Nonlinear and Switching Control for the HVAC System

Nonlinear and Switching Control for the HVAC System

Ming-Li Chiang

and Li-Chen Fu

1 Department of Electrical Engineering, National Taiwan University

Taipei, Taiwan, R.O.C.

2 Department of Electrical Engineering, Department of of Computer Science

and Information Engineering, National Taiwan University

Taipei, Taiwan, R.O.C.

Abstract

In this paper, we integrate nonlinear control and supervisory control theory to design a controller of a heating, ventilating and air conditioning (HVAC) system. First we design the nonlinear controllers by Lyapunov theory and backstepping. Then we create the discrete abstraction of this system and thus derive the discrete event system model from the continuous plant. Finally we design the supervisory controller of the discrete event model and thus complete the design of a hybrid control system. With the switching logic controlled by the supervisor and the good performance of the nonlinear controller, we can achieve many specifications with this systematic design.

Key Words: HVAC system, nonlinear control, switching system.

I. INTRODUCTION

HVAC systems for buildings are major consumers of electrical energy through the world [1]. Improving the energy efficiency of HVAC system while maintaining the comfort is the primary goal in the development of controller design. With increasing complexity of modern HVAC systems, how to control and optimize the operation with guaranteed perfor-mance, stability and reliability becomes a challenging issue since the air conditioning process is highly non-linear; the interaction between the temperature and humidity control loops is significant and the constraints imposed by non-ideal actuator behavior are considerable. To control such systems efficiently and effectively in the presence of dynamic interaction and random disturbance so as to conserve energy while maintaining the desired thermal comfort level requires more than a conventional methodology. A good controller for the air-handling units (AHUs) is extremely desirable for human comfort and energy saving. Classical HVAC control techniques such as the ON/OFF controllers (thermostats) and the proportional-integral-derivative (PID) controllers are still very popular because of their low cost [2]. Some advanced control method such as adaptive control and predicative control are also studied in literatures [3], [5], [4]. In [6], the actuators’ dynamics is considered and the feedback linearization approach is adopted to design the controller. However in the long run, these controllers are expensive since they operate at a very low-energy efficiency.

Hybrid systems theory [7] which combine the continuous-variable dynamical systems(CVDS) and discrete event sys-tems(DES) and the interactions of them is a promising approach to improve the control of HVAC systems. CVDS, as the name, are usually modelled by differential equations and theoretically measured by norms. DES are systems characterized with event-driven and discrete states that use different performance measurement due to discrete dynam-ics, and therefore the space of interest are often not metric [8]. Because of the different nature of CVDS and DES, we can not directly use conventional design methodologies for hybrid systems. Control of hybrid systems usually involves with multi-layer controller architecture which has better per-formance then the single controller by the effect of switching [9]. For the multi-layer controller, the logical control in the upper layer is used to decide the switching rules. The servomechanism control in the lower layer is used to satisfy the performance specification or constraints of the CVDS. In this paper we design a satisfactory nonlinear controller in the lower layer for the heating and cooling subsystems and a supervisor in the upper layer for decision making.

From discrete abstraction, we can extract a DES model from the hybrid system with some properties preserved [10], [11]. In the DES model, the occurrence of discrete events cause the transitions between discrete states (e.g., HeatingOn-CoolingOff, HeatingOff-CoolingOff, ... etc.). The supervisory controller oversees the event sequence and con-trol the on/off behaviors of system. When the system op-eration stays in one mode, the continuous state trajectory (temperature) with the corresponding differential equation will be controlled by the nonlinear controller. The process completes the control of hybrid systems. Briefly, DES and CVDS may affect each other and causing the system highly complex and difficult to be controlled. We discuss the HVAC system, both the DES and CVDS, and then design the supervisor and nonlinear controller for it.

The rest of this paper is organized as follows: In section 2, we give some preliminaries about the hybrid system model and some concepts of supervisory control. In section 3 we introduce the model of a HVAC system and apply Lya-punov method with the backsteeping technique to design the nonlinear controller for the heating and cooling subsystems. In section 4, discrete abstraction of the continuous plant is performed and the supervisor is designed. Thus we complete

the design of a hybrid control system. Finally, section 5 concludes the paper.

II. PRELIMINARIES

In this section, we introduce the structure of hybrid system models and fundamental concepts of supervisory control theory.

