Building temperature control: A passivity-based approach

Building Temperature Control: A Passivity-Based Approach Sumit Mukherjee1 , Sandipan Mishra2 , John T. Wen1 Abstract— This paper focuses on the temperature control in a multi-zone building. The lumped heat transfer model based on thermal resistance and capacitance is used to analyze the system dynamics and control strategy. The resulting thermal network, including the zones, walls, and ambient environment, may be represented as an undirected graph. The thermal capacitances are the nodes in the graph, connected by thermal resistances as links. We assume the temperature measurements and temperature control elements (heating and cooling) are collocated. We show that the resulting input/output system is strictly passive, and any passive output feedback controller may be used to improve the transient and steady state performance without affecting the closed loop stability. The storage functions associated with passive systems may be used to construct a Lyapunov function, to demonstrate closed loop stability and motivates the construction of an adaptive feedforward control. A four-room example is included to illustrate the performance of the proposed passivity based control strategy. I. INTRODUCTION With the soaring energy cost, there is increasing emphasis on energy conservation. Building is a major source of energy consumption, accounting for close to 40% of the energy usage in the US [1]. One of the major energy consumers in buildings is the heating, ventilation, and air conditioning (HVAC) system. Numerous approaches have been proposed on the control of HVAC systems. Many focus on optimal energy utilization by taking into account variable electricity rates (peak vs. non-peak) and power consumption [2]–[5]. Active and passive thermal energy storage has been proposed to reduce energy cost by shifting the major energy consumers away from the peak hours [4], [5]. Recent work draws on more detailed building dynamical models, including weather forecast, to implement on-line optimization such as model predictive control (MPC) [6]–[10]. Models of building heat transfer vary greatly in complexity, ranging from finite element method [11] to lumped energy and mass transfer between subdivided zones [12], [13]. Though these methods lead to a more accurate representation of rooms, they are computationally inefficient for online feedback control of temperature in multiple rooms. An alternate approach is to model the temperature control problem using the electrical analogy [14]–[16]. For multiple interconnected zones, the heat transfer model becomes an equivalent electrical circuit network. This allows the application of graph theory to simplify the network control problem 1 S. Mukherjee and J.T. Wen are with the Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, New York mukhes3@rpi.edu, wenj@rpi.edu 2 S. Mishra is with the Mechanical, Aerospace, and Nuclear Engineering Department, Rensselaer Polytechnic Institute, Troy, New York mishrs2@rpi.edu [17]. This model may also be modified to include the effect of occupancy, load changes and even room dynamics [7]. The model may be further expanded to include the dynamics of heating equipments [18]. Recent work in [19] recognizes that the rich literature of graph-theoretic approaches in formation control may be applied to thermal network [20]–[22]. The nodes in the graph are zone temperatures, and the links are wall thermal impedance (including thermal resistance and capacitance). This gives rise to the challenging problem of controlling a graph with dynamic links. We adopt a similar approach in this paper, but represent all thermal capacitances as nodes, connected by thermal resistance. This results in a standard undirected graph, with collocated temperature measurements and heat input at nodes corresponding to the rooms. We show that the resulting system is strictly passive. Any passive negative output feedback controller would be stabilizing. This is not surprising, as passivity based control has long been used for electrical circuits and mechanical structures with collocated input and output. The storage functions of the passive interconnection may be combined and used as a Lyapunov function to analyze the closed loop stability and extends the controller to include adaptive feedforward. The passive controller may be additionally designed to minimize energy usage, transient performance, and disturbance rejection (using e.g., H2 or H∞ optimization subject to