AND CONTROL

RELATED APPLICATION INFORMATION

[0001] This application claims priority to provisional application serial number 61/885,564 filed on October 2, 2013, incorporated herein by reference.

BACKGROUND

Technical Field

[0002] The present invention relates to strategies for optimal placement of sensors and actuators for temperature control, and more particularly to the placement of sensors for temperature and climate measurements and the placement of air conditioning devices for a given room.

Description of the Related Art

[0003] Traditional sensor and actuator deployment for climate control depends almost solely on heuristic rules. Existing technology rarely deals with how to best place the sensors and actuators, but instead focuses on temperature monitoring and control with sensors and actuators having already been placed. Literature on optimal sensor and actuator placement for HVAC system design is mainly on the theoretical analysis of dynamic models comprised of partial differential equations (PDEs). From a practical view of point, these studies, due to their theoretical complexity, are too complicated for application to typical consumer applications.

[0004] Sensor and actuator placement arises in other areas besides climate control, such as sensors for vibrational control, and especially control of flexible structures. Relevant studies are also control-theory-aided. However, the solutions developed are either heuristic or rather complicated involving optimization methods, such as large-scale nonlinear integer programming. Thus computationally less expensive and easy-to- implement methods are in great need.

SUMMARY

[0005] The present disclosure is directed to the positioning of sensors and actuators in climate control applications. In one embodiment, the method for determining the location of actuators and sensors for climate control includes providing a model of temperature and airflow within a room. The model includes a plurality of temperature and time transition states in a grid corresponding to a geometry of the room. A matrix for the placement of sensors is calculated with a processor from the model using a Lyapunov equation in which a variable for the Lyapunov equation includes a matrix for the transition state of temperature obtained from the model of temperature and airflow within the room. A trace of the matrix for the placement of sensors is maximized to provide optimum placement of the sensors within the room. A matrix for calculating the placement of actuators within the model using the Lyapunov equation is also calculated in which a variable for the Lyapunov equation includes the matrix for the transition state obtained from the model of temperature and airflow within the room. A trace of the matrix for the placement of actuators is maximized to provide optimum placement of the actuators within the room.

[0006] In another embodiment, the present disclosure provides a system for

determining the location of actuators and sensors for climate control that includes a modeling module configured to provide a model of temperature and airflow within a room. The model may include a plurality of temperature and time transition states in a grid corresponding to a geometry of the room. The system further includes a sensor placement module that is configured to determine with a processor a maximized trace of an optimization problem for the placement of sensors using a Lyapunov equation. A variable for the Lyapunov equation includes a matrix for the transition state obtained from the model of temperature and airflow within the room. The maximized trace of the matrix for the placement of sensors provides optimum placement of the sensors within the room. The system may further include actuator placement module configured to determine a maximized trace of an optimization problem for the placement of actuators using the Lyapunov equation. A variable for the Lyapunov equation includes the matrix for the transition state obtained from the model of temperature and airflow within the room. The maximized trace for the placement of actuators provides optimum placement of the actuators within the room.

[0007] In another embodiment, the present disclosure provides a non-transitory computer program product comprising a computer readable storage medium having computer readable program code embodied therein for performing a method for determining the location of actuators and sensors for climate control. The method may include providing a model of temperature and airflow within a room. The model includes a plurality of temperature and time transition states in a grid corresponding to a geometry of the room. A matrix for the placement of sensors is calculated from the model using a Lyapunov equation in which a variable for the Lyapunov equation includes a matrix for the transition state of temperature obtained from the model of temperature and airflow within the room. A trace of the matrix for the placement of sensors is maximized to provide optimum placement of the sensors within the room. A matrix for calculating the placement of actuators within the model using the Lyapunov equation is also calculated in which a variable for the Lyapunov equation includes the matrix for the transition state obtained from the model of temperature and airflow within the room. A trace of the matrix for the placement of actuators is maximized to provide optimum placement of the actuators within the room.

[0008] These and other features and advantages will become apparent from the following detailed description of illustrative embodiments thereof, which is to be read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF DRAWINGS

[0009] The disclosure will provide details in the following description of preferred embodiments with reference to the following figures wherein:

[0010] Fig. 1 is a block/flow diagram of a method for determining the location of actuators and sensors for climate control, in accordance with one embodiment of the present disclosure.

[0011] Fig. 2 is a block/flow diagram of a method for determining the location of actuators and sensors for climate control, in accordance with another embodiment of the present disclosure.

[0012] Fig. 3 shows an exemplary system to perform the methods for optimizing the location of actuators and sensors for climate control, in accordance with the present disclosure. DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

[0013] The present principles are directed to strategies for optimal placement of sensors (for temperature or climate measurements) and actuators (such as air conditioning (A/C) devices) for a given room. In some embodiments, the strategies disclosed herein optimally place sensors and actuators in a large space such that the temperature can be better monitored and regulated. The strategies can be constructed within a solid theoretical framework and have practical significance for HVAC system design with manageable computational cost.

[0014] It should be understood that embodiments described herein may be entirely hardware or may include both hardware and software elements, which includes but is not limited to firmware, resident software, microcode, etc.

[0015] Embodiments may include a computer program product accessible from a computer-usable or computer-readable medium providing program code for use by or in connection with a computer or any instruction execution system. A computer-usable or computer readable medium may include any apparatus that stores, communicates, propagates, or transports the program for use by or in connection with the instruction execution system, apparatus, or device. The medium can be magnetic, optical, electronic, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. The medium may include a computer-readable storage medium such as a semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk, etc. The medium may include a non-transitory storage medium. [0016] A data processing system suitable for storing and/or executing program code may include at least one processor, such as a hardware processor, coupled directly or indirectly to memory elements through a system bus. The memory elements can include local memory employed during actual execution of the program code, bulk storage, and cache memories which provide temporary storage of at least some program code to reduce the number of times code is retrieved from bulk storage during execution.

Input/output or I/O devices (including but not limited to keyboards, displays, pointing devices, etc.) may be coupled to the system either directly or through intervening I/O controllers.

[0017] As used herein, the term "actuator" means a type of motor that is responsible for moving or controlling a mechanism of a system. The actuator is typically operated by a source of energy, such as an electric current, hydraulic fluid pressure, or pneumatic pressure, and converts that energy into motion. An actuator is the mechanism by which a control system acts upon an environment. For example, in an HVAC system, the actuator typically controls valves and dampers to control the flow of air and liquids.

[0018] As used herein, the term "sensor" means a device to measure and monitor a variable, such as temperature, pressure and humidity of ambient air. The sensors consistent for use with the present disclosure may be of electronic control or pneumatic control. Pneumatic sensors typically sense pressure. Resistance sensors, such as resistance temperature devices (RTDs), may be used for measuring temperature. Voltage sensors can be used for measuring temperature, humidity and pressure. Current sensors may be used to measure temperature, humidity and pressure. [0019] In some embodiments, the disclosed methods, apparatus and systems provide a control- theory-based method to determine the best locations of sensors and actuators. More specifically, in some embodiments, the sensors and actuators are placed through maximizing variables related with the observability and controllability of a certain system. The problem can be solved in an analytical manner, obtaining closed-form solutions.

[0020] Compared to previous methods, the advantages provided by the approach disclosed herein are as follows: First, the solution is not only based on optimal design, but is an easily comprehendible solution for consumers, users and installers. The solutions disclosed herein are inspired by control theories and achieved via solving an optimization problem. A well-designed, but straightforward method, is established to compute the solution. Second, the speed of obtaining the solution is fast and fully manageable compared to previous approaches. In some embodiments, the most time- consuming part of the disclosed approach is solving a matrix equation, which can be handled by many numerical algorithms embedded in software. One example of a type of algorithm that is suitable for solving the matrix equation is a Lyapunov equation. In control theory, the discrete Lyapunov equation may be in the form of:

Equation 1. AXA ^H - X + Q = 0 wherein Q is a hermitian matrix and A is the conjugate transpose of A. The continuous

Lyapunov equation is of form:

Equation 2. AX + XA ^H + Q = 0

[0021] It is noted that the above-described Lyapunov equation is only one example of an algorithm that is suitable for solving the matrix problem in accordance with the present disclosure. Other algorithms may also be suitable for use with the present disclosure.

[0022] As will be described in greater detail below, the methods, systems, and computer program product that are disclosed herein provide a computationally faster strategy for determining sensor and actuator placement when compared to previous sensor and actuator deployment strategies while retaining the rigor of the solutions. The methods disclosed herein are applicable to energy management scenarios, such as data centers and large commercial spaces, which is facilitated through the improved computational speed of the approach that is disclosed herein. The improved sensing and actuation possibilities provided by the methods, systems and computer products that are disclosed herein can lead to a reduction of energy consumption (and hence a reduction in operating costs) through efficient placement and operation of air conditioning

component.

[0023] Fig. 1 depicts one embodiment of the sensor and actuator placement approach in accordance with the present disclosure. The sensor and actuator placement approach that is illustrated in Fig. 1 may be model-based. In some embodiments, at step 10 of process flow depicted in Fig. 1 models are first prepared to describe the dynamic behavior of the airflow and heat transfer in a room. The model begin with partial differential equation (PDE) based models at step 10, which are converted to the space state form at step 20.

[0024] In one embodiment, the airflow model at step 10 of the sensor and actuator placement approach that is depicted in Fig. 1 may employ a Navier-Stokes equation, which characterizes the motion of fluids. The motion of fluid can described by the equations of mass, energy and momentum balance, and this set of equations is often referred to as the Navier Stokes equations (NS). In the case of the Newtonian fluid they can be written as:

Equation 3. (dp/dt) + V· (pv) = 0 (mass)

Equation 4. (5(pe)/5t) + V· (pvh) = V· (kVT) (energy)

Equation 5. (9(pv)/9t) + V· (pvvT) + Vp = V · (μ νν) + f (momentum)

[0025] where the scalars p, T, e, h, p, k and μ are respectively the fluid pressure, temperature, specific energy, specific enthalpy, density, thermal conductivity and dynamic viscosity; the vectors v and f are the fluid velocity and the external forces only, such as gravity, acting on the fluid.

