**Research & Publications by L. Gerard van Willigenburg**

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Introduction / Subjects / Publications / Submitted publications / Software

**Short Curriculum:**

Assistant professor (b. 1958) at Wageningen University (1990) who received his M.Sc. in Electrical Engineering from Delft University of Technology (1983) and a Ph.D. degree from Delft University of Technology (1991) for his thesis entitled "Digital optimal control of rigid manipulators".

**Professional and private research interests:**

*Professional*: Structural properties of dynamic systems, digital optimal control, reduced-order control, adaptive control and model predictive
control (receding horizon control). The application areas are indoor climate
control (greenhouses, stables, buildings, stadiums), robot control, automatic
guidance of agricultural field machines the control of processes in the
food industry (bioreactors, sterilization, drying, membrane filtration) and systems biology.

*Private*:
fundamentals of physics (thermodynamics, relativity and quantummechanics)

**Appropriate control system design methodologies**

Control is an *applied *science. Control
theorists have developed an impressive amount of sophisticated control methodologies
under all kinds of different assumptions. These assumptions are hardly ever all
satisfied in practice. Therefore control practitioners face the problem of selecting
an appropriate control system design methodology which is not an entirely scientific
or mathematical issue. An interesting line of research is to investigate to what
extent science and mathematics can help and how to bridge the remaining gap. A first
attempt in this direction has been published [L.G. Van Willigenburg et al. Preprints
IFAC World Congres, Prague, 3-8 July, 2005,
04926.pdf
]. For a visualisation of this result see
appropriate
poster.pdf .

Introduction / Subjects /
Publications / Submitted publications
/ Software

**Digital optimal control: Short introduction **

For non-linear systems digital optimal controllers are synthesized
that explicitly take into account the inter-sample (continuous-time) behaviour of
the controlled system, the digital nature of the controller, and the way in which
the system is sampled. Synchronous (conventional), *asynchronous *and *
random *sampling schemes are considered [R.E.
Kalman and J.E. Bertram, 1959, "A unified approach to the theory of sampling
systems",
Journal
of the Franklin Institute, 67, 405-436 ]. Based on an accurate mathematical
model of the system and a mathematical criterion describing the control objectives,
a digital optimal control sequence and a digital optimal LQG compensator
realizing feedback are computed
off-line. Both the digital optimal control sequence and the digital optimal LQG
compensator are stored in the controller memory. The only on-line computations that
need to be performed by the controller are a small number of matrix vector multiplications
to compute control corrections from the on-line measured data. This type of control
of non-linear systems is approximately optimal if the model structure and its parameter
values are well established and if the remaining uncertainty is small [M. Athans,
1971, "The role and use of the stochastic Linear-Quadratic-Gaussian problem in control
system design", IEEE Trans.
Aut. Contr., AC-16(6), 529-552 ]. If the model structure is well established
but some of the parameter values are not, and the on-line data contain information
regarding these parameter values, actively adaptive controllers are required. As
opposed to other (passively) adaptive controllers, actively adaptive controllers
take into account the need to identify the parameters in selecting the control [Y.
Bar-Shalom, 1981, "Stochastic dynamic programming: Caution and probing",
IEEE Trans.
Aut. Contr., AC-26(5), 1184-1195 ]. Since most of our research regarding digital
optimal control and LQG compensation of non-linear systems has been completed actively
(dual) adaptive controllers have become a new subject of our research [L.G. Van
Willigenburg et al. Preprints IFAC World Congres, Prague, 3-8 July,
04926.pdf
].

The application areas are indoor climate control (greenhouses, stables, storage buildings, stadiums), the control of mechanical systems (agricultural field machines, a tomato picking robot), processes in the food industry (bioreactors, sterilization, drying, membrane filtration) and economics.

**Systems theory & physics: Short introduction**

Systems theory lies at the heart of any type of mathematical
modeling. Therefore modeling the fundamentals of physics may largely benefit from
systems theory. In building models physicists often presume systems to be in equilibrium.
Although generally for very good reasons the equilibrium assumptions and their justification
may sometimes lead to erroneous models or unprovable mathematical requirements.
A good example is the equilibrium assumptions that underly statistical mechanics
and thermodynamics. In our opinion statistical mechanics, and in fact any type of
statistical modeling, is a "last resort" if no a-priori knowledge or insight is
available. Starting from very simple linear dynamics at the micro-scale of physical
systems we showed how the second law of thermodynamics emerges from them. No statistical
mechanics or equilibrium conditions are needed to obtain this result [48].
Motivated by this insight we are currently investigating the further consequences
of this result for thermodynamics. In addition we are investigating the modeling
and understanding of possible energy transfer mechanisms at the level of fundamental
physical particles, particularly electrons and photons. This issue appears to be
connected to the question why the speed of light is always observed to be constant.
This constant velocity, independent of observer motion, is a widely accepted postulate
of special relativity but not all well understood.