positive realness constraint [23], [24]) without compromising stability. An outer loop may also be added to adjust the temperature set points to balance between human comfort and energy consumption, using, e.g., model predictive control. A four-room example similar to the one in [19] has been included to illustrate the performance of passivity-based control. II. SYSTEM MODEL The focus of this manuscript is on temperature regulation in a building consisting of interconnected zones. We consider a lumped heat transfer model using thermal resistance and capacitance. Thermal resistance models the heat flow based on temperature difference: Q = ∆T /R, where Q is the heat transferred across the resistance (W), ∆T is the temperature difference (K), and R is the thermal resistance (K/W). Thermal capacitance (or thermal mass) models the ability of a space to store heat: C d∆T dt = Q, where C has the unit J/K. A single zone is modeled as a thermal capacitor, while a wall is modeled as an RC network, e.g., a standard 3R2C model [15] as shown in Figure 1 (Note that C1 and C2 denote the room capacitances – the wall is characterized by three thermal resistors and two thermal capacitors. Enumerating all thermal capacitors in the system, and Fig. 1. 3R2C of heat transfer across walls letting Ti be the temperature of the ith capacitor, we may write the equation governing Ti as X (e) (c) −1 Rij (Ti − Tj ) + Qi + Qi Ci Ṫi = − (1) j∈Ni where Ni denotes all resistors connected to the ith capacitor, (e) Qi is the external heat input:  −1 −Ri0 (Ti − T∞ ) if adjacent to ambient (e) , Qi = 0 otherwise (2) (c) and Qi is the heat input:  ui if node is heated/cooled (c) . (3) Qi = 0 otherwise As in [19], this interconnected system may be viewed as a graph consisting of nodes representing temperatures at thermal capacitors (and a reference node for the ambient) connected by thermal resistors as links. In contrast to [19], every thermal capacitor is included as a node (including all wall capacitors), instead of only those for the controlled zones. Denote the undirected graph corresponding to the thermal network by G. Let N be the number of thermal capacitors in this network, then there are N + 1 nodes in G (including the ambient temperature as the reference node). Let L be the number of thermal resistors in the system, then there are L links in G. Enumerate the nodes from 0 to N , with node 0 denoting the ambient. We may assign an orientation G by considering one of the two nodes of each − link to be the positive end. Denote by L+ i (Li ) the set of links for which node i is the positive (negative) end. Denote the complete incidence matrix (for a graph with a reference node) of G as D1 ∈ RN +1×L where The value of D1 is determined by the graph structure as follows:   +1 if j ∈ L+ i (4) D1ij = −1 if j ∈ L− i  0 otherwise. We assume that G is a connected graph, i.e., there is a path connecting any one node to any other node (i.e., no room is thermally isolated). From Property 1.5 in [22], rank of D1 is N and the null space of D1T is spanned by 1N +1 , an RN +1 column vector consisting of all 1’s. Since the reference node temperature is externally imposed, we separate it out in the complete incidence matrix:   d0 (5) D1 = D where d0 ∈ R1×L and D ∈ RN ×L is the incidence matrix with the reference node removed. The following result shows that D is of full row rank, i.e., the null space of DT is the zero vector. Proposition 1: Given a connected undirected graph G with incidence matrix D1 ∈ RN ×L . The reduced incidence matrix D as defined from (5) is of full row rank, N . Proof: Suppose η ∈ N (DT ), where N (·) denotes null space. Then η ∈ span(1N ) since the graph is connected. From Property 1.5 in [22], we also have N (D1T ) = span(1N +1 ). Hence, dT0 + DT 1N = 0. We have already shown that DT 1N = 0. Therefore, dT0 = 0. This contradicts the connectedness assumption of G. Hence, η must be the zero vector.  