[0026] In some embodiments, the heat transfer model at step 10 of the sensor and actuator approach that is depicted in Fig. 1 can be described by the convection-diffusion equation.

[0027] One example of a convection-diffusion equation for use with the heat transfer model may include:

Equation 6. ^ = V ^■ (£>Vc) - V ^■ vc) + R

[0028] where c is the variable of interest (species concentration for mass transfer, temperature for heat transfer), D is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, and v is the average velocity that the quantity is moving. R describes "sources" or "sinks" of the quantity c. V represents gradient and V represents divergence. In a common situation, the diffusion coefficient is constant, there are no sources or sinks, and the velocity field describes an incompressible flow (i.e., it has zero divergence). Then the formula simplifies to: Equation 7. ^ = DV ² c— v ^■ Vc.

[0029] The following description of equations 8, 9, 10 and 11 represents one preferred embodiment of step 10 of the method depicted in Fig. 1, in which the Navier-Stokes equations for describing the conservation of momentum and mass for incompressible airflow is given, respectively, as follows:

Equation 8 p [dV/dt + (V· V)V] = pg-Vp + μ V ² V

V-V = 0

[0030] where g is the gravity vector, Vp the pressure gradient, μ the dynamic viscosity. In this example, a steady- state airflow is assumed in this study, i.e., dY/dt = 0, because the model is to represent the steady- state large-scale behavior of the indoor airflow field and is intended to reduce the complexity of analysis. Consistent with this embodiment, for a time-varying temperature field T(x,y,z,t), the heat transfer via convection-diffusion is given by:

Equation 9. pc _P + V · VT ) - V · (KVT) = h

[0031] where p, c _p and κ denote, respectively, the density, specific heat and thermal conductivity of air, and h represents the heat generated or removed ('sources' or 'sinks' of T in terms of heat transfer). For equations 8 and 9, the following boundary condition is applied:

Equation 10. -n-V = Vb,

[0032] where n is the unit outward normal vector at a point on the space domain boundary, and Vb is assumed to be zero at static boundaries and non-zero at non-static ones. In some scenarios, when Vb 6 is not equal to 0, its value is known or can be determined directly from certain sensors, e.g., real-time pressure sensors. The flow of heat in the direction normal to the boundary is specified by:

Equation 11. -n- (kVT) = q + aT,

[0033] where q results from the power of the heating or cooling sources at the boundaries and a is a coefficient.

[0034] The model obtained from step 10 includes of a set of partial differential equations (PDEs). To apply control theoretic approaches, the model composed of partial differential equations may be converted into a state- space form by applying the numerical method of lines on a uniformly gridded space in step 20 of the process flow that is described in Fig. 1. The uniformly gridded space provides a temperature and airflow distribution for a series of grid points in the model of airflow and temperature within a particular room. For example, there can be grid points at every 50 cm within a room. Each grid point within the uniformly gridded space provides a temperature variable within the room.

[0035] A state space form is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

Linearization, i.e., the method of lines (MOL), is finding the linear approximation to a function at a given point. The method of lines (MOL) approximates the spatial derivatives by a finite-difference-based discretization, with the resulting ordinary differential equations (ODEs) established over the time domain. For example, the MOL is applied to equation 9 along the boundary condition of equation 11 to obtain the ordinary differential equations (ODEs) and subsequently the state-space form to describe the temperature dynamics. [0036] Considering a uniformly gridded three-dimensional space. The number of grid points along each axis is Nx, Ny, Nz, respectively. The state vector x is the collection of temperature values at all grid points, and the input vector u is a collection of the heat sources or sinks on the grid, that is

Equation 12. x(t) = T(i, j, k, t) u(t) = h(i,j, k, t)

«XI

[0037] The dimension of x is n _x = NxxNy xNz, and the dimension of u is the number of sources and sinks in the system, denoted as n _u . In some embodiments, n _u «n _x . The state-space equation is:

Equation 13. x(t) = Ax(t) + Bu(t)

[0038] The matrices A G R ^{Ux x Ux} and B G R ^{Ux x n} " are determined by equations 9 and 11. B indicates the placement of sources or sinks, i.e., actuators. It has a sparse binary structure— each element is 0 or 1 (after normalization), and only one element of each column can be 1 as the actuators are assumed to be point sources. That is,

Equation 14. ¾ G {0,1 } Vi, j, B _j = 1, 2, · · · , nu.

[0039] The measurement vector y G ^ny has a dimension equal to the number of sensors, and n _y « n _x . The output equation representing the sensor measurements are as follows:

Equation 15. y(t) = Cx(t),

[0040] where C is also a sparse binary matrix representing sensor locations with: Equation 16. Ci,j G{0,1 } Vi,j, _x C = 1 for i = 1, 2, · · · , ny. [0041] Together equations 13 and 16 represent the state- space model for heat transfer in accordance with the present disclosure. It is a linear, time-invariant and high- dimensional system, as a result of the PDE reduction. In the following process flow for optimal sensor and actuator deployment, the sparse binary structure of B and C will be fully utilized to alleviate the difficulty of analysis and design. Equations 13 and 15 can be expressed together as follows: = Ax + Bu,

Equation 17.

y = Cx

[0042] where x represents the state of the system, e.g., temperature at grid points, u is the input to the system, e.g., cool air from the air conditioner, and y is the system output, e.g., the temperature measurements at locations where sensors are deployed. The matrix A determines the state transition. The state transition provided by matrix A describes the dynamic change in temperature in the room over time. Matrix A is provided by the model at step 10 of the process flow that is depicted in Fig. 1. Matrix A takes into account the geometry and size of the room in which the air condition is being applied, the open space, as well as the equipment that is present in the room. Matrix B is related to the positioning of the actuators within the room. For example, matrix B provides for the positioning of the source of air conditioning, e.g., cool air, within the room. Matrix C provides for the positioning of the sensors within the room for measuring the changes in temperature. Through matrix B the input excites the state, and with matrix C certain states are directly measured. For example, matrix B, i.e., related to the positioning of the actuators, provides for changes in temperature within a particular room, and matrix C, i.e., related to the positioning of the sensors, provides for the direct measurement of temperature within the room. In the above state space model, point source actuators, steady- state airflow field and negligible humidity effects have been considered. In the following process flow for optimal sensor and actuator deployment, Matrix B and C are to be mathematically found via actuator and sensor placement within the model, respectively.

[0043] For example, in one embodiment, sensor placement may be mathematically formulated using the following optimization problem, in which max_C represents the best placement, i.e., optimum placement, of the sensors within a room that is being air conditioned:

Equation 18. max_C trace (Wo)

[0044] The goal of the optimal sensor deployment strategy is maximizing the trace of the observability Gramian. Since the system described in equations 13, 15 and 17 is physically stable, A is a stable matrix, and the observability Gramian, Wo, is defined as:

Equation 19. W ₀ = /" e ^AT C ^T e ^At dt,

[0045] where the optimal sensor locations are determined via selecting C to maximize the trace of W ₀ , as follows:

Equation 20.

m ^ax _c tr [W ₀ (C)]

n _x

s. t. Ci = £ {0,1} Vi,j ^ Ci = 1 for i = 1, 2,···, n _y ,

7 = 1

[0046] where Q _j = 1 when the sensor / is placed at the j-th point in the gridded domain and Ci _j =0 otherwise. The Observability Gramian is a Gramian used in optimal control theory to determine whether or not a linear system is observable, i.e., a measure for how well internal states of a system can be inferred by knowledge of its external outputs a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability represents the ability to estimate the internal state variable using the input and output of the system. In some embodiments, its Gramian W ₀ has important implications regarding the system and state estimation. The following summarizes one interpretation of W ₀ , which may begin with determining the amount of information that the output contains about the state, because the observed energy in the output can be written as:

Equation 21. ||y||l = ζ ' y ^T (τ)γ(τ)άτ = x ^T (0)W _o x(0),

[0047] where x(0) is the initial state. Thereafter, the J-C ₂ norm of the system G from equations 13 and 15 is a weighed trace of W ₀ , which can be expressed as:

Equation 22. || G \\ ₂ = tr(B ^T W ₀ B),

[0048] The Gramian W ₀ affects the state estimation accuracy when the output is measured with noise. Taking the example, when measurements have been corrupted by additive noise v _t , the equation becomes y(t) = Cx(t) + v(t). The least- squares estimation of x(0) given for y(t) for 0 < t <∞ is:

Equation 23. x(0) = x(0) + W ₀ ^_1 /" e ^ATT C ^T v(x)dx.

[0049] In some embodiments if {v(t)} is a continuous-time wide-sense-stationary (wss) Gaussian white noise process with autocovariance function R _v (x) = r5(r)I, then the estimation error covariance will be rW ₀ ^_1 . In some embodiments, as a measure of the observability, tr(W ₀ ) can be vital, because larger values correspond to an increase in the overall observability of the system. It is also related with the rank maximization of W ₀ .

[0050] A nonsingular W ₀ can guarantee complete observability. However, in some instances, W ₀ can be rank-deficient if the system is only detectable. This may happen when a limited number of sensors are deployed. In such a case it would be valuable to deploy sensors to obtain a C such that the rank of W ₀ is maximized:

Equation 24. max_C rank(W ₀ ) .

[0051] Solving this rank maximization problem (globally) can be difficult, and is known to be computationally non-deterministic polynomial-time hard (NP-hard). One heuristic is to replace the rank objective with the trace, in order to solve the following:

Equation 25. max_C tr(Wo)

[0052] Because tr(W ₀ ) =∑" _{=1 t} (W ₀ ) , where X _t (W ₀ )s for i = 1 , 2, · · · , n _x are the eigenvalues of W ₀ , maximizing tr(W ₀ ) typically results in a high rank matrix.

