scispace - formally typeset

Journal ArticleDOI

A convex approach to a class of non-convex building HVAC control problems: Illustration by two case studies

15 Apr 2015-Energy and Buildings (Elsevier)-Vol. 93, pp 269-281

AbstractIn this paper, a convexification approach is presented for a class of non-convex optimal/model predictive control problems more specifically applied to building HVAC control problems. The original non-convex problems are convexified using a convex envelope approach. The approach is tested on two case studies: a benchmark building HVAC system control problem from the literature and control of a hybrid ground-coupled heat pump (HybGCHP) system. For the first application, convexified model predictive control was used and results were compared with fuzzy and adaptive control results. For the HybGCHP system, convexified optimal control was applied and the results were compared with dynamic programming based optimal control. In the first case superior performance was observed over the corresponding fuzzy and adaptive control results from the literature. For the HybGCHP system the associated convexified optimal control gave almost global optimal results in terms of responses and cost criteria. The suggested method is especially useful for optimal building HVAC control/design problems which include non-convex bilinear or fractional terms. Since a polynomial expression can be recursively expressed as a system of bilinear equations, the proposed method, in principle, can be applied to all systems where polynomial non-convexities exist.

Topics: Optimal control (60%), Adaptive control (58%), Convex optimization (55%), Model predictive control (55%), Convex hull (53%)

Summary (4 min read)

1. Introduction

  • In the context of energy-efficient buildings HVAC control has gained increasing attention in recent years.
  • This leads to the risk of designing a non-working controller on the real system or a working controller with suboptimal results.
  • De Ridder et al. and Verhelst [12, 5] used mathematical model-based control methods for HybGCHP systems, which allow global optimization.
  • There xist analytical formulas for a bilinear function or a rational function of two varibles.

2. PART I: Theoretical Foundations

  • First, the authors introduce a class of non-convex control p blems which include the optimal/model predictive control of HybGCHP systems.
  • Next, the proposed convex relaxation method for the given non-convexcontrol problems is detailed.
  • Since in Section 3.2 the proposed convex relaxation is applied to optimal control of HybGCHP systems and the results are compared with its dynamic programming-based control results, the authors also give a short oveview of the dynamic programming at the end of this part.

2.1.2. Non-convex Model Predictive Control Problem Class

  • The rest of descriptions of variables and functions are the same as in Section 2.1.1.
  • In MPC control, the control input over the prediction horizon is calculated at every time step and the first element of the control input vector is applied at the current time step.
  • At the next time step, the same calculation procedure is peated.
  • Due to this implementation scheme, MPC is sometimes called receding horizon control.

2.2.1. Overview of convex optimization

  • This means that if the authors take any two points inS and draw a line segment between these two points, then every point on that line segmentalso belongs toS. Definition 2.2 (Convex Function [13]).
  • This definition means that if the authors take any two points on the graph of a convex functionf and draw a straight line between them, then the portion of thefunction between these two points will lie below this straight line.
  • Armed with the above definitions of convex sets and functions, the authors are now ready to define a convex optimization problem.
  • Definition 2.3 (Convex Optimization Problem: the most general form [13]).
  • Given a convex functionf , a convex setS and the decision variable vectorx, the associated convex optimization problem is defined as minimizef(x) subject tox ∈ S. Definition 2.4 (Convex Optimization Problem: less general form [13]).

2.2.2. Convex Envelope

  • Next, the concept of convex envelope is defined.
  • Given a continuous functionk(x), its convex envelope, denoted byconvk(x), over a convex setS is defined as the pointwise supremum of all convex functions which are majorized byk(x): convk(x) = sup{r(x)| r convex andr(y) < k(y) ∀y ∈ S}.
  • The overall system of relaxed constraints consists of 18 scalar convex constraints, which can be compactly represented as Efrw ≥ gfr-c(x1, x2), (12) whereEfr is a constant vector andgfr-c(x1, x2) is a convex function with respect to the variablesx1, x2.
  • The derivation, although simple, is lengthy and hence was skipped.

