Optimal Sensor and Actuator Deployment for HVAC Control System ...
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Optimal Sensor and Actuator Deployment for HVAC Control System Design Huazhen Fang, Ratnesh Sharma and Rakesh Patil Abstract— This paper studies control-theory-inspired optimal sensor and actuator deployment to improve the temperature monitoring and control performance of HVAC (heating, ventilation and air conditioning) systems in buildings. The deployment strategies are based on maximizing observability- and controllability-based metrics such as the respective Gramians. Our solution approach has the following benefits compared to previous work. First, an analytical, closed-form solution is developed. Second, the computational cost is low to ensure practical use of our approach. The effectiveness of our deployment strategies is demonstrated via simulation examples and by calculating different metrics of relevance.
I. INTRODUCTION This paper focuses on novel strategies for optimal Sensor and Actuator Deployment (SAD). Control-theory-inspired strategies are developed in order to improve temperature monitoring and control performance of HVAC (heating, ventilation and air conditioning) systems in buildings. SAD for buildings has relied on either heuristic rules or the other extreme of computationally burdensome CFD modeling [1]. There is limited research on control-theory-based SAD for HVAC applications. In [2, 3], authors decide the suitability of sensor and actuator locations for thermal systems using a linear advection partial differential equation (PDE). They explore different criteria based on observability and controllability Gramians. However, these elegant theoretical analyses are difficult to translate into practice. The SAD problem has been studied for some other applications using different metrics to evaluate the quality of the deployment. Gramian-based approaches have been quite popular due to the interpretation of the Gramians as an indicator of a system’s control and monitoring performance, [4–9]. In addition, direct control-oriented SAD has also been studied, for example, H2 control in [10, 11] and H∞ control in [12, 13]. We observe that the essential treatment of SAD in most of the above literature is to solve a formulated optimization problem. The solutions are expressed in terms of computation-based optimization procedures. These approaches are fundamentally approximate and require significant computing power, especially for largescale systems [14]. In our work, we provide insights for SAD using observability- and controllability-based metrics especially for Huazhen Fang is with the Department of Mechanical Engineering, University of California, San Diego, CA 92093, USA, and currently a research intern with the Energy Management Department at NEC Laboratories America, Inc., Cupertino, CA 95014, USA [email protected]. Ratnesh Sharma and Rakesh Patil are with the Energy Management Department at NEC Laboratories America, Inc., Cupertino, CA 95014, USA {ratnesh,rakeshmp}@nec-labs.com.
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temperature monitoring and control. Specifically, we propose SAD strategies that analytically maximize the traces of the observability and controllability Gramians. Our approach presents two benefits over previous work. First, we produce analytical closed form solutions. Second, these solutions are computationally cheaper and lend themselves well to practical applications. The basic approach is further improved by incorporating spatial constraints on locations of sensors and actuators. The improved approach further enhances the value and performance of our approach, as demonstrated through simulation examples using different configurations. The remainder of the paper is organized as follows. Section II introduces PDE-based airflow and heat transfer models which are then reduced to the state-space form. Section III develops our novel SAD strategies through the maximization of the trace of the system’s observability and controllability Gramians. Simulation examples are presented in Section IV to illustrate the value of our approaches. II. 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 and apply control-theory-based sensor and actuator deployment strategies. A. Construction of Airflow and Heat Transfer Models The Navier-Stokes equations describing the conservation of momentum and mass for incompressible airflow are given by (1) and (2), respectively [15]: ∂V + (V · ∇)V = ρg − ∇p + µ∇2 V, (1) ρ ∂t ∇ · V = 0. (2) where g is the gravity vector, ∇p the pressure gradient, µ the dynamic viscosity. A steady-state airflow is assumed in our study, i.e., ∂ V/∂t = 0, as we are interested in steady-state large scale behavior and intend to reduce the complexity of the analysis [16]. For a time-varying temperature field T (x, y, z,t), the heat transfer via the convection-diffusion equation is given by ∂T + V · ∇T − ∇ · (κ∇T ) = h, (3) ρc p ∂t where ρ, 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) [16].
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where C shows sensor locations with a structure like B:
For (1)-(2), the following boundary condition is applied: −n · V = Vb ,
(4)
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. We suppose that, when Vb 6= 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 · (k∇T ) = q + αT,
B. 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 finite-difference-based discretization, with the resulting ODEs established over the time domain [17]. The MOL is applied to (3) along with the boundary condition (5) to obtain the ODEs and subsequently the state-space form to describe the temperature dynamics. Further details 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 Nx , Ny and Nz , respectively. The state vector x is the collection of temperature values at each grid point, and the input vector u is a collection of the heat sources or sinks on the grid, that is, .. .. . . h(i, j, k,t) T (i, j, k,t) x(t) = , u(t) = . .. .. . . n ×1 n ×1 u
The dimension of x is nx = Nx × Ny × Nz , and the dimension of u, nu , depends on the number of sources and sinks in the system, denoted as nu . The state-space equation is x˙ (t) = Ax(t) + Bu(t).
nx
∑ Bi, j = 1 for
i=1 nu
j = 1, 2, · · · , nu , (7)
∑ Bi, j = 1 for i = 1, 2, · · · , nx .
j=1
The measurement vector y has a dimension ny equal to the number of sensors. The output equation representing the sensor measurements are as follows: y(t) = Cx(t),
j=1
(9)
ny
j = 1, 2, · · · , nx .