Combining the above results and expressions, the overall heat transfer model is: C Ṫ = −DR−1 DT T + B0 T∞ + Bu (6) where C is a diagonal, positive definite matrix consisting of the wall capacitances, R is a diagonal, positive definite matrix consisting of the link thermal resistances, D is as in (5), B0 = −DR−1 dT0 is a column vector with non-zero elements as the thermal conductance of nodes connected to the ambient, T∞ is the ambient temperature, u is the heat input into each zone, and B is the corresponding input matrix. Note that since D is full row rank, DR−1 DT is positive definite. We address temperature regulation of the zones that are directly affected by active heating/cooling devices. Therefore, the output of interest is y = B T T. −1 (7) T We have shown DR D is positive definite. Therefore, the open loop system (with u = 0) is exponentially stable. If T∞ is a constant, then the steady state temperatures are given by Tss = (DR−1 DT )−1 B0 T∞ . (8) III. PASSIVITY BASED CONTROL A. Output Set Point Control Consider the output set point control problem: Given T∞ , find u based on feedback of T to drive y to ydes . We first find the steady state solution. The following shows that a unique solution may always be found. Proposition 2: Given (T∞ , ydes ), the unique steady state solution (T ∗ , u∗ ) of (6)–(7) satisfies    ∗   −B0 T∞ −DR−1 DT B T . (9) = ydes u∗ BT 0 and is given by T∗ = T T T T T +B ⊥ (B ⊥ AB ⊥ )−1 B ⊥ B0 T∞ u ∗ = + T (I − B ⊥ (B ⊥ AB ⊥ )−1 B ⊥ A)B + ydes + ∗ B (AT − B0 T∞ ) (10) (11) where B is the Moore-Penrose pseudo-inverse (in this case, the left inverse) of B, B ⊥ is the annihilator of B, and A := DR−1 DT . Proof: The steady state condition follows directly from (6)– (7) by setting Ṫ to 0. Denote the matrix in (9) by Λ. Let (T, u) be in the null space of Λ: −DR−1 DT + Bu T B T = 0 = 0. (12) (13) ⊥T The second equation is equivalent to T = B ξ where B ⊥ B = 0. Substitute into the first equation and pre-multiply by B ⊥ , we get T B ⊥ DR−1 DT B ⊥ ξ = 0 which implies ξ = 0. It then follows the null space is the zero vector. Hence the steady state solution is unique. To solve for the steady state solution, first use the second equation in (9) to express T ∗ as T T T ∗ = B + ydes + B ⊥ ξ ∗ where ξ ∗ is to be determined. Substitute into the first equation in (9), we obtain the stated result.  Once we have (T ∗ , u∗ ), we can form the error system as Cδ Ṫ = −DR−1 DT δT + Bδu, δy = B T δT (14) where δT := T − T ∗ , δu := u − u∗ , δy := y − ydes . The stability of the first order system and collocation of the input and output immediately suggests inherent passivity of the system. A system with state x, input u and output y is passive if there exists a continuously differentiable storage function V (x) ≥ 0 such that The derivative along solution is V̇ = −δT T DR−1 DT δT + δT Bδu. Substituting in the controller (15), we get V̇ = −δT T DR−1 DT δT − δT T BK(B T δT ) ≤ −γV. The last inequality follows from the positive definiteness of DR−1 DT and the passivity of K. Hence, the zero equilibrium of (14) is exponentially stable. Since δy = B T δx, it follows that y → ydes exponentially.  This result states that we have a large class of stabilizing controller to draw from in building control, with virtually no model information requirement. (We do need model information to compute the feedforward, u∗ , however, error in u∗ , while influencing the steady state, does not affect stability.) Any available model information may be used to design K towards an optimization objective, e.g., energy efficiency, while preserving the passivity structure. For example, the H2 optimization problem subject to positive realness constraint may be posed as a convex optimization and solved using linear matrix inequality (LMI) approach [23], [24]. B. Adaptive Control If u∗ is unknown, we may replace it by an estimate as in adaptive control. Theorem 2: Consider the passive controller with adaptive feedforward u û˙ ∗ T V̇ ≤ −W (x) + u y for some function W (x) ≥ 0. If W (x) is positive definite, then the system is strictly passive. The notion of passivity is motivated by physical systems that conserve or dissipate energy, such as passive circuits and mechanical structures, where V (x) corresponds to an energy function. Passivity is a useful tool for nonlinear stability analysis and control design, particularly for large scale interconnected systems as in network flow control and formation control. Indeed, the celebrated Passivity Theorem states that, if two passive systems H1 and H2 with positive definite and radially unbounded storage functions V1 (x) and V2 (x) respectively, are interconnected as in a negative feedback interconnection, then the equilibrium of the interconnection is stable in the sense of Lyapunov. The follow result on the class of stabilizing control law then follows immediately. Theorem 1: Given u = u∗ − K(y − ydes ) (15) where K is a passive (possibly dynamic) system and u∗ satisfies (9), the equilibrium T ∗ , in (9), is a globally exponentially stable equilibrium, and y → ydes exponentially. Proof: Consider the Lyapunov function candidate for (14): V (δT ) = 1 T δT CδT. 2 (16) = = û∗ − K(y − ydes ) −Γ(y − ydes ) (17) (18) where K is strictly passive and Γ > 0. Then y → ydes asymptotically as t → ∞. Proof: Let δu∗ = û∗ − u∗ . Modify the Lyapunov function in (19) to include δu∗ : V (δT, δu∗ ) = 1 T 1 δT CδT. + δu∗ T Γ−1 δu∗ 2 2 (19) where Γ > 0. Using (17)–(18), the derivative along solution is V̇ = −δT T DR−1 DT δT − δT T BK(B T δT ) +δy T δu∗ + δu∗ T Γ−1 δu∗ = −δT T DR−1 DT δT. By using Barbalat’s Lemma [25], we have δT → 0 asymptotically.  