2.4. Recap of Dynamic Programming

  • Dynamic programming is a closed-loop, global optimal contrl method (global optimal up to approximations due to state-input gridding and interpolations).
  • It is based on the “principle of optimality” [16] which simply says that in a multi-stage process whatever the previous states are, the remaining decisions must be optimal with regard to the state following from the current state.
  • This principle allows the optimal control problem of aK-stage process to be recursively formulated starting from the last stage.
  • For dynamic programming-based control methods, the most important issue is to have an accurate model with minimum number of states and inputs due to the famous curse of dimensionality problem [16, 17].
  • The reader is referred to references [16, 17] for details on dynamic programming.

3. PART II: Applications

  • The purpose of this section is to test the proposed convexification-based control methods first on a benchmark HVAC building control system case study from the literature to which the authors apply the convexified MPC control method and compare these control results with the adaptive and fuzzy control results from the literature.
  • Next, the optimal control of a HybGCHP system is considered,which is the main application example for the developed methods.
  • For the HybGCHP system, the convexified optimal control results are compared with the dynamic programming control results of the same system to assess the performanceof th convexified optimal control method.

3.1. Application to a Benchmark Problem

  • The benchmark case study considered in this paper is the building heating control system, as described by Calvino et al. [18] and Chaudry and Das [19].
  • The model parameter values used in Eq.(14) are taken from Chaudhry and Das [19] and are given in Table 1.
  • Figure 3 shows the corresponding results when convexified MPC is usedwith Np = 10.

3.2.1. HybGCHP System Description

  • It is assumed that the heat demand (Q̇h) is provided by the heat pump and the boiler and the cold demand (Q̇c) is provided by the passive cooler (using a heat exchanger instead of an active chiller) and the chiller.
  • The coefficients of performance given by the above expressions depend on the temperatures of the source and the emission system, as expressed by COPhp = fhp(Tf , Tsw,h), COPpc = fpc(Tf , Tsw,c), COPch = fch(Ta, Tsw,c), whereTsw,h, Tsw,c represent the supply water temperature for heating and supply water temperature for cooling, respectively.
  • Next, the authors will discuss these constraints and present their expressions.

3.2.2. Heat & Cold Demand Satisfaction

  • The building heating and cooling demands should be satisfiedwith some acceptable violation margins: Q̇h(t)−.
  • Note that the margins are taken to be time-dependentto allow different degrees of flexibility over time.
  • During critical demand load periods these margins can be set very strictly.
  • It is assumed thatQ̇h andQ̇c are given and hence building modeling is not included in the optimization.

3.2.3. Circulating Fluid Temperature Bounds

  • The cooling of a building requires heat injection into the ground during summer.
  • This increases the ground temperature towards winter,which, in turn, increases COPhp.
  • The ground temperature, which is represented indirectly byTf , should be kept below the supply water temperature,Tsw,c, for passive cooling of the building.
  • Similarly, heating of a building requires heat extraction from the ground.

3.2.5. Borehole Dynamics

  • In the equivalent diameter approach, the heat transfer from the Utube is approximated by the heat transfer from a single pipe with a hypothetical diameter through which the heat exchanging fluid circulates.
  • T are soil nodal temperatures andp is the known parameter vector including thermal, physical and other parameters of the system (diffusivitiesαg, αs, conductivitieskg, ks, different radii rfg, rgs, discretization step sizes,etc.) andunet is the net heat power injected to the per borehole length.
  • POD is a flexible model-order reduction method compared to other model order reduction methods, also known as Remark 2.
  • Here, the authors consider optimal control which is an open-loop control method where control actions are c lculated off-line and the accuracy of these control actions is depending on the“simulation performance” of the model, not on its “prediction performance” because in classical optimal control measured values are not used and everythingis done off-line.
  • A too simple model cannot have a good simulation performance over a long period (like one year or multiple years), as shown in Figure 6.