[0053] The observability Gramian W ₀ is a measure of observability based on the system structure A (see e ^At ) and the state observing structure C. Using the above noted equations, the sensor placement is equivalent to finding the matrix C such that the trace of W ₀ is maximized. Such a metric is used, because W ₀ determines the amount of information that the output contains about the state and the system' s robustness to measurement noise.

[0054] In accordance with some embodiments of the present disclosure, the optimization problem described above with respect to equation 13, 15, and 17 is solved analytically via a three-step procedure. First, a Lyapunov equation, +XA ^T = — I , is solved using numeric computing software, at step 30 of the process flow illustrated in Fig. 1. X is the solution of the Lyapunov equation and A is the state matrix, and -/ is the identity matrix.

[0055] Then the diagonal elements of the solution X are sorted, and the matrix £" is determined, with 0 or 1 assigned to each element at step 40 of the process flow illustrated τ in Fig. 1. The optimization metric trace (W ₀ ) can be written equivalently as trace(XC C), where X is the solution to the Lyapunov matrix. Due to the binary structure of C, C C plays the role of picking some diagonal elements of X. To maximize the considered metric, the largest diagonal elements of X are selected. More specifically, the diagonal elements of X sorted first, the large ones are found, and then the corresponding values in C are set to be 1 and the others to be 0. Further details are provided in the following description.

[0056] One computationally attractive solution to equation 20, which maximizes the tr (W ₀ )under the structural constraints of C can be developed, and written, as follows:

Equation 26. tr[W ₀ (C)] = tr ( f e ^AT TC ^T Ce ^A ^dx) = f tr(e ^AT -rC ^T Ce ^A *) dx = J ^" " tr(e ^A ^ ^AT TC ^T C)dx = tr (/" e^e^ drC ^T C) e ^τ e dx is the unique solution of the Lyapunov equation:

Equation 27. AX + XA ^T = -I.

[0058] In addition, L = C C is a binary diagonal matrix. Each of its diagonal elements, L _jj , is 0 or 1 for j = 1, 2, · · · ,n _x ; LJJ = 1 if a sensor is located at the j-th grid point.

Therefore, to maximize tr(W ₀ ) = tr(XL), only the largest diagonal elements n _y need to be found (sort operation), by determining the rows they belong to, and assigning 1 to the corresponding elements in C. That is, after searching through the diagonal elements of X, the set S ={s _k : k = 1, 2,···, n _y ) is obtained such that X _jj > ¾ for j E S and / 4- S;

whereinC _{i si} = 1 by placing a sensor at the Si-the point for i = 1, 2, · · · n _y _

[0059] Through this design, the sensor placement strategy maximizes an important metric closely related with system observability, helping improve the system monitoring and control performance. In addition, its implementation is fast and computationally feasible compared to previous methods.

[0060] In summation the optimal sensor deployment strategy may be summarized as follows:

Step 1 : Solve AX +XA ^T ₌ -I

Step 2: Find the indices of the n _y largest diagonal elements of X and determine the index set S = [s _k : k = 1, 2, ··· , n _y ) with ¾· > ¾_ _/ for j E S and I jk S;

Step 3: set the (i,¾)-th element of C to 1 for / = 1, 2, ···, n _y and other elements to 0, or equivalently, Qj = 1 if j=Si and otherwise, Q ₇ = 0;

Step 4: place sensors accordingly.

[0061] A variation of the algorithm to avoid dense deployment is also developed, by introducing the constraint that each sensor effectively covers a certain area or region. One example of a constraint is that in a room of a data center some portions of the room may be occupied by the equipment within the room, such as servers. This represents a constraint, because the space occupied by the equipment can not also be occupied by a sensor or an actuator. As a result, the sensors are spatially deployed to ensure considerable observability as well as accurate temperature field reconstruction. Another consideration is that the above described optimum sensor deployment strategy is that it may yield an undesired dense or clustered sensor deployment, i.e., multiple sensors deployed within a relatively small area. Additionally, it is desirable to integrate practitioner' s experience and industry guidelines into the decision process.

[0062] To overcome the above noted disadvantages, embodiments have been contemplated in which a observability map has been built that shows the distribution tr(Wo) over the space. The information it offers can be used with awareness of spatial limitations and inclusion of expert experience to decide sensor locations. To construct the map, a single sensor is placed at a grid point. In this case, C £ E ^{1 x i½} , where the element corresponding to this grid point will take 1 and the others 0. Then tr(W ₀ ) is calculated to quantify the observability if a sensor is placed here. By analogy, a map illustrating the relationships between tr(W ₀ ) and each spatial location can be generated. In some embodiments, the computation only relies on solving the Lypanov equation (equation 26) for X, because the diagonal elements of X are equivalents of tr(W ₀ ) with a single sensor placed on the corresponding locations. To show this, an assumption is made that a sensor is placed at the i-th grid point, implying the i-th element of C is 1, i.e.,: Equation 29. C = [0 ··· 0 1 0 - 0] _{x χ ΐ1χ}

[0063] Then it follows that:

Equation 30. tr(W ₀ ) = tr(XC ^T C) = ¾.

[0064] In general, an area in the map should be given more weight during sensor placement if it has larger tr(W ₀ ). This information can be easily fused with prior experience and knowledge at the practitioners level. In view of the above, an improved optimal sensor deployment can be summarized as follows:

Step 1: Solve AX + XA ^T = -I.

Step 2: Extract the diagonal elements of X and rearrange them with respect to the spatial locations to build the observability map.

Step 3: Decide the best locations of sensors with the aid of the map information, practitioner's experience and knowledge and industry guidelines.

Step 4: Place sensors accordingly. [0065] The above process flow including process steps 10, 20, 30 and 40 provides for sensor placement of an HVAC system.

[0066] The actuator deployment problem is a dual of the sensor deployment problem if actuators are considered as point sources. For actuator placement, in some embodiments, the following problem is established and considered, in which max _B represents the best placement, i.e., optimum placement, of the actuators within a room that is being air conditioned:

Equation 30: max_B trace (Wc)

[0067] where the observability Gramian equation is:

Equation 31: Wc = /" e ^At BB ^T e ^ATt dt.

[0068] In the observability Gramian equation, Wc represents the controllability based on the system structure A and control structure B. The actuator placement problem is similar to the above described sensor placement problem. But, in the actuator placement problem, the actuator placement is equivalent to finding the matrix $ such that the trace of is maximized. For example,

Equation 31

m tr[W _c (B ]

s. t. B _y = G {0,1} Vi,; ^ Bi = l for; = 1, 2,-, n _u ,

[0069] Equation 32 is a dual of equation of equation 24 of the optimum actuator deployment scheme. The controllability Gramian from equation 29 is chosen as the measure of control authority for a dynamic system in accordance with the present disclosure according to the following observations. [0070] First, W _c is closely related with minimum energy control. Consider driving a system from x(0) = 0 to x(t) = x using the lowest amount of control energy:

Equation 32: ⁿ ^ ⁿ E(t) s. t. x(t) = Ax(t) + Bu(t), x(0) = 0, x(t) = x,

[0071] where E(i) = U ^t (T) u(x)dx. The resulting control input is:

Equation 33: U(T) = B ^T e ^AT(t - ^T) W _c ^_1 (t)x, 0 < τ < t.

[0072] Hence, the control energy over an infinite time horizon is E(∞) = x ^T W ₀ ^_1 x. Second, the J-C ₂ norm of G is also a weighted trace of the controllability Gramian:

Equation 34: \\ G \\ ₂ = tr(CW _c C ^T ).

[0073] Finally, in some embodiments, a larger W _c can be a factor that helps suppress the influence of process noise. For example, if the input u is corrupted by an additive

Gaussian white noise with a covariance Q = ql:

Equation 35: x(t) = Ax(t) + B [u(t) + w(t)] .

[0074] Suppose the control objective is to drive the state to x. By optimal control theory, irrespective of how the control input u is chosen, the state x, will not be precisely achieved due to the effects of the noise w. The state covariance will be:

Equation 36: E[(x(∞)— x) (x(∞)— x) ^T ] = qW _c ^_1 ,

[0075] which is inversely W _c . Thus, in some embodiments, a larger W _c may contribute to noise suppression. The rank of the controllability matrix is relevant to the rang of W _c . When a system is only stability due to the small number of actuators, the rank of the controllability matrix can be increased by placing the actuators in the best positions. Therefore, it is advantageous to solve max _B rank (W _c ). Similar to equation 23, this is an NP-hard problem. The trace heuristic can hence be used to solve this problem, i.e., maxs

[0076] Similar to the sensor placement problem, the optimization problem for actuator placement may be a three-step process. First, a Lyapunov equation +XA ^T = — I , is solved using numeric computing software, at step 50 of the process flow illustrated in Fig. 1. Then the diagonal elements of the solution X from the Lyapunov equation are sorted, and the matrix B is determined, with 0 or 1 assigned to each element at step 60 of the process flow illustrated in Fig. 1. The optimization metric trace(Wc) can be written equivalently as trace(XBB ), where X is the solution to the Lyapunov matrix. Due to the binary structure of B, B B indeed plays the role of picking some diagonal elements of X. To maximize the considered metric, the largest diagonal elements of X are selected. For example, the diagonal elements of X are first sorted, the large ones diagonal elements are then determined, and the corresponding values in B are set to be 1 and the others to be 0. In this way, the positions of actuators, which depend on B, are determined.

[0077] In one example, the optimal actuator deployment strategy may be summarized as follows:

Step 1: solve A ^T X + XA = -I.

