Non-square Systems Research Articles

Helicopter translational rate command using individual channel analysis and design

Translational rate command (TRC) in helicopter flight involves controlling the earth-referenced forward and side velocities, using the helicopter’s cyclic control. Handling qualities requirements for TRC systems suggest that the design should comprise an inner attitude command-attitude hold (ACAH) loop, with TRC control as an outer loop. This forms a non-square system, and a method to determine the implications of this structure for stability robustness is proposed here. Control synthesis was performed using the method of individual channel analysis and design. This technique is a neo-classical, frequency-domain control analysis and design method for multivariable systems. The control law obtained is very simple, and the handling qualities, assessed by non-linear simulation, are predicted to be at ‘Level 1’ (satisfactory without improvement). The conditions for stability robustness are satisfied at frequencies of significance for the handling qualities.

Simultaneous uniform disturbance rejection and decoupling for nonsquare linear time-dependent analytic systems using restricted state feedback

In this paper, we consider, for the first time, the combined problem of uniform disturbance rejection and decoupling of nonsquare linear time-dependent analytic systems via restricted static state feedback. The problem is treated on the basis of an algebraic approach, whose main feature is that it reduces the problem to that of solving a homogeneous algebraic system of equations. On the basis of this fundamental system of equations, necessary and sufficient conditions of a simple algebraic nature are derived for the existence of a restricted static state-feedback controller that rejects the disturbances and simultaneously decouples the closed-loop system. Furthermore, simple expressions for a special admissible controller pair are derived analytically.

ROBUST DECENTRALIZED CONTROL OF NON-SQUARE SYSTEMS

The independent design procedure for robust decentralized controllers proposed by Hovd and Skogestad (1994) is extended to non-square systems with more inputs than outputs. The proposed design method is applied to a non-square mixing tank example. The results further justify that non-square systems should be controlled in thier original non-square form instead of squaring them by adding or eliminating variables.

Disturbance Rejection With Simultaneous Input-Output Linearization and Decoupling Via Restricted State Feedback

The disturbance rejection with simultaneous input-output linearization and decoupling problem of nonsquare nonlinear systems via restricted state feedback is investigated in this paper. The problem is treated on the basis of an algebraic approach whose main feature is that it reduces the determination of the admissible state feedback control laws to the solution of an algebraic and a first order partial differential systems of equations. Verifiable necessary and sufficient conditions of algebraic nature based on these systems of equations are established for the solvability of the aforementioned problem. Moreover, an explicit expression for a special admissible restricted state feedback controller is analytically derived. [S0022-0434(00)02101-8]

A Multivariate Self-Tuning Controller for Run-To-Run Process Control under Shift and Trend Disturbances

Many manufacturing systems are controlled with PID-type controllers. In some industries, such as semiconductor manufacturing, specifications or changing conditions impose a need for adjusting such controllers on a run-to-run basis. This need has originated a collection of techniques called run-to-run process control. Self-tuning control, where model estimation is done on-line, has been shown to be a feasible tool for run-to-run control in single input–single output systems. This paper presents a self-tuning multiple input-multiple output controller for run to run control. The controller compensates for a variety of disturbances frequently found in semiconductor manufacturing, such as offsets or shifts and trends. The controller also compensates for autocorrelated responses or noise terms, for coupled responses, and for non-square systems (i.e., the number of inputs may be greater than the number of outputs). A sensitivity analysis is presented to show the performance of the controller under various system...

Positivity embedding for noncolocated and nonsquare flexible systems

Positivity based control design for flexible structures provides closed-loop stability regardless of parameter variations and unmodeled dynamics. The present framework requires the plant to be square and the actuators colocated with rate sensors. These constraints severely limit achievable performance in control systems using the positivity approach. In this paper, a dynamic embedding is derived to render a nonsquare plant with noncolocated actuators and sensors positive real. The dynamic embedding is parametrized for general flexible structures. A numerical algorithm is developed to compute the embedding parameters. The Draper tetrahedral truss structure is used to demonstrate the design of dynamic embeddings. Relaxing the colocation constraint in a positivity based design significantly improves the closed-loop performance.

Perfect output control for nonsquare generalised state space systems

The problem of perfect output control, via proportional restricted state feedback, is studied for the case of nonsquare generalised state space systems. The necessary and sufficient conditions for the problem to have a solution are established. Furthermore, the necessary and sufficient conditions for the solution of the problem of command matching are derived, using proportional restricted state and output feedback.