∑ Ci, j = 1 for
i=1
Together (6) and (8) represent the state-space model for heat transfer in our study. III. OPTIMAL DEPLOYMENT STRATEGIES Optimal sensor and actuator deployment strategies are developed in this section. A. Optimal Sensor Deployment The optimal sensor deployment strategy is build upon the observability concept. For the system in (6) and (8) and a stable A, the observability Gramian, Wo , is defined as Z ∞
Wo =
(8)
>
eA τ C> CeAτ dτ,
(10)
0
1) Inside Observability Gramian: The goal of the sensor deployment strategy is to obtain the “best” temperature description of the system. Wo is used as an indication of the temperature description due to its following interpretations related to state estimation. If a system has initial state x(0), the observed energy in the output can be written as kyk22 =
Z ∞ 0
y> (τ)y(τ)dτ = x> (0)Wo x(0),
Thus, the larger Wo is, the more information the output contains about the state. The H2 norm of a system can be written as kGk2 = tr B> Wo B ,
(6)
The matrices A and B are determined from (3) and (5). It should be noted that B indicates the placement of actuators. It has a sparse structure — each element is 0 or 1 (after normalization). Only one element of each column and row can be 1 as the actuators are assumed to be point sources: Bi j ∈ {0, 1} ∀i, j,
∑ Ci, j = 1 for i = 1, 2, · · · , ny ,
(5)
where q results from the power of the heating or cooling sources at the boundaries and α is a coefficient.
x
nx
Ci j ∈ {0, 1} ∀i, j,
which is a weighted trace of Wo [18]. A larger H2 norm leads to better suppression of output measurement noise. It is known that a nonsingular Wo guarantees observability. However, Wo will be rank-deficient if the system is only detectable rather than being observable. 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 Wo can be increased: maxC rank(Wo ). Solving this rank maximization problem (globally) is rather difficult, known to be computationally NP-hard [19]. A widely used heuristic is to replace the rank objective with the trace, so we would solve maxC tr(Wo ). Because tr(Wo ) = ∑ni=1 λi (Wo ), where λi (Wo ) are the eigenvalues of Wo for i = 1, 2, · · · , nx , maximizing tr(Wo ) tends to result in a highrank matrix [19].
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2) Optimal Sensor Deployment Strategy: Based on the above discussion, the optimal sensor locations are determined via selecting C to maximize tr(Wo ): max tr [Wo (C)] C
nx
∑ Ci, j = 1 for i = 1, 2, · · · , ny ,
s.t. Ci j ∈ {0, 1},
j=1
(11)
to physical limitations. Second, ‘dense’ or ‘clustered’ sensor deployment — multiple sensors deployed within a relatively small area — should be avoided, since each sensor can cover a certain region for sensing or monitoring. Equivalently, one has to consider a zone of influence of each sensor and avoid overcrowding of sensors. As a consequence, an improved OSD, or iOSD strategy is proposed as follows: Improved optimal sensor deployment — iOSD strategy
ny
∑ Ci, j = 1 for
j = 1, 2, · · · , nx .
Step 1 solve AX + XA> = −I; Step 2 find the largest diagonal element of X; if its corresponding location is in an acceptable region (defined by the system’s designer), mark its index as s1 , set C1,s1 = 1 and place the sensor, otherwise find the next largest one and repeat this step; Step 3 find the next largest diagonal element of X; if its corresponding location is in an acceptable region and this sensor’s monitoring region (previously defined) does not overlap with the regions of the sensors placed before, mark its index as si , set Ci,si = 1 and place the sensor, otherwise find the next largest one and repeat; Step 4 repeat Step 3 for i = 2 until i = ny ; then set the remainder of the elements of Ci, j = 0 for i = 1, 2, · · · , ny and j 6= si .
i=1
where Ci, j = 1 when sensor i is placed at the j-th point in the gridded domain and Ci, j = 0 otherwise. This optimization can be solved as an integer programming problem. However, we develop a more computationally attractive solution. This solution strategy described below is a major contribution of this paper. We can write Z ∞ > eA τ C> CeAτ dτ tr [Wo (C)] = tr 0 Z ∞ > tr eA τ C> CeAτ dτ = Z0 ∞ > tr eAτ eA τ C> C dτ = 0 Z ∞ Aτ A> τ > e e dτC C . = tr 0
B. Actuator Deployment
R > We note that, because A is stable, X = 0∞ eAτ eA τ dτ is the
unique solution of the Lyapunov equation AX + XA> = −I. In addition, L = C> C is a binary diagonal matrix. Each of its diagonal elements, L j, j , is 0 or 1 for j = 1, 2, · · · , nx ; L j, j = 1 if a sensor is located at the j-th point. Therefore, to maximize tr(Wo ) = tr(XL), we simply find the ny 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, · · · , ny } such that X j, j > Xi,i for any j ∈ S and i ∈ / S ; we then let Ci,si = 1 by placing a sensor at the si -th point for i = 1, 2, · · · , ny . The optimal sensor deployment strategy ( OSD), is summarized as follows: Optimal sensor deployment — OSD strategy Step 1 solve AX + XA> = −I; Step 2 find the indices of the ny largest diagonal elements of X and determine the index set S = {sk : k = 1, 2, · · · , ny } with X j, j > Xi,i for j ∈ S and i∈ / S; Step 3 set the (i, si )-th element of C to 1 for i = 1, 2, · · · , ny and other elements to 0, or equivalently, Ci, j = 1 if j = si and otherwise, Ci, j = 0.
The actuator deployment problem is a dual of the sensor deployment problem if the actuators are considered point sources. The trace of the controllability Gramian Wc is maximized in this case, where Z ∞
Wc =
>
eAτ BB> eA τ dτ,
(12)
0
1) Inside Controllability Gramian: The controllability gramian Wc is chosen as the measure of control authority for a dynamic system in our study due to its following interpretations. Consider driving a system from zero initial state x(0) = 0 to a final state x¯ , i.e., x(t) = x¯ , using minimum control energy: min E(t) u
s.t. x˙ (t) = Ax(t) + Bu(t), x(0) = 0, x(t) = x¯ , where E(t) =
Rt > 0 u (τ)u(τ)dτ. The resulting control input is
u(τ) = B> eA
> (t−τ)
W−1 x, 0 ≤ τ ≤ t. c (t)¯
Hence, the control energy over an infinite time horizon is ¯ . Thus, a larger Wc results in lower control E(∞) = x¯ > W−1 c x energy. A larger Wc 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 = qI:
Further improvements are made to the OSD strategy to account for some practical issues. Two considerations worth noting in practical sensor deployment are: First, certain areas may be inaccessible or unavailable for sensor placement due
x˙ (t) = Ax(t) + B [u(t) + w(t)] . Suppose the control objective is to drive the state to x¯ . From optimal control theory, we know that irrespective of how the
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control input u is chosen that the state x, will not be precisely achieved due to the effects of the noise w. The covariance of the state will be i h E (x(∞) − x¯ ) (x(∞) − x¯ )> = qW−1 c , which is inversely proportional to Wc . Thus a larger Wc may contribute to noise suppression. The rank of the controllability matrix is related to the rank of Wc . 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. The trace heuristic maxB tr (Wc ) can be used in this case. 2) Optimal Actuator Deployment Strategy: Based on the above discussion, a possible way to define optimal actuator deployment would be max tr [Wc (B)] B
nx
s.t. Bi j ∈ {0, 1} ∀i, j,
∑ Bi, j = 1 for
j = 1, 2, · · · , nu ,
i=1
(13)
nu
∑ Bi, j = 1 for i = 1, 2, · · · , nx .