Introduction / Subjects / Publications / Submitted publications / Software

**Digital optimal control: Subjects **

**Systems theory & physics: Subjects **

- Equipartition and the second law of thermodynamics
- Connecting and unmasking relativity and quantum theory

Introduction / Subjects / Publications / Submitted publications / Software

**Asynchronous and aperiodically sampled digital
control systems, multirate control systems**

Asynchronous and aperiodic sampling very
often occur in digital control systems. This may be due to the fact that the computer
is time-shared or part of a computer network or due to technical imperfections of
the instrumentation. Asynchronous and aperiodic sampling is preferred over synchronous
sampling if some measurements are cheap and some are expensive. Asynchronous and
aperiodic sampling may be applied *intentionally *to eliminate hidden oscillations,
to reduce the influence of intelligent disturbances, or to increase stabilizability.
Asynchronous and aperiodic sampling occurs in economic and biological systems. The
theory and several related algorithms concerning the digital optimal control and
LQG compensation of non-linear systems have been generalized from synchronous to
asynchronous and aperiodic sampling. Publications: [9],
[10], [11], [23],
[41].

Introduction / Subjects / Publications / Submitted publications / Software

**Randomly sampled digital control systems, randomized
digital optimal control**

Random sampling occurs in computer controlled
systems if the computer is time-shared or is part of a computer network. Random
sampling is also caused by technical imperfections in the instrumentation. Random
sampling occurs in economic and biological systems and when a stochastic process
like a radar or sonar echo governs the sampling . Random sampling may be applied
*intentionally *to eliminate hidden oscillations, to reduce the influence
of intelligent disturbances, or to increase stabilizability. Theory and algorithms
to synthesize digital optimal full and reduced-order LQG controllers for randomly
sampled linear systems have been developed. Publication: [24].

Introduction / Subjects / Publications / Submitted publications / Software

**Reduced order LQG control, fixed order dynamic
compensation, optimal projection equations**

The conventional optimal (full-order) LQG
compensator (controller) for systems with deterministic parameters is very well
known and very well established computationally and is a very powerful tool to control
both linear and non-linear systems. Especially
if the dimension of the system model is large and the sampling intervals are small
reducing the order of the LQG compensator (controller) becomes vital. Then optimal
full-order LQG design becomes optimal reduced-order LQG design. For this more difficult
problem strengthened discrete-time optimal projection equations have been developed,
which in contrast to the conventional version, are equivalent to first-order necessary
optimality conditions and the minimality of the compensator. Based on these equations
numerical algorithms have been developed to compute optimal reduced-order LQG compensators
for time-invariant and time-varying discrete-time systems, with either deterministic
or white parameters. The algorithms need further investigation, optimization and
proof. Publications: [13], [15],
[18], [19], [23],
[25], [39].
The optimal full-order
compensator for systems with white parameters (multiplicative white noise) generalises
the LQG compensator result and therefore is more powerful while it is also well
established computationally [W.L. De Koning, 1992, "Compensatability and optimal
compensation of systems with white parameters",
IEEE Trans.
Aut. Contr., AC-37(5), 579-588 ]. Nevertheless it is not very well known despite
the fact that systems with white parameters arise in the design of digital control
systems if some of the parameters are white, such as the parameters of the plant
or the sampling period. Furthermore parameters may be assumed to be white to design
non-conservative robust control systems.

Introduction / Subjects / Publications / Submitted publications / Software

**Robust control, systems with white parameters
(mutiplicative white noise), compensatability & optimal compensation, delta operator
and U-D factored algorithm formulations**

White system parameters (multiplicative white noise) may destabilize
a system, as opposed to additive white noise. White parameters may therefore be
used to design robust controllers. Also they may result from stochastic sampling
of a continuous-time system
[24]. LQG like controllers may be
designed for these systems. The optimal LQG like output feedback controller is called
an optimal compensator computed from an LQG like optimal compensation problem.
Compensatability and the optimal full-order
compensator for systems with white parameters generalises the LQG result and therefore
is more powerful while it is also well established computationally [W.L. De Koning,
1992, "Compensatability and optimal compensation of systems with white parameters",
IEEE Trans.
Aut. Contr., AC-37(5), 579-588 ]. Nevertheless it is not very well known despite
the fact that systems with white parameters arise in the design of digital control
systems if some of the parameters are white, such as the parameters of the plant
or the sampling period. Furthermore parameters may be assumed to be white to design
non-conservative robust control systems. Formulation of the full and
reduced-order control and compensation problems in the delta domain offers the possibility
to unify continuous and discrete-time results. It also enables transferring discrete-time
results to continuous-time. Finally it allows for more efficient and accurate control
designs, especially if the sampling interval becomes very small. These opportunities
have been exploited by us in the papers [19], [51],
[52], [54].

U-D factorization has the potential to improve computational efficiency and accuracy.
A formulation of the full and reduced-order LQG problem by means of Lyapunov equations
offers alternative ways for U-D factorization [39].
For discrete-time systems with white stochastic parameters (multiplicative white
noise) U-D factorization of the algorithm for full-order and reduced-order compensation
is new and associated with the minimal realization of matrix valued white stochastic
processes [55],[60].