Multivariable system simplification using moment matching and optimisation

A generalised formulation of the moment matching problem, which encompasses previous formulations and is applicable to non-square continuous and discrete systems, is presented. The formulation entails points about which moments can be matched. These points can be specified either by trial and error or by means of optimisation, so that the reduced model has desirable properties such as stability and accuracy. Novel properties of Pade approximants for non-square systems are elucidated, and details of the approach are illustrated by means of examples.

Model Accuracy for Economic Optimizing Controllers: The Bias Update Case

In many advanced control applications the numbers of manipulated and controlled variables are not the same. A combination of steady-state economic optimization and model predictive control has been employed to exploit the optimization opportunities presented by such non-square systems. The economic optimization subsystem uses a process model and is usually coupled with a model updating scheme to compensate for plant/model mismatch. Care must be exercised during the design of such a system to ensure the model-based optimization yields operating conditions which are optimal for the true plant. We present a rigorous criterion for determining under what conditions the model-based optimization embedded within the model predictive controller, using bias update, is capable of finding the plant optimum despite plant/model mismatch. The model accuracy criterion can be applied by solving an appropriate nonlinear programming problem. Further, it is shown that when the embedded model-based optimization problem has linear constraints, the bias update method will ensure convergence to the plant optimum, providing the model accuracy criterion is satisfied and an appropriate filter is included in the feedback path. Discussions are concluded with a demonstration of the methods on a real-time gasoline blending control and optimization problem

State estimation based model predictive control applied to shell control problem: a case study

In this paper, we demonstrate how a practical control problem with multiple control/optimization objectives and various operating constraints is formulated in the theoretical framework of state estimation based model predictive control (SEMPC) proposed by Lee . We use the shell control problem (SCP) of a heavy oil fractionator as case study. The shell control problem embodies most of the critical elements of challenging industrial process control problems (e.g. unmeasured disturbances, model uncertainty, input/output constraints, optimization objective conflicting with control requirements, failure-prone sensors, secondary measurements, nonsquare system, etc.) and therefore serves as a good test problem for investigation of potential benefits and pitfalls of the new technique. We demonstrate in the case study that, while the theory for the new model predictive control (MPC) technique is rigorously laid out, it is nontrivial for practicing engineers to formulate various practical objectives correctly within the theoretical framework to realize all the potential performance improvements of SEMPC over conventional MPC. By formulating and analyzing a series of different SEMPC controller designs for SCP, this paper highlights some of the possible difficulties that engineers may encounter in applying SEMPC to practical control problems and shows how these difficulties are overcome most efficiently.

Model predictive control with linear models

AbstractThis article discusses the existing linear model predictive control concepts in a unified theoretical framework based on a stabilizing, infinite horizon, linear quadratic regulator. In order to represent unstable as well as stable multivariable systems, the standard state‐space formulation is used for the plant model. The incorporation of a nominally stabilizing constrained regulator eliminates the current requirement of tuning for nominal stability. Output feedback is addressed in the well‐established framework of the linear quadratic state‐estimation problem. This framework allows the flexibility to handle nonsquare systems, noisy inputs and outputs, and nonzero input, output, and state disturbances. This formulation subsumes the integral control schemes designed to remove steady‐state offset currently in industrial use. The online implementation of the controller requires the solution of a standard quadratic program that is no more computationally intensive than existing algorithms.

Singular-value-decomposition approach to multivariable generalised predictive control

A change of basis, from the standard set to the set of eigenvectors, provides the means for the decomposition of a multivariable problem into a set of scalar problems. This idea was deployed in an earlier paper to embed scalar generalised predictive control into the multivariable framework. Eigen-decompositions, however, can be sensitive to perturbations and cannot be applied to nonsquare matrices. The paper shows how an analoguous approach to multivariable predictive control can be based on a singular-value decomposition, and illustrates its applicability to nonsquare systems as well as demonstrates its superior sensitivity properties by means of two numerical examples.

Chemometric methods for process monitoring and high‐performance controller design

AbstractA huge amount of data is collected by computer monitoring systems in the chemical process industry. Such tools as principal component analysis and partial least squares have been shown to be very effective in compressing this large volume of noisy correlated data into a subspace of much lower dimension than the original data set. Because most of what is eliminated is the collinearity of the original variables and the noise, the bulk of the information contained in the original data set is retained. The resulting low dimensional representation of the data set has been shown to be of great utility for process analysis and monitoring, as well as in selecting variables for control. These types of models can also be used directly in control system design. One way of approaching this is to use the loading matrices as compensators on the plant. Some advantages of using this approach as part of the overall control system design include automatic decoupling and efficient loop pairing, as well as natural handling of nonsquare systems and poorly conditioned systems.