j=1
Analogous to OSD, the optimal actuator deployment (OAD) strategy for actuator placement is summarized as follows.
in our approach is solving the Lyapunov equation, which can be handled in O(n3x ) operations and O(n2x ) memory with [20]. By comparison, the integer programming used in the literature is proven to be NP-hard, often requiring computation in polynomial time [21]. Remark 2: The metric optimized for OSD (OAD) is the trace of the observability (controllability) Gramian. Hence, OSD (OAD) is optimal only in the sense of this metric. Other relevant metrics could be related explicitly to the estimation errors (e.g. norm of estimation error) or the control energy etc. and result in different deployment strategies. However, OSD (OAD) retain their importance for two reasons. First, our approach presents a closed-form solution and the ease of computation lends to practical implementation, as discussed in Remark 1. Second, it provides guidance and vital clues (as a first step) to practitioners on where to deploy sensors (actuators). Remark 3: The strategy developed as an improvement to OSD (OAD), i.e. iOSD (iOAD), has a strong practical appeal. The improved approach accounts for physical constraints. In addition, it enhances estimation (control) by spatially spreading the sensors (actuators). This is better demonstrated by simulation examples in Section IV. However, it should be understood that iOSD (iOAD) does not strictly minimize the trace of the observability (controllability) Gramian.
Optimal actuator deployment — OAD strategy IV. SIMULATION RESULTS
Step 1 solve A> X + XA = −I; Step 2 find the indices of the nu largest diagonal elements of X and determine the index set S = {sk : k = 1, 2, · · · , nu } with X j, j > Xi,i for j ∈ S and i ∈ / S; Step 3 set the (s j , j)-th element of B to 1 for j = 1, 2, · · · , nu and other elements to 0, or equivalently, Bi, j = 1 if i = s j and otherwise, Bi, j = 0.
We present a simulation study of the OSD, iOSD, OAD and iOAD strategies in this section. The first two examples illustrate sensor and actuator deployment, respectively, for 2D heat transfer, and the third considers sensor deployment for a 2D data center.
Similar to sensor deployment, practical limitations in actuator of deployment have to be taken into account. The OSD strategy is improved by incorporating spatial constraints, resulting in improved OAD or iOAD strategy similar to the iOSD strategy presented above. We omit the presentation of iOAD strategy as it is easy to deduce from the OAD and iOSD strategies. C. Remarks on the Deployment Strategies The following remarks are presented to give a better perspective on the deployment strategies developed above: Remark 1: Compared to existing sensor deployment methods that have used the observability Gramian Wo , our approach promises two important benefits. First, rather than being near-optimal based on numerical methods, our approach produces a truly optimal solution, maximizing 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
Example 1. Consider a 2D square domain D with dimensions of 5m × 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 boundaries will be constantly 20◦ C, i.e., T (t)|∂ D = 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. The placement is conducted by OSD, iOSD and at random locations (using rand function in MATLAB). We assume that each sensor has a 1m radius of influence. This does not mean that the sensors can measure everything in that radius. It is chosen merely to avoid a clustered deployment of sensors. The Kalman filter is then applied to the state-space model obtained from (3) in each case of deployment for temperature estimation. Fig. 1 shows the sensor deployment and compares the true and estimated temperature fields at t = 1s, 10s and 20s. For OSD, the sensors are densely placed in the central area, largely because the temperature in this part is the hardest
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tr(Wo ) 1.62 1.45 1.26
tr(Σ) 105.54 99.55 103.74
δ 4711.27 3287.57 4085.02
OAD iOAD Random deployment
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Fig. 1 Example 1: change of temperature field from t = 1s to 20s. First row: true temperature field; second row: estimated temperature field using measurements from sensors deployed using the OSD strategy; third row: estimated temperature field using measurements from sensors deployed using the iOSD strategy; fourth row: estimated temperature field using measurements from sensors deployed randomly. Sensor locations are indicated by white dots in all cases.
OSD iOSD Random deployment
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Fig. 2 Example 2: change of temperature field from t = 1s to 80s. Top: temperature field due to actuators deployed using OAD strategy; middle: temperature field due to actuators deployed using iOAD strategy; bottom: temperature field due to actuators deployed randomly. The locations of actuators are indicated by white dots.
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ε 69.17 25.19 56.04
TABLE I Example 1: quantitative comparison of temperature estimation with sensors deployed by OSD, iOSD and randomness.
to estimate as only the boundary temperatures are exactly known at each instant. By comparison, iOSD spreads out the sensors accounting for each sensor’s 1m zone of influence. Since the system is stable, the estimated temperature fields gradually approach the truth for all cases over time. However, the estimation accuracy tends to be best with iOSD. Table I compares tr(Wo ), tr(Σ), δ and ε. Σ is the state estimation error covariance of the KF, δ and ε are the cumulative and final-step estimation errors, respectively, 12 Z t¯ 2 kx(t) − xˆ (t)k dt δ= , ε = kx(t¯) − xˆ (t¯)k2 , 0
where xˆ is the estimate of x and t¯ is the simulation time.
tr(Wc ) 2.287 × 1012 2.287 × 1012 2.287 × 1012
ζ 513.59 585.63 607.91
ξ 1545.63 1161.76 1324.40
σ 8.67 2.31 10.09
TABLE II Example 2: quantitative comparison of temperature control with sensors deployed by OAD, iOAD and randomness.
