U-D factorisation of the strengthened discrete-time optimal projection equations

Abstract

Algorithms for optimal reduced-order dynamic output feedback control of linear discrete-time systems with white stochastic parameters are U-D factored in this paper. U-D factorisation enhances computational accuracy, stability and possibly efficiency. Since U-D factorisation of algorithms for optimal full-order output feedback controller design was recently published by us, this paper focusses on the U-D factorisation of the optimal oblique projection matrix that becomes part of the solution as a result of order-reduction. The equations producing the solution are known as the optimal projection equations which for discrete-time systems have been strengthened in the past. The U-D factored strengthened discrete-time optimal projection equations are presented in this paper by means of a transformation that has to be applied recursively until convergence. The U-D factored and conventional algorithms are compared through a series of examples.

Keywords:

• systems with state and control-dependent noise
• optimal reduced-order controller design
• compensatability and optimal compensation
• multiplicative white noise
• stochastic parameters
• UDU factorisation

Additional information

Notes on contributors

L. Gerard Van Willigenburg

L. Gerard Van Willigenburg was born in Leiden, The Netherlands, in 1958. He received his MSc in electrical engineering and his PhD degree in control engineering at Delft University of Technology, The Netherlands, in 1983 and 1991, respectively. From 1986 to 1991, he was a research engineer at the process dynamics and control group in the department of applied physics. Since 1991, he is an assistant professor of the Systems and Control Group (currently called Biomass Refinery and Process Dynamics Group) at Wageningen University. His professional research interests include digital optimal control, reduced-order control, model predictive control and adaptive dual control. The application areas range from indoor climate control (storage rooms, greenhouses, stables and buildings), robot control, automatic guidance, to process control (fermentation) and the control of economic and biological systems (plants). The modelling of fundamental physics (thermodynamics, quantum mechanics and relativity) has become a private research interest.

Willem L. De Koning

Willem L. De Koning was born in Leiden, The Netherlands, in 1944. He received his MSc in electrical engineering and his PhD degree in control engineering at Delft University of Technology, The Netherlands, in 1975 and 1984, respectively. From 1969 to 1975, he was a research engineer of power electronics and control at the department of electrical engineering. From 1975 to 1987, he was an assistant professor of Process Dynamics and Control in the department of Applied Physics. From 1987 to 2006, he was an associate professor of mathematical system theory at the department of mathematical system theory in the department of technical mathematics and informatics. He has also held a visiting position at the Florida Institute of Technology, Melbourne. His research interests include control of distributed parameter systems, robust control, adaptive control, reduced-order control and digital control. The main application areas are the process industry and mechatronics. Although he retired in 2006, he continues some of his research on a personal basis.