3.2.6. Non-convex Optimal Control Problem for Total EnergyCost Minimization

  • (23j) With this formulation, the non-convexity of the optimal control problem comes from Eq.(23j): a bilinear term.
  • Note that the rational term includingTa in the cost function does not create any non-convexity becauseTa is not a decision variable or a function of decision variables.
  • In the next subsection, Eq.(23j) will be replaced by its convex approximation.

3.2.7. Convexified Optimal Control Problem and Comparison with Dynamic Programming

  • Note that all other constraints are already/or can be easilyput into the form given in Eq.(7).
  • To apply dynamic programming for assessing the performanceof onvexified optimal control, the optimization problem given by Eq.(23)is equivalently reformulated with two inputs instead of four by assuming zero violat n margins (see Eq.(17)) to alleviate the curse of dimensionality problem in dynamic programming.
  • This is done for allfe sible gridded states.
  • Figures 7(a)-7(b) show the time evolution of the mean temperature ofthe circulating fluid (Tf ), Figures 7(c)-7(d) show the heat pump power and heating demand (Q̇hp, Q̇h), Figures 8(a)-8(b) show the chiller power and cold demand (Q̇ch, Q̇c) and finally Figures 8(c)-8(d) show the evolution of accumulated cost (J).

3.3. HybGCHP System Control Using Convexified MPC

  • The non-convex model predictive control (NMPC) problem forHybGCHP system is shortly described as follows.
  • Here, the authors do not present the results for convexified MPC, they just wanted to show how the proposed idea can be used in the context of NMPC.
  • In practice this will cause the following problems.
  • Firstly, the large-scale model with 506 states is not observable since there are 506 states and only one input-output pair.

4. Conclusion

  • In this paper a convexification approach using convex envelopes for hard-tosolve non-convex optimization problems involving rational and/or bilinear terms of decision variables was proposed.
  • For the first case study of an HVAC building control system, the performance of convexified MPC was compared to the performance of fuzzy and adaptive control from the literature.
  • For the second application, the total energy costminimization of buildings with a HybGCHP system, convexified optimal control was used and its results were compared to dynamic programming based control results, which is a closedloop, global optimal control method (global optimal up to approximations in the gridding of states/inputs and used interpolations).
  • The overall message of this paper is that given a non-convex optimization/control problem of thermal systems, the first step should be to analyse the given system in terms of the non-convex terms and then investigate whether convex envelopes for the associated terms exist or not.
  • The accuracy degree of the approximation of the original non-convex optimization problem by a convex one may be very case dependent.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

A Convex Approach to a Class of Non-convex Building
HVAC Control Problems: Illustration by Two Case
Studies
Ercan Atam
, Lieve Helsen
Dept. of Mechanical Engineering, KU Leuven, Celestijnenlaan 300 box 2421, Leuven 3001,
Belgium.
Abstract
In this paper, a convexification approach is presented for a class of non-convex
optimal/model predictive control problems more specifically applied to building
HVAC control problems. The original non-convex problems are convexified using
a convex envelope approach. The approach is tested on two case studies: a bench-
mark building HVAC system control problem from the literature and control of a
hybrid ground-coupled heat pump (HybGCHP) system. For the first application,
convexified model predictive control was used and results were compared with
fuzzy and adaptive control results. For the HybGCHP system, convexified optimal
control was applied and the results were compared with dynamic programming
based optimal control. In the first case superior performance was observed over
the corresponding fuzzy and adaptive control results from the literature. For the
HybGCHP system the associated convexified optimal control gave almost global
optimal results in terms of responses and cost criteria. The suggested method is
especially useful for optimal building HVAC control/design problems which in-
clude non-convex bilinear or fractional terms. Since a polynomial expression can
be recursively expressed as a system of bilinear equations, the proposed method,
in principle, can be applied to all systems where polynomial non-convexities exist.
Keywords: HVAC control, Hybrid ground-coupled heat pumps, Convex
optimization, Convex envelope, Optimal control, Nonlinear model predictive
control
Corresponding author
Email address: Ercan.Atam@kuleuven.be (Ercan Atam)
Preprint submitted to Energy and Buildings February 8, 2015