It is seen that OSD, as expected, yields the largest tr(Wo ). However, the corresponding estimation performance is better for iOSD, due to the concentrated deployment in OSD. This is not surprising since OSD does not guarantee the best estimation but only the largest tr(Wo ), as discussed in Remark 2. Compared to OSD and random sensor placement, iOSD strikes a good balance in improving the estimation accuracy at a mild sacrifice of tr(Wo ). Example 2. This example considers a 2D square domain D with dimensions of 5m × 5m and zero velocity field. The heat transfer process is modeled by (3), with the boundary condition n · ∇T = 0. 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., T (0)|D = 18, and the desired temperature is 20◦ C denoted as Td = 20. Here, 5 actuators will be used to steer the temperature. We deploy the actuators in three ways: OAD, iOAD and random placement. It is assumed that the temperature field, i.e., state information, is completely known. The linear quadratic regulator (LQR) is used to obtain the optimal actuator control inputs. The actuator placement is shown in Fig. 2. OAD places the actuators in the four corners and a similar theme of dense
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deployment is observed. iOAD places an actuator at each corner and the fifth one in the center due to the constraint avoiding dense deployment. From Fig. 2, it is easily observed that the desired final temperature Td = 20 is more evenly reached for the iOAD case. A quantitative comparison is provided in Table II. The values of tr(Wc ) are similar in all three cases. OAD still results in tr(Wc ) insignificantly larger than the other two and is not shown in the table. It also leads to the least amount of input energy ζ , where
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However, as for control accuracy, iOAD obviously outperforms the other two if we compare the values of ξ and ρ, where 12 Z t¯ , σ = kx(t¯) − Td k. kx(t) − Td k2 dt ξ= 0
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 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. Housing computer systems and associated telecommunication and storage systems, data centers have been key facilities in the digital information era. The normal operation of a data center relies on effective HVAC control. This motivates us to consider the data center example and improve monitoring strategies for HVAC control. We consider sensor deployment for a simplified 2D data center. The floor plan of the data center is shown in Fig. 3a. It occupies a rectangular region of 10m × 10.8m. There are four rows of racks, where the IT equipment such as servers, storage and network devices are mounted. 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 (1)-(2). The flow velocity at the inlet and outlet side of an ACU is assumed 5m/s and it is 0.05m/s at the IT equipment. A uniform interval of 0.2m is used to grid the 2D space. Note that the interior space of ACUs and racks is excluded from the solution domain. The airflow field is illustrated in Fig. 3b. The heat transfer is described by (3). Suppose that the initial temperature within the room is 20◦ C. The cooled air blown by ACUs is assumed to be at 13◦ C, and the
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Fig. 3 Example 3: (a) floor plan of the 2D data center; (b) steadystate airflow field.
temperature difference between the exhaust and inlet sides of the IT equipment is assumed to be 5◦ C. After transforming (1)-(3) into the state-space form as discussed in Section II, we deploy 15 sensors by iOSD and randomly. Then the KF is used in each case to reconstruct the temperature field map. The initial temperature estimate is 16◦ C. The sensor placement is shown in Fig. 4. In the iOSD case, the sensor locations are approximately symmetric, as a result of the symmetric layout of the data center. In addition, more sensors are placed at the corners and cold aisles. This is because the temperature at the hot aisles directly depends on the cold aisles as we assume a constant temperature difference between the exhaust and inlet sides of IT equipment. As the temperature of the cooled air blown by ACUs is known, no sensors are deployed in the proximity of the ACU’s exhaust side. Since the system is stable, estimation of the temperature field will converge to the truth in both cases. However, the estimation converges faster and is more accurate for iOSD. It is observed that the estimated temperature fields with iOSD are closer to the true ones at 20s and 60s, as illustrated in Fig. 4. Table III presents a quantitative evaluation of the results, showing that the iOSD has larger tr(Wo ) as well as smaller estimation errors compared to random deployment. To summarize the results, the OSD (OAD) maximizes the trace of observability (controllability) Gramian. However, iOSD (iOAD) results in a more practical solution avoiding clustered deployment and provides improved estimation error performance. In all cases, our solutions perform better than a random case. The practical value of the approach is demonstrated through its application to a simplified data center example. V. CONCLUSION In this paper we developed optimal sensor and actuator deployment strategies and applied it to improve temperature monitoring and control performance in HVAC systems. The traces of the observability and controllability Gramians are chosen to represent the temperature monitoring and control
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Fig. 4 Example 3: change of temperature field from t = 1s to 60s. Top: temperature field controlled by actuators deployed using the iOSD strategy; bottom: temperature controlled by actuators deployed randomly. Sensor locations are indicated by black dots.
iOSD Random deployment
tr(Wo ) 9.23 6.40
tr(Σ) 93.50 102.77
δ 1581.68 1853.46
ε 11.73 26.60
TABLE III Example 3: quantitative comparison of temperature estimation with sensors deployed by iOSD and randomness.
performance, respectively, due to a variety of relevant interpretations of these metrics. These Gramian-based metrics are maximized to yield the desired sensor and actuator locations. The major contribution of our work is the analytical closed form solution strategy developed to obtain these optimal locations. Our solution approach utilizes the solution of the Lyapunov equation and converts the maximization problem to a sorting problem which is computationally less expensive. In addition, an improved version of the solution to handle spatial constraints related to the possible locations is proposed. This improved solution strategy has a better practical appeal as it avoids clustered deployment of the sensors and actuators and results in improved estimation and control performance. The simulation examples include a simplified data center for which the optimal sensor deployment is obtained in a manner that can be used in a real-world situation.
