Dr. Mark A. Hopkins, Professor (retired)

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Dr. Hopkins in 2014.

E-Mail: mark.hopkins@rit.edu

PZgui Toolbox for Matlab     (Pole/Zero Graphical user interface)

Education and Interests

Publications, Patents, etc.

Prof. Hopkins retired from RIT on July 1, 2017.

Education and Interests:

Ph.D., 1988, Virginia Tech (Virginia Polytechnic Institute and State University), Blacksburg VA
System Identification
Control Systems
Research Interests:
Model identification of flexible structures: Typically, the order of the model to be identified is at least 600, because the system it represents has considerably more than 300 (often closely-spaced) lightly-damped vibrational modes that are distributed over a frequency range of at least 4.5 orders-of-magnitude, with a dynamic range of about 120 dB. The identified model ideally has bode magnitude error less than 2 dB and bode phase error less than 10 degrees at every frequency over the full range of interest (when compared to the empirical data from the real system). This is a very tough problem, particularly the very wide bandwidth, not only in terms of system identification, but also in terms of numerical stability. I made very good progress on this modeling problem in the late 1990's in work at Eastman Kodak Company, Commercial & Government Systems Division (now part of Harris Corp.), and developed a technique, called the Pick And Place Algorithm (P.A.P.A.), to create models this good.
Extending the P.A.P.A. modelling technique to MIMO systems: During a sabbatical (2001-02) at C&GS, I successfully extended P.A.P.A. to multiple-input, multiple-output (MIMO) systems. The crux of this problem is converting a matrix of transfer functions into a state-space model in modal-canonic form, by way of partial fraction expansions. Getting this to work involved creating a new algorithm, called Residue Rank Reduction (R-cubed), that iteratively reduces the rank of all the MIMO Cauchy-residue-arrays to rank one. That is necessary in order to avoid having any multiple poles at the same location in the state-space model. The feasibility proof of the method is a two-input, two-output, 850-pole state-space model of a large flexible structure. This model was created from four P.A.P.A. transfer functions, and has extremely good frequency response accuracy from 0.2 Hz to 10 kHz.
More recently (2009-18), I have solved the problem of creating very high-order discrete-time MIMO state-space models that agree quite well with empirical data over a very broad range of frequencies ( > 4.5 orders-of-magnitude). Over those years, and particularly since retiring from RIT, I developed a method that successfully decouples individual modes, to the greatest extent possible, so that they can more easily be identified. This led to a very successful modeling technique that I call FORCANTER (Frequency Observability Range Context And Nelder-Mead To Estimate Residues). This method relies on four fundamental tools, (1) The FORSE algorithm, published in 1996 by Liu, Jacques, and Miller in ASME JDSMC, (2) the mode-decoupling technique mentioned above, (3) other linear model components that, collectively, comprise the "subsystems" that model the effects of non-modal (i.e., real-valued) poles, such as integration, ac-coupling, and delay; and (4) the Nelder-Mead tuning algorithm. Some colleagues at Harris have dubbed this method "FORSE-Fed P.A.P.A.", a name that has some merit. Early results were published in 2013 at the SPIE International Symposium on Smart Structures and Materials, in San Diego, CA.
Control of flexible structures: Apply advanced control techniques to high-order models generated by the model identification. This problem involves model-order reduction techniques, as there are significant numerical problems involved in designing a controller from models larger than 600-th order, and no possibility of implementing in hardware a real-time controller of that size at fast sample rates. In the past several years, I have been developing a method that tunes an initial LQR-based controller/observer design to achieve damping, isolation, and robustness goals, for the class of infinite-dimensional, reasonably linear, flexible structures that are my main interest. This method relies on having excellent high-order as well as reduced-order models, and uses the Nelder-Mead Simplex Method, and other similar algorithms, to tune the two feedback matrices. It also depends upon distributed computing to make it feasible to achieve good results in a reasonable amount of time. The initial work was funded by ITT Geospatial Systems (now Harris Corp.).
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Publications, Patents, etc.