Note that, not surprisingly, the adaptive feedforward is simply the integral control. If additional dynamics of u∗ is known, it is straightforward to incorporate into the adaptive controller. The inherent passivity in the system implies robust stability even when the operating condition changes. For example, if windows or doors are open and changing the thermal resistance, the closed system would remain stable. C. Optimization for Energy Efficiency The passivity of the system suggests an inner-outer loop architecture for thermal/energy management. As mentioned, if room heat transfer model is available, the passive controller may be designed to optimize the energy usage while preserving the passivity structure. Similarly, if the variation of the ambient temperature is known, e.g., through weather forecast, it may also be taken into account in the passive feedback controller design or in the feedforward calculation of u∗ . In addition, the desired temperature ydes may be manipulated to achieve additional objective such as balancing user comfort and energy efficiency. If T∞ is measured, the steady state relationship in (10)–(11) may be used to optimize ydes . elements is N = 4 + 2 ∗ 8 = 20. There are 27 thermal resistance elements, so L = 27. Hence, the dimension of the incidence matrix D (without the ambient node) is 20 × 27. For the purpose of numerical simulation we assume that the dimensions of two larger rooms are 3m×4m, and the two smaller ones are 3m×3m. The passages between the rooms have a width of 1m, the rooms are all 2.5m high and the walls are assumed to be 15cm thick. Further, we assume that the insulation (thermal resistance) of the material used for walls of room 2 is poorer than the other rooms, to simulate for example, a glass paneled room. Using values of volumetric heat capacities from [26] and values of thermal resistances from [27], we have a model of the form as in (6)–(7). D. HVAC System Model Heat input control is provided through the building heating-ventilation-air-condition (HVAC) system. There are numerous architectures and design choices. For example, a model suggested in [6]–[8] consists of a central heating/cooling unit, zone heating/cooling coils, zone dampers, and fan. The heat input then becomes ui = θi ṁs (Tsi − Ti ) (20) Fig. 2. where ṁs is the totalP supply air mass flow, θi is the damper position for zone i, i θi = 1, and Tsi is the supply air temperature into zone i given by Tsi = δTr + (1 − δ)T∞ + ∆Thi + ∆Tc Layout of the four-room example (21) where δ is the return-air/outside-air ratio, ∆Thi the zone heater input, and ∆Tc the central heat input. The return air temperature is assuem to be the mix of the zone P temperature weighted by the damper position: Tr = i θi Ti . In this case, we potentially have a large number of control variables, {θi , ∆Thi } for each zones and (ṁs , δ, ∆Tc ). We can use all these variables to implement the passive controller while minimizing instantaneous power consumption and satisfy the constraints. The controller, however, will no longer be decentralized. IV. SIMULATION EXAMPLE To illustrate results in the paper, consider a four-room temperature control example as shown in Figure 2. This example is taken from [19], with the added heat transfer to the ambient for all the rooms. Each double headed arrow in the diagram represents a thermal connection (either between two rooms through a wall/open door or between a room and outside atmosphere). The connection between two rooms through a wall are represented using a 3R2C wall model (same for a room and the outside atmosphere), while connections between two rooms through an open door are represented by a single resistance (corresponding to the inverse of the convective heat transfer coefficient of air). The corresponding thermal network graph is shown in Figure 3. Nodes 1, 4, 10, 7 correspond to the room thermal capacitance. Other nodes denote the wall capacitive elements. For the 4 rooms and 8 walls, the