R EFERENCES [1] 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. [2] 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. [3] S. Sinha, U. Vaidya, and R. Rajaram, “Optimal placement of actuators and sensors for control of nonequilibrium dynamics,” in Proceedings of the European Control Conference, 2013, pp. 1084–1088. [4] D. Georges, “The use of observability and controllability gramians or functions for optimal sensor and actuator location in finite-dimensional systems,” in Proceedings of the 34th IEEE Conference on Decision and Control, 1995, pp. 3319–3324. [5] B. Marx, D. Koenig, and D. Georges, “Optimal sensor/actuator location for descriptor systems using Lyapunov-like equations,” in Proceedings of the 41st IEEE Conference on Decision and Control, 2002, pp. 4541–4542. [6] T. H. Summers and J. Lygeros, “Optimal sensor and actuator placement in complex dynamical networks,” 2013, http://arxiv.org/abs/1306.2491. [7] 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. [8] H. R. Shaker, “Optimal sensor and actuator location for unstable systems,” Journal of Vibration and Control, vol. 19, no. 12, pp. 1915– 192, 2012. [9] 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. [10] K. K. Chen and C. W. Rowley, “H2 optimal actuator and sensor placement in the linearised complex ginzburglandau system,” Journal of Fluid Mechanics, vol. 681, pp. 241–260, 2011. [11] R. Potami, “Optimal sensor/actuator placement and switching schemes for control of flexible structures,” Ph.D. dissertation, Worcester Polytechnic Institute, 2008. [12] 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. [13] A. Arabyan and S. Chemishkian, “H∞ -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. [14] 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. [15] Y. Joshi and P. Kumar, Eds., Energy Efficient Thermal Management of Data Centers. Springer US, 2012. [16] V. L´opez and H. F. Hamann, “Heat transfer modeling in data centers,” International Journal of Heat and Mass Transfer, vol. 54, no. 53065318, 2011. [17] W. E. Schiesser and G. W. Griffiths., A Compendium of Partial Differential Equation Models. Cambridge University Press, 2009. [18] T. Chen and B. A. Francis, Optimal Sampled-Data Control Systems. Springer-Verlag, 1995. [19] 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. [20] P. Benner, J.-R. Li, and T. Penzl, “Numerical solution of largescale lyapunov equations, riccati equations, and linear-quadratic optimal control problems,” Numerical Linear Algebra with Applications, vol. 15, no. 9, pp. 755–777, 2008. [21] C. H. Papadimitriou, “On the complexity of integer programming,” Journal of ACM, vol. 28, no. 4, pp. 765–768, 1981.
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Optimal Sensor and Actuator Deployment for HVAC Control System Design Huazhen Fang, Ratnesh Sharma and Rakesh Patil Abstract— This paper studies control-theory-inspired optimal sensor and actuator deployment to improve the temperature monitoring and control performance of HVAC (heating, ventilation and air conditioning) systems in buildings. The deployment strategies are based on maximizing observability- and controllability-based metrics such as the respective Gramians. Our solution approach has the following benefits compared to previous work. First, an analytical, closed-form solution is developed. Second, the computational cost is low to ensure practical use of our approach. The effectiveness of our deployment strategies is demonstrated via simulation examples and by calculating different metrics of relevance.
I. INTRODUCTION This paper focuses on novel strategies for optimal Sensor and Actuator Deployment (SAD). Control-theory-inspired strategies are developed in order to improve temperature monitoring and control performance of HVAC (heating, ventilation and air conditioning) systems in buildings. SAD for buildings has relied on either heuristic rules or the other extreme of computationally burdensome CFD modeling [1]. There is limited research on control-theory-based SAD for HVAC applications. In [2, 3], authors decide the suitability of sensor and actuator locations for thermal systems using a linear advection partial differential equation (PDE). They explore different criteria based on observability and controllability Gramians. However, these elegant theoretical analyses are difficult to translate into practice. The SAD problem has been studied for some other applications using different metrics to evaluate the quality of the deployment. Gramian-based approaches have been quite popular due to the interpretation of the Gramians as an indicator of a system’s control and monitoring performance, [4–9]. In addition, direct control-oriented SAD has also been studied, for example, H2 control in [10, 11] and H∞ control in [12, 13]. We observe that the essential treatment of SAD in most of the above literature is to solve a formulated optimization problem. The solutions are expressed in terms of computation-based optimization procedures. These approaches are fundamentally approximate and require significant computing power, especially for largescale systems [14]. In our work, we provide insights for SAD using observability- and controllability-based metrics especially for Huazhen Fang is with the Department of Mechanical Engineering, University of California, San Diego, CA 92093, USA, and currently a research intern with the Energy Management Department at NEC Laboratories America, Inc., Cupertino, CA 95014, USA [email protected]. Ratnesh Sharma and Rakesh Patil are with the Energy Management Department at NEC Laboratories America, Inc., Cupertino, CA 95014, USA {ratnesh,rakeshmp}@nec-labs.com.
978-1-4799-3271-9/$31.00 ©2014 AACC
temperature monitoring and control. Specifically, we propose SAD strategies that analytically maximize the traces of the observability and controllability Gramians. Our approach presents two benefits over previous work. First, we produce analytical closed form solutions. Second, these solutions are computationally cheaper and lend themselves well to practical applications. The basic approach is further improved by incorporating spatial constraints on locations of sensors and actuators. The improved approach further enhances the value and performance of our approach, as demonstrated through simulation examples using different configurations. The remainder of the paper is organized as follows. Section II introduces PDE-based airflow and heat transfer models which are then reduced to the state-space form. Section III develops our novel SAD strategies through the maximization of the trace of the system’s observability and controllability Gramians. Simulation examples are presented in Section IV to illustrate the value of our approaches. II. 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 and apply control-theory-based sensor and actuator deployment strategies. A. Construction of Airflow and Heat Transfer Models The Navier-Stokes equations describing the conservation of momentum and mass for incompressible airflow are given by (1) and (2), respectively [15]: ∂V + (V · ∇)V = ρg − ∇p + µ∇2 V, (1) ρ ∂t ∇ · V = 0. (2) where g is the gravity vector, ∇p the pressure gradient, µ the dynamic viscosity. A steady-state airflow is assumed in our study, i.e., ∂ V/∂t = 0, as we are interested in steady-state large scale behavior and intend to reduce the complexity of the analysis [16]. For a time-varying temperature field T (x, y, z,t), the heat transfer via the convection-diffusion equation is given by ∂T + V · ∇T − ∇ · (κ∇T ) = h, (3) ρc p ∂t where ρ, 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) [16].
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where C shows sensor locations with a structure like B:
For (1)-(2), the following boundary condition is applied: −n · V = Vb ,
(4)
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. We suppose that, when Vb 6= 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 · (k∇T ) = q + αT,
B. 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 finite-difference-based discretization, with the resulting ODEs established over the time domain [17]. The MOL is applied to (3) along with the boundary condition (5) to obtain the ODEs and subsequently the state-space form to describe the temperature dynamics. Further details 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 Nx , Ny and Nz , respectively. The state vector x is the collection of temperature values at each grid point, and the input vector u is a collection of the heat sources or sinks on the grid, that is, .. .. . . h(i, j, k,t) T (i, j, k,t) x(t) = , u(t) = . .. .. . . n ×1 n ×1 u
The dimension of x is nx = Nx × Ny × Nz , and the dimension of u, nu , depends on the number of sources and sinks in the system, denoted as nu . The state-space equation is x˙ (t) = Ax(t) + Bu(t).
nx
∑ Bi, j = 1 for
i=1 nu
j = 1, 2, · · · , nu , (7)
∑ Bi, j = 1 for i = 1, 2, · · · , nx .
j=1
The measurement vector y has a dimension ny equal to the number of sensors. The output equation representing the sensor measurements are as follows: y(t) = Cx(t),
j=1
(9)
ny
j = 1, 2, · · · , nx .