A. Hybrid System Model

Hybrid control systems consist of three parts [11]: the “plant” contains all the continuous dynamics, including the continuous controller. The “supervisor” is a discrete event system which controls the logic. The supervisor is usually described as a finite automaton and designed by the DES theory. The “interface” plays a key role in hybrid systems, it connects the continuous and discrete dynamics, like a A/D, D/A converter, see Fig. 1. Supervisory control of hybrid systems is based on the discrete event system theory. Discrete

abstraction is used to approximate the continuous plant into

a DES model and thus the techniques of supervisory control can be applied. It is a important step in this approach. Researches of discrete abstraction are active in the field of hybrid systems.

Fig. 1. A switching control system model with the supervisor.

B. Ramadge-Woham Supervisory Control Theory

The Ramadge-Woham supervisory control theory provides a mathematical framework for the design of discrete event systems. Here we follow the definitions and notations in [12] and give a brief introduction to the theory. A discrete event system can be described by an automaton:

G= {X,Σ,δ, x0, Xm}

where X is the set of discrete states,Σis the set of events (or

alphabets),δ: X×Σ→ X is the state transition function , x0

is the initial state, and Xm is the set of the marked states. Let

Σ∗ _{denote the set of all finite strings from the event set Σ,}

e.g., u=σ1σ2∈Σ∗where σ1,σ2∈Σ. A language is the set

of all sample paths that demonstrate the possible behaviors of G. That is

L(G) = {t ∈Σ∗|δ(x0,t)!}

where δ(x0,t)! means that there exists a state x′∈ X such

that δ(x0,t) = x′. Let Γ(x) be the feasible events at state x, i.e., events that could happen at state x. Now we can start to discuss the control of DES.

The main concept of supervisory control is to restrict the “illegal” behaviors that we do not want. We care about whether the states start from the initial states can be transited to the marked states by available event sequences. Generally the specification of the DES is defined by a language K, and what the supervisor need to do is to make the language

L(S/G) belongs to the specification, that is, L(S/G) ⊆ K,

where S/G means that G is supervised by the supervisor S.

We can regard the supervisor S : L(G) → 2Σ as a function

that maps a event sequence of L(G) to the feasible event set.

For any event sequence t that G generated so far, the feasible

event set function of controlled DES S/G is

ΓS/G(δ(x0,t)) = S(t) ∩Γ(δ(x0,t))

The procedure of supervisory control of a discrete event system is summarized as follows:

1) According to the problem we concerned, label the possible “illegal” states in set X . The illegal states mean the states that violate our requirement, e.g., precedence constraints or dead locks.

2) Let H be the automaton obtained from removing the illegal states from G.

3) Let the system specification K= L(H) and design the

supervisor to make the system behave well.

III. PLANTOFTHEHVAC SYSTEM

As we mentioned above, the “plant” represents all the con-tinuous dynamics of the hybrid control system, including the nonlinear controllers which are the candidates of switchings.

A. HVAC system

The air in the room is assumed to have a uniform tem-perature distribution and the heat loss between components is neglected. The system operates as shown in Fig. 2.

The outdoor air enters the system at temperature T0(t) and

volumetric flow rate f0(t). Air with temperature T0 and

flow rate f(t) passes through the heat exchanger where an

amount of heat is exchanged with the air. Since we have the assumption of perfect mixing, the air temperature within

and exiting the heat exchanger is T2(t), which represents the

supply air temperature. After being cooling or heating in

the heat exchanger, the air at temperature T2passes into the

thermal space with the help of fan and the air temperature

in the thermal space is T3(t). The heat load in the room

is included as QO. Air leaving the thermal space is drawn

through the fan and some portion of it excluded from the system whereas the reminder is recirculated to mix with the fresh air from outdoor.

Fig. 2. Model of the HVAC system [4]

The schematic layout of a HVAC system is illustrated in Fig. 3, where a group of components working together to

move heat to somewhere (heating), and to remove heat from

somewhere it is not wanted (cooling) and to put it there as it is un-subjectable (ventilating)[13]. The HVAC system is complex and there are some researches working on it with supervisory control, e.g., [14].

1150MM x 1150MM Cooling Tower EA OA RAF RA ATC Damper Filters Cooling Coil Heating Coil SAF Cooling CV Chilled CV Boiler Condenser Evaporator Compressor CHWP CWP Cooling Tower HWP Heating CV HWR HWS CHWR CHWS CWR CWS AHU

Fig. 3. HVAC schematic layout.