Step 2: find the indices of the Q _u largest diagonal elements of X and determine the index set S = {s _{k :} k = 1, 2, · · · , n _a } with ¾ > X _fjf for j e S and i φ R ^{l x rix} S.

Step 3: set the (¾/)-th element of B to 1 for j = 1, 2, · · · , n _u and other elements to 0, or equivalently, ¾ = 1 if i = SjWL ^{1 χ 71χ} and otherwise, ¾ = 0.

[0078] Through this design, the actuator placement strategy maximizes a metric closely related with system observability, helping improve the system monitoring and control performance. In addition, its implementation is fast and computationally feasible compared to previous methods. Similar to the above described optimized sensor deployment, the optimized actuator deployment may be improved by taking into account multiple decision criteria, including the controllability map, awareness of physical spatial constraints and expert experience.

[0079] Fig. 2 depicts another embodiment of the sensor and actuator placement method and system in accordance with the present disclosure. The method of sensor and actuator placement may begin with preparing a models to describe the dynamic behavior of the airflow and heat transfer process in the room for climate control, at step 70. Step 70 of the process flow depicted in Fig. 2 has been described above with reference to steps 10 and 20 in Fig. 1. The approach leads to two strategies, one for sensor deployment 80 and the other for actuator deployment 90. The sensor deployment can be conducted using the metric of observability Gramian at step 80. In some embodiments, a meaningful metric is the trace of the Gramian to be maximized at step 110. One feature of the methods, systems and computer products disclosed herein is to transform the problem in order to obtain an analytical solution for this maximization problem. This maximization is achieved via solving a Lyapunov equation at step 120. The solution of the Lyapunov equation is used to find the optimal locations of sensors to obtain the best picture of the states in the room, therefore obtaining the most accurate temperature picture of a room. The steps employing the observability Gramian, maximized trace of the Gramian, and solving the Lyapunov equation for the sensor optimization have been described above in steps 30 and 40 of the process flow described above with reference to Fig. 1. [0080] Referring to Fig. 2, further improvement can be made by incorporating a constraint on the spatial distribution of sensors to avoid dense deployment at step 130. For example, a constraint on the spatial distribution of sensors can include removing from the analysis the locations at which equipment is present within the room. For example, in a room of a data center, the space in the room that is occupied by servers can be removed from the analysis, because the sensors and actuators cannot occupy the same space as the physical equipment within the room.

[0081] In other cases, it might be desirable to use other metrics related to the observability Gramian matrix, such as norm eigenvalue at step 140 or maximum eigenvalue at step 150, or metrics related with the state estimation error covariance specified by an algebraic Riccati equation at steps 160 and 170 of Fig. 2. In addition, the number of sensors needed for optimum climate control, e.g., temperature and airflow, can also be determined by analyzing the observability Gramian at steps 180 and 190.

[0082] The actuator placement strategy at step 90 is a dual problem of the sensor placement. In some embodiments, the controllability Gramian at step 200 can be used in a way similar to the above discussion on the observability Gramian at step 100. For example, in some embodiments, a meaningful metric is the trace of the Gramian to be maximized at step 210. One feature of the methods, systems and computer products disclosed herein is to transform the actuator optimization problem in order to obtain an analytical solution for this maximization problem. This maximization is achieved via solving a Lyapunov equation at step 220. The solution of the Lyapunov equation is used to find the optimal locations of actuators to obtain the best picture of the states like effectuating the most efficient temperature changes as a function of time for the room. The steps employing the observability Gramian, maximized trace of the Gramian, and solving the Lyapunov equation for the actuator optimization have been described above in steps 50 and 60 of the process flow described above with reference to Fig. 1, as well as equation 10.

[0083] Similar to the strategy for sensor deployment, the strategy for actuator deployment can be modified by taking into account the spatial constraints at step 230. Other metrics can be applied to develop placement strategies for actuator in a way similar to sensor placement, such as norm eigenvalue at step 240 or maximum eigenvalue at step 250, or metrics related with the state estimation error covariance specified by an algebraic Riccati equation at steps 260 and 270. In addition, the number of actuators needed for optimum climate control, e.g., temperature and airflow, can also be determined by analyzing the observability Gramian at steps 280 and 290.

[0084] Fig. 3 depicts one embodiment of a system to perform methods for optimizing the location of actuators and sensors in climate control systems. In one embodiment, the system 300 preferably includes one or more processors 118, such as hardware processors, and memory 308, 316, such as non-transitory memory, for storing applications, modules and other data. In one example, the one or more processors 118 and memory 308, 306 may be components of a computer, in which the memory may be random access memory (RAM), a program memory (preferably a writable read-only memory (ROM) such as a flash ROM) or a combination thereof. The computer may also include an input/output (I/O) controller coupled by a CPU bus. The computer may optionally include a hard drive controller, which is coupled to a hard disk and CPU bus. Hard disk may be used for storing application programs, such as some embodiments of the present disclosure, and data. Alternatively, application programs may be stored in RAM or ROM. I/O controller is coupled by means of an I O bus to an I/O interface. I/O interface receives and transmits data in analog or digital form over communication links such as a serial link, local area network, wireless link, and parallel link.

[0085] The system 300 may include one or more displays 314 for viewing. The displays 314 may permit a user to interact with the system 300 and its components and functions. This may be further facilitated by a user interface 320, which may include a mouse, joystick, or any other peripheral or control to permit user interaction with the system 300 and/or its devices, and may be further facilitated by a controller 312. It should be understood that the components and functions of the system 300 may be integrated into one or more systems or workstations. The display 314, a keyboard and a pointing device (mouse) may also be connected to I/O bus of the computer.

Alternatively, separate connections (separate buses) may be used for I/O interface, display, keyboard and pointing device. Programmable processing system may be preprogrammed or it may be programmed (and reprogrammed) by downloading a program from another source (e.g., a floppy disk, CD-ROM, or another computer).

[0086] The system 300 may receive input data 302 which may be employed as input to a plurality of modules 305, including at least a modeling module 306, sensor placement module 308, and an actuator placement module 310. The system 300 may produce output data 322, which in one embodiment may be displayed on one or more display devices 314. It should be noted that while the above configuration is illustratively depicted, it is contemplated that other sorts of configurations may also be employed according to the present principles. [0087] In one embodiment, the modeling module 306 is configured to provide a model of temperature within a room. The model that is provide by the modeling module may be calculated from equations to characterize the motion of fluids, such as a Navier-Stokes equation, and equations to provide a heat transfer model, such as the convection-diffusion equation. Further details regarding providing the model of temperature and airflow in the room have been provided above in the description of steps 10 and 20 of Fig. 1.

[0088] In one embodiment, the sensor placement module 308 is configured to provided the optimized placement of sensors for direct measurement of temperature within the room. The sensor placement module 308 can determine optimum sensor deployment using the metric of observability Gramian. For example, the metric may include a trace of the Gramian to be maximized. Additionally, maximization may be achieved via solving a Lyapunov equation. The solution of the Lyapunov equation can provide the optimal locations of the sensors to provide the best picture of the states, therefore obtaining the most accurate temperature picture of the room. Further details regarding functionality of the sensor placement module are provided in the description of steps 100, 110, 120 and 130 of Fig. 2, and steps 30 and 40 of Fig. 1 including equation 9.

[0089] In one embodiment, the actuator placement module 310 is configured to provided the optimized placement of actuators for effectuating changes in states, such as temperature and airflow, within the room. The actuator placement module 310 can determine optimum actuator deployment using the metric of observability Gramian. For example, the metric may include a trace of the Gramian to be maximized. Additionally, maximization may be achieved via solving a Lyapunov equation. The solution of the Lyapunov equation can provide the optimal locations of the sensors to provide the best picture of the states, therefore obtaining the most accurate temperature picture of the room. Further details regarding functionality of the sensor placement module are provided in the description of steps 200, 210, 220 and 230 of Fig. 2, and steps 50 and 60 of Fig. 1.

[0090] The methods, systems and computer program products disclosed herein provide analytical and closed-form solution for sensor and actuator location in climate control applications, such as HVAC. Prior technologies depend on heuristic rules to place the sensors and actuators. The strategies disclosed herein proposes an analytical solution through the use of the Lyapunov equation to maximize the trace of the observability Gramian, as described above with reference to steps 110, 120, 210 and 220 in Fig. 2. Compared to previous heuristic or approximate solutions, the strategies described herein are more rigorous and can result in improved system design, e.g., improved positioning of sensors and actuators, especially for HVAC systems.

[0091] The methods, systems and computer program products disclosed herein provide for optimized sensor and actuator location in climate control applications with relatively low computational cost. Though its development results from rigorous theoretical analysis, the strategies disclosed herein are computationally practical and can be conveniently addressed using generic scientific computing software.

[0092] The foregoing is to be understood as being in every respect illustrative and exemplary, but not restrictive, and the scope of the invention disclosed herein is not to be determined from the Detailed Description, but rather from the claims as interpreted according to the full breadth permitted by the patent laws. Additional information is provided in an appendix to the application entitled, "Additional Information". It is to be understood that the embodiments shown and described herein are only illustrative of the principles of the present invention and that those skilled in the art may implement various modifications without departing from the scope and spirit of the invention. Those skilled in the art could implement various other feature combinations without departing from the scope and spirit of the invention.

APPENDIX TO THE SPECIFICATION

Closed-form Optimal Sensor and Actuator Deployment for Building

System Application

Abstract

In this paper, we study optimal sensor and actuator deployment from a control-theoretic perspective, aiming to improve temperature monitoring and control performance of HVAC (Heating, Ventilation and Air Conditioning) systems in buildings. HVAC systems are indispensable to modern buildings and a major source of worldwide energy consumption. There is a need for a rigorous yet practically useful approach to determine the best spatial locations of sensors (for monitoring) and actuators (HVAC devices). We develop deployment strategies through maximizing observability- and controllability-based metrics such as their respective Gramians. The proposed novel strategies possess significant practical value due to the following advantages over previous work. First, they are built upon analytical, closed-form solutions to the formulated optimization problems, introducing a rigor absent in previous intuitive approaches. Second, the computational cost of our strategies is relatively low and affordable for even large-scale systems, allowing easy use in the real world. An exhaustive simulation-based study demonstrates the effectiveness and the generic nature of the proposed deployment strategies.

