Yongdong Zhao

Feedforward controllers and tracking accuracy in the presence of plant uncertainties

In the absence of plant parameter uncertainty feedforward controllers can be synthesized to achieve perfect continuous tracking. When plant has uncertainties it is, in general, impossible to achieve such perfect tracking. Investigated in this paper is the role played by feedforward controllers in the presence of plant uncertainties. We show that the use of feedforward controllers cannot improve the tracking error beyond what is achievable with a properly designed feedback loop, over all plant uncertainties. Including preview in the feedforward will not alter the situation either. We present two methods of designing robust compensators so that the tracking error due to uncertainties will be made small in some sense in the frequency domain and will have zero error in the steady state.

On the Generation of QFT Bounds for General Interval Plants

Considered in this paper is the question of synthesizing a fixed compensator for an interval plant so that (i) robust stability and (ii) robust disturbance rejection in the sense of Quantitative Feedback Theory (QFT) are attained. As expected, the problem reduces to one of loop shaping a nominal transfer function that avoids a set of frequency dependent forbidden regions while simultaneously stabilizing the nominal plant. It is shown that for the class of interval plants considered, the QFT boundaries (i.e., the magnitude restrictions on the nominal loop transfer function at each phase for a given frequency) can be explicitly computed by solving a number of simultaneous inequalities at each frequency. These results yield computationnally efficient algorithms mainly because the need for the usual one dimensional search on the magnitude of the nominal loop transfer function has been completely removed

Robust Stability of Plants With Mixed Uncertainties and Quantitative Feedback Theory

Quantitative Feedback Theory (QFT) has often been criticised for lack of a rigorous mathematical theory to support its claims. Yet it is known to be a very effective design methodology. In this paper we re-examine QFT and state several results that confirm the validity of this highly effective framework proposed by Horowitz. Also provided are some additional insights into the QFT methodology that may not be immediately apparent. We consider three important fundamental questions: (i) whether or not a QFT design is robustly stable, (ii) does a robust stabilizer exist and (iii) does a controller assuring robust QFT performance exist. The first two are obvious precursors for synthesizing controllers for performance robustness. We give necessary and sufficient conditions that unambiguously resolve the question of robust stability under mixed uncertainty, thereby, confirming that a properly executed QFT design is automatically robustly stable. Also given is a sufficiency condition for a robust stabilizer to exist which is derived from the well known Nevanlinna-Pick theory in classical analysis. Finally, we give a sufficiency theorem for the existence of a QFT controller and deduce that when the uncertain plant set is minimum phase with no unstructured uncertainty there always exists a controller satisfying robust performance specifications in the sense of QFT.

A New Formulation of Multiple-Input Multiple-Output Quantitative Feedback Theory

The QFT robust performance problem in its entirety may be reduced to an H∞ problem by casting each specification as a frequency domain constraint on the nominal sensitivity function and the complementary sensitivity function. It is shown that the existence of a solution to a standard H∞ problem guarantees a solution to the QFT problem whereas the existence of a QFT solution does not necessarily guarantee an H∞ solution. A solution obtained via this formulation for the QFT problem is in general more conservative when compared to what may be obtained from classical QFT loopshaping. However, one does not have to restrict the QFT controller to be diagonal as is usually done in MIMO-QFT. In addition, a simple constructive approach is provided for the design of a prefilter matrix for MIMO systems. In the standard QFT approach, the synthesis of a prefilter matrix for the MIMO case is much more involved than that of the SISO case.

Robust stability of closed-loop systems resulting from nonsequential MIMO-QFT design

Considered in this paper is the question of whether a compensator realized by the MIMO-QFT nonsequential method robustly stabilizes the entire plant family. In order to establish out results, first the classic small gain theorem for robust stability is modified to allow uncertain plant families with poles arbitrarily crossing the imaginary axis, or equivalently, plants in which the number of unstable poles does not remain fixed over all uncertainties. The conventional assumption that the number of unstable poles remain fixed over all uncertainties can be quite restrictive, especially, in the case of plants with structured uncertainties. It is shown that to assure robust stability of the closed loop, resulting from the MIMO-QFT nonsequential approach, one more requirement must be added to the procedure. The needed extra condition can be quite naturally incorporated during the execution of the nonsequential technique. As a result, the proposed condition does not significantly alter the basic MIMO-QFT nonsequential procedure.