Introduction / Subjects / Publications / Submitted publications / Software

**Temporal linear system structure**

LQG perturbation feedback controllers for non-linear systems are
often based on the linearised dynamics about optimal control and state trajectories.
These linearised dynamics constitute a time-varying linear system that may be
*temporarily *uncontrollable or *temporarily *unreconstructable. This
highly relevant information for the LQG feedback controller design is not detected
by any of the four conventional Kalman decompositions. Temporal uncontrollability
is associated to differential controllability in continuous-time and to N-step controllability
is discrete-time. The description and detection of temporal uncontrollability and
temporal unreconstructability in both continuous and discrete time requires the
introduction of systems with time-varying (state) dimensions. These type of systems
moreover enable a well rounded realization theory for time-varying linear systems
as already suggested by Kalman in 1969. The description of systems with time-varying
dimensions in combination with suitable Kalman decompositions does enable the description
and detection of temporal uncontrollability and temporal unreconstructability [43],
[45], [47].
Associated to temporal uncontrollability and temporal unreconstructability are the
properties temporal and differential stabilizability and detectability [52],
[57], [58].
They determine whether temporal uncontrollability and temporal unreconstructability
cause temporal instability of the closed loop system. In case of perturbation
feedback control of nonlinear systems about trajectories causes and fixes of
temporal uncontrollability of the linerized dynamics about the trajectory have
been determined. These are linked to the structure of the nonlinear system
dynamics as well as properties of the trajectory to be tracked [S2016-1].

Introduction / Subjects / Publications / Submitted publications / Software

**Optimal sampling rates and LQG control**

The existence of optimal sampling rates and a way to compute them
have been established for digital time-varying LQG tracking problems with an exogenous
component and costs associated to taking measurements. They have been published
in an economic context. Currently the extension to unconventional sampling schemes
and the digital optimal control of non-linear systems is under investigation. Publications:
[3], [4], [8], [15],
[18], [23].

Introduction / Subjects / Publications / Submitted publications / Software

**Actively (dual) adaptive receding horizon
controllers**

Digital optimal control and LQG compensation of non-linear systems is approximately optimal only if the model structure and its parameter values are well established and if the remaining uncertainty is small and may be represented by small additive white noise. If the model structure is well established but some of the parameter values are not, while the on-line data contain information regarding these parameter values, actively adaptive controllers are required. As opposed to other (passively) adaptive controllers, actively adaptive controllers take into account the need to identify the parameters in selecting the control [Y. Bar-Shalom, 1981, "Stochastic dynamic programming: Caution and probing", IEEE Trans. Aut. Contr., AC-26(5), 1184-1195 ]

Based on an extensive literature search an actively adaptive digital controller structure has been developed which involves a least squares parameter estimator and a receding horizon optimal controller. Besides being actively adaptive the advantage of this structure is that it consists of two well established parts in the control literature being the least squares parameter estimator and the receding horizon optimal controller. Due to the receding horizon optimal controller this structure computes a reasonably accurate "cost to go" and due to the least squares parameter estimator it exploits fully all the currently collected data to estimate the uncertain parameters. A disadvantage is that this controller structure is computationally expensive, both in real-time and in simulation. In real-time this problem can be relaxed by selecting larger sampling periods which does not pose a problem since our digital controller design procedure explicitly considers the inter-sample behaviour. The current application areas are greenhouse climate control and the control of drying and sterilization processes where initially some of the model parameters are not very well known. Related publication: [40]

Introduction / Subjects / Publications / Submitted publications / Software

**Minimal realization of time-varying systems
with time-varying dimensions**

When designing perturbation feedback controllers for non-linear systems based on linearized dynamics the linearized dynamics may be temporarily uncontrollable/unreconstructable [1]. To detect this as well as to obtain a satisfactory realization theory for time-varying continuous-time systems piecewise constant rank systems are introduced. They constitute continuous-time systems with time-varying dimensions. In combination with a Kalman decomposition based on differential controllability and differential reconstructability they enable the description and detection of temporal system structure as well as temporal uncontrollability and/or unreconstructability [43], [45], [47].

Introduction / Subjects / Publications / Submitted publications / Software

**Applications of receding horizon controllers
**

Receding horizon controllers are especially appropriate for control problems where the data provide accurate state information but poor or no information concerning possible model improvements, such as the values of certain parameters. The current applications areas are greenhouse climate control and the control of sterilization and drying processes. Publications: [11], [12], [16], [21], [22].

Introduction / Subjects / Publications / Submitted publications / Software

**Digital time-optimal control of mechanical
systems**

Algorithms for the digital time-optimal control of mechanical systems, subjected to actuator constraints, have been developed. It concerns either systems that travel a path in space or systems that travel from a specified initial to a specified final configuration. For n-dimensional non-linear systems, linear in the control variables, it has been shown that non-singular solutions to time-optimal control problems with bounded control and fixed initial and final states generally have no more than a total of n-1 switch times. Publications: [1], [2], [5], [29] .

Introduction / Subjects / Publications / Submitted publications / Software

**Solution of optimal control problems with
state constraints using Chebyshev polynomials**

A known numerical algorithm based on Chebyshev polynomials [J. Vlassenbroeck, "A Chebyshev polynomial method for optimal control with state constraints", 1988, Automatica , vol. 24 (4), pp. 499-506 ] used to solve continuous-time optimal control problems, including state and control constraints, has been investigated and programmed in Matlab to be applicable to general problems.