Nomenclature
Variables Description Unit
COP coefficient of performance [-]
c
e
electricity price Euro/(kWh)
c
g
gas price Euro/(kWh)
C
a
heat capacity kJ/K
H
t
global heat transfer coefficient of the building enve-
lope
W/K
J total energy-use cost Euro
k conductivity W/(mK)
N
c
control horizon length [-]
N
p
prediction horizon length [-]
r
fg
grout region inner radius cm
r
gs
grout region outer radius cm
t time sec.
˙
P
ch
electrical power used by chiller W
˙
P
gb
electrical power used by gas boiler W
˙
P
hp
electrical power used by heat pump W
˙
P
pc
electrical power used by passive cooler W
˙
Q
c
cooling load demand W
˙
Q
ch
thermal power extracted from the building through
active cooling
W
˙
Q
ext
thermal power extracted from ground W
˙
Q
gb
thermal power supplied to the building by gas boiler W
˙
Q
gain
internal gains W
˙
Q
h
heating load demand W
˙
Q
hp
thermal power supplied to the building by the heat
pump
W
˙
Q
inj
thermal power injected to ground W
˙
Q
net
net thermal power injected to ground W
˙
Q
pc
thermal power extracted from the building through
passive cooling
W
T temperature
C
α
g
diffusivity; exponent m
2
/s; []
η
gb
gas boiler efficiency [-]

Subscripts
a ambient air
aff affine
c convex
ch chiller
f fluid
fr fractional
g grout
gb gas boiler
hp heat pump
i indoor air; inlet
max maximum
min minimum; minimize
p pipe
pc passive cooler
s soil
Abbreviations
DP dynamic programming
GCHP ground-coupled heat pump
HVAC heating, ventilation and air conditioning
HybGCHP hybrid ground-coupled heat pump
NMPC nonlinear model predictive control
OC optimal control
POD proper orthogonal decomposition
1. Introduction
In the context of energy-efficient buildings HVAC control has gained increas-
ing attention in recent years. Especially, future worries about the shortage of fuel
sources and the requirement of reduction in greenhouse gas emission levels neces-
sitate building HVAC control systems to be more efficient. HVAC devices and the
building itself are often modeled using physical principles of heat transfer, ther-
modynamics and fluid mechanics. These models usually include nonlinearities
and non-convexities which pose difficulties for controller design. Although it is
not the aim to list all nonlinearities and non-convexities encountered in building
HVAC control systems, among them the bilinear and fractional terms are the most

dominant ones. An example of a bilinear term in building HVAC applications is
the mass flow rate times temperature. The coefficient of performance of a heat
pump, which is the ratio of the thermal power delivered to the building over the
electrical power used is an example of a fractional expression in building HVAC
applications.
Once the building HVAC control system includes a bilinear or fractional term,
the underlying system is a nonlinear system from the control point of view and
it is a non-convex system from the optimization point of view. If the controller
design is based on optimization (like optimal or model predictive control), then
the controller design task basically involves solution of a non-convex optimiza-
tion problem. It is very hard to solve non-convex control problems over longer
control periods due to a large number of decision variables and the possibility
of divergence. Even in case of a solution, a global minimum cannot be guaran-
teed. Existing solvers cannot handle non-convex optimization problems with a
large number of decision variables. The simplest solution to such a non-convex
control problem is to linearize the model around some operating point and using
linear optimization. However, this leads to the risk of designing a non-working
controller on the real system or a working controller with suboptimal results. As
a result, linearization is not desirable and should be avoided whenever alternative
controller design options are available.
A challenging HVAC control application where bilinear/fractional terms ex-
ist is the control of ground-coupled heat pumps (GCHP) and hybrid ground-
coupled heat pump systems combined with low-exergy heat emission systems
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. The attractivity of such systems comes from having
the potential to reduce the primary energy use related to space heating and cooling
by 70% compared to conventional heating and cooling systems [11]. For GCHP
systems with vertical borehole heat exchangers (BHE), however, the large invest-
ment cost of the borefield represents a major bottleneck. This explains the trend
towards compact, hybrid GCHP systems which combine smaller borefields with
supplementary heating or cooling devices such as gas-fired boilers and chillers.
Although the design of a compact HybGCHP system is often driven by cost con-
siderations to limit the drilling cost without compromising thermal comfort in the
building, sometimes other reasons may also lead to HybGCHP systems, such as
limited drilling area for boreholes, the specific ground characteristics, regulation
or too high imbalance of the thermal load.
De Ridder et al. and Verhelst [12, 5] used mathematical model-based con-
trol methods for HybGCHP systems, which allow global optimization. However,
they are based on some simplifications and/or some unrealistic assumptions in-

