Robust control theory

Chiang, R.Y. (1988) Modern Robust Control Theory, PhD Dissertation, USC. 

Over the last two decades, there has been increasing interest in probabilistic, or stochastic, robust control theory. Monte Carlo simulation methods have been used to synthesize and analyze controllers for uncertain systems [170,255], First- and second-order reliability methods were incorporated to compute the probable performance of linear-quadratic-regulator... 

Andre L. C. Araujo was bom in Fortaleza, Brazil, in 1973. He is currently an Associate Professor with the Department of Telecommunications Engineering, Federal Institute for Education, Science, and Technology of Ceara (IFCE), Fortaleza, Brazil. He works toward the Ph.D. degree in Robust Control Theory with the Federal University of Ceara (UFC), Fortaleza, Brazil. His research interests include telemetry, wireless sensor networks, embedded systems, and data communications. 

New research advances in control theory that are bringing it closer to practical problems are promising dramatic new developments and attracting widespread industrial interest. One of these advances is the development of "robust" systems. A robust control system is a stable, closed-loop system that can operate successfully even if the model on which it is based does not adequately describe the plant. A second advance is the use of powerful semiempirical formalisms in control problems, particularly where the range of possible process variables is constrained. 

Linear control theory will be of limited use for operational transitions from one batch regime to the next and for the control of batch plants. Too many of the processes are unstable and exhibit nonlinear behavior, such as multiple steady states or limit cycles. Such problems often arise in the batch production of polymers. The feasibility of precisely controlling many batch processes will depend on the development of an appropriate nonlinear control theory with a high level of robustness. 

In view of this, a robust scheme based on the Hoo control theory [24] is developed in the present work. The algorithm guarantees both stability and performance for a family of perturbed plants with model uncertainties and exogenous inputs (i.e., chamber disturbances and sensor noises) over a wide range of operating conditions, an advantage especially desired for combustion dynamics problems. 

The system dynamics uncertainty A(s) contains parametric and model uncertainties, and its L2 gain bounded as A(s) oo < 1/7- Based on the L2"gain control theory, the first task of a robust controller for stabilizing perturbed plants is to endow the closed-loop system with the following property ... 

These issues of model accuracy and model uncertainty were not dealt with at all in the 1960s and 1970s. They are absent from all process control textbooks, an exception being the recent textbook by Seborg et al. [13]. The ability of classical control theory to explain the demonstrated effects and difficulties is very limited. In the past 10 years, much progress has been made in the area of robust control. This progress is of obvious practical value. The main results are derived [8] and summarized elsewhere [12, 14]. 

A robust mathematical tool is needed to perform an optimisation. We have found that optimal control theory provides such a tool. The system is constrained by the requirement to balance energy, momentum and mass. These constraints must be specified for each particular case. In optimal control terminology there are two classes of variables. The first class are the state variables, for instance the temperature, T z,t), the pressure, p z,t), and the concentrations, Cj z,t), in a tubular reactor. The second class are the control variables, which are determined from the outside. An example is the temperature, T (z,t), on the outside along the tubular reactor. Optimal control theory in this case provides a general method to obtain T (z,t) such that the total entropy production is minimal, given certain constraints. 

A robust experimental concept potentially apphcable to all time domains is the feedback-optimized control [21] which requires no knowledge of the underlying mechanism in effecting control. Optimal control theory [187] provides theoretical and computational description of the process to help understand the control mechanisms in effect. 

While most of the schemes described above have been demonstrated experimentally, a robust and efficient control of molecular processes is often difficult to achieve experimentally. In this context, the inclusion of robustness in optimal control theory to achieve such a robust and efficient control currently represents a very active field of research [70]. One of the main difficulties is that the interaction of a laser field with a sample of molecules is necessarily averaged with respect to the random orientation of the individual molecules of the sample. To overcome this issue, several... 

In previous chapters, Laplace transform techniques were used to calculate transient responses from transfer functions. This chapter focuses on an alternative way to analyze dynamic systems by using frequency response analysis. Frequency response concepts and techniques play an important role in stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979). We introduce a simplified procedure to calculate the frequency response characteristics from the transfer function of any linear process. Two concepts, the Bode and Nyquist stability criteria, are generally applicable for feedback control systems and stability analysis. Next we introduce two useful metrics for relative stability, namely gain and phase margins. These metrics indicate how close to instability a control system is. A related issue is robustness, which addresses the sensitivity of... 

Frequency response concepts and techniques play an important role in control system design and analysis. In particular, they are very useful for stability analysis, control system design, and robustness analysis. Historically, frequency response techniques provided the conceptual framework for early control theory and important applications in the field of communications (MacFarlane, 1979). 

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