Introduction / Subjects / Publications / Submitted publications / Software

**Application of digital controllers in economics**

The application of digital optimal fixed and reduced-order LQG controllers, especially those for systems with white parameters, has large potentials in economics. Therefore the transformation of digital control problems, involving systems with stochastic parameters, into discrete-time equivalents, in the case of unconventional sampling, is under investigation. Publications: [8], [17].

Introduction / Subjects / Publications / Submitted publications / Software

The digital optimal control algorithm for non-linear systems [10] has been successfully implemented, as a receding horizon optimal controller, in a greenhouse optimal control experiment [12]. Difficulties concerning different time-scales of the system have been partially solved but remain under investigation. Publications: [6], [7], [11], [12], [16a], [34], [37].

Introduction / Subjects / Publications / Submitted publications / Software

**Mechanical design and digital optimal control
of a tomato harvester**

Sponsored by DISC (Dutch Institute of Systems and Control) the mechanical design and digital control of a tomato-picking robot is under investigation. An end-effector to pick all the fruits on a single truss has been designed is currently being manufactured. Currently the tomato picking robot is the MK-II of Eshed Robotec. Software to control both the end-effector and the robot is being developed and will be implemented using MATLAB and dSPACE control equipment. Publications: [22a], [29], [36], [42].

Introduction / Subjects / Publications / Submitted publications / Software

**Automatic guidance and high precision
control of agricultural field machines**

The automatic guidance and high precision control of agricultural field machines is under investigation. The research is performed in corporation with IMAG-DLO (Van Zuydam). Among the sensors used is GPS. Also the design of a digital control system to control the movement of a plow is under investigation. This research is conducted in corporation with Rumpstadt, an agricultural field machine manufacturer. Publication: [14] .

Introduction / Subjects / Publications / Submitted publications / Software

Digital LQG tracking controllers have been successfully implemented on the "ball and plate" laboratory set-up which constitutes an almost linear two-input system. A "helicopter type" laboratory set-up has been manufactured, which constitutes a non- linear multivariable two-input system. The laboratory set-ups are used for education and for evaluating digital optimal control algorithms designed for different sampling strategies. Software, based on MATLAB and dSPACE control equipment, has been developed, that allows for the implementation of arbitrary sampling strategies (asynchronous, random).

Introduction / Subjects / Publications / Submitted publications / Software

**Evolutionary algorithms for optimal control
**

Several non-linear optimal control problems
in the chemical and food industry as well as in the area of climate control, have
local solutions. The majority of optimal control algorithms are gradient based and
are therefore inclined to find local solutions. Evolutionary algorithms have the
potential of locating global solutions but there efficiency is generally poor. The
work of Lopez Cruz [26], [27],
[28], [32], [33]
has (finally) convinced me that differential evolution algorithms, an efficient
type of evolutionary algorithm, can compete and outperform existing optimal control
algorithms developed to find global solutions.

Introduction / Subjects / Publications / Submitted publications / Software

**Equipartition and the second law of thermodynamics**

Having to teach a small course on thermodynamics raised our interest
in the foundations of the second law of thermodynamics that are still debated among
scientists. In a recent paper [48]
we investigated these foundations starting from the hypothesis that the second law
should have a "Darwanian explanation" that entirely follows from the simple reversible
Hamiltonian dynamics of particles at the micro-scale. This work was also inspired
by a recent book that takes a dynamical systems approach to thermodynamics (Haddad,
Chellaboina, Nersesov, 2005, Thermodynamics A dynamical systems approach, Princeton
University Press) and by some recent publications on equipartition of energy (Bernstein
& Bhat, Rapisarda & Willems) also from a dynamical systems perspective.

Introduction / Subjects / Publications / Submitted publications / Software

**Connecting and unmasking relativity and quantum theory**

Many people tell you not to even try to understand physics, especially Einsteinâ€™s relativity and most of all quantum theory. Even among physicists themselves the advice not to try to understand quantum theory is heard. The good news is that it is actually quite simple to understand both! To see this it is necessary to cut of the mask of both relativity and quantum theory. Both are not entirely without error and these errors have obscured what can now be simply understood. From an appropriate inspection of the history of relativity and quantum theory modifications of both are proposed. Also by inspection of the literature part of these modifications can be found, but only separately. When combined these provide a conceptually very simple model of physics that is largely equivalent to acoustic waves moving in air. Particles do not exist, they are local manifestations of wave structures so matter has a wave structure (WSM: wave structure of matter) immediately explaining wave-particle duality. At the very bottom of this simple model describing wave propagation lies a computational difficulty. That probably explains why all kinds of approximate models have been proposed leading to many paradoxes as it comes to understanding fundamental physics. Although the understanding of fundamental physics through our proposed model becomes easy the challenge shifts towards explaining how all known physics emerges from this simple model. Wave interference, wave modulation, and the Doppler effect will be prominent parts of these explanations. A paper describing these ideas and results has been published [63].

Mail comments, suggestions or questions to: Gerard van Willigenburg

[0] L.G.
Van Willigenburg, 1989, "True digital tracking for an orthogonal robot
manipulator",
Proceedings. ICCON '89. IEEE International Conference on Control and
Applications.

[1] L.G.
Van Willigenburg, 1990, "First-order controllability and the time optimal control
problem for rigid articulated arm robots with friction",
Int.
J. Contr., Vol. 51, no 6, pp. 1159-1171.