For the system operates with k different modes, we model it as the equation of this form:

˙

x=Σkfk(x, uk,Θk)

where Θk indicates which mode is operating now. Θk is a

time-varying variable which has binary value 0 or 1 and is determined by the supervisor. Here 0 and 1 means OFF and ON respectively. Consider the single-zone thermal system in Fig. 3 The following thermal dynamic equations [15] are derived from the principle of conservation of energy

ρcpVhe dT2 dt = fρcp(T1− T2) +ΘHηHC(T2, T3,ΘH) −ΘCηCC(T2, T3,ΘC) ρcpVts dT3 dt = fρcp(T2− T3) + QL (1)

where T3 is the air temperature in the building, T2 is the

resulting temperature of heat exchanger and Tois the

temper-ature outside. T1= T3+f_fo(To− T3) is the mixed temperature

obtained from To and T3, where f_fo means the

system-to-fresh-air volumetric flow-rate ratio. ΘH and ΘC indicate

the heating and cooling operation mode. ηH and ηC are

heating and cooling constants, respectively. C(T2, T3,ΘH) and

C(T2, T3,ΘC) are our controller used for heating and cooling.

System variables and parameters are described in Table I and the parameter values within are adopted from [16].

TABLE I

HVAC SYSTEMVARIABLES ANDPARAMETERS

ρ air density 1.19 Kg/m3

cp constant pressure specific heat of air 1005 J/Kg◦C

Vhe effective heat exchanger volume 1.719 m3

Vts effective thermal space volume 1655.115 m3

fo volumetric ventilation airflow rate 2 m3/s

f volumetric circulate airflow rate 8.0231 m3_/s

t time sec

Ti temperature at location i ◦C

QL thermal load Watt

B. Nonlinear Controller Design For The Plant

Let the continuous states x= [x1, x2]T= [T2, T3]T and make

a substitution of T1to (1), then we have

˙ x1=V1he[(To− x2) f0+ (x2− x1) f +ΘHηHC(x1,x2,ΘH)−ΘCηCC(x1,x2,ΘC) ρcp i ˙ x2=_V1_ts h (x1− x2) f +_ρQ_cL_p i (2)

The objective of the nonlinear controller is to regulate the

state x2 = T3 to a set point x∗2 = T3∗. In the following

procedure we will use the concept of backstepping control to complete the task. Define the error e2= x2− x∗2, then

˙ e2= ˙x2= 1 Vts  (x1− x2) f + QL ρcp  (3) and (2) can be rewritten as

˙ x1 = f(x1, e2) + 1 Vheρcp (ΘHηHC(x1, e2+ x∗2,ΘH) −ΘCηCC(x1, e2+ x∗2,ΘC)) (4) ˙ e2 = x˙2= h(x1, e2) (5) where f(x1, e2) = − f Vhe x1+ f− fo Vhe e2+ Tofo Vhe + f− fo Vhe x∗₂ h(x1, e2) = f Vts x1− f Vts e2− f Vts x∗₂+ QL Vtsρcp

Suppose that e2 can be stabilized by a smooth state

feedback control when x1=φ(e2). If there is a Lyapunov

function V1(e2) that satisfies the inequality ˙

V1=

∂V1

∂e2

[h(φ(e2), e2)] ≤ −W (e2) (6)

where W(e2) is always positive, then e2 will converges to

zero and this means x2→ x∗2. Let z= x1−φ(e2) as the error of x1, then

˙z = x˙1− ˙φ(e2) ˙

From above equations we will stabilize e2by control z. Now define a Lyapunov function V2(x1, e2) = V1(e2) +12z2and let ˙z= v, then ˙ V2 = ∂ V1(e2) ∂e2 ˙ e2+ z˙z = ∂V1(e2) ∂e2 [h(z +φ(e2), e2)] + zv Choosing v= −∂V1(e2) ∂e2 ( ∂h