number of capacitive Fig. 3. Thermal graph representation of the four-room example Open loop Behavior Given a constant T∞ , the room and wall temperatures will all reach T∞ at the steady state. For the initial temperature at 12◦ C and T∞ = 10◦ C, the room temperatures all converge slowly to 10◦ C with the longest time constant around 15 hours. Proportional Output Feedback We now set the target room temperature at 20◦ C and apply the proportional feedback of the room temperature error (which is a special case of (15)) and the feedforward, u∗ , from (10)–(11). With the proportional feedback gain set at Kp = 500, the room temperature converges to the desired setting.However, such model information is typically unavailable, and the usual implementation simply ignores the feedforward. In this case, there will be a steady state error, the size of which is inversely proportional to Kp . The response is shown in Figure 4. With u∗ , which represents the feedforward portion of the control, the total energy usage over 3 days with u∗ is 199MJ with peak power at 4.7KW. Without u∗ , i.e., the controller is feedback only, the 3-day energy usage is lower (as the attained temperatures are lower than the target) at 167MJ with peak power at 4.0KW. Fig. 5. Closed loop temperature response with proportional and integral feedback Fig. 4. Closed loop temperature response with proportional feedback only Proportional Integral Output Feedback We next apply the adaptive feedforward controller, which, with K chosen as the proportional controller, is just the proportional-integral control. The room temperature and heat input responses are as shown in Figures 5–6. The proportional gain may be set much smaller at 1, and the integral gain Γ at 0.1. The 3-day energy usage is about the same at 199MJ with peak power at 2.4KW. Since the system is open loop stable, it may be shown that the instantaneous power may be capped and still achieve closed loop stability, while sacrificing the transient performance. The temperature and heat input response with peak power capped at 1KW is shown in Figure 7–8. The overall energy consumption remains the same, but the peak demand is reduced. Room 2 is the most energy consuming room because of its poorer insulation. If the set point for room 2 is reduced by just 1◦ C to 19◦ C, the 3-day energy consumption of this room is reduced by a whopping 43%. The temperature and heat input responses are shown in Figures 9–10. Note that more improvement is possible as all room temperatures overshot the set point in the first day before settling back to the target value. A model based optimization scheme, such as model predictive control, would be able to improve the transient, but more model information would be needed. When the ambient temperature varies with time, the stability is not affected, but the room temperature shows the corresponding fluctuation (which may be reduced by increasing the proportional feedback gain). The room temperature and heat input for T∞ = 5 sin(2πt/T ) + 5◦ C, T =24hrs, are shown in Figures 11–12. The overall energy usage is increased due to the periodic increasing heating need. V. CONCLUSION This paper presents a passivity perspective on building thermal control. By including all thermal capacitances as nodes, we show that the building thermal control problem is inherently passive. This allows a large class of stabilizing controller to be considered, possibly including additional criteria such as energy minimization. Current work includes the design of the outer loop to adjust temperature set point based Fig. 6. Closed loop heat input response with proportional and integral feedback Fig. 7. Closed loop temperature response with proportional and integral feedback Fig. 8. Closed loop heat input response with proportional and integral feedback and input saturation constraint at 1KW on energy consumption/cost and human comfort, and the inclusion of the dynamics of the heating/cooling elements. ACKNOWLEDGMENT This work was supported in part by the Center for Automation Technologies and Systems (CATS) under a block grant Fig. 9. Closed loop temperature response with proportional and integral feedback and lower room 2 temperature set point Fig. 10. Closed loop heat input response with proportional and integral feedback and lower room 2 temperature set point Fig. 11. Closed loop temperature response with proportional and integral feedback under sinusoidal ambient temperature Fig. 12. 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