Keywords: Sensor deployment, actuator deployment, observability Gramian,

controllability Gramian, HVAC system

1. INTRODUCTION

This paper focuses on developing control-theory-inspired strategies for spatial deployment of sensors and actuators for HVAC (Heating, Ventilation and Air Conditioning) systems in buildings to improve temperature monitoring and control performance. Optimal

* Corresponding author. Tel. +1-408-863-6065.

Email addresses: hzf ang@ucsd . edu (Huazhen Fang), ratnesh@nec-labs . com (Ratnesh Sharma) , rakeshmpSnec-labs . com ( ^" Rakesti PaXil)

Preprint submitted to Control Engineering Practice July 15, 20 APPENDIX TO THE SPECIFICATION

sensor and actuator deployment (SAD) will bring the following potential benefits to HVAC systems. First, it provides opportunities to reduce energy consumption through system-level design. This is particularly important because residential and commercial buildings account for about 40% of total energy use in the United States [1] . The second benefit is better thermal regulation and comfort from improved monitoring and estimation performance. Since HVAC system research has been directed primarily toward control design, see [2-5] and the references therein, there is ample scope to enhance control performance and energy efficiency through optimal SAD design [6J .

Literature review: Traditionally, SAD for buildings has relied on heuristic rules or the other extreme of computationally burdensome solutions such as computational fluid dynamics (CFD) modeling [7]. A formal control-theory-based synthesis has been found challenging due to the complexity of temperature and airflow dynamics in buildings. Existing methods, though rather few, can be divided into two categories, direct and indirect. Methods of the former kind are based on metrics directly measuring estimation and control performance. For instance, the sensor locations are determined in [8] via minimizing the estimation error covariance of Kalman filter (KF). In the latter case, metrics quantifying a system's observability and controllability are considered. In [9, 10] , sensors are placed using criteria based on observability and controllability Gramians for thermal systems described by a linear advection partial differential equation (PDE).

The SAD problem has been studied in other application areas from the perspectives of direct and indirect design. Direct SAD has been studied for vibration reduction and the Ginzburg-Landau fluid system on the basis of Ή ₂ control, Ή _∞ control and Linear Quadratic Gaussian (LQG) control , see [1 1—1 5] . For indirect approaches, metrics associated with the observability and controllability Gramians, e.g., their maximum (or minimum) eigenvalues and traces, have been used to construct SA D methods for descriptor systems, power grids, vibration control of flexible structures [16-21] . The treatment of SAD in most of the above literature is to solve formulated optimization problems. The solutions, expressed in terms of computation-based optimization procedures, are essentially approximate and require substantial computing power [22], thus limiting their application to large-scale systems like buildings.

Statement of contributions. In this work, wc develop novel SAD strategies for temperature monitoring and control. The design is intended to maximize the traces of the observability and controllability Gramians of a building thermal system. Compared with previous work, the strategies promise important benefits. First, the development is based on APPENDIX TO THE SPECIFICATION

analytical and closed-form solutions. Second, the less computational cost makes them more competitive than their counterparts in the literature. In addition, they are derivative-free and avoid challenges faced in calculating derivatives for optimization. Due to these benefits our design lends itself very well to practical applications. The basic SAD strategies are improved to account for on-thc-ground experience, prior knowledge, industry guidelines and physical limitations in the decision- making process, further enhancing the practical value and performance of our methodology. Their connections with and potential to improve ASHRAE guidelines for sensor deployment in data centers |23j will be revealed. The effectiveness of the proposed strategies are demonstrated through simulation examples of heat transfer and data, centers. This work builds upon our earlier study in [24] , with discussion and justification in further detail.

Organization : The remainder of the paper is organized as follows. Section 2 introduces PDE-based airflow and heat transfer models which are then reduced to the state-space form. Section 3 develops our novel SAD strategies through maximizing the trace of the system's observability and controllability Gramians. Simulation examples are presented in Section 4 to illustrate the value of the proposed strategies. Section 5 summarizes our results and ideas for future work.

2. AIRFLOW AND HEAT TRANSFER MODELING

In this section, PDE-based models describing airflow and heat transfer are briefly presented. Then, these models are converted to the state-space form in order to develop control- theory-based SAD strategies.

2.1. Construction of Airflow and Heal Transfer Models

The Navier-Stokes equations describing the conservation of momentum and mass for incompressible airflow are given respectively, as follows:

V · V = 0. (2) where g is the gravity vector, Vf> the pressure gradient, μ the dynamic viscosity. A steady- state airflow is assumed in our study, i.e., dV/dt = 0, as we are interested in the steady- state large-scale behavior of the indoor airflow field and intend to reduce the complexity of analysis [25] . APPENDIX TO THE SPECIFICATION

For a time- varying temperature field T(x, y, z, t) , the heat transfer via convection-diffusion is given by

where p, c _p and κ denote, respectively, the density, specific heat and thermal conductivity of air, and h represents the heat generated or removed ('sources' or 'sinks' of T in terms of heat transfer) .

For (l)-(2) , the following boundary condition is applied:

n · V = V _b , (4) where n is the unit outward normal vector at a point on the space domain boundary, and V _¾ is assumed to be zero at static boundaries and non-zero at non-static ones. We suppose that, when _¾ 0, its value is known or can be determined directly from certain sensors, e.g. , real-time pressure sensors.

The flow of heat in the direction normal to the boundary is specified by

n · (fcVT) = q + aT, (5) where q results from the power of the heating or cooling sources at the boundaries and is a coefficient.

2.2. Conversion to State-Space Form

The state-space model for the thermal dynamics is obtained through the method of lines (MOL) . MOL approximates the spatial derivatives by a finitc-diffcrcncc-bascd discretization, with the resulting ODEs established over the time domain |26J . The MOL is applied to (3) along with the boundary conditions (5) to obtain the ordinary differential equations (ODEs) and subsequently the state-space form to descri be the temperature dynamics. Further detai ls on the state-space form are presented below as understanding the importance of the entities in the state space equations is essential to understanding the deployment strategies.

Consider a uniformly gridded three-dimensional space. The number of grid points along each axis is N _x , N _y and N _z , respectively. The state vector x is the collection of temperature values at all grid points, and the input vector u is a collection of the heat sources or sinks on the grid, that is, x(t) T(i, j, k, u(t) h i, j, k, t) (6) APPENDIX TO THE SPECIFICATION

The dimension of x is n _x = N _x x N _y x N _z , and the dimension of u is the number of sources and sinks in the system, denoted as n _u . In most practical cases, n _u < n _x . The state-space equation is

x(i) = Ax(t) I Bu(i). (7)

The matrices A G ] ^{rax rax} and B G R™ ^{x X} ™ ^u are determined from (3) and (5) . It should be noted that B indicates the placement of sources or sinks, i.e. , actuators. It has a sparse binary structure— each element is 0 or 1 (after normalization) , and only one element of each column can be 1 as the actuators are assumed to be point sources. That is,

B _j G {0, 1} Vi , ∑ Bi _j = 1 for j = 1 , 2, · · · , n _u . (8) i=l

The measurement vector y G has a dimension equal to the number of sensors, and n _y < r?, _x . The output equation representing the sensor measurements are as fol lows:

y(t) = Cx(i) , (9) where C is also a sparse binary matrix representing sensor locations with d _j G {0, 1 } ∑ d _j = 1 for i = 1, 2, · · · , n _y . ( 10) i=i

Together (7) and (9) represent the state-space model for heat transfer in our study. It is a linear, time-invariant and high-dimensional system, as a result of the PDE reduction. In the following quest for optimal SAD, the sparse binary structure of B and C will be fully utilized to alleviate the difficulty of analysis and design.

Remark 1. In above, we have considered point source actuators, steady-state airflow field and negligible humidity effect. Such assumptions are necessary to simplify the modeling and ensuing analysis. It is noteworthy that the approximation-induced errors will not prevent us from capturing the system's overall large-scale behavior. Similar assumptions can be found in a large body of previous work, e.g. , 8 10, 25] , to name a, few. As an example, a steady mean velocity field is used to replace the time- varying one in j9 l .

3. OPTIMAL SENSOR AND ACTUATOR DEPLOYMENT STRATEGIES

Optimal SAD strategies are developed in this section. The choice of observability- and controllability- Gramian-based metrics is first justified. Then, the basic strategies for deployment are derived first, and then the relevance of these strategies is improved by some relaxation . APPENDIX TO THE SPECIFICATION

3.1. Optimal Sensor Deployment

The goal of the optimal sensor deployment strategy is maximizing the trace of the observability Gramian. Since the system in (7) and (9) is physically stable, A is a, stable matrix, and the observability Gramian, W ₀ , is defined as

W ₀ = e ^ATr C ^T Ce ^AT dr, (11)

The optimal sensor locations are determined via selecting C to maximize tr(W ₀ ):

maxtr [W ₀ (C)J

(12) s.t. d _j G {0, 1} Vi,j, ∑ d _j = 1 for i = 1, 2, · · · , n _y ,

i=i

where d = 1 when the sensor i is placed at the j-th point in the gridded domain and Ci = 0 otherwise.

3.1.1. Inside Observability Gramian

Observability represents the ability to estimate the internal state variable using the input and output of a system. Its Gramian W ₀ has important implications regarding the system and state estimation. Hence, it can be used as an indirect yet tremendously instrumental indicator of the temperature estimation capabilities in our study. The following summarizes the interpretations of AV ₀ .