  • Hopkins, M. A., and A. M. Kibbe, "Open-source hardware in controls education," ASEE Computers in Education Journal, Vol. 5, No. 4, October 2014, pp 62-70.
  • Hopkins, M. A., and A. M. Kibbe, "Open-source hardware in controls education," Proceedings of 121st ASEE Annual Conference and Exposition, Ref. #8778, Indianapolis IN, 2014.
  • Hopkins, Mark A., "Identification of flexible structures by frequency-domain observability range context," Proceedings of SPIE, International Symposium on Smart Structures and Materials, San Diego CA, 2013, v.8688, pp.O1-O9.
  • Hopkins, Mark A., "Relating continuous-time and discrete-time in the classroom," Proceedings of ASEE 2008 Annual Conference and Exposition, AC-2008-2741, Pittsburgh PA, June 22-25, 2008.
  • Hopkins, Mark A., "Improved selection of state-weighting matrices for LQR MIMO-controller design," Proceedings of IASTED, 10th International Conference on Intelligent Systems and Control, L. M. Sztandera, ed., Vol. 592-800, Cambridge MA, November, 2007.
  • Hopkins, Mark A., "Demonstrating in the classroom the ideas of frequency response," Proceedings of ASEE Engineering Teaching and Learning Practices, Joan Dannenhoffer, ed., Toronto ONT, October, 2007.
  • Rowen, D., and M.A.Hopkins, "H-infinity controller design for structural damping," Proceedings of SPIE, 13th International Symposium on Smart Structures and Materials, Damping and Isolation Conference, Volume 6169, pp. 22-33, February 26 - March 3, 2006.
  • Hopkins, M.A., D.A. Smith, P. Vallone, and R. Sandor, "Hybrid multivariable controller architecture," Proceedings of SPIE Vol. 5760, 12th International Symposium on Smart Structures and Materials, Damping and Isolation Conference, March 6-10, 2005, pages 390-401.
  • Hopkins, M.A., and H.F. VanLandingham, "Optimal Nonlinear Estimation of Linear Stochastic Systems: The Multivariable Extension," ASME Journal of Dynamic Systems, Measurement and Control, Vol. 118, No. 2, June 1996, pp. 350-353.
  • Vallone, Phillip, and Mark Hopkins, "Experimental Verification of a Novel System ID Technique Called PLID using a 3-D Flexible Structure," Proc. of the SPIE North American Conf. on Smart Structures and Materials, San Diego CA, February 1995.
  • Hopkins, M.A., and H.F. VanLandingham, "Optimal Nonlinear Estimation of Linear Stochastic Systems," ASME Journal of Dynamic Systems, Measurement and Control, Vol. 116, No. 3, September 1994, pp. 529-536.
  • Lyon, Bruce, and Mark Hopkins, Genetic Algorithm Design of a Fuzzy Logic TRC Controller, Xerox/WRC Monograph, January 1994.
  • Hopkins, Mark A., Report on a Three-Input, Three-Output Fuzzy Logic Tone Reproduction Curve (TRC) Controller, Xerox/WRC Monograph, May 1993.
  • Hopkins, Mark A., Report on an Adaptive Fuzzy Logic Controller, Xerox/WRC Monograph, November 1992.
  • Hopkins, Mark A., A New Method to Predict Photoreceptor Dark Decay Voltage Transients, Xerox/WRC Monograph, August 1991.
  • Hopkins, Mark A., Photoreceptor Dark Decay Potential Control in a Prototype Color Printer, Xerox/WRC Monograph, July 1990.
  • Hopkins, M.A., and H.F. VanLandingham, "Optimal Joint Parameter and State Estimation of Linear Stochastic MIMO Systems," Proceedings of the 1988 American Control Conference, Atlanta, GA.


  • European Patent # EP0682294, Method of Controlling Print Quality for an Electrophotographic Printer, Inventor: Mark A. Hopkins, Assignee: Xerox Corp., issued June 23, 1999.
  • U.S. Patent # 5,760,812, Two-Input, Two-Output Fuzzy Logic Print Quality Controller for an Electrophotographic Printer, Inventor: Mark A. Hopkins, Assignee: Xerox Corp., issued June 2, 1998.
  • U.S. Patent # 5,390,004, Three-Input, Three-Output Fuzzy Logic Print Quality Controller for an Electrophotographic Printer, Inventor: Mark A. Hopkins, Assignee: Xerox Corp., issued February 14, 1995.
  • U.S. Patent # 5,355,197, Method and Apparatus for Predicting the Cycle-Down Behavior of a Photoreceptor, Inventor: Mark A. Hopkins, Assignee: Xerox Corp., issued October 11, 1994.

Professional Affiliations:

  • Senior Life-Member Institute of Electrical and Electronics Engineers (IEEE)
  • Member Society for Industrial and Applied Mathematics (SIAM)
  • Life Member Tau Beta Pi
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MATLAB  Toolbox Shareware to Download

PZgui (Pole/Zero graphical user interface)

A MATLAB Add-On Tool for the Study of Linear Systems and Control

This is the ultimate interactive tool for studying SISO transfer-functions.

60,000 lines of code developed over a period of 23 years.

Now includes "The Hopkins Demos", which cover topics from the voltage/current relationship in inductors to convolution (the world's best convolution demo), to eigenvalues and singular values.

It is Matlab 2014b+ Graphics Ready.

       Currently in Version 8.2.09   released June 6, 2018.

( Runs in ANY   MATLAB Version 2008a or later, the Student Version (it includes the Controls Toolbox), or the Professional Version with Controls Toolbox.

Go to download page for zipped PZgui M-files

    This highly interactive, GUI-based program helps students and engineers to comprehend the complicated relationships among the following:

  • Pole/zero maps
  • Open-loop and closed-loop bode plots
  • the Nichols chart, with phase-lead/phase-lag design tool, and PID design tool
  • the Nyquist contour and the corresponding Nyquist plot
  • Output sensitivity function
  • Root locus
  • Open-loop and closed-loop time responses, with a selection of standard inputs
  • Both continuous-time domain and discrete-time domain
  • Zero-Order Hold equivalent models, and bilinear/Tustin transform models

    Completely updated comprehensive 55-page User's Manual.

    Includes special tools to study PID and Lead-Lag design.

    All features are available both for continuous-time and discrete-time systems, enhancing the ability to study relationships between them.

    Correctly handles models that have hundreds of poles and zeros.

    Can generate large random "flexible-structure-like" models.

The program consists of more than 60,000 lines of code in 75 Matlab M-files,
with a 55-page PDF User's Guide.

            Click here for more info about PZgui (the User's Manual)
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e-mail Mark Hopkins: mark.hopkins@rit.edu

This page was updated 06-Jun-2018