∂e2) − kz with a positive constant

k, then

˙

V2(x1, e2) ≤ W (e2) − kz2< 0

By Lyapunov analysis we know that x1and e2are

asymp-totically stable. Since v= ˙x1− ˙φ(e2) and z = x1−φ(e2), from (4) and apply v=∂V1(e2) ∂e2 ( ∂h ∂e2) − kz we have ˙ x1 = v + ˙φ(e2) =  −∂V1(e2) ∂e2 (∂h ∂x1 ) − k(x1−φ(e2))  +∂φ ∂e2 h(x1, e2) = f(x1, e2) + 1 Vheρcp (ΘHηHC(x1, e2+ x∗2,ΘH) −ΘCηCC(x1, e2+ x∗2,ΘC))

and thus yields the controller

C(x1, x2,ΘH) = Vheρcp ηH  −∂V1(e2) ∂e2 ( ∂h ∂x1) −k(x1−φ(e2)) +_∂∂φ_e2h(x1, e2) − f (x1, e2)  when ΘH= 1, ΘC= 0, and C(x1, x2,ΘC) = Vheρcp ηC  −∂V1(e2) ∂e2 ( ∂h ∂x1) −k(x1−φ(e2)) +_∂∂φ_e 2h(x1, e2) − f (x1, e2)  when ΘH= 0, ΘC= 1.

C. Simulations of the Nonlinear Controller

Using the system parameter values in Table I and suppose

the reference set point of T₃∗= 21.66◦C and the outdoor

temperature To= 29.44◦C, then we have

Vheρcp = −4.864 × 10−4

f(x1, e2) = −4.667x1+ 3.503(e2+ x∗2) + 110.145

h(x1, e2) = 4.847 × 10−3x1− 4.847 × 10−3(e2+ x∗2)

−0.148

Note that e2= x2− x∗2= T3− T3∗. By the analysis in previous subsection, we choose V1(e2) =1₂(e2+ x∗2)2 and

φ(e2) =

−k2(e2+ x∗2) + 4.847 × 10−3(e2+ x∗2) − 0.148

4.847 × 10−3

with a positive constant k2, then ˙V1(e2) = −k2(e2+

x∗₂)2 _{= −k}

2x22 < 0. Hence the controller C(x,Θ) =

C(x1, x2,ΘH)ΘH+C(x1, x2,ΘC)ΘC is derived from previous

section. Note that the value ofΘHandΘCwill be determined

by the supervisor which will be discussed in next section. Fig. 4 shows the cooling performance of the nonlinear controller withΘC= 1, T3∗= 21.66◦C, To= 29.44◦C and the

thermal load QL= 84970W . Fig. 5 shows the temperature

tracking to 25◦C and 35◦C consecutively with To= 30◦C and

the thermal load QL= 8000 × (To− T3)W . From the figures

we can see that the performance is satisfactory with respect to the responding time and the transient response.

Fig. 4. Cooling performance.

0 500 1000 1500 2000 20 25 30 35 40 T3,T2 s Cesus degree T3 T2 0 500 1000 1500 2000 20 25 30 35 40 T3,ref s Cesus degree T3 ref 0 500 1000 1500 2000 −2 −1 0 1 2 indicators s indicator Heating Cooling 0 500 1000 1500 2000 −4 −2 0 2 4x 10 5 Outpowers s Watt Heating Cooling

Fig. 5. Temperature tracking.

Since the number of switchings are always finite in a finite time (nonzeno) and each subsystem is asymptotically stable, it can be shown that the Lyapunov like function is decreasing at every instants when the mode switches in and hence by the analysis of multiple Lyapunov function [17], we know that the overall system is stable under switchings.

IV. DISCRETE ABSTRACTIONANDSUPERVISORY

CONTROLOF THE HVAC SYSTEM

Once we designed a set of nonlinear controllers for the continuous plant, the problem is how to determine the switching mechanism or the control modes. This is the main objective of the supervisor. To design supervisory controller for the hybrid systems, we have to approximate our continuous plant into a discrete event model and thus the Ramadge-Woham framework can be applied.