∑ Ci, j = 1 for
i=1
Together (6) and (8) represent the state-space model for heat transfer in our study. III. OPTIMAL DEPLOYMENT STRATEGIES Optimal sensor and actuator deployment strategies are developed in this section. A. Optimal Sensor Deployment The optimal sensor deployment strategy is build upon the observability concept. For the system in (6) and (8) and a stable A, the observability Gramian, Wo , is defined as Z ∞
Wo =
(8)
>
eA τ C> CeAτ dτ,
(10)
0
1) Inside Observability Gramian: The goal of the sensor deployment strategy is to obtain the “best” temperature description of the system. Wo is used as an indication of the temperature description due to its following interpretations related to state estimation. If a system has initial state x(0), the observed energy in the output can be written as kyk22 =
Z ∞ 0
y> (τ)y(τ)dτ = x> (0)Wo x(0),
Thus, the larger Wo is, the more information the output contains about the state. The H2 norm of a system can be written as kGk2 = tr B> Wo B ,
(6)
The matrices A and B are determined from (3) and (5). It should be noted that B indicates the placement of actuators. It has a sparse structure — each element is 0 or 1 (after normalization). Only one element of each column and row can be 1 as the actuators are assumed to be point sources: Bi j ∈ {0, 1} ∀i, j,
∑ Ci, j = 1 for i = 1, 2, · · · , ny ,
(5)
where q results from the power of the heating or cooling sources at the boundaries and α is a coefficient.
x
nx
Ci j ∈ {0, 1} ∀i, j,
which is a weighted trace of Wo [18]. A larger H2 norm leads to better suppression of output measurement noise. It is known that a nonsingular Wo guarantees observability. However, Wo will be rank-deficient if the system is only detectable rather than being observable. 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 Wo can be increased: maxC rank(Wo ). Solving this rank maximization problem (globally) is rather difficult, known to be computationally NP-hard [19]. A widely used heuristic is to replace the rank objective with the trace, so we would solve maxC tr(Wo ). Because tr(Wo ) = ∑ni=1 λi (Wo ), where λi (Wo ) are the eigenvalues of Wo for i = 1, 2, · · · , nx , maximizing tr(Wo ) tends to result in a highrank matrix [19].
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2) Optimal Sensor Deployment Strategy: Based on the above discussion, the optimal sensor locations are determined via selecting C to maximize tr(Wo ): max tr [Wo (C)] C
nx
∑ Ci, j = 1 for i = 1, 2, · · · , ny ,
s.t. Ci j ∈ {0, 1},
j=1
(11)
to physical limitations. Second, ‘dense’ or ‘clustered’ sensor deployment — multiple sensors deployed within a relatively small area — should be avoided, since each sensor can cover a certain region for sensing or monitoring. Equivalently, one has to consider a zone of influence of each sensor and avoid overcrowding of sensors. As a consequence, an improved OSD, or iOSD strategy is proposed as follows: Improved optimal sensor deployment — iOSD strategy
ny
∑ Ci, j = 1 for
j = 1, 2, · · · , nx .
Step 1 solve AX + XA> = −I; Step 2 find the largest diagonal element of X; if its corresponding location is in an acceptable region (defined by the system’s designer), mark its index as s1 , set C1,s1 = 1 and place the sensor, otherwise find the next largest one and repeat this step; Step 3 find the next largest diagonal element of X; if its corresponding location is in an acceptable region and this sensor’s monitoring region (previously defined) does not overlap with the regions of the sensors placed before, mark its index as si , set Ci,si = 1 and place the sensor, otherwise find the next largest one and repeat; Step 4 repeat Step 3 for i = 2 until i = ny ; then set the remainder of the elements of Ci, j = 0 for i = 1, 2, · · · , ny and j 6= si .
i=1
where Ci, j = 1 when sensor i is placed at the j-th point in the gridded domain and Ci, j = 0 otherwise. This optimization can be solved as an integer programming problem. However, we develop a more computationally attractive solution. This solution strategy described below is a major contribution of this paper. We can write Z ∞ > eA τ C> CeAτ dτ tr [Wo (C)] = tr 0 Z ∞ > tr eA τ C> CeAτ dτ = Z0 ∞ > tr eAτ eA τ C> C dτ = 0 Z ∞ Aτ A> τ > e e dτC C . = tr 0
B. Actuator Deployment
R > We note that, because A is stable, X = 0∞ eAτ eA τ dτ is the
unique solution of the Lyapunov equation AX + XA> = −I. In addition, L = C> C is a binary diagonal matrix. Each of its diagonal elements, L j, j , is 0 or 1 for j = 1, 2, · · · , nx ; L j, j = 1 if a sensor is located at the j-th point. Therefore, to maximize tr(Wo ) = tr(XL), we simply find the ny 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, · · · , ny } such that X j, j > Xi,i for any j ∈ S and i ∈ / S ; we then let Ci,si = 1 by placing a sensor at the si -th point for i = 1, 2, · · · , ny . The optimal sensor deployment strategy ( OSD), is summarized as follows: Optimal sensor deployment — OSD strategy Step 1 solve AX + XA> = −I; Step 2 find the indices of the ny largest diagonal elements of X and determine the index set S = {sk : k = 1, 2, · · · , ny } with X j, j > Xi,i for j ∈ S and i∈ / S; Step 3 set the (i, si )-th element of C to 1 for i = 1, 2, · · · , ny and other elements to 0, or equivalently, Ci, j = 1 if j = si and otherwise, Ci, j = 0.