troduced during the controller design. For example, De Ridder et al. [12] used
dynamic programming. Dynamic programming is a powerful method since it is
a closed-loop, global optimal control algorithm. However, the model used by De
Ridder et al. [12] for dynamic programming is a very simple first-order model
for the ground mean temperature. The chosen control time step for the system
is one week, which is very long since typical control actions for buildings may
require control time steps in the order of minutes or hours. Moreover, the real-
ization of the designed controller requires the measurement of the underground
field temperature, for which measurement may be either difficult or non-accurate.
As a result, the approach of De Ridder et al. [12] involves both some modeling
simplifications and a hard-to-realize implementation. Verhelst [5] applied a linear
optimal control method. The simplification made in this work is that the coeffi-
cients of performance (COP) for heat pump and chiller were taken to be constant,
in contrast to being functions of source and sink temperatures. COP values were
taken to be constant to avoid a non-convex optimization problem, which cannot
be solved over an horizon of a couple years especially when short control time
steps are considered. Although a mathematical model-based optimal control was
considered, the simplifications of taking the mentioned COPs as constant values
without a formal justification restricts the work of Verhelst [5]. Moreover, the
model used for control and emulator were the same, which neglects the impact of
model mismatch and therefore limits the generality of the approach followed.
The objective of this paper is to present and illustrate a convex relaxation
method for a class of non-convex optimal control and non-convex model predic-
tive control problems applied to two case studies, among which the control of a
HybGCHP system to minimize the total energy cost is a special case. The con-
vex relaxation method is based on the use of convex envelopes for bilinear and
fractional terms. The convex envelope of a function is the largest convex function
majorized by that function. Approximation of the non-convex terms by their con-
vex envelopes will transform the optimization problem to an approximate problem
which is convex and for which the global minimum can be found, if the problem
is feasible. In convex optimization problems, a local minimum is a global mini-
mum. Although the calculation of a convex envelope for a general multi-variable
function is non-deterministic polynomial-time hard, there exist analytical formu-
las for a bilinear function or a rational function of two variables. Moreover, it is
recursively possible to represent a polynomial non-convexity as a system of bilin-
ear equations and hence the proposed convexification method, in principle, can be
used for all systems where polynomial non-convexities exist.
Before testing the proposed convexification method on a HybGCHP system,

Citations
More filters

Journal ArticleDOI
Abstract: In this paper, a compact overview of the state-of-the-art in modeling of ground-coupled heat pump (GCHP) systems and an in-depth review of their optimal control along with the associated research challenges are given. The main focus is on optimal control but since design of an optimal controller may require a model, a relatively short literature review of modeling approaches is also discussed. Adopting the adage “a picture is worth a thousand words”, we tried to include a minimal number of representative schematics and result figures for some of the reviewed studies for clarity and a better understanding of the presented material. In addition to the literature review, we included our comments, points of view, alternative solutions and some potential future directions. This review paper is useful both for engineers and researchers involved in modeling and optimal control of GCHP systems. The second part of the paper, “Ground-Coupled Heat Pumps: Part 2 – Literature Review and Research Challenges in Optimal Design”, focuses on the literature review on optimal design and the associated design challenges for GCHP systems.