[2] L.G. Van Willigenburg, 1991, "Computation of time-optimal
controls applied to rigid manipulators with friction",
Int.
J. Contr., Vol. 54, no 5, pp. 1097-1117.

[3] L.G. Van Willigenburg, W.L. De Koning, 1992, "The digital
optimal regulator and tracker for stochastic time-varying systems",
Int.
J. Syst. Sci., Vol 23, no 12, pp. 2309-2322.

[4] L.G. Van Willigenburg, 1992, "Computation of the digital
LQG regulator and tracker for time-varying systems",
Opt.
Contr. Appl. Meth., Vol. 13, pp. 289-299 , request associated Matlab software:
ml4.tar

[5] L.G. Van Willigenburg, 1993, "Computation and implementation
of digital time-optimal feedback controllers for an industrial X-Y robot subjected
to path, torque and velocity constraints",
Int. J.
Rob. Res., Vol. 12, no 5, pp. 420-433.

[6] M. Tchamitchian, L.G. Van Willigenburg, G. Van Straten, 1993,
"Optimal control applied to tomato crop production in a greenhouse",
Proceedings
of the 2nd European Control Conference, Groningen, The Netherlands, June 28-July
4, 1993, pp. 1348-1352.

[7] R.F. Tap, L.G. Van Willigenburg, G. Van Straten, 1993, "Optimal
control of greenhouse climate: computation of the influence of fast and slow dynamics",
Proceedings 12th IFAC World Congress, Sydney Australia, 18-23 July 1993.

[8] J.C. Engwerda, L.G. Van Willigenburg, 1995, "Optimal Sampling-Rates
and Tracking Properties of Digital LQ and LQG Tracking Controllers for Systems with
an exogenous component and Costs Associated to Sampling".
Computational
Economics 8, pp. 107-125, 1995., request associated Matlab software:
ml8.tar

[9] L.G. Van Willigenburg, W.L. De Koning, 1995, "Derivation
and computation of the digital LQG regulator and tracker in the case of asynchronous
and aperiodic sampling",
C-TAT,
Vol. 10, no 4, part 5, pp. 2083-2098, request associated Matlab software:
ml9.tar

[10] L.G. Van Willigenburg, 1995, Digital optimal control and
LQG compensation of asynchronous and aperiodically sampled non-linear systems",
Proceedings
3rd European Control Conference, Rome, Italy, September 1995, Vol. 1, pp. 496-500.

[11] R.F. Tap, L.G. Van Willigenburg, G. Van Straten, 1996, "Receding
horizon optimal control of greenhouse climate using the lazy man weather prediction",
Proceedings of the 13th IFAC World Congress, San Francisco, USA, 30 June-5 July,
1996, paper 4a-01 3.

[12] R.F. Tap, L.G. Van Willigenburg, G. Van Straten, 1996,
"Experimental results of receding horizon optimal control of greenhouse climate",
Acta
Horticulturae, 406, pp. 229-238.(Proceedings of the Second IFAC/ISHS Workshop
on Mathematical and Control Applications in Agriculture and Horticulture, Silsoe
UK, 12-15 sept. 1994).

[13] W.L. De Koning, L.G. van Willigenburg, 1998, "Numerical
algorithms and issues concerning the discrete-time optimal projection equations
for systems with white parameters",
Proceedings
UKACC International Conference on Control '98, 1-4 Sept. 1998, University of Swansea,
UK, Vol. 2, pp.1605-1610. , request associated Matlab software:
ml13.tar

[14] H.L. Dijksterhuis, L.G. Van Willigenburg, R.P. Van Zuydam,
1998, "Centimetre-precision guidance of moving implements in the open field: a simulation
based on GPS measurements", Computers and Electronics in agriculture, 20, pp. 185-197.

[15] L.G. van Willigenburg, W.L. De Koning, 1999, "Optimal
reduced-order compensators for time-varying discrete-time systems with deterministic
and white parameters",
Automatica,
35, 129-138 , request associated Matlab software: ml15.tar

[16] Z.S. Chalabi, L.G. van Willigenburg, G. van Straten, 1999, "Robust optimal
receding horizon control of the thermal sterilization of canned food",
Journal
of Food Engineering, 40, pp. 207-218.

[16a] G. Van Straten, R.F. Tap, L.G. Van Willigenburg, 1999,
"Sensitivity of on-line RHOC of greenhouse climate to adjoint variables for the
crop",
Proceedings
14th IFAC World Congres, Beiing, China, July 5-9, 1999, Paper no. K-4a-01-1.

