The output control of linear time-invariant multivariable systems with unmeasurable arbitrary disturbances

Necessary and sufficient conditions are derived for a minimal order linear time-invariant differential feedback control system to exist for a linear time-invariant multivariable system with unmeasurable arbitrary disturbances of a given class occurring in it, such that the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs, independent of the disturbances occurring in the system, and such that the closed-loop system is controllable. The feedback gains of the control system are obtained so that the dynamic behavior of the closed-loop system is specified by using either an integral quadratic optimal control approach or a pole assignment approach. The result may be interpreted as being a generalization of the single-input, single-output servomechanism problem to multivariable systems or as being a solution to the asymptotic decoupling problem.     

A generalization of the output control of linear multivariable systems with unmeasurable arbitrary disturbances

The results of a previous paper [1] on the regulation and tracking of a servomechanism problem are extended to include control inputs on the outputs of the system and for the case that the state is not available for measurement.     

The feedforward control of linear multivariable time-invariant systems

Necessary and sufficient conditions are derived for a stable feedforward control system to exist for a multivariable linear time-invariant system so that: (1) any measurable disturbances occurring in the system are asymptotically rejected, and (2) the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs. The result may be interpretated as being a generalization of the single-input, single-output feedforward servo problem to multivariable systems or as being a solution to the asymptotic decoupling problem. Some numerical examples varying from 2nd to 9th order are included to illustrate the result.     

Pole assignment in linear time-invariant multivariable systems with constant disturbances

Necessary and sufficient conditions are derived for a minimal order realizable, linear, time invariant differential feedback control system to exist for a linear time-invariant system with constant unknown disturbances, such that the eigenvalues of the closed-loop system take on pre-assigned values in the left hand part of the complex plane and such that the outputs of the system tend to zero as t → ∞. Some numerical examples are included to illustrate the result.     

The systematic design of control systems for large multivariable linear time-invariant systems

A systematic design procedure for determining realistic feedforward-feedback control systems for large multivariable linear constant systems is outlined in this paper. A numerical example of a boiler system and a distillation column is included and a comparison of the simulated response of the controlled systems obtained by the proposed method is made with the corresponding conventional control systems. The new control systems obtained are relatively simple, are realistic to implement, and appear to be a significant improvement over the conventional ones.     

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

Necessary and sufficient conditions are found for there to exist a robust controller for a linear, time-invariant, multivariable system (plant) so that asymptotic tracking/regulation occurs independent of input disturbances and arbitrary perturbations in the plant parameters of the system. In this problem, the class of plant parameter perturbations allowed is quite large; in particular, any perturbations in the plant data are allowed as long as the resultant closed-loop system remains stable. A complete characterization of all such robust controllers is made. It is shown that any robust controller must consist of two devices 1) a servocompensator and 2) a stabilizing compensator. The servocompensator is a feedback compensator with error input consisting of a number of unstable subsystems (equal to the number of outputs to be regulated) with identical dynamics which depend on the disturbances and reference inputs to the system. The sorvocompensator is a compensator in its own right, quite distinct from an observer and corresponds to a generalization of the integral controller of classical control theory. The sole purpose of the stabilizing compensator is to stabilize the resultant system obtained by applying the servocompensator to the plant. It is shown that there exists a robust controller for "almost all" systems provided that the number of independent plant inputs is not less than the number of independent plant outputs to be regulated, and that the outputs to be regulated are contained in the measurable outputs of the system; if either of these two conditions is not satisfied, there exists no robust controller for the system.     

Interval observers for linear time-invariant systems with disturbances

It is shown that, for any time-invariant exponentially stable linear system with additive disturbances, time-varying exponentially stable interval observers can be constructed. The technique of construction relies on the Jordan canonical form that any real matrix admits and on time-varying changes of coordinates for elementary Jordan blocks which lead to cooperative linear systems. The approach is applied to detectable linear systems.     

On transmission zeros and zero directions of multivariable time-invariant linear systems with input-derivative control

The purpose of this paper is to investigate the problem of finding inputs which generate zero outputs in linear systems whose state equation contains the derivative of the input. The method developed makes use of the simplest matrix generalized inverse, the {1}-inverse.     