First, it decides the amount of information of the output contains about the state, because the observed energy in the output can be written as llylli = ^T (T)y(r)dr = x ^T (0)w _o x(0), (i?>)

where x(0) is the initial state.

Second, the Ή ₂ norm of the system G in (7)- (9) is a weighted trace of W ₀ [27], which can be expressed as

||G|| ₂ = tr (B ^T W ₀ B) . (14)

Finally, it affects the state estimation accuracy when the output is measured with noise. Suppose that the measurements are corrupted by additive noise ^. Then the measurement equation becomes y(t) = Cx(i) + v(i). The least-squares estimation of x(0) given y(i) for 0 < t < oo is

x(0) = x(0) + W ^"1 / e ^ATr C ^T v(r)dT. (15)

Jo APPENDIX TO THE SPECIFICATION

Furthermore, if {v (t) } is a continuous-time wide-sense-stationary (wss) Gaussian white noise process with autocovariance function R, _v (r) = rS(r)l, then the estimation error covariance will be rW ^{" 1} .

As a measure of the observability, tr(W _D ) is vital — larger values correspond to an increase in the overall observability of the system. It is also related with the rank maximization of W ₀ . It is known that a nonsingular W ₀ guarantees complete observability. However, W ₀ will be rank-deficient if the system is only detectable. This may happen when a limited number of sensors are deployed. In such a case it would be valuable to deploy sensors to obtain a C such that the rank of W„ is maximized: max rank(W _o ). ( 16)

Solving this rank maximization problem (globally) is rather difficult, known to be computationally non-deterministic polynomial-time hard (NP-hard) [28] . A widely used heuristic is to replace the rank objective with the trace, so we would solve: max tr(W ₀ ) . ( 1 7)

Because tr(W ₀ ) = ^™ ₌₁ ^(W,,) , where Ai(W ₀ )s for % = 1 , 2, · · · , n _x are the eigenvalues of W ₀ , maximizing tr(W _Q ) tends to result in a high-rank matrix [28] .

3.1.2. Optimal Sensor Deployment Strategy

Previous work on observability-Gramian-based sensor deployment strategies usually involve computationally burdensome optimization procedures such as integer programming. In this section a computationally attractive solution to (12) which maximizes tr(W ₀ )) under the structural constraints of C is developed.

We can write tr [W ₀ (C)] = tr Qf °° e ^ATT C ^T Ce ^AT dr^ = tr e ^ATT C ^T Ce ^A ") dr

= jT tr (e ^Ar e ^ATr C ^T c) dr = tr . ( 18)

Because A is stable, X = β ^Ατ β ^{ΑΤ τ} (1τ is the unique solution of the Lyapunov equation

AX + XA ^T -I. ( 19)

In addition, L = C ^T C is a binary diagonal matrix. Each of its diagonal elements, L _j , is 0 or 1 for j— 1 , 2, · · · , 7?, _x ; _jj — 1 if a sen sor is l ocated at t he j-tb grid point. Therefore, APPENDIX TO THE SPECIFICATION

to maximize tr(W _G ) = tr(XL) , we only need to find the n _y largest diagonal elements (sort operation) , determine the rows they belong to, and assign 1 to the corresponding elements in C. That is, after searching through the diagonal elements of X, we obtain the set S = {si '. i = 1 , 2, · · · , n _y } such that X _jj > X _{i i} for any j e S and i ^ S; we then let = 1 by placing a sensor at the -th point for i = 1, 2, · · · , n _y .

The optimal sensor deployment strategy ( OSD) , is summarized as follows:

Optimal Sensor Deployment — OSD strategy

Step 1 solve AX + XA ^T - -I;

Step 2 find the indices of the n _y largest diagonal elements of X and determine the index set S = {s^ : k = 1, 2, · · · , n _y } with X _jj > X.^- for j€ S and i φ S;

Step 3 set the («, s _¾ )-th element of C to 1 for i = 1, 2, · · · , n _y and other elements to 0, or equivaleritly, Ci = 1 if j = s.; and otherwise, C^ _j = 0;

Step 4 place sensors accordingly.

It is seen from above that OSD results from a, straightforward conceptual formulation and an analytical solution. Translating OSD into practice calls for certain flexibility to accommodate various practical considerations. In other words, additional real-life value can be offered to practitioners if we make improvements to OSD for better applicability. One issue that can be encountered in practice is spatial constraints, since certain areas may be inaccessible or unavailable for sensors due to physical limitations. Another issue is that OSD may yield undesired 'dense' or 'clustered ' sensor deployment — multiple sensors deployed within a relatively small area. Moreover, a primary concern that often arises is how to integrate the practitioner's experience and existing industry guidelines into the decision process.

Responding to such needs, we propose to build an observability map, which shows the distribution tr(W _c ) over the space. The information it offers can be used with awareness of spatial limitations and inclusion of expert experience to decide the sensor locations. To construct the map, we place a single sensor at a grid point. In this case, C€ Μ. ^{1 κ} " ^χ , where the clement corresponding to this grid point will take i and the others 0. Then tr(W ₀ ) will be calculated to quantify the observability if a sensor is placed here. By analogy, a map illustrating the relationships between tr(W ₀ ) and each spatial location can be generated. We note that the computation only relies on solving the Lyapuno equation (19) for X, because the diagonal elements of X are equivalents of tr(W ₀ ) with a single sensor placed on the corresponding locations. To show this, l t us assume that a sensor is placed at the i-th APPENDIX TO THE SPECIFICATION

grid point, implying the i-th. element of C is I , i.e.,

C 0 0 1 0 0

Then it follows that tr(W ₀ ) = tr(XC ¹ C) = X _tl . (21)

In general, an area in the map should be given more weight during sensor placement if it has l arger t.r(W ₀ ) . Such information can be easily fused wi th prior experience and knowledge at the practitioner's level. As a. consequence, an improved OSD, referred to as iQSD, is proposed as follows:

Improved Optimal Sensor Deployment— iOSD strategy

Step 1 solve A X -f X Λ ^' — I :

Step 2 extract the diagonal elements of X and rearrange them with respect to the spatial locations to build the observability map;

Step 3 decide the best locations of sensors via with the aid of the map information, practitioner's experience and knowledge and industry guidelines, and place the sensors.

ensors accordin

3.2. Actuator Deployment

The actuator deployment problem is a dual of the sensor deployment problem if the actuators are considered as point sources. The controllability Gramian W _c is considered in this case, where

We define the optimal actuator deployment as follows: max tr [W _C (B)]

which is dual to (12) . APPENDIX TO THE SPECIFICATION

3.2.1. Inside Controllability Gramian

The controllability Gramian W _c is chosen as the measure of control authority for a dynamic system in our study due to its following interpretations.

First, W _c is closely related with minimum energy control. Consider driving a system from x(0) = 0 to x(i) = x using the lowest amount of control energy:

min E(i)

u (24) s.t. x(t) = Ax(t) + Bu(t), x(0) = 0, x(t) = x,

where E(t) = f* u ^T (r)u(r)dr. The resulting control input is u(r) = B ^T e ^{AT (i} -^W _c - ¹ (i)x, 0 < r < t. (25)

Hence, the control energy over an infinite time horizon is E(oo) = x ^T W ^~1 x.

Second, the Ή ₂ norm of G is also a weighted trace of the controllability Gramian:

|| G || ₂ = tr (CW _C C ^T ) . (26)

Finally, a, larger W _c can also help suppress the influence of process noise. Suppose that the input u is corrupted by an additive Gaussian white noise w with covariance Q = ql:

±(t) = Ax(i) + B [u(t) + w(i)] . (27)

Suppose the control objective is to drive the state to x. By optimal control theory, irrespective of how the control input u is chosen, the state x, will not be precisely achieved due to the effects of the noise w. The state covariance will be

E (x(oo) (x(oo) (28) which is inversely proportional to W _c . Thus a larger W _c may contribute to noise suppression.

The rank of the controllability matrix is relevant to the rank of W _c . When a system is only stabilizable due to the small number of actuators, one can try to increase the rank of the controllability matrix by placing the actuators in the best positions. Thus wc would want to solve max _B rank(W _c ). Similar to (16), it is an NP-hard problem. The trace heuristic can hence be used to solve this problem, i.e., maxe tr (W _c ).

3.2.2. Optimal Actuator Deployment Strategy

The optimal actuator deployment (OAD) strategy can be developed analogously to OSD, which is summarized as follows. APPENDIX TO THE SPECIFICATION

Optimal Actuator Deployment — OAD strategy

Step 1 solve A ^T X + XA = -I;

Step 2 find the indices of the n _u largest diagonal elements of X and determine the index set S = {s _k : k = 1, 2, · · · , ra _u } with X _{J 3} > X _l for j E S and i ^ S

Step 3 set the (s _{J 7} j)-th element of B to 1 for j = 1 , 2, · · · , n _u and other elements to 0, or equivalently, Z¾ = 1 if i = s _j and otherwise, B{ = 0.

Along similar lines to sensor deployment, the OAD strategy can be improved by taking into account multiple decision criteria,, including the controllability map, awareness of physical spatial constraints and expert experience. This will result in improved OAD or iOAD strategy similar to the iQSD strategy presented above. We omit the presentation of iOAD strategy as it is easy to deduce from the QAD and iQSD strategies.

3.3. Remarks on the Deployment Strategies

The following remarks are presented to give a complete perspective on the deployment strategies developed above:

Remark 2. Compared with the existing sensor (actuator) deployment methods that have used the observability (controllability) Gramian W ₀ (W _c ) , our approach promises two important benefits. First, rather than being near-optimal based on numerical methods, we constructed the deployment strategies on the basis of truly optimal solution to maximization of the considered reward function through a straightforward theoretical approach. Second, the implementation is easy and fast as there is no need for a specific numerical optimization algorithm, making the computation manageable even for large-scale dynamic systems. The computational bottleneck in our design is solving the Lyapunov equation, which can be handled in 0 (n^) operations and 0(n^) memory with [29] . By comparison, the integer programming used in the literature is shown to be NP-hard [30] .