[17] L.G. van Willigenburg, W.L. De Koning, 2000, "The equivalent
discrete-time optimal control problem for time-varying continuous-time systems with
white stochastic parameters",
International
Journal of System Science, 31, 4, pp. 479-487., request associated Matlab software:
ml17.tar

[18] L.G. Van Willigenburg, W.L. De Koning, 2000, "Numerical
algorithms and issues concerning the discrete-time optimal projection equations",
European
Journal of Control, 6, 1, pp. 93-110, request associated Matlab software:
ml18.tar

[19] L.G. van Willigenburg, W.L. De Koning, 2000, "Finite
and infinite horizon fixed-order LQG compensation using the delta operator",
Proceedings
UKACC International Conference on Control 2000, 4-7 September, Cambridge, UK, (paper002.pdf)

[20] E.J. Quirijns, L.G. Van Willigenburg, A.J.B. van
Boxtel, 2000, "New perspectives for optimal control of drying processes",
Proceedings
ADCHEM 2000 International Symposium on Advanced Control of Chemical Processes, pp.
437-442, 14-16 June, Pisa,Italy

[21] L.G. van Willigenburg, E.J. Van Henten, W.Th.M.
Van Meurs, "Three time-scale receding horizon optimal control in a greenhouse with
a heat storage tank",
Proceedings
of the Agricontrol 2000 conference, 12-14 July 2000 , Wageningen, The Netherlands

[22] M. Timmerman, L.G. van Willigenburg, A. Van 't
Ooster, "Automatic receding horizon optimal control of the natural ventilation in
cattle barns", MRS report 2000-11,
Proceedings
of the Agricontrol 2000 conference, 12-14 July 2000., Wageningen, The Netherlands

[22a] E.J. van Henten, G. van Dijk, M.C. Kuypers, B.A.J.
van Tuijl, L.G. van Willigenburg, 2000, "Motion planning for a cucumber picking
robot",
Proceedings
of the Agricontrol 2000 conference, 12-14 July 2000, Wageningen, The Netherlands,
pp. 39-44.

[23] L.G. van Willigenburg, W.L. De Koning, 2001, "Synthesis
of digital optimal reduced-order compensators for asynchronously sampled systems",
International Journal of System Science, 32, 7, pp. 825-835 , request associated
Matlab software: ml23.tar

[24] W.L. De Koning, L.G. van Willigenburg, 2001, Randomized
digital optimal control,
Chapter
12 in Non Uniform Sampling Theory and Practice , Kluwer Acadamic/Plenum Publishers,
ISBN 0-306-46445-4, Edited by Farokh Marvasti, request associated Matlab software:
ml24.tar

[25] L.G. Van Willigenburg, W.L. De Koning, 2002, "Minimal
and non-minimal optimal fixed-order compensators for time-varying discrete-time
systems",
Automatica,
38, 1, pp. 157-165.

[26] I.L. Lopez Cruz, L.G. van Willigenburg, G. van Straten,
2000, "Evolutionary algorithms for optimal control of chemical processes", Proceedings
of the IASTED International Conference on Control Applications, May 24-27, Cancun
Mexico, pp. 155-161.

[27] I.L. Lopez Cruz, L.G. van Willigenburg, G. van
Straten, 2001, "A parameter control strategy inside differential evolution algorithms
for optimal control", Proceedings of the IASTED International Conference on Artificial
Intelligence and Soft Computing, May 21-24, Cancun, Mexico, pp. 211-216.

[28] I.L. Lopez Cruz, L.G. van Willigenburg, G. van
Straten, 2001, "Optimal control of nitrate in lettuce by gradient and differential
evolution algorithms", Proceedings 4th IFAC workshop on Artificial Intelligence
in Agriculture, Budapest, Hungary, 6-8 June, pp. 123-128.

[29] C.W.J. Hol, L.G, van Willigenburg, E.J. van Henten,
G. van Straten, 2001, "A new optimization algorithm for singular and non-singular
digital time-optimal control of robots",
Proceedings IEEE International Conference
on Robotics and Automation (ICRA), May 21-26, Seoul, Korea, vol. 2, pp. 1136 -1141.

[30] L.G. van Willigenburg, W.L. De Koning, 2002, "Minimality,
canonical forms and storage of finite-horizon discrete-time compensators",
Preprints
IFAC World Congress, Barcelona, 21-26 July, 2002, paper 984.

[31] G. van Straten, L.G. van Willigenburg, R.F. Tap,
2002, "The significance of crop co-states for receding horizon optimal control of
greenhouse climate",
Control
Engineering Practice, 10, 625-632.

[32] I.L. Lopez Cruz, L.G. van Willigenburg, G. van Straten,
2003. Efficient evolutionary algorithms for multimodal optimal control problems,
Journal of Applied Soft Computing 3 (2): 97-122.

[33] I.L. Lopez Cruz, L.G. van Willigenburg, G. van Straten,
2003. Optimal control of nitrate in lettuce by a hybrid approach: differential evolution
and adjustable control weight gradient algorithms,
Computers and Electronics in Agriculture 40 (1-3): 179-197.

[34] R.J.C. van Ooteghem, J.D. Stigter, L.G. Van Willigenburg,
G. Van Straten, 2004. "Optimal control of a solar greenhouse",
European Control
Conference 2003, University of Cambridge, Cambridge, United Kingdom, September 1-4,
2003.

[35] L.G. van Willigenburg, J. Bontsema, W.L. De Koning,
L. Valenzuela, C. Martinez, "Digital optimal reduced-order control of a solar power
plant",
Proceedings
of the UKACC Control 2004, Paper 205, Bath, UK, 6-9 September 2004.

[36] L.G. van Willigenburg, C.W.J. Hol, E.J. van Henten,
2004, "On-line near minimum time path planning and control of an industrial robot
picking fruits",
Computers
and Electronics in Agriculture, 44, 3, 223-237.