Integrity against arbitrary feedback-loop failure in linear multivariable control systems

This paper is concerned with the integrity problem in linear multivariable control systems. System integrity is defined as the property whereby the closed-loop system remains stable against arbitrary feedback-loop failures.^§ Introducing a new class of matrices, called the U-matrix, necessary and sufficient conditions for integrity are established with a stable plant. The results show that integrity is ensured if and only if all principal minors of the return difference matrix or the sensitivity matrix are units, with the controller being stable. Some useful sufficient conditions for integrity are also derived by utilizing interesting properties of the U-matrix. In particular, many previously known results concerning integrity are unified and several are extended. Moreover, the set of all proper controllers ensuring integrity can be characterized with the U-matrix. Based on this characterization, a procedure for synthesizing a controller which ensures integrity, and attains desirable input-output responses with decoupling, is proposed.     

Norm-bounded integrity conditions of uncertain multivariable linear time-invariant systems under decentralized control with integral action

Decentralized controllers with integral action represent a large portion of multivariable controllers currently employed in the process industry. The integrity of these systems with respect to loop failures has been relatively well researched. In particular, important results have been found with respect to the relative gain array (RGA). However, less has been done on the treatment of the integrity of uncertain systems. This paper considers the integrity conditions of uncertain systems with norm-bounded uncertainties, and extends four existing integrity conditions to these uncertain systems. These are: integral stabilizable (IS), integral controllable (IC), integral controllable with integrity (ICI), and decentralized integral controllability (DIC). For the first three, necessary and sufficient conditions are derived. For the DIC condition, a sufficient condition has been derived. These conditions impose a limit on the norm of the allowed perturbations, below which the closed-loop system is guaranteed to maintain integrity.     

Stabilization of linear time-invariant systems with periodic communication constraints by output sample hold control

This article considers feedback control systems wherein the control loops are closed through a real-time network, and expresses the linear time-invariant system with the constraint in an input or output as a periodic discrete time system. It is shown that this system is stabilized by using output sample hold contol. This method has the merit that the capacity of a sensor-controller communication bus is small. This work was presented, in part, at the 8th International Symposium on Artificial Life and Robotics, Oita, Japan, January 24–26, 2003     

A differential representation for multivariable linear systems with disturbances

It is shown how to compute a differential representation for a multivariable linear system with disturbancesdot{x}(t)=Ax(t)+Bu(t)+ W_{x}w(t)y(t)=Cx(t)+ Eu(t)+ W_{y}w(t). Explicit formulas forM_{y}(D)andM_{z}(D)in a differential equivalent representationP(D)z(t)=u(t)+M_{z}(D)w(t)y(t)=R(D)z(t)+M_{y}(D)w(t)are presented in this paper.     

Feedback control for linear time-invariant systems with state andcontrol bounds in the presence of disturbances

The problem of the stabilizing linear control synthesis in the presence of state and input bounds for systems with additive unknown disturbances is considered. The only information required about the disturbances is a finite convex polyhedral bound. Discrete- and continuous-time systems are considered. The property of positive D -invariance of a region is introduced, and it is proved that a solution of the problem is achieved by the selection of a polyhedral set S and the computation of a feedback matrix K such that S is positively D-invariant for the closed-loop system. It is shown that if polyhedral sets are considered, the solution involves simple linear programming algorithms. However, the procedure suggested requires a great amount of computational work offline if the state-space dimension is large, because the feedback matrix K is obtained as a solution of a large set of linear inequalities. All of the vertices of S are required     

Computation of invariant zeros of linear,time-invariant,multivariable systems

In this paper a numerical method is presented for computing the invariant zeros of a controllable linear, time-invariant, multivariable system described by the 4-tuplo (A, B, C, D) or the triple (A, B, C). The method is based on the fact. that a controllable system can be made maximally unobservable by means of state variable feedback, thereby causing the cancellation of the invariant zeros by an equal number of the system poles. The invariant zeros are obtained as the eigenvalues of a matrix of the same dimension as the number of invariant zeros. The method is applicable to both multivariable as well as single-input, single-output systems. Examples are given to illustrate the use of the method.     

Design of multivariable linear time-varying tracking systems with stochastic disturbances

A new designing method of multivariable linear time-varying tracking systems with stochastic disturbances for polynomial command inputs is proposed. It has been shown that it is possible to choose the matrices of gain elements placed in feedbacks and parallel paths of the closed-loop system so that the output y(t) tracks the polynomial command input v(t) with (E expectation operator).     

Stabilization of linear multivariable systems by output feedback

Absfract-A method is developed for improving the stability of linear multivariable systems using output feedback. The technique, which utilizes a gradient approach, has been mechanized in a digital computer program. Illustrative results are given for a seven-state two-feedback model of the Saturn V booster.