Remark 3. The metric optimized for 0SD (OAD) is the trace of the observability (controllability) Gramian. Hence, 0SD (OAD) is optimal only in the sense of this metric. Although other metrics directly evaluating estimation (control) performance has been used in the literature to develop deployment strategies, 0SD (OAD) retain their importance for several reasons. First, our approach presents a closed-form solution, and the ease of computation lends itself to practical implementation, as discussed in Remark 2. Second, as observability (controllability) is crucial for estimation (control) , the Gramian-based strategies would help APPENDIX TO THE SPECIFICATION

improve the estimation (control) performance, though in an indirect way. Finally, independent of estimators (controllers), it can be easily applied to any dynamic system and allows easy modification or improvement, which may be needed when some practical constraints arise. In general, it provides guidance and vital clues as a first step to practitioners on where to deploy sensors (actuators).

Remark 4. As an improvement to OSD (OAD), iOSD (iOAD) has a strong practical appeal, incorporating various practical considerations into the determination of measurement and actuation locations. This is better demonstrated by simulation examples in Section 4. However, it should be understood that iOSD (iOAD) maximizes the trace of the observabi lity (controllability) Gramian limited by the chosen constraints and does not necessarily result in the largest possible value of the trace of the observability (controllability) Gramian.

Remark 5. Although the proposed strategies are motivated by temperature monitoring and control in rooms and buildings, they have a much wider potential with possible application in wireless sensor networks, active vibration control of flexible structures, environmental monitoring, etc.

4. RESULTS

An exhaustive simulation-based study is presented in this section to evaluate the effectiveness of the proposed OSD, iOSD, OAD and iOAD strategies. The first two examples illustrate SAD for 2D heat transfer in a square room, and the third considers sensor deployment for 2D data centers. While 2D settings are considered for simplicity, the strategies can be applied to 3D cases without modification. The performance of the optimization strategics is quantified by the traces of the observability and controllability Gramians for all the examples, in addition, the celebrated Kalman filter (KF) and linear quadratic regulator (LQR) will be used to examine estimation and control performance due to different SAD. Concerning state estimation state-feedback-based control, respectively, they are closely linked with the observability a d controllability concepts, thus making useful tools for compari g SA ! ) strategies. We would also like to caution the reader that the proposed strategies can help improve estimation and control but do not guarantee best results. This is because their design is positioned to maximize tr(W ₀ ) and tr(W _c ).

Example 1. Consider sensor deployment for a 2D square domain D with dimensions of 5m x 5m. The velocity field is assumed zero. The temperature at all points is— 10°C at the initial moment, i.e. , T(t = 0) \D =—10. When t > 0, the temperature at the four wall APPENDIX TO THE SPECIFICATION

Figure 1 Observability map boundaries will be constantly 20°C, i.e., T(t) \ _dD = 20 for t > 0. The temperature field changes through time as a result of heat diffusion.

Suppose that the initial temperature is not known and that the heat transfer process is corrupted by noise. We intend to deploy 5 sensors in the domain and then estimate the temperature field in real time using sensor measurements. An observability map is constructed and il l ustrated in Figure 1 to show the spatial di stribution of tr(W ₀ ) . A gradual decline of the value from center to periphery is noted, indicating that the central area should be given priority attention. Consequently, the sensors are densely placed by OSD in the center, as shown in Figure 2. Physically, this is because the temperature in this part is the hardest to estimate as only the boundary temperatures are exactly known at each instant. To avoid clustered deployment, we assume each sensor has a lin radius of influence. This does not mean that the sensors can accurately measure the states in that radius. It is merely chosen to maintain a minimum separation of sensors. Then iOSD places one sensor in the exact center and iOSD spreads out the other sensors accounting for each sensor's 1m zone of influence. The sensors are symmetrically distributed as a result of the symmetric spatial setting. For a comparison, the sensors are placed in randomly selected locations as the the third case.

The KF is then applied to the state-space model obtained from (3) in each case of deployment for temperature estimation. Figure 2 compares the resultant estimated temperature fields with the true ones at t— Is, 10s and 20s. Since the system is stable, the estimated temperature fields gradually approach the truth over time for all cases. However, the estimation accuracy tends to be the best with iOSD. Table 1 provides an examination of APPENDIX TO THE SPECIFICATION

Figure 2 First row: true temperature fields at Is, 10s and 20s; second to fourth rows: estimated temperature fields with sensors deployed 0SD, iOSD and random selection, respectively. Sensor locations are indicated by white dots in all cases. tr(W ₀ ) , tr(∑) , δ and e. ∑ is the estimation error covariance of the KF, and δ and e are the APPENDIX TO THE SPECIFICATION

Table 1 Quantitative comparison of temperature estimation with sensors deployed by OSD, iOSD and random selection, respectively.

Figure 3 Controllability map cumulative and final-step estimation errors, respectively,

where x is the estimate of x and t is the simulation time. Tt is seen that QSD, as expected, yields the largest tr(W ₀ ). However, the estimation performance, as indicated by tr(∑), δ and e, is better in the case of iOSD, which is in agreement with visual observation. This is because OSD guarantees the largest tr(W ₀ ) but not necessarily the best estimation performance, as discussed in Remark 3. This shows that a sparse deployment of sensors, as determined by iOSD, is favorable for estimation. Compared with OSD and random sensor placement, iOSD achieves good estimation accuracy at a mild sacrifice of tr(W ₀ ).

Example 2. This example studies actuator placement for a 2D square domain D with dimensions of 5m x 5m and zero velocity field. The heat transfer process is modeled by (3) , with the boundary condition -n-VT = OLT . Note that heat loss occurs at the wall boundaries in this setting. The temperature on D at the initial time instant is 18°C, i.e., (0) |o — 18, APPENDIX TO THE SPECIFICATION

Figure 4 First to third rows: time-based temperature fields with actuators deployed by OAD, iOAD and random selection, respectively. The locations of actuators are indicated by white dots. and the desired temperature is 20°C denoted as T _d = 20. Here, 5 actuators will be used to steer the temperature. We deploy the actuators in three ways: OAD, iOAD and random placement. The LQR is used to obtain the optimal actuator control inputs. It is assumed here that the temperature field, i.e. , state information, is completely measured at each time step.

We set out by establishing the controllability map is established in the first place, as shown in Figure 3. The actuator placement is depicted i Figure 4. The value of tr(W„) increases from central to the peripheral, reaching maximum at the four corners. Then OAD places all the actuat ors in the corners and a. similar theme of dense deployment is o bserved. APPENDIX TO THE SPECIFICATION

Table 2 Quantitative comparison of temperature control with actuators deployed by OAD, iOAD and random selection, respectively.

To account for the radius of influence of each actuator, iOAD places an actuator at each corner and the fifth one in the center. From Figure 4, it is easily seen that the desired final temperature _d ----- 20 is the most evenly reached for the iOAD case.

A quantitative comparison is provided in Table 2. The values of tr(W _c ) are simi lar in all three cases. OAD still results in tr(W _c ) slightly larger than the other two cases (not observable in Table 2 due to the insignificant difference) . It also leads to the least amount of input energy ζ, where

However, as for control accuracy, iOAD outperforms the other two if wc compare the values of ξ and p, where ξ = ||x(t) - T _d \\ ² dt ) , σ = ||χ(ϊ) - T _d || ² . (31)

Even thoughiOAD results in improved control performance (observing σ) , it is important to note that it consumes more energy than the OAD case. However, both OAD andiOAD consume less energy and result in overall improved control performance as compared to the random deployment case. Thus, actuator placement based on both OAD or iOAD can potentially help with the reduction of control energy while iOAD is a more practical solution.

Example 3. This example investigates sensor deployment for small data centers and its influence on temperature estimation and control. Data center energy consumption doubled from 2000 to 2006 in the United States, accounting for 1.5% of the total electricity use in 2006 and with rapid growth expected in the foreseeable future [31] . We will consider simplified 2D data centers to demonstrate our strategies, with the discussion involving ASHRA E TC9.9 Facility and Equipment Thermal Guidelines for Data Center and other Data, Processing Environments [23] . APPENDIX TO THE SPECIFICATION

(a) (b) (c)

Figure 5 (a) floor plan of a 2D data center; (b) steady-state airflow field; (c) observability map of the data center.

Let us begin with a data center with its floor plan shown in Figure 5a. It occupies a rectangular region of 10m x 10.8m. There are four rows of racks, where the IT equipment such as servers, storage and network devices are located. The configuration is based on the well-known alternating hot-aisle/cold-aisle rack layout— the inlet side of the IT equipment faces a cold aisle and the exhaust (outlet) side faces a hot aisle. Four air conditioning units (ACUs) are placed at the left and right sides, each extracting surrounding air and then blowing cooled air into the room. Heat transfer is not restricted at the walls of the room.

The steady-state airflow within the data center is derived using (l)-(2). The flow velocity is assumed to be 5m/s at the inlet and outlet side of an ACU, and 0.05m/s for the IT equipment. A uniform interval of 0.2m is used to grid the 2D space. Note that the interior space APPENDIX TO THE SPECIFICATION

of ACUs and racks are excluded from the solution domain. The airflow field is illustrated in Figure 5b. The heat transfer in the room is described by (3). Suppose that the initial temperature within the room is 20°C, and the temperature difference between the exhaust and inlet sides of the IT equipment is assumed to be 5°C.