[37] Ooteghem, R.J.C. van; Stigter,
J.D.; Willigenburg, L.G. van; Straten, G. van (2004 )
Receding Horizon Optimal Control of a Solar Greenhouse.
GreenSys2004, Leuven, 2004 september 12-16, In: GreenSys2004
Sustainable Greenhouse Systems : Greensys2004, Leuven, 12-16 September 2004.

[38] L.G. van Willigenburg, J. Bontsema, W.L. De Koning,
L. Valenzuela, C. Martinez, 2004, "Direct reduced-order digital control of a solar
power plant", In: The improving human potential programme: access to research infrastructure
activities. research results at Plata Forma Solar de Almeria within the year 2003
access campaign, pp. 9-16, ISBN 84-7834-474-8.

[39] L.G. van Willigenburg, W.L. De Koning, 2004, "UDU
factored discrete-time Lyapunov recursions solve optimal reduced-order LQG problems",
European
Journal of Control, 10, pp. 588-601., request associated Matlab software:
ml39.tar. Discussions on this paper (one of them by D.C.
Hyland who discovered the optimal projection equations) and final comments from
us,
European
Journal of Control, 10, pp. 602-613.

[40] L.G. van Willigenburg, W.L. De Koning, Z.S. Chalabi,
M. Tchamitchian, 2005, "On the selection of appropriate control system design methodologies",
Preprints IFAC World Congres, Prague, 3-8 July,
04926.pdf.

[41] L.G. van Willigenburg, W.L. De Koning, 2006, "On
the synthesis of time-varying LQG weights and noises along optimal control and state
trajectories",
Optimal
Control Applications and Methods, 27, 137-160. , request associated Matlab software:
ml41.tar

[42] Henten, E.J. van, Slot, D.A. van 't, Hol, C.W.J.,
Willigenburg, L.G. van, 2006, "Optimal design of a cucumber harvesting robot", Proceedings
of AgEng 2006, Bonn, Germany, 3-7 september 2006,
Paper
nr. 802, 6 pp.

[43] L.G. van Willigenburg, W.L. De Koning, "A Kalman
decomposition to detect temporal linear system structure", Proceedings European
Control Conference 2007, Kos, Greece, July 2-7,
Paper
no. 78, 6 pp, request associated Matlab software:
ml43.tar

[43a] G van Straten, L.G. van Willigenburg, 2008, "On Evaluating Optimality
Losses of Greenhouse Climate Controllers",
Proceedings of the 17th World
Congress, The International Federation of Automatic Control, Seoul, Korea,
July 6-11.

[44] L.G. van Willigenburg, W.L. De Koning, 2008, "Linear
systems theory revisited,
Automatica,
44, 1669-1683, request associated Matlab software:
ml44.tar

[45] L.G. van Willigenburg, W.L. De Koning, 2008, "Temporal
linear system structure",
IEEE
Transactions on Automatic Control, 53, 5, 1318-1323, request associated Matlab
software: ml45.tar

[46] L.G. van Willigenburg, W.L. De Koning, 2008, "How
non-zero initial conditions affect the minimality of linear discrete-time systems",
International Journal of System Science, 39, 10, 969-983, request associated
Matlab software: ml46.tar

[47] L.G. van Willigenburg, W.L. De Koning, "Temporal
linear system structure: The discrete-time case",
Proceedings of the ECC 2009, 23-26 August, 2009, Budapest, pp. 225-230,
request associated Matlab software: ml47.tar

[48] L.G. van Willigenburg, W.L. De Koning, 2009, "Emergence
of the second law out of reversible dynamics",
Foundations of Physics, 39, 1217-1239,
Open
Access , request associated Matlab software: ml48.tar

[49] E.J. Van Henten, D.A. van 't Slot, C.W.J. Hol,
L.G. van Willigenburg, 2009, Optimal manipulator design for a cucumber harvesting
robot,
Computers
and Electronics in Agriculture 65 (2), pp. 247-258.

[50] E.J. Van Henten, E.J. Schenk, L.G. van Willigenburg,
J. Meuleman and P. Barreiro, 2010, "Collision-free inverse kinematics of the redundant
seven-link manipulator used in a cucumber picking robot",
Biosystems Engineering, 106, 2, 112-124.

[51] L.G. van Willigenburg, W.L. De Koning, 2010, "Compensatability
and optimal compensation of systems with white parameters in the delta domain",
International
Journal of Control, 83, 12, 2546-2563, request associated Matlab software:
ml51.tar

[52] L.G. van Willigenburg, W.L. De Koning, 2012, "Temporal
and differential stabilizability and detectability of piecewise constant rank systems",
Optimal Control Applications & Methods, 33, 302-317, request associated Matlab
software: ml52.tar

[53] G. van Straten, L.G. van Willigenburg, R.J.C. van
Ooteghem, E.J. van Henten, 2011, Optimal Control of Greenhouse Cultivation, CRC
Press, ISBN 978-1-4200-5961-8, request associated Matlab
software: ml53.tar

[54] L.G. Van Willigenburg, W.L. De Koning, 2014, "Equivalent
optimal control problem in the delta domain for systems with white stochastic parameters",
International Journal of System Science, 45, 3, 509-522, request associated Matlab
software: ml54.tar

