the proposed algorithm to illustrate its effectiveness and reliability.

A note on invertibility and relative degree of MIMO LTI systems

The property of system invertibility is important both from a theoretical standpoint and an important consideration in left and right inversion problems that deal with input estimation and tracking control respectively. Nuances associated with the notion of relative degree in multi-input, multi-output (MIMO) systems on account of input–output delays pose challenges in system inversion problems and are more acute in the case of non-square systems. In this note, using the LTI discrete-time MIMO systems framework, we review existing notions of relative degree and develop new definitions and generalizations for relative degree to help address some of the gaps in system inversion problems. In doing so, we also show through simple but elegant results that existence of well-defined relative degree for an LTI MIMO system implies its invertibility. Furthermore, we also highlight the utility of these ideas in practical tracking control and input estimation situations and problems.

Denominator Assignment, Invariants, and Canonical Forms Under Dynamic Feedback Compensation in Linear Multivariable Nonsquare Systems

In this article, we generalize previously reported results for linear, time-invariant, stabilizable multivariable systems described by a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">strictly</i> proper transfer function matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> with number of outputs greater than or equal to the number of inputs. By making use of a special kind of a left generalized inverse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)_{\alpha }^{\oplus }$</tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> , we define and examine the equivalent relation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {R}$</tex-math></inline-formula> relating <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> with the members of the equivalence class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[P(s)]_{R}$</tex-math></inline-formula> of the closed loop-transfer function matrices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{C}(s)$</tex-math></inline-formula> obtainable from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P(s)$</tex-math></inline-formula> by the use of a proper compensator <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C(s)$</tex-math></inline-formula> in the feedback path. For <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {R}$</tex-math></inline-formula> , we establish a set of complete invariants and a canonical form. These results give rise to a simple algorithmic procedure for the computation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">proper</i> internally stabilizing and denominator assigning compensators <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C(s)$</tex-math></inline-formula> for the class of plants with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p=m$</tex-math></inline-formula> and having no zeros in the closed right half complex plane: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {C}^{+}$</tex-math></inline-formula> and in the case when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$p&gt;m$</tex-math></inline-formula> plants characterized by right polynomial matrix fraction descriptions with a polynomial matrix numerator having at least one subset of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> rows that give rise to a nonsingular polynomial matrix with no zeros in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {C}^{+}$</tex-math></inline-formula> .

The Use of the Moore-Penrose Pseudoinverse for Evaluating the RGA of Non-Square Systems

A recently-derived alternative method for computing the relative gain array (RGA) for singular and/or non-square systems has been proposed, which provably guarantees unit invariance. This property is not offered by the conventional method that uses the Moore-Penrose (MP) pseudoinverse. In this paper, we note that the absence of the scale-invariance property by the conventional MP-RGA does not *necessarily* imply a practical disadvantage in real-world applications. In other words, while it is true that performance of a controller should not depend on the choice of units via its input and output variables, this does not necessarily imply that the resulting MP-RGA measures of component interaction lead to different controller-design input-output pairings. In this paper we consider the application of the MP-RGA to a realistic transfer function relating to a Sakai fractional distillation system. Specifically, for this transfer function we assess whether or not the choice of unit, which in this case relates to temperature, affects the choice of loop pairings implied by the resulting RGA matrix. Our results show that it does, thus confirming that unit-sensitivity of the MP-RGA undermines its rigorous use for MIMO controller design. Index Terms— Control systems, Moore-Penrose pseudoinverse, Process control, Relative Gain Array (RGA), UC inverse, Unit consistency.

Fault-tolerant control and interval operability for non-square faulty systems

Fault-tolerant control and interval operability for non-square faulty systems

Design of centralized controller for multivariable process using MOPSO algorithm

Objective: To estimate centralized PID controller parameters for 4 outputs and 5 inputs crude distillation non-square system with RHP zeros process. Methods/Analysis: The Multi- Objective Particle Swam Optimization (MOPSO) algorithm is applied to determine the PID controller parameters for the considered distillation column process. Findings: The performance of the proposed controller is compared with two centralized controller schemes, Davison’s and Tanttu and Lieslehto methods. The Integral Square Error (ISE), Integral Absolute Error (IAE) and Integral of Time Absolute Error (ITAE) are chosen as performance indices. The simulation results prove that MOPSO tuned centralized controller gives the best performance when compared to other analytical techniques for both set point tracking and in disturbance rejection environment. Novelty: In practice, conventional PID controllers are tuned using classical methods, which require complex numerical calculations. In this paper, an attempt is made to fine tune the PID controller for a MIMO process using Multi Objective optimization technique and obtained challenging results as compared to conventional methods. Keywords: Nonsquare system; Centralized control; Multi Objective Particle Swam Optimization; PID controller

