Linear Control Theory with an ℋ∞ Optimality Criterion

This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods

Properties of the mixed μ problem and its bounds

Upper and lower bounds for the mixed μ problem have recently been developed, and here we examine the relationship of these bounds to each other and to μ. A number of interesting properties are developed and the implications of these properties for the robustness analysis of linear systems and the development of practical computation schemes are discussed. In particular we find that current techniques can only guarantee easy computation for large problems when μ equals its upper bound, and computational complexity results prohibit this possibility for general problems. In this context we present some special cases where computation is easy and make some direct comparisons between mixed μ and “Kharitonov-type” analysis methods

Contrasting Views of Complexity and Their Implications For Network-Centric Infrastructures

There exists a widely recognized need to better understand and manage complex “systems of systems,” ranging from biology, ecology, and medicine to network-centric technologies. This is motivating the search for universal laws of highly evolved systems and driving demand for new mathematics and methods that are consistent, integrative, and predictive. However, the theoretical frameworks available today are not merely fragmented but sometimes contradictory and incompatible. We argue that complexity arises in highly evolved biological and technological systems primarily to provide mechanisms to create robustness. However, this complexity itself can be a source of new fragility, leading to “robust yet fragile” tradeoffs in system design. We focus on the role of robustness and architecture in networked infrastructures, and we highlight recent advances in the theory of distributed control driven by network technologies. This view of complexity in highly organized technological and biological systems is fundamentally different from the dominant perspective in the mainstream sciences, which downplays function, constraints, and tradeoffs, and tends to minimize the role of organization and design

Robustness and performance trade-offs in control design for flexible structures

Linear control design models for flexible structures are only an approximation to the “real” structural system. There are always modeling errors or uncertainty present. Descriptions of these uncertainties determine the trade-off between achievable performance and robustness of the control design. In this paper it is shown that a controller synthesized for a plant model which is not described accurately by the nominal and uncertainty models may be unstable or exhibit poor performance when implemented on the actual system. In contrast, accurate structured uncertainty descriptions lead to controllers which achieve high performance when implemented on the experimental facility. It is also shown that similar performance, theoretically and experimentally, is obtained for a surprisingly wide range of uncertain levels in the design model. This suggests that while it is important to have reasonable structured uncertainty models, it may not always be necessary to pin down precise levels (i.e., weights) of uncertainty. Experimental results are presented which substantiate these conclusions

Identification of flexible structures for robust control

Documentation is provided of the authors' experience with modeling and identification of an experimental flexible structure for the purpose of control design, with the primary aim being to motivate some important research directions in this area. A multi-input/multi-output (MIMO) model of the structure is generated using the finite element method. This model is inadequate for control design, due to its large variation from the experimental data. Chebyshev polynomials are employed to fit the data with single-input/multi-output (SIMO) transfer function models. Combining these SIMO models leads to a MIMO model with more modes than the original finite element model. To find a physically motivated model, an ad hoc model reduction technique which uses a priori knowledge of the structure is developed. The ad hoc approach is compared with balanced realization model reduction to determine its benefits. Descriptions of the errors between the model and experimental data are formulated for robust control design. Plots of select transfer function models and experimental data are included

ℋ∞ control of nonlinear systems via output feedback: controller parameterization

The standard state space solutions to the ℋ∞ control problem for linear time invariant systems are generalized to nonlinear time-invariant systems. A class of local nonlinear (output feedback) ℋ∞ controllers are parameterized as nonlinear fractional transformations on contractive, stable nonlinear parameters. As in the linear case, the ℋ∞ control problem is solved by its reduction to state feedback and output estimation problems, together with a separation argument. Sufficient conditions for ℋ∞-control problem to be locally solved are also derived with this machinery

H∞ control of nonlinear systems: a convex characterization

The nonlinear H∞-control problem is considered with an emphasis on developing machinery with promising computational properties. The solutions to H∞-control problems for a class of nonlinear systems are characterized in terms of nonlinear matrix inequalities which result in convex problems. The computational implications for the characterization are discussed

H∞ Control of Nonlinear Systems: A Class of Controllers

The standard state space solutions to the H∞ control problem for linear time invariant systems are generalized to nonlinear time-invariant systems. A class of nonlinear H∞-controllers are parameterized as nonlinear fractional transformations on contractive, stable free nonlinear parameters. As in the linear case, the H∞ control problem is solved by its reduction to four simpler special state space problems, together with a separation argument. Another byproduct of this approach is that the sufficient conditions for H∞ control problem to be solved are also derived with this machinery. The solvability for nonlinear H∞-control problem requires positive definite solutions to two parallel decoupled Hamilton-Jacobi inequalities and these two solutions satisfy an additional coupling condition. An illustrative example, which deals with a passive plant, is given at the end