Osita D. I. Nwokah

Algebraic and topological aspects of quantitative feedback theory

The current interest in robust control has called into question the applicability of the Quantitative Feedback Theory (QFT) robust design method introduced by Horowitz. A number of issues have been raised regarding inherent restrictions of both the design method and the uncertain plant set. Using a multivariable root locus technique extended to uncertain systems, this paper shows that the QFT assumptions are indeed not restrictive and are in fact equivalent to other well-known conditions for robust stabilisability. Because QFT is one of the very few methods to specifically address the quantitative robust performance issue, these results should lead to better methods of developing new QFT-design software as well as improved robust control methods to satisfy a priori quantitative performance bounds.

Parametric robust control by quantitative feedback theory

The problem of performance robustness, especially in the face of significant parametric uncertainty, has been increasingly recognized as a predominant issue of engineering significance in many design applications. Quantitative feedback theory (QFT) is very effective for dealing with this class of problems even when there exist hard constraints on closed loop response. In this paper, SISO-QFT is viewed formally as a sensitivity constrained multi objective optimization problem which can be used to set up a constrained H¿ minimization problem whose solution provides an initial guess at the QFT solution. In contrast to the more recent robust control methods where phase uncertainty information is often neglected, the direct use of parametric uncertainty and phase information in QFT results in a significant reduction in the cost of feedback. An example involving a standard problem is included for completeness.

On multivariable stability in the gain space

Abstract General existence conditions under which the stability of the individual loops of a multivariable system in a given compact and bounded gain space imply the global asymptotic stability of the multivariable system in the same space are very useful in practice and essentially indicate when in principle it is possible to obtain stable closed-loop performance of a multivariable system by tuning every loop separately. Such conditions are particularly useful for fault tolerant control of multivariable systems. This paper gives the required necessary and sufficient conditions and effectively unifies all previous work on diagonal stabilizability which includes as special cases: the diagonal dominance, the H matrix, the GKK matrix and the decentralized integral controllability (DIC) conditions reported previously in the literature. Some application examples are included.

Strong robustness in uncertain multivariable systems

A review is presented of the basis of the Horowitz quantitative feedback theory (QFT). It is shown that the basic assumptions of (QFT) have mathematical justification. However, necessary and sufficient conditions for the solvability of the QFT problem are presently not known. By reposing the QFT problem as an M-matrix design problem, it is possible to develop the desired existence conditions as well as simplify the design of the controllers which solve the QFT problem. Some preliminary results are presented and form the basis of ongoing work into the automation of the QFT design process.<<ETX>>

Multivariable decentralized integral controllability

Abstract General existence conditions under which the stability of the individual loops of a multivariable system in a given compact and bounded gain space imply the global asymptotic stability of the multivariable system in the same space are very useful in practice and essentially indicate when in principle it is possible to obtain stable closed loop performance of a multivariable system by tuning every loop separately. Such conditions are particularly useful for fault tolerant control of multivariable systems. This paper gives the required necessary and sufficient conditions but proceeds from there to derive the stronger existence conditions for stability in the gain space when every loop contains an integrator. Such conditions are called decentralized integral controllability and are needed to guarantee not only robust stability in the gain space but also zero steady state performance errors in the same space, thus assuring system fault tolerance.

A note on decentralized integral controllability

A concept of decentralized integral controllability (DIC) defined on a given gain space ϕ is clarified and related to the original definition given by Morari and Zafirou (1989). This leads to a simple proof of the existence of DIC on ϕ from which can be routinely deduced existence conditions for DIC in the sense of Morari and Zafirou.

Frequency Response Specifications and Sensitivity Functions in Quantitative Feedback Theory

Within the scope of control of uncertain systems, the problem of performance robustness, especially in the face of parametric uncertainty, has been increasingly recognized as a predominant issue of engineering significance in many design applications. Quantitative Feedback Theory (QFT), a frequency response-based method introduced by Horowitz, has been shown advantageous in many cases where performance specifications for such systems, in terms of hard constraints on closed loop response, are to be met. In this paper, these traditional QFT design criteria are contrasted with a relaxed, sensitivity-based formulation for single input single output (SISO) feedback systems. The advantage of the latter is a greater degree of mathematical commonality with alternative frequency domain methods, thus laying the groundwork for future benchmark studies in control design. The methodology is demonstrated by application to a lateral autopilot design problem for the C-135 aircraft, both to the traditional QFT design specifications as well as to the relaxed sensitivity based criterion.

The Benchmark Problem Solution by Quantitative Feedback Theory

In this paper we employ quantitative feedback theory (QFT) to synthesize compensators, satisfying various design specifications stipulated in the benchmark problem (Wie and Bernstein [1990]). Minimizing, the cost of feedback or the amount of controller bandwidth is the main objective in QFT. It emphasizes the fact that feedback is only necessary because of uncertainty and that the amount of feedback should therefore be directly related to the extent of plant uncertainty and external disturbances. QFT, quantitatively formulates these two factors in the form of (a) sets TR = {TR} of acceptable command or tracking input-output relations and TD = {TD} of acceptable disturbance input-output relations, and (b) a set P = { P } of possible plants. The control objective is to guarantee that the closed loop transfer function TR = Y/R is a member of TR and TD = Y/D is a member of TD, for all P in P.

On multivariable stability in the grain space

General existence conditions under which the stability of the individual loops of a multivariable system in a given compact and bounded gain space imply the global asymptotic stability of the multivariable system in the same space are very useful in practice, and they essentially indicate when in principle it is possible to obtain stable closed loop performance of a multivariable system by tuning every loop separately. Such conditions are particularly useful for fault tolerant control of multivariable systems. The authors give the required necessary and sufficient conditions and effectively unify all previous work on diagonal stabilizability which includes as special cases the diagonal dominance, the H-matrix, the GKK-matrix, and the decentralized integral controllability (DIC) conditions reported previously in the literature. Some application examples are included.<<ETX>>

Pseudo-Derivative Feedback Control

The extensive presence of simple PID controllers all over the process and manufacturing industries attests to their continuing and lasting effectiveness in dealing with all but the most complex and demanding industrial control problem. Problems do arise with PID controllers however. In particular high frequency sensor noise problems can become severe in some applications, due to the presense of the D component of the PID controller. By altering the controller structure slightly, it is possible to obtain the attendant benefits of derivative action, without taking the derivative of the error function. This is called pseudo-derivative feedback and overcomes many of the major problems of derivative control. The structure and ramifications of this scheme is examined here, for both multivariable and single input single output systems.