[55] L.G. Van Willigenburg, W.L. De Koning, 2013, "Minimal
representation of matrix valued white stochastic processes and U-D factorization
of algorithms for optimal control",
International Journal of Control, 86, 2, 309-321, request associated Matlab
software: ml55.tar

[56] L.G. Van Willigenburg, W.L. De Koning, 2013, "Temporal and one-step stabilisability and detectability of discrete time linear systems",
IET Control Theory & Applications, 7, 1, 151-159, request associated Matlab
software: ml56.tar

[57] L.G. Van Willigenburg, W.L. De Koning, 2013,
"Temporal and one-step stabilizability and detectability
of time-varying discrete-time linear systems",
System Modeling and Optimization, IFIP Advances in Information and Communication Technology,
Volume 391, 306-317, request associated Matlab software: ml57.tar

[58] L.G. Van Willigenburg, W.L. De Koning, 2014,
"Theoretical and numerical issues concerning temporal stabilisability and detectability",
Preprints ACODS 2014 Conference, Kanpur, India, 13-15 March,
paper 117, 368-375, request associated Matlab software: ml58.tar

[59] B.A. Vroegindeweij, L.G. van Willigenburg, P.W.G. Grootkoerkamp, E.J. van Henten, 2014,
"Pathplanning for the autonomous collection of eggs on floors",
Biosystems Engineering,121,186-199.

[60] L.G. Van Willigenburg, W.L. De Koning, 2014, "U-D factorization of the strengthened discrete-time optimal projection equations",
International Journal of System Science, http://dx.doi.org/10.1080/00207721.2014.911388.

[61] L.G. Van Willigenburg, H.M. Vollebregt, R.G.M. van der Sman, 2015, "Optimal adaptive scheduling and control of beer membrane filtration",
Control Engineering Practice, http://dx.doi.org/10.1016/j.conengprac.2014.10.004.

[62] L.G. Van Willigenburg, W.L. De Koning, 2015, "Temporal stabilizability and compensatability of time-varying linear discrete-time sytems with white stochastic parameters",
European Journal of Control, http://dx.doi.org/10.1016/j.ejcon.2015.01.005.

[63] W.L. De Koning, L.G. Van Willigenburg, 2015, "Connecting and unmasking relativity and quantum theory",
Physics Essays, 28, 3, 392-398.

[64] L.G. Van Willigenburg, W.L. De Koning, 2016, "Improvements and corrections concerning UD factorisations of algorithms for optimal full and reduced-order output feedback"
Journal of Control Engineering and Technology, 6, 2, 14-18, request associated Matlab software:ml64.tar

[65] E.A.Y. Amankwah, K. A. Dzisi, G. van Straten, L.G. van Willigenburg, A.J.B. van Boxtel, 2017, "Distributed mathematical model supporting design and construction of solar collectors for drying"
Drying Technology, https://doi.org/10.1080/07373937.2016.1269806.

**Submitted publications**

[S2016-1] L.G. Van Willigenburg, "Causes and fixes of temporal and conventional uncontrollability about control and state trajectories of nonlinear systems".

[S2017-1] L.G. Van Willigenburg, W.L. De Koning, "Wave structure of matter explaining time dilation and length contraction in classical physics".

[S2017-2] Dan Xu, Shangfeng Du, L.G. Van Willigenburg, "Adaptive two time-scale receding horizon optimal control for greenhouse lettuce cultivation".

[S2017-3] Dan Xu, Shangfeng Du, L.G. Van Willigenburg, "Optimal control of Chinese solar greenhouse cultivation".

[S2017-4] J.D. Stigter, L.G. Van Willigenburg, J. Molenaar, "An Efficient Method to Assess Local Controllability and Observability for Non-Linear Systems",
IFAC Conference MATHMOD 2018, Februari 21-23, Vienna, Austria (accepted for
publication and oral presentation)

**Software associated to the publications available
upon request **

Part of the software is also available through Matlab Central: http://www.mathworks.nl/matlabcentral/fileexchange/authors/31018

[04] Associated Matlab software ml4.tar

[08] Associated Matlab software ml8.tar

[09] Associated Matlab software ml9.tar

[13] Associated Matlab software ml13.tar

[15] Associated Matlab software ml15.tar

[17] Associated Matlab software ml17.tar

[18] Associated Matlab software ml18.tar

[23] Associated Matlab software ml23.tar

[24] Associated Matlab software ml24.tar

[39] Associated Matlab software ml39.tar

[41] Associated Matlab software ml41.tar

[43] Associated Matlab software ml43.tar

[44] Associated Matlab software ml44.tar

[45] Associated Matlab software ml45.tar

[46] Associated Matlab software ml46.tar

[47] Associated Matlab software ml47.tar

[48] Associated Matlab software ml48.tar

[51] Associated Matlab software ml51.tar

[52] Associated Matlab software ml52.tar

[53] Associated Matlab software ml53.tar

[54] Associated Matlab software ml54.tar

[55] Associated Matlab software ml55.tar

[56] Associated Matlab software ml56.tar

[57] Associated Matlab software ml57.tar

[58] Associated Matlab software ml58.tar

[64] Associated Matlab software ml64.tar