Disturbance rejection in multivariable systems by state feedback controller

In this paper, we analyse some fundamental structural properties of linear time-invariant multivariable systems in the controller canonical form and present a direct method for the computation of bases and associated friends for output-nulling, input-containing and reachability subspaces in terms of the parameters of the system and the invariant zero structure, both in the nondefective and in the defective case. Using this analysis, it is possible to express the solvability conditions of important control and estimation problems in terms of easily checkable conditions on the system matrices.

The root-locus-approximation procedure is extended to provide for the complex-plane analysis of equations which contain a term quadratic in the variable parameter in addition to the usual linear term. Such an extension will have many applications; in the paper, a discussion is given concerning the design of linear time-invariant multivariable feedback systems. It is shown that, particularly for 2-input/2-output problems, where the algebra involved in obtaining the equations to be analysed remains fairly simple, much useful information is available by straightforward hand calculation. Brief suggestions are given for further generalisations of the techniques (inclusion of a cubic gain term etc.)

In this paper, Singular Perturbation Margin (SPM) and Generalized Gain Margin (GGM) are used as the classical Phase Margin (PM) and Gain Margin (GM) like stability metrics for multivariable Linear Time-Invariant (LTI) systems established from the view of the singular perturbation and the regular perturbation, respectively. The bijective relationship between the SPM and PM of a multivariable LTI system is established as corresponding lemmas and theorems, which is a generalisation of the single input single output cases. The analysis process for SPM and GGM are provided here. All the theoretical results are shown by a two-input-two-output LTI system (a simple model of a two-body spacecraft with non-collocated sensors and actuators using position feedback) as an application example. Such results are one essential step towards the big map of establishing a set of both theoretically rigorous and physically meaningful stability margin metrics for NLTV systems.

A numerical technique for achieving multiple design constraints for linear time-invariant multivariable control systems is presented. The technique uses singular value gradient information to improve the singular value frequency responses of system transfer matrices. An example is given using the technique for improvement of disturbance rejection and certain measures of stability robustness for a large space structure subsystem. >

It is shown that the problem of pole assignment by gain output feedback in linear time-invariant multivariable systems may be reduced to an algebraic optimisation problem, using the concept of virtual model following. The method requires no transformation of the open-loop system to a canonical form. A number of related problems involving combinations of pole placement with a secondary design objective are also discussed.

Model reduction techniques are simplification methods based on mathematical approaches employed to realize reduced models for the original high order systems. Some existing classical model reduction techniques for multivariable system are considered and compared for their performances. Interlacing property and coefficients matching (IPCM) method gives overall minimum integral square error (ISE), integral absolute error (IAE) and integral time absolute error (ITAE) values compared to other methods. Though the IPCM method is efficient, it may not guarantee for minimization of all objective functions simultaneously. In this paper, model reduction approach based on objectives like ISE, IAE and ITAE using multi-objective differential evolution (MODE) method is proposed for reducing the numerator and the denominator is reduced by interlacing property. MODE method minimizes the small, normal and large errors persisting for long time between original and reduced models. This multi-objective approach is applied for model reduction of 10th order multivariable linear time invariant power system model. Simulation results are demonstrated for single and multi-objective model reduction and compared with multi-objective particle swarm optimization (MOPSO) method to prove the validity of proposed MODE technique.

A simple algebraic approach is used to derive necessary and sufficient conditions for the existence of a state feedback matrix to localise disturbances and which some of the closed-loop-system eigenvectors must satisfy for disturbance localisation. This latter result provides valuable insight as to how the state feedback matrix may be synthesised.

Given the state space realization <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{A,B,C\}</tex> of a linear time-invariant multivariable dynamic system, a computational algorithm for calculating elements of the corresponding transfer function matrix is presented. The algorithm neither requires cofactor calculation, nor applies any state transformation. The only requirement is calculation of the coefficients of characteristic equations of given matrices. Also, the Fortran subroutine based on the presented algorithm is included.

Robust controller design methodology for multivariable chemical processes: Structured perturbations

Addendum to "The robust control of a servomechanism problem for linear time-invariant multivariable systems"

Uncontrollability of the tandem connection and of the feedback connection of two linear time-invariant multivariable systems, each having the Jordan canonical form representation, is considered. Sufficient conditions for uncontrollability of those systems are given.

This paper is concerned with the nonlinear static state feedback stabilization of time-invariant linear multivariable continuous-time systems with friction driven backlash in the control inputs. A time-invariant static nonlinear state feedback compensator is proposed, easily obtainable solving an algebraic Riccati equation. The global asymptotic stability of the resulting closed-loop nonlinear system is proved using the Lyapunov's direct method and the La Salle's invariance principle. A simulation study shows that the proposed nonlinear state feedback compensator achieves an overall stabilization performance significantly better than the known backlash inverse compensator.

This article provides a framework for robust performance analysis of linear time-invariant uncertain multivariable systems obtained by classical system identification methods. The performance measures are in terms of H 2 or H ∞ norm of a closed-loop transfer matrix. An upper bound for the performance analysis criterion is computed via an LMI-based optimisation problem. The LFT description is used as a tool for uncertainty modelling. The proposed performance conditions are derived based on using the parameter-dependent Lyapunov functions and are deduced via a parametrisation for the set of multipliers corresponding to the ellipsoidal uncertainty set delivered by system identification procedure. The effectiveness of the proposed analysis approach is demonstrated by a numerical example.

Adaptive control of linear multivariable systems using dynamic regressor extension and mixing estimators: Removing the high-frequency gain assumptions

In this paper, we examine the problems of existence and computation of proper and stable stabilising compensators in the feedback path of a special class of linear time-invariant multivariable systems characterised by a strictly proper transfer function matrix. For the class of systems that are ‘square’, i.e. have the same number of inputs and outputs and have no zeros in the closed right half complex plane and the class of ‘non-square’ systems that include a ‘square’ subsystem that has no zeros in the closed right half complex plane we present a sufficient condition for the existence and computation of proper stabilising (pole placing) compensators which in certain cases that depend on the transfer function matrix of the system can be chosen to be stable.