80 citations


Journal ArticleDOI
Abstract: Driven by the opportunity to harvest the flexibility related to building climate control for demand response applications, this work presents a data-driven control approach building upon recent advancements in reinforcement learning. More specifically, model-assisted batch reinforcement learning is applied to the setting of building climate control subjected to dynamic pricing. The underlying sequential decision making problem is cast into a Markov decision problem, after which the control algorithm is detailed. In this work, fitted Q-iteration is used to construct a policy from a batch of experimental tuples. In those regions of the state space where the experimental sample density is low, virtual support tuples are added using an artificial neural network. Finally, the resulting policy is shaped using domain knowledge. The control approach has been evaluated quantitatively using a simulation and qualitatively in a living lab. From the quantitative analysis it has been found that the control approach converges in approximately 20 days to obtain a control policy with a performance within 90% of the mathematical optimum. The experimental analysis confirms that within 10 to 20 days sensible policies are obtained that can be used for different outside temperature regimes.

75 citations


Posted Content
TL;DR: Model-assisted batch reinforcement learning is applied to the setting of building climate control subjected to dynamic pricing and it is found that within 10 to 20 days sensible policies are obtained that can be used for different outside temperature regimes.
Abstract: Driven by the opportunity to harvest the flexibility related to building climate control for demand response applications, this work presents a data-driven control approach building upon recent advancements in reinforcement learning. More specifically, model assisted batch reinforcement learning is applied to the setting of building climate control subjected to a dynamic pricing. The underlying sequential decision making problem is cast on a markov decision problem, after which the control algorithm is detailed. In this work, fitted Q-iteration is used to construct a policy from a batch of experimental tuples. In those regions of the state space where the experimental sample density is low, virtual support samples are added using an artificial neural network. Finally, the resulting policy is shaped using domain knowledge. The control approach has been evaluated quantitatively using a simulation and qualitatively in a living lab. From the quantitative analysis it has been found that the control approach converges in approximately 20 days to obtain a control policy with a performance within 90% of the mathematical optimum. The experimental analysis confirms that within 10 to 20 days sensible policies are obtained that can be used for different outside temperature regimes.

62 citations


Cites background from "A convex approach to a class of non..."

  • ...When considering building climate control, MPC has received considerable attention in the recent literature [6, 7, 8, 26]....

    [...]


Journal ArticleDOI
Abstract: The residential and commercial building sector is known to use around 40% of the total end-use energy and, hence, is considered to be the largest energy consumer sector in the world [1]. Approximately half of this energy is used for heating/cooling, ventilation, and air-conditioning (HVAC), and this usage is increasing 0.5?5% per year in developed countries [2]. The distribution of energy use percentages within the building for the United States is shown in Figure 1. This trend is similar for the rest of the world.