A. Discrete Abstractoin

Discrete abstraction is the technique that used to reduce

a continuous dynamic system into a discrete event system while preserving the important dynamics we care about in

the original continuous system. The direct way to extract the DES model from the continuous plant is partitioning the state space by several hypersurfaces [11]. In our interest of the HVAC system, we would like to keep the temperature

x2≡ T3 in some fixed range. So we divide the continuous

state space into three parts

x2< 20, 20 < x2< 25, x2> 25 This can be done by defining the smooth functions

h1(x) = x2− 20

h2(x) = 25 − x2

We collect the continuous states in each state space parti-tion as the DES plant states ˜p1, ˜p2 and ˜p3, respectively. To define the abstracted DES model, we introduce the transition relation of it. When the continuous state cross form one discrete state ˜pi to another state ˜pj, a plant event ˜xi j is

generated. Thus when a plant event occurs, it means there might be a state transition (see Fig. 6). The transited DES plant state is effected by the previous DES state and the

the control signal Θ, hence the whole DES plant model

could be characterized as an automaton G= ( ˜P, ˜X,Θ,ψ,λ),

where ˜P is the set of DES states, ˜X is the set of plant

events,Θ is the set of control signals which is given by the

supervisor, ψ: ˜P×Θ→ 2P˜ _{is the transition relation of DES}

states and plant events, andλ : ˜P× ˜P→ 2x˜_{is the transition} relation of DES states and control signals. The relation can be summarized as follows

˜

p[n − 1] ×Θ−→ ˜p[n]ψ

˜

p[n − 1] × ˜p[n]−→ ˜x[n]λ

Note the causality of the transitions between each compo-nents.

Fig. 6. Partition of the continuous plant state space.

For the setup of heating-cooling control mode, the system will identify the initial DES plant state. If the initial state is

˜

p1,ΘC will be zero for all the times (i.e., cold days, heating

mode on). Conversely, if the initial state is ˜p3, ΘH will be

zero.

How to judge the occurrence of the plant events is an important part of discrete abstraction. The popular method to determine whether the continuous state cross the

hy-persurface hi is by performing the gradient analysis at the

boundaries of adjacent partitions on the continuous state space [11].The concept can be interpreted by Fig. 7. Suppose the DES plant states ˜pband ˜pcare adjacent, if the trajectory

cross the hypersurface hi(x) = 0 from ˜pb to ˜pc, then the

following conditions must be satisfied: when ˜pb is in the

place of hi(x) < 0 and h(ξ) = 0, then ∇xh(ξ) · f (ξ,C) > 0.

And if ˜pbis in the place of hi(x) > 0, then∇xh(ξ)· f (ξ,C) <

0. This is clear from Fig. 7. After defining the partitions and

Fig. 7. The gradient analysis on the hypersurface.

transition relations, the DES model of the continuous plant is derived and the DES model automaton is shown as Fig. 8.

Fig. 8. DES plant model of the HVAC system.

B. Supervisory Controller

The supervisory controller S oversees the HVAC system

and determines the value of Θ for the nonlinear controller

˜

S is the supervisory controller states and the transition flow

is

( ˜x[n] × ˜s[n − 1])−→ ˜s[n]δ −→φ Θ

Fig. 9 shows the the state transition diagram of the supervisor

and to meet the specification 20∼ 25, we define

φ( ˜s1) = ΘH= 1 ⇐⇒ (heating mode on)

φ( ˜s2) = ΘH= 0 ⇐⇒ (heating mode off)

φ( ˜s3) = ΘC= 1 ⇐⇒ (cooling mode on)

φ( ˜s4) = ΘC= 0 ⇐⇒ (cooling mode off)

With the DES model and the supervisor, the nonlinear

32 x 23 x 21 x 12 x S1 S2 on off 12 x 21 x 23 x 32 x S3 S4 on off heating cooling

Fig. 9. Supervisor for the HVAC system.

controller designed in section 3 will be supervised and the

temperature will always keep in 20◦C to 25◦C. We can

save energies with this design because we can turn off the control in some situation which is specified by supervisor (e.g., ˜s2 and ˜s4). Of course, we can only use the nonlinear controller to regulate the desired temperature, this depends on the demand. Control of the whole system, including the continuous plant and the discrete logic, is thus completed.

V. CONCLUSIONS

In this paper, we discuss the HVAC system with both the continuous and discrete dynamics and complete a systemati-cal design for this hybrid control system. First we design the nonlinear controller for the heating and cooling subsystems inside HVAC system. Then we extract a discrete event system model from the continuous plant by discrete abstraction. A supervisor for the DES plant is then designed and hence both the continuous and discrete phases of the hybrid system are controlled. A practical HVAC system contains many complex behaviors that we do not describe in this paper. There are still many problems to be considered such as the interactions between components other then the heating and cooling exchangers, and the the discrete abstraction of them . We will consider more working components of this system and include analysis of the nondeterminism of discrete abstraction in the future work.

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