EDC2002 Department of Mechanical Engineering A Graphical User Interface for Computer-aided Robust Control System Design J.F. Whidborne, S.J. King P. Pangalos, Y.H. Zweiri

EDC2002 Department of Mechanical Engineering Introduction Graphical classical control design tools (Bode, Nyquist etc) developed before advent of efficient numerical computation - good qualitative information Early quantitative methods (linear-quadratic optimal control) developed in 1950’s before availability of graphical input and output devices Multivariable computer-based graphical methods (inverse Nyquist array, characteristic locus array) do not exploit numerical capabilities of modern digital computer & suffer from curse of dimensionality Modern frequency-based approaches, (H ,  ) exploit graphical & numerical potential of modern computers - suffer curse of dimensionality less - but lack of supporting GUI-based tools.

EDC2002 Department of Mechanical Engineering McFarlane & Glover’s Loop Shaping Design Procedure (LSDP) modern H  -optimization approach (H  -norm is max magnitude of frequency response) multivariable (many inputs and outputs) robust (stability guaranteed in the face of plant perturbations & uncertainty) based on concepts from classical Bode plot methods - graphical frequency domain method number of graphical plots required is max(n,m)+n+m (Inverse Nyquist Array requires nxm)

EDC2002 Department of Mechanical Engineering LSDP - Step 1 augment plant G with weighting functions W 1 and W 2 G(s) W 2 (s)W 1 (s) Augmented Plant G s (s)

EDC2002 Department of Mechanical Engineering W 1 and W 2 chosen so weighted plant has “good” shape high gain at low freq Low gain at high freq Singular values close at cross over Roll-off < 20 dB/dec max sing. value min sing. value freq Singular values of G s (dB)

EDC2002 Department of Mechanical Engineering LSDP - Step 2 G(s)W 2 (s)W 1 (s) K s (s) optimal controller check design index - if  > 5 return to step 1 synthesize H  -optimal controller to robustly stabilize shaped plant

EDC2002 Department of Mechanical Engineering G(s) W 1 (s)K s (s)W 2 (s) LSDP - Step 3 Final controller K(s) = W 1 (s) K s (s) W 2 (s) K(s)

EDC2002 Department of Mechanical Engineering LSDPTOOL - A Graphical User Interface M ATLAB © Toolbox Features –main GUI for designing weighting functions W 1 and W 2 –GUI for input and editing model G(s) –window for displaying design index  and step responses –full M ATLAB © help system –load, save, print options

EDC2002 Department of Mechanical Engineering Case Study - A Maglev System magnetic levitation of a ball bearing controller detector light z open loop unstable electromagnet current, i, varied by controller vertical displacement of ball, z, measured by light emitter & detector F mg i

EDC2002 Department of Mechanical Engineering Maglev Controller Design Small deviations of system from equilibrium gives linearised system state description where LSDPTOOL used to design controller

EDC2002 Department of Mechanical Engineering

EDC2002 Department of Mechanical Engineering

EDC2002 Department of Mechanical Engineering

EDC2002 Department of Mechanical Engineering

EDC2002 Department of Mechanical Engineering

EDC2002 Department of Mechanical Engineering

EDC2002 Department of Mechanical Engineering Design index value  = 4.23 indicates a good design Weighting functions: System simulated in S IMULINK © Controller Design

EDC2002 Department of Mechanical Engineering Step Responses (Non-linear Model)

EDC2002 Department of Mechanical Engineering Maglev Laboratory Rig

EDC2002 Department of Mechanical Engineering Toolbox Availability Available on WWW at or through MATLAB CENTRAL on Mathworks WWW site at