Stabilizability of Unstable Linear Plants Under Bounded Control

Considered in this paper is the problem of stabilizing a linear unstable plant under bounded control. It is shown that a linear strictly unstable plant can never be globally stabilized under bounded control, and a linear unstable plant can never be globally stabilized under bounded linear feedback control. Hence, any stabilizing controller (be it linear or nonlinear) for a strictly unstable plant can only be locally stabilizing. Consequently, it is important to estimate the domain of attraction achievable with a locally stabilizing control under a prespecified control bound. Since, it is difficult to construct the domain of attraction, we use the domain within which the control is not saturated to approximate it. The relation between these two domains are characterized and computable expressions for the biggest stability ball and the biggest stability polytope that lie inside the domain of non saturating control are given

An H∞ Formulation of Quantitative Feedback Theory

The QFT robust performance problem in its entirety may be reduced to an H∞ problem by casting each specification as a frequency domain constraint either on the nominal sensitivity function or the complementary sensitivity function. In order to alleviate the conservative nature of a standard H∞ solution that is obtainable for a plant with parametric uncertainty we develop a new stability criterion to replace the small gain condition. With this new stability criterion it is shown that the existence of a solution to the standard H∞ problem guarantees a solution to the QFT problem. Specifically, we provide an explicit characterization of necessary frequency weighting functions for an H∞ embedding of the QFT specifications. Due to the transparency in selecting the weighting functions, the robust performance constraints can be easily relaxed, if needed, for the purpose of assuring a solution to the H∞ problem. Since this formulation provides only a sufficient condition for the existence of a QFT controller one can then use the resulting H∞ compensator to initiate the QFT loop shaping step.

Frequency Domain Necessary and Sufficient Conditions for Robust Stability of Interval Plants

Considered in this paper is the robust stabilization of a special family of single-input single-output, interval plants in which only the denominator polynomial or the plant poles are uncertain. Two frequency domain necessary and sufficient conditions are derived for the robust stability of the closed loop system. The first stability criterion reduces to a question of whether or not a specially constructed polar plot intersects the box [-0.5,0.5] × [-0.5,0.5] in the complex plane and the second reduces to a question of whether or not the polar plot of the nominal loop transfer function intersects a specially constructed frequency dependent domain in the complex plane. Both criteria can be used in synthesizing controllers for the special class of interval plants considered.

A frequency domain criterion for robustly stabilizing a family of interval plants

Considered in this paper is the problem of robust stabilization of a family of interval plants by a single fixed compensator. A frequency domain sufficient condition determines the closed-loop stability. In particular if a specially constructed frequency function does not, at any frequency ω ∈ [0, ∞], intersect the box [−1,1] × [−1,1] in the complex plane then the entire closed loop family is stable. Also identified is a class of interval plants for which the sufficient condition in fact does become necessary

Frequency domain criteria for synthesizing robustly stabilizing compensators for a class of interval plants

Considered in this paper is the robust stabilization of a special family of single-input single-output, interval plants in which only the denominator polynomial or the plant poles are uncertain. Two frequency domain necessary and sufficient conditions are derived for the robust stability of the closed-loop system. The first stability criterion reduces to a question of whether or not a specially constructed polar plot intersects the box [-1, 1] x [-1, 1] in the complex plane and the second reduces to a question of whether or not the polar plot of the nominal loop transfer function intersects a specially constructed frequency dependent domain in the complex plane. Both criteria can be used for synthesizing controllers for the special class of interval plants considered. A loop shaping technique is proposed for the synthesis of a robustly stabilizing compensator. For the special class of interval plants considered, the polar plot of the nominal loop transfer function must not intersect a frequency dependent parallelogram. The four corners of the parallelogram can be explicitly computed at each frequency.