This paper deals with the robust performance problem of a linear time-invariant control system in the presence of robust controller uncertainty. Assuming that plant uncertainty is modeled as an additive perturbation, a geometrical approach is followed in order to find a necessary and sufficient condition for robust performance in the form of a bound on the magnitude of controller uncertainty. This frequency domain bound is derived by converting the problem into an optimization problem, whose solution is shown to be more time-efficient than a conventional structured singular value calculation. The bound on controller uncertainty can be used in controller order reduction and implementation problems.

In feedback control applications, the plant to be controlled is often modeled as a fixed linear time-invariant system. The modeling process typically neglects some of the plant’s dynamics which results in model uncertainties. In spite of such differences between the plant and its corresponding nominal model, the designed controller must be able to control the plant, that is, it should guarantee robust stability under the new circumstances caused by the plant uncertainties [

Often, stability is not the only property of a closed-loop system that must be robust to plant perturbations. Tracking or regulation errors are caused by exogenous disturbances acting on the system, and plant perturbations can cause these errors to increase greatly. In such cases, the closed-loop performance can become unacceptable. Hence, it is necessary to check for both robust stability and robust performance of the system [

Based on the above observations, a robust controller is often designed and implemented in order to maintain the robust stability and performance of the system. But the controller itself may also have perturbations caused by a nonideal implementation, for example, discretization, order reduction, fixed point implementation of the controller, or environmental conditions such as temperature changes that can affect an analog controller. Therefore, it is reasonable to assume that the controller has uncertainty too [

Some control design techniques result in having a finite impulse response (FIR) controller which is easy to implement [

Previous work using the gap metric takes both plant and controller uncertainties into account, explaining the trade-off between the uncertainties in order to maintain robust stability. This is done by providing upper bounds on admissible uncertainties which preserve closed-loop stability and a specified small tolerance on the input-output behavior of a feedback system [

The presence of controller uncertainty in the closed-loop system requires a revision of robust stability and performance conditions. In the following problem formulation, the nominal plant and controller are given, and plant perturbation is modeled as additive uncertainty. Assuming that the controller perturbation is also modeled as an additive uncertainty, a frequency domain bound on the magnitude of the controller’s uncertainty is derived. This bound provides information on how large a perturbation the controller can handle and, for real-time implementations, when to redesign the controller. In other words, it acts as a necessary and sufficient condition for maintaining robust performance, comparing to a previously derived sufficient condition [

This article is organized as follows. Section

A typical control loop is depicted in Figure

Typical control loop with nominal controller and plant.

Expanded control loop with additive uncertainties and fictitious performance connection.

The block

Let

The controller implementation problem is posed as follows. What is the maximum range of variation for the controller’s additive uncertainty representing the discrepancy between the designed controller and its implemented version, as given by an upper bound for the uncertainty weighting function

Assume that

Since the normalized uncertainties appear in conjunction with their respective weighting functions in (

The bound in (

Nominal system characteristics, arbitrary aggregated uncertainty set and the sufficient upper bound.

The dashed contour bounds all possible occurrences of

Therefore, in order to find a necessary and sufficient condition, the magnitude of the aggregated uncertainty set should be verified for a variety of border locations and not just at the peak point A. In fact, the maximum magnitude calculation is only necessary for a certain area facing the solid circle of radius

(a) The critical range for maximum possible magnitude verification, (b) limit calculation regarding

The robust performance of the system is guaranteed if and only if for all

Referring to Figure

On the other hand, based on the definition of the uncertainty, it is preferable to do the calculations with respect to

Changing the geometric coordinates by normalizing (

Under the assumptions given in Lemmas

The innermost optimization in (

The intermediate search level is done with respect to

The outermost search level is performed on all possible

Regarding the innermost search level in (

The optimization routine in

The numerical search required in order to find the solution for (

The proof is given in the appendix.

Considering the definitions given in Theorem

Consider the following system setup:

The calculation is performed over a set of 200 logarithmically equidistant points distributed over [1e^{−2} Hz, 1e^{5} Hz] frequency span meaning that the search for optimum bound is performed 200 times. Both the sufficient upper bound given by (

The robust performance bounds derived for the given system configuration.

The sufficient bound is obviously more conservative than the necessary and sufficient bound. In order to verify the accuracy of the necessary and sufficient bound, an eleventh-order weighting function is approximated in order to serve as

(a) The robust performance bound and the approximated weighting function

In order to verify the bound more closely, the same weighting function approximation and

(a) The robust performance bound and the approximated weighting function

In order to compare the proposed method with conventionally used ones, a benchmark test is done in order to observe the time efficiency of our method. The reference method used in this comparison employs the structured singular value calculation directly, relying on well-developed

With the same system configuration given at the beginning of the section, the above routine is run at the same frequency points fed through the proposed method, and the resulting controller uncertainty bound is shown in Figure

The average computation time needed for bound calculation.

Method | Proposed method | |
---|---|---|

Time (sec) | 88.9 | 434.8 |

Partitioning of search domain of

LHS | RHS | |
---|---|---|

0 | + | |

+ | 0 | |

0 | 0 | + |

+ | 0 | |

0 | + |

The necessary and sufficient robust performance bounds for controller uncertainty derived by the proposed method and using

Clearly, the proposed method works 4-5 times faster than the

In this article, a geometric approach is used in order to find a necessary and sufficient robust performance condition for a single-input single-output LTI system in the form of an upper bound on the magnitude of additive controller uncertainty. The bound is derived numerically using a three-level optimization which is made efficiently based on the propositions discussed about the problem formulation. The provided example demonstrates the accuracy of the proposed algorithm in determining the maximum possible size of controller uncertainty maintaining robust performance. The algorithm is shown to be more time-efficient when compared to the conventional structured singular value calculation tools under the same accuracy requirements.

For the case where

For the case

The angle boundaries versus primary search regions.

The equality

As a result, there is no need to search for the maximizing solution of