60 citations


Journal ArticleDOI
TL;DR: This paper provides a unified framework for model predictive building control technology with focus on the real-world applications and presents the essential components of a practical implementation of MPC such as different control architectures and nuances of communication infrastructures within supervisory control and data acquisition (SCADA) systems.
Abstract: Author(s): Drgoňa, J; Arroyo, J; Cupeiro Figueroa, I; Blum, D; Arendt, K; Kim, D; Olle, EP; Oravec, J; Wetter, M; Vrabie, DL; Helsen, L | Abstract: It has been proven that advanced building control, like model predictive control (MPC), can notably reduce the energy use and mitigate greenhouse gas emissions. However, despite intensive research efforts, the practical applications are still in the early stages. There is a growing need for multidisciplinary education on advanced control methods in the built environment to be accessible for a broad range of researchers and practitioners with different engineering backgrounds. This paper provides a unified framework for model predictive building control technology with focus on the real-world applications. From a theoretical point of view, this paper presents an overview of MPC formulations for building control, modeling paradigms and model types, together with algorithms necessary for real-life implementation. The paper categorizes the most notable MPC problem classes, links them with corresponding solution techniques, and provides an overview of methods for mitigation of the uncertainties for increased performance and robustness of MPC. From a practical point of view, this paper delivers an elaborate classification of the most important modeling, co-simulation, optimal control design, and optimization techniques, tools, and solvers suitable to tackle the MPC problems in the context of building climate control. On top of this, the paper presents the essential components of a practical implementation of MPC such as different control architectures and nuances of communication infrastructures within supervisory control and data acquisition (SCADA) systems. The paper draws practical guidelines with a generic workflow for implementation of MPC in real buildings aimed for contemporary adopters of this technology. Finally, the importance of standardized performance assessment and methodology for comparison of different building control algorithms is discussed.

60 citations


References
More filters

Book
01 Mar 2004
Abstract: Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

33,299 citations


Book
01 May 1995
TL;DR: The leading and most up-to-date textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under uncertainty, and discrete/combinatorial optimization.
Abstract: The leading and most up-to-date textbook on the far-ranging algorithmic methododogy of Dynamic Programming, which can be used for optimal control, Markovian decision problems, planning and sequential decision making under uncertainty, and discrete/combinatorial optimization. The treatment focuses on basic unifying themes, and conceptual foundations. It illustrates the versatility, power, and generality of the method with many examples and applications from engineering, operations research, and other fields. It also addresses extensively the practical application of the methodology, possibly through the use of approximations, and provides an extensive treatment of the far-reaching methodology of Neuro-Dynamic Programming/Reinforcement Learning.

10,491 citations


"A convex approach to a class of non..." refers background or methods in this paper

  • ...The reader is referred to references [16, 17] for details on dynamic programming....

    [...]

  • ...For dynamic programming-based control methods, the most important issue is to have an accurate model with minimum number of states and inputs due to the famous curse of dimensionality problem [16, 17]....

    [...]


Proceedings ArticleDOI
02 Sep 2004
TL;DR: Free MATLAB toolbox YALMIP is introduced, developed initially to model SDPs and solve these by interfacing eternal solvers by making development of optimization problems in general, and control oriented SDP problems in particular, extremely simple.
Abstract: The MATLAB toolbox YALMIP is introduced. It is described how YALMIP can be used to model and solve optimization problems typically occurring in systems and control theory. In this paper, free MATLAB toolbox YALMIP, developed initially to model SDPs and solve these by interfacing eternal solvers. The toolbox makes development of optimization problems in general, and control oriented SDP problems in particular, extremely simple. In fact, learning 3 YALMIP commands is enough for most users to model and solve the optimization problems

7,174 citations


"A convex approach to a class of non..." refers methods in this paper

  • ...The convexified optimal control problem is coded using YALMIP [30] in MATLAB and calling CPLEX [31] as solver....

    [...]


Book
01 Jan 1970

3,279 citations


Journal ArticleDOI
TL;DR: For nonlinear programming problems which are factorable, a computable procedure for obtaining tight underestimating convex programs is presented to exclude from consideration regions where the global minimizer cannot exist.
Abstract: For nonlinear programming problems which are factorable, a computable procedure for obtaining tight underestimating convex programs is presented. This is used to exclude from consideration regions where the global minimizer cannot exist.

1,755 citations


Frequently Asked Questions (1)
Q1. What are the contributions in "A convex approach to a class of non-convex building hvac control problems: illustration by two case studies" ?

In this paper, a convexification approach is presented for a class of non-convex optimal/model predictive control problems more specifically applied to building HVAC control problems. The suggested method is especially useful for optimal building HVAC control/design problems which include non-convex bilinear or fractional terms.