The actuator deployment problem is a dual of the sensor deployment problem if the actuators are considered point sources. The trace of the controllability Gramian Wc is maximized in this case, where Z ∞
Wc =
>
eAτ BB> eA τ dτ,
(12)
0
1) Inside Controllability Gramian: The controllability gramian Wc is chosen as the measure of control authority for a dynamic system in our study due to its following interpretations. Consider driving a system from zero initial state x(0) = 0 to a final state x¯ , i.e., x(t) = x¯ , using minimum control energy: min E(t) u
s.t. x˙ (t) = Ax(t) + Bu(t), x(0) = 0, x(t) = x¯ , where E(t) =
Rt > 0 u (τ)u(τ)dτ. The resulting control input is
u(τ) = B> eA
> (t−τ)
W−1 x, 0 ≤ τ ≤ t. c (t)¯
Hence, the control energy over an infinite time horizon is ¯ . Thus, a larger Wc results in lower control E(∞) = x¯ > W−1 c x energy. A larger Wc 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 = qI:
Further improvements are made to the OSD strategy to account for some practical issues. Two considerations worth noting in practical sensor deployment are: First, certain areas may be inaccessible or unavailable for sensor placement due
x˙ (t) = Ax(t) + B [u(t) + w(t)] . Suppose the control objective is to drive the state to x¯ . From optimal control theory, we know that irrespective of how the
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control input u is chosen that the state x, will not be precisely achieved due to the effects of the noise w. The covariance of the state will be i h E (x(∞) − x¯ ) (x(∞) − x¯ )> = qW−1 c , which is inversely proportional to Wc . Thus a larger Wc may contribute to noise suppression. The rank of the controllability matrix is related to the rank of Wc . 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. The trace heuristic maxB tr (Wc ) can be used in this case. 2) Optimal Actuator Deployment Strategy: Based on the above discussion, a possible way to define optimal actuator deployment would be max tr [Wc (B)] B
nx
s.t. Bi j ∈ {0, 1} ∀i, j,
∑ Bi, j = 1 for
j = 1, 2, · · · , nu ,
i=1
(13)
nu
∑ Bi, j = 1 for i = 1, 2, · · · , nx .
j=1
Analogous to OSD, the optimal actuator deployment (OAD) strategy for actuator placement is summarized as follows.
in our approach is solving the Lyapunov equation, which can be handled in O(n3x ) operations and O(n2x ) memory with [20]. By comparison, the integer programming used in the literature is proven to be NP-hard, often requiring computation in polynomial time [21]. Remark 2: The metric optimized for OSD (OAD) is the trace of the observability (controllability) Gramian. Hence, OSD (OAD) is optimal only in the sense of this metric. Other relevant metrics could be related explicitly to the estimation errors (e.g. norm of estimation error) or the control energy etc. and result in different deployment strategies. However, OSD (OAD) retain their importance for two reasons. First, our approach presents a closed-form solution and the ease of computation lends to practical implementation, as discussed in Remark 1. Second, it provides guidance and vital clues (as a first step) to practitioners on where to deploy sensors (actuators). Remark 3: The strategy developed as an improvement to OSD (OAD), i.e. iOSD (iOAD), has a strong practical appeal. The improved approach accounts for physical constraints. In addition, it enhances estimation (control) by spatially spreading the sensors (actuators). This is better demonstrated by simulation examples in Section IV. However, it should be understood that iOSD (iOAD) does not strictly minimize the trace of the observability (controllability) Gramian.
Optimal actuator deployment — OAD strategy IV. SIMULATION RESULTS
Step 1 solve A> X + XA = −I; Step 2 find the indices of the nu largest diagonal elements of X and determine the index set S = {sk : k = 1, 2, · · · , nu } with X j, j > Xi,i for j ∈ S and i ∈ / S; Step 3 set the (s j , j)-th element of B to 1 for j = 1, 2, · · · , nu and other elements to 0, or equivalently, Bi, j = 1 if i = s j and otherwise, Bi, j = 0.
We present a simulation study of the OSD, iOSD, OAD and iOAD strategies in this section. The first two examples illustrate sensor and actuator deployment, respectively, for 2D heat transfer, and the third considers sensor deployment for a 2D data center.
Similar to sensor deployment, practical limitations in actuator of deployment have to be taken into account. The OSD strategy is improved by incorporating spatial constraints, resulting in improved OAD or iOAD strategy similar to the iOSD strategy presented above. We omit the presentation of iOAD strategy as it is easy to deduce from the OAD and iOSD strategies. C. Remarks on the Deployment Strategies The following remarks are presented to give a better perspective on the deployment strategies developed above: Remark 1: Compared to existing sensor deployment methods that have used the observability Gramian Wo , our approach promises two important benefits. First, rather than being near-optimal based on numerical methods, our approach produces a truly optimal solution, maximizing 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
Example 1. Consider a 2D square domain D with dimensions of 5m × 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 boundaries will be constantly 20◦ C, i.e., T (t)|∂ D = 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. The placement is conducted by OSD, iOSD and at random locations (using rand function in MATLAB). We assume that each sensor has a 1m radius of influence. This does not mean that the sensors can measure everything in that radius. It is chosen merely to avoid a clustered deployment of sensors. The Kalman filter is then applied to the state-space model obtained from (3) in each case of deployment for temperature estimation. Fig. 1 shows the sensor deployment and compares the true and estimated temperature fields at t = 1s, 10s and 20s. For OSD, the sensors are densely placed in the central area, largely because the temperature in this part is the hardest
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tr(Wo ) 1.62 1.45 1.26
tr(Σ) 105.54 99.55 103.74
δ 4711.27 3287.57 4085.02
OAD iOAD Random deployment
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Fig. 1 Example 1: change of temperature field from t = 1s to 20s. First row: true temperature field; second row: estimated temperature field using measurements from sensors deployed using the OSD strategy; third row: estimated temperature field using measurements from sensors deployed using the iOSD strategy; fourth row: estimated temperature field using measurements from sensors deployed randomly. Sensor locations are indicated by white dots in all cases.
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Fig. 2 Example 2: change of temperature field from t = 1s to 80s. Top: temperature field due to actuators deployed using OAD strategy; middle: temperature field due to actuators deployed using iOAD strategy; bottom: temperature field due to actuators deployed randomly. The locations of actuators are indicated by white dots.