Computation of Zeros of Generalized State-Space Systems

Abstract The paper presents an efficient and numerically stable algorithm for the computation of finite zeros of generalized state-space systems. The proposed algorithm reduces the system matrix pencil to a regular pencil whose finite generalized eigenvalues are the finite zeros of the system. The reduction is performed using unitary (in complex case) or orthogonal (in real case) pencil similarity transformations. The method is general, it can handle both square and non-square systems, degene-rate systems as well as singular systems. It exploits efficiently the particular structure of the system matrix pencil. The proposed algorithm can be viewed as an extension of a similar algorithm for stan-dard state-space systems. Numerical properties and algorithmic details are discussed. Numerical examples are given for illustrating the performances of the algorithm.

An approach to microwave imaging using a multiview moment method solution for a two-dimensional infinite cylinder

An approach based on a multiview solution to the inverse-scattering problem of a two-dimensional infinite cylinder is developed in a space-frequency domain. Microwave imaging is simulated by a computer algorithm using the moment method. To overcome ill-conditioning and solve nonsquare systems, a pseudoinverse transformation is employed. The equivalent current density and the complex conductivity are considered as object functions for image formation. The results of some numerical simulations in a noisy environment are reported. and a discussion of monoview and multiview imaging techniques for a space-frequency domain is presented. >

New algorithm for non-square internal model control systems

A new algorithm for non-square internal model control systems based on the pole placement approach has been developed. The algorithm can handle any n-input and m-output minimum- or non-minimum-phase systems. The process factorization and inversion are combined into the derivation of the control law so that these procedures can be avoided. A tuning gain is introduced in the algorithm to improve the performance, and can be determined in a stable region by an optimization technique. The offset is eliminated through the integral action of the errors in the algorithm. Moreover, the overall characteristic behaviour of the control system consists of the feedforward and feedback control behaviour so that it has both tracking and disturbance rejection abilities. Comparisons with classical internal model control (IMC) and simplified model predictive control design for m × m square systems are also discussed.

Multivariable controller design based on shrinking process poles

A systematic method based on the pole placement approach is proposed to design non-square multivariable internal model control systems. The control law is a kind of the classical IMC controller for square MIMO systems, but process model factorization and inversion are combined into the derivation of the approach. The control law can be extended to handle any minimum or non-minimum phase n input and m output systems. The absolute value of the implemented tuning parameters is restricted to (0,1) to guarantee internal stability of the control system. If all the parameters are set equal to one, the resulting closed-loop responses follow the conservative behaviour in simplified model predictive control. Alternatively, if the parameters are shrunk to zero, the proposed control law becomes multivariable deadbeat control.

The relative gain for non-square multivariable systems

Generally speaking, chemical processes in nature are non-square systems with unequal numbers of inputs and outputs. However, only limited tools, e.g. singular-value decomposition (SVD), are capable of analyzing non-square multivariable systems directly. In this paper, Bristol's relative gain array (RGA) is extended to non-square multivariable systems. The non-square relative gain array (NRG) is defined as the ratio of the open-loop gain, when all loops are without any control, to the closed-loop gain, when all loops other than the loop explored are under perfect control in the least-square sense. Properties of NRG are rigorously derived. Similar to the square RGA, the NRG can be used to assess the performance of non-square control systems based on steady-state information. Furthermore, the NRG can be served as a criterion to choose a square subsystem from a non-square system if a square control system is preferred. Two distillation examples are presented to illustrate the use of the NRG.

STEADY-STATE TWO-DIMENSIONAL INVERSE HEAT CONDUCTION

A method for analysis of multidimensional steady-state inverse heat conduction problems is presented. The method employs an adjoint formulation to approximate a set of sensitivity coefficients that relate temperature and heat flux observations to unknown surface conditions. The resulting nonsquare system of equations is regularized and subsequently solved in a least-squares sense. A technique is presented for evaluating the accuracy of the estimated surface conditions in terms of resolution and variance. The method is applied to example problems in two dimensions for cases in which limited information about the surface condition is available.

Interaction measures for nonsquare decentralized control structures

AbstractA block relative gain measure for nonsquare systems has been derived. Evaluated at steady state, this measure serves as a performance‐related tool for evaluating control structures prior to controller design. Both the nonsquare dynamic block relative gain and the relative sensitivities have been presented as dynamic interaction measures that depend on the controller tuning and are consequently applicable to design of nonsquare decentralized controllers. Rigorous relationships between the dynamic block relative gain and the performance and stability of a closed‐loop system have been established.