Λ state-space representation can be obtained to describe the dynamic behavior of the temperature according to Section 2. Then following iOSD, we construct the observability map, which is shown in Figure 5c. As is seen, the cold aisles and four corners have much higher values than other areas. Since the corners are of much less interest in practical temperature monitoring, the three cold aisles should be emphasized by iOSD. This is in agreement with. ASIIRAE TC9.9 guidelines, which suggest installation of sensors in cold aisles only. Then according to the ASHRAE recommendations and industrial practice, iOSD can place the sensors in the cold aisles, mounting them on the intakes of equipment or in the middle of the aisles for ambient temperature measurement. A physical interpretation for preference towards cold aisles is that the temperature at the hot aisles directly depends on the cold ones. Similarly, because the temperature of the cooled air blown by ACUs is known, no sensors are deployed in the proximity of an ACU's outlet side.

We can also show that both iOSD and j 23 j will both recommend sensor placement in in the cold aisles in many other data center layouts. Or in other words, iOSD will yield results that corroborate t he validity of [231 in all such cases. However, a question that arises is: will it be wise to establish the measurement points only in cold aisles in any problem setting?

To resolve this question, let us turn to another data center floor plan, which is shown in Figure 6a. It has dimensions of 10m x 9m. There are three rows of racks, and two ACUs are positioned to blow cool air of 13°C into cold aisles. The resul ting airflow velocity field is given in Figure 6b. Suppose 10 sensors will be placed to monitor the temperature. To handle this, the observability map for this layout is computed and illustrated in Figure 5c. It is seen that the value of tr(W ₀ ) is mostly small in the cold aisles. Yet the right-bottom and left-top areas lie in the high part of the value range.

If following the ASHRAE guidelines, one still has to choose locations in cold aisles as measurement points. As a way of contrast, iOSD will place sensors in all aisles, and especially, install some in the areas with large value in the map. Let us look into both cases by examining the KF-bascd estimation performance. Figure 7 shows the two placement results and compares the estimation with the truth.

The change of the temperature fields through time is illustrated in the first row of Figure 7. The estimated temperature fields are shown in the second and third rows. Since APPENDIX TO THE SPECIFICATION

Figure 7 First row: True temperature fields through time; second row: KF- estimated temperature fields with sesnors deplyed via, iOSD; third row: KF-estimated temperature fields with sensors deployed according to 123] . the overall system is stable, estimation will converge to the truth in both cases. However, we see faster and more accurate estimation for i OSD by comparing the first row with the second and third of Figure 7, respectively. Table 3 outlines a quantified comparison. We see APPENDIX TO THE SPECIFICATION

Table 3 Quantitative comparison of temperature estimation for the data center with sensors deployed by iOSD and ASHRAE TC9.9 [23], respectively. that the iOSD results in larger tr(W _G ) as well as smaller estimation errors. This evaluation implies that sensor deployment via iOSD will favor temperature estimation at least within the defined problem setting.

By this example, the iOSD strategy offers general guidance rather than specific instructions and shows the promise for improvement of temperature estimation by placing sensors in well-selected locations. Further revelation implies that it has the potential to amend the ASHRAE guidelines, especially when a control-oriented sensor deployment is needed and a good model is available for describing the temperature dynamics.

5. CONCLUSION

In this paper we have developed optimal sensor and actuator deployment strategies and applied them to improve temperature monitoring and control performance in HVAC systems. The traces of observability and controllability Gramians are chosen as optimization metrics, due to their important implications for a dynamic system. These Gramian-based metrics are maximized to yield the best sensor and actuator locations. The major contribution of our work is the proposed solution to the considered problem. Not only is it analytical and closed-form , but also reduces the computational burden compared to previous approaches.

An improved version of the solution is constructed in order to incorporate expert experience, industry guidelines and practical constraints into the decision-making process. The obtained strategies are practically more significant as they account for on-the-ground experience, meet spatial limitations and result in better estimation and control performance. A set of simulation examples highlight the effectiveness of our solution. In particular, examples applying the deployment strategies to simplified data center models demonstrate the real-world use of our approach through a comparison with ASHRAE guidelines. Future work will focus on applying these strategies to diverse real- world scenarios and on improving theoretical aspects of this methodology to consider other metrics of interest for optimization. APPENDIX TO THE SPECIFICATION

References

[1] U.S. EIA, "Annual energy review 2011 ," U.S. Energy Information Administration, Tech. Rep., Sep.

2012.

[2] C. P. Underwood, HVAC Control Systems: Modelling, Analysis and Design. Taylor & Francis, 2003.

[3] R. McDowall, in Fundamentals of II VAC IP Book. Oxford: Elsevier Science Ltd, 2006.

[4] Y. Ma, F. Borrclli, B. Hcnccy, R. Coffey, S. Rcngca, and P. Haves, "Model predictive control for the operation of building cooling systems," IEEE Transactions on Control Systems Technology, vol. 20, no. 3, pp. 796-803, 2012.

[5] C. Rash, C. Patcl, and R. Sharma, "Dynamic thermal management of air cooled data centers," in The Tenth Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronics Systems, 2006, pp. 445-452.

[6] R. Patil, Z. Filipi, and F. H., "Computationally efficient combined plant design and controller optimization using a coupling measure," ASME Journal of Mechanical Design, vol. 134, no. 7, pp. 071 008— 071 008-8, 2012.

|7J D. Wang, E. Arens, T. Webster, and M. Shi, "How the number and placement of sensors controlling room air distribution systems affect energy use and comfort," in Proceedings of International Conference for Enhanced Building Operations, 2002.

[8] J. A. Burns, J. Borggaard, E. Cliff, and L. Zietsman, "An optimal control approach to sensor/actuator placement for optimal control of high performance buildings," in Proceedings of International High

Performance Buildings Conference, 2012, p. Paper 76.

[9] U. Vaidya, R. Rajaram, and S. Dasgupta, "Actuator and sensor placement in linear advection PDE with building system application," Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 213-224, 2012.

[1 0] S. Si nha, U. Vaidya, and R . Rajaram, "Opti mal placement οΓ actuators and sensors Tor control of nonequilibrium dynamics," in Proceedings of the European Control Conference, 2013, pp. 1084—1088.

[11] K. K. Chen and C. W. Rowley, "¾2 optimal actuator and sensor placement in the linearised complex ginzburglandau system," Journal of Fluid Mechanics, vol. 681, pp. 241—260, 2011.

[12J R. Potami, "Optimal sensor/ actuator placement and switching schemes for control of flexible structures," Ph.D. dissertation, Worcester Polytechnic Institute, 2008.

[13] K. Hiramoto, H. Doki, and G. Obinata, "Optimal sensor/actuator placement for active vibration control using explicit solution of algebraic riccati equation," Journal of Sound and Vibration, vol. 229, no. 5, pp. 1057-1075, 2000.

[14] A. Arabyan and S. Chemishkian, '"¾ _∞ -optimal mapping of actuators and sensors in flexible structures," in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 1, 1998, pp. 821—826.

[15] C. Coiburn, D. Zhang, and T. Bcwlcy, "Gradient- based optimization methods for sensor & actuator placement in LTI systems," 2011, http://fccr.ucsd.edu/pubs/CZB_placement.pdf.

[1 ] Ό. Georges, "The use οΓ observabil ity and controllabil ity gram iaris or fu nctions Tor opti mal sensor and actuator location in finite- dimensional systems," in Proceedings of the 3 th IEEE Conference on

Decision and Control, 1995, pp. 3319—3324.

[17] B. Marx, D. Koenig, and D. Georges, "Optimal sensor/actuator location for descriptor systems using APPENDIX TO THE SPECIFICATION

Lyapunov-like equations," in Proceedings of the J^lst IEEE Conference on Decision and Control, 2002, pp. 4541-4542.

[18] T. H. Summers and J. Lygeros, "Optimal sensor and actuator placement in complex dynamical networ ks," 2013, http:/7arxi v.org/abs/ l 306.2491 .

[19] D. Halim and S. R. Moheimani, "An optimization approach to optimal placement of collocated piezoelectric actuators and sensors on a thin plate," Mechatronics, vol. 13, no. 1, pp. 27—47, 2003.

[20] FT. R. Shaker, "Opti mal sensor and actuator location for u nstable systems," Journal of Vibration and Control, vol. 19, no. 12, pp. 1915-192, 2012.

[21] Z.-C. Qiu, X.-M. Zhang, H.-X. Wu, and H.-H. Zhang, "Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate," Journal of Sound and Vibration, vol. 301, no. 3-5, pp. 521-543, 2007.

[22] S. L. Padula and R. K. Kincaid, "Optimization strategies for sensor and actuator placement," National Aeronautics and Space Administration, Tech. Rep. NASA/TM-1999-209126, 1999.

[23] T. S. ASIIRAE TC9.9 Mission Critical Facilities and E. Equipment, "Facility and equipment thermal guidelines for data center and other data processing environments," ASHRAE, Tech. Rep., 09 2003.

[24] H. Fang, R. Sharma, and R. Patil, "Optimal sensor and actuator deployment for HVAC control system design," in Proceedings of the American Control Conference, under review, 2014.

[25] V. Lopez and H. F. Hamann, "Heat transfer modeling in data centers," International Journal of Heat and Mass Transfer, vol. 54, no. 5306-5318, 2011.

[26] W. E. Schiesser and G. W. Griffiths., A Compendium of Partial Differential Equation Models. Cambridge University Press, 2009.

[27] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. Springer London, 1995.

[28] M. Fazel, H. Hindi, and S. Boyd, "Rank minimization and applications in system theory," in Proceedings of the American Control Conference, 2004, pp. 3273—3278.

[29] P. Benner, J.-R. Li, and T. Penzl, "Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems," Numerical Linear Algebra with Applications, vol. 15, no. 9, pp. 755-777, 2008.

[30] C. H. Papadimitriou, "On the complexity of integer programming," Journal of ACM, vol. 28, no. 4, pp. 765-768, 1981.

[31 ] U.S. EPA, "Report to Congress on server and data center energy efficiency," U.S. Environmental Protection Agency, Tech. Rep., 2007.