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ε 69.17 25.19 56.04
TABLE I Example 1: quantitative comparison of temperature estimation with sensors deployed by OSD, iOSD and randomness.
to estimate as only the boundary temperatures are exactly known at each instant. By comparison, iOSD spreads out the sensors accounting for each sensor’s 1m zone of influence. Since the system is stable, the estimated temperature fields gradually approach the truth for all cases over time. However, the estimation accuracy tends to be best with iOSD. Table I compares tr(Wo ), tr(Σ), δ and ε. Σ is the state estimation error covariance of the KF, δ and ε are the cumulative and final-step estimation errors, respectively, 12 Z t¯ 2 kx(t) − xˆ (t)k dt δ= , ε = kx(t¯) − xˆ (t¯)k2 , 0
where xˆ is the estimate of x and t¯ is the simulation time.
tr(Wc ) 2.287 × 1012 2.287 × 1012 2.287 × 1012
ζ 513.59 585.63 607.91
ξ 1545.63 1161.76 1324.40
σ 8.67 2.31 10.09
TABLE II Example 2: quantitative comparison of temperature control with sensors deployed by OAD, iOAD and randomness.
It is seen that OSD, as expected, yields the largest tr(Wo ). However, the corresponding estimation performance is better for iOSD, due to the concentrated deployment in OSD. This is not surprising since OSD does not guarantee the best estimation but only the largest tr(Wo ), as discussed in Remark 2. Compared to OSD and random sensor placement, iOSD strikes a good balance in improving the estimation accuracy at a mild sacrifice of tr(Wo ). Example 2. This example considers a 2D square domain D with dimensions of 5m × 5m and zero velocity field. The heat transfer process is modeled by (3), with the boundary condition n · ∇T = 0. 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., T (0)|D = 18, and the desired temperature is 20◦ C denoted as Td = 20. Here, 5 actuators will be used to steer the temperature. We deploy the actuators in three ways: OAD, iOAD and random placement. It is assumed that the temperature field, i.e., state information, is completely known. The linear quadratic regulator (LQR) is used to obtain the optimal actuator control inputs. The actuator placement is shown in Fig. 2. OAD places the actuators in the four corners and a similar theme of dense
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deployment is observed. iOAD places an actuator at each corner and the fifth one in the center due to the constraint avoiding dense deployment. From Fig. 2, it is easily observed that the desired final temperature Td = 20 is more evenly reached for the iOAD case. A quantitative comparison is provided in Table II. The values of tr(Wc ) are similar in all three cases. OAD still results in tr(Wc ) insignificantly larger than the other two and is not shown in the table. It also leads to the least amount of input energy ζ , where
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However, as for control accuracy, iOAD obviously outperforms the other two if we compare the values of ξ and ρ, where 12 Z t¯ , σ = kx(t¯) − Td k. kx(t) − Td k2 dt ξ= 0
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 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. Housing computer systems and associated telecommunication and storage systems, data centers have been key facilities in the digital information era. The normal operation of a data center relies on effective HVAC control. This motivates us to consider the data center example and improve monitoring strategies for HVAC control. We consider sensor deployment for a simplified 2D data center. The floor plan of the data center is shown in Fig. 3a. It occupies a rectangular region of 10m × 10.8m. There are four rows of racks, where the IT equipment such as servers, storage and network devices are mounted. 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 (1)-(2). The flow velocity at the inlet and outlet side of an ACU is assumed 5m/s and it is 0.05m/s at the IT equipment. A uniform interval of 0.2m is used to grid the 2D space. Note that the interior space of ACUs and racks is excluded from the solution domain. The airflow field is illustrated in Fig. 3b. The heat transfer is described by (3). Suppose that the initial temperature within the room is 20◦ C. The cooled air blown by ACUs is assumed to be at 13◦ C, and the
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Fig. 3 Example 3: (a) floor plan of the 2D data center; (b) steadystate airflow field.
temperature difference between the exhaust and inlet sides of the IT equipment is assumed to be 5◦ C. After transforming (1)-(3) into the state-space form as discussed in Section II, we deploy 15 sensors by iOSD and randomly. Then the KF is used in each case to reconstruct the temperature field map. The initial temperature estimate is 16◦ C. The sensor placement is shown in Fig. 4. In the iOSD case, the sensor locations are approximately symmetric, as a result of the symmetric layout of the data center. In addition, more sensors are placed at the corners and cold aisles. This is because the temperature at the hot aisles directly depends on the cold aisles as we assume a constant temperature difference between the exhaust and inlet sides of IT equipment. As the temperature of the cooled air blown by ACUs is known, no sensors are deployed in the proximity of the ACU’s exhaust side. Since the system is stable, estimation of the temperature field will converge to the truth in both cases. However, the estimation converges faster and is more accurate for iOSD. It is observed that the estimated temperature fields with iOSD are closer to the true ones at 20s and 60s, as illustrated in Fig. 4. Table III presents a quantitative evaluation of the results, showing that the iOSD has larger tr(Wo ) as well as smaller estimation errors compared to random deployment. To summarize the results, the OSD (OAD) maximizes the trace of observability (controllability) Gramian. However, iOSD (iOAD) results in a more practical solution avoiding clustered deployment and provides improved estimation error performance. In all cases, our solutions perform better than a random case. The practical value of the approach is demonstrated through its application to a simplified data center example. V. CONCLUSION In this paper we developed optimal sensor and actuator deployment strategies and applied it to improve temperature monitoring and control performance in HVAC systems. The traces of the observability and controllability Gramians are chosen to represent the temperature monitoring and control
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Fig. 4 Example 3: change of temperature field from t = 1s to 60s. Top: temperature field controlled by actuators deployed using the iOSD strategy; bottom: temperature controlled by actuators deployed randomly. Sensor locations are indicated by black dots.
iOSD Random deployment
tr(Wo ) 9.23 6.40
tr(Σ) 93.50 102.77
δ 1581.68 1853.46
ε 11.73 26.60
TABLE III Example 3: quantitative comparison of temperature estimation with sensors deployed by iOSD and randomness.
performance, respectively, due to a variety of relevant interpretations of these metrics. These Gramian-based metrics are maximized to yield the desired sensor and actuator locations. The major contribution of our work is the analytical closed form solution strategy developed to obtain these optimal locations. Our solution approach utilizes the solution of the Lyapunov equation and converts the maximization problem to a sorting problem which is computationally less expensive. In addition, an improved version of the solution to handle spatial constraints related to the possible locations is proposed. This improved solution strategy has a better practical appeal as it avoids clustered deployment of the sensors and actuators and results in improved estimation and control performance. The simulation examples include a simplified data center for which the optimal sensor deployment is obtained in a manner that can be used in a real-world situation.
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