Applying negative imaginary systems theory to non-square systems with polytopic uncertainty

This paper aims to apply Negative Imaginary (NI) systems theory to non-square LTI systems, for example, flexible structure systems with redundant sensors and actuators. The paper proposes static pre- and post-compensation schemes to transform stable fat (i.e. no. of inputs more than the no. of outputs) and tall (i.e. no. of outputs more than the no. of inputs) LTI plants into the class of Strongly Strict Negative Imaginary (SSNI) systems, a subset of the Strictly Negative Imaginary (SNI) systems. The proposed compensators can also stabilise a non-square or a non-NI square plant in a positive feedback loop. Moreover, the compensators can be utilised to design a simple constant reference tracking scheme exploiting the integral controllability propoerty of SSNI systems. Subsequently, observer-based pre- and post-compensators are developed to handle the cases when some of the system states are not available for direct measurement. Then, the compensation schemes are extended to stable, non-square or non-NI square systems with polytopic uncertainty. Finally, the post-compensation technique is applied to stabilise a class of tall/square uncertain LTI plants preceded by a slope-restricted nonlinearity.

Economic performance tracking for nonsquare MPCs based on a two‐layer approach

AbstractRange tracking for nonsquare systems is frequently adopted in process industries and its implementation is developed by model predictive control technologies. However, these approaches differ from those most considered in academia, with set‐point tracking and the same number of controlled and manipulated variables. In this scope, real‐time optimization (RTO) emerges as a diffused technology to improve the economic performance considering process, safety, and environmental constraints. In this work, a way to integrate economic aspects in industrial model predictive controllers (MPCs) by treating the nonlinear economic function as an output of the process model is proposed. The tracking error of the cost function is monitored, and its real value is estimated by a state estimator. The approach was applied to a nonsquare range system, exemplifying a continuous stirred‐tank reactor (CSTR) with Van de Vusse kinetics, and showed that it is capable of tracking the minimum cost operation robustly. The paper also compares the proposed strategy to the traditional RTO implementation, which provides optimized targets, and presents slight improvement in the steady‐state operation regarding the optimal cost seeking.

MPC model monitoring and diagnosis for non-square systems

Many industrial model predictive control applications, called non-square systems, have more variables to be controlled than manipulated variables available. At these cases, the control objectives are related to keep the controlled ones within a range, instead in reference value (setpoint). Assessing the model quality of these controllers is fundamental, however, most available MPC assessment methods are setpoint dependent, providing misleading results when these are unavailable. The methods proposed by Botelho et al. (2015, 2016a, 2016b) allow the model assessment of linear MPCs independently from the setpoints. Even though such methodologies have this advantage, there is not a procedure to assess this kind of controller. Therefore, this paper presents a guideline to apply Botelho et al. (2015, 2016a, 2016b) methods in non-square MPCs. The Shell Heavy Oil Process is used as a case study. The results demonstrate that the procedure is capable of estimate the effect of modeling problems and indicate the associated controlled variable, as well as whether the problem is due to a model-plant mismatch or unmeasured disturbance in non-square controllers.

Time domain modeling and control of complex non-linear chemical processes using relay feedback test

This study analyzes with the technology of auto tuning using relay feedback test for non-square MIMO system through process modeling, identification, input-output pairing and control strategies. However, the control configuration selection based on conventional steady-state Relative Gain Array (RGA) matrix sometimes degrades the loop performance and it needs attention. This study also deals with the real challenges in time domain modeling and appropriate pairing of loops for non-linear chemical processes. The choice of input-output pairing for square system has been extended to non-linear chemical processes. This ensures the system’s stability by selecting an appropriate manipulated-controlled variable pairing in non-linear chemical processes and the same is also tested for benchmark of non-linear chemical processes. In this study, two types of non-square systems are considered: one is systems with excess input than output variables and the other is systems with excess output than input variables. Some benchmark non-linear chemical processes, such as processes with mild nonlinearity, moderate to high nonlinearity and highly nonlinearity, are also taken for this study. Three standard benchmark processes are used in analysis of nonlinear chemical processes, namely: (1) Continuous Stirred Tank Reactor (CSTR) process; (2) hydrogen-ion- concentration (pH) process and (3) distillation column and this procedure is illustrated via simulation of 3-by-2 and 2-by-3 non-linear chemical processes.

A Modified Functional Observer-Based EID Estimator for Unknown Continuous-Time Singular Systems

This paper presents the design of a linear quadratic analog tracker (LQAT) based on the observer–Kalman-filter identification (OKID) method and the design of a modified functional observer-based equivalent input disturbance (EID) estimator for unknown square–non-square singular analog systems with unknown input and output disturbances. First, an equivalent mathematical model of the singular analog system is presented to simulate the time response of continuous-time linear singular analog systems to arbitrary inputs via the model conversion method. Then, for the unknown singular analog system, it constructs a linear quadratic analog tracker with state feedback and feed-forward gains based on the off-line OKID method. Furthermore, it extends the design methodology of the EID estimator for strictly proper regular systems with unknown matched–mismatched input and output disturbances to proper regular systems. It is important to mention that the newly developed modified functional observer for proper systems is used to estimate the unknown EID of singular analog systems and that the constraints on the dimensions of unknown disturbances can be eliminated by using the newly proposed EID estimation method. The contributions of this paper can be listed as follows: (1) based on both the OKID method and the discrete-to-continuous model conversion, the simulation of the time responses of the continuous-time linear singular models (which are not feasible using existing MATLAB toolboxes) become feasible; (2) for effective control of the unknown singular analog system, an off-line OKID method is proposed to design an LQAT with state feedback and feed-forward gains; and (3) based on the newly developed modified functional observer for the reduced-order proper regular system, the original EID estimator in the literature is newly extended to estimate the EID from the unknown strictly proper singular analog system, without the original dimensional constraints of the disturbances. It is important to mention that the disturbances of interest can be unknown matched–mismatched input and output disturbances.

Two-Layered Model Predictive Control Strategy of the Cut Tobacco Drying Process

In this article, a two-layered model predictive control strategy is proposed for the nonsquare system of nonlinear cut tobacco drying process. The control objective is to optimize the drum dryer temperature, hot air temperature, and cut tobacco outlet temperature meet the process constraints while meeting the moisture content of cut tobacco. Firstly, the tobacco drying process system was introduced, and the nonsquare system model and performance index function were established. Then a nonlinear moving horizon estimator (NMHE) and real-time optimization (RTO) are designed. NMHE provides state and parameter estimation for the controller, and RTO provides an optimal operating setpoint for the controller. Subsequently, a two-layered model predictive control (SSTO-MPC) design integrated with a steady-state target optimization layer (SSTO) is proposed for the nonsquare system of nonlinear cut tobacco drying process. Extensive simulations under different scenarios illustrate the effectiveness of the proposed SSTO-MPC design compared with the conventional MPC.

Sensor blending and Control allocation for non-square linear systems to achieve negative imaginary dynamics

This paper deals with the design of static pre- and post- compensators to transform stable, non-square LTI systems into the class of strongly strict negative imaginary systems. The pre-compensator plays the role of a control allocator while the post-compensator does sensor blending in order to make a non-square system square along with satisfying the strongly strict negative imaginary property. A specific structure of the post-compensator is also given that guarantees a feasible solution of the LMI conditions when applied to systems with number of outputs greater than or equal to number of inputs. The proposed pre- and post- compensators can also stabilize a non-square plant in closed-loop upon satisfying a particular DC-gain condition and furthermore, they can be utilized to develop a simple constant input tracking framework for non-square systems. The LMI-based design methodology offers a numerically tractable solution framework and hence the easy implementation of the proposed scheme in practical applications. Illustrative examples are provided throughout the paper to demonstrate the usefulness of the proposed results in widening the scope of the negative imaginary theory to non-square LTI systems (e.g. safety-